An Introduction to Dynamics of Colloids - Jan K. G. Dhont

STUDIES IN INTERFACE SCIENCE

An Introduction to Dynamics of Colloids

S T U D I E S IN I N T E R F A C E S C I E N C E

SERIES E D I T O R S D. M 6 b i u s and R. M i l l e r

Vol. I Dynamics of Adsorption at Liquid Interfaces

Theory, Experiment, Application by S.S. Dukhin, G. Kretzschmar and R. Miller

Vol. II An Introduction to Dynamics of Colloids

by J.K.G. Dhont

An Introduction to Dynamics of Colloids

JAN K.G. DHONT van 't Hoff Laboratory

for Physical and Colloid Chemistry University of Utrecht

Utrecht, The Netherlands

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Ohon t , Jan K. O. An i n t r o d u c t i o n to dynamics o f c o l l o i d s / Jan K.G. Dhon t .

p. cm. - - ( S t u d i e s tn i n t e r f a c e s c i e n c e ; v o l . 2) I n c l u d e s b i b l i o g r a p h i c a l r e f e r e n c e s (p . ISBN 0 - 4 4 4 - 8 2 0 0 9 - 4 ( a c i d - F r e e pape r ) 1. C o l l o i d s . 2. R o l e c u l a r dynamics .

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To my mother

In memory of my father

This Page Intentionally Left Blank

PREFACE

This book is a self-contained treatment of the fundamentals of a number of aspects of colloid physics. It is intended to bridge the gap that exists between more or less common knowledge to researchers in this field and existing textbooks for graduate students and beginning researchers. For many aspects of the theoretical foundation of modern colloid physics one has to resort to original research papers, which are not always easy to comprehend. This book is aimed to provide the theoretical background necessary to understand (most of) the new literature in the field of colloid physics. Needless to say that the topics treated in this book are biased by my own interests (this is especially true for the last two chapters).

There are roughly two kinds of theoretical considerations to be distinguished �9 those aimed to predict equilibrium properties and equilibrium microstructure of suspensions, and those concerned with dynamical behaviour. The present book is concerned with dynamical behaviour. The treatment of static properties is brief and is concerned only with those quantities that are relevant as an input for theories on dynamics. Some knowledge on equilibrium thermodynamics and statistical mechanics is therefore assumed.

Both chemists and physicists are active in colloid science. In many cases the mathematical background of chemists is less developed than for physicists. To make this book accessible also for those with a chemistry background, the first chapter contains a section on the mathematical techniques that are frequently used. Complex function theory is worked out in relative detail, since this is a subject that is often missing in mathematics courses for chemists. More complicated mathematical steps in derivations are always worked out in appendices or in exercises. In addition, for the same reason, the first chapter contains a section on fundamental notions from statistical mechanics.

I tried to write each chapter as independently from others as possible. Results from previous chapters, when needed, are quoted explicitly, and in most cases explained again in an intuitive way. This offers the possibility to combine a limited number of chapters for a graduate course, taking quoted results with their intuitive interpretation from chapters that are not included for granted.

In the main text, little reference is made to literature. At the end of each chapter I added a self-explanatory section "Further Reading and References", in which some literature is collected. It is virtually impossible, nor is it my

vii

intention, to provide each chapter with a complete list of references. I must apologize to those not referred to, who contributed significantly to subjects treated in this book.

I am grateful to my colleaques at the van 't Hoff laboratory for giving me the opportunity to write this book. Special thanks go to Arnout Imhof, Luis Liz-Marz~, Henk Verduin and Anieke Wierenga, who made a number of suggestions for improvement of most of the chapters. I am especially grateful to Gerhard N~igele (University of Konstanz), not only for his constructive criticism, but also for providing me with some additional exercises.

Many of the weekends I could have spent together with my wife were used to work on this book. I would not have managed to finish this book without her continuous encouragement.

Utrecht, 4 January 1996 Jan K.G. Dhont

viii

C O N T E N T S

C H A P T E R S :

1 : INTRODUCTION 2 : BROWNIAN MOTION OF

NON-INTERACTING PARTICLES 3 : LIGHT SCATTERING 4 : FUNDAMENTAL EQUATIONS OF MOTION 5 : HYDRODYNAMICS 6 : DIFFUSION 7 : SEDIMENTATION 8 : CRITICAL PHENOMENA 9 : PHASE SEPARATION KINETICS

1-68

69-106 107-170 171-226 227-314 315-442 443-494 495-558 559-634

CHAPTER 1 : INTRODUCTION 1-68

1.1 An Introduction to Colloidal Systems 1.1.1 Definition of Colloidal Systems 1.1.2 Model Colloidal Systems and Interactions 1.1.3 Properties of Colloidal Systems

1.2 Mathematical Preliminaries 1.2.1 Notation and some Definitions 1.2.2 Integral Theorems 1.2.3 The Delta Distribution 1.2.4 Fourier Transformation 1.2.5 The Residue Theorem

The Cauchy-Riemann relations Integration in the complex plane Cauchy's theorem The residue theorem An application of the residue theorem and Fourier transformation

1.3 Statistical Mechanics

2 2 5 11 13 13 16 17 19 22 22 24 25 26

28 31

ix

1.3.1 Probability Density Functions (pdf's) Conditional pdf's Reduced pdf's The pair-correlation function

1.3.2 Time dependent Correlation Functions 1.3.3 The Density Auto-Correlation Function 1.3.4 Gaussian Probability Density Functions

Appendix Exercises Further Reading and References

31 33 35 37 40 43 46 49 51 64

CHAPTER 2 : BROWNIAN MOTION OF NON-INTERACTING PARTICLES 69-106

2.1 Introduction 2.2 The Langevin Equation 2.3 Time Scales 2.4 Chandrasekhar 's Theorem 2.5 The pdf on the Diffusive Time Scale 2.6 The Langevin Equation on the Diffusive Time Scale 2.7 Diffusion in Simple Shear Flow 2.8 Rotational Brownian Motion

2.8.1 Newton's Equations of Motion 2.8.2 The Langevin Equation for a Long

and Thin Rod 2.8.3 Translational Brownian Motion of a Rod 2.8.4 Orientational Correlations

Exercises Further Reading and References

70 70 74 79 80 81 83 88 88

91 96 97 102 105

CHAPTER 3 : LIGHT SCATTERING 107-170

3.1 Introduction 3.2 A Heuristic Derivation 3.3 The Maxwell Equation Derivation 3.4 Relation to Density Fluctuations

108 109 113 122

3.5 Static Light Scattering (SLS) 3.6 Dynamic Light Scattering (DLS) 3.7 Some Experimental Considerations

The Dynamical Contrast The Finite Interval Time Ensemble Averaging and Time Scales

3.8 Light Scattering by Dilute Suspensions of Spherical Particles

3.8.1 Static Light Scattering by Spherical Particles 3.8.2 Dynamic Light Scattering by Spherical Particles

3.9 Effects of Polydispersity 3.9.1 Effects of Size Polydispersity

Static Light Scattering Dynamic Light Scattering

3.9.2 Effects of Optical Polydispersity 3.10 Scattering by Rigid Rods

3.10.1 The Dielectric Constant of a Rod 3.10.2 Static Light Scattering by Rods 3.10.3 Dynamic Light Scattering by Rods


125 132 135 135 138 140

141 141 143 144 145 145 147 149 153 153 154 158 160 169

CHAPTER 4 : FUNDAMENTAL EQUATIONS OF MOTION

4.1 Introduction 4.2 A Primer on Hydrodynamic Interaction 4.3 The Fokker-Planck Equation 4.4 The Smoluchowski Equation 4.5 Diffusion of non-Interacting Particles

4.5.1 Linear Fokker-Planck Equations 4.5.2 Diffusion on the Brownian Time Scale 4.5.3 Diffusion on the Fokker-Planck Time Scale

4.6 The Smoluchowski Equation with Simple Shear Flow 4.6.1 Hydrodynamic Interaction in Shear Flow 4.6.2 The Smoluchowski Equation with Shear Flow 4.6.3 Diffusion of non-Interacting Particles in

Shear Flow

171-226

172 177 179 183 186 187 189 191 195 196 197

199

xi

4.7 The Smoluchowski Equation with Sedimentation 204 4.7.1 Hydrodynamic Interaction with Sedimentation 204 4.7.2 The Smoluchowski Equation with Sedimentation 206

4.8 The Smoluchowski Equation for Rigid Rods 4.8.1 Hydrodynamic Interaction of Rods 4.8.2 The Smoluchowski Equation for Rods 4.8.3 Diffusion of non-Interacting Rods


208 209 212 218 220 225

CHAPTER 5 : HYDRODYNAMICS 227-314

5.1 Introduction 5.2 The Continuity Equation 5.3 The Navier-Stokes Equation 5.4 The Hydrodynamic Time Scale

Shear Waves Sound Waves

5.5 The Creeping Flow Equations 5.6 The Oseen Matrix 5.7 Flow past a Sphere

5.7.1 Flow past a Uniformly Translating Sphere 5.7.2 Flow past a Uniformly Rotating Sphere

5.8 Leading Order Hydrodynamic Interaction 5.9 Faxen's Theorems 5.10 One step further : the Rodne-Prager Matrix 5.11 Rotational Relaxation of Spheres 5.12 The Method of Reflections

5.12.1 Calculation of Reflected Flow Fields 5.12.2 Definition of Mobility Functions 5.12.3 The First Order Iteration 5.12.4 Higher Order Reflections 5.12.5 Three Body Hydrodynamic Interaction

5.13 Hydrodynamic Interaction in Shear Flow 5.13.1 Flow past a Sphere in Shear Flow 5.13.2 Hydrodynamic Interaction of two Spheres in

Shear Flow

228 229 231 234 235 237 238 241 244 245 248 250 253 255 257 258 262 266 267 268 273 276 277

278

xii

5.14 Hydrodynamic Interaction in Sedimenting Suspensions

5.15 Friction of Long and Thin Rods 5.15.1 Translational Friction of a Rod 5.15.2 Rotational Friction of a Rod

Appendix A Appendix B Appendix C Appendix D Appendix E Exercises Further Reading and References

281 282 285 286 288 294 295 296 300 302 311

CHAPTER 6 : DIFFUSION 315-442

6.1 Introduction 6.2 Collective Diffusion

The zero wavevector limit Short-time and long-time collective diffusion Light scattering

6.3 Self Diffusion Short-time and long-time self diffusion

6.4 Diffusion in Stationary Shear Flow 6.5 Short-time Diffusion

6.5.1 Short-time Self Diffusion 6.5.2 Short-time Collective Diffusion 6.5.3 Concluding Remarks on Short-time Diffusion

6.6 Gradient Diffusion 6.7 Long-time Self Diffusion

6.7.1 The Effective Friction Coefficient 6.7.2 The Distorted PDF 6.7.3 Evaluation of the Long-time Self Diffusion

Coefficient 6.8 Diffusion in Stationary Shear Flow

6.8.1 Asymptotic Solution of the Smoluchowski Equation

The inner solution- K < v/Pe ~

316 317 321 323 324 324 327 329 331 332 339 349 351 356 356 359

360 363

366

366

xiii

The outer solution" K > x/Pe ~ Match of the inner and outer solution and structure of the boundary layer An experiment

6.9 Memory Equations 6.9.1 Slow and Fast Variables 6.9.2 The Memory Equation 6.9.3 The Frequency Functions 6.9.4 An Alternative Expression for the

Memory Functions 6.9.5 The Weak Coupling Approximation 6.9.6 Long-Time Tails

6.10 Diffusion of Rigid Rods

368

369 372 372 373 374 380

381 383 388 392

6.10.1 The Intensity Auto-Correlation Function (IACF) 392 The effect of translational and rotational coupling

6.10.2 Rotational Relaxation The equation of motion for P (fi 1, t)

Appendix A Appendix B Appendix C Appendix D Appendix E Exercises Further Reading and References

398 400 405

Evaluation of h(k, IAI1,62) and TI (1~!1,1~12) 407 Solution of the equation of motion for P(fi 1, t) 409 Mean field approximation fortheT-coefficients 410 Evaluation of the scattered intensity 412

415 416 418 420 421 424 437

CHAPTER 7 : SEDIMENTATION 443-494

7.1 Introduction Sedimentation at infinite dilution

7.2 Sedimentation Velocity of Interacting Spheres 7.2.1 Probability Density Functions (pdf's) for

Sedimenting Suspensions

444 445 446

447

xiv

7.2.2 The Sedimentation Velocity of Spheres 7.2.3 Sedimentation of Spheres with Hard-Core

Interaction 7.2.4 Sedimentation of Spheres with very Long

Ranged Repulsive Pair-Interactions 7.3 Non-uniform Baektlow

The effective creeping flow equations Solution of the effective creeping flow equations

7.4 The Sedimentation-Diffusion Equilibrium 7.4.1 Barometric Height Distribution for

Interacting Particles 7.4.2 Why does the Osmotic Pressure enter eq.(7.70)?

7.5 The Dynamics of Sediment Formation A simple numerical example of sediment formation The sedimentation velocity revisited


450

457

459 461 462

465 468

469 472 473

476 479 481 490

CHAPTER 8 : CRITICAL PHENOMENA 495-558

8.1 Introduction 8.2 Long Ranged Interactions

8.2.1 The Ornstein-Zernike Approach Asymptotic solution of the Omstein-Zernike equation

8.2.2 Smoluchowski Equation Approach 8.2.3 A Static Light Scattering Experiment

8.3 The Ornstein-Zernike Static Structure Factor with Shear Flow

Scaling Correlation lengths of the sheared system

8.4 The Temperature and Shear Rate Dependence of the Turbidity

The definition and an expression for the turbidity

496 501 501

505 508 513

515 520 523

525

525

XV

A scaling relation for the turbidity 8.5 Collective Diffusion 8.6 Anomalous Behaviour of the Shear Viscosity

8.6.1 Microscopic expression for the Effective Shear Viscosity

8.6.2 Evaluation of the Effective Viscosity The contribution ~c ~ The contribution q~ The contribution q ~ The contribution ~ A scaling relation for the non-Newtonian shear viscosity

Appendix A Appendix B Exercises Further Reading and References

527 530 535

536 538 539 541 541 543

545 548 549 550 555

CHAPTER 9 : PHASE SEPARATION KINETICS 559-634

9.1 Introduction 9.2 Initial Spinodai Decomposition Kinetics

9.2.1 The Cahn-Hilliard Theory 9.2.2 Smoluchowski Equation Approach 9.2.3 Some Final Remarks on Initial Decomposition

Kinetics

561 567 567 572

577 The mechanism that renders a system unstable 579

9.3 Initial Spinodal Decomposition of Sheared Suspensions 580

9.4 Small Angle Light Scattering by Demixing Suspensions 586

9.5 Demixing Kinetics in the Intermediate Stage 590 9.5.1 Decomposition Kinetics without

Hydrodynamic Interaction 591 Evaluation of the ensemble averages in terms of the static structure factor 594 Simplification of the equation of motion 596 Shift of kin(t) and k~(t) with time 597

xvi

The dimensionless equation of motion 9.5.2 Contribution of Hydrodynamic Interaction 9.5.3 Solution of the Equation of Motion 9.5.4 Scaling of the Static Structure Factor

9.6 Experiments on Spinodal Decomposition Appendix A Appendix B Appendix C Appendix D Exercises Further Reading and References

598 599 602 605 607 612 615 617 618 622 630

INDEX 635

xvii

Chapter I

INTRODUCTION

2 Chapter 1.

This introductory chapter consists of three sections. The first section introduces colloidal systems. The various common kinds of pair-interaction potentials of mean force are discussed. In further chapters the various pair- interaction potentials between the colloidal particles are modelled by simple expressions. The origin of these interactions is discussed in the present chapter on a heuristic level. Some of the phenomena exhibited by concentrated colloidal systems are discussed as well. A mathematical section is added for the benefit of those readers who feel that their mathematical background is insufficient. This section contains an exposition of the most important mathematical techniques that are used in this book. It has been my intention here to provide a concise treatment of those topics that may not have been part of mathematics courses of readers with a physical-chemistry education. In courses on mathematics for chemists, the residue theorem is often not included. Special attention is therefore given to that theorem, which is derived in a more or less self-contained manner. The third section is on basic notions from statistical mechanics and introduces the concept of probability density functions and time dependent correlation functions. Although this book is concerned with dynamical aspects, equilibrium probability density functions play an important role. For explicit evaluation of non-equilibrium and dynamical quantities, in most cases, the input of equilibrium probability density functions is required. Therefore, some properties of equilibrium probability density functions are discussed. In addition, Gaussian variables are discussed in some detail, since these play an important role in this book.

1.1 An Introduction to Colloidal Systems

1.1.1 Definition of Colloidal Systems

Colloidal systems of gold particles were already known many centuries ago, and their nature, being "extremely finely divided gold in a fluid", was recognized as early as 1774 by Juncher and Macquer. The year 1861 marks the beginning of systematic research on colloidal systems by publications of Thomas Graham. Graham made a distinction between two kinds of solutions �9 solutions of which the dissolved species is able to diffuse through a membrane, and solutions where no diffusion through a membrane is observed. Graham named the latter kind of solution "colloids". 1 Colloids do not diffuse

1The word "colloid" stems from the Greek word for glue, "kolla".

1.1. Colloidal Systems

through a membrane, simply because the dissolved species is too large, that is, their linear dimension is larger than the pores of the membrane. These large particles are nowadays referred to as colloidal particles. Before Graham's publications, in 1827, the Botanist Robert Brown observed irregular motion of pollen grains in water, which grains happen to have a colloidal size. There has been a considerable disagreement about the origin of this irregular motion, which played an important role in the establishment of the molecular nature of matter. The irregular motion observed by Brown is referred to as Brownian motion, and is the result of random collisions of solvent molecules with the colloidal particles. The molecular nature of the solvent is thus observable through the irregular Brownian motion of colloidal particles. Although it was generally accepted around 1910 that molecules were more than the theorists invention, the experimental work of Jean Perrin (1910) definitely settled this issue. He confirmed the earlier theoretical predictions of Einstein (1906) and Langevin (1908), and verified that colloidal particles are nothing but "large molecules". Their irregular motion is then identified with thermal motion, common to all molecules, but only visible by light-microscopic techniques for colloidal particles. Graham's colloids are solutions of such large molecules exhibiting Brownian motion, so that colloidal particles are also referred to as Brownian particles. The interested reader is referred to the section Further Reading and References for detailed accounts on the history of colloid science.

Colloidal systems are thus solutions of "large molecules". The large molecules are the colloidal or Brownian particles. These should be large compared to the solvent molecules, but still small enough to exhibit thermal motion (in the present context more commonly referred to as Brownian motion). Particles in solution are colloidal particles when "they are large, but not too large". The lower and upper limits for the size of a particle to be classified as a colloidal particle are not sharply defined.

The minimum size of a colloidal particle is set by the requirement that the structure of the solvent on the molecular length scale enters the interaction of the colloidal particle with the solvent molecules only in an averaged way. Many solvent molecules are supposed to interact simultaneously with the surface of a single colloidal particle. The interaction of the colloidal particle and the solvent molecules can then be described by macroscopic equations of motion for the fluid, with boundary conditions for the solvent flow on the surface of the colloidal particle. Brownian motion is then characterized through macroscopic properties of the solvent (such as its viscosity and temperature).

4 Chapter 1.

This is feasible when the size of the colloidal particle is at least about ten times the linear dimension of a solvent molecule. The minimum size of a Brownian particle is therefore ~ 1 rim.

The maximum size of a colloidal particle is set by the requirement that it behaves as "a large molecule", that is, when it shows vivid thermal motion (=Brownian motion). Thermal motion is relevant only when thermal displacements are a sizable fraction of the linear dimension of the particle during typical experimental time ranges. A brick in water (before it sunk to the bottom of the container) shows thermal motion also, but the displacements relative to its own size on a typical experimental time scale are extrememly small. Thermal motion of bricks in water is irrelevant to the processes in such systems. As soon as thermal motion is of importance to processes in solutions of large objects, these objects are classified as colloidal or Brownian particles. This limits the size of colloidal particles to ~ 10 #m. Besides the very small thermal excursions of the position of a brick due to thermal collisions with solvent molecules, it also moves to the bottom of a container in a relatively short time. This may also happen for smaller objects then a brick (and is then referred to as sedimentation) in a time span that does not allow for decent experimentation on, for example, Brownian displacements. This provides a more practical definition of the upper limit on the size of an object to be classified as a Brownian particle" displacements under the action of the earth's gravitational field should be limited to an extent that allows for experimentation on processes for which Brownian motion is relevant. For practical systems this sets the upper size limit again to about 10 #m, and sometimes less, depending on the kind of experiment one wishes to perform (see also exercise 1.1). Clearly, without a gravitational field being present, the latter definition of the upper limit for the colloidal size is redundant.

Colloidal solutions are most commonly referred to as suspensions or dispersions, since here solid material (the colloidal material) is "suspended" or "dispersed" in a liquid phase. There are roughly three kinds of dispersions to be distinguished, depending on the properties of the single colloidal particles: (i) the colloidal particles are rigid entities, (ii) they are very large flexible molecules, so-called macromolecules, and (iii) they are assemblies of small molecules which are in thermodynamic equilibrium with their environment. Examples of the second kind of colloids are polymer solutions, solutions of large protein molecules, very long virusses (like fd-virus). Polymer solutions may behave as dispersions of the first kind, when the polymer chain in a poor


solvent is shrunk to a rigid spherically shaped object. An example of colloids of the third kind are micro-emulsions, which mostly exist of droplets of water (or some apolar fluid) in an apolar fluid (or water) together with stabilizing surfactant molecules which are nested in the interface between the droplets and the solvent. The droplets consitute the colloidal particles which can exchange matter with each other.

In this book the first kind of suspensions will be discussed. Furthermore, the discussion is limited to spherical, and to some extent, to rigid rod like Brownian particles. This may seem a severe restriction, and indeed it is, but these seemingly simple systems have a rich dynamic (and static) behaviour, about which many features are still poorly understood. The things that can be learned from these seemingly simple systems are a prerequisite to the study of more complicated colloidal systems of the second and third kind mentioned above. There are many industrial colloidal systems of the first kind which are extremely complicated due to the variety of colloidal particles that is present in the suspension, and due to the complicated interactions between the colloidal particles (for example as the result of an inhomogeneous charge density on the surfaces of the colloidal particles or their complicated anisometric geometry). In this book, relatively simple colloidal systems are treated, where the colloidal particles are mostly assumed identical and the interaction is modelled by simple functions. Again this is a severe restriction, but a quantitative treatment of most of the complicated industrial systems is as yet hardly feasible. The theories discussed in this book can be, and in some cases have been tested, using model dispersions which are chemically prepared specially for that purpose. The behaviour of industrial systems can often be understood on the basis of these model experiments and calculations, although on a qualitative level.

1.1.2 Model Colloidal Systems and Interactions

There are many colloidal model systems consisting of metallic particles, such as gold, silver, copper, lead, mercury, iron and platinum particles. Examples of non-metallic colloidal systems are carbon, sulfur, selenium, tellurium and iodine particles. ~ There are many different methods to prepare these kinds of particles, including chemical, electrochemical and mechanical methods.

2Most of these particles have a radius larger than 10 #m, which is actually beyond the maximum size of what we would classify nowadays as colloidal.

6 Chapter 1.

The two most widely used spherical model particles, in order to understand the microscopic basis of macroscopic phenomena, are latex and amorphous silica particles. Latex particles consist ofPMMA (poly-[methylmethacrylate]) chains. In water, which is a poor solvent for PMMA, these particles are compact rigid spheres, while in for example an apolar solvent like benzene, which is a good solvent for PMMA, the particle swells to a soft and deformable sphere. In the latter case the individual polymer PMMA chains must be chemically cross-linked (with for example ethylene glycol dimethacrylate) while otherwise the particles fall apart and one will end up with a solution of free polymers. The silica model particles consist of a rigid amorphous Si02 core. The solubility in particular solvents depends on the surface properties of these particles, which can be modified chemically in various ways. Different chemical modifications of the surface give rise to different kinds of interaction potentials between the colloidal particles.

Two forces that are always present are the attractive van der Waa/s force and a repulsive hard-core interaction. The destabilizing attractive van der Waals force is of a relatively short range and can be masked by longer ranged repulsive forces due to charges on the surface of the particles, polymer chains grafted on the surface or a solvation layer (for example, silica particles in water are surrounded by a 3 nm thick structured water layer, which makes these particles relatively insensative to van der Waals attractions). The strength of these van der Waals forces is related to the refractive index difference between the particle cores and the solvent. The refractive index difference at the frequency of light is usually chosen small in order to be able to perform meaningful light scattering experiments. In most cases this minimizes the van der Waals forces. 3 For large particles or for particles with a large refractive index difference with the solvent, van der Waals forces can lead to irreversible aggregation of the colloidal particles. The repulsive hard-core interaction is simply due to the enormous increase in energy when the cores of two colloidal particles overlap. This is an interaction potential that is zero for separations between the centers of the two spherical colloidal particles larger than twice their radius, and is virtually infinite for smaller separations. For spheres "with a soft core", such as swollen latex particles in a good solvent, the repulsive interaction increases more gradually with decreasing distance between the colloidal particles (compare figs. l b and c).

aThe van der Waals force is actually related to a sum of the refractive index over all frequencies, so that minimizing the refractive index at one particular frequency does not necessarily imply small van der Waals forces.

1.1. ColloidalSystems

The surface of a colloidal particle may carry ionized chemical groups. The core material of the colloidal particles itself may carry such charged groups, or one can chemically attach charged polymers to the surface of the particles when it is favorable to use more apolar solvents (for example silica particles coated with TPM (3-methacryloxypropyltrimethoxysilane)). The charged surfaces of such colloidal particles repel each other. The pair- interaction potential of such charged colloidal particles is not a Coulomb repulsion (,,~ 1/r, with r the distance between the centers of the two spherical colloidal particles), but is screened to some extent by the free ions in the solvent. When the surface of a colloidal particle is negatively charged, free ions with a negative charge are expelled from the region around the particle while positive ions are attracted towards the particle. In this way a charge distribution is formed around the colloidal particle, the so-called double layer, which partly screenes the surface charges. The asymptotic form of the pair- interaction potential for large distances, where the potential energy is not too large, is a screened Coulomb potential, or equivalently, a Yukawa potential, ,~ e x p { - ~ r } / r , where ~ measures the effectiveness of screening, that is, the extent of the double layer. Screening is more efficient (n is larger) for larger concentrations of free ions, and addition of salt can diminish the double layer repulsion such that van der Waals forces become active, which can lead to aggregation of the colloid. When the potential energy is large, the Yukawa form for the pair-interaction potential no longer holds, and is a more complicated function of the distance. The total potential, being the sum of the van der Waals energy and the interaction energy due to the charges on the surfaces, including the role of the free ions in solution, is commonly referred to as the DVLO-potential, where DVLO stands for Derjaguin-Verwey-Landau- Overbeek, the scientists who established the theory concerning these kind of interactions. For low concentrations of free ions in the solvent, and negligible van der Waals attractions, the DVLO pair-interaction potential is a long ranged repulsive interaction as sketched in fig. 1.1 a.

The surfaces of the colloidal particles may be coated with polymer chains, where the polymer chains are either chemically attached to the surface ("grafted polymers") or physically adsorbed. Examples are silica particles coated with stearylalcohol and latex particles coated with PHS (poly-[ 12-hydroxy stearic acid]). The length of these polymer chains is usually very small in comparison to the size of the core of the colloidal particles. When the solvent is a good solvent for the polymer, the polymer brushes on two colloidal particles are repulsive, since the polymer rather dissolves in solvent than in its own melt.

Chapter 1.

|174 �9 | �9 * - | o e _ ~ o & _|174 | | . ~ .

o.'1: "l ~ . ' . ; ; �9 O

| �9

I r ( ) '

|

|

Figure 1.1" The most common kinds of pair-interaction potentials for spherical colloidal particles: (a) the screened Coulomb potential, that is, the DVLO potential with negligible van der Waals attraction, (b) an almost ideal hard-core interaction, (c) steric repulsion of long polymers in a good solvent, grafted on the surface of the colloidal particles, ((t) short ranged attraction of polymers in a marginal solvent.

These kind of interactions are referred to as steric repulsion. The interaction is then an almost ideal hard-core repulsion, as sketched in fig. lb. In practice such steric repulsions are often essential to screen the destabilizing van der Waals attractions. For very long polymers (such as poly-[isobutylene]), the range of the repulsive interaction is of course larger, and resembles that of swollen latex particles in a good solvent. This longer ranged repulsive potential is sketched in fig.l.lc. If on the other hand the solvent is a marginal solvent for the polymer, the energetically more favorable situation is overlap of two polymer brushes. This then results in a very short ranged attractive pair-interaction potential, superimposed onto a hard-core repulsion, as sketched in fig.l.ld. An example of such a system is a dispersion of silica particles coated with stearyl alcohol with benzene as solvent. The strength of the attraction may be increased by lowering the quality of the solvent for the polymer at hand, for example by changing the temperature, and may lead to phase separation.

Attractive interactions of short range can also be induced by the addition of free polymer under theta-conditions (such as polystyrene in cyclohexane at 34.5~ The origin of this attraction is that free polymer is expelled from regions between nearby colloidal particles, for geometric and entropic

_• ::~"~:~:~: .. ::i~i!~. :~i::!:;:;i!~i:.~ :~ii~':i~: :~:~i~i~:,~ :~i!!i!:~:;~i! :.~ ..iiii

~ ~!! ii~ ~ ii i ~ i ~ ~%i !: !~ ii~ ~ii ~ i : ~ ' . . ~ ~ ~ : . : : : ~::~:'~ .......... :,: :::~:~ ~ ......... ............ ~: :~ 5 ]- I-~ ~ - i " ~ I ' ' ~ , . , ..:::;;:_ ............... i;ii.. ...... ~ ~ i : : ~ ; ; ; . . . . 1 O0 120 140 160 180

Diameter [nm] Figure 1.2: An electron micrograph of silica particles (a) and the histogram of the size distribution (b). The horizontal bar corresponds to 100 rim.

reasons, leading to an uncompensated osmotic pressure that drives the colloidal particles towards each other. This so-ca!led depletion attraction is of a range that is comparable to the size of the polymers, and a strength that depends on the concentration of the polymers. These attractions can be strong enough to give rise to phase separation.

The potentials described above may be treated on a quantitative level, where the sometimes complicated dependence of the pair-interaction potential on the distance between the colloidal particles is derived. On several occasions in this book we will use simple expressions for the pair-interaction potential. For example, for charged particles we will use a Yukawa potential and for particles coated with polymers in a marginal solvent we use a simple square well potential, the depth of which is considered as a variable parameter. We will not go into the derivation of precise formulas for pair-interaction potentials. The section Further Reading and References contains a list of some of the books that deal with these subjects in detail.

The above mentioned model systems do have a certain degree of polydispersity, that is, there is a certain spread in size and optical properties. A typical example is given in fig.l.2. Fig.l.2a is an electron micrograph of some particles, showing the almost perfect spherical geometry of the cores, although for smaller particles (say < 10 nm radius) the spherical geometry can be less perfect. Fig.l.2b shows a histogram of the size distribution of the same particles as determined from electron micrographs like the one in fig. 1.2a. The mechanism of the chemical reaction that underlies the synthesis

10 Chapter 1.

of colloidal particles is mostly such that the relative spread in size decreases as the reaction proceeds, that is, as the average size of the particles increases. Typically the relative spread in size is about 5 - 10%.

Model rigid rod like particles are much more difficult to prepare than the above mentioned spherical particles. Rigid rod like colloidal particles that are most frequently being used for experimentation up to now is TMV-virus (where TMV stands for Tobacco Mosaic Virus, which is a plant virus). These are charged hollow cylindrical particles with a length of 300 nm and a diameter of 18 n m. Another virus that is used is the so-called fd-virus, which is a very long and thin particle. This is not really a rigid rod, but has a considerable amount of flexibility. The advantage of these virus systems is that they are quite monodisperse. A considerable effort is needed to isolate larger amounts of these virusses and fresh samples must be prepared about every two weeks. Rod like particles of latex can be synthesized by stretching elastic sheets which contain deformable spherical latex inclusions. In this way almost identical charged rods with a well defined shape are obtained. The amount of colloidal material is however very small. Classical examples of inorganic colloidal rods are vanadiumpentoxide and iron(hydr)oxide colloids. Recently, rigid rod like particles with a core consisting of boehmite (A1OOH) have been synthesized. These particles can be coated with polymers, like the spherical silica particles mentioned above. The disadvantage here is the relatively large spread in size, and the, up to now, poorly understood interactions between the rods that play a role.

Besides the potential interactions, which also exist in molecular systems, there are interactions which are special to colloidal systems. As a colloidal particle translates or rotates, it induces a fluid flow in the solvent which affects other Brownian particles in their motion. These interactions, which are mediated via the solvent, are called hydrodynamic interactions or indirect interactions. Potential interactions are most frequently referred to as direct interactions. The dynamics of Brownian motion of interacting colloidal particles is affected not only by direct interactions, but also by these hydrodynamic interactions. Since, by definition, colloidal particles are large in comparison to the size of the solvent molecules, the analysis of hydrodynamic interaction is actually a macroscopic hydrodynamic problem, that is, the colloidal particles may be viewed as macroscopic objects as far as their interaction with the fluid

1.1. CoHoidalSystems 11

is concerned. 4 For colloidal systems one cannot simply speak of"interactions" without specifying the kind of interaction, direct or indirect, that is, potential interaction or hydrodynamic interaction.

As a result of the large size difference between the Brownian particles and the solvent molecules (and free ions and possibly small polymers that may be present), the time scale on which the colloidal particles move is much larger than those for the solvent molecules. That is, during a time interval in which Brownian particles have hardly changed their positions, the solvent molecules are thermally displaced over distances many times their own size) This means that the fluid (free ions and polymers) are in instantaneous equilibrium in the field generated by the Brownian particles on a time scale that is relevant for the subsystem of Brownian particles. The pair-interaction potential for Brownian particles is, by definition, proportional to the reversible work needed to realize an infinitesimal displacement of one colloidal particle relative to a second colloidal particle. Due to the above mentioned separation in time scales, the solvent molecules (free ions, polymers) may be assumed in equilibrium with the field generated by the colloidal particles during their displacement. This reversible work is then equal to the change of the Helmholtz free energy of the total system of two Brownian particles and the solvent (free ions, polymers), and therefore consists of two parts : a part due to the change of the total internal energy of the system of two Brownian particles and the solvent, plus a change related to the change in entropy of the solvent (free ions, polymers). This free energy change, which is the relevant energy on the forementioned time scale, is usually referred to as the potential of mean force. The above discussed pair-interaction potentials for colloidal particles are such potentials of mean force.

1.1.3 Properties of Colloidal Systems

Since colloidal particles are nothing but large molecules, exhibiting thermal motion, colloidal systems undergo phase transitions just as molecular systems do. For example, colloidal systems can crystallize spontaneously, where the Brownian particles reside on lattice sites around which they exert thermal motion. The solvent structure on the other hand remains unaffected during and

4Hydrodynamic interaction is treated in chapter 5. 5Such a separation in time scales is discussed in detail in chapter 2 on Brownian Motion

of non-Interacting Particles.

12 Chapter 1.

after crystallization of the Brownian particles. It is the subsystem of colloidal particles that undergoes the phase transition while the solvent is always in the fluid state. Since the lattice spacing is now of the order of the wavelength of light, Bragg reflections off the crystal planes are visible. White light, for example, is Bragg reflected into many colours, depending on the lattice spacing and the angle of observation. For molecular crystals, Bragg reflection can be observed indirectly for example by means of X-ray experiments. Investigations on the structure of colloidal fluids can be done by means of light scattering for the same reason �9 structures extend over distances of the order of the wavelength of visible light. Besides crystallization, many other types of phase transitions in colloidal systems are observed that also occur for molecular systems. Fluid-gas phase separation (into a concentrated and dilute colloidal fluid) can occur in case of attractive interactions. Also, thermodynamically meta-stable states exist, like gel states, where colloidal particles are permanently but reversibly attached into strings which span the entire container, or glass states of large concentration where the colloidal particles are "structurally arrested", that is, where rearrangements of particle positions are not possible due to mutual steric hinderence. Besides thermodynamic instabilities, mechanical instabilities can occur in case of very strong attractive interactions, which lead to agglomeration of colloidal particles into more or less compact flocs, referred to as flocculation or aggregation.

Some of the further topics of interest concerning the first kind of colloidal systems mentioned in subsection 1.1.1 are the effect of interactions on translational and orientational Brownian motion, sedimentation, optical properties, response of microstructural arrangements to external fields such as electric and magnetic fields or an externally imposed shear flow, critical behaviour, visco-elastic behaviour, and phase separation kinetics.

All these phenomena are affected by interactions between the colloidal particles, both direct and indirect, that is, both energetically and hydrodynamically. The question then is how these phenomena can be described and how predictions can be made on the basis of a given pair-interaction potential and hydrodynamic interaction functions. This is roughly the question with which statistical mechanics is concerned.

It is the aim of the present book to establish, in a self-contained manner, the statistical mechanical theory for dynamical phenomena of interacting colloids. Needless to say that a detailed treatment all the above mentioned topics is not feasible in a single book. I had to make a choice, which is to a large extent dictated by the aim to write an introductory text, and is of course

1.2. MathematicalPreliminaries 13

also biased by my own interests. This book treats translational and rotational Brownian motion, sedimentation, light scattering, effects of shear flow, critical phenomena, and to some extent the kinetics of phase separation.

1.2 Mathematical Preliminaries

The purpose of this mathematical section is to provide a concise treatment of subjects that may not have been part of mathematics courses of readers with a physical-chemistry background. Special attention is given to the residue theorem. For those of you with a more physics oriented education this section is probably superfluous. You should be able to solve the mathematical exercises at the end of this chapter.

1.2.1 Notation and some Definitions

Vectors and matrices are always denoted by boldfaced symbols, while their indexed components, which are real or complex numbers, are not boldfaced. For example, the position in three dimensional space ~3 is a vector r with three components rj, with j - 1,2 or 3, where rl is the z-coordinate, r2 the y-coordinate and ra the z-coordinate" r - (ra, r2, ra) - (x, y, z). A vector may have more than just three entries. The number of entries is the dimension of the vector. The length of a vector a - (a a , . . . , aN) of dimension N is

given by the Pythagorian formula ~/~jU 1 [aj [2, and is simply denoted by a

non-boldfaced a or by [a 1. The length of the forementioned position vector , ,

is thus r - x/'x 2 + y2 .q_ z 2.

A hat ^ is used on vectors to indicate that they are unit vectors, that is, vectors with a length equal to 1. The unit vector in the direction of some given vector a is simply equal to fi - a/a.

More generally, a matrix M represents an ordered set of real or complex numbers Mja,...,j,, with jm - 1, 2 , . . . N for all m - 1, 2 , . . . , n (although different ranges N for each j~ are also admissible). The number of indices n is the indexrank of the matrix, and N is the dimension of the matrix. Vectors can thus be regarded as matrices of indexrank 1, since the components of a vector carry only one index. For example, the above mentioned position vector r can be regarded as a matrix of indexrank 1 and of dimension 3, since each index can take the values 1, 2 and 3.

14 Chapter 1.

The transpose M T of a matrix with elements Mij is the matrix with elements Mji, that is, the indices are interchanged. The elements above the "diagonal", where i - j , are thus interchanged with their "mirror" elements relative to the diagonal, and vice versa,

T

a l l a 1 2 a 1 3 �9 �9 �9 a l N a l l a 2 1 a 3 1 �9 �9 �9 aN1

a 2 1 a 2 2 a 2 3 �9 �9 �9 a2N a 1 2 a 2 2 a 3 2 " " �9 aN2

a 3 1 a 3 2 a 3 3 �9 �9 �9 a3N - - a 1 3 a 2 3 a 3 3 �9 �9 �9 aN3

aN1 aN2 aN3 " ' ' a N N a l N a2N a3N "�9 a N N

.(1.1)

A special matrix is the identity matrix or unit matrix I, which has elements 6ij - 1 for i - j , and 6ij - 0 for i ~ j . The 6ij is the so-called Kronecker delta. Thus, the elements of I on the diagonal, where i - j are equal to 1, while the off-diagonal elements, where i ~ j are all equal to 0. This matrix leaves vectors unchanged, that is, I . a = a for any vector a.

Vectors can be multiplied with other vectors in several ways. Two vectors a and b can be multiplied to form a matrix of indexrank 2, which matrix is denoted as ab, and has per definition components (ab)ij - aibj. Such a product is referred to as a dyadic product. Similar products of more than two vectors are referred to as polyadic products. The so-called inner product

N . a . b is defined as ~ j = l ajbj, where * denotes complex conjugation, and is itself a scalar quantity (a real or complex number). The inner product of a vector with itself is nothing but its squared length. Two vectors are said to be perpendicular when their inner product vanishes. In case a and b are 3-dimensional vectors, the outer product a x b is defined as the vector perpendicular to both a and b, with a direction given by the cork screw rule, and a length equal to ab I sin{ 0 } [, with 0 the angle between a and b. The three components of this vector are a2b3 - aab2, aabl - a l b 3 and a l b 2 - a 2 b l .

The usual multiplication of a vector a by a matrix M is denoted as M �9 a, where the dot indicates summation with respect to adjacent indices. M �9 a is thus a vector with the jth component equal to ~N= 1 Mj~a~. Summation over adjacent indices also occurs when two matrices, say A and B, are multiplied" (A B)i j N �9 - ~n=l A~,~Bnj. Such summations over adjacent indices can be generalized to more than simply one index. For example, A �9 B denotes the summation over two indices, indicated by the two dots, A �9 B - N A,~mBm~. Such summations are generally referred to as E r r , m - - 1

contractions. The number of indices with respect to which the contraction


ranges is indicated by the number of vertical dots. The contraction symbol | is often used to indicate contraction with respect to the maximum possible number of indices. For example, let A denote a matrix of indexrank n and B of indexrank m, with m > n, then,

A | B - y~ Aj,...j2 j~ Bi~ j2...j, J,+a...jm, (1.2) jl ""in

which is a matrix of indexrank m - n. Notice the order of the indices.

Let X - (x x, x ~, �9 �9 �9 x N) denote a N-dimensional vector. Functions of the variables x~, �9 �9 �9 XN can be interpreted as being functions of the vector X. The most common examples are functions of the position vector X - r - (x, y, z) in 3-dimensional space. Functions of vectors which are real or complex valued are called scalar fields or simply scalar functions. Functions of vectors which are vectors or matrices are called vector tields. For example, f ( X ) - X is a scalar field, while F ( X ) - X X is a vector field. Vector fields are usually (but not always) denoted by a capital boldfaced letter.

The gradient operator V x is a differential vector operator defined as V x - (O/OXa, O/OX2,''', O/OXN). Products of this operator with (scalar or vector) fields are much the same defined as the above described products of vectors and matrices, except that differentiation with respect to the components of X is understood. The gradient V x f ( X ) of a scalar field f is thus a vector field with entries Of(X)/Oxj. Similarly, the dyadic product V x F ( X ) is a matrix with the ijth-element equal to OFj(X)/Oxi. The divergenceof a vector field of indexrank 1 is a scalar field equal to the inner product of the gradient operator and the vector field" V x F ( X ) - U �9 ~j=l OFj(X)/Oxj. Analogous to a dyadic product of two vectors, the dyadic product V x V x is a matrix operator with components 02/OxiOxj. The first few terms of the Taylor expansion of a scalar field f ( X + A) around A -- 0 can thus be written in terms of contractions of polyadic products of the gradient operator and A as,

1 f ( X + ~ ) - f ( X ) + ~ . V x f ( X ) + - ~ A A ' V x V x f ( X )

- 1

+ 6 ~ A & A ' V x V x V x f ( X ) + . . . . (1.3)

Contractions are defined as before for vectors and matrices, except that here differentiation is understood. For example,

N a z 'v v.v f(x) - E

l,n,m---1

0 3

Am A n A t OXtOXnOXm f ( X ) .

16 Chapter 1.

A specially important operator is the Laplace operator V~c, which is a short- hand notation for Vx �9 Vx - ~Y=I 02/Ox~. In case X is the 3-dimensional position vector and F(X) is a 3-dimensional vector field with indexrank 1, the outer product V x F(X) is defined in analogy with the outer product of two vectors, where again differentiation is understood.

We always use square brackets to indicate to which part in an expression the action of a differentiation is limited. For example, the action of the first gradient operator in the combination V x f ( X ) �9 V xg(X) is ambiguous without specifying whether it acts only on f or also on #. When the first gradient operator is understood to operate on f only, this is indicated by square brackets as [Vxf (X) ] . [Vxg(X)] (square brackets are put around Vxg also for esthetical reasons). When the first gradient operator is understood to operate on both f and g, this is denoted as V x . [f(X) Vxg(X)].

1.2.2 Integral Theorems

Two very important theorems are the integral theorems of Gauss and Stokes. Let W be some volume in the N-dimensional space NN. Gauss's integral theorem states that for continuous differentiable N-dimensional vector fields r(x),

fw dX V x . F(X) - ~ w dS- F ( X ) , (1.4)

where the integral on the right hand-side ranges over the surface OW that encloses the volume W, and dS is the N-dimensional vector with a length equal to an infinitesimally small surface area on 014;, and with a direction perpendicular to that surface, pointing away from the volume. In eq.(1.4), dX is an abbreviation for dxl dx2"" dxN, an infinitesimally small volume element in NN.

Stokes's theorem states that, again for continuous differentiable fields,

fs dS. (V x F(r)) - ~ s dl. F ( r ) , (1.5)

where S is a surface in ~3, OS its boundary, and dl is a vector with a length equal to an infinitesimal length segment on the curve OS and a direction that is related to the direction of dS by the cork screw rule. Volume and surface integrals are thus expressed in terms of integrals ranging over their boundaries.


The proof of these two theorems can be found in standard texts on mathematics, and should be part of the mathematics education of any physical-chemist.

Two further integral theorems, referred to as Green's integral theorems, are an almost immediate consequence of Gauss's integral theorem. The vector field F in Gauss's integral theorem (1.4) is now chosen as F (X) - f ( X ) V x g ( X ) , with f and g scalar functions. Using that,

V x . [ f(X)Vxg(X)] - f (X)V~cg(X ) + [ V x f ( X ] . [Vxg(X)] ,

immediately yields Green's first integral theorem,

:wdX { f ( X ) V ~ c g ( X ) + [ V x f ( X ) ] . [Vxg(X)]} - ~owdS. f ( X ) V x g ( X ) . (1.6)

Interchanging f and g in the above equation and subtraction leads to Green's second integral theorem,

fw dX { f ( X ) V ~ g ( X ) - g ( X ) V ~ f ( X ) } (1.7)

- ~owdS �9 { f ( X ) V x g ( X ) - g ( X ) V x f ( X ) } .

These integral theorems play an important role in the various mathematical aspects of dynamics of colloids.

1.2.3 The Delta Distribution

On several occasions we will make use of an "infinitely sharply peaked" scalar function with a normalized surface area. This function is zero everywhere except in one particular point x - x0 in ~ where it is infinite in such a way that its integral equals 1. Being zero everywhere except in one point seems in contradiction with the condition that its integral is non-zero. Indeed this is not a function in the usual sense but belongs to the class of so-called generalized functions, or equivalently, distributions. In this subsection we will not give the general definition of a distribution but rather specialize to the delta distribution, since this is the only distribution that is used in this book.

Consider a sequence of scalar functions Cn(x), n - 1, 2 , . - - , with the properties,

L : lim,~_..~ f-~oo dx t~,(x) f(x)

- 1 , for all n , } - f(zo), (1.8)

18 Chapter 1.

Xo X Figure 1.3: A sketch o f a delta sequence together with a test-function f . The test-function is essentially equal to f (xo) in the range of x-values where ~n (x) for large n is non-zero.

for any well behaved function f.6 Such a sequence of functions is referred to as a delta sequence, centered at xo. The probably simplest example of a delta sequence is,

1 1 - < x < z o + - - Cn(x) n , for xo 2n 2 n '

= 0 , e lsewhere . (1.9)

The first condition in (1.8) is trivially satisfied. around z - xo yields,

Taylor expansion of f ( z )

F lim dx Cn(x) f ( x ) n---+ oo co

_ ~ f(~)(Xo) lim dx (bn(x)(x - xo) TM

m-'O m . c~

co ( 1 ) m + i f (~)(xo) ~+1 = E { ~ 7 1 ) i [1 - ( -1 ) ] U moon

m--O

where f(m) (Zo) is the ruth derivative of f (x) in x - zo. Only the term with m - 0 survives the limit where n ~ oc, so that also the second condition (1.8) is satisfied. Hence, the sequence (1.9) is a delta sequence. General- ly a delta sequence can be recognized by observing that the functions are increasingly sharply peaked around some x0. As sketched in fig. 1.3, for large

6The functions f for which this property is assumed to hold are referred to as test-functions, and are most commonly assumed to be infinitely continuous differentiable, with a compact support, meaning that they are zero everywhere except in a closed and bounded subset of !l~.

n, the functions Cn become so sharply peaked that f (x) ~ f(x0) over the entire range of integration where ~n (x) contributes to the integral.

For compact notation and without the necessity to specify a particular delta sequence of functions, the delta distribution ~(x - xo) is written as,

" l im" Cn(x) - 6 ( x - xo) , (1.10) n.--+ oo

and the property (1.8) reads,

f ~ dx 6(x - xo) f (x) - f (xo) . oo

(1.11)

Notice that the limit lim~__.oo r (z) does not exist in the usual sense. That is why in eq.(1.10) we used the notation" lim" : it means that integrals should be evaluated first for finite n's, after which the limit where n ~ co is taken. Such a limit is called a distributional limit.

Two somewhat more complicated delta sequences are discussed in exercise 1.3. The particular sequence in exercise 1.3a plays an important role in the theory of Fourier transformation, while the sequence in 1.3b is important in relation to Brownian motion.

The delta distribution 6(X - X0) in higher dimensions is simply defined as a product of the above defined 1-dimensional delta distributions,

~ ( X - - X o ) -- ~(X 1 - - X l O ) X " ' " X t~(X N - - X N O ) , (1.12)

with Xo - (Xl 0 , " " " , X N 0 ) . Equation (1.11) immediately carries over to the N-dimensional case,

/ dX 6(X - X o ) f ( X ) - f ( X o ) , (1.13)

where the integration range is the entire ~N. Instead of scalar functions f , vector fields may be integrated similarly.

1.2.4 Fourier Transformation

It is often convenient to decompose functions into sinusoidally varying functions. Consider first a scalar function f of the scalar x. The decomposition in sine and cosine functions can be written as,

f0 c<) f ( x ) - dk [f~(k)sin{kx} + f~(k)cos{kx}] . (1.14)

20 Chapter 1.

The so-called wavenumber or wavevector r k is equal to 27r/A, with A the wavelength of the particular sinusoidal contribution. The functions f~(k) and f~ (k) are the sine and cosine Fourier transforms of f(x), respectively. These functions measure the contribution of the particular sine and cosine contributions to f(x). If for example f~(ko) is relatively large for a particular wavevector k - ko, the function f (x ) has much the character of cos{ kox}. The above decomposition in sine and cosine functions can be written more compactly as,

1 f_" dk f (k) exp{ikx} (1 15) f ( x ) - 2---~ o~ " "

Contrary to the sine and cosine transforms f, and f~, the so-called Fourier transform f (k) of f ( z ) is a complex valued function, s Using that the complex exponential exp{ikx} is equal to cos{kx} + i sin{kx}, it is easily seen that the two above formulas (1.14,15) are equivalent, with f~(k) - [ f (k)+ f ( - k ) l / 2 r and f , (k) - i[f(k) - f(-k)]/27r. The prefactor 1/2~" in eq.(1.15) is introduced for later convenience. Although eqs.(1.14) and (1.15) are completely equivalent, the form in eq.(1.15) is more compact and mathematically more easy to handle. In exercise 1.4 you are asked to show, using the delta sequence of exercise 1.3a, that the Fourier transform can be expressed in terms of the function f (x) itself as,

f (k) - dz f ( z ) e x p { - i k x } . (1.16) O 0

Would we have introduced in eq.(1.15) a prefactor different from 1/2r, a prefactor different from unity would have been found here. This expression for the Fourier transform can be used to calculated f(k) of a given function f (x ) , provided of course that the integral exists. Calculation of f(k) from f ( x ) is referred to as Fourier transformation, while the inverse operation, calculation of f (x) from f (k), is called _Fourier inversion.

The above decomposition into sinusoidally varying functions can be generalized to functions of N-dimensional vectors X - (x~, x2 , . . . , XN). First decompose the xi-dependence of f (X) in sinusoidal functions as discussed above, with x = xi and k = kl,

1 dkx f(k~,x2,x3, . , xN)exp{ikxxl} . f ( z i , z 2 , z 3 , . . . , z N ) -

7Although k is a scalar, it is nevertheless often referred to as a wavevector. SWe frequently use the same symbol ( f in this case) for different functions, where the

argument is understood to indicate which function is meant.

1.2. Mathematical Preliminaries 21

Regard the right hand-side now as a function of z2 , . . . , XN, and decompose the x2-dependence, with z = x2 and k = k2, to obtain,

f(xl,x2, x3,' ' . ,XN)

J_ i? _-- 1 dkl dk2 f (k l k2, x 3 , " " XN)exp {i [klXl -[- k2x2]} . (27r) 2 ~ ~ ' ,

This procedure is repeated N times, leading to,

1 / / (X) - (27r)N dkf (k) exp{ik. X} , (1.17)

with the wavevector k equal to (kl, k~,..., kN). The integral in understood to range over the entire N-dimensional k-space NN. Successively applying eq.(1.16) N times yields the Fourier transform in terms of the function itself,

f(k) - f d X / ( X ) e x p { - i k . X}. (1.18)

The Fourier transform of a vector field is simply defined by the vector of which each scalar component is Fourier transformed as discussed above. Thus, the jth component of F(k) is simply the above introduced Fourier transform of the scalar function Fj(X).

Fourier transformation is not only a physically appealing thing to do, it is also a useful mathematical technique to solve differential equations. To appreciate this, consider as an example the Fourier transform of Vx �9 F(X),

f d X [Vx. F(X)] exp{- ik . X} =

]" dX Vx" IF (X)exp{- /k . X}] + ik. f dX F ( X ) e x p { - i k . X} -

f d S . F ( X ) e x p { - i k . X } + i k . F ( k ) .

In the second equation we used Gauss's integral theorem (1.4). Since the volume integrals range over the entire space, the surface integral ranges over a spherical surface with a radius that tends to infinity. Since, for finite radii R of the spherical surface, we have that (with maxlxl=R(...) denoting the maximum value of (. . .) for all X with length R ),

I f dS. F ( X ) e x p { - i k . X} i_< lids. F(x)exp{-ik. X}!

<_ maxlxl=R I F ( X ) I dS - 2 i r ( X ) l, maxlxl=R J

22 Chapter 1.

the surface integral is zero when the product of the maximum value of I F(X) I on spherical surfaces with very large radii R and the surface area of the spherical surface tends to zero as R ~ ~ . For such vector fields we see that the Fourier transform of Vx. F(X) is equal to ik. F(k). Exercise (1.5) contains some more examples of Fourier transformation of derivatives. It is always found that the gradient operator is replaced, after Fourier transformation, by ik, and of course fields are replaced by their Fourier transform. In this way, linear differential equations with coefficients that are independent of X reduce upon Fourier transformation to simple algebraic equations. Fourier inversion according to eq.(1.17) then yields the solution of the differential equation. Fourier inversion often relies on evaluation of integrals using the residue theorem, which is discussed in the next subsection. In the next subsection we also give an example where a differential equation is solved by means of Fourier transformation, which example is relevant to the interaction of two charged colloidal particles.

1.2.5 The Residue Theorem

Here we will consider integrals of complex valued scalar functions of a complex variable, so-called complex functions. The complex variable is denoted by z = x + i y, with x its real part and y its imaginary part. The function itself is generally complex valued, and can also be written as a sum of its real and imaginary part,

f (z) - u (z ) + i v ( z ) . (1.19)

Both real valued functions u and v may be regarded as functions of x and y, and we can also write,

f ( z ) - u(x, y) + iv(x, y). (1.20)

For example, in case f (z) - z 2, we have u(x , y) - x 2 - y2 and v(x, y) - 2xy. The complex number z may be visualized as the point (x, y) in ~2, which 2- dimensional space is in the present context referred to as the complex plane.

The Cauchy-Riemann relations

The derivative of a complex function is defined as for real functions of a real variable as,

f ' ( z ) - lira f ( z + h ) - f ( z ) (1.21) h..-.O h '


provided that this limit exists. The new feature over differentiation of functions of a real variable is that the point z can now be approached from various directions (see fig.l.4). For example, when z is approached along a line parallel to the x-axis, then the complex number z + h can be written as z + A = x + A + i y, with A a real number tending to zero as h goes to zero. Alternatively, z can be approached along a line parallel to the y-axis, in which case z + h - z + i A - x + i(y + A). A necessary condition for the existence of the derivative is that these two ways of taking the limit in eq.(1.21) yield the same result,

l i m f ( X + A + i Y ) - f ( x + i y ) = l i m f ( X + i ( Y + A ) ) - f ( x + i y ) A-.~o A a~o i A

The left hand-side is equal to d f ( z ) / d x , while the right hand-side is equal to d f ( z ) / d ( i y ) . Decomposing f in its real and imaginary part (see eq.(1.20)), and equating the real and imaginary parts of the above equality yields,

Ou(x,y) Ov( ,y) Oy Ox

a u ( x , y ) a v ( x , y )

Ox Oy " (1.22)

These are the Cauchy-Riemann relations, which are conditions under which the limit in eq.(1.21) taken in the two forementioned directions are equal. These relations thus provide necessary conditions for differentiability, that is, when the above relations are found invalid for a given function f at some point z, then that function is not differentiable at that point. An example of a function that is not differentiable can be found in exercise 1.6.

The converse can also be proved, provided that the derivatives in the Cauchy-Riemann relations are continuous. That is, when the Cauchy-Riemann

Figure 1.4: A point z in the complex plane can be approached from different directions.

r ~ , ' X "r

24 Chapter 1.

I

I t t I f t I i ' I

a b X

Figure 1.5: Curves in the complex plane. The curve in (a) defines y as a function of x. The curve in (b) must be split into the curves " ~ 1 and 72, each of which is described by y as a function of x. The curve in (c) is an example of a closed curve. The dashed area is the interior 7 i'u of the closed curve.

relations hold and all partial derivatives are continuous, then f is continuous differentiable. For the derivation of the residue theorem we do not need this converse statement and we therefore do not go into its proof.

Integration in the complex plane

Functions f ( z ) can be integrated over curves 7 in the complex plane. These integrals can be defined in terms of integrals that one is used to in 1 dimension. In case of curves 7 as depicted in fig.l.5a, which defines y as a function of x, the integral f.~ ranging over the curve 7 is simply defined as (with y'(x) - dy(x) /dx) ,

f~ dz f (z) - f (dx + idy) [u(x, y ) + iv(x, y)] rl

- (1 + + iv( , (1 .23)

The points a and b mark the smallest and largest value of x on 7, as indicated in fig. 1.5a. In this example, y is regarded as a function of x. This is not possible for the curve sketched in fig. 1.5b. The flick is now to write 7 as a sum of, in this example, two curves ~1 and 72, which separately allow to regard y as a function of x. The integral f.y is now the sum of f.y~ and f-n, each of which


may be evaluated as for the example of fig.l.5a. When convenient, one can of course interchange the roles of x and y, express x as a function of y, and integrate with respect to y. You are asked in exercise 1.7 to evaluate a few integrals explicitly.

Of particular interest are closed curves (such as the one sketched in fig.l.5c). Integrals ranging over such closed contours can again be written in terms of integrals with respect to x or y as discussed above for the example of fig. 1.5a.

Cauchy's theorem

Cauchy's theorem is basically a simplified version of Stokes's integral theorem (1.5). Suppose that the surface S is located in the (x, y)-plane of Na (see fig.l.6). The vector dS is then equal to dx dy (0, 0, 1), and dl points in the anti-clockwise direction. Consider fields F(r) of the form (-u(x, y), v(x, y), 0), with u and v continuous differentiable functions. Since in this case, V x F(r) - (0, 0, Ov/Ox + Ou/Oy), Stokes's integral theorem reduces to,

fs dx dy { Ov(x, y) Ox

+ ~o,~ {-dx u(x, y) + dy v(x, y)} .

Replacing v by u and u by - v gives,

{Ou(z,y) fs dx dy -O-x Ov(x' Y) } - ~o {dx v(x y) + dy u(x y)} .

b y S ~

When u and v satisfy the Cauchy-Riemann relations (1.22), the left hand-sides

dS

•

Figure 1.6: The special choice of the surface S in Stokes's integral theorem.

26 Chapter 1.

of the two above equations vanish. Hence,

o - g ~ {d~ ~(~, y) - dy v(~, ~ ) } ,

0 - ~s {dxv(x,y) + dyu(x,y)} .

Identifying the (x, y)-plane with the complex plane, OS with a closed curve 7, and writing f - u + iv, it follows that,

~ dz f(z) - ~(d~ + i~y) (~(~, y) + iv(~, y))

{d~ ~(~, y) - d~ v(~, y)} + i ~ {d~ v(~, y) + dy u(~, y)} - O.

We thus found that when f(z) is a continuous differentiable function, that,

~ dz f (z) - O . (1.24)

This is Cauchy's theorem. Continuous differentiability of f is not required throughout the complex plane, but only within a set that contains 7 together with its interior .),int. 9

Cauchy's theorem can be used to show that a continuous differentiable complex function is infinitely continuous differentiable, meaning that all higher order derivatives are continuous. Such functions are also called analytic functions. We do not go into the proof of this statement.

The residue theorem

We are now in the position to derive the residue theorem. Only the simplest version of this theorem is used in this book, where only so-called first-order poles are encountered. The discussion of the residue theorem is therefore limited here to that simplest form.

Consider the following integral over a closed contour 7,

~ dz a(z) Z - - ZO

with g continuous differentiable (or equivalently, analytic) within a set that contains 7 and its interior 7 int. When z0 is not in ,,lint nor on 7, the integrand

9The interior of 7 is the entire region that is enclosed by 7 (the dashed area fig. 1.5c).


Figure 1.7" Deformation of the integration contour 7 (solid curve) to the contour "7 ~t , which includes the curves "~1, "~2 and the circle C~(zo) with arbitrary small radius e around Zo, with dock-wise orientation.

f ( z ) - g ( z ) / ( z - zo) satisfies the conditions that go with Cauchy's theorem. Hence,

~ dz g(z) = 0 , for zo r T U Tint . (1.25) Z -- Zo

When zo is in 7 int, however, the integrand is generally infinite at z - zo and therefore certainly not analytic. We can, however, exclude the point z0 from the interior of an extended contour, by deformation of the integration countour as depicted in fig.l.7. At some arbitrary point on 7, the contour is extended towards zo, by a curve 71 say, a circle C~(zo) with an arbitrary small radius e encloses z0, and the curve 0'2 closes the contour again. The superscript c on CC(zo) is used to indicate that the circle is traversed in clockwise direction. When no such superscript is added, an anti-clock-wise orientation is understood. For example, "7 is traversed anti-clock-wise, while 7 ~ is the same curve traversed in clockwise direction. The extended contour ,.[ext __,.[ _~_ 71 "~- 72 "~- CCe ( z o ) does not contain the point z0 in its interior nor is Zo o n ,,/ext, SO that according to Cauchy's theorem,

dz g(z) = O. ext Z -- Zo

The two curves "/1 and 72 are arbitrary close to each other. The integrals ranging over these two curves then cancel, since they are traversed in opposite

28 Chapter 1.

direction. It thus follows that,

d z g (z )_ _ d z .g (z )_ . Z - Zo ,(zo) Z - Zo

(1.26)

Notice that C, (zo) here is the circle traversed in anti-clock-wise direction. The integral on the right hand-side is evaluated by noting that points on the circle can be written as z - zo + e exp{iqo}, with 0 < ~ < 27r. Since e is arbitrarily small, we have,

fo dz 9(z) = lim ieexp{i~o}d~ 9(zo + e exp{i~}) = 27rig(zo) ,(~o) z - Zo ~lo ~ ~ �9 e exp{iq;}

= d z

where it is used that g is continuous, implying that lim, lo g(zo + e exp{i~}) - 9(Zo). We thus arrive at the residue theorem (in its simplest form),

g ( z o ) - 27ri g(z)

Z - - Z 0

, for all zo E 7 i" (1.27)

provided that 9(z) is continuous differentiable for all z in a set that contains both 7 and ?int. In this simple form, the residue theorem is also commonly referred to as Cauchy's formula.

An application of the residue theorem and Fourier transformation

The example treated here is relevant to the screened Coulomb interaction potential described in section 1.1. The differential equation that we arc going to solve here describes the electrostatic potential around a small charged colloidal particle. This potential is not simply the Coulomb potential since ions in the solvent are attracted or repelled by the colloidal particle, so that a charge distribution around the particle is formed, which is reffcrcd to as the double layer. This charge distribution screens the charge of the colloidal particle to some extent, giving rise to an electrostatic potential that goes to zero faster than the Coulomb potential at large distances from the colloidal particle. The differential equation for the electrostatic potential is derived in exercise 1.9a. In exercise 1.9b the solution that will be obtained below is used to calculate the interaction potential between two charged colloidal particles, which turns out to be the Yukawa potential mentioned earlier.

The differential equation for the electrostatic potential ~(r) reads,

V 2 ~ ( r ) - x 2 ~ ( r ) - Q---6(r), (1.28)


with a a constant with dimension m -1, c the dielectric constant of the solvent, and 5(r) the 3-dimensional delta distribution centered at r - O. The delta distribution describes the presence of the small charged particle, carrying a total charge Q, which is assumed to be located at the origin. As we have seen in the subsection on Fourier transformation, V is replaced by ik upon Fourier transformation (see also exercise 1.5). This implies that V 2 - V �9 V is replaced by - k 2. Fourier transformation of the above differential equation thus gives,

(k2 + n2)r - Q ,

where it is used that the Fourier transform of the delta distribution centered at the origin is equal to 1 (see eq.(1.13) with X - r and f - e x p { - i k �9 r} ). Fourier inversion thus leads to,

Q exp{ik, r} (I)(r) - (27r)3c f d k ~2 ~ xi . (1.29)

We now transform the integration to spherical coordinates (k, O, q;). Jacobian of this standard transformation is k 2 sin{ 0 }, so that,

The

Q fo~176 ~ r (2~)3c k2 j

k ~ + x2 dl~ exp{ikl~, r} ,

where tr - k/k, the unit vector with the same direction as k, and ~ dk the integral with respect to the spherical angular coordinates O and ~, which is an integral ranging over the unit spherical surface,

J dkexp{ ikk r} fo 2~ fo ~ �9 - dT dO sin{O} exp{ik, r} .

A little thought shows that this integral is independent of the direction of the position vector r. We can therefore choose that vector along the z-direction, so that k . r - kr cos { O }. Hence, with x - cos{ O },

/~ld 2rr [exp{ikr} - exp{- ikr}] . dl~ exp{ikl~, r} - 27r x e x p { i k r x } - ~ r

We thus obtain,

(I)(r) - 87r2e i r Q f_~~ dk - - - 7 - k 2 k tr 2 [exp{ i kr } - e x p { - i k r }] , (1.30)

30 Chapter 1.

Y

R*

X

|

Figure 1.8: Closing integration contours in the upper (a) and lower (b) half of the complex plane.

where we used that the integrand is an even function of k to replace f o dk by 1 oo ~_~ dk. Consider the integrals,

f_,o k 1+ - oo dk k2 + tc 2 exp{4-ikr}. (1.31)

The potential is then equal to �9 - 8~, i~ [I+ - I_]. We are going to evaluate both integrals I+ with the help of the residue theorem. The first step is to transform the integrals into integrals ranging over a closed contour in the complex plane. This can be done by interpreting the integration over ( - o o , +oo) as integration over the real axis in the complex plane, and by adding integrals over semi circles with infinite radii in the upper or lower part of the complex plane (see fig.l.8). Let Cn+ denote the semi circle of radius R in the upper (+) or lower ( - ) complex plane. In exercise 1.10 you are asked to prove Jordan's lemma, which states that for r > 0,

lim ~ d z f ( z ) exp{+izr} - 0 , R----~oo R4-

when lim max~eca, I f ( z ) I - 0 . R---,oo

(1.32)

This lemma can be understood intuitively by noting that all complex numbers z on Cn+ can be written as, z = R[cos{~p} + i sin{~p}], with 0 < 9~ < 7r, hence [ exp{+izr} l= exp { - R r sin{~}}, so that the integrand tends to zero exponentially fast as R ~ oo. On Cn_ on the other hand, z - R[cos{qo} - i sin{~v}], with 7r < ~o < 27r, so that, [ exp{ - i z r} 1- exp { - R r sin{~o} }, and again the integrand tends to zero exponentially fast as R --o c~. We can thus

1.3. Statistical Mechanics 31

add integrals over the semi circles at infinity since they are zero. The result is an integral over a closed contour in the complex plane, allowing for an application of the residue theorem to evaluate that integral. Such a procedure is called closing of the integration contour in the upper (or lower) complex half plane. Let "7+ denote the closed contour in the upper (+) or lower ( - ) complex half plane, as depicted in fig. 1.8a and 1.8b respectively (note that 7+ is anti-clockwise, but that "7- is clockwise). The integrals in eq.(1.31) are thus equal to,

I• - ~+ dz Z

(z + ix)(z - i ~ ) exp{+izr} - 4-7ri e x p { - x r } ,

where the residue theorem (1.27)is used with g ( z ) - z exp{:kizr}/(z+ix) and z0 - +ix. The solution of the differential equation (1.28) is thus found to be equal to,

Q exp{ -x r} (1.33) r 47re r "

The parameter x -1 is the distance over which the charge of the colloidal particle is effectively screened, and is referred to as the screening length, or sometimes the Debye screening length. This result is used in exercise 1.9b to calculate the potential of mean force between two charged colloidal particles.

The above procedure of closing a contour in order to be able to apply the residue theorem is used in this book on several occasions. Details are usually given either in exercises or appendices. A few exercises are added to this chapter to get used to these kind of calculations.

1.3 Statistical Mechanics

1.3.1 Probability Density Functions (pdf's)

It is not feasible nor meaningful to solve Newton's equations of motion for a collection of many particles" the problem is too complicated and the initial values for the position coordinates and momenta that must be specified are not known when an experiment is performed. This is where statistical approaches are useful, where one asks for the probability that, for example, the position coordinates and momenta take certain specified values, to within a certain accuracy, at some specified time. In particular one can ask for the probability that certain initial conditions occur.

32 Chapter 1.

Imagine a collection of macroscopically identical systems (for example, colloidal suspensions). Thermodynamic variables for each system are the same, but of course microscopically each of the systems is generally in a different state, that is, the position coordinates and momenta of the particles in each system at a certain instant in time are generally different. Such a collection of macroscopically identical systems is referred to as an ensemble. The phase space for spherical particles is defined as the 6N-dimensional space spanned by the position coordinates rl, �9 �9 �9 rN and momenta Pl, �9 �9 �9 PN of all N particles in each system. The instantaneous values of positions and momenta specify the microstate of a system, and is represented by a single point in phase space. The evolution of positions and momenta in a system is described by a curve in phase space. Now suppose that we made a photograph of the entire ensemble, and that the microstate of each system in the ensemble is determined from that photograph. 1~ In this way a single point in phase space is assigned to each of the systems, resulting in a point distribution for the ensemble. The density of points is proportional to the probability of finding a single system in that microstate at that particular time. The probability density function (abbreviated hereafter as pdf) P(X, t) of X -- (rl," �9 �9 rN, Pl," �9 ",PN) is now defined as,

P ( X , t ) d X the probability that positions and momenta

are in ( X , X + dX) at time t . (1.34)

Here, (X, X + dX) denotes an infinitesimal neighbourhood of X of extent dX - dry . . , drN dp~. . , dpN. The pdf is normalized in the sense that,

f dXP(X,t) - 1. (1.35)

Consider a function f - f (X) of position coordinates and momenta. Such functions are referred to as phase functions, and may be scalar functions or vector fields. Phase functions are the microscopic, thermally fluctuating counterparts of macroscopic variabales. Frequently, phase functions, and also (a subset of) the phase space coordinates themselves, are alternatively referred to as stochastic variables. The macroscopic variable corresponding to a phase function is obtained by ensemble averaging, and is given by,

< f > -- / dX P(X, t) f ( X ) . (1.36)

10Fo r the determination of the momenta one should actually make two photographs.

The brackets < --- > are nothing but a short-hand notation for the integral on the right hand-side. This average is the ensemble average of f. Alternatively one may introduce the pdf P (f, t) for a stochastic variable f instead of X, by rewriting the above equation as,

< f > - / d f P ( f , t ) f . (1.37)

This pdf is equal to,

P( f , t) - / dX P(X, t) 5 ( f - f (X)) , (1.38)

as is easily verified by substitution into eq.(1.37), noting that f d f 6 ( f - f (X)) f = f (X) . The above expression for P( f , t) is simply a counting of the extent of the subset in phase space where f (X) attains a particular numerical value f , weighted with the local point density.

Other more complicated pdf's can be defined. For example, P (X, t, Xo, to) is the pdf for X to occur at time t and Xo at some earlier time to, or more presicely,

P (X, t, X0, to)dXdXo the probability that positions and momenta

are in (X, X + dX) at t ime t (1.39)

and in (Xo, Xo + dXo) at t ime to < t .

By definition, the connection with the earlier defined pdf is,

P(X, t) - f dXo P(X, t, Xo, to). (1.40)

Equivalently, one may define pdf's like P ( f , t, g, to) where f and g are phase functions. Just as above, we have that, P ( f , t) - f dg P ( f , t, g, to).

Two stochastic variables f and g are said to be statistically independent when P( f , t, g, to) - P( f , t)P(g, to). An ensemble average like < f g > is then simply equal to the product of the averages < f > and < g >. For very large time differences t - to, phase functions always become statistically independent.

Conditional pdf's

Consider again the photograph of the ensemble discussed earlier, which allows for the determination of the microstate of each of the systems in the

34 Chapter 1.

f fo

t to Figure 1.9: Two possible realizations of the time evolution of the phase function f , given that at time to the phase function had a particular value fo. The smooth curve is the conditional ensemble average < f > fo.

ensemble. Now consider only those systems which at a certain earlier time to < t were in a particular microstate Xo. This subset of systems in the ensemble is an ensemble itself, and pdf's may be defined as above for this new ensemble. This new ensemble is an ensemble of systems which are prepared in microstate Xo at time to. The pdf's for X are pdf's with the constraint that at an earlier time to the system was in the microstate Xo. Such pdf's are called conditionalpdf's, and are denoted as P(X, t[Xo, to). Hence,

P(X, t l Xo, to)dX the probability that positions and momenta

are in (X, X + dX) at t ime t , given (1.41)

that their values were Xo at t ime to < t .

Similarly, conditional pdf's of phase functions f , given that the phase function had a particular value fo at an earlier time may be defined as,

P( f , t l fo, to)df the probability that the phase function is

in ( f , f + df ) at t ime t , (1.42)

given that its value was fo at t ime to < t .

By definition, the connection between conditional pdf's and the earlier discussed pdf's (sometimes referred to as unconditional pdf's) reads,

P(X, t[ Xo, to) -- P(X, t, Xo, to), (1.43) P(Xo, to)

and similarly for pdf's of phase functions. The conditional ensemble average of a phase function f , given that f - f0 at some earlier time to, is denoted as

< f >f0,

< f > f 0 - /dfP(f, tlfo, to)f. (1.44)

This ensemble average is in general a function of the time t. The phase function evolves in time for each system in the ensemble differently, since there are many different microstates Xo that satisfy fo - f(Xo). Two such different realizations are depicted in fig.l.9. The conditional ensemble average is the average of all those possible realizations.

One can of course define time independent conditional pdf's. For example, one may ask for the probability that particles 3, 4 , . . . , N have positions ra, r4 , . - - , rN, given that particles 1 and 2 have fixed positions rl and r2, respectively. That conditional pdf is, in analogy with eq.(1.43), equal to,

P ( r l , . . . ,rN) (1.45) P ( r3 , . . - , r N [ r l , r2) -- P 2 ( r l , r2) '

where P2(ra, r2) is the pdf for (ra, r2), which pdf will be discussed in more detail later.

To determine an ensemble average experimentally, there is no need to actually construct a collection of many macroscopically identical systems. When an experiment on a single system is repeated independently many times, the average of the outcome of these experiments is the ensemble average. In many cases only a single experiment is already sufficient to obtain the ensemble average. When the system is so large that the quantity of interest has many independent realizations within different parts of the system, an ensemble average is measured in a single experiment that probes a large volume within the system.

Reduced pdf's

We shall often encounter ensemble averages of stochastic variables which are functions of just one or only two particle position coordinates. The ensemble average of a phase function of just two position coordinates, rl and r2 say, is,

< f > = f d r l . . . f d r N P ( r x , . . . , r N , t)f(rx,r2)

= f dr1 f dr2 P2(rx, r2, t ) f ( rx , r2), (1.46)

36 Chapter 1.

where,

P2(rl, r2, t) - f dr3 . . . f drNP(r l , . . . , rN, t) . (1.47)

P2 is referred to as the reduced pd f of order 2, the two-particle pd f or simply as the second order pdf. This equation can be regarded as a special case of P ( f , t) - f d # P ( f , t, #, to), with to - t, f - (rx, r2) and g - ( r3 , . . . , rN). Similarly, ensemble averages of phase functions of just one position coordinate are averages with respect the first order reduced pdf,

P~(rl,t)- f dr2...f drNP(rl, ,rN, t). (1.48)

Higher order reduced pdf's (such as Pa(r~, r2, ra, t) ) are similarly defined. The probability of finding a particle at some position r at time t is pro-

portional to the macroscopic number density p(r, t), which is the average number of particles per unit volume at r and at time t. Normalization sets the proportionality constant,

1 Pl(r, t) - ~ p(r, t ) . (1.49)

A similar relation for P2 will be discussed later, when the pair-correlation function is introduced.

When the system is in thermal equilibrium, the time independent pdf for the position coordinates is proportional to the Boltzmann exponential of the total potential energy r �9 �9 �9 rN) of the assembly of N particles,

P ( r l , . - . , rN) - - e x p { - - f l ( X ) ( r l , - - . , rN)} Q ( N , T , V )

, ( 1 . 5 0 )

with/~ - 1/kB T (kB is Boltzmann's constant and T is the absolute temperature) and Q(N, T, V) is the configurational partition function,

Q ( N , T , V) - / drl . . . / d r N exp{--flr �9 (1.51)

When the total potential energy ~ is known, the reduced pdf's can thus be calculated in principle for systems in equilibrium, except that the integrals in eqs.(1.47,48) are too complicated. Finding good approximations for the first few reduced pdf's for systems in equilibrium, either from eqs.(1.47,48) or by other means, is the principle goal of equilibrium statistical mechanics. These


equilibrium pdf's are often a necessary input for explicit evaluation of non- equilibrium ensemble averages also. Since this book is on non-equilibrium and dynamical phenomena we will not go into the various approximate methods to calculate these equilibrium pdf's, but merely mention some of their properties together with definitions of related functions.

The pair-correlation function

When particles do not interact with each other, all reduced pdf's are products of Pl'S. In particular, P2(ra, r~, t) - P1 (rl, t) P1 (r2, t). Interactions can formally be accounted for by an additional factor g(ra, r2, t), the so-called pair-correlation function,

1 P2 (rl, r2, t) - P, (r,, t) e , (r2, t) g(ra, r2, t) - ~-sp(r , , t) p(r2, t) g(r l , r2, t ) .

(1.52) Similarly, the three-particle correlation function g3 "corrects" for the effect of interactions for the third order pdf P3,

P3(rl, r2, r3, t) - /91 (rl, t) Pl(r2, t) Pl(r3, t) g3(rl, r2, r3, t) (1.53) 1

- N3 p(rl, t) p(r2, t) p(r3, t) g3(rl, r2, r3, t ) .

For large distances [r~ - r 2 I between two particles, the pair-correlation function attains its value without interactions, which is 1 by definition. The three-particle correlation function becomes equal to i when all three particles are well separated.

In case of homogeneous and isotropic fluids in equilibrium, the pair- correlation function is a function of r - I r~ - r21 only, and can be expanded in a power series of the number density fi - N / V as,

g(r) -- g o ( r ) + /~gl(r) + ~2 g2(r) + ' " . (1.54)

The leading term go describes interactions between two particles without the intervening effects of other particles. This then is nothing but the pair- correlation function for a system containing just two particles. It is the relevant pair-correlation function for systems which are so dilute that events where three or more particles interact simultaneously are unlikely. According to eqs.(1.49-51), with/91 - 1 /V, we thus obtain,

go(r - [ ra - r2 I) - V2 exp{-/3V(r)} f drx f dr2 e x p ( - ~ V ( r ) } '

38 Chapter 1.

where V(r) is the potential energy of an assembly of just two particles, the pair-interaction potential. Now noting that,

f dr1 f dr2 exp{-/3V(r)} - f dr2 f d ( r a - r2 )exp{- f lV( r )}

= V ( f d r [ e x p { - f l V ( r ) } - 1 ] + V} ~ V 2 ,

since the integral in the last equation is of the order R~,, with Rv the range of the pair-interaction potential, it is found that,

go(r) -- exp{ - f lV(r )} . (1.55)

In this book we will use the phrase "on the pair-level ", whenever interactions between three or more (colloidal) particles simultaneously are disregarded. Hence, eq.(1.55) is the pair-correlation function on the pair-level, and can be used to calculate ensemble averages for dilute systems. In general, the pair-correlation function does include "higher order interactions", that is, it includes the intervening effects of the remaining particles on the interaction between two given particles. A systematic approach where the expression (1.50), after substitution into the definition (1.47) for P2, is expanded in terms of Mayer-functions, leads to,

f g l (r) --I r l - - r2 [) - exp{-/3V(I ra - r21)}/dr3 f(I rx - r3 [) f([ r2 - r3 I),

i I J

(1.56) where f ( r ) is the Mayer-function f ( r ) - exp{-/3V(r)} - 1. The derivation of this result can be found in most standard texts on statistical mechanics, a few of which are collected in the section Further Reading and References at the end of this chapter.

In exercise 1.12, 9x is calculated explicitly for hard-sphere interactions, with the result (the subscript "hs" stands for "hard-spheres"),

g h s ( r ) - - go( r ) -~- /~g l ( r ) - 1, for r > 4 a ,

[ ] = l+qo 8 - 3 - + for re[2a 4a) , a ~ ~

= O, for r < 2 a , (1.57)

where a is the radius of the hard-core and r ~aa/~ is the fraction of the total volume that is occupied by the cores of the particles, the so-called volume fraction. This pair-correlation function is plotted in fig. 1. lOa for ~ - O. 1. At


1

0.5 ~ _ iI- I-J- 2 4 2 4 21+

Figure 1.10: The pair-correlation function to first order in concentration for hard-spheres (see eq.(1.57)) with qo - O. 1, (a), a sketch for hard-spheres at larger concentrations (b), and for charges spheres with a long ranged repulsive pair-interaction potential (c).

larger concentrations, the pair-correlation function develops a large contact value (defined as the value of g at r = 2a + e with e arbitrary small), and peaks appear at larger distances, as depicted in fig. 1. lOb. The pair-correlation function behaves quite differently in case of long ranged and strongly repulsive interacting particles, as depicted in fig. 1.10c. This may be the case for charged colloidal particles in de-ionized solvents. First of all, the contact value of g is zero : the probability that two particles touch is zero due to their strong repulsive interaction. Secondly, the peak position shifts to smaller distances for higher concentrations. This is due to the tendency of the particles to remain far apart from each other so as to minimize their (free) energy. The peak position varies approximately as 1/fi 1/3 for such systems.

Consider a colloidal particle at the origin. One may ask about the average density around that particle, which density is a function of the distance from the particle due to interactions. This density is N PI, as in eq.(1.49), with the additional condition that there is a particle in the origin. According to eq.(1.43) (with t = to, X0 = 0 = the position of the particle at the origin and X - r) this conditional probability is equal to P2(r, r' - O, t)/P~ (r' - O, t). Hence, from the definition (1.52) of the pair-correlation function,

Number density at r with a particle at the origin -

N P 2 ( r ' r ' - O ' t ) r' / 9 1 ( r ' - O , t ) = p(r , t ) g(r, - O , t ) . (1.58)

40 Chapter 1.

Well away from the origin, where interaction with the particle at the origin is lost so that g(r, r' - 0, t) - 1, this is simply the macroscopic density p(r, t), as it should. The peaks in the figures 1.10b,c thus imply enhanced concentrations around a given particle at those distances. For hard-core interactions there is also an enhanced concentration close to contact. This enhancement is due to depletion : particles are expelled from the gap between two nearby particles leaving an uncompensated repulsive force from particles outside the gap that drives the two particles together. Each colloidal particle, charged or uncharged, is thus surrounded by a "cage" of other particles.

The "effective interaction potential" veff(r) can be defined for isotropic and homogeneous systems in equilibrium as,

g(r) - exp{- f lV ~ff (r)} . (1.59)

According to eq.(1.55) this effective potential is equal to the pair-interaction potential on the pair-level. The average force F ~ff (r) between two particles for arbitrary concentrations can be shown to be equal to - V V ~ff (r) (see exercise 1.11), and includes the effects of intervening particles. Hence, by definition, Feff(r) _ /~-1~7 ln{g(r)} -- ~-1~. dln{g}(r)/dr, so that there is an attraction for those distances where dg(r)~dr < 0. For hard-spheres near contact there is thus attraction, the depletion mechanism for which was already explained above. Around the peak in the pair-correlation function the effective force changes from strongly repulsive to attractive. Multi particle interactions may thus lead to attractions even if the pair-interaction potential is purely repulsive.

1.3.2 Time dependent Correlation Functions

Consider the conditional ensemble average,

< g >So - fdgP(g, tlfo, to)g. (1.60)

This ensemble average is a time dependent function, also for systems in equilibrium. It describes the average evolution of the phase function #, given that at time to < t the value of the phase function f was fo. When this conditional average is subsequently averaged with respect to fo, the result is simply the unconditional ensemble average < g > �9 since P(g, t I f o, to) - P(g, t, fo, to)/P(fo, to) we have,

< < g >/o > -- f dfo P(fo, to) f dg P(g, t I fo, to) g

= - < >

The second pair of brackets < ..- > on the left hand-side denotes ensemble averaging with respect to the initial condition fo. This ensemble average is time independent for systems in equilibrium. In an experiment one usually measures an unconditional ensemble average, that is, the system is not prepared in a certain state before the experiment is started. The most simple unconditional ensemble average that contains information concerning the dynamics of stochastic variables also for systems in equilibrium is the correlation function o f f and g, defined as,

<< g >So fo > - / d f o P ( f o , to)fdgP(g, tlfo, to)gfo. (1.61)

Alternatively, the correlation function may be written in terms of pdf's of phase space coordinates X. Using eqs.(1.38,43) it is easily shown that, 11

< f(X(to)) g(X(t)) > - f xf xo f(Xo) g(X) P(Xo, to) P (X, t l Xo, to)

= f xf xo f ( X o ) g ( X ) P ( X , t, Xo, to), (1.62)

where the left hand-side is nothing but a more transparent notation for the correlation function < < g >fo fo >. The correlation function is a function of t and to. For equilibrium systems, however, in which there is no preferred instant in time, the correlation function depends only on the difference t - t o .

For very large time differences t - to, the dynamics of g becomes independent of whatever value f had at time to. Formally this means that

P(Xo, t) P(X, t l Xo, to) - P (X, t, Xo, to) ~ P(X, t) P(Xo, to).

The correlation function is thus seen to tend to < f > < g > as t - to ~ oc. The time required to render f and g statistically independent, to within some degree, is referred to as the correlation time for f and g.

For colloidal systems, a statistical description is feasible on a time scale that is large compared to correlation times for the solvent. This is a description where quantities are averaged over a time interval that is large compared to

::The natural extension of eq.(1.38) to be used here is, f dX f dXo P(X, t, Xo, to) di(g - g(X)) 5(]'0 - f(Xo)).

P(g, t, ./'o, to) =

42 Chapter 1.

the correlation time of the solvent. On such a coarsened time scale there is an accompanied coarse graining of phase space coordinates, corresponding to the changes of position and momenta during that time interval.

An alternative expression for correlation functions can be obtained from equations of motion for pdf's of phase space coordinates. Chapter 4 is devoted to the derivation of such equations of motion. These equations are of the form,

o_ P(X, t) - s P(X t) Ot ' '

(1.63)

where/~ is the time evolution operator (mostly a differential operator) that acts on the phase space variables X. At time to the phase space variables are supposed to be equal to Xo. The pdf is thus infinitely sharply peaked around X - No at time t - to. From the normalization (1.35) it thus follows that,

P(X, t - to) - 5(X - Xo), (1.64)

with 6 the delta distribution. Note that the solution of the equation of motion (1.63) with this initial condition is actually the conditional pdf P(X, t lXo, to). The formal solution reads,

P(X, t] Xo, to) - exp{/~(t - to)} 6(X - Xo), (1.65)

where the operator exponential is defined by the Taylor series of the exponential function,

OO

exp{/~(t - to)} - ~ ( t - to) n/~n n = 0 n [ "

(1.66)

Here, for n > 0,/~'~ - ~ / ~ . . . ~ , while/~o _ ~-, the identity operator which n •

leaves phase functions unaltered, that is, 2 f - f for any phase function f. That the formal expression (1.65) is indeed the solution of eqs.(1.63,64) follows from differentiating term by term,

0 exp{/~( t - to)} Ot

O 0 -- E ( t - - t o ) n-1 /~n

n = l ( n - 1 ) !

_ s ( t - to)"

n-"O

/~n _ ~ exp{/~(t - to)}.

Substitution into eq.(1.62) and integrating with respect to X0, using the definition (1.13) of the delta distribution (with the roles of X and X0 interchanged) yields,

< f(X(to))g(X(t))>- f dXg(X)exp{/~(t to)} [ f (x ) P(X, to)].

(1.67) For systems in equilibrium, where P is time independent, this expression shows explicitly that the correlation function is a function of the time difference t--to only. The advantage of this expression over eq.(1.62) is that the conditional pdf does not appear explicitly. In principle this expression can be evaluated once the operator s in the equation of motion (1.63) is known. A drawback on eq.(1.67) is that each term in the operator exponential must be evaluated to obtain the correlation function, and this is in general technically not feasible. Since the n th term in the definition (1.66) of the operator exponential is ,-~ ( t - to) n, evaluation of the first few terms in the expansion leads to an expression that is valid for short times, where t is not much larger than to. Such expansions are referred to as short-time expansions. A special case for which eq.(1.67) can evaluated explicitly for arbitrary times is given in exercise 1.14.

1.3.3 The Density Auto-Correlation Function

A particularly important stochastic variable is the microscopic number density p of colloidal particles, which is defined as (5 is the delta distribution introduced in subsection 1.2.3),

N

p ( X l r ) - r N I r ) - - 5 ( r j -- r ) . j = l

(1.68)

The summation ranges over all the colloidal particles in the suspension. In- tegrating this phase function with respect to r over some volume A V yields the number of colloidal particles in that volume for the particular choice of position coordinates. This can be seen as follows. Let x(r) - 1 for r E A V, and 0 otherwise, the so-called characteristic function of AV. The integral of the microscopic density over A V can then be written as,

N N N

~_~/zxvdr ~ ( r j - r) - ~ / dr ~ ( r j - r ) x ( r ) - ~ x(rj) �9 j=l j=x j=x

44 Chapter 1.

In the last step we used the definition (1.13) of the delta distribution with X - r and X0 - rj and f ( r ) - x(r). Since, by definition, x ( r j ) i s 1 for rj E A V and 0 otherwise, the right hand-side in the above equation is precisely the number of particles in AV. The ensemble average of this phase function is the macroscopic number density p(r, t) at position r and time t,

N P / *

p(r, t) - J dra-.. J P(r l , . - . , t) E (rj - r) - N Pl(r, t), j = l

in accord with eq.(1.49). The microscopic density may be decomposed into sinusoidally varying

components by Fourier transformation (see subsection 1.2.4). Fourier transformation of eq.(1.68) with respect to r yields,

N

p ( X l k ) - p ( r l , . . . , r N [k) - ~ e x p { - i k - r j} . (1.69) j = l

Consider the correlation function of two Fourier components of microscopic densities, the so-called density auto-correlation function, which is denoted here as S(k', k, t - to) for brevity,

1 S(k', k, t - to) - ~ 

N

_ 1 ~ <exp { - i lk ' ri(to) + k r j ( t ) ]}> N i,j=~

The prefactor 1/N is added for later convenience. When the system under consideration is homogeneous, the density auto-correlation function is zero, unless k - - k ' . This can be seen as follows. Without loss of generality, all particles may be assumed identical. All terms in the summation in the above expression for the correlation function with i - j are then equal, and all terms with i ~ j are equal, so that,

S(k', k, t - to) - <exp {- i [k ' . r l ( t0) + k.rx(t)]} >

+ ( N - l ) <exp { - i [ k ' . r l ( t o ) + k.r2(t)]} > .

The ensemble averages are now written, according to eq.(1.62) as,

1/ / S(k', k, t - to) = ~ dr dro exp { - i [ k ' . ro + k . r]} P~(r , t [ ro, to)

1) / dr / dro exp {- i [k ' �9 ro + k . r]} P12(r, t[ ro, to). + (g-

The indices 11 and 12 on the conditional pdf's are used to indicate that r and ro are the position coordinates of one and the same or two different particles, respectively. For example, P~l(r, t I ro, to) is the conditional pdf for the position coordinate of a particle at time t, given that the same particle is at ro at time to. In the homogeneous system, both these pdf's are functions of the relative separation R_ - r - ro, This property is referred to as translational

1 invariance. Now using the mathematical identity (with R+ - 7(r + ro) ),

1 k ' - ro + k . r - (k' + k ) . R+ + ~ ( k ' - k) . R _ ,

yields,

1 f dR+ exp{- i (k ' + k ) . R+} S(k ' , k, t - to) =

f i k' k ) R _ } [ P I ~ ( R _ , t t o )+(N 1)P12(R-t to)] • d R _ e x p { - ~ ( . . . . , - �9

An obvious short-hand notation for the pdf's is used here. The crucial thing to note is that the integral f dR+ exp{- i (k ' + k) �9 R+ } is precisely equal to (27r)36(k ' + k) (see exercise 1.3a). Hence, unless k' = - k , the density auto-correlation function is zero. a2 For translationally invariant systems the meaningful density auto-correlation function is therefore S~(k, t - to) - S(k' - - k , k, t - to), or,

1 S~(k, t - to) = ~ < p* (X(to) Ik) p (X(t) lk) >

1 N -- N i ~,3---1 <exp { ik - ( r i ( to ) - r j ( t ) )} > . (1.70)

The superscript * stands for complex conjugation. This function is referred to as the collective dynamic structure factor. In the absence of interactions, for very dilute suspensions, the collective dynamic structure factor becomes equal to 1 for t = to, as shown in exercise 1.15. This is why in its definition the prefactor 1/N is added.

The collective dynamic structure factor only depends on k -1 k l when the system under consideration is also rotationally invariant, meaning that the pdf's Pll (R_, t - to) and P12(R-, t - to) depend only on the magnitude R_ - I R - I of the relative separation and not on its direction.

12Note that for k' - - k we have ~ f dR+ exp{- i (k ' + k) . R+ } - 1.

46 Chapter 1.

We will also encounter the so-called self dynamic structure factor, which is defined as,

S ~ ( k , t - to) - <exp{ ik . (rl(to) - r i( t))} > . (1.71)

This correlation function described the dynamics of the position coordinate of a single particle (particle number 1). Although the dynamics of the position coordinate of a single particle is probed, the self dynamic structure factor does depend on interactions, via the pdf with respect to which the ensemble average is taken. Obviously, the dynamical behaviour of a single particle is affected by interactions.

For equal times t = to, the auto-correlation function (1.70) reduces to,

s ( k ) - s ~ ( k , t - t o - o ) = 1 N

~ < exp {ik. ( r i - r j )} > i , j=l

= 1 + fi / dRg(R) exp(ik . R} (1.72)

sin{kR} 1 + 47r~f~176 R 2 g(R)

kR dO

where spherical angular integrations are performed, ~;dRexp{ik. R} - 47r s in{kR}/kR, precisely as in the mathematical subsection 1.2.5 to transform eq.(1.29) to eq.(1.30). Rotational invariance has been assumed here to write the pair-correlation function as a function of R - I R [ . This equal time correlation function is the so-called static structure factor, and is essentially the Fourier transform of the pair-correlation function. Notice that the equal time self dynamic structure factor is trivially equal to 1.

The self- and collective dynamic structure factor and the static structure factor will be analysed in detail later on in this book.

1.3.4 Gaussian Probability Density Functions

A stochastic variable is called a Gaussian variable when its pdf is a Gaussian pdf. When the stochastic variable X is a scalar quantity, its Gaussian pdf is defined as,

P ( X , t ) - ~/2~- < (X- < X >)2 >

1 ( x - < x > ) 2 } exp - 2 < ( X - < X > ) 2 > ,(1.73)

where < X > and < X 2 > are generally time dependent averages. When the stochastic variable X is a N-dimensional vector, its Gaussian pdf is a generalized version of this definition, namely,

P ( X , t ) - ,/(2r~Ydetf, D1 exp - ( X - < X > ) . D -1 . ( X - < X > ) , V k / t . J

(1.74) and is usually referred to as a multivariate Gaussian pds Here, det{D} is the determinant of the matrix D, which matrix is referred to as the covariance matrix, and D - ~ is the inverse of that matrix. The covariance matrix is equal to the following average of a dyadic product,

D - < ( X - < X > ) ( X - < X > ) > . (1.75)

When the components Xj of X are statistically independent, meaning that < XiXj > - < Xi > < Xj > for any i # j , the covariance matrix reduces to a matrix with non-zero entries only on the diagonal. In that case it is easily seen that eq.(1.74) reduces to a product of Gaussian pdf's (1.73), with X equal to one of the components of X. That D is indeed equal to the average in eq.(1.75) can be seen as follows. Consider the integral,

I (h , t) - f d ( X - < X >) P (X, t) exp{h �9 ( X - < X > ) } , (1.76)

where it is understood here and in the following that the variables can take any value in ~N. This integral is evaluated in the appendix, with the result,

I ( h , t ) - e x p { 1 - - h . D . h } 2

(1.77)

From eq.(1.76) it follows that,

~2 < (Xi- < Xi >) (X j - < Xj >) > - lim I (h , t ) .

h---~O Oh~ i)hj (1.78)

Substitution of eq.(1.77) into the right hand-side immediately confirms eq.(1.75). Notice that the Fourier transform of a Gaussian pdf with respect to the

difference X - < X > is found from eq.(1.76), by replacing h by - i k , to be equal to,

{1 } P ( k , t ) - exp - ~ k . D . k . (1.79)

48 Chapter 1.

The importance of Gaussian pdf's lies in the fact that a sum of many stochastic variables is a stochastic variable with a Gaussian pdf. This is true, independent o f the form of the pdf of the original variables. This is roughly the contents of what is known as the central limit theorem. More precisely,

n

m ! 1 ~ x j , with xjs stochastic variables with identical Let X - V/- ff j= l

pdf's , with < xj > - 0 and with x}s statistically independent ,

that i s , < x~xj > = 0 for i 7~ j . The pdf of X is then the

Gaussian pdf in eq.(1.73) in the l imit where n --, e~ . (1.80)

The prefactor 1/v/-ff in the definition of X is added to assure that the covariance of X is independent of n, and equal to the covariance of xj's, meaning that,

2 < X 2 > - < x j > for all j , which follows from the statistical independence of the xj's. The condition < xj > - 0 is not a restriction, since one can always subtract the average from a stochastic variable to obtain a new stochastic variable with zero average. The proof of the central limit theorem is as follows. Since the xj's are statistically independent, eq.(1.38) with f = X and X - (x 1, �9 �9 �9 xn) gives,

P ( X , t) - f dxl . . . f P(x l , t) . . . P(x , t) 6 ( X 1. )

Fourier transformation of P (X, t) with respect to X yields a product n Fourier transforms of P (x, t), where the index on x is omitted,

P(k, t) - dx P(x , t) exp - - ~

The exponential function is now Taylor expanded, and limn~oo (1 + a /n ) " = exp{a} is used to obtain, for large n's,

i k2x i k3x3 P(k, t) - I v/'ff 2 n 6 nal 2

[ - 1 2 n n-2

1k2 X exp{- exp{

Appendix 49

Now, according to eq.(1.79) in one dimension, this is precisely the Fourier transform of a Gaussian pdf. The conclusion is that the pdf of X is the Gaussian pdf (1.73). The central limit theorem is easily generalized to the case where both X and the xj's are vector quantities.

An important property of Gaussian variables is that averages of products of variables can be reduced to averages of only two variables. In particular, an average of four variables can be written in terms of averages of products of only two variables. This property is referred to as Wick's theorem,

Let X - ( X a , . " , X~) be a Gaussian variable with zero average.

Then, <XpXqX~X~ > - (1.81)

< XpXq >< X~X~ > + < X~X~ ><XqX~> + < XpX~ >< XqX~ >.

The zero average is no restriction, as before, since one may simply define a new stochastic variable X - < X > in case the average is non-zero. Wick's theorem then applies to this new variable. This theorem follows from the observation that,

0 4

< > - Oh,Oh O O . I(5. t).

with I (h) the integral defined in eq.(1.76). Substitution of eq.(1.77) for I (h ) into the right hand-side of the above identity, and noting that averages of products of two variables are obtained similarly, for example < XpXq > = limh--,o -- 021(h)/OhpOhq, immediately verifies Wick's theorem. This theorem can be extended to averages of arbitrary many variables. Averages of products of an odd number of variables are zero (see exercise 1.16), while products of an even number of variables are equal to products of two variables, where, as in Wick's theorem, the summation is over all possible permutations of the indices (p, q, r and s in Wick's theorem).

Another important property of Gaussian variables is that a sum of such variables is itself a Gaussian variable. This statement is proved in exercise 1.17.

Appendix In this appendix it is shown that the integral I (h , t), defined in eq.(1.76), is equal to the expression given in eq.(1.77). The integral to be evaluated is (with

50 Append ix

Y - X - < X > ) ,

1 I { ' I (h , t) - q ( 2 7 r ) N d e t { D ~. d Y exp h . Y - 7 Y" D - ' Y} This integral may be evaluated by recasting the combination in the exponential into the form,

1 h . Y - ~ Y . D -~ 1 ( Y - D . h ) . D -1 "Y - - 7

1 �9 ( Y - D . h ) + 7 h . D - h .

This identity may be verified, using that in an expression like x . M . x, the a [ M q- M T] q- matrix M can always be assumed symmetric, since M -

!2 [ M - M T], and x . [ M - M T ] . x -- 0, while M + M T is a symmetric

matrix (see eq.(1.1) for the definition of the transpose M T of the matrix M). II. J . . I

In particular, both D and its inverse may be taken symmetric. One must also use that, for a symmetric matrix, x . M . y - [M. x] . y, for all vectors x and y. This is easily verified by writing the inner products in terms of the components of the vectors and the matrix (see exercise 1.2b). Introducing the new integration variable Z = Y - D �9 h thus yields,

I (h , t) - q ( 2 r ) N d e t { D }

1 1 D_ 1 . exp { ~ h - D . h } / d Z exp Z �9 �9 Z}

Now, D can be transformed to a diagonal matrix, meaning that there is a matrix S, with det { S } - 1 and S-X _ sT, such that,

S -1 �9 D -1 �9 S -

d~ -1 0 . . . 0 0 d~ 1 . . . 0 �9 : : �9 .

0 0 . . .

Hence, with A - S- ~ Z, so that dA - det { S-1 } dZ - dZ,

1 x.i1 ) / } /(h,t) = r 7h. D. h dA exp -T j= l ~ d;1A]

= e x p { l h . D . h } 2

where it has been used that f dAj exp{ -l~dj-1A~} - q27rdj, and dl x d2 x

�9 .. x dN -- det { D }. This is the result quoted in eq.(1.77).

Exercises Chapter I

Exercises

51

1.1) As will be shown in chapter 7 on sedimentation, the sedimentation velocity 0 of a colloidal sphere with radius a in very dilute suspensions is equal to, V s

1 0 _ ~ F e x t

v~ 6rrr/oa

with r/0 the shear viscosity of the solvent and F ~t the (external) force acting on the colloidal particle. Verify that this external force in the earth's gravitational field g, corrected for buoyancy forces is given by,

47ra3

g T ps), (1.82)

with pp (p f) the specific mass of the colloidal material (the solvent). The magnitude of the earth's acceleration is g - I g I - 9.8 m / s 2. Calculate the maximum size of a colloidal silica particle in water to be able to perform experiments during 1 minute, such that the particle displacement due to sedimentation is not larger than its own radius. The viscosity of water is 0 .001Ns/m 2 and the specific mass of water and amorphous silica are 1.0g/ml and ,,~ 1.8 g/ml, respectively. (The answer is" 574 nm.)

1.2) * (a) Show that for any vectors a, b and c, ( a b ) - c - a ( b . c). Verify

that (h~). b is the projection of b onto a. Conclude that [ I - tiff], b is the projection of b onto the plane perpendicular to a.

(b) Show that a . M �9 b - [M T �9 a]- b for any vectors a and b and any matrix M.

A matrix M is called anti-symmetric when Mij - -Mji . Show that for such a matrix a . M �9 a - 0.

Show that for two matrices A and B, (A . B) T - B T �9 A T. (c) Verify that I | I - N, with N the dimension of the identity matrix.

Show that a a ~ . ~ | b b ~ _ b - (a . b) '~. nX nX

1.3) * Two delta sequences (a) Consider the sequence of functions,

1 f~ dz e x p { - i z ( x - xo)} =

1 sin{n(x Xo)}

- 7r x - x 0

52 Exercises Chapter I

Transform to the integration variable n(x - xo) to verify that this is a delta sequence centered at xo. This result is conveniently abbreviated as,

( ~ ( x - Xo) -- - - 1 / _ ' ~ xo)} . 2~ ~

According to eq.(1.12), the natural generalization of this equation to N- dimensions is,

1 / 6 ( X - Xo) - (2~r)N dZ e x p { - i Z . ( X - Xo)}.

(Hint" Use that f-~oo dz sin{z} = 71". ) z

(b) Show that the sequence of functions,

n r -- ~ e x p { - n 2 ( x - Xo)2},

is a delta sequence centered at xo. (Hint" Transform to the integration variable z - n(x - x0) and use that

f-~oo dz exp { - z 2 } - V ~ . )

1.4) * Fourier inversion, Parseval's theorem and the convolution theorem (a) The Fourier inversion formula (1.18) can be verified by substitution of

that equation into eq.(1.17). Verify that this substitution leads to,

f (X) = 1

(2r) n f dX' f ( X ' ) f dk exp{ - ik - (X - X ' )} ,

where in eq.(1.18) the integration variable is renamed as X' to make the distinction with the variable X in eq.(1.17). Now use the result of exercise 1.3a (with Z - k and Xo - X') to verify that the right hand-side of this equation is indeed equal to f(X).

(b) Show that,

/ dX f (X)g(X) - 1 /

(27r)g dk f ( k )g* (k ) .

To this end, substitute the Fourier transforms of f and g into the right-hand side of this equation and use the delta distribution of exercise 1.3a.

This equation is known as Parseval's theorem.

Exercises Chapter I 53

(c) Consider the Fourier transform of the following integral,

I(X) - / dX' f (X - X ' )g(X' ) .

The integration range is the entire ~N. Show that the Fourier transform with respect to X is the product of the Fourier transforms of the functions f and g,

I(k) - f(k) g(k).

This result is known as the convolution theorem. (Hint" Use exp{ - ik . X} - exp{- ik . (X - X')} e x p { - i k . X'}, and

transform to the integration variables X - X' and X'. )

1 . 5 ) *

(a) Choose g(X) - exp{ - ik . X} in Green's second integral theorem (1.7), to show that,

fw dX {k 2 f ( X ) e x p { - i k - X } + exp{- ik . X}V~cf(X)} - 0

when,

lim [R2maxlxl:R If(X)l] R---+oo

- - 0 - ' - lim [R2max,x,=R IVxf(X) l ] . R--+r

Conclude that the Fourier transform of V~f(X)is equal to - k 2 f(k). (b) Choose F(X) - a g(X) in Gauss's integral theorem (1.4), with a an

arbitrary but constant vector, to show that,

f / .

a . J w dX Vxg(X) - a . ~ow dS g(X).

Show that it follows that,

fw dX Vxg(X) - ~ w dS g(X).

Now choose g(X) - exp{- ik . X} f(X) to show that,

/w dX { - i k exp{- ik . X}f(X) + exp{- ik . X}Vxf(X)} - 0,

when,

lim [R2maxlxl=n If(X)[] - O. R---+oo


Conclude that the Fourier transform fo V x f ( X ) is equal to i k f ( k ) . (c) Choose F( r ) - a f ( r ) in Stokes's integral theorem (1.5) and show,

similar to exercise (b), that,

fs dS x V f ( r ) - J/os dl f ( r ) .

Notice that when S is a closed surface, this integral is 0 since then its boundary OS is empty.

1.6) * A non-differentiable complex function Consider the function f (z) - x - iy. Verify that this function does not

satisfy the Cauchy-Riemann relations (1.22). This function is therefore not differentiable as a complex function, where the point z = x can be approached from various directions. Calculate the integral of this function over the closed unit circle, where x 2 + y2 _ 1, in two ways" first by noting that on that circle x - cos {qo} and y - sin{cp} and integrating with respect to q;, and then by writing the circle as a sum of two curves, on each of which y may be written as a function of x, and integrating with respect to x. Provided you performed the integration correctly, you will find that the integral is zero. This shows that the converse of Cauchy's theorem is not true, that is, when f (z ) is a non-analytic function, its integral over a closed contour is not necessarily non-zero.

1.7) * Integrations in the complex plane (a) Show from the definition (1.23) that,

f dz Izl 5 5(1 + 2i1,

where [z 12= x 2 + y2, and with 7 the straight line that connects z - 0 with the point z - 1 + 2i.

(b) Show from the definition (1.23) that,

f d z zexp{z ) - i e x p { l + i } ,

with "7 the straight line that connects z = 1 with z = 1 + i. The exponential function of a complex number is equal to,

exp{z} - exp{x} exp{iy} - exp{x)[cos{y} + i s i n { y ) ] .

(Hint" Use that f dz z exp{z} - [o f dz exp{ az }] I~=~ . )

Exercises Chapter 1 55

1.8) * Show that for the function f ( z ) - z exp{z},

- exp{x} [x cos{y}- y sin{y}] ,

- exp{x} [y cos{y} + x sin{y}] .

Verify that these functions satisfy the Cauchy-Riemann relations (1.22) in the entire complex plane.

Use Cauchy's theorem to show that the function F(z) - fo dw f (w) is unambiguously defined, that is, is independent of the integration path that connects the origin and the point z.

1.9) Interaction of two charged colloidal spheres (a) Consider a small charged colloidal particle, located at the origin, in a

solvent that contains free ions. The electrostatic potential ~(r) is related to the free charge density p(r) by Poisson's equation,

- p ( r )

with e the dielectric constant of the solvent, which is assumed equal to that of the colloidal material. The charge density is a sum of two contributions. First, argue that there is a charge density equal to Q 6(r) due to the presence of the colloidal particle at the origin (Q is its total charge and 6(r) is the 3-dimensional delta distribution centered at the origin). Second, there is a charge density p,(r) in the solvent due to unequal concentrations of free ions. Hence,

V2tb(r ) = p,(r) Q 5(r) . s s

We have to relate p, to the potential to obtain a closed differential equation for r Let ezj be the charge carried by an ion of species j , with e > 0 the electron charge, and let the mean number density of that species be equal to

0 The interaction of an ion at position r with the remaining ions (and the pj. colloidal particle) is now approximated here in two ways �9 the electrostatic interaction is approximated by ezj~(r), which is the energy of that ion in the average electrostatic field generated by the remaining ions and the colloidal particle, and other kinds of interactions are neglected (for example, hard-core interactions between the ions are not accounted for). In this "mean field


approximation", the number density of such "point-like" ions of species j at r is equal to the Boltzmann exponential,

0 pj(r) - pj exp{-/3ezjO(r)}.

Conclude from this result that,

_ _ o e x p { - f l e z j O ( r } - Q 5(r) V2O(r) _ 1 E ezjpj 7 " e j

The summation runs over all free ion species in solution. This is the closed equation for the electrostatic potential that we were after, which is known as the non-linear Poisson-Boltzmann equation. This non-linear equation cannot be solved in closed analytical form. When the electrostatic potential is not too large in comparison to kBT, which is the case for larger distances from the colloidal particle, the exponential functions in the above equation may be linearized, using that exp{x} ~ 1 + x for small x. Convince yourself that electroneutrality demands that,

zjp ~ - Q 0 j V '

for a large volume V of the system. Linearize the Poisson-Boltzmann equation to show that,

V2O(r) - ~2O(r) - Q--- 5(r), s

with,

2 0 e 2 ~ j zj pj

- kBTe

This is the differential equation (1.28), and its solution is given in eq.(1.33). (b) The Helmholtz free energy of a system of two colloidal particles and the

free ions in the solvent is the pair-interaction potential. The pair-interaction force between the two colloidal particles is equal to,

F - - V [ U - T S ] ,

with V differentiation with respect to the position coordinate of a colloidal particle, U the total potential energy, and S the entropy of the free ions in


solution. Within the linearization approximation, and for colloidal particles with a fixed charge Q, the total electrostatic potential Or(r) is the sum of the potentials in eq.(1.33) of each of the separate colloidal particles,

Or(r) - O([ r - R 1 I )+ O(I r - R2 I),

with R1, 2 the position coordinates of the two colloidal particles. The electro- 1 2 static energy density is equal to 7e I Vr I, hence,

u - dr IVr f k 2 2 2(27r) 3 dk I (I),(k) [

Verify the second equation (use Parseval's theorem of exercise 1.4b). The entropy is equal to,

S - - k s / d r 1 . . . f d r u P ( r l , ' " , r / ) l n { P ( r l , . . . , r M ) } ,

where the integrations range over the position coordinates of all the M ions in solution, and P is the corresponding pdf. To within the approximations discussed in (a), this pdf is equal to,

P ( r l , " - , r M ) -- exp{ fl M - Ej=l ezjOt(rj)}

Q(Ni, . . . , Nm, V, T)

with Q(N1, �9 �9 �9 Nm, V, T) the configurational partition function (see eq.( 1.51 )), which now depends on the number of ions Nj of ion species j - 1 , . . . , m, in solution,

M

Q ( N 1 , ' " , Nm, V, T) - f dr1.., f dru exp{-/3 y~ ezjCgt(rj)}. j--1

Expand up to quadratic order with respect to the electrostatic potential, using 1 2 1 ~ 1 + X + X 2 and In{1 - x} ~ - x -- 1 272 that exp{x} ,,~ 1 + x + 7x , 1-x 2 '

and show that,

S = VM }-t 2 V M 2 I2 '

where,

) Ii - / d r l . . . / d r M ~ y ~ ezj(bt(rj) j - 1

, with i - 1 , 2 .


Since we are only interested in changes of the entropy as the relative position of the colloidal particles is changed, the term V M in{ V M } is of no concern to us here. Furthermore, f drj dgt(rj) is also a constant, independent of the position of the colloidal particles. Use this to verify, by substitution of the expressions for 11,2, that the relevant expression for the entropy reads,

i f ex2f 12 - T S - ~.ex 2 drr - 2(27r) 3 dk Jet(k) .

As for the electrostatic energy, Parseval's theorem must be used to arrive at the last expression. Conclude that the pair-interaction potential is equal to,

V ( I R 1 - R 2 I ) - U - T S = s

2(27r) 3 f dk (k 2 + ~2) i Or(k)12 .

Now show from eq.(1.33) by Fourier transformation that,

~t(k) = [exp{ik. R1) + exp{ik. R2}] s

1 l f o ~ x ~ ~ dr exp{-~r} [exp{ikr} - exp{-ikr}]

_ 1

Qe [exp{ik. tl4} + exp{ik-R2}] k~ + ~2"

Verify that, apart from terms which do not depend R1,2,

V ( I R 1 - R2 1) - Q2 1 e x p { i k - ( R 1 - R2)} e (27r) a f dk )~2 ~ ~

The integral here is precisely the expression (1.29) (with r - R 1 - R 2 , and Q replaced by Q2) that we evaluated with the help of the residue theorem. Conclude that,

V ( I R , - R2 l) - Q 2

47re I R1 - R21

exp{-tr [RI - R2 l)

This is the screened Coulomb or Yukawa potential referred to in section 1.1. From the above analysis it is clear that the validity of this expression

for the pair-interaction potential is limited to larger separations between the colloidal particles, where the electrostatic potential is small compared to kB T. On closer approach, the full non-linear Poisson-Boltzmann equation should generally be considered.

We also assumed a constant total charge on the colloidal particles, independent of their relative separation. This is the case when the degree of ionization of the chemical groups on the surfaces of the colloidal particles is close to 100 %. For partial de-ionization, the local electrostatic potential affects the ionization equilibrium and thereby the charge on the colloidal particle. In those cases a more appropriate condition is a constant surface potential rather than a constant charge.

1.10) * Jordan's/emma (a) Fourier inversion often relies on the evaluation of integrals of the

form f-~oo dk f(k)exp{ikr}, where r is either a positive or negative number. Suppose that r > 0. As discussed in the example in subsection 1.2.5, such integrals may be written as integrals over a closed contour in the complex plane, by identifying the integration range of the integral as the real axis of the complex plane, and by adding an integral ranging over the semi circle Ca+ in the upper half of the complex plane, with a radius R tending to infinity (see fig.l.8). Jordan's lemma states that for r > 0,

lim [ dz f(z) exp{+izr} - 0 R-.,,oo JCR+

when lim max~ecR+ If(z)I ---~ 0

and similarly when all + 's are replaced by - ' s (see eq.(1.32)). First show, by noting that z - R exp{i~} on CR+, with 0 < ~ < r , that,

- - I dz f(z) exp{+izr}[< fc Idzlif(z)llexp{+izr}l R+ _re+

= R d~, [ I ( R exp{i~}) [ e x p { - R r sin{~o} }.

= [dz[

Now pick an arbitrary small number e, and choose R so large that,

maxzecn+ I f (n exp { itp } ) I < e .

Verify that,

In+ <_ eR [~ dcp exp{ Rr sin{~}} 2eR [~/2 - - dcp e x p { - R r sin{T} }. dO JO

Convince yourself that for T E [0 7r/2] 2 , , ~r < sin{ ~ }, and hence that,

In+ <_ 2eR [~/~ 7r

d~ e x p { - R r 2 ~ / ~ } - - c [1 - e x p { - R r } ] . J 0 r

Hence,

71" lim I R + < _ - e .

R---+cx~ r

Since e can be made arbitrary small, limn_.oo 1R+ must be equal to zero. This concludes the proof of Jordan's lemma.

(b) Show by contour integration that, with a > 0,

cos{ } dx = - exp{ -a} ,

oo x 2 q - a 2 a

fo '~ x sin{x} 7r dx x2+a2 = ~exp{- -a} .

(Hint �9 Write the cosine and sine function as a sum of two exponential functions, and use Jordan's lemma twice in each case, once by adding an integral ranging over Cn+ and an integral ranging over Cn_. )

1.11) The effective interaction potential The effective interaction potential V ~ff is defined in eq.(1.59) for a trans-

lational and rotational invariant system as,

g(r) -- exp{-/~V ~H(r)}.

Use the definitions (1.46,51), with P1 (r, t) - 1 /V , together with the expression (1.50) for the pdf of all the positions coordinates, to show that (with

- I r l - I),

f dra.., f drN exp{--/~(~(rl,.-., rN)} g(r) -- ~ Q ( N , V , T )

Use this expression to obtain,

--Vrl V e r Y ( r ) - /~-lVrl ln{g(r)}

f d r a . . , f d r g [--V~ (I)(ri , - - . , rg)] exp{ - - f l r rN)}

f d r 3 . - , f d r N exp{-- f l~(r l , . . . , rN)}

rN)] P ( r l , ' " , r N ) ' P2(ri, r2)

where V~ is the gradient operator with respect to r~. According to eq.(1.45) this is the force on particle 1 with position coordinate rl, with a fixed position of

61

particle 2, averaged over the position coordinates of the remaining particles. This is why V ~ff is also referred to as the potential o f mean force. The distance over which the pair-correlation function tends to 1 is also the range of the effective interaction between two particles, where the intervening effects of the remaining particles is included.

1.12) gl ( r ) [or hard-spheres

The hard-sphere pair-interaction potential Vh, (r) is formally defined as,

Vh,(r) - 0 , for r >__ 2 a ,

= cr , for r < 2 a ,

fh~(r) -- exp{-flVh,(r)} - 1

with a the radius of the hard-core. Verify that the Mayer-function for this pair-potential is equal to,

/ d r 3 f([ r l - r3 [) f([ r 2 - - r 3 1)

- 0 , for r > 2 a ,

= - 1 , for r < 2 a .

Conclude that the integral in eq.(1.56) for g l (1 rl - r2 [) is equal to the overlap volume of two spheres with radii 2a, as depicted in fig.l.11. This overlap volume is non-zero only for r < 4a. Verify the following steps for the integration with respect to spherical coordinates, as indicated in fig. 1.11 (with

- cos{O}),

- 2 x d~p dO

dO f2~ /2r

Figure 1.11" The overlap vo lume

o f two spheres with

radii 2a.

d R R 2 sin{ 0 }

I I

Exercises Chapter 1

Z

fr I fr 2a - 47r dx d R R 2 14~ 12~

47ra318 3 r_ 1 (at-)3 ] = V

Verify eq.(1.57).

, for r < 4 a .

1.13) N u m b e r densi ty fluctuations

A measure for the amplitude of the fluctuations of the microscopic density is its standard deviation,

a2( r , r ' , t ) - < [pro(r)- < pro(r)] [pm(r')- < pm(r')] > ,

where pro(r) is a short-hand notation for the microscopic number density p ( r a , �9 �9 � 9 rN [ r). Show with the help of eqs.(1.46,52) that,

a2(r, r', t) - p(r, t) 5(r - r') + p(r, t)p(r', t)[g(r, r', t) - 1] .

Define the phase function, N - f v dr p( r l , - . - , rN Jr) - f v dr Pm (r), which is the number of particles contained in the volume V. Supposed that the linear dimensions of the volume V are much larger than the distance over which the pair-correlation function attains the value 1. Integrate the above expression with respect to r and r ~ over V and show that for a homogeneous isotropic equilibrium fluid (with fi = < N > / V ) ,

< ( N - < N > ) 2 >

< N > = 1 + f i f d R h ( R ) - 1 + 4 ZfdRR h(R). The volume integral of the so-called total-correlation funct ion h = g - 1

thus measures the amplitude of fluctuations of the number of particles in a large volume (large compared to the range of the total-correlation function). Conclude that the relative standard deviation <(N-cJV>)~> r goes to zero when the volume becomes infinitely large.

1.14) As will be shown in chapter 4, the conditional pdf P(r , t) for the position coordinate r of a non-interacting Brownian particle at time t, given that the particle was in the origin at time to - 0, satisfies the following equation of motion,

O P(r , t) - DoV 2 P(r , t) Ot

with Do the diffusion coefficient. The initial condition is, P( r , t = 0) - 5(r). We are going to evaluate the collective dynamic structure factor (1.70) for this case. The time evolution operator is now equal to,

/~ - DoV 2 "

First note that non-interacting particles are by definition statistically independent, so that, for i r j ,

< exp{ik. ( r i ( t - O ) - r j ( t ) ) } > - < exp{ik, r i ( t - 0 ) > < exp{ik, rj(t)} > .

Each of the averages on the right hand-side is with respect to Px - 1 / V . Show that, for large volumes V, these averages are proportional to 5(k), so that the "cross terms" with i r j in eq.(1.70) are zero for k r O. Verify that, for non-interacting and identical colloidal particles, the collective dynamic structure factor reduces to,

S~(k, t) - < exp{ik. (rl(t - O) - rl(t))} > .

The collective dynamic structure factor thus becomes equal to the self dynamic structure factor in case of non-interacting particles.

Show that Z~ exp{ik �9 r} - -D o k 2 exp{ik �9 r}, and hence,

/~n exp{ik, r} - ( -D o k2)" exp{ik, r} .

Use this in the definition (1.66) of the operator exponential to show that,

exp{/~t} exp{ik- r} - e x p { - D o k 2 t } exp{ik- r} .

Use the expression (1.67), with P(r , t) - l /V , to verify that,

S~(k, t) - S~(k, t) - e x p { - D o k 2 t } .

1.15) For non-interacting particles, the pair-correlation function is identically equal to 1. Conclude from the middle equation in (1.72) that the static structure factor is equal to 1 for k r 0. Show this also from the first equation in (1.72), following the reasoning of the previous exercise.

1.16) Follow the reasoning in the proof of Wick's theorem (1.81) to show that,

<xpxqx >-<xpxq><x > + < xpx ><xq>+<xqx > < x p > .

64 Further Reading

Since the (reduced) pdf for a single component of X is a Gaussian pdf, which is an even function, the averages of the single components are all 0, and hence, < XpXqX~ > - 0. In fact, the average of any product of an odd number of components is 0.

1.17 Sums of Gaussian variables Let z j, j - 1 , . . . n, denote statistically independent Gaussian variables

with zero average. Define the stochastic variable X - ~ j ~ zj. Apply eq.(1.38) to the present case to show that the pdf of X is equal to,

( n ) P ( X , t) -- f dxl . . . / dxn Pl (Xl, t) " " Yn(Xn, t)~ X - E xJ '

j= l

where Pj denotes the Gaussian pdf of zj. It follows from eq.(1.79) that the Fourier transform Pj ( k, t) of Pj (z j, t) is equal to,

Pj(k, t ) - exp - k 2 < z j > .

Use this to show that the Fourier transform of P(X, t) is equal to,

P(k, t l = P l ( k , t ) x . . . x P , ( k , t ) = e x p - k z [ < z a 2 > + . - . + < z . > ] .

Conclude that X is a Gaussian variable. This conclusion holds for any finite value of n. The central limit theorem states that for infinite n the zj 's need not be Gaussian.

Further Reading and References

Some of the well known books which contain both historically interesting facts and scientific details known at that time, are,

�9 J. Perrin, Die Brown'sche Bewegung und die Wahre Existenz der Molectile, SonderAusgabe aus Kolloidchemische Beihefte, Verlag von Theodor Steinkopff, Dresden, 1910.

�9 J. Perrin (translated by L1. Hammick), Atoms, Constable & Company, London, 1916.

Further Reading 65

�9 R. Zsigmondy (translated by E.B. Spear), The Chemistry of Colloids, volumes 1,2, John Wiley & Sons, New York, 1917.

�9 Wo. Ostwald, Grundriss der Kolloidchemie, Verlag von Theodor Steinkopff, Dresden/leipzig, 1917.

�9 Wo. Ostwald (translated by M.H. Fischer), Theoretical and Applied Col- loid Chemistry (original title" Die Welt der Vemachl/~ssigten Dimensionen), John Wiley & Sons, New York, 1922.

�9 T. Svedberg, Colloid Chemistry, The Chemical Catalog Company, New York, 1924.

�9 R. Zsigmondy, P.A. Thiessen, Das Kolloidale Gold, Akademische Ver- lagsgesellschaft M.B.H., Leipzig, 1925.

�9 H. Freundlich, New Conceptions in Colloidal Chemistry, Methuen & Company, London, 1926.

�9 H. Freundlich, Kapillarchemie, volumes 1,2, Akademische Verlagsge- sellschaft M.B.H., Leipzig, 1932.

�9 A.W. Thomas, Colloids, McGraw-Hill, New York, 1934. �9 J. Alexander (ed.), Colloid Chemistry, volumes 1-6, Reingold Publishing

Corporation, New York, 1946. �9 H.B. Weiser, Colloid Chemistry, volumes 1-3, John Wiley & Sons, New

York, 1949. �9 H.R. Kruyt (ed.), Colloid Science, volumes 1,2, Elsevier Publishing

Company, New York, 1949. �9 J. Stauff, Kolloidchemie, Springer Verlag, Berlin, 1960. �9 K.J. Mysels, Introduction to Colloid Chemistry, Interscience Publishers,

New York, 1967.

More recent textbooks, which discuss the origin of interactions in detail and which describe many phenomena that are not treated in the present book, are,

�9 R.D. Void, M.J. Void, Colloid and Interface Chemistry, Addison-Wesley Publishing Company, London, 1983.

�9 W.B. Russel, The Dynamics of Colloidal Systems, The University of Wisconsin Press, London, 1987.

�9 T.G.M. van de Ven, Colloidal Hydrodynamics, Academic Press, London, 1989.

�9 W.B. Russel, D.A. Saville, W.R. Schowalter, Colloidal Dispersions, Cambridge University Press, Cambridge, 1991.

�9 R.J. Hunter, Foundations of Colloid Science, volumes 1,2, Clarendon Press, Oxford, 1991.

66 Further Reading

�9 J.N. Israelachvili, Intermolecular and Surface Forces, Academic Press, London, 1991.

�9 R.B. McKay (ed.), Technological Applications of Dispersions, Surfactant Science Series volume 52, Marcel Dekker, New York, 1994. An informative overview concerning several properties of colloidal systems of spherical particles is,

�9 P.N. Pusey, in Liquids, Freezing and the Glass Transition, Les Houches Lectures 1989, part 1, North Holland, Amsterdam, 1991.

Interactions between charged particles has been described, independently, by, �9 E.J.W. Verwey, J.Th.G. Overbeek, Theory of Stability of Lyophobic

Colloids, Elsevier, Amsterdam, 1948. �9 B.V. Derjaguin (translated by R.K. Johnston), Theory of Stability of

Colloids and Thin Films, Consultants Bureau, New York, 1989.

The synthesis of latex particles is described in, �9 E.B. Bradford, J.W. Vanderhoff, T. Alfrey, J. Colloid Sci. 11 (1956) 135. �9 J.W. Vanderhoff, E.B. Bradford, TAPPI, 39 (1956) 650. �9 A. Kotera, K. Furusuwa, Y. Takeda, Kolloid-Z. u. Z. Polymere 239

(1970) 677. �9 R.M. Fitch (ed.), Polymer Colloids, Plenum Press, New York, 1971. �9 H. Ono, H. Saeki, Colloid & Polymer Sci. 253 (1975) 744. �9 E Candau, R.H. Ottewill (eds.), Scientific Methods for the Study of Poly-

mer Colloids and their Applications, Kluwer Academic Publishers, Dordrecht, 1988. Preparation of silica particles is based on a reaction discovered by St6ber, and is nowadays usually referred to as "the St6ber synthesis",

�9 W. St6ber, Kolloid-Z. 147 (1956) 131. A standard reference on the preparation of silica particles is,

�9 R.K. Iler, The Chemistry of Silica, John Wiley & Sons, New York, 1979. There is a large body of literature on surface modification of silica and latex particles as well as the synthesis of other kinds of colloidal particles. Overviews on these subjects are,

�9 J. Th. Overbeek, Adv. Colloid Int. Sci., 15 (1982) 251. �9 E. Matijevi6, Chem. Mater. 5 (1993) 412. �9 A. Vrij, A.P. Philipse, NATO Advanced Research Workshop on Fine

Particles Science and Technology from Micro to Nanoparticles, Acquafredda di Maratea, july 15-21, 1995.

Further Reading 67

The preparation of ellipsoidal latex particles is described in, �9 M. Nagy, A. Keller, Polymer Communications 30 (1989) 130. �9 C.C. Ho, M.J. Hill, J.A. Odell, Polymer Papers 34 (1993) 2019. �9 S. Wang, J.E. Mark, Macromolecules 23 (1990) 4288.

The preparation of colloidal boehmite rods is discussed in, �9 P.A.Buining, C. Pathmamanoharan, A.P. Philipse, H.N.W. Lekkerkerker,

Chem. Eng. Sci. 48 (1993) 411. Extraction and purification of TMV particles is described in,

�9 H. Boedtker, N.S. Simmons, J. Am. Chem. Soc. 80 (1958) 2550.

Textbooks on statistical mechanics which may be consulted for more detailed information concerning the subjects discussed in section 1.3 in this chapter ale,

�9 T.L. Hill, Statistical Mechanics, McGraw-Hill, New York, 1956. �9 D.A. McQuarrie, Statistical Mechanics, Harper & Row, New York, 1976. �9 J.P. Hansen, I.R. McDonald, Theory of Simple Liquids, Academic Press,

London, 1976. �9 R.K. Pathria, Statistical Mechanics, Pergamon Press, Oxford, 1977. �9 J.P. Boon, S. Yip, Molecular Hydrodynamics, Dover Publications, New

York, 1980. �9 L.D. Landau, E.M. Lifshitz, Statistical Physics, volumes 1,2, Pergamon

Press, Oxford, 1982. �9 M. Toda, R. Kubo, N. Sait6, Statistical Physics I, Equilibrium Statistical

Mechanics, Springer Verlag, Berlin, 1983. �9 R. Kubo, M. Toda, N. Hashitsume, Statistical Physics II, Nonequilibrium

Statistical Mechanics, Springer Verlag, Berlin, 1985. �9 H.S. Wio, An Introduction to Stochastic Processes and Nonequilibrium

Statistical Physics, World Scientific Publishing, Singapore, 1994.

Chapter 2

BROWNIAN MOTION OF NON-INTERACTING PARTICLES

69

70 Chapter 2.

2.1 Introduction

As discussed in the previous chapter, a colloidal particle exerts so-called Brownian motion due to thermal collisions with solvent molecules. This eratic motion can be described on the basis of Newton's equations of motion, where the interactions of the Brownian particle with the solvent molecules are taken into account by a rapidly fluctuating force. The statistics of Brownian motion can be studied in this way when reasonable approximations for the statistical properties of the fluctuating force can be made. We analyse in this chapter the translational Brownian motion of a single sphere, also in the presence of an externally imposed shear flow, and the translational and rotational motion of a long and thin rigid rod like particle.

A particular advantage of this approach is that it allows for a clear distinction of several time scales. As will turn out, the (angular) momentum coordinate of a Brownian particle relaxes to thermal equilibrium with the heat bath of solvent molecules within a time interval over which its position and orientation hardly change. This is a key feature of Brownian motion that offers the possibility to describe the statistics of displacements without involving the momentum coordinate. Especially for the treatment of interacting particles in later chapters this will turn out to be a very pleasant feature.

In the present chapter we are considering Brownian motion of non- interacting particles, that is, of Brownian particles which do not interact with other Brownian particles. This is the case for very dilute dispersions. In- teractions of the Brownian particle with the solvent molecules must be fully accounted for, however, since these interactions drive the Brownian motion.

2.2 The Langevin Equation

Relaxation times for fluids are known experimentally to be of the order 10 -14 S.

As will be established shortly, relevant time scales for Brownian particles are at least 10 -gs. This separation in time scales is the consequence of the very large mass of the Brownian particle relative to that of a solvent molecule, and is essential for the validity of the Langevin description.

The interaction of the spherical Brownian particle with the solvent molecules is separated into two parts. First of all, there is a rapidly varying force f(t) with time t as the result of random collisions of solvent molecules with the Brownian particle. This force fluctuates on the forementioned solvent

2.2. Langevin Equation 71

time scale of 10-14,S. Secondly, as the Brownian particle attains a velocity v - p / M (p is the momentum coordinate of the Brownian particle and M its mass), there is a friction force due to systematic collisions with the solvent molecules. When the volume of the Brownian particle is much larger than that of the solvent molecules, this systematic force equals the hydrodynamic friction force of a macroscopically large sphere. For not too large velocities, that friction force is directly proportional to the velocity of the Brownian particle, and the proportionality constant 3' is the friction constant: friction f o r c e - - 'Y p / M . The friction coefficient of a macroscopically large sphere is shown in chapter 5 on hydrodynamics to be equal to,

"7 - 67rr/oa, (2.1)

with r/0 the shear viscosity of the solvent and a the radius of the Brownian particle. The friction coefficient in eq.(2.1) is commonly referred to as Stokes's friction coefficient. Newton's equation of motion for a spherical Brownian particle is thus written as,

dp /d t - - T P / M + f(t). (2.2)

The position coordinate r of the Brownian particle is, by definition, related to the momentum coordinate as,

d r ~ d r - p / M . (2.3)

Since the systematic interaction with the solvent molecules is made explicit (the first term on the right-hand side of eq.(2.2)), the ensemble average of the fluctuating force f is equal to zero,

< f(t) > - O. (2.4)

Due to the forementioned large separation in time scales, it is sufficient for the calculation of the thermal movement of the Brownian particle to use a delta correlated random force in time, that is,

< f ( t ) f ( t ' ) > - G 6(t - t'), (2.5)

where 6 is the delta distribution and G is a constant 3 • 3-dimensional matrix, which may be regarded as a measure for the strength of the fluctuating force, and is referred to as the fluctuation strength. Such a delta correlated random

72 Chapter 2.

force limits the description to a time resolution which is large with respect to the solvent time scale of 10-14s.

Equation (2.2) is Newton's equation of motion for a macroscopic particle with a fluctuating random force added to account for the thermal collisions of the solvent molecules with the Brownian particle. Such an equation is called a Langevin equation. It is a stochastic equation of motion in the sense that the momentum coordinate of the Brownian particle, as well as its position coordinate, are now stochastic variables. It makes no sense to ask for a deterministic solution of eqs.(2.2,3), since only ensemble averaged properties of the random force f are specified. The effort should be aimed at the calculation of the conditional probability density function for p and r at time t, given their initial values at time t - 0. Hereafter, "probability density function" is abbreviated as pdf. The solution of the Langevin equation is the specification of the pdf for the stochastic variable (p, r). Note that eq.(2.2) is mathematically meaningless as it stands without the specifications (2.4,5) of the statistical properties of the random force f.

Integration of eq.(2.2) yields,

f ' p(t) - p ( 0 ) e x p { - ~ t } + Jo d t ' f ( t ' ) e xp{ . . . - t ' ) } . (2.6)

Now let r be a time interval much larger than the solvent time scale of 10-14s. The random force evolves through many independent realizations during that time interval. On the other hand, let r be so small that e x p { - T t / M } is almost constant over times of the order of r, that is, we take r << M/7. With this choice of r , eq.(2.6) can be rewritten as,

N-1 fj(j+l)r "7 7 p(t) -- p(0) e x p { - ~ t } + j=o y~ e x p { - - - ~ ( t - j r ) } ~ dt' f(t ') , (2.7)

where N - t / r . Since the random force evolves through many independent realizations during the time interval r, each integral in eq.(2.7) is a Gaussian variable with a mean equal to zero. This is a consequence of the central limit theorem, as each integral may be regarded as a sum of many statistically equivalent and independent terms (the central limit theorem is formulated in (1.80) in the introductory chapter). Furthermore, since a sum of independent Gaussian variables is also a Gaussian variable, as shown in exercise 1.17 in chapter 1, it follows from the representation (2.7) that,

xl - p(t) - p(O) e x p { - ~ t } ,

2.2. Langev in Equat ion 73

is a Gaussian variable. What about the position coordinate r of the Brownian particle? The above

reasoning is easily extended to include the position coordinate. Using eq.(2.3), integration of eq.(2.6) yields (see exercise 2.1),

r(t) - r(O) + p ( O ) [ 1 - e x p { - 7 ] "y (2.8/

ljf0t [ 7 ( t t' ] + - d t ' f ( t ' ) 1 - e x p { - ~ - )} . 7

Following the same reasoning as before shows that,

x2 - r ( t ) - r(O) p(0) [ 1 - e x p { - - ~ t } ] , 7

is a Gaussian variable with a mean equal to zero. Consider the stochastic variable (p, r). Let us define the variable,

X = (Xl, x2)

fo - ( d t ' f ( t ' ) e x p { - - - - ~ ( t - t ' ) }

,-L [ ] 1 dt' f(t') 1 - e x p { - ~ ( t - t')} ) , 7

(2.9)

where eqs.(2.6) and (2.8) are used in the second line. According to the above discussion, X is a Gaussian variable. For given initial values p(0) and r(0) of the momentum and position coordinates, the pdf of the variable X is clearly identical to the pdf of (p, r). Hence, the pdf of (p, r) is given by (for notation, see subsection 1.3.1 on conditional pdf's in the introductory chapter),

1 P ( p , r , t l p ( O ) , r ( O ) , t - O ) - "27r'~/2 " "et-" ( ) x /a ll)

1 exp{-~X. 9 -1. X}, (2.101

with,

< XlXl > D - < X X > - < x 2 x ~ >

< XlX2 > ) < x 2 x 2 > " ( 2 . 1 1 )

d e t D denotes the determinant of D, and n - 6 is the dimension of X. Note that each of the matrices < xixj > , i, j = 1, 2, is 3 x 3-dimensional, so that D is 6 x 6-dimensional. Using eqs.(2.4,5), the ensemble averages < xixj >

74 Chapter 2.

are easily calculated,

MG [X-exp(-2 ] < XxXx > = 27 ~ t } , (2.12)

< x l x 2 > - < x 2 x l > - ~ 1 - e x p { - t} , (2.13)

M G 7 l [ e x p { _ 2 7 ] t}-i

- 2 [1 - e x p { - M t } ] ) .

X 2 X 2 ~ " -

(2.14)

It is now possible to identify the matrix G, using the equipartition theorem, which states that (see exercise 2.2),

^ M lim < p(t)p(t) > - I /~ (2.15) t--+oo

where/3 - 1/kB T, with kB Boltzmann's constant and T the temperature, and the unit matrix. The fluctuation strength now follows from the definition of

the variable xx (below eq.(2.7)) and the ensemble average (2.12). For times t >> M/7, eq.(2.12) reduces to,

M - G - - . (2.16)

27

Comparison with eq.(2.15) identifies the fluctuation strength,

G - i 2._~_7. (2.17)

This relation is often referred to as a fluctuation dissipation theorem, because it connects the fluctuation strength with the friction coefficient, which determines the dissipation of kinetic energy into heat. With the identification of the fluctuation strength G, the pdf of the Gaussian variable (p, r) is completely specified.

2.3 Time Scales

In an experiment, the time scale is set by the time interval over which observables are averaged during a measurement. For example, taking photographs

2.3. Time Scales 75

of a Brownian particle is an experiment on a time scale which is set by the shutter time of the camera. Subsequent photographs reveal the motion of the Brownian particle averaged over a time interval equal to the shutter time. Any theory considering the motion of the Brownian particle obtained in such a way should of course be aimed at the calculation of observables, averaged over that time interval. A time scale is thus the minimum time resolution of an experiment or theory, and observables are averaged over the time interval that sets the time scale.

We have already introduced the solvent time scale in the previous section. The solvent time scale is of the order of the relaxation times for solvent coordinates, and is of the order 10 -14 s. The Langevin equation, together with the specifications (2.4,5) for the ensemble averages of the random force, is an equation that is valid on a time scale that is much larger than the solvent time scale. One might be tempted to set the random force f in the Langevin equation (2.2) equal to zero, since the average of f over a time interval equal to many times the solvent time scale is zero. However, the correlation function of f in eq.(2.5) is delta correlated, so that averages of products of the random force that appear on using the Langevin equation (2.2) cannot be set equal to zero. Thus, the random force on the right-hand side of eq.(2.2) must be retained. The coarsening in time is made explicit in the ensemble averages (2.4,5), while the original equations of motion (2.1,2) remain intact. The smallest time scale on which the specifications (2.4,5) for the averages of the random force make sense, is much larger than the solvent time scale. This time scale is usually referred to, for historical reasons, as the Fokker-Planck time scale, which we shall denote as TFp.

At the end of the previous section we have seen that the ensemble average attains its equilibrium form for times t >> M/7. The momentum coordinate p thus relaxes on a time scale >> M/.y. Consider now the full time dependence of < p(t)p(t) >. An explicit expression follows immediately from the definition of the variable xx in the previous section (below eq.(2.7)) and the expression (2.12) for the average < x~x~ >, together with the identification (2.17) of the fluctuation strength G,

< p( t )p( t ) > - I - ~ 1 - e x p { - ~ t } + p(O)p(O) e x p { - ~ t } , (2.18)

where, as before, I is the unit matrix. For small times << M/7 this becomes,

< p( t)p( t) > - p(O)p(O). (2.19)

76 Chapter 2.

Hence, for these small times the Brownian particle did not yet change its velocity due to collisions with solvent molecules.

Let us now analyse the mean squared displacement as a function of time. The time dependent mean squared displacement follows immediately from the definition of the variable x2 in the previous section Oust below eq.(2.8)) and the expression (2.14) for the average < x2x2 >, together with the identification (2.17) of G,

< ( r ( t ) - r (O) ) ( r ( t ) - r(O)) > -

2M 7 1 [ + i '/372 ( ~ t - ~ .exp{-

For times t >> M/7, this becomes,

2 < ( r ( t ) - r ( 0 ) ) ( r ( t ) - r(0)) > - i -s--t. p,-/

p(O)p(O) 7 i] 2 .y2 [exp{-~t} - (2.20)

"r }]1 M - ~ T t } - 1 ] - 2 [ 1 - e x p { - ~ t .

(2.21)

The mean squared displacement thus varies linearly with time. This is quite different for ballistic motion, where the mean squared displacement would be proportional to t 2. The interpretation of this result is, that the Brownian particle suffered many random collisions with the solvent molecules, leading to many random changes of its velocity and thus reducing its displacement with time as compared to ballistic motion. Ballistic motion is observed for small times t << M/7,

p(O)p(O)t _ v(O)v(O) t (2.22) < ( r ( t ) - r ( 0 ) ) ( r ( t ) - r(0)) > - M2

where v is the velocity of the Brownian particle. This equation is in accordance with eq.(2.19) �9 the velocity is not yet affected by collisions with solvent molecules for these small times, so that the displacement of the Brownian particle is simply linear with time.

For time scales >> M/7, the momentum coordinate is thus in equilibrium with the solvent, and the position coordinate changes, on average, proportional to V~. This time scale is usually referred to as the Brownian, Diffusive, or Smoluchowski time scale, which shall be denoted as TD. On that time scale a statistical description for the motion of the Brownian particle is feasible, without involving the momentum coordinate. We thus come to the following ordering of time scales,

10-14s - - Tsolvent << TFP <'~ M/7 << T D . (2.23)

2.3. Time Scales 77

Ir(o)- r(r)12

0 t

-L; Figure 2.1" The mean squared displacement <1 r(t) - r(O) 12> as a function of time.

Using typical values for the mass and friction coefficient of a Brownian particle, one finds that M/7 ~ 10-9s (see exercise 2.3).

A statistical description on the solvent time scale involves the position and momentum coordinates of both the solvent molecules and the Brownian particles. On the Fokker-Planck time scale, the solvent coordinates are long relaxed to thermal equilibrium, and only the momentum and position coordinate of the Brownian particle need to be considered. Finally, on the Brownian or diffusive time scale, in addition, the momentum coordinate of the Brownian particles relaxed to equilibrium with the heat bath of solvent molecules, and a statistical description involving just the position coordinate of the Brownian particle is feasible.

A coarsening of the time scale implies a coarsening of the length scale. On the diffusive time scale the spatial resolution is not better than the distance over which the Brownian particle moves during a time interval equal to the diffusive time scale. The ensemble average of that distance, the diffusive length scale 1D, is easily obtained from eqs.(2.2,4). From these equations it

78 Chapter 2.

follows that, "7 - p(O) e x p { - ~ t } ,

so that,

[~176 1 [ 1D Jo M

(2.24)

= p(O) (2.25)

A typical value for [ p(0) [ is obtained from the equipartition theorem,

I p(0) X/ - ~/3MksT. (2.26)

The diffusive length scale is thus estimated as,

lD ~ ~ / 3 M k B T / 7 - ~/3MkBT/67ryoa. (2.27)

Typical values yield (see exercise 2.3),

l--D-D ~ 10 - 4 - - 10 -3, (2.28) a

where a is the radius of the Brownian particle. The important conclusion is, that on the diffusive time scale the coarsening of the spatial resolution is only a tiny fraction of the size of the Brownian particle. For the study of processes where a significant displacement of the Brownian particle is essential, a statistical description on the diffusive time sca/e is therefore sufficient.

The results of the present section are summarized in fig.2.1, where the mean squared displacement is plotted as a function of time. For small times, that is, on the Fokker-Planck time scale, the mean squared displacement is proportional to t 2, eq.(2.22), whereas for large times, on the diffusive time scale, the mean squared displacement is linear in t, eq.(2.21). The linear curve in the diffusive regime intercepts the mean squared displacement axis for zero time at - l ~ .

For non-interacting Brownian particles the diffusive time scale is the largest time scale of interest. As soon as interactions amongst Brownian particles come into play there are two further time scales. These two time scales are related to direct and hydrodynamic interactions between the Brownian particles and are referred to as the interaction time scale and the hydrodynamic time scale, respectively. These time scales are discussed in the chapters 4 and 5 on interacting particles and on hydrodynamics, respectively. The hydrodynamic time scale is of the same order as the diffusive time scale discussed here, while the interaction time scale can be much larger.

2.4. Chandrasekhar's Theorem 79

Before the pdf of the position coordinate is constructed (on the diffusive time scale) the method of solving the Langevin equation as discussed in section 2.2 is generalized in the following section.

2.4 C h a n d r a s e k h a r ' s T h e o r e m

Chandrasekhar's theorem is a generalization of the analysis of section 2.2 to arrive at the expression (2.10) for the pdf of X. Instead of repeating the analysis of section 2.2 for each of the Langevin equations which are considered in the following sections, we discuss the general solution of these equations here once, and apply the resulting theorem to these special cases.

Let X be a m-dimensional stochastic variable, which obeys the following integrated Langevin equation,

o•0 t

X(t) - @(t) + dt' @(t - t'). F(t ') . (2.29)

and the force F are both m-dimensional vectors and ~ is a m x m- dimensional matrix. Both �9 and ~ are deterministic and known functions of time. The stochastic force F is characterized by,

< F(t) > - O, (2.30)

and, < F( t )F( t ' ) > - H 5(t - t'), (2.31)

with H a constant m x m-dimensional matrix. The conditional pdf of X at time t, given that its value is @(t - 0) at time t - 0, is then given by,

1 P(X, t I ~(0) , t - 0) =

(27r)m/2 ~detM(t)

• e x p [ - 2 ( X - ' ( t ) ) . M - l ( t ) . ( X - , ( t ) ) ] ,

(2.32)

where the m • m-dimensional covariance matrix M(t ) is defined as,

M ( t ) - fo t dt' a . (2.33)

80 Chapter 2.

The dots here denote contraction of adjacent indices, that is, the ij th element of M(t) is,

m f0t Mij(t) - ~ dr' ~Pip(t')Hpq (~T)qj( t ' ) , (2.34) p,q=l

and the supersrcipt "T" stands for the "transpose o f ". It is assumed here that the inverse of M(t) exists. This statement-Chandrasekhar's theorem- is established in precisely the same way as the expression (2.10) in section 2.2. According to eqs.(2.29) and (2.31), X - @(t) is a Gaussian variable for all times t (large enough, however, to ensure the validity of the delta correlation (2.31) of F). The pdf for X can then be written down immediately, provided that the inverse of the matrix M (t) exists, since the pdf of X - @ (t) is identical to that of X.

For the case considered in section 2.1, we have, X - (x~,x2), F = (f, f), and both �9 and @ follow immediately by comparison of the integrated Langevin equation (2.9) with eq.(2.29).

2.5 The pdf on the Diffusive Time Scale

The pdf of r on the diffusive time scale, where the momentum coordinate is in thermal equilibrium with the solvent, is obtained from Chandrasekhar's theorem and the integrated Langevin equation (2.8). Comparison of eq.(2.8) with eq.(2.29) and using eq.(2.17) for the fluctuation strength, yields, for times t>> M / 7 ,

X u r~

- r(O)+ ~'p(,O____~), "7

F - f, (2.35)

- i 1~ [ 1 - e x p { - ~ t ~ ' }],

n - ]: 2--7-7.

The dimension m is 3 in this case. Note that the exponential time dependence of �9 must be retained, even on the diffusive time scale, since in the integrated Langevin equation ~ occurs as a function of t - t', and t' ranges from 0 to t.

2.6. Diffusive Time Scale 81

According to the above equalities and eq.(2.33), on the diffusive time scale, the matrix M (t) is given by,

M(t ) - i2Dot, (2.36)

where the diffusion coefficient Do is defined as,

1 kBT Do = ~-~ = 67rr/o'------~" (2.37)

Such a relation between a diffusion coefficient and a friction coefficient is commonly referred to as an Einstein relation, and when an explicit expression for the friction coefficient is substituted it is referred as a Stokes-Einstein relation. Einstein and Stokes-Einstein relations apply also to rotational and translational diffusion of rigid rods, as will be seen in section 2.8. Chandrasekhar's theorem (2.32) thus yields,

1 [ Ir-r(0)-P(~ 2] P ( r , t I r (0) , t - 0) - (47rDot)3/2 exp - 4Dot " (2.38)

In the previous section we have seen that on the diffusive time scale, the length scale is much larger than I p(0) [ /7. The corresponding term in the exponential in the pdf of r is therefore meaningless, and should be omitted. For future reference we display here the more appropriate expression,

P( r , t I r(0), t - 0)) - 1 [ ,r r/0/,2]

(47rDot)3/2 exp - 4Dot " (2.39)

The physical meaning of the diffusion coefficient is that it sets the time required for significant displacements of the Brownian particle (see exercise 2.3).

2.6 The Langevin Equation on the Diffusive Time Scale

In arriving at the integrated Langevin equation (2.8) for the position coordinate r of the Brownian particle, we had to perform two integrations : a first integration of the equation of motion (2.2) for the momentum coordinate, and a second integration of the resulting integrated Langevin equation (2.6). The question is whether it is possible to coarsen the time scale right from

82 Chapter 2.

the beginning, on the level of the differential form of the Langevin equation. If possible, this would save the extra work involved in performing a second integration. For more complicated Langevin equations, like for rigid rod like Brownian particles (see the following sections), such a coarsening directly from the start saves a lot of work.

Since on the diffusive time scale the momentum coordinate is in thermal equilibrium with the solvent, one might guess that a coarsening at the level of the differential form of the Langevin equation (2.2) can be established simply by setting,

dp/d t - O, (2.40)

that is, inertia of the Brownian particle is unimportant. It then follows that,

p / M - f(t)/7, (2.41)

so that a Langevin equation involving only the position coordinate is obtained from eq.(2.3),

dr~dr - f(t)/7. (2.42)

The corresponding integrated Langevin equation is thus simply,

lfo r(t) - r(O) + ~ dt' f(t'). (2.43)

Applying Chandrasekhar's theorem to this integrated Langevin equation immediately reproduces the pdf in eq.(2.39).

Equation (2.40) can be justified by simply rescaling the Langevin equation with respect to the coarsened time and length scales. The time scale we wish to work with here is the diffusive time scale rD >> M / 7 , and the length scale is the diffusive length scale 1D as given in eq.(2.27). Defining the rescaled time and position,

t t - - t / T D , (2.44)

r' = r/1D, (2.45)

the Langevin equations (2.2,3) are written as,

1 M

TD dp ' /d t ' - - p ' + f', (2.46)

1 _ pt dr' /dt ' ~ , (2.47)

2.7. Diffusion in Shear Flow 83

where the rescaled momentum and stochastic force are defined as,

p t ~ TD lD p' (2.48)

f, _ _ M TD f. (2.49) "y 1D

The primed variables are the variables in which we are interested when going to the coarsened description. The only thing we have done is to express time and position in new units, corresponding to the minimum resolution in the coarsened description. The factor that multiplies dp' /d t ' in eq.(2.46) is very small, since rD >> M / 7 . Therefore, the left hand-side of eq.(2.46) may be set equal to zero. This is the justification for eq.(2.40).

In the following sections, diffusion of spheres in shear flow and of rod like Brownian particles are considered. The corresponding Langevin equations are coarsened to diffusive time and length scales as described above, saving the considerable effort of solving the full Langevin equations.

2.7 Diffusion in Simple Shear Flow

Consider two flat plates with solvent contained in between. The plates are oppositely displaced, by means of external forces, with a constant speed (see fig.2.2). For not too large velocities of the plates, this induces a spatial linearly varying velocity of the solvent. For the coordinate system sketched in fig.2.2, the fluid flow velocity Uo is equal to,

uo(r) - r . r , (2.50)

with, /010/ r - - ~ 0 0 0 , (2.51)

0 0 0

where ~ is called the shear rate, which is proportional to the velocity of the plates. The matrix 1" is the velocity gradient matrix. The fluid flow velocity as defined by eqs.(2.50,51) is called a simple shear flow.

Consider a Brownian particle immersed in a solvent which is in simple shear flow. The friction force is now not just equal to --),p/M. Instead of the absolute velocity of the Brownian particle, we have to use the velocity relative

84 Chapter 2.

--~V

V~ Figure 2.2:

Simple shear flow.

to the local velocity of the solvent. The friction force is thus, - 3 ' ( ~ - F . r). The Langevin equations (2.2,3) thus change to,

ep/et - --~ ~ - r . r + f(t) ,

e r / e t - p/M.

(2.52)

(2.53)

(2.55)

To this end it is desirable to rewrite the Langevin equation in terms of an equation of motion for ~ - F . r. Using that F . F - O, combination of the two Langevin equations (2.52,53) readily yields,

d(M~ -- F" r) 7^ F) P f(t___)) dt = - ( ~ I + . ( ~ - F . r ) + M"

Integration gives,

p(t) M

Fr(t) -y (p(O) e x p { - ( ~ I + r ) t } . M r . r(O))

+

(2.56)

(2.57)

1 t '7"

< f(t)f(t ') > - G5 6(t - t').

The strength of the fluctuating force, GS, may be different from that given in eq.(2.17), where no shear flow is applied. Also the equipartition theorem changes : the fluctuating velocity is the total velocity minus the local velocity of the solvent,

( ) ( ) 1 lim < p(t) r . r(t) p(t) r . r(t) > - i/3~/r (2.54) t - ~ M M

Let us first calculate the fluctuation strength for the sheared system, which is defined as before,

The exponential function of a matrix is formally defined by the Taylor series expansion of the exponential function,

1 B , ~ t,~ exp{Bt) - ~ n! " n = 0

(2.58)

Differentiating the sum term by term, it is easily shown that,

dexp{Bt}

dt - B exp{Bt}. (2.59)

This property of the matrix exponential is used to solve eq.(2.56) in a similar ^

way as if B - - ( ~ I + F) were a scalar quantity. Since r n - 0 for n > 1, it follows from the definition of the matrix exponent (2.58), that,

e x p { - F t } - i - r t. (2.60)

Using this in the evaluation of the ensemble average in the equipartition theorem (2.54) from eq.(2.57), leads to,

( )( ) 1 lim < p(t) r . r(t) p(t) F . r(t) > - I t iM (2.61) t--.oo M M

-- 2M71 G~ - ~ r (G+. + r . G+) + 2 ( N ) ~ r . G s . r ~ .

The superscript "T" on a matrix stands for "the transpose o f " that matrix. The simplest possible guess for the symmetric solution is,

G+ - + ( r +

The real numbers ao and ax can indeed be chosen such that this form solves eq.(2.61). Using that F . F . F T - 0, and, F . F T. F T - 0, it is found that ao - 2"7/fl and al - M//3. Hence,

[ ] G~ = 27 ~:+ + F T) (2.62)

Having determined the fluctuation strength, we are now in the position to use Chandrasekhar's theorem. Here, we calculate the pdf for the position r of the Brownian particle on the diffusive time scale. At time t - 0, the position coordinate is at the origin" r(0) - 0. On the diffusive time scale t >> M / 7 ,

86 Chapter 2.

the left hand-side of eq.(2.56) may be set equal to zero (see section 2.6) so that the Langevin equations (2.53,56) for the position reduce to,

( M ) f(t) -1 f(t) ,~ F . r + - - . (2.63) dr/ dt - r . r + i + F M ,7

In the above approximation we assumed small shear rates, such that,

1/+ >> M / 7 . (2.64)

For these small shear rates, the mass of the Brownian particle drops out. In fact, on the diffusive time scale, where inertia effects are of no importance, we can only consider such small shear rates. Considering larger shear rates involves the mass explicitly, so that the analysis should be performed on the Fokker-Planck time scale. Integration, with the initial condition that r(0) - 0, gives,

lfo' r(t) - dr' e x p { r ( t - t ' )}. f(t '). (2.65)

The identification with the quantities appearing in Chandrasekhar's theorem sets,

and,

X - r~

= O,

F - f, 1

- - exp{rt}, 7

(2.66)

(2.67)

(2.68)

(2.69)

H - G+ = --~- i-l- ~-7- 7

For shear rates as small as specified in eq.(2.64), the mass of the Brownian particle drops out, as it should,

H - G ~ ,~ G - 2 7 i . ( 2 . 7 1 )

The fluctuation strength is thus not affected by the shear flow, provided that the inequality (2.64) is satisfied.

The matrix M(t) in eq.(2.33) is found to be equal to,

1 1 FT t2 ) M(t) - 2Dot i + + r r l t + -~ r . (2.72)

(a} (b)

" i i \ ~= _- ,,,

F.r = E.r + O . r

Figure 2.3: Decomposition of a simple shear flow with positive shear rate in an ex-

l ( r + r T) and a rotational flow (b), with tensional flow (a), with E - ~ -

where the diffusion coefficient Do is defined in eq.(2.37). For the calculation of the pdf, according to eq.(2.32), we need the inverse of M(t) , which is easily found to be equal to,

1 - ! -~ t 0 / 2

1 1 "~t 1 + 1 ,~2t2 0 M - ' ( t ) - 2Dot (1 + ~'~2t2) - 7 5

0 0 (1 + 1 ,~2t2 ) (2.73)

According to Chandrasekhar's theorem, the pdf of r, given that the particle at time t - 0 is at the origin, is,

1 P ( r , t I r - 0 , t - 0 ) - ~/(4~Dot) 3 (1 + 1-!5 "~2t2)

[ ] z 2 x 2 + y2 xy ";It-

• exp 4Dot 4Dot (1 + 1,~2t2 ) + 4Dot(1 + 1-~'~ 2t2) "

(2.74)

For small times, "-~ t << 1", for which the Brownian particle just started to move away from the origin, the pdf is larger in comparison to the pdf without shear flow in directions where ~xy is positive,

P(r , t I r - 0, t - 0) ~f(47rDot) 3

r 2 - xy';tt] "'~t << 1" exp - 4Dot '

88 Chapter 2.

We have put the condition here between quotes, because the strict condition is a bit more complicated, and involves also the position coordinate. The physical meaning of the approximation is clear, however, and we shall not display the strict conditions under which the above approximation is valid. The above result can be understood, when the simple shear flow is decomposed as,

1 (r 1 ( r + r T ) . r + - F T) r. u ( r ) - r . r - (2.75)

The first term on the right hand-side is a so-called extensional fluid flow, which is sketched in fig.2.3(a), while the second term represents a purely rotational flow, as sketched in fig.2.3(b). In this figure the shear rate is chosen positive. Clearly, the extensional flow tends to displace a Brownian particle away from the origin for xy > 0, while for xy < 0, the particle is pushed back to the origin. The result is a larger probability to find the particle at a certain distance from the origin in the regions where xy > 0, and a smaller probability where xy < 0. The rotational component of the shear flow just takes the Brownian particle from regions with positive values of xy to regions where xy is negative, and vice versa. For larger times, the effect of the shear flow is much more complicated, and is described by the expression (2.74) for the pdf.

2.8 Rotational B rownian Motion

In this section, Brownian motion of rigid rod like particles is considered. For these anisometric particles, translational Brownian motion couples to rotational motion. The Langevin equation for translational motion must now be supplemented with a Langevin equation for rotational motion. Before these Langevin equations are stated, the next subsection contains a refresher of the Newtonian equations of motion for rigid non-spherical objects.

2.8.1 Newton's Equations of Motion

Let us first recall Newton's equations of motion for non-spherical rigid particles. The rigid body contains a large number of molecules, with positions r,~, momenta p,~, and masses m,~; n = 1, 2, 3 , . . . . The positions of the molecules are fixed relative to each other, that is, the body is rigid as a result of the inter molecular interactions. The velocity v,~ of molecule n is composed of two

2.8. Rotational Motion 89

Figure 2.4: Motion o f a rigid body. 12 is the angular velocity and v~ is the translational velocity of the reference poin t r~.

s

I !, !

z

-u

parts" the rigid body can rotate and translate. To make the distinction between the two contributions, the velocities are written as,

v~ - f t x ( r = - r ~ ) + v~, (2.76)

where r~ is an arbitrary point inside the rigid body with a translational velocity v~, and ~ is the angular velocity with respect to the point r~ (see fig.2.4). The equation of motion for the total momentum p is,

dp d d-t Y~Pn

n

dfl

n n

+ M dv~ = F dt '

(2.77)

where F is the total external force on the particle, and M - ~ n mn is the total mass of the particle. With the following choice for the point r~,

- too, (2.78) n n

which is the center of mass of the rigid body, eq.(2.77) becomes similar to Newton's equation of motion for a spherical particle,

dp~/dt - F, (2.79)

90 Chapter 2.

where p~ - Mv~. The rotational motion of the particle is characterized by the angular momentum J,

C t2 J - ~ r . x p. , (2.80) n

where the superscript c refers to coordinates relative to the center of mass coordinate (r~ - r,~ - r~ and p~ - pn - p~). The equation of motion of the angular momentum J follows simply by differentiating the defining equation (2.80), and using Newton's equation of motion for each molecule separately,

dd/dt - ~ r~ x F. - T, (2.81) n

with F,~ the force on the n th molecule. The last equality in this equation defines the torque T on the particle. Eqs.(2.79) and (2.81) are Newton's equations of motion for translational and rotational motion, respectively.

Notice that the angular momentum is a linear function of the angular velocity 12, since, according to eqs.(2.80,76)

a - ~ m n r : x (~ x r : ) . (2.82) n

The right hand-side can be written as a matrix multiplication of f~,

J - V-fl, (2.83)

with I ~ the inertia matrix, the ij th component of which is,

n

(2.84)

with 6ij the Kronecker delta (6ij - 0 for / ~ j , and 6~j - 1 for i - j). The torque, angular momentum, angular velocity and inertia matrix may be considered the rotational counterparts of force, momentum, translational velocity and mass, respectively.

For the calculation of fluctuation strengths via the equipartition theorem, we need an expression for the kinetic energy, Ekin. Using eqs.(2.76,78,84), one finds,

1 =

n


- ~--~ ~m.1 [f~ x r: + v~] �9 [f~ x r: + v~] n

1 2 1 = ~ -~m,~v~ + ~ ~m~(a x r~). (a x r~) n n

1 2 1 )2 )2] = + - ( a . r:

n

1 f t . i~" f~ (2.85) - Mv~ + -~ .

The first term on the right-hand side in the last line is the translational kinetic energy, the second term is the kinetic energy associated with rotation about the center of mass.

2.8.2 The Langevin Equation for a Long and Thin Rod

Clearly, thermal collisions of solvent molecules with the Brownian particle result in both stochastic motion of the center of mass as well as the angular momentum. The Langevin equations are now obtained from the above equations of motion, by simply replacing the external force and torque by their fluctuating counterparts plus a friction term that accounts for systematic collisions with solvent molecules once the particle attains a certain velocity and angular momentum.

In the following, we specialize to a long and thin cylindrically symmetric rod (see fig.2.5a). For such a long and thin rod, the rotational motion around the cylinder axis of symmetry need not be considered. The components of the inertia matrix related to rotational motion around the long cylinder axis are very small in comparison to its remaining components, and may be disregarded. In the following, the angular velocity [2 is therefore understood to denote the component of the angular velocity perpendicular to the cylinder axis of symmetry, as depicted in fig.2.5a.

Denoting the fluctuating force by f, as before, and the fluctuating torque by T, the complete set of Langevin equations for such a particle is (we omit the superscripts "c" in the following),

dp/dt - r~ - M " p + f(t), (2.86)

d r / d t - p / M , (2.87)

dJ /d t - - % Ft + T(t), (2.88)

I . f~ - J. (2.89)

92 Chapter 2.

A

.C). ;" U :b U i

i

Ca) Fh - liV

Figure 2.5: (a) The long and thin cylindrically symmetric rod. (b) The translational friction coefficients '711 and 7•

F

(b)

For the long and thin rod, a little consideration shows that the friction force due to rotational motion is directed along -f~. For not too large angular velocities, the friction torque is proportional to -f~. The proportionality constant 7~ is the rotational friction coefficient. Furthermore, the friction of such a particle due to pure translational motion depends on the orientation of the rod. Let "711 denote the friction coefficient as the rod translates parallel to its long axis, and 7• for translation perpendicular to its long axis (see fig.2.5b). For arbitrary directions of the velocity v, the friction is a simple linear combination of these two friction forces, provided that the hydrodynamic equations governing the fluid flow around the rod are linear. In chapter 5 on hydrodynamics, this will turn out to be the case when the dimensions of the rod are not too large (more precisely : the Reynolds number must be small, with the thickness of the rod as the typical length scale). The unit vector fi pointing in the direction of the long axis of the rod is referred to as the the orientation of the rod. Since then tiff-v is the velocity parallel to the long axis and ( I - riO)-v its perpendicular component, the hydrodynamic friction force F h is,

F h = - l " f .v , (2.90)

with, l"f = 7ll tiff + 7• [I - Off]. (2.91)

Because the translational friction coefficient is orientation dependent, the translational equation of motion (2.86) is coupled to the rotational equation of motion (2.88).


In chapter 5 on hydrodynamics, explicit expressions are derived for the three friction coefficients in terms of the length L and thickness D of the rod. In a simple approximation the friction coefficients are,

{L} % = 7rr/0L3/3 In ~ , (2.92)

711 - 27r~oL/ln ~ , (2.93)

7- - 2711. (2.94)

These expressions are good approximations for very large values of L/D. For rods with a large aspect ratio L/D, the inertia matrix is easily calcu-

lated, replacing the sum over molecules by an integral. For a constant local mass density p of the rod material, the inertia matrix in eq.(2.84) becomes,

_ f dr, ( D ) [�89

,.~ r p d l / 2 [ i - tiff] J_IL

= 1 M L 2 [ i _ tiff] 12

(2.95)

The typical magnitude for the inertia matrix is thus 1ML2. The component of the angular velocity perpendicular to the rods long axis,

f~, is simply expressed in terms of orientational variables (see fig.2.5a). Since,

dfi = f~ x fi, (2.96)

dt

it follows that, dfl

12 - fi x dr" (2.97)

There are two equipartition theorems to be considered here" for the translational velocity and for the angular velocity. First consider the translational velocity. Integration of eq.(2.86) yields,

l ' f p(t) - e x p { - ~ - t } - p ( 0 ) (2.98)

+ Jotdt ' exp{-~-~-f ( t - t ' )}. f(t ').

94 Chapter 2.

The definition of the matrix exponential was already discussed in section 2.7 (see eq.(2.58)). It follows by induction that,

r7 - vl~aa + v ~ [ i - aa] , (2.99)

and hence, from the defining expression for the matrix exponential,

e x p { - ry (t t' 711 ( exp{ "7• t' ~ - -- )} = exp{ t - t ' ) } f i f i + (t - ) } [i -- rift] - ~ , - ~ . (2.100)

Eq.(2.98) can thus be written as,

p(t) - Pll( t)+ p l ( t ) , (2.101)

with,

jfo t 711 t' P l i ( t ) - exp{ 7lit} Pll(0)+ d t ' exp{ - - -~ ( t - )}fll(t') - M

7• f0 t p• - exp{ - ~ t } p • + dt'exp{ --~7• (t - t ' ) )fz (t') ,

(2.102)

(2.103)

where the random force parallel and perpendicular to the rods orientation are defined as,

fll(t) - fi(t)fi(t), f(t), (2.104)

f• - [ i - fi(t)fi(t)], f(t), (2.105)

and similarly for Pll(0) and P.L (0). Since the random force is delta correlated in time and < fll(t), f• (t) > - 0, it follows that there are two independent fluctuation strengths for the random force parallel and perpendicular to the rods orientation,

< fll(t), f,,(t') >

< f~(t), f• (t') > - GII 5(t - t'), (2.106) - G• 5 ( t - t'). (2.107)

Notice that we are working here with inner products instead of dyadic products as for the spherical particle, so that both Gll and G• are scalars. Since,

< pll(t)" p 3 . ( t ) > - O . (2.108)

2.8. Rotational Mot ion 95

the Hamiltonian of the Brownian rod is a sum to two quadratic terms related to the perpendicular velocity and a single quadratic term related to the parallel velocity. From the equipartition theorem (exercise 2.2) it is thus found that,

lim < Pll(t). Pll(t) > - M / f l , (2.109) t----+oo

lim < p• p• > - 2 M / f t . (2.110) t---~oo

The scalar fluctuation strengths are now obtained from eqs.(2.101-103) and (2.106,107) as,

GII - 2 % 1 / / 7 , (2.111) G.L -- 4,'),• (2.112)

This concludes the determination of the translational fluctuation strengths, which will be used to investigate the translational Brownian motion of the rod.

Before performing a similar analysis for the angular velocity, we have to find the rotational analogue of the equipartition theorem (2.15). To this end we return to the Langevin equations (2.86-89). Since 12 is perpendicular to fi, it follows from eq.(2.95) that,

1 I . 12 - -:-~ M L 2 f t . (2.113)

1 2

Substitution of eq.(2.89) into (2.88) and integration then gives,

~( t ) - 12(0) exp{- 127,. M L 2 t } (2.114)

12 L t + M L 2 dt' T(t ') e x p { - - - - - 12- , ( t - t'))

M L 2

The fluctuating torque due to random collisions of solvent molecules with the rod has an ensemble average equal to zero, while,

< T ( t ) T ( t ' ) > - G~ 6(t - t'), (2.115)

with G~ the rotational fluctuation strength. Using this in eq.(2.114) gives,

6 lim < 12(t)12(t) > - G~ (2.116) t~oo % M L 2 "

On the other hand, from eq.(2.113) and the expression for the rotational kinetic energy (the last term in eq.(2.85)), one finds,

E ~Ot~uon~ _ 1 ML2f~2( t ) (2.117) ki~ 2--4 "

96 Chapter 2.

Remember that 12 is the component of the angular velocity perpendicular to the long axis, so that the contribution of the angular velocity along the long axis to the kinetic energy is omitted here. Thus, 1-12 is the sum of two independent quadratic terms, so that the equipartition theorem states that,

lg i?ro ta t iona l < ~-"kin > =ksT (see exercise 2.2). Hence,

kBT lim < f~(t)f~(t) > - 12iML 2.

t--.~ oo (2.118)

Combining eqs.(2.116) and (2.118) identifies the rotational fluctuation strength,

G~ - i 27~ /3 " (2.119)

This is the rotational analogue of eq.(2.17) for the translational fluctuation strength.

Having determined the fluctuation strengths we are now in the position to analyse the statistics of translational and rotational displacements.

2.8.3 Translational Brownian Motion of a Rod

Rcscaling the Langcvin equation (2.86) as discussed in section 2.6 shows that the diffusive time scale for translational motion is much larger than both M/711 and M/7• The Langevin equation for the position coordinate of the center of mass of the rod on the diffusive time scale is,

dr/dt - r~ ~ f( t) . (2.120)

The inverse of the friction matrix (2.91) appearing here is easily calculated,

1 1 r 7 x - - -off + [ i - off]. (2.121)

711 7•

Using this result, the Langcvin equation (2.120) can be written in terms of the parallel and perpendicular components of the random force (see eqs.(2.104,105)),

1 1 dr/dr- ~fl l(t)+ f• (2.122) "711 ")'•

The integrated Langevin equation is,

2.8. Rotat ional Mot ion 97

which can be used, just as for the spherical particle, together with eqs.(2.106, 107) for the ensemble averages of the parallel and perpendicular components of the random force and the expressions (2.111,112) for the fluctuation strengths, to calculate the mean squared displacement,

<l r ( t ) - r ( 0 ) 1 2 > - 6Dt, (2.124)

where, - 1

D - ~ (DII + 2D+). (2.125)

Here we introduced the translational diffusion coefficient for parallel and perpendicular motion, in analogy with the definition (2.37) of the diffusion coefficient for a spherical particle, by the Einstein relations,

D I I - 1//3,Yll , (2.126) D• - 1//37• (2.127)

The expression (2.124) for the mean squared displacement is identical to that for a spherical particle. The center of mass of the rod thus diffuses as if the particle were spherically symmetric. It should be realized, that the ensemble average in eq.(2.124) is also with respect to the orientations of the rod. For the free diffusing rod considered here, each orientation has equal probability, so that the mean diffusion coefficient (2.125) is a weighted average of the two diffusion coefficients for parallel and perpendicular translational motion.

2 . 8 . 4 O r i e n t a t i o n a l C o r r e l a t i o n s

The rotational Langevin equation (2.88) may be coarsened to a diffusive time scale, using the rescaling procedure discussed in section 2.6. The rotational inertial term dd/dt in the Langevin equation (2.88) may be set equal to zero

M L 2 I on the time scale TD, which is much larger than -iT-/%. This can be seen as follows. First, it follows from eq.(2.95) for the inertia matrix for the long and thin rod, and eq.(2.89) for the angular velocity, that,

1 J - I . 12 - - - M L 2 f ~ . (2.128)

12

The Langevin equation (2.88) thus becomes,

1 - ~ M L 2 d f~ /d t - - % 1 2 + T(t). (2.129)

98 Chapter 2.

ML2 /"~'r , The rescaling arguments of section 2.6 show that on a time scale >> --iT this reduces to,

dfl 1 f~ - f i x = - -T( t ) . (2.130)

dt %

Here we are concerned with the statistics of the orientation ft. As a first step, the differential equation (2.130) should then be solved for fi(t) in terms of the fluctuating torque T. To this end, eq.(2.130) is rewritten as,

dfi/dt - 1T( t ) x ft. (2.131) %

To integrate this equation, the right hand-side is written as a matrix multiplication,

dfi/dt - A(t) . fi, (2.132)

with, I 0 -T~(t) T:(t) l 1 T3(t) 0 -T~(t) , (2.133)

A( t ) - ~ -T2(t) Tl(t) 0

where Tj is the jth component of T. The differential equation (2.132) is equivalent to the integral equation,

f0 t ,a(t) = ,a(0)+ dt 'A(t ' ) . ,a ( t ' ) , (2.134)

which is solved by iteration,

,a(t) fi(0) + ~ dtl dt2 dt3. . , dt~_a dtn n=l dO

A(ta)" A(t2) . . . . . A(tn) . s (2.135)

For the calculation of the ensemble average of fi(t), the ensemble averages of the multiple integrals over products of A's must be evaluated explicitly. From the definition of the matrix A it follows immediately that,

1 A(t)-f i (0) = - -T ( t ) x fi(0), (2.136)

% 1

A2(t) �9 fi(0) = ~-~2T(t) x (T(t) x fi(0))

1 = 7--~ [-T2(t) i + T( t )T( t ) ] . fi(0). (2.137)

I

L ~ t 2

tl Figure 2.6: Integration of the correlation function of the torque over half the domain of its argument.

Since the ensemble average of the random torque, and hence of A, is zero, and its correlation function is delta correlated in time, the first two terms in the ensemble averaged iterated solution (2.135) are found from eqs.(2.136,137),

fotdtl < A(tl) > .fi(0) -

f0' jo O, (2.138)

2 t fi(0). (2.139)

Here we used that, 1

o t~ dt2 t~(ta - t2) - ~ . (2.140)

Since tl is not in the interior of the integration range here, this integral is 1 not equal to 1. That its value is equal to 7 can be seen as follows. On the

smallest time scale, the correlation function < T (t 1) T (t 2) > of the random torque, and hence of A, is a symmetric function of the difference t l - t2. The integral with respect to t2 in eq.(2.139) ranges over half of the symmetric

1 correlation function (see fig.2.6), and is thus equal to 7 x the integral ranging 1 over the entire range of the argument. This explains the value of 7 of the

integral in eq.(2.140). Mathematically speaking one could say that the delta distribution here is the limit of a sequence of symmetric functions, and the integration ranges only over half of the domain of its argument. To evaluate the ensemble averages over higher order products of A in the iterated solution (2.135), we use that, on the diffusive time scale, T, and hence also A, is a

1 O0 Chapter 2.

Gaussian variable. On the diffusive time scale, T is an average over many independent realizations, so that, according to the central limit theorem, it is a Gaussian variable (for exactly the same reason that the random force f in the Langevin equation (2.2) for a sphere may be considered Gaussian). All the ensemble averages of products of an odd number of A's are thus zero (see exercise 1.16). The ensemble averages of products of an even number of A's can be written as a sum of products of averages of only two A's (see the introductory chapter on Gaussian variables, in particular Wick's theorem (1.81)). Consider for example the ensemble average of the n = 4 term in the iterated solution (summation over the repeated indices p, q, r, s is understood here, Aij is the ij th component of A and ~t~(0) is the s th component of fi(0)),

dtl dt2 dt3 dt4 < Aip(ta)Apq(t2)Aq~(t3)A~,(t,) > ~t,(O)-

dta dt2 dta dt4[< Aiv(tl)Apq(t2) > < Aq~(ta)A~,(t4) > ~z,(O)

+ < Aiv(tl)Aq~(ta) > < Avq(t2)A~(t4) > ft,(O) + < Aiv(tl)A~(t4) > < Avq(t2)Aq~(ta) > ~(0)] .

For the respective products of ensemble averages in the above equation we need to evaluate the following integrations over delta distributions,

fo dt fot~ t2 t3 t 1 dr2 fo dr3 fo dr4 ~ ( t l - t2) ~( t3 - t4)

dtl dt2 dt3 dt4 5(tl - t 3 ) 5 ( t 2 - t4 ) ,

and,

The first of these four-fold integrals is equal to,

dtl dt2 dtz fo dt4 5(tl - t2) 5(tz - t4) - (g 2

1 )2 originates from integration of delta functions ranging where the factor (g over half the domain of their arguments, as explained above. By inspection, the other two four-fold integrals turn out to be zero, because the arguments of the delta functions are non-zero in the entire integration range. Only products

with the consecutive time ordering t~ ~ t2 ~ t3 --* . " ~ tn contribute. Using the expression (2.119) for the rotational fluctuation strength, we thus arrive at the following result,

fo t dtl fo < A( t l ) . A(t2). A(t3). A(t4) > . fi(0)

4 )2 1)2 l t2fl(0) =

In the next higher order terms in the ensemble average of the iterative solution (2.135), the product with the consecutive time ordering is likewise the only surviving one. Along similar lines one shows that, for even n's,

fo t dtl "" fo t"-~ 4 )~/2 1)n/2 l t n / 2 I. dt~ < t ( t l ) . . . . . t ( t n ) > - ( - -~ r r (2 2"

The iterative solution is thus,

< a( t )> 1 )n t,,] - ~ ~ (-2D~ fi(0) n-'0

= exp{-2D~t} fi(0), (2.141)

where, in analogy with the definition of the translational diffusion coefficient for a sphere, eq.(2.37), the rotational diffusion coefficient D~ is defined by the Einstein relation,

D~ = 1/fl%. (2.142)

The mean squared rotational displacement is thus equal to,

<1 fi(t) - fi(0) 12> - 2 (1 - exp{-2D~t}). (2.143)

For small times this result is quite similar to eq.(2.21) for the mean squared displacement of a sphere,

<1 f l ( t ) - f i (0 )12>- 4D~t , 2D~t << 1. (2.144)

Rotational Brownian motion may be visualized as a point on the unit spherical surface, representing the tip of the unit vector fi, which exerts Brownian motion. For small times this is Brownian motion on a two dimensional fiat surface. The result (2.144) for small times is thus equivalent to the mean squared displacement of a sphere in two dimensions, except that the diffusion

102

t ll

Exercises Chapter 2

t"= t'

Figure 2.7" The integration range in the (t', t") -plane.

coefficient is different. For larger times the tip experiences the curvature of the unit spherical surface, leading to the more complex behaviour as described by eq.(2.143).

Exercises

2.1) * Integration of eq.(2.6) leads to the following double integral,

fo fo "r (t" t dt" *" dt' f ( t ' ) e x p { - ~ - t')}.

The integration range in the (t', t")-plane is sketched in fig.2.7. Interchange the order of integration to reduce this double integral to the single integral in eq.(2.8).

2.2) The equipartition theorem Let X be a n-dimensional vector. Suppose that the Hamiltonian H is of

the form,

n

H - C + } ' ~ X ~ - C + X . X , m--- -O

with C independent of X. The equilibrium pdf P~q of X is then proportional

tO,

P~q ,-, exp{-fl X . X}.

Verify by integration that,

1 < > -

z~p

with 6~j the Kronecker delta.

2.3) A spherical Brownian particle with a radius of 100nm and a mass density of 1.8 g / m l is immersed in water, with a viscosity equal to 0.001 N s / m 2. Use eq.(2.1) for the friction coefficient "7 to calculate the time M / 7 and the diffusive length scale l o. Calculate the time at which the mean squared displacement is equal to a 2, with a the radius of the Brownian particle.

2.4) Brownian motion in an externa/force field A constant force F is applied to a spherical Brownian particle (an example

of such a force is the earth's gravitational force). Solve the Langevin equations for the position coordinate on the diffusive time scale to show that,

1 [ r - r ( O ) - ~ t I P(r , t [ r(O), t - O) - (47rDot)a/~ exp - 4Dot

As a first step, you should verify that the fluctuation strength is unaffected by this force. Interpret the result.

(Hint" For long times, the particle attains a constant velocity. The equipartition theorem (2.15) must now be taken with respect to the coordinate frame attached to the particle, that is, in the left hand-side of eq.(2.15), the momentum coordinate p(t) must be replaced by p ( t ) - < p(t) >.)

2.5) Brownian motion in shear flow Calculate the mean position < r(t) > and the mean squared displacement

< r(t)r(t) > for a Brownian particle in simple shear flow, with its position at an arbitrary point r(0) at time zero. Interpret the results.

2.6) Consider a Brownian particle which can occupy only discrete positions which are indexed by the integer n E { " . , -3 , - 2 , - 1 , O, 1, 2, 3 , . . .} . Suppose that the probability per unit of time for a single step to the left or the

104 Exercises Chapter 2

right is equal to a. Let P(n, t [ no, t - 0) denote the pdf for the position n of the Brownian particle, given that at t - 0 the particle was at the position no. Interpret the various terms in the following equation of motion for P,

O R ( n , t I n o , t - o)

Ot c~ [P(n + 1, t [ no, t - 0) + P(n - 1, t I no, t - 0) - 2P(n , t ] no, t - 0)].

Derive from this equation the equations of motion for < n - no > and the mean squared displacement < (n - no) 2 >. Show that < n - no > - 0 and < (n - no) 2 > = 2czt. Compare this with eq.(2.21).

2.7) Translational velocity of a rod Let fi be the fixed orientation of a long and thin rod. A constant force F is

applied to its center of mass. Calculate the angle between F and the ensemble averaged velocity which the rod attains in the steady state in terms of 711, 7• and ft.

2.8) The diffusive angular scale Use the Langevin equation (2.129) to estimate the ensemble averaged

angular displacement of a rod during a time interval comparable to the diffusive M L 2 time scale >> --i7-/7~, similar to the analysis at the end of section 2.3. This

is the coarsened angular resolution, similar to the diffusive length scale for translational motion.

2.9) Suppose that a rod is aligned in a very strong external field that exerts a torque but not a net force on the rod. The orientation of the rod is along the z-axis and Brownian motion of the orientation is fully suppressed by the external field, that is, the orientation is a given constant. Analyse the Langevin equations (2.86,87) to show that,

< > -

< ( z ( t ) - > -

< ( y ( t ) - y(O)) 2 > - 2D•

2Diit.

Here, x, y and z are the three components of the position coordinate of the center of mass.

(Hint" Since fi is now fixed along the z-direction, the different components of the displacement in eq.(2.123) can be analysed separately.)

Further Reading 105


Early papers on Brownian motion of non-interacting particles are, �9 A. Einstein, Investigations on the Theory of the Brownian Motion, Dover

Publications, 1956. �9 M. von Smoluchowski, Ann. Phys. 21 (1906) 756. �9 M.P. Langevin, C.R. Acad. Sci. Paris 146 (1908) 530. �9 S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1.

The book of Wax contains several interesting early papers on Brownian motion, including the above mentioned paper by Chandrasekhar,

�9 N. Wax (ed.), Selected Papers on Noise and Stochastic Processes, Dover Publications, New York, 1954.

The theory of stochastic differential equations, of which the Langevin equations discussed here are simple examples, is a separate discipline of mathematics. More about stochastic differential equations can be found in,

�9 M. Lax, Rev. Mod. Phys., 38 (1966) 541. �9 N.G. van Kampen, Stochastic Processes in Physics and Chemistry, North

Holland, Amsterdam, 1983. �9 C.W. Gardiner, Handbook of Stochastic Methods, Springer-Verlag,

Berlin, 1983.

Chapter 3

LIGHT SCATTERING

107

108 Chapter 3.

3.1 Introduction

Light scattering by colloidal suspensions is a major experimental tool to study the statistical properties of these systems. In further chapters we present experimental light scattering results, so that knowledge of this important experimental technique is, at least, desirable. What makes light scattering such an important experimental tool, is that the scattered electric field strength is directly proportional to a certain Fourier component of the instantaneous microscopic density. The Fourier component that is probed is set by the direction in which the scattered light is detected. This enables the study of density fluctuations, which are the result of Brownian motion of the colloidal particles. For example, the predictions about the dynamics of non-interacting particles, obtained in the previous chapter, can be verified by light scattering in an experimentally straightforward manner. The same holds for interacting Brownian particles, which are considered in later chapters.

Let us first try to understand intuitively why the scattered electric field strength is related to the microscopic density. Consider an assembly of points, fixed in space. Suppose a plane wave of monochromatic light impinges onto this assembly of points, each of which scatters light without changing its wavelength nor its phase. The total electric field strength that is scattered in a certain direction is the sum of the electric fields scattered in the same direction by the individual points. Clearly, the phase difference of the scattered light from two points depends on their relative positions, as well as on the direction in which the electric field strength is measured (see fig.3.1). As the two points change their relative position, the phase difference of the electric field strengths scattered by these two points changes, so that the measured total electric field strength changes. A measurement of the electric field strength (or the scattered intensity) thus contains information concerning the relative positions of the points, that is, on the instantaneous realization of the fluctuating microscopic density of the assembly of points. Information about different Fourier components of the density is obtained, in principle, by measuring the scattered intensity in different directions. The scattered intensity contains this structural information only, when the distance between the points is of the order of the wavelength of the scattered light. Only in those cases, changes in positions give rise to phase changes of the light which lead to appreciable changes in constructive and destructive interference.

The size of colloidal particles, as well as their nearest neighbour distances are of the order of the wavelength of visible light. When we then imagine the

3.2. HeuristicDerivation 109

k 0

I

V'X r w

j ~ I

I

' k . / I I [ . . . .

�9 �9

O

A B

r,z-x . I / / / v N I

, ~ �9 I / �9 . / ' X/o . . . . . . . . . .

' ~ z �9

~ k

Figure 3.1" A schematic representation of the scattering of light by an assembly of point- like particles (.). Each of the Brownian particles can comprise many of the ooint-like scatterers.

assembly of points to represent an assembly of Brownian particles (where a ~ingle Brownian particle may comprise many points), it becomes obvious that ~cattering data contain both information concerning the internal structure of individual particles and their mutual separations. In this chapter we quantify :hese ideas.

Before actually solving the Maxwell equations in order to obtain the scat- :ered electric field strength in terms of particle sizes, orientations and position ;oordinates, we shall first, in the next section, continue on the above discussed qualitative considerations.

3.2 A Heuristic Derivation

think of each Brownian particle as being composed of infinitesimally small volume elements. A single volume element may be identified with a point

110 Chapter 3.

scatterer as described in the introduction. The incident field is a monochromatic plane wave. Let us first calculate the phase difference of electric field strengths scattered by two volume elements, with position coordinates r and r' say, into a direction that is characterized by an angle O, (see fig.3.1). O, is the angle between the propagation direction of the incident plane wave and the direction in which the scattered field is detected, and is be referred to as the scattering angle. The incident wavevector ko is the vector pointing in the propagation direction of the incident field, and its magnitude is 27r/A, where )~ is the wavelength of the light. Similarly, k, is the scattered wavevector" its magnitude k, - I k~ I is equal to that of the incident wavevector,

k o - k, - 2~r/A. (3.1)

It is thus assumed that the interaction of the electric field with the material of the Brownian particles is such that the wavelength is not affected. Since the energy of each photon is then the same before and after the scattering process, such a scattering event is called elastic. The photon is bounced off the scattering material without any transfer of energy to that material (see, however, exercise 3.1). We thus limit ourselves here to what is usually referred to as elastic light scattering.

The phase difference A~ of the electric field strengths scattered by two points located at r and r' under a scattering angle O, is equal to 27r A/)~, where A is the difference in distance traversed by the two photons : A = A B § B C (see fig.3.1). Now, A B - ( r ' - r) . ko/ko, and B C - ( r - r ' ) . k~/k~. Hence, using eq.(3.1),

2xr - ( r ' - r). ( k o - k~). (3.2)

To every volume element at a position r, we can thus associate a phase equal to r . (ko - k~). The total scattered electric field strength E~ is the sum of exp{ir . (ko - k,)} over all volume elements, weighted by the scattering strength of the volume elements, which is proportional to the fraction of the incident field strength that is actually scattered. The scattering strength per unit volume of a volume element at r is denoted here as f(r) . The scattering strength of a volume element with volume dr is then f ( r ) dr. Replacing the sum over volume elements by an integral yields,

E~ - fv, dr f ( r ) exp{i(ko- k~). r} Eo, (3.3)

where Eo is the incident field strength. In the next section it is shown that the scattering strength f(r) is related to the dielectric constant of the colloidal

3.2. Heuristic Derivation 111

in phase se

f (b)

Figure 3.2: The difference in the phase of fields traversed through a colloidal particle and the solvent (a) and the refraction at the interface solvent~colloidal particle (b).

material at r, relative to that of the solvent. The integration range V~ in eq.(3.3) is the illuminated volume from which scattered light is detected. This volume is the scattering volume.

In the derivation of eq.(3.3) it is assumed that the phase of the incident field is simply related to the position r in space as 27rr. ko/k0A. The refractive index of the colloidal material, however, is generally different from that of the surrounding fluid, so that the wavelength of the light inside the colloidal particles and in the fluid differ. Since part of the incident light, before it is scattered, traverses through the material of colloidal particles and in part traverses through the fluid, there are phase differences in the incident field (see fig.3.2a). The phase difference of the electric field that traverses through

t/, a a colloidal particle and the field that traverses through the fluid is 27r[~ - ~] , with a the radius of the colloidal particle and Ap (A f) the wavelength of the light in the particle (fluid). Let A0 denote the wavelength of the light in vacuo, so that Ap - Ao/np and A, - Ao/n,, with np (n~) the refractive index of the colloidal particle (fluid). The phase difference is thus equal to 27r[np - n f] ~o" This phase shift should be small (say < 0.1) in order that eq.(3.3) is a good approximation,

a 27r I n p - n f I-- r < 0.1. (3.4)

A o

A second assumption implicitly made to derive eq.(3.3), is that the incident field is not attenuated, either by scattering or by absorption. Therefore, only a small fraction of the incident light may be scattered, and both the fluid and colloidal particles are assumed not to absorb light.

A third assumption is that the direction of the incident field is the same

112 Chapter 3.

everywhere in the scattering volume. Refraction of light at the interface fluid/colloidal particle is thus neglected (see fig 3.2b). This is justified when,

[ n p - n s l < 0.1. (3.5)

Furthermore, multiple light scattering is neglected. That is, light being scattered once is not scattered a second, third, .-. time, neither within the colloidal particles nor between distinct particles, before reaching the detector. These higher order scattering events are negligible when only a small fraction of incident light is scattered.

The integral (3.3) may be rewritten in order to make the distinction between interference of light scattered from volume elements within single colloidal particles and from distinct particles. Since the scattering strength is only non- zero within the colloidal particles, eq.(3.3) can be written as a sum of integrals ranging over the volumes Vj, j - 1, 2, . . . , N, occupied by the N colloidal particles in the scattering volume,

N

E, - j~l iv, dr f ( r ) exp{i (ko- k , ) . r) Eo. (3.6)

The integration range Vj is the volume that is occupied by the jth colloidal particle. For non-spherical particles this volume depends on the orientation of the particle, and for any kind of particles, also for spherical particles, Vj depends on the location of the jth particle. Let rj denote a fixed point inside the jth particle, which is referred to as its position coordinate. The position coordinate dependence of Vj can easily be accounted for explicitly, by changing for each j the integration variable to r' = r - - rj. The new integration range Vj ~ is the volume occupied by the particle with its position coordinate at the origin. For spherical particles, with their positions chosen at the center of the spheres, Vj ~ is a sphere with its center at the origin. For non-spherical particles Vj ~ depends on the orientation of particle j. In terms of these new integration variables eq.(3.6) reads,

N

- - fy dr' f ( g ) e x p { i ( k o - k~). r')} Eo. (3.7) E, ~ e x p { i ( k o k , ) . r j ) } o j = l

The exponential functions containing the position coordinates rj describe the interference of light scattered from different colloidal particles, while the integral describes interference of light scattered from different volume elements within single particles.

3.3. Maxwe11Equation Derivation 113

The scattering theory in which the above mentioned conditions are assumed is usually referred to as the Rayleigh Gans Debye scattering theory. Although these conditions seem quite restrictive, scattering data of many colloidal systems can be interpreted with the help of the expression (3.7) for the scattered electric field strength.

In the following section, the result (3.7) is obtained from the Maxwell equations, leading to an expression for the scattering strength f ( r ) in terms of the dielectric properties of the colloidal particles and the solvent. In fact, the scattering strength is in general a matrix f(r), since the scattering material inside a given volume element may be optically anisotropic. The scattered intensity is then proportional to f ( r ) . Eo, and has a polarization direction that can be different from the incident field E0. In the above equations, the scattering strength should therefore be replaced by a matrix that multiplies the incident field strength Eo.

Those readers who are satisfied with the above heuristic derivation may prefer to skip the following section. Just take notice of the more precise expressions (3.44,45,33) for the scattered electric field strength.

3.3 Th e M a x w e l l E q u a t i o n Der iva t ion

The incident electric field strength Eo(r, t) at a point r and at time t is a plane wave with wavevector ko and frequency w,

Eo(r, t) - Eo exp{i[ko �9 r - wt]}, (3.8)

where Eo is the amplitude of the incident field strength. The total electric field strength E(r, t), which is the sum of the incident field and the scattered field, satisfies the Maxwell equations,

0 V x E(r, t) = -0-~B(r, t ) , (3.9)

0--D , v x H(r,t) = Ot (r,t) (3 10)

with B the magnetic induction, H the magnetic field strength and D the electric displacement. The colloidal material and the fluid surrounding the colloidal particles are assumed to behave as linear dielectric materials with a magnetic permeability equal to that of vacuum, #o,

D(r , t ) - C(r). E ( r , t ) , (3.11)

B(r , t ) - #o H ( r , t ) . (3.12)

114 Chapter 3.

Here E(r) is the dielectric constant at the point r. This dielectric constant is in general a matrix. All matter is supposed not to exhibit magnetic properties which affect the scattering process. Furthermore, for positions r outside the colloidal particles, the dielectric constant is equal to that of the fluid, E(r) - I of, which is assumed isotropic (I is the identity matrix). The fluid is also assumed homogeneous, so that e f is a constant, independent of the position r. Fluctuations of the dielectric constant of the fluid, resulting from density fluctuations, are neglected here. Otherwise ef should be replaced by a position dependent dielectric constant, describing a particular realization of the fluctuating dielectric constant. Scattering from the solvent is thus neglected here. Only scattered intensity due to inhomogeneities in the dielectric constant due to the presence of the colloidal particles is considered.

Taking the curl of eq.(3.9), using that V x (V x E) - V(V. E) - V2E and substitution of eqs.(3.10,11,12) yields a single equation for the total electric field strength,

~2 V ( V . E(r, t)) - V2E(r, t) - -#o g'(r) �9 ~-~ E(r, t) . (3.13)

In case the scattering process is elastic and the colloidal material reacts instantaneously on the incident electric field (so that there is no discrete phase shift when scattering occurs), the total electric field strength is of the form,

E(r , t ) - E(r) exp{-iwt}. (3.14)

Substitution into eq.(3.13) gives,

V(V. E ( r ) ) - V2E(r) - #oW 2 g'(r) �9 E(r). (3.15)

Fourier transformation with respect to r gives (replace V in eq.(3.15) by ik, as discussed in the introductory chapter at the end of subsection 1.2.4),

[k 2 ] : - kk]- E(k) - #o OJ2 / dr g'(r). E(r) e x p { - i k , r}, (3.16)

with,

E(k) - / dr E(r) exp{- ik �9 r)}, (3.17)

the Fourier transform of E(r) and k the conjugate Fourier variable of r, which is referred to as the wavevector. In eq.(3.16), kk is a dyadic product (see the introductory chapter on notation, subsection 1.2.1).

3.3. Maxwe11Equa t ion Derivat ion 115

In order to solve the integral equation (3.16) by iteration to first order in g(r) - i e f, it will turn out to be convenient to subtract #ow 2efE(k) from both sides,

[(k 2 - # o w 2 e f ) i - kk] .E(k) - #ow 2 / dr ( g ' ( r ) - i e f ) .E( r ) e x p { - i k - r } .

(3.18) Since for an uncharged solvent, V �9 E(r , t) - 0 for positions in the solvent, eq.(3.13) reduces to the following wave equation,

[o2] V 2 - #o ef ~ 7 E(r , t) - 0 , (3.19)

showing that the speed of light cf in the fluid equals,

c f - 1/x/#0e ) . (3.20)

On the other hand, the frequency u of the light multiplied by its wavelength Af is its velocity cf �9 v A f - cf . Using that w - 27rv and ko - 27r/Af one finds,

cf ko = w . (3.21)

Combination of eqs.(3.20,21) gives,

ko - ~/#o w 2 e l . (3.22)

With the help of this relation, eq.(3.18) can be written as,

- 1 i - �9 E ( k ) = ~ dr �9 E ( r ) e x p { - i k . r}, j=l ef

(3.23) where we also used that ~'(r) - [ ef - 0 outside the colloidal particles, so that the integral in eq.(3.18) is a sum over integrals ranging over the volumes Vj, j - 1, 2 , . . . , N, occupied by the assembly of N colloidal particles.

As a first step in solving eq.(3.23), the matrix on the left hand-side must be inverted. The determinant of that matrix, however, is zero for k = rkko, so that for these wavevectors the inverse of the matrix does not exist. To avoid this problem, suppose now that the solvent is slightly absorbing the incident light. In that case, according to eq.(3.8), ko is a complex number, with a small imaginary component. The matrix can then be inverted for any real value of k. We therefore replace ko by ko + ia , with both ko and a real numbers. In the

116 Chapter 3.

solution of the integral equation (3.23) we will let a tend to zero. The inverse of the matrix is (see exercise 3.2),

[( k2 ) kk ]1 ,ko+ o, ti kk __ (k0 +ia) 2 ] ( k o + i a ) 2 - 1 i - ( k o ~ i a ) 2 - k 2 - ( k o + i a ) 2 "

(3.24) The a-dependence of the numerator is of no importance, and we may set a equal to zero there. It is the a-dependence of the denominator which is essential to be able to invert the matrix for all real valued wavevectors. Therefore, only the a-dependence in the denominator in eq.(3.24) is kept. The integral equation (3.23) is thus converted to,

N

E(k) - ko 2 T~(k) �9 ~ fy~ dr j = l

E(r)- ies ef

�9 E ( r ) e x p { - i k . r} , (3.25)

where,

Substitution of E(r) in terms of its Fourier transform E(k),

(3.26)

1/ E(r) - (27r)a dk 'E(k ' ) exp{ik' , r}, (3.27)

leads to the following integral equation for E(k),

E(k) = T~(k ) .~ ] dk' dr e x p { i ( k ' - k ) . r} .E(k').

(3.28) Suppose that the dielectric constant of the colloidal particles is equal to

that of the solvent. Then eq.(3.28) predicts that E(k) - 0, which is incorrect since in this case the electric field strength should be equal to that of the incident field. That is, if E(r) - I el, then we should have,

E(k) - / dr [Eo exp{iko �9 r}] e x p { - i k , r} - (27r) 3 Eo 6(k - ko), (3.29)

with 6 the delta distribution. What is missed in the derivation of eq.(3.28) is the nullvector of the matrix on the left hand-side of eq.(3.23). The nullvector

3.3. Maxwell Equation Derivation 117

of that matrix is precisely the incident field strength. With the use of eq.(3.29) it is easily verified that,

k ~ - I i - . E o $ ( k - k o ) - 0 , (3.30)

since ko _1_ Eo. Before inversion of the matrix in eq.(3.23), this nullvector should be subtracted from E(k) in order to assure that the solution of eq.(3.28) equals the incident field strength in case E(r) - ef i. Eq.(3.28) should thus read,

E ( k ) - (2 r )3Eot~(k-ko)

k~ T~(k ) . ~ f d k ' dr j=l

g ( r ) - i e I

ef

(3.31)

exp{i(k' - k) . r}/ E(k'). J

Each of the integrals here, ranging over a volume Vj occupied by the jth colloidal particle, is depending on the position rj of that particle. Integration with respect to r' - r - rj instead of r transforms the integral into,

i fvdrg(r) exp{i(k' - k) . r} (3.32) s

el

g(r ') - i ef = exp{ i (k ' - k) . rj} fv ~ dr' ef exp{i(k' - k ) . r'},

where Vj ~ is the volume that is occupied by the jth colloidal particle with its position coordinate at the origin. The integral over Vj ~ repeatedly occurs, so that we give it its own abbreviation,

~C(r') t s B j ( k ' - k) - / .IV

exp{i(k' - k) . r '} . (3.33)

This matrix is referred to as the scattering amplitude of the jth colloidal particle. It depends entirely on the optical properties of that particle and, for non-spherical particles, on its orientation. With eqs.(3.32,33) the integral equation (3.31) is finally written as,

E(k) - (2~) 3 Eo ~(k - ko) (3.34) N

+ (2r) 3k~ T~(k) �9 ~ f dk' exp{ i (k ' - k) . r j}Bj (k ' - k ) - E ( k ' ) j--1

118 Chapter 3.

The first term is the incident field contribution, the second term arises from scattering of the colloidal particles.

Eq.(3.34) is a complicated integral equation which is not easily solved in closed analytical form. Suppose, however, that the scattered electric field is only a small fraction of the incident field. This is the case when the difference in the dielectric constant of the fluid and the Brownian particles is not too large. The total electric field is then approximately equal to the incident field. The scattering contribution in eq.(3.34) can then be approximated by taking E(k') in the integral equal to the Fourier transform of the incident field, E(k') - (27r)aEo6(k ' - ko). This leads to,

E(k) - (27r) 3 Eo 8(k - ko) (3.35) N

+ ko 2 T~(k) �9 ~ exp{i(ko- k) . r j } B j ( k o - k) . Eo. j=l

This is the first term in the iterative solution of the integral equation (3.34). In literature on scattering such an approximation is usually referred to as a first order Born approximation. The higher order terms in the iterative solution represent higher order scattering events. Only first order scattering is accounted for in the first order Born approximation, that is, multiply scattered intensities are not considered.

The total electric field within the colloidal particles is approximately equal to the incident field when both its direction and phase are not too much affected by the colloidal material. The conditions (3.4,5) assure that this is indeed the case.

The final step in the calculation of the electric field strength is the Fourier inversion of the solution (3.35) (see eq.(3.27)). This can best be done by resubstitution of the definitions of the scattering amplitude B in eq.(3.33) and the matrix T~ in eq.(3.26), and retransformation to the integration variable r " - r ~ + r j ,

k~ r"} E(r) - Eo exp{iko, r} + (27r)-------- ~ f dr" exp{iko. (3.36)

[ ( V V ) f exp{ik. ( r - r")} ] g ( r " ) - • i + ko dk "

The gradient operators V are with respect to r. We used here that,

�9 E o .

VV exp{ik, r} - - k k exp{ik- r ) .

3.3. Maxwe11Equation Derivation 119

Im k

Im k - k~ -~ - , Rek

Figure 3.3: The integration contours for the evaluation of the k-integral in eq.(3.36).

The k-integral on the right hand-side of eq.(3.36) is calculated with the help of the residue theorem after integration over the angular spherical coordinates of k �9 the relevant closed integration contours in the complex k-plane are sketched in fig.3.3. This integration is worked out in exercise 3.3. After a lengthy calculation it is found that,

1 -q- k o l r - r" l

( V V ) f exp{ik �9 (r - r")}

{ [ , , _, i 1 ] r"l 2 2r 2 l + k o l r _ r , , l - k ~ l r - ~ -

3i

(3.37)

3 ] ( r - r " ) ( r - r " ) } e x p { i k o l r - r " l } ko 2 I r - r " 12 ] r - r " 12 I r - r" I

The experimental situation is always such that the position r of the detector is at a distance from the scattering volume that is very much larger than the wavelength of the light. Since r" lies inside the scattering volume, it follows that ko [ r - r" I>> 1. The two expressions between the square brackets in eq.(3.37) thus reduce to unity. Secondly, the distance between the detector and the scattering volume is always very much larger than the dimensions of the scattering volume, that is, r >> r". Therefore we can Taylor expand [ r - r" I around r" = O,

r. r" 1 ( ! rr) .r , ,r , , l r - r " l - ~ ~ ~ i - 7~ + . . - . (3.38/

120 Chapter 3.

It follows that in case, ko(r")2/r << 1, (3.39)

the exponential function on the right hand-side of eq.(3.37) may be approximated as,

r �9 r I I exp{iko I r - r" I} ~ exp{ikor} exp{-iko }. (3.40)

r

Since r" is of the order of the linear dimension l, of the scattering volume and r is of the order of the distance ld of the detector from the scattering volume, the inequality (3.39) is satisfied when,

kol~/ld << 1, (3.41)

which is a constraint on the experimental geometry. Typical values for ld and ko are, ld ,~ 0.3 m and ko ,.~ 2 �9 10 r m -~. The inequality (3.41) then implies that I, << 120 #m. In practice this is very difficult to realize. The point here is that, when measuring the scattered intensity, that intensity is the sum of many statistically independent contributions from regions in the scattering volume" each region contains particles which interact, while (the majority of) particles in different regions do not interact which each other. The linear dimension of such "clusters of particles" is a few times the distance at which the pair- correlation function has its main peak. In effect, the linear dimension of the scattering volume l, in eq.(3.41) can be replaced by the linear dimension of the statistically independent regions. The latter is indeed very much smaller than 120 #m.

The expression (3.37) may thus be replaced by its far field approximation,

( V V ) f exp{ik �9 (r - r")} lim I + dk = (3.42) k0 - (ko +

2r 2 { i - ~-~rr} exp{ikor}r exp{-ik~ r" r" } " r

A further consequence of the fact that the linear dimensions of the scattering volume are much smaller than the distance between the scattering volume and the detector is that, r/r ,~ ks/k~, with ks the scattered wavevector. Since k, - ko we therefore have, kor/r ~ k,. Using this in eq.(3.42) and subsequent substitution into eq.(3.36), transforming again the integration variable from r" to r' = (r" - rj), finally leads to,

E(r) - Eo exp{ik0, r} + E , ( r ) , (3.43)

3.3. Maxwell Equation Derivation 121

with E, the scattered electric field strength,

N

E~(r, ko-k~) - 47r k~ exp{ikor}r T(k,).~--~ Bj(ko-k~)exp{i(ko-k , ) . r j} .Eo j = l

(3.44) where we abbreviated,

T(k,) - I - k~l~, (3.45)

with k, - k,/ko the unit vector in the direction of k~. The dependence of the scattered field strength on the detector position r and ko - k, is denoted explicitly in eq.(3.44). The prefactor exp{ikor}/r represents a spherical wavefront, with an electric field amplitude ,-, 1/r. The intensity then varies like ,~ 1/r 2. This expresses conservation of energy, since the surface area of a sphere with radius r is ,-~ r 2.

Apart from the factors multiplying the summation over colloidal particles and the polarization dependence of the scattered field through the matrix T(k,) , eq.(3.44) is identical to the intuitively derived result (3.7) (with an anisotropic scattering strength). According to the definition of the scattering amplitude Bj, eq.(3.33), the scattering strength per unit volume f(r) that was introduced in the previous section, is proportional to,

i f(r) - E(r) - e l . (3.46) e/

The matrix on the right hand-side of (3.37) is proportional to the electric dipole propagation matrix, which connects the dipole moment to the electric field strength resulting from it. The constitutive relations (3.11,12) thus imply that the incident field induces electric dipoles in each of the volume elements inside a colloidal particle, which emit an electric field : the sum of these dipole fields is the scattered electric field strength E,. In the first order Born approximation, subsequent scattering of these emitted dipole fields into the detector by other volume elements is neglected.

For isotropic optical properties of the colloidal particles, Bj is proportional to the unit matrix I. The scattered intensity is then proportional to T(k~) �9 E0. The matrix T(k,) in (3.45) is the projection onto the plane which is perpendicular to k, (see fig.3.4). Thus, T (k , ) . E0 is the component of the incident field perpendicular to the detection direction ,-, k,. Since the polarization direction of the incident field is in the same direction as the induced dipoles in this case, the matrix T(k~) takes into account that at certain

122 Chapter 3.

Figure 3.4: The matrix T (k ~) is the projection operator onto the plane perpendicular to k,.

T(ks}.E o Eo

scattering angles only part of the dipole fields radiate towards the detector. If, for example, Eo I[k,, the scattered field strength is zero. In this case the detector is looking "right on top of the heads of the dipoles".

3.4 Relation to Density Fluctuations

The scattered electric field strength (3.44) is calculated for a fixed configuration of colloidal particles. In reality these particles exhibit Brownian motion. Brownian motion is so slow, however, that many photons are scattered in a time interval during which the configuration of Brownian particles did not change to an extent that the phases of the scattered fields are seriously affected (you are asked to verify this in exercise 3.4). The measured instantaneous intensity on the diffusive time scale, and also on the Fokker Planck time scale, is therefore the intensity averaged over many time intervals 27r/w (with w the frequency of the light), and is equal to,

i(t) - -~ ~00(E~(t).fi~)(E:(t).fi~) - ~ ~00l(E,(t ) .h~) 12 , (3.47)

where * denotes complex conjugation, and the unit vector h, is the polarization direction of the detected light. This particular polarization direction is selected with the use of a polarization filter placed in front of the detector. The time dependence of the scattered electric field strength, which is denoted here explicitly, refers to the Brownian motion of the particles �9 E, depends on time, according to eq.(3.44), both via the orientation of the Brownian particles (through the scattering amplitudes Bj) and the positions of the particles (through the exponential functions containing the positions). A change in

3.4. Density Fluctuations 123

i l t l

T < -

Figure 3.5: The fluctuating intensity.

the configuration of the Brownian particles (reorientation and/or translation) changes the interference of the scattered electric field strength and thereby the instantaneous scattered intensity i(t). The intensity thus fluctuates due to Brownian motion (rotational and/or translational) around a mean value 1, as depicted in fig.3.5.

For spherical particles, for which the scattering amplitude is independent of the orientation, the instantaneous scattered intensity is, according to eq.(3.44) or (3.7), proportional to,

N

i(k, t) ,~ ~ exp{ik. ( r i ( t ) - r/(t))} - I p(k, t)! 2 , (3.48) i,j=l

where we abbreviated k = ko - k, (k should not be confused with the wavevectors that we used in the previous section), and we suppressed the dependence on the detector position r. Furthermore, p(k, t) is the Fourier transform of the microscopic density, which is discussed in subsection 1.3.3 in the introductory chapter,

N

p(r, t) - y~ di(r - r j ( t )) . j = l

(3.49)

For spherical particles it is thus possible to study, by light scattering, the statistics of density fluctuations. These fluctuations determine to a large extent the equilibrium and transport properties of the system of Brownian

124 Chapter 3.

/ / /

/

ko

\ \ kS

A=~TT/K I ~ I

I I ',

[~s- k

Figure 3.6: Light scattering visualized as "diffuse Bragg scattering" by sinusoidal density variations. Each little dot represents a Brownian particle.

particles. The Fourier component of the density that is probed in a light scattering experiment is set by the wavelength 3, of the light and the scattering angle, since (see exercise 3.5),

4r Os k - I k o - k , I - T sin{T)" (3.50)

Light scattering can thus be visualized as "diffuse Bragg scattering" by planes of sinusoidal density variations with the wavelength A that fits the wavevector k,

i - 2~/k. (3.51)

This is sketched in fig.3.6. The intensity at the particular wavevector is proportional to the squared amplitude of the corresponding Fourier component of the instantaneous density. For non-spherical particles, the scattering amplitudes contribute in addition to the time dependence of the scattered intensity via

3.5. Static Light Scattering 125

rotational Brownian motion. In those cases this simple picture is no longer valid.

There are two types of experiments to distinguish" measurement of the ensemble averaged intensity 1 (see fig.3.5) and measurement of the actual time dependence of the fluctt, ating intensity. The first type of experiments is usually referred to as static light scattering, the second type as dynamic light scattering. In a static light scattering experiment the ensemble averaged properties of density fluctuations are measured, while in a dynamic light scattering experiment the dynamics of density fluctuations is probed. The theoretical background for these two types of experiments, for spherical Brownian particles, is established in the next two sections. Non-spherical particles are considered later in this chapter.

3.5 Static Light Scattering (SLS)

In a static light scattering experiment, the mean intensity,

I (k) = < i(k, t) >, (3.52)

is measured as a function of the scattering angle, which, at some fixed wavelength, sets the magnitude of the wavevector k = k 0 - k,, as given in eq.(3.50). The brackets < . . . > denote ensemble averaging over the orientations and positions of the Brownian particles. In an experiment this ensemble average is obtained as a time average �9 the intensity should be collected over a time interval that is much larger than the time required for the Brownian particles to probe all accessible configurations. For spherical and optically isotropic particles we have (see the definition (3.33)),

Bj(k) - i Bj (k ) , (3.53)

with,

Bj(k) - Jy dr' e ( r ' ) - e] exp{ik, r'} o ~S

f ~ ' ,2 e ( r ' ) - el sin{kr'} - 47r dr'r

Jo ey kr ~ , (3.54)

where e(r) is the isotropic dielectric constant of the jth particle with radius aj. In the second line in this equation the angular integrations are performed

126 Chapter 3.

explicitly. According to eqs.(3.44,47), the mean scattered intensity is now given by,

I = y , Io k ] r2 (47r)2/~ (fi," rio (3.55)

1 g x -~ ~ Bi(k)B;(k) < exp{ik. ( r i - rj)} > ,

i,j=l

where fi0 - Eo/E0 is the polarization direction of the incident light and - N/V~ the number of Brownian particles per unit volume. Furthermore,

lo is the incident intensity,

1 ~ey I o - ~ ~ o E g .

In eq.(3.55) we used that,

f i , . T(k~)- rio - fi," rio,

which follows from the fact that T(k , ) is symmetric and fi, Z k,. The scattering amplitudes Bj can be taken outside the ensemble averaging in eq.(3.55), since for the spherical particles under consideration they are constants, independent of orientation. The factor V~ I o / r 2 on the right hand-side of the expression (3.55) is an apparatus constant. The so-called Rayleigh ratio, defined as,

R - I r 2/Io V~, (3.56)

is independent of apparatus constants, and is determined solely by the properties of the system of Brownian particles. In the following, we express the scattered intensity in terms of the Rayleigh ratio.

Suppose now that the Brownian particles are identical. The indices on Bj can then be omitted, and the sum over ensemble averages for i ~ j is equal to the number of terms in that sum, N(N - 1), times the ensemble average for a single term (say i - 1 and j - 2),

ko 4 2 B 2 n ( k ) - (47r)2/~ (fi~. rio) (k) [1 + ( N - 1) < exp{ik. (rl - r2)} >] .

(3.57) Let us first express the ensemble average as an integral of the probability density function (pdf) for the positions, P ( r l , r2). For a homogeneous system (both translational and rotational invariant), this pdf is a function of the

difference coordinate ! rl - r2 ],

P ( r l , r 2 ) - P ( I r l - r2 1). (3.58)

Hence (with r - r l - r2),

< exp{ik . (ra - r2)) > -- Iv, dr1 fu, dr2 P ( r l , r 2 ) e x p { i k . ( r l - r2)}

= V~ fv~ dr P(r)exp{ik. r} . (3.59)

The integrations range over the scattering volume V~ from which scattered light is detected. The pair-correlation function g was introduced in subsection 1.3.1 of the introductory chapter. In the homogeneous system its definition in eq.(1.52) reduces to,

1 P(r) - V~ 9(r) . (3.60)

Furthermore we define the form factor P(k), not to be confused with the pdf in the above expression, as the squared scattering amplitude normalized to unity at zero wavevector,

P(k) -[B(k)/B(k - O)[2 (3.61)

I fo dr r2 ~(~)-~l sin{a~}12 __ e f k r

fo dr r 2 e( r ) -e l e l

where a is the radius of the Brownian particles. Introducing also the volume averaged dielectric constant of the particles,

1 ~ 47r fO a 2s ) (3.62) ~p - ~pp dre(r ) - -~p drr ,

4 where Vp - g rra a is the volume of a Brownian particle, the scattering amplitude at zero wavevector can be written as,

I B(k - 0 ) 1 2 - Vp 2 ! ep - ef 12 . (3.63) e]

Substitution of eqs.(3.59-63 ) into eq.(3.57) gives,

kg 2 R(k)- ho) (3.64)

~s I e (k ) 1 + # dr e(~) e x p { i k - r } .

128 Chapter 3.

Since g(r) ~ 1 for r ~ c~, the integral over V~ here is divergent as V, ~ ~ . We therefore write the integral as a sum of two integrals,

fiJv. drg(r) exp{ik, r} - fiJv. d r ( g ( r ) - 1) exp( ik , r}

+ fi fv~ dr exp{ik, r}. (3.65)

The first of the integrals on the right hand-side here is independent of the size of V~ when the linear dimension V)/3 of the scattering volume is larger than the distance over which the pair-correlation function decays to unity. The second integral tends to a delta distribution of k as V~ ~ c~ (see the discussion at the end of this section). Hence, for k 5r 0, eq.(3.64) can be written as,

k o 4 2 - R(k) - (47r)2 # (fi, . rio) Vp 2 [ e] 12 P ( k ) S ( k ) , (3.66)

where the structure factor S(k) is defined as,

1 N S(k) - ~ ~ < e x p { i k - ( r i - r i ) } >

i,j=l

= 1 + f i f d r ( g ( r ) - 1) exp{ik, r}

= 1 -t- 47r~ fo ~ dr r 2 (g(r) - 1) sin{kr}kr ' (3.67)

where the integral extends over the entire ~3, that is, the limit V, ~ c~ is taken here.

The form factor in eq.(3.66) for the Rayleigh ratio describes the interference of the scattered electric fields from different volume elements within single particles, while the structure factor accounts for the interference of fields scattered from different Brownian particles. The structure factor is the interesting quantity" it is the Fourier transform of the pair-correlation function (minus 1), with which many thermodynamic properties of the colloidal system at hand can be predicted.

The scattered intensity is equal to zero at scattering angles where the interference of fields scattered from volume elements within single particles is destructive, that is, at angles where the form factor is zero. For optically homogeneous particles (particles with a dielectric constant which does not depend on the position in the particle" e(r) - constant) the form factor is easily calculated from eq.(3.61),


0 InP(k)!~ tn P(k) : '

0

i fl

Figure 3.7" The aogaritnm of) the form factor/'or optically homogeneous spheres as a function of ka. The straight line/'or smM1 ka-values is re/erred to in subsection 3.8.1 in connection with the Guinier approximation.

p(k) _ [3 ka c~ - sin{ka} ] 2 (ka) 3 . (3.68)

This function is plotted in fig.3.7. For large enough particles the scattered intensity vanishes at certain scattering angles due to destructive interference. At these scattering angles, no information about the structure factor can be obtained experimentally.

The factor,

C = I ev - ef 12 , (3.69) e]

in the expression (3.66) for the Rayleigh ratio is usually called the optical contrast of the system of Brownian particles. It can be varied by means of temperature or changes in the composition of the solvent. In the derivation of the Rayleigh Gans Debye approximation (3.44) it was assumed that only a small fraction of the incident field is scattered. This assumption can be made to hold true by choosing a solvent, such that the optical contrast is small, but of course still large enough to perform accurate measurements. To a good approximation, the dielectric constant e of the composite system

130 Chapter 3.

of Brownian particles and the solvent is simply the volume average of the dielectric constants of the particles and the solvent,

, - + ( i - r

with ~ - ~a3t~ the volume fraction of Brownian particles. Hence,

ds d~ = ep - e l .

The optical contrast C can thus be written as,

. (3.70)

The refractive index n of a substance with a magnetic permeability equal to

that of vacuum (#o) is defined as, n - ~/c/eo (Co is the dielectric constant of vacuum). Note that it follows from this definition and the expressions (3.20,21), that the wavelength A in a medium with refractive index n is A = Ao/n, with Ao the wavelength in vacuum. This was already used in the heuristic derivation of the conditions (3.4,5) for the validity of the Rayleigh Gans Debye approximation. We can thus rewrite eq.(3.70) in terms of the experimentally quite easily accessible refractive indices,

2n dn ) 2 C = n~ dr (3.71)

where the approximation is valid to leading order in the small difference of the refractive indices n and nf. This expression offers the possibility to determine the optical contrast experimentally : plot the refractive index of the dispersion as a function of the concentration of colloidal particles, and calculate its slope dn/d~.

To leading order in the difference of the dielectric constant of the solvent and the Brownian particles, the optical contrast (3.69) can also be rewritten in terms of the volume averaged refractive index h v of the Brownian particles and that of the solvent,

C - 4 [ n v - n f [ 2 . (3.72) nf

Similarly, the form factor (3.61) can be written in terms of refractive index differences as,

~ ~r ~- ~ ~-~j--Z] , (3.73) JO n /

with np(r) the refractive index inside the Brownian particle at a distance r from its center. Here, we used that,

n~(,-) - n~, - (n, , ( , ' ) + n ] ) ( n ~ ( , ' ) - ns ) ~ 2 h i ( u p ( r ) - ns ) ,

whenever np (r) and n S are not too different. This concludes the basic features of static light scattering of spherical

particles. To conclude this section, let us return to the validity of the neglect of the last integral on the right hand-side of eq.(3.65). One may ask whether the neglect of that integral/5 fv, dr exp{ik, r} in eq.(3.65) is allowed for practical situations, since in an experiment V~ is not infinitely large. In fact, the first integral in the right hand-side of eq.(3.65) is much smaller than the second integral in the strict limit k ~ O, for any finite scattering volume V~ with a linear dimension much larger than the range of the pair-correlation function, since then,

lim p [ drg(r) exp{ik, r} ~ lim/~ [ dr exp{ik, r} - /~ V~ - N . k---+O JVs k---,O Jr,

This limit is reached for values of the wavevector, so small, that the exponential is approximately unity over the entire scattering volume, that is, for k/2rr << Vf I/3, with V, I/3 the linear dimension of the scattering volume. In this case one thus finds from the definition of the structure factor (first line in eq.(3.67)),

lim S(k) - N . k---,0

On the other hand, for any non-zero wavevector, the last integral on the right hand-side of eq.(3.65) is zero as V~ ~ oc, so that,

fi f dr g(r) exp{ik, r} = ~ f dr (g(r) - 1) exp{ik �9 r} lim Y$ ---.t. o o

In this case eq.(3.67) for the static structure factor is recovered. It is clear that one cannot interchange the two limits,

lim lim p [ . d rg ( r ) exp{ ik r} ~ [ v,-~oo k-~o Jr, " k-~o y,-~oolim lim /~ JV, dr g(r) exp{ik, r}

132 Chapter 3.

The interesting order of taking limits is first the thermodynamic limit (V, ~ oe with # constant) and then the limit k ~ 0. In this order the structure factor is equal to the Fourier transform of g(r) - 1, which is the interesting quantity. In the opposite order, the static structure factor is simply equal to the number of Brownian particles in the system. Therefore, in an experiment, one should determine the intensity in the zero wavevector limit by extrapolating to k = 0 from a series of measurements performed at wavevectors for which the corresponding wavelength A - 27r/k fits many times into the scattering volume, or, equivalently, k/27r >> Vs "1/3 (see exercise 3.6 for a further exploration of this matter).

3.6 Dynamic Light Scattering (DLS)

In a dynamic light scattering experiment the objective is to measure the time dependence of the fluctuating intensity rather than the mean intensity as measured in a static light scattering experiment. The simplest function that characterizes the fluctuations of the intensity is the intensity auto-correlation function (IACF), defined as,

gi(k, t) - < i(k, to) i(k, t + to) > . (3.74)

For an equilibrium system, the IACF is independent of to (see subsection 1.3.2 in the introductory chapter on correlation functions), which we henceforth set equal to zero. In terms of the scattered electric field strength E, (t), the IACF is a four-point average,

1 e I (3.75) gi(k, t) - 4 #o

• < (E , (0) . f i , ) (E:(0) , f i , ) (E , ( t ) , f i , ) (E : ( t ) , fi,) > .

According to eq.(3.44) or (3.7), the scattered electric field strength is a sum over N >> 1 terms. This sum can be written as a sum over many statistically independent terms, where each term itself is a sum over "clusters" of interacting particles. The linear dimension of a cluster is the distance over which the pair-correlation function tends to unity. These clusters of particles are statistically independent. The central limit theorem (1.80) therefore implies that the scattered electric field strength is a Gaussian variable (with zero average), provided that the scattering volume contains a large number of such independent clusters of particles. According to Wick's theorem (1.81),

3.6. Dynamic Light Scattering 133

the four-point ensemble average in eq.(3.75) can thus be written as a sum of products of two-point averages (henceforth we simply write E~ (0) instead of E ~ ( t - 0)),

1 ef (3.76) gl(k, t) - 4 #o

• [< (E, (0) . f i , ) (E:(0) , fi,) > • < (E , ( t ) . f i , ) (E: ( t ) , fi,) >

+ < (E~(0). f i , ) (E , ( t ) , ft,) > • < (E: (0) . f i , ) (E:( t ) - f t , ) >

+ < (E , (0) . f i , ) (E: ( t ) , f i , ) > • < (E: (0) . f i~)(E,(t) , f t , ) > ] .

The first of these terms is nothing but 12, where I is the mean scattered intensity (see the previous section), which is independent of time for an equilibrium system. Defining the electric field auto-correlation function (EACF) gE as,

gE(k, t ) - ~ ~00 < (E, (0) . fi~)(E:(t) , ft,) > , (3.77)

the third term in eq.(3.76) is equal to I gE [ 2. This will turn out to be the interesting quantity in DLS. The second term in eq.(3.76) is equal to zero for non-zero wavevectors. This can be seen as follows. The second term consists of ensemble averages of the following form,

< exp{ik. (ri(O)+ rj(t))} > ,

where i and j are either different or equal. Let P (rj, t [ ri, 0) be the conditional pdf for the position rj of particle j at time t, given that the position of particle i at time t = 0 is ri. This pdf is only a function of the difference coordinate ri - rj for homogeneous systems �9 P(r j , t [ ri, t - 0) - P(ri - rj, t). The ensemble average is then equal to (with r' - ri(t - 0) and r - rj(t)),

< e x p { i k - ( r i ( t - 0) + rj(t))} >

- fv~ dr ' /v~ dr P ( r ' - r, t ) P ( r ' ) exp{ik. (r' + r )} ,

where P(r ' ) is the pdf for the position coordinate. Since P( r ' ) - 1/V~ for the homogeneous equilibrium system considered here, this can be written, in the thermodynamic limit (where V, ~ c~ and/~ constant) as,

xrlim ] / Lv~..-,.~ ~ d ( r / + r) exp{ ik - ( r ' + r)) x d ( r ' - r ) P ( r ' - r, t ) ,

134 Chapter 3.

where the factor 1/8 is the Jacobian of the transformation,

(r', r) --, (r' + r, r ' - r ) .

The integral with respect to (r' - r) is well behaved, since the pdf is a normalized function. The integral between the square brackets is equal to unity for k = 0, and is zero for k ~ 0, since that integral is the delta distribution divided by the volume, which itself tends to infinity. Hence, the ensemble average is zero for non-zero wavevectors, so that the second term in eq.(3.76) does not contribute. In a real experiment the scattering volume is not infinite. The same considerations as at the end of the SLS section apply here. The scattering volume is so large in comparison to 27r/k, that in an experiment the limit of an infinite volume is established.

The IACF can thus be written in terms of the mean scattered intensity and the EACF (3.77),

gz(k, t) - 12 + [ g E ( k , t) [ 2 . (3.78)

This equation is usually referred to as the Siegert relation. It is convenient to rewrite the Siegert relation in terms of normalized correlation functions,

~i(k, t ) - g l ( k , t ) / I 2 , (3.79)

~E(k,t) -- g E ( k , t ) / I . (3.80)

By definition,

~E(k, t -- 0) -- 1 , ~I(k, t - 0) - 2. (3.81)

We hereafter abbreviate these normalized correlation functions in the text also as EACF and IACF, respectively. The S iegert relation can now be written as,

~t(k, t ) - 1 + [ ~E(k,t) 12 . (3.82)

The fact that the scattered electric field strength is a Gaussian variable, enables this simple connection between the IACF and the EACF, the latter of which is nothing but the (normalized) density auto-correlation function.

An expression for the EACF for spherical particles, in terms of an ensemble average over the phase space coordinates of the particles is obtained by substitution of the expression (3.44) into the definition (3.77,80),

I 1 N

OE(k,t) -- S(k) N ~ i,j=l

< exp{ik. (ri(O) - rj(t))} > . (3.83)

3. Z Experimental Considerations 135

For dilute suspensions, where Brownian particles do not interact with each other, this ensemble average may be calculated from the time dependent pdf's which were calculated on the basis of the Langevin equation in the previous chapter. For spherical particles this is done in section 3.8. Rigid rod like particles are considered later in the present chapter.

3.7 Some Experimental Considerations

The Dynamical Contrast

What is observed on a screen on which the scattered intensity is collected is an assembly of bright spots, rapidly appearing and disappearing, separated by dark regions (see fig.3.8). These bright spots, "speckles", are the result of (partly) constructive interference of light scattered from all Brownian particles in the scattering volume, and the dark regions correspond to scattering directions in which destructive interference occurs. As the configuration of Brownian particles changes in time, due to Brownian motion, the positions of the speckles and dark regions change. The detected light at a certain scattering angle is collected via a pinhole (see fig.3.8). For a DLS experiment the size of that pinhole should not exceed the size of the speckles. When the size of the pinhole is large compared to the size of the speckles, so that the total intensity of several speckles is detected, the amplitude of fluctuations in the measured intensity is less than in case only one (or a fraction of a) speckle is measured. In the limit that many speckles are detected simultaneously, one would measure the ensemble averaged intensity I at each instant of time. The experimentally determined IACF therefore depends on the size of the detector pinhole relative to the speckle size. For an infinitely small pinhole, according to eq.(3.81), t)i(k, t - 0) - 2. For pinholes with some finite size, the measured amplitude of fluctuations is always smaller than this optimum value. The experimental value of ~z(k, t - 0) is commonly referred to as the dynamical contrast.

The size of the speckles is determined by the experimental geometry. How should this geometry look like to obtain a dynamical contrast close to 2? Consider two points rt and r~ on the left and right side in the scattering

1 V 1/3, where V 1/3 is volume, as depicted in fig.3.9, with [ r~ - rt l - the linear dimension of the scattering volume. Suppose that k0 is along the line connecting the points rt and r~. Let Ol +) and O! +) denote the

136

I

ko sampte

Chapter 3.

[speck Figure 3.8" The instantaneous speckle pattern of scattered light. The circular hole in the screen is the detector pinhole.

scattering angles for which the light that is scattered from the pionts rt and r~, respectively, interferes constructively (see fig.3.9). These scattering angles define the location of a speckle on the screen. Let Ol -) and O(~-) denote the scattering angles at which the intensity of the speckle dropped to a small value. The phases (I)I +) ((I)(~ +)) of the scattered light from rt (r~) are related to these scattering angles. For example (see also the heuristic section 3.2),

r +) - r , . (ko k (+),,, r t~o ok~ (ko ~-,,t,"(+)~ [1 cos - ) - - - r , k o - , ! l e+)j

where k (+) is the scattering wavevector corresponding to the point rt and the s,l

angle O} +). Similarly,

(I)! +) - r~ ko [1 - cos{O!+)}] ,

(I)} -) - rt ko [1 -cos{O}-)}] ,

(I)! -) - r~ k0 [1 - cos{O!-)}] .

By definition we have,

r r - 2 r n ,

3.7. Experimental Considerations 137

v

4"

{ - } . , {_)

kl+l ~ / A . . / . . . , s ~ l+ l "" " s , t ~ / 2" ' ...,A~ vT-,~'t k~+~ .~__J /~ , , , " v-- i r~c*l "s,r [ ~---~ ~swj-. /~",,..~r _

I.

" ' ~-- ~0

Figure 3.9: Definition of the points rt,~ and the scattering angles ~(~) ~ . l l l 7, �9

intensity profile is that of a speckle. The sketched

1 r ~}-) -- 27r(n 4- ~) ,

with n an integer. Hence,

(~!-~_ ~I-~)_ (~+~_ ~I+~) - +~. (3.84)

Let O, denote the scattering angle associated with the detector pinhole, which is in between Ol +) and 0~-). For small differences between the scattering angles, ai(• ~t,, can be Taylor expanded to leading order around ~(+) - O,

~ . . I / ~ T �9

Substitution of these leading order expansions into eq.(3.84) leads to,

I r, kosin{O,} (0~ +) - O~ -)) - r, ko sin{O,} (01 + ) - 01-)) I - 7r.

The size of a speckle is,

138 Chapter 3.

with ld the distance between the detector and the scattering volume. Since 1 /3 [ r , - rt 1- 7 V) , we finally find the following constraint for the size of the

pinhole for a good dynamical contrast,

Ipinhole < l speckle "-" 2 r, la 1 1

V 1/3 ko [ sin{O~} I. (3.85)

A dynamical contrast close to 2 thus requires a small scattering volume. For the typical values la - 30cm, V)/3 - 200 #m and ko - 210 r m -1, it is found

1 that for O, - 7r/2, l,p~kte ~ ~ ram. Therefore, the pinhole diameter should not be larger than about 0.1 - 0.3 ram. For O, --o 0 or ~ 7r, the above expression for the speckle size is incorrect for two reasons : (i) higher order terms in the Taylor expansion of the scattering angles around O, must now be included, and (ii) V~ x/3 diverges at these limiting angles.

The Finite Interval Time

Besides a finite detector area, also the time interval over which photons are collected to obtain the "instantaneous" intensity is finite. The experimentally determined instantaneous intensity is always the average of the true instantaneous intensity over the time interval during which photons are collected,

1 [ t+~ - dt' i(k, t ' ) . (3.86) i~

The superscript "exp" refers to experimental quantities. The time span r is referred to as the interval time. The experimental IACF is then,

g lexp( k, t) - - < i~*P(k, O)i~*'(k, t) >

_ 1 dt~ dt2 < i(k, t~)i(k, t2) > 47 .2 r at -~

1 f dta d t ' g z ( k t ' ) (3.87)

= .r i t + r - t 1

47 .2 - , a t - r - t l ' '

where in the last line we used that the IACF is a function of the time difference t' = t2 - t~ only. The integration range in the (t', t~)-plane is the dashed area in fig.3.10a. Interchanging the order of integration, the above integral can be written as a sum of integrals over the two triangular domains in fig.3.10a,

1 dtl dt' gi(k, t')

r i t + r - t 1

47 -2 r dt-~--t l

3.7. ExperimentalConsiderations 139

t'

I {a) , . . . . . ~ 2"~

x ~ , , . , -) T"

_.; r 4 t-,',~-t ,~'-- ,

~lt ' - t l

1,/2a-~

I I q / ,r

- 2 2 " 22"

{b}

Figure 3.10: (a) The integration range in the (t', t 1 )-plane, for the calculation of the experimental IACE (b) The triangular function.

t '-t

l [ z t + 2 r f t + r - t ' Z t Z r ] 4r 2 dt' dtt + dt' dtl gt(k, t') a -r -2r -r-t'

- d t ' + dt ' g I ( k t ' ) a t 4 T 2 2 r 4T-2 ' " (3.88)

Let us define the triangular function A ( t' - t ) ,

2~+(t'-t)~2~= , for A ( t ' - t) - 2~'--(,'-t) for (2.-)2

0 , for

(t ' - t) E ( - 2 r , 0)

( t ' - t ) E ( 0 , 2 r )

( t ' - t ) ~ [ - 2 r , 2 r ] .

This is a triangle, centered around t ' - t - 0, with a surface area equal to 1 (see fig.3.10b). The sum of the integrals in eq.(3.88) can now be written as (with t" - t ' - t),

-~(k, t) ~/'+=~ ~'J_* t"/ (k, t + t"). - d t ' A(t ' - t)gt(k, t ') - d t" A( g1 ~1I Jt-2r 2r (3.89)

140 Chapter 3.

The experimental IACF is thus equal to a weighted average (weighted with respect to the triangular function) of the true IACE To obtain a proper experimental estimate of the IACF, the interval ( -2 r , 2r) must not be too large in comparison to the decay time of the IACE To obtain an estimate of how large the interval time r may be chosen, let us approximate the IACF around a time t by a second order polynomial, that is, we Taylor expand the IACF up to the second order term,

gi(k, t + t") ~ g/(k, t) + dgt(k,dt t ) t "+ ~ d2gz(k'dt 2 t)t,,~

Substitution of this approximation into eq.(3.89) gives,

1 d2gi(k, t) gl~'(k, t) - gt(k, t) + g dt 2

, - 2 r _ t" _ 2T.

v 2 , (3.90)

Notice that the linear term in t" does not contribute to the error in the experimental estimate of the IACE It is not the slope, but rather the second order derivative of the IACF that sets the error. In the particular (but not unimportant) case that the IACF is an exponential in time ,,~ exp{-t / r0}, where To is the relaxation time of the IACF, eq.(3.90) gives,

( k , t ) - gr(k, t) - ~ . (3.91)

For a relative error smaller than 0.01, the interval time r must therefore be smaller than 0.17 x the relaxation time To. Furthermore, for a positive second derivate, which is usual, the experimental estimate is always larger than the true IACE

Ensemble Averaging and Time Scales

Both in SLS and DLS experiments, the ensemble averaged quantities are obtained as time averages,

I = ~ dtoi(k, to),

< i(k, O)i(k, t) > = ~ dto i(k, to)i(k, t + to),

where T is so large, that all accessible configurations occurred many times during that time interval. For the SLS experiment, as discussed above, the

3.8. Scattering by Dilute Systems 141

measuring time T may be shorter when the detector collects the intensity of more speckles simultaneously. Use is made of the fact that the system is in equilibrium, in which case both I - < i(k, to) > and < i(k, to) i(k, t + to) > are independent of to. Since the experimental "instantaneous" intensity is an average over the time interval 2T (see eq.(3.86)), the above integral for the IACF is in practice a sum,

m

< i(k, 0)i(k, t - tn) > - - - ~ i~P(k, tj)i~'(k, t~ + tj), m j = l

where ti - (2i + 1) x r and m is a number such that 2m x T ~ T. Experimental values for the IACF are thus obtained at discrete values tn - (2n + 1) x T.

To conclude this section, a remark on time scales should be made. The detectors which are used (Photo Multiplier Tubes or Diodes) respond to light with a time resolution which is as large as the diffusive time scale. An experiment with standard detectors, and colloidal particles which are not too large, should therefore be interpreted on the basis of theories which are coarsened to the diffusive time scale.

3.8 Light Scattering by Dilute Suspensions of Spherical Particles

In dilute suspensions, where at each instant the separation between the vast majority of Brownian particles is very much larger than their own dimensions, the effects of interactions between these particles can be neglected. In the next subsections we discuss the scattering properties of such dilute systems, consisting of spherical Brownian particles. The polarization directions fi, and rio are chosen equal here, so that (fi, . rio) - 1.

3.8.1 Static Light Scattering by Spherical Particles

According to eq.(3.67), the Rayleigh ratio (3.66) in the dilute limit/~ ~ O equals,

R ( k ) - (47r)2 e] [ 2 P(k). (3.92)

Suppose that the difference in the refractive index of the Brownian particles and the solvent is large in comparison to the variation of the refractive index

142 Chapter 3.

within the particles. The difference np(r)-n f in expression (3.73) for the form factor may then be replaced, to a good approximation, by a constant hp - n f, that is, the Brownian particles may be considered as optically homogeneous. The form factor for homogeneous spheres is given in eq.(3.68), and is plotted in fig.3.7. The form factor P(k) is defined as the squared scattering amplitude of a single colloidal particle, normalized to unity at zero wavevector. For such optically homogeneous particles, the particle radius a can be obtained from scattering angle dependent light scattering measurements, by fitting the intensity to expression (3.68) for the form factor. When there is a close match between the refractive indices of the Brownian particles and the solvent, the optical inhomogeneities of the particles affect the scattering angle dependence of the intensity and invalidate eq.(3.68) for the form factor.

For small particles, only the initial decay of the function (3.68) can be obtained experimentally. Since the maximum attainable wavevector is k = 2ko, see eq.(3.50), the first minimum of the form factor of homogeneous particles is outside the experimental range when k0 a < 2. The size of the particles can now best be obtained from a so-called Guinier plot. For these small particles, the product kr in the integral (3.73) for the form factor is small throughout the integration range. Taylor expansion of the form factor with respect to kr gives,

1 k2 ] 1 k2 2 e(k) - 1 - -~ R~ + O ((ka) 4) ,.~ exp{ -~ Rg}, (3.93)

where the optical radius of gyration Rg is defined as,

I fo dr r 4 '~p(")-'V 1 , n !

Rg - fo dr r 2 '~p(~)-'V n y

1/2

(3.94)

Expression (3.93) for the form factor is commonly referred to as the Guinier approximation. The approximation of the truncated Taylor expansion by an exponential function in eq.(3.93) is of some practical convenience �9 it turns out, experimentally, that a plot of In{ 1} versus k 2 is linear over a somewhat larger wavevector range than a plot of I versus k 2. The logarithmic plot is commonly referred to as the Guinier plot. The slope of this plot gives the radius of gyration. The inset in fig.3.7 shows that the range of validity of the

1 Guinier plot is ka < 2~. Notice that for very small particles (say a < 20 nm), the decrease in intensity over the entire accessible wavevector range is too small for an accurate SLS determination of the radius. In such cases DLS

3.8. Scattering by Dilute Systems 143

is the more appropriate experimental technique. For optically homogeneous particles the radius is related to the radius of gyration as,

a - ~ R g . (3.95)

For a close match of the refractive indices, the radius of gyration is not so simply connected to the geometrical radius, and may even be an imaginary number (R~ can be negative).

The Guinier plot can also be used for non-spherical particles to obtain information about their size. See exercise 3.12b for long and thin rods.

3.8.2 Dynamic Light Scattering by Spherical Particles

Since the interactions between the Brownian particles are neglected here, use can be made of the time dependent pdf's as calculated from the Langevin equation in the previous chapter, in order to obtain explicit expressions for the EACE Let us first express the EACF (3.83) in terms of these pdf's.

The "cross terms" i ~ j in eq.(3.83) are zero for non-interacting particles. Since different particles are statistically independent in the dilute limit, we have, for i ~ j,

< e x p { i k - ( r , ( 0 ) - rj(t))} > -

< exp{ik �9 r,(0)} > < exp{ik �9 rj(t)} > .

Since in the equilibrium situation considered here the pdf of the position coordinate of a single Brownian particle equals l /V , with V the volume of the system, each of the averages of the exponents on the right hand-side is a delta distribution (when taking the thermodynamic limit),

< exp{ik, ri(0)} > = < exp{ik, rj(t)} > - lim 1 fv v ~ V dr exp{ik, r}.

(3.96) For non-zero wavevectors these averages are therefore zero. Only the "diagonal terms" i - j in eq.(3.83) for the EACF survive for non-interacting Brownian particles. Furthermore, the static structure factor in eq.(3.67) is equal to 1 for non-interacting particles.

Let P(r , t I ro, t - 0) - P ( r - r0, t) denote the conditional pdf for the Brownian particle position r at time t, given that its position at time t - 0 was

144 Chapter 3.

ro ( P ( r - ro, t) should not be confused with the form factor, for which we also used the symbol P). Since the pdf for ro is 1/V, the EACF is equal to,

~E(k, t) -- f dr' P(r ' , t) exp{ik- r '} , (3.97)

with r' = r - ro. It is assumed here that all Brownian particles are identical. The EACF is thus the Fourier transform of the Gaussian pdf (2.39) that we calculated in the previous chapter. According to what has been said in subsection 1.3.4 in the introductory chapter on Fourier transformation of Gaussian pdf's, we obtain,

1 N

i,j=l < exp{ik. ( r i ( 0 ) - rj(t))} > - exp{-D0 k 2 t}. (3.98)

Exercise 1.14 provides an alternative derivation of this result. The diffusion coefficient Do is simply related to the radius of a Brownian particle through the Stokes-Einstein relation (2.37), so that this expression offers the possibility to determine the size of Brownian particles from DLS experiments. A radius determined in this way is commonly referred to as the hydrodynamic radius.

3.9 Effects of Polydispersity

As was already mentioned in the introductory chapter, two colloidal particles are never exactly identical. Even for model systems, where care is taken to prepare monodisperse particles, the relative deviations of their radii is at least of the order 0.02 - 0.05. In addition, there may be a certain degree of polydispersity in optical properties. Since the scattering amplitude of a Brownian particle depends on the size of the particle as well as its optical properties, both experimental SLS and DLS data are affected by polydispersity.

The two extreme situations of only size polydispersity (without optical polydispersity) and only optical polydispersity (wihout size polydispersity) are discussed in the following subsections. Size polydispersity is discussed only for very dilute systems, for which interactions between the Brownian particles can be neglected. As will turn out, optical polydispersity can be exploited to study experimentally different kinds of diffusion processes. Therefore, effects of optical polydispersity are discussed for more concentrated dispersions, where interactions between Brownian particles are important.

3.9. Polydispersity Effects 145

3.9.1 Effects of Size Polydispersity

Static Light Scattering

For very dilute suspensions, where interactions of the Brownian particles may be neglected, the ensemble averaged scattered intensity is simply the sum of the scattered intensities of the individual Brownian particles. The sum over particles can be written as an integral, weighted with the pdf for the polydispersity parameters. Here we assume that the particles only differ in size. The polydispersity in the dielectric constant can be neglected when the difference in the dielectric constant of the particles with that of the solvent is much larger than the spread of the dielectric constants between the Brownian particles. Let Po(a) be the pdf for the radius a of the Brownian particles. The measured "polydisperse" Rayleigh ratio for a dilute suspension is then,

R'~ - da Po(a) R(k, a) , (3.99)

where R(k, a) is the Rayleigh ratio of a spherical particle with radius a. For dilute suspensions (for which the structure factor is equal to 1), and equal polarization direction of the incident and scattered light, the monodisperse Rayleigh ratio for optically homogeneous particles, R - R(k , a), follows from eqs.(3.66,68),

-- K * a 6P(k) = K * a 6 [ 3 k a c ~ sin{ka}]: R ( k , a ) [ (ka) 3 ]

where a constant K* is introduced,

, (3.100)

K* k4 e-P - ~f k4 9 fi C , (3.101)

with C the optical contrast which is defined in eq.(3.69). The pdf for the size distribution is often well represented by the log-normal distribution, defined as,

1 Po(a) - x/~)r~2a: exp{-( ln{a/ao}) 2/2/32}, (3.102)

where ao is the most probable radius, that is, the pdf has its maximum at a - ao, and the parameter fl is related to the standard deviation a, relative to the mean radius a, as,

- i + (3.103)

146 Chapter 3.

- 5

- 1 0

Figure 3.11"

[~)p ol

tn (k) 0

0 . 2 - - -

- 0 . 1

001 V \ , ,,,, . . . . - , , I - , - , - , , , , , I

0 5 kcto 10

The logarithm of the polydisperse form factor (3.104) versus kao, for various values o f the relative standard deviation a/~, as indicated by the numbers attached to the different curves.

The polydisperse form factor is defined as the intensity normalized to unity at zero wavevector, just as for monodisperse systems,

PP~ - Rr'~176 - 0). (3.104)

The experimental form factor as calculated by numerical integration from eqs.(3.99,102,104) is plotted in fig.3.11 as a function of the wavevector for various degrees of polydispersity. The most striking effect of polydispersity is that the minima in the form factor disappear. Since for each radius the minima of the form factor are located at different wavevectors, the sum of all the scattered intensities from individual particles is no longer equal to zero at particular wavevectors.

For pdf's that are sharply peaked, first order expansions with respect to the standard deviation are sufficient to describe the effects of polydispersity. Such an expansion is discussed in exercise 3.7.

pot

1.05

I f

0.95 0.05-"

0 . 9 ' ' ' I'0 0 5 k% Figure 3.12: The polydisperse diffusion coefficient, relative to the monodisperse diffusion coefficient Do ( ~), versus kao for various values of the relative standard deviation a/?z, as indicated by the numbers attached to the different curves.

Dynamic Light Scattering

The polydisperse EACF is found from the expression (3.44) for the scattered electric field strength,

< (E , (0) - f i , ) (E*(t) , fi,) > ~ ~ =

< (E~. f i , ) (E : , fi~) >

N * exp{ik (ri (t Y~i,j=l Bi (k)Bj (k) < �9 (0) - rj ))} > N . Ei,j=l B i (k )B d (k) < exp{ik . (ri - rj)} >

g exp{ik Ej=I I Bj(k)12 < �9 (rj(0) - r j ( t))} >

Ej=I I S~(k)I: where in the last line the "cross terms" i ~ j are set equal to zero, which is allowed for the dilute suspensions under consideration here. According to eq.(3.98), the ensemble averages are equal to,

< exp{ ik - ( r j ( 0 ) - r j(t))} > - exp{-Do(aj)k2t} ,

where the radius dependence of the Stokes-Einstein diffusion coefficient,

D o ( a j ) - kBT/67r~?oaj,

148 Chapter 3.

is denoted explicitly. Assuming polydispersity in size only, the summations over particles may be replaced by a weighted integral with respect to the radius. Substitution of the two above expressions into the general expression

p, pol for SE then gives,

~P~ t ) - f o da Po(a) B2(k, a) exp{-Do(a) k2t} E f o da Po(a) B2(k, a)

, (3.105)

where the radius dependence of the scattering amplitudes B is denoted explicitly.

The initial slope of ln{[IVE ~ as a function of time can be used to define the polydisperse diffusion coefficient. For small times, eq.(3.105) gives, upon Taylor expansion of the exponential function,

~pol (k,t) 1 - k2t f o da Po(a) B2(k, a) Do(a) + ... f o da Po(a) B2(k, a)

exp { 13P~

where the polydisperse diffusion coefficient is introduced,

DpOt f o da Po(a) B2(k, a) Do(a) o (k) - f o da Po(a) B2(k, a)

(3.106)

r~pot is plotted as a function of the The polydisperse diffusion coefficient ~0 wavevector for various degrees of polydispersity in fig.3.12. The log-normal size distribution (3.102) is used here to produce these numerical results. First of all, at small wavevectors, the polydisperse diffusion coefficient is found to be smaller than Do(~). The reason for this is, that for small wavevectors, the larger particles scatter more light than the small particles. Formally, this can be seen from eq.(3.63) �9 for small wavevectors the scattering amplitudes B(k, a) are proportional to the volume of the corresponding particle, so that in the integral in the numerator of eq.(3.105), more weight is given to the larger particles. A second feature of fig.3.12 is, that the polydisperse diffusion coefficient "oscillates' around the mean value D0(~). This oscillatory behaviour can be understood as follows. Consider two particles with somewhat different radii, al and a2, with a2 > al. Suppose that the scattering angle is such that ka2 - 4.49-.. , so that the form factor of the particle with radius a2 is zero (see fig.3.7). In that case, the detected scattered intensity comes entirely from the particle with radius a~, and the measured diffusion coefficient is in this

case D~ ~ - D0(al). For a larger scattering angle, such that kal - 4.49-. . , r~vot _ Do(a2). Since D0(al) > Do(a2), the measured diffusion likewise, ~-0

coefficient thus decreases with increasing wavevector in the neighbourhood of the two form factor minima. In case of a continuous distribution of sizes, the scattered intensity of the smaller particles dominates for wavevectors just on the left of the minimum in the (polydisperse) form factor, so that a relatively large diffusion coefficient is measured. Just on the right of the minimum, likewise, the diffusion coefficient is relatively small. This explains the "oscillatory" behaviour of the diffusion coefficient as a function of the wavevector.

For sharply peaked pdf's, the polydisperse EACF can be expanded with respect to the standard deviation in the size. To leading order, the resulting approximation is usually referred to as the second cumulant approximation. Exercise 3.8 is a discussion of the second cumulant approximation.

3.9.2 Effects of Optical Polydispersity

For concentrated suspensions, where interactions between the Brownian particles are important, there are two fundamental correlation functions which are of interest. The so-called collective dynamic structure factor S~, which is defined as,

1 N S~(k,t) - ~ ~ < exp{ik. (ri(O) - rj(t))} > , (3.107)

i,j=l

and the self dynamic structure factor S~, which is defined as, t

S~(k, t) - < exp{ik. ( r l ( O ) - rl(t))} > . (3.108)

The collective dynamic structure factor is (apart from the factor l / N ) the time dependent correlation function of the Fourier transform of the density (see also subsection 1.3.3 in the introductory chapter). Its time dependence describes the dynamics of sinusiodal density fluctuations with wavelength A - 27r/k. Since a density fluctuation involves simultaneous movement of many particles, the function (3.107) is connected with collective phenomena. Notice that at time t - 0, the dynamic collective structure factor is just the structure factor introduced in eq.(3.67). To make the distinction between the latter structure

1The collective and self dynamic structure factor are also referred to as the collective and self intermediate scattering function, respectively.

150 Chapter 3.

factor and the dynamic collective structure factor more explicit, the structure factor (3.67) is also referred to as the static structure factor. The self dynamic structure factor (3.108) characterizes the dynamics of a single particle (particle 1 in this case). The dynamics of a single particle is of course affected by the interactions with all other particles. In exercise 3.9 the connection between the self dynamic structure factor and the mean squared displacement is derived. In chapter 6 on diffusion, both the collective and self dynamic structure factor are considered in detail.

The dynamic structure factors can both be measured by means of dynamic light scattering. For a monodisperse system, according to eq.(3.83), DLS measures the collective dynamic structure factor. The self dynamic structure factor can in principle be obtained by mixing two suspensions, each of which is monodisperse. In the mixture, one of the species should be very dilute, such that these particles -the "tracer particles "- do not interact with each other. They may, however, interact with the particles of the other species -the "host particles". Suppose now that the difference in optical properties of the two species is such, that the few tracer particles scatter all (or most of) the light. To achieve this, the refractive index of the host particles should match the refractive index of the solvent quite closely, since their concentration is much larger than that of the tracer particles, and yet, the tracer particles should scatter most of the light. In the expression (3.44) for the scattered electric field strength, only the scattering amplitudes of the tracer particles survive. Since the tracer particles do not interact with each other, the experimental EACF is precisely equal to the self dynamic structure factor, as "cross terms" i ~ j in the general expression (3.83) are zero.

In practice it is difficult to prepare such a tracer system, since some self- contradictory conditions must be satisfied" a few tracer panicles, in a suspension containing many host particles, must scatter the major fraction of the total scattered intensity. As is shown below, the finite degree of polydispersity in optical properties, that is always present in practice, enables the measurement of both the self and collective dynamic structure factor, provided there is no polydispersity in size (or only a very small polydispersity in size).

Suppose that all particles are equally sized, and moreover, have identical pair-interaction potentials. All particles are then statistically equivalent. The optical properties, however, are assumed different, that is, the scattering amplitudes Bj (k) are generally different for different j 's. Since the particles are

statistically equivalent, we have,

< e x p { i k - ( r i ( 0 ) - rj(t))} > - s (k, t) - s , ( k , t)

N - 1 i C j ,

so that the normalized polydisperse EACF can be written as,

ot(k,t) - E < (E,(0) . f i , ) (E: ( t ) , fi,) >

< (E , . f i , ) (E; , fi,) >

-N1 2i,j=lN Bi(k)Bj (k ) < exp{ik. ( r i ( 0 ) - rj(t))} > 1 N -N 2i,j=l Bi(k)BJ(k) < exp{ik. ( r i - rj)} >

1 N 2 1 N --N E j - 1 Bj Ss -~ -~ E(ii/ : j)_ 1 BiBj (S~- S.)/(N- 1)

1 N 2 1 N - E j = a B j + Bi -- / ( N ) U ~ 7~(ir 1 Bj (S 1) - 1

Defining the particle number averages,

1 ~ Bj(k), -

j= l (3.109)

and, 1 N

j--1

the above expression can be written as (in case N >> 1),

(3.110)

~pol E (k, t) - A,(k) S,(k, t) + A~(k) S~(k, t), (3.111)

where the mode amplitudes A~,~ are equal to,

A,(k) - [B2(k) - ~2(k)] , (3.112)

A~(k) - . (3.113)

For the monodisperse case, for which, ~2 _ B2 _ B2 ' eq.(3.111) simply p, pol reduces to ~E (k,t) - S~(k, t ) /S(k) , in accordance with eq.(3.83). The

expression (3.111) for the EACF is the optically polydisperse generalization of the expression (3.83) for the monodisperse EACE

152 Chapter 3.

I " I

[ngE . - 2 " ",,,

- 4

-6 .01 .02 .03 f[s]

Figure 3.13" The logarithm of the polydisperse EACF for silica particles coated with octade- cyl alcohol chains, dispersed in cyclohexane. The volume fraction is approximately 0.35. On increasing the temperature from T - 20 o 6' to T - 35 o C, the mean refractive index of the particles approaches that of the solvent cyclohexane. The solid curves are fits to the data points with a sum of two exponentials. This figure is taken from Kops-Werkhoven, Fijnaut (1982).

There is thus an extra contribution (proportional to the self dynamic structure factor) to the EACF in comparison to the monodisperse case. Intuitively this contribution may be understood as follows. When two optically distinct particles interchange their positions, the microscopic density remains unchanged, but nevertheless, the scattered intensity changes. In the monodisperse case, the intensity can only vary due to changes of the microscopic density. Hence, compared to the monodisperse case, there is an additional mechanism (interchange of optically distinct particles) that contributes to the time dependence of the fluctuating intensity. This is the origin of the first term in eq.(3.111).

The expression (3.111) for the polydisperse EACF shows that the measured time dependence of the EACF is a sum of two modes, corresponding to the self and collective dynamic structure factors. In case the time scales on which the self and collective dynamic structure factor decay are sufficiently different, both these structure factors can be obtained from a single EACF measurement. Some experimental results for a silica dispersion are given in fig.3.13. The

3.10. Scattering by Rigid Rods 153

curves drawn through the data points are fits to a sum of two exponentials. This makes sense whenever both the self and collective dynamic structure factors are well described by single exponential functions of time. Here, the optical contrast (3.69) is varied by varying the temperature. For a large optical contrast (T - 20 0 C) the EACF is almost single exponential. In this case, the polydispersity in optical properties is small compared to the optical contrast. The collective mode amplitude A~ is much larger than the self mode amplitude A, in this case. For this temperature, to a good approximation, only the collective dynamic structure factor is measured. On the other hand, for a close match of the mean refractive index of the particles and the solvent (T - 35 0 C), the mode amplitudes A~ and A~ are of the same order of magnitude (see exercise 3.10 for a more detailed discussion on this matter). As can be seen from fig.3.13, in this case the decay is approximately a sum of two exponentials. According to eq.(3.111), one of these exponentials may be identified with the self dynamic structure factor and one with the collective dynamic structure factor.

3.10 Scattering by Rigid Rods

In this section we discuss the general features of light scattering by rigid rod like Brownian particles. An explicit evaluation of these general expressions for correlation functions (like eq.(3.98) for spherical particles) is given in chapter 6 on diffusion. The considerations in chapter 2 on rods are not sufficient for such an explicit calculation.

As a first step, the dielectric properties of a rod are specified in the following subsection. SLS and DLS is the subject of the two subsequent subsections.

3.10.1 The Dielectric Constant of a Rod

The anisotropic molecular structure of a long and thin cylindrically symmetric rod-shaped Brownian particle gives rise, in most cases, to a different polari- zability perpendicular and parallel to the rod. The dielectric constant is then different for polarization directions of the incident electric field perpendicular and parallel to the rod. Let e• and ell denote the respective dielectric constants. The incident electric field strength E0 can be decomposed in its component along the rod, Ell, and its component perpendicular to the rod, Ex,

Eo - Ell + E z ,

154 Chapter 3.

Ell - f i f i ' E o ,

where the unit vector ~ is the orientation of the rod (see fig.2.5a for the definition of the orientation of the rod). Hence,

E" Eo - Cl lEl l + e~ E•

= <ii fi f i . Eo + c.l. [ i - f i f i ] . Eo.

The dielectric constant of the rod minus that of the solvent is therefore depending on the orientation of the rod as,

[ 1] E - i ,s - + A , ,a,a - i , (3.114)

where,

_ 2 e • + ell _ e y , ( 3 . 1 1 5 ) e - - 3

is the (weighted) average of the dielectric constants relative to that of the solvent, and,

Ae - e l l - c• (3.116)

is the difference between the two dielectric constants. The variation of the dielectric constants ell and cj. within the rod cannot

be neglected when there is a close match with the dielectric constant of the solvent. When the variation of ~ll,-t within the rod is smaller or comparable to ~ - el, the dielectric matrix cannot be taken outside the integration in the expression (3.33) for the scattering amplitude. In the following it is assumed that the dielectric constants are independent of the position in the rod material.

3.10.2 Static Light Scattering by Rods

The scattering amplitude of a rigid rod may be calculated from the expression (3.114) for the dielectric constant. Substitution of eq.(3.114) into expression (3.33) for the scattering amplitude gives,

B(k, flj) - Bj(k) _ 1

3

(3.117)

,-~ ~ i --[- As f l j f l j -- 5 X r f ~ dl exp{ik, fij l} -~L

) where L is the length of the rod, D its diameter, and,

jo(x) - sin{z} . (3.118) z

In the first line in eq.(3.117) the rods are assumed identical, so that the particle number index j enters only through the orientation of that rod. Furthermore, in the last two lines on the right hand-side of eq.(3.117), it is assumed that k D << 1 (say k D < 0.2), so that the exponential function hardly changes on varying the position r ~ perpendicular to the rod's long axis. In exercise 3.11 the scattering amplitude for a thicker rod (for which k D is not small) is calculated.

Notice that the scattering amplitude of a rod depends on the orientation of that rod, only if 1Lk- fij is not small, say > 0.5. The relative phases of the scattered field strength from different elements in a single rod do not change significantly on rotation of the rod when �89 fij < 0.5, which inequality is

1 kL < 0.5 For such short rods, c.q. small wavevectors, the satisfied when 7 jo-function in eq.(3.117) is equal to 1.

Since the scattered intensity of an assembly of rods not only depends on the positions of the rods but also on their orientations, the intensity is now averaged with respect to both the position and the orientation coordinates. The average scattered intensity, expressed in terms of the Rayleigh ratio (3.56), follows immediately from eqs.(3.44,45,47,117),

R - IoV~ 2 <1 E , . f i , [2>_ 7r L (3.119)

k~ ~p(k) {~ ,') k) 2 ~x~ s(',o) ~ k) • (4~)~ s(' ( + (k) + (~x~) s(o,o)( } .

The form factor P(k) is defined as,

(1 ) P(k) - < j o 2 ~ L k . f i > , (3.120)

where the subscript on fi is omitted. For thicker rods, the above formula for the Rayleigh ratio is still valid, except that the jo-functions have to be replaced

156 Chapter 3.

by a different expression (as for example, the expression derived in exercise 3.11). The ensemble average (3.120) is explicitly evaluated in exercise 3.12a.

There are three structure factors introduced here, which are distinguished by the double superscripts i for isotropic and a for anisotropic, referring to the isotropic part of the dielectric constant (3.114) (proportional to ~) and the anisotropic part (proportional to A e), respectively. These structure factors are defined as,

N

1 E (h~. rio) 2 S(i'i)'k'() - N i,j=l

jo (�89 fi,)jo (�89 fij) exp{ik (ri rj)} > , (3.121) X < �9 - -

< >

N 1

jo ( ILk . fii)jo (}Lk. fij) x cos{k. ( r i - rj)} > , (3.122/

< jo >

1 Y (fi," - - ~ J : ] ' f i o ) ( f i , ' [ f i j f i j - - ~ i ] ' f i o ) S(~'")(k) -- N E < [fiifi, 1 1 i,j=l

j0 (1Lk" fi,)j0 (1Lk" fii) exp{ik (ri rj)} > . (3.123) X �9 - -

1Lk fi] <jx( �9 > ]

In obtaining these expressions, use is made of,

fi~. T(k~). a - fi~. a ,

for any vector a. This follows from the definition (3.45) of T(k,) and the fact that ft, _1_ k,.

Notice that the average scattered intensity is a complicated mix of orientational and translational correlations. The intuitive "diffuse Bragg scattering" picture, as discussed in section 3.4 for spherical particles, no longer applies for rigid rods. Instead of diffuse Bragg scattering from the Brownian particles, one should now interpret the scattered intensity as being the result of diffuse Bragg scattering from segments, which are the elementary volume elements of which each rod is composed. Since the relative positions of segments within each rigid rod are fixed by the rod's orientation, fluctuations of the segment

density are determined by both fluctuations in the orientations and positions of the Brownian rods.

Due to their complexity, the above expressions are of little practical value. There are two special cases in which the above complicated formulas reduce to simpler forms, which are often used in experimental reality. Let us discuss these simpler forms.

Case (i) " A e / ~ << 1.

For many systems of practical importance, the difference A e is much smaller than ~. In those cases the anisotropy of the dielectric constant may be neglected and only the isotropic structure factor survives,

ko 4 ~2 (i,i) k) (3 24) R - (4r)2 # 7r L P (k ) S ( . .1

This is a good approximation for Ae/~ < 0.1. Notice that S (i'0 ,~ (fi~ �9 rio)2, so that this approximation only makes sense when the polarization direction of the incident and scattered light are not perpendicular.

For small wavevectors, such that �89 k L < 0.5, the jo-functions are equal to 1, and the above expression simplifies to,

ko 4 ( ( 0 ) 2 )2 i N R - (47r)~ fi r L ~2(fi .f io)2P(k) ~ ~ < e x p { i k . ( r , - r j ) } > .

i,j=l (3.125)

The scattered intensity now takes a form which is identical to that for spherical particles (see eqs.(3.66,67)).

Case (ii)" fi, _1_ rio.

In this case only the anisotropic structure factor S (~,~) survives,

R - ( 4 7 r ) 2 # 7r -~ L P ( k ) ( A e ) 2 S ( ~ ' ~ ) ( k ) , (3.126)

and the structure factor reduces to,

1 N i,j--1

158 Chapter 3.

1Lk ' f i J ) exp{ik (ri r j ) } > . (3.127) jo (5 " .

1Lk fi~ > ]

The scattered intensity is now strongly dependent on orientational correlations, much more than in case (i). Translational correlations, however, do play a role as well via the exponential function, except for small wavevectors. Even for small wavevectors, for which the jo-functions are equal to 1, a strong orientational correlation dependence remains, in contrast to case (i).

The time dependent anisotropic structure factor for a dilute suspension in which the rods are allowed to relax from an aligned configuration to the isotropic state is evaluated in exercise 4.7. Effects of interactions on such an orientational relaxation process are analysed in subsection 6.10.2 in the chapter on diffusion.

3.10.3 Dynamic Light Scattering by Rods

In the derivation of the Siegert relation (3.82), no assumption was made concerning the nature of the Brownian particles. The Siegert relation is equally valid for spherical and for rigid rod like Brownian particles. The ensemble averages for rods, however, are with respect to a probability density function of both the positions and the orientations.

For spherical particles the field auto-correlation function (EACF) is given in eq.(3.83). For rod like Brownian particles this is a much more complicated function in which orientational variables play a role. The normalized EACF is obtained by substitution of eqs.(3.44,45,117) and the expression (3.119)for the average scattered intensity into the definitions (3.77,80),

^ ~2 S(i,i)(k, t) + 2 ~Ae S(i'~)(k, t) + (Ae) 2 S(~,~)(k, t) (3.128) g~(k, t) - ~ S(',')(k) + 2 ~ZX~ S(',~ + (~X~)~ S(~176 '

where the following dynamic structure factors are introduced (we abbreviate ri(0) - ri(t - 0), and fii(0) - fii(t - 0) ),

1 N s(', ')(k, t) - -~ ~ ( ~ . ~o) ~

i,j=l

jo ( Lk •

Lk. h) >

(3.129)

exp{ik. (ri(0) - rj(t))} > ,

( [ 1] ) _ 1 ~ ( f i . r io)< fi~" f i i ( 0 ) f i i ( 0 ) -g i "rio (3.130) S(i 'a)(k, t) i i , j = l

jo ( iLk" fi,(0))jo ( iLk" fij(t)) cos{k (r,(0) rj(t))} > , i " - -

< jo ( Lk. a) > i N ( [ 1[] )

S(~ '~) (k , t ) - ~ ~ < fi~. fi,(0)fi,(0)- 5 "rio (3.131) i,j=l

1

jo (1Lk" fii(0))jo (�89 fij(t)) exp{ik (ri(0) rj(t))} > . X " - -

< jo ( Lk . ) > The dynamic structure factors reduce to their static counterparts in r 123) at time t = 0.

The two special cases considered in the previous section lead to simplified expressions for the EACF which are of experimental relevance. In case A e/~ < 0.1, the EACF is well approximated as,

[lE(k, t) -- S(i'i)(k, t)/S(i'i)(k) . (3.132)

l kL < 0 5, this expression reduces to that For small wavevectors, such that 7 for spherical particles (see eqs.(3.83,67)). Orientational correlations do not play a role in this case. In case the polarization direction of the incident and detected light are perpendicular, the EACF is given by,

gE(~, t) -- s(a'a)(~, t ) / s (a 'a ) ( k ) . (3.133)

As for static light scattering, this EACF is more sensitive to orientational correlations than the isotropic EACF (3.132). Even for small wavevectors the anisotropic EACF (3.133) remains sensitive to orientational correlations.

Little is known about the explicit time and wavevector dependences of both static and dynamic light scattering characteristics of suspensions of rod like Brownian particles. In principle, as a first step in the calculation of the EACF, the time dependent probability density function for the positions and orientations of the rods should be obtained from its equation of motion, the so-called Smoluchowski equation, which is derived in the next chapter. The Smoluchowski equation is used in subsection 6.10.1 in the chapter on diffusion to calculate the isotropic EACF in eq.(3.132) explicitly for non- interacting rods. It turns out that, even for these dilute systems, the EACF is

not single exponential, as for spherical particles (see eq.(3.98)), but equals a sum of many exponentials. The number of exponentials that need to be taken into account depends on the numerical value of k L. For larger values of k L

rotational motion becomes more important, and the EACF consists of more exponentials. As we have seen above, for small scattering angles such that l k L < 0.5 rotational motion does not affect the isotropic dynamic structure 2 factor S (~,0, and translational correlations can be studied by light scattering just as for spherical Brownian particles (in fact, this will turn out to be true for k L < 5).

Exercises

3.1) Consider a photon, moving on a common line towards a Brownian particle with a zero velocity. The mass of the Brownian particle is M. The wavevector of the photon before the collision is ko, its frequency wo and its mass too. Let k, denote the wavevector after the collision (which is colinear with the incident wavevector ko), w, its frequency and m, its mass. Show that,

1 Wo-W~ k o - k ~ _ 7(mo4-m~)

w0 + w, k0 + k, M

Estimate the relative frequency shift for a typical mass M - 10 -15 g and a typical wavelength of light of 500 rim.

(Hint �9 use conservation of momentum, moc = - m s c + M y , with c the velocity of light (= 300000 k m / s ) and v the velocity of the Brownian particle

1 M732, where after the collision, and conservation of energy, moc 2 - re ,c2+

the classical expression for the kinetic energy of the Brownian particle is used. Also use the relations hwo,~ - mo,,C 2 for the energy of the photon, and mo,,c - h ko,, for the momentum of the photon.)

The frequency of a photon is thus not exactly equal before and after interaction with the Brownian particle. The relative frequency change is, however, extremely small, and is neglected in the present chapter. Since there is always a small frequency shift upon scattering, so that the energy hw of the photon changes, the light scattering process considered here is sometimes referred to as quasi elastic l ight scattering.


3.2) * Calculate the inverse of the matrix,

which appears on the left hand-side of eq.(3.23), for complex valued ko. To this end, try a matrix of the from,

kk A i + B k2 ~ ,

and determine the functions A and B.

3.3) * Consider the integral,

f dk exp{ik. (r- r")} k: -(ko u i.):

which appears on the right hand-side of eq.(3.36). First perform the spherical angular integrations and show that the integral is equal to,

fo ~ k 2 s i n { k l r - r" I} 47r dk k2 _ (/Co + ic~) 2 k I r - r " l

Note that the integrand is an even function in k, so that the integral may be 1 oo written as fo ~176 ( . . . ) - 7 f-oo ( ' " ) . The integral can thus be written as the sum

of two integrals as follows,

- dk i oo

k 2 e x p { i k l r - r " 1}

k 2 - ( k o + ia) 2 k l r - r" I

/ ? k ~ exp{ - ik [ r - r" l} ] - oodk k 2 _ (ko +ic~) 2 k l r - r"[ "

Each of the two integrals can now be evaluated by means of the residue theorem, after closing the integration range in the upper complex half plane (for the first integral on the right hand-side in the above equation) or the lower half plane (for the second integral), as discussed in subsection 1.2.5 in the intro- ductorty chapter. The integration paths that need to be considered here, are depicted in fig.3.3. Perform the integrations and evaluate the differentiations on the left hand-side of eq.(3.37).

3.4) In this chapter, the scattered electric field strength is calculated for a fixed configuration of Brownian particles. This is a valid procedure only if Brownian motion is so slow that two distinct particles have hardly displaced relative to each other during the time interval that light needs to propagate over the distance between the two particles. To assess the validity of this procedure, estimate the change of the phase of the electric field strength due to Brownian motion during the time interval that light takes to traverse a distance of 1 cm, which is a typical size for cuvettes.

(Hint : estimate the displacement 1 of a particle from its mean squared dis-

placement/,~ ~/<[ r(0) - r(t) 12> - x/6Dot. See chapter 2 for a derivation of this result.)

3.5) Use [ k0 I-I k, I to show that,

4r sin{ O, I k o - k ~ l - -~ --f-},

with A the wavelength of the light in the dispersion.

3.6) In the derivation of the expression (3.67) for the static structure factor, it was assumed that,

# fy, dr exp{ik, r} << 1 ,

with V~ the scattering volume, that is, the volume in the cuvette from which scattered light is collected. A typical linear size of the scattering volume is 0.5 mm and a typical value for the wavevector ko is 2 �9 10 r m -~. Calculate the integral for a rectangular shaped scattering volume with sides of length 0.5 ram. A typical value for the number density fi of Brownian particles is 1019 m -3. Conclude from your result that the above inequality is not satisfied at all. The resolution of this problem is as follows. The incident intensity I0 is not constant throughout the scattering volume, but is more or less Gaussian shaped. When the spatial variation of the incident intensity profile is smooth over distances of the order of the correlation length (that is, the distance over which the pair-correlation function g tends to unity), the above integral is to be replaced by,

f Io(r) ] dr exp{ik, r} << 1

I0

where the integral ranges now over the entire ~3, since the intensity profile itself defines now the extent of the scattering volume. The intensity Io(r) at the position r in the cuvette is equal to,

Io(r)- Ioexp{-( 7 },

with 1 - 0.5 ram. The above inequality now reads,

i k212 } << 1.

Conclude that the inequality is satisfied in this case. The conclusion is that it is essential that the edges of the scattering volume

are not very sharp. Such sharp edges give rise to large wavevector contributions of the above integral, in which case the scattering pattern would exhibit very intense circular "scattering rings". A smooth incident intensity profile is essential for a light scattering setup.

3.7) Small size polydispersity and static light scattering Consider Brownian particles which are polydisperse in size. (a) Use eqs.(3.104,99) to show that for sharply peaked pdf's Po(a) around

a = a ,

PP~ = f ~ da Po(a) a 6 P(k , a)

f o da Po(a) a 6

- (a )2[ 1 02 (~6/3(k)) '~ P(k ) + 2a 4 O~ 2 ] - 15P(k) ,

where P (k, a) is the form factor of a sphere with radius a and/5 _ p (k, ~) the form factor of a sphere with the average radius,

~0 ~176 ?z - da Po(a) a .

For optically homogeneous particles, P(k, ?z) is given by eq.(3.68). standard deviation in sizes, a, is defined as,

The

0. 2 ~ ~0 ~176 da Po(a) (a - &)2

For those wavevectors for which the term in the square brackets in the above expression for ppol(k) is equal to zero, there is no effect of polydispersity.


These are the wavevectors at which the curves in fig.3.11, for a small polydispersity, intersect.

(Hint" Use that, for example,

1 0 2 a6e(k, a) ~ ~t6P(k) + (a - a)--~a -~ ~ .

(b) For small wavevectors the polydisperse radius a p~ is defined in analogy with eqs.(3.93,95) as,

1 k2 aVOt)2 1 k2 2 pr, ot(k) ~ exp { - g ( } ,~, 1 - g (a r'~

This is the radius that one would obtain experimentally, analysing the scattering data as if the system were monodisperse (and assuming optically homogeneous Brownian particles). Use the expression for ppol(k) as derived in (a), together with,

- 1 k2 a2 P(k) ~ 1 - g ,

to show that, to leading order in the polydispersity,

aV~ = a l 1 + 1 3 ( ~ /

This relation can be used to correct the measured radius to obtain the true mean radius ~. An estimate of the relative standard deviation in sizes can be obtained from electron micrographs or accurate dynamic light scattering measurements, as discussed in the next exercise.

3.8) Small polydispersity and dynamic light scattering Second cumulant analysis

Consider the EACF (3.105) for sharply peaked pdf's Po(a). Taylor expand the exponential function in the numerator of eq.(3.105) around Do(a) - D~ ~ (k), up to second order, with the polydisperse diffusion coefficient defined in eq.(3.106). Show that,

~pol E (k,t)

[1 2] n'~ k2t} 1 + (k 2 exp{ -~o ~ t ) rr~

{ o,o, 1 exp (k)k t + 5(k t)

where aD is the standard deviation of Do(a) with respect to D~ ~ (k),

f o da Po(a) B2(k, a) (Do(a) - D~ ~ (k)) ~ a2D -- f ~ de Po(a) B2(k, a) "

Notice that the polydisperse EACF contains a quadratic term in the time. Now show from eq.(3.106) by Taylor expansion around a = a that, to leading order in polydispersity,

D~)~ - Do(a)

+ 2B2(k, ?z) ~ (B2(k' a)Vo(a)) - Vo(a)~a2 (k, a) ,

with a the standard deviation of the radius (see the previous exercise). Use this result in the above expression for cry9 to verify that (also make use of the fact that ODo(?Z)/Oa - -Do(?z)/~),

Use that B(k ~ 0, a) ,-~ a 3 and Do (a) ,-~ 1/a, to verify that for small wavevectors (say k~ < 1/2),

We thus finally obtain,

{ [ (~)2] ID (o.) 2 (k2t) 2 } ~~ ~ O,t) = exp -Do(~) 1 - 5 k2t + -~ o(a) ?z "

Experimental data for the IACF on slightly polydisperse samples should thus be fitted to an equation of the form

constant1 § constant2 • exp { - 2 a ( k ) k 2 t + 2 f l ( k ) (k 2t)2} .

The mean radius and the relative standard deviation follow from our final result as,

Do(a) - a (k ~ 0) + 103(k ---, 0) , ( ~ ) ~ _ 2 ~ ( k ~ 0 ) ~ ( k ~ 0)

a - ~(k --. 0 ) + ~0~(k --. 0) ~ 2 . ( k --. 0)

It is thus possible to measure with dynamic light scattering both the mean radius and the polydispersity of a slightly polydisperse system.

The above analysis of scattering data is called a second cumulant analysis. The coefficient/~ is commonly referred to as the second cumulant.

3.9) Expand the self dynamic structure factor (3.108) up to second order in k, and show that,

1 k2 2 S,(k,t) -- 1 - g <1 r l ( O ) - ra(t) I + OI I0 �9

A measurement of self dynamic structure factor for small wavevectors thus allows the experimental study of the (time dependent) mean squared displacement. We return to this fact in the chapter 6 on diffusion.

(Hint: use that, for example,

< (Xl(0) -- Xl( t ))(yl(0) -- yl( t)) > --

< ( X l ( 0 ) - X l ( t ) ) > < ( y l ( 0 ) - y l ( t ) ) > - 0,

where rl - - (Xl, Yl, Z l ) " )

3.10) Contrast variation Consider a binary mixture of optically distinct particles which all have the

same size. The mixture is a i �9 1 mixture of the two species of particles. Let C~'2 denote the relative difference (~p - ey)/el of the dielectric constant of particles of species 1 or 2. Suppose that the particles may be considered as optically homogeneous and take, for the sake of simplicity, the static structure factor equal to unity. Verify that the mode amplitudes (3.112,113) are equal tO,

A,(k) -

A~(k) =

1 [ C [ - C;] 2

(Ct ) + '

1 [C~ -1- C~] 2 2 + (c;)

The dielectric constant ef of the solvent can be varied by varying the temperature or by variation of the solvent composition. Conclude from the above expressions that, in case ef is equal to dielectric constant of either one of the species of particles, A~ - A~. On the other hand, when ef is very different

from the dielectric constants of the two species, the collective mode amplitude is much larger than the self mode amplitude.

3.11) Form factor of a thick rod The following integral, ranging over the volume of a cylinder with its

geometrical center at the origin,

I - 1 fy d r e x p { i k r ) , V o

was calculated in subsection 3.10.2 for a thin cylinder (kD < 0.2). Now suppose that k D is not small. Without loss of generality the orientation of the cylinder may be taken along the z-axis, and the wavevector oriented in the x, z-plane. For an arbitrary orientation of the rod and the wavevector, simply

replace k~ by k . fi, and k~ by k • ~/1 - ( k . f i lk) ~'. Evaluate the integral to show that,

(1 i l ~ , i:l/k) 2 ) , 2J1 7 kD (k sin{7 L k . fi} I = •

1 keV[1 (k. ~. l tk , ) 1 L k . 1.1 ' 2

where J~ is a Bessel function. (Hint" use the definition of the Bessel function of order n,

J~(x) - lr fo '~ dqo [cos{x sin{~} } cos{nq~} + sin{x sin{T} } sin{nqo}] ,

- ) and the recurrence relation,

3.12) Form factor of a thin rod In a very dilute suspension of rod like Brownian particles, the (normalized)

pdf for the orientation of a rod is a constant equal to 1/47r. (a) Show that in that case the form factor of a thin rod is given by,

< j~o(1Lk �9 fi) > = (sin,z )

kL z

Since the integrand is positive, this function is never zero, in contrast to the form factor of spherical particles. This can be understood as follows. At each instant in time different rods have different orientations. For each different orientation the wavevector where complete destructive interference occurs is


different. Hence, the scattering of a set of randomly oriented rods may be viewed as a system of polydisperse scatterers.

(b) Use the above expression for the form factor to calculate the initial slope of a Guinier plot (a plot of In{/} versus k 2 ) for such a dilute dispersion

1 L 2 of thin rods. (The answer is" - ~ .)

3.13) Heterodyne dynamic light scattering In a so-called heterodyne light scattering experiment, the scattered light

is mixed with incident light (directed towards the detector). The detected electric field strength is thus,

Eh~t(t) -- Et~ + E~(t),

with E~ the field scattered by the particles and E l~ the incident field strength which is mixed with the scattered field. The latter field is usually referred to as the local oscillator field strength. The DLS experiment as described in the main taext of the present chapter is referred to in literature as homodyne light scattering to make the distinction with heterodyne DLS.

The detected intensity is now equal to,

lCe_~o EgO~ 2 i (k, t) - ~ x [ + E,(t) I

Start with the definition (3.74) of the IACF and use the Siegert relation (3.78) for the homodyne correlation functions to show that the heterodyne IACF is equal to ( ~e stands for "the real part of"),

gh~t(k,t ) - ( I ' O ~ ) 2 + 211~ + 12 + 21'~ ~egE(k, t) + 12 19E(k, t)12

with I t~ the local oscillator intensity and I the mean scattered intensity by the Brownian particles. Conclude that for I t~ > 50 x 1, the heterodyne IACF is essentially equal to the homodyne EACE

3.14) Consider a dilute system of Brownian particles, where to each particle a constant force F is applied (for example, a gravitational force or a force due to an electric field). Show that in a heterodyne DLS experiment, with I t~ >> 1, the IACF is given by,

2,lot, I1 cos k x exp O0 k2t ]

Further Reading 169

Is the homodyne IACF affected by the extra velocity that the particles attain due to the external field ?

(Hint" use eq.(3.97) together with the pdf that was calculated in exercise 2.4 and the expression for the heterodyne IACF of the previous exercise.)


The data shown in fig.3.13 are taken from, �9 M.M. Kops-Werkhoven, H.M. Fijnaut, J. Chem. Phys. 77 (1982) 2242.

The original papers on the effect of optical polydispersity are, �9 M.B. Weismann, J. Chem. Phys. 72 (1980) 231. �9 D.L. Cebula, R.H. Ottewill, J. Ralston, P.N. Pusey, J. Chem. Soc. Trans.

177 (1981) 2585. �9 P.N. Pusey, H.M. Fijnaut, A. Vrij, J. Chem. Phys. 77 (1982)4270.

The effects of polydispersity on the measured static structure factor are analysed, on the basis of the Ornstein-Zernike equation, in,

�9 R.J. Baxter, J. Chem. Phys. 52 (1970) 4559. �9 L. Blum, G. Stell, J. Chem. Phys. 71 (1979)42. �9 A. Vrij, J. Chem. Phys. 69 (1978) 1742, 71 (1979) 3267, 72 (1980)

3735. �9 P. van Beurten, A. Vrij, J. Chem. Phys. 74 (1981) 2744. �9 P. Salgi, R. Rajagopalan, Adv. Coll. Int. Sci. 43 (1993) 169. �9 G. N~igele, T. Zwick, R. Krause, R. Klein, J. Coll. Int. Sci. 161 (1993)

347.

These theoretical predictions are compared with computer simulations in, �9 D. Frenkel, R.J. Vos, C.G. de Kruif, A. Vrij, J. Chem. Phys. 84 (1986)

4625.

In this chapter, the discussion is limited to the Rayleigh Gans Debye scattering theory. The so-called Mie scattering theory is less restrictive. More about Rayleigh Gans Debye and Mie scattering theory, along with other theoretical aspects of light scattering, can be found in for example,

170 Further Reading

�9 H.C. van de Hulst, Light Scattering by Small Particles, Dover Publica- tions, New York, 1981.

�9 M. Kerker, The Scattering of light and Other Electromagnetic Radiation, Academic Press, New York and London, 1969.

�9 B.J. Berne, R. Pecora, Dynamic Light Scattering, Wiley Interscience, New York, 1976.

�9 K.S. Schmitz, An Introduction to Dynamic Light Scattering by Macro- molecules, Academic Press, New York, 1990.

There are a number of books on light scattering which, in addition, contain technical information concerning experimental set ups,

�9 B. Chu, Laser Light Scattering, Basic Principles and Practice, Academic Press, London, 1991.

�9 R. Pecora (ed.), Dynamic Light Scattering, Applications of Photon Cor- relation Spectroscopy, Plenum Press, New York, 1985.

�9 W. Brown (ed.), Dynamic Light Scattering, The Method and Some Applications. Oxford Science Publications, Clarendon Press, Oxford, 1993. A collection of classic papers on light scattering by various kinds of macromolecules can be found in,

�9 D. Mclntyre, F. Gomick (eds.), Light Scattering from Dilute Polymer Solutions, International Science Review Series volume 3, Gordon and Breach Science Publishers, New York, 1964.

Chapter 4

FUNDAMENTAL EQUATIONS OF MOTION

171

172 Chapter 4.

4.1 Introduction

One approach to the theoretical study of the dynamics of Brownian systems was developed in chapter 2 for non-interacting Brownian particles. The starting point there is a stochastic equation of motion for the phase space coordinates of a Brownian particle, the so-called Langevin equation. Such an approach is also feasible for systems of interacting particles. However, the Langevin equations for interacting particles are non-linear in the phase space coordinates, via the interaction terms in the equations of motion, which gives rise to fundamental problems in defining the statistical properties of the stochastic forces (see van Kampen (1983)). An alternative route towards a theory for the dynamics of colloidal systems is via equations of motion for the probability density function of relevant phase space coordinates of the particles (such as positions, momenta, orientations and angular velocities). As we have seen in section 1.3 in the introductory chapter, once the equation of motion for the probability density function (pdf) of the phase space coordinates is known, time dependent correlation functions (such as the important density auto-correlation function) can be expressed in terms of either an operator exponential or in terms of an explicit solution of the equation of motion.

There are several ways of obtaining equations of motion for the probability density function (pdf) of the phase space coordinates. Probably the most fundamental approach is to start with the Liouville equation for the pdf of the phase space coordinates of all the particles in the system �9 both the fluid molecules and the colloidal particles. This Liouville equation can be integrated over the rapidly fluctuating phase space coordinates and subsequently coarsened to the time scale under consideration. See Mazo (1969), Murphy and Aguirre (1971) and Deutch and Oppenheim (1972) for such an approach. This approach is rather technically involved, and is not pursued here. Alternatively, the above mentioned Langevin equations can be shown to be equivalent to so-called Fokker-Planck equations, which are the equations of motion for the corresponding pdf's which we are after here. For the equivalence of stochastic differential equations and Fokker-Planck equations, see for example, Lax (1966), van Kampen (1983) and Gardiner (1983). As mentioned above, there are fundamental problems with this approach for the particular case we are interested in here. A stochastic differential equation approach, similar in spirit, is based on the Navier-Stokes equation for the fluid in which the colloidal particles are immersed, with a fluctuating stress matrix which plays a similar role as the fluctuating force in the Langevin equations

4.1. Introduction 173

as discussed in chapter 2. The statistical properties of the fluctuating stress matrix must be specified, just as for the fluctuating force in the Langevin equation. The resulting stochastic differential equations can then be analysed to obtain the statistical properties of the Brownian particles (see Bedeaux (1974) and Noetinger (1990)).

The approach chosen here is a well known and rather direct method to derive the fundamental equations of motion. The idea of this approach is as follows. Suppose one is interested in the equation of motion for the pdf of a stochastic variable X, which is an m-dimensional vector. One can think of X as the set of momentum and position coordinates of the Brownian particles (on the Fokker-Planck time scale) or as the set of position coordinates only (on the Brownian time scale). The pdf of X is denoted as P(X, t), which is a function of time in general. Consider an ensemble of (infinitely) many macroscopically identical systems, that is, systems of which the macroscopic parameters are specified (such as the temperature, pressure, volume,.. .) , which are the same for each system. At each instant of time the microscopic variable X has a different value for each of the systems in the ensemble, despite the fact that they are macroscopically identical. Think, for example, of two dispersions with identical temperature, pressure, concentration..., and of X as the set of position coordinates of the Brownian particles. Photographs of the two systems, taken at equal times, to determine the instantaneous positions of the Brownian particles, will be different for the two systems, although the two systems are macroscopically identical. There are many "microscopic realizations" of a single macroscopic state. The stochastic variable X is a function of time, which is set by the interactions between the particles and the initial state of the system. The microscopic variable X is represented by a single point in the m-dimensional space, usually referred to as the phase space. This point describes a curve in that space as time proceeds (see fig.4.1a). An instantaneous microscopic state of the ensemble is thus represented by a set of (infinitely) many points in the m-dimensional space. Now, the probability for a system to be in a microscopic state pertaining to some given value X0 of X, is proportional to the number of systems in the ensemble having that particular value of X. The density of points in the neighbourhood of a specific value Xo of X, at a given time t, is thus proportional to P ( X - X0, t) (fig.4. lb). Let W be an arbitrary volume in the m-dimensional space to which X belongs, and let c3W denote the (closed) boundary of W. The change of the "number of points" inside the volume W is determined by the flux of points through the boundary 0W, which is the integral of the "point current density" j - (dX/dt) P(X, t)

X

Chapter 4. 174

m axes |

SMALL LARfiE

: : : . . . - . . . .~.~. .: .;~,. . 0 .

�9 ~

| Figure 4.1" (a) The stochastic variable X is a point in m-dimensional phase space, and describes a curve in that space with time. (b) Each point is an instantaneous realization of X for a single system in the ensemble of systems. The point density is proportional to the pdf P(X, t) at the particular time t considered. For a non-equilibrium system the point density changes with time.

ranging over the surface OW. Formally,

~-~ dX P(X, t) = - w dS �9 ~-~X P(X, t) , (4.1)

where dS is an infinitesimally small, (m - 1)-dimensional surface element, outward-normal to 0W (see fig.4.2). The minus sign on the right hand-side of eq.(4.1) is added, since the number of points inside W decreases when dX/dt is parallel to dS.

Now suppose that it is possible to relate dX/dt to the instantaneous value of X, using a physical model for the particular variable at hand. That is, suppose there is a relation of the form,

d x ( t ) - H(X( t ) ) , (4.2)

where H may be a function, functional or any other type of operator (which does not contain time derivatives) working on X.

The time derivative on the left hand-side of eq.(4.1) can be taken inside the integral and the integral on the right hand-side can be written as a volume integral over ]41 with the use of Gauss's integral theorem (in m dimensions).


dS

_--j

~W

%-

Figure 4.2: W is an arbitrary volume in the m-dimensional space with a closed boundary OW. dS is an (m - 1)-dimensional vector with infinitesimal size dS, normal to OW pointing outward of the volume W. The current density j of points is equal to (dX/dt) P(X, t).

This gives, together with eq.(4.2),

--~ P(X'ot t ) -

- fw dX V ~ - [ H ( X ) P ( X , t)] , (4.3)

where V~ is the m-dimensional gradient operator with respect to X. Since W is an arbitrary volume, the integrands in the above equation must be equal. This can be seen by choosing W centered at an arbitrary point X with a diminishing volume (for example an m-dimensional sphere with its center at X and a radius that tends to zero). Then both integrals in the above equation reduce to the value of the integrands at X multiplied by the volume of W. It thus follows that,

0 P (X t) - Z~ P(X, t) Ot '

(4.4)

where the operator/~ acts on the variable X, and is given by,

s - - V ~ . [ H ( X ) ( . . - ) ] , (4.5)

where the dots (-..) stand for an arbitrary function of X.

176 Chapter 4.

Once a suitable H in eq.(4.2) is found, the equation of motion for the pdf of X follows immediately from eqs.(4.4,5). The conditional pdf P(X, t [ Xo, to) to find a value X at time t, given that at time t - to the value was X0, is the solution of the equation of motion (4.4) subject to the initial condition,

P(X, t - to) - 5 ( X - Xo), (4.6)

with (5 the m-dimensional delta distribution. To obtain the correlation function < f (X(to)) g (X(t)) > for two func-

tions f and # of X, the explicit solution of the equation of motion with the initial condition (4.6) can be used in the expression,

< f (X(to)) g (X(t)) >=fdXfdXo f(Xo)g(X) P(Xo, to) P(X, t I Xo, to), (4.7)

where P(Xo, to) is the pdf for an instantaneous value Xo at time to. In an equilibrium system this pdf is independent of time.

Altematively, the following operator exponential expression can be used,

< f (X(to)) g (X(t)) >-[dXg(X) exp{/~ (t - to)} [ f (X)P(X, to) ] , i t /

(4.8) for which only the form of the operator/~ in eq.(4.5) needs be known.

For a derivation of the above two expressions for the correlation function of f and g, see subsection 1.3.2 in the introductory chapter. Remember that the ensemble averages here also involve averaging with respect to initial conditions.

The equation of motion for the pdf of the position and momentum coordinates of spherical Brownian particles on the Fokker-Planck time scale, and for the pdf of the position coordinates on the diffusive time scale are derived in sections 4.3 and 4.4, respectively. The equations of motion are solved in section 4.5 for non-interacting particles. The effects of shear flow and sedimentation are analysed in sections 4.6 and 4.7, respectively. Section 4.8 is concerned with the dynamics of rigid rod like Brownian particles.

In the derivation of the fundamental equations of motion, interaction forces between the Brownian particles play an essential role through the relation that specifies the function H in eq.(4.2). For colloidal systems there is an essential contribution to the total interaction forces which is not present in molecular systems. Apart from direct interactions which are also present in molecular systems, arising from a position coordinate dependent potential energy,

4.2. Hydrodynamic Interaction 177

there are interactions between the Brownian particles which are mediated via the fluid in which the Brownian particles are immersed. This so-called hydrodynamic interaction is discussed in the following section.

4.2 A Primer on Hydrodynamic Interaction

Besides direct interactions, which are also present in molecular systems, there are so-called hydrodynamic interactions between Brownian particles immersed in a fluid. The origin of the interactions is not difficult to understand. A Brownian particle that attained a velocity at a certain time induces a fluid flow in the solvent. This fluid flow propagates through the solvent and encounters other Brownian particles (see fig.4.3), which are thus affected in their motion, giving rise to an interaction which is determined by both their velocities and positions.

These interactions can be described on the basis of the Navier-Stokes equation for the solvent, provided that the Brownian particles are very large in comparison to the solvent molecules, so that they may be treated as macroscopic bodies. This is a difficult hydrodynamic problem to which chapter 5 on hydrodynamics is devoted. In the present section, only generic features of hydrodynamic interaction are discussed. Moreover, the discussion here is restricted to Brownian particles with a spherical geometry. Rod like Brownian particles are considered in section 4.8.

First of all, it turns out that the propagation of fluid flow disturbances, created by the movement of a Brownian particle, is so fast, that the phase space coordinates of the Brownian particles hardly change during the time interval that a disturbance takes to reach other Brownian particles. It is then a good approximation to assume that the hydrodynamic interaction is instantaneous. The fluid flow resulting from the movement of a Brownian particle can be

Figure 4.3" Hydrodynamic interaction. Particle A induces a fluid flow in the solvent which affects particle B in its motion.

178 Chapter 4.

thought of as existing, without any time delay, throughout the entire fluid. In this approximation the fluid flow at a given time is a function of the velocities and positions of all the Brownian particles at that particular instant only. As a result, the hydrodynamic interaction forces are functions of the instantaneous momentum and position coordinates. Thus, the force F h that the solvent exerts on the i th Brownian particle is set by the instantaneous values of the momentum and position coordinates of all the Brownian particles,

F h -- F h (pl(t) ,""", pN(t ) , r l ( t ) , ' " , rg ( t ) ) .

The superscript "h" here stands for "hydrodynamic". In section 5.3 in the chapter on hydrodynamics it is shown that the pro-

pagation velocity of shear- and sound waves is not large enough to ensure the validity of instantaneous hydrodynamic interaction on the Fokker-Planck time scale. On the Fokker-Planck time scale the above approximation in questionable. On the Brownian time scale, however, the validity of instantaneous hydrodynamic interaction is beyond doubt.

Secondly, it turns out that the Navier-Stokes equation, for the typical sizes and velocities of the Brownian particles and a typical shear viscosity of the solvent, can be linearized with respect to the fluid flow velocity. More precisely, the Reynolds number for the hydrodynamic problem considered here is small, which allows the neglect of the non-linear inertial terms in the Navier-Stokes equation (see section 5.5 in the chapter on hydrodynamics). As a result, the hydrodynamic forces are linear functions of the velocities v j of the Brownian particles. Hence, the above expression reduces to a linear form,

N

Fh -- -- E T i j ( r l , r 2 , ' " , r N ) " v j . j = l

(4.9)

Here, the 3 x 3-dimensional microscopic friction matrices Tij are introduced. They are functions of all the position coordinates, but independent of the velocities. The mathematical problem concerned with hydrodynamic interaction is the explicit calculation of these matrices. A minus sign is added to the right hand-side of the above expression because the hydrodynamic force on a particle tends to be directed in the opposite direction to its velocity.

In very dilute suspensions, where distances between (the majority of) Brownian particles are very large, hydrodynamic interaction is unimportant, just as direct interaction. In that case eq.(4.9) reduces to,

F~ - - T v i , (4.10)

4.3. Fokker-Planck Equation 179

with 7 the friction coefficient of an isolated sphere, which is shown in subsection 5.7.1 in the chapter on hydrodynamics to be equal to,

7 - 67rr/oa , (4.11)

with ~7o the shear viscosity of the solvent and a the radius of the Brownian particles.

The "off-diagonal" friction matrices, Tij with i ~ j , describe the hydrodynamic interaction of particle j with particle i. The "diagonal" friction matrices T , also depend on hydrodynamic interaction, and are not just equal to the friction coefficient of an isolated sphere, as one might think on first sight. The fluid flow that is the result of movement of the i th sphere "reflects" from the other Brownian particles back to the i th sphere, thus exerting a force on that particle in addition to the friction force of an isolated particle.

Eq.(4.9) is used in the present chapter without specifying the explicit position coordinate dependences of the microscopic friction matrices. These are established in sections 5.8,10,12 in the chapter on hydrodynamics.

4.3 The Fokker-Planck Equation

The Fokker-Planck equation, in the present context, is the equation of motion for the pdf of the momentum and position coordinates of all the Brownian particles in the system. This equation of motion is valid on the Fokker-Planck time scale, where the phase space coordinates of the solvent molecules are long relaxed (see chapter 2).

In colloid science the phrase "Fokker-Planck equation" refers explicitly to the equation of motion for the pdf of the momentum and position coordinates. In more general texts this nomenclature is usually reserved for a whole class of equations of motion, to which all of the equations of motion that are derived in the present chapter belong. Here, we reserve the name "Fokker-Planck equation" to the equation of motion derived in the present section, and ascribe to each equation of motion that is treated in subsequent sections its own name.

The stochastic variable here is,

X -- (P l , P 2 , ' " , PN, rl, r 2 , . . . , rN) , (4.12)

with pj (rj) the momentum (position) coordinate of the jth Brownian particle. This is a 6N-dimensional vector, with N the number of Brownian particles in the system.

180 Chapter 4.

As we have seen in the introduction, once the function H in eq.(4.2) can be specified, the equation of motion can be written down immediately using eqs.(4.4,5). Now, d p i / d t - F~, and dr~/dt - p i / M , with M the mass of a Brownian particle and Fi the total force on the i th particle. Hence,

d X(t) - (Fl( t ) , F2(t) FN(t), pl(t) p2(t) _ _ �9 . �9

dt ' ' M ' M ' pN(t))M " (4.13)

The trick is thus to express the total forces Fi in terms of the momentum and position coordinates.

The total force on the i t h Brownian particle is the sum of three forces. The hydrodynamic force which the solvent exerts on the particle is given in terms of momentum and position coordinates by eq.(4.9) (notice that vj - p j /M) . The direct force is equal to -V,~ (I), where V,~ is the gradient operator with respect to r~, and (I) is the total potential energy of the assembly of Brownian particles, which is a function of the position coordinates. As we are considering a description on a coarsened time scale (the Fokker-Planck time scale), there may be additional forces, the form of which is, as yet, unknown. That such additional forces must be present can be seen as follows. Consider a very dilute suspension in which interactions between the Brownian particles are absent, that is, in which both hydrodynamic and direct interactions are absent. Suppose that the suspension is inhomogeneous in density. Despite the absence of both hydrodynamic and direct interactions, the system evolves towards a state with a homogeneous density. The driving force for that process is missing when only hydrodynamic and direct forces are considered. The additional force is denoted here as F*. Hence,

N Pj

Fi = - ~ T i j ( r l , . . . , r N ) " M j--1

V ~ , r + F*. (4.14)

Now using that,

N - E [ % , . ] ,

i=1

with Vp~ the gradient operator with respect to pi, the equation of motion for the pdf P - P ( P l , �9 �9 � 9 P N , r l , - �9 �9 r N , t ) follows from eqs.(4.4,5) as,

O__ - ~g [ _ M p ' " V~,P - Vp,. (F* P) i--1

4.3. Fokker-Planck Equation 181

{( }] +Vp,. [V~,r + y~ Tij" P �9 j=l

(4.15)

The additional force F~ can now be specified as follows. For very long times the system attains equilibrium and the pdf is proportional to the exponential Boltzmann distribution. That is,

lira P ( P I , " ' , r N , t) ,,~ exp --/3 ~ + ~ 2 M j ' t---*oo /=1

(4.16)

where/3 - 1/kBT (kB is Boltzmann's constant and T is the temperature). Since the time derivative on the left hand-side of eq.(4.15) is zero in equilibrium, the right hand-side must also be equal to zero in case the pdf is given by the above Boltzmann form. As is easily verified this condition is satisfied for the following form of F~,

N

F* - -/~BT E Ti j . X7p~ In{P}. j=l

(4.17)

Substitution of this result into eq.(4.15) finally gives the equation of motion that we set out to derive,

O P ( P l , " ' , r N , t) -- / ~ F P P ( P l , ' ' ' , r N , t ) ,

where ~,FP is the Fokker-Planck operator,

(4.18)

s N[ p, y~ - ~ . V~,(...) (4.19) i=1

{( )) }] + Vp,. [V,,r + y~ T i j . --~ + k s T Vp, ( . . . ) . j = l

As it stands, this is a quite complicated equation. It can be rewritten in a somewhat more elegant form by introducing the so-called super vector notation. The "super vectors" are the two 3N-dimensional vectors,

P -- (Pl , P 2 , ' " , PN) , (4.20) r - ( r a , r2 , . . . , rN) �9 (4.21)

182 Chapter 4.

The gradient operators with respect to these super vectors are 3N-dimensional gradient operators,

Vp - (Vv,, Vp~, . . . , VpN), (4.22)

V~ - (V~,V~ 2 , - . - ,V~ N ) . (4.23)

The microscopic friction matrix in super vector notation is,

T l l "~e'12 "'" T I N T21 ~'22 "'" "~2N

T - . . . . (4.24)

"rN1 "~N2 "'" TNN

The Fokker-Planck operator now reads,

,~FP(''') = P V,.(.-.) (4.25) M

p / { oj +-,- + +

The explicit notation of indices is lost in the super vector notation. The Fokker-Planck equation can be used to describe Brownian motion

(on the Fokker-Planck time scale) of a single particle in an external potential. Hydrodynamic interaction is absent in this case, and the potential energy �9 is now the potential due to the external field. The Fokker-Planck equation now reduces to,

O p ( p , - r, t) / ~ e P (P r, t ) , (4.26)

with r the position coordinate of the particle, p its momentum, and,

/~ 'v ( ' " ) = P V,.( . . .) (4.27) M

P +

Here, 7 is the friction coefficient of a sphere (see eq.(4.11)). The superscript "0" on the Fokker-Planck operator here refers to the neglect of interactions between the Brownian particles. For a freely diffusing particle, where the external potential ~ is zero, this equation should confirm the results that were obtained in chapter 2 on the basis of the Langevin equation (see section 4.5). The Fokker-Planck equation (4.26,27) for a single particle in an external potential is also referred to as Kramer's equation.

4.4. Smoluchowski Equation 183

4.4 The Smoluchowski Equation

The Smoluchowski equation is the equation of motion for the pdf of the position coordinates of the Brownian particles and applies on the Brownian (or diffusive) time scale. The momentum coordinates of the Brownian particles are relaxed to thermal equilibrium with the heat bath of solvent molecules on this time scale. As a consequence, the total force on each Brownian particle is zero (see chapter 2), that is, the friction force which the fluid exerts on a Brownian particle is balanced by the other forces acting on that particle.

The stochastic variable is now the set of position coordinates,

X -- ( r l , r 2 , . . . , r N ) . (4.28)

Hence, d x (Pl P2 PN) dt (t) - M ' M " " ' M "

(4.29)

To find the function H in eq.(4.2), the momentum coordinates must be expressed in terms of the position coordinates. This relation is set by the balance of the hydrodynamic forces and the other forces, that is, the total force F i on the left hand-side of eq.(4.14) is equal to zero on the Brownian time scale,

N Pj

0 - - ~ T i j ( r l , . . . , r N ) . M j--1

V~i(I)(rl,... , r N ) + F B~ , (4.30)

where the additional force F~, in the present context, is usually referred to as the Brownian force, which is denoted as F~ ~. In order to express the momentum coordinates in terms of the positions, the above equation must be rewritten in the super vector notation that was introduced in the previous section. Using the notations (4.20-24) and,

F B ~ - (F1B~, F2 B~, . . . , F B~) , (4.31)

eq.(4.30) is rewritten as,

0 - - T ( r ) P M

V ~ ( r ) + F B~ . (4.32)

Hence, P

M - r - X I r / � 9 + (4.33)

184 Chapter 4.

where T -1 is the inverse of T. At this stage it is convenient to introduce the 3 • 3-dimensional microscopic diffusion matrices Dij, which are defined as,

Dll D~2 . - - D I N

D21 D22 "" D2N T -1 - / 3 D - / 3 . . . . (4.34)

DN~ DN2 "" DNN

Notice that/~ Dij is not simply the inverse of Tij " in the ij th microscopic diffusion matrix all microscopic friction matrices mix up, since D is the inverse of the entire matrix T in eq.(4.24).

Eqs.(4.29,33) identify the function H in eq.(4.2). We thus immediately obtain, from eqs.(4.4,5),

(9 P(r , t) - V~. ~D(r ) . [([V~(I)] - F n~) P(r , t ) ] . Ot

(4.35)

The Brownian force is now determined, just as for the Fokker-Planck equation, from the equilibrium form of the pdf,

lim P(r , t) ,-~ exp {-/~r . (4.36) t--*c~

In equilibrium, the term within the round brackets in eq.(4.35) is thus equal to zero for the particular form (4.36) of the pdf. From this requirement the following form for the Brownian force is found,

F s" = - k B T V ~ ln{P}. (4.37)

The Smoluchowski equation is thus finally found, in super vector notation, as,

0 ~-~P(r, t) - s P(r , t ) , (4.38)

where/~s is the Smoluchowski operator,

s ") - V,-D(r)-[/~[V,r + V,(--.)] . (4.39)

In terms of the original position coordinates, this equation reads,

ff-~tP(rl,-",rN, t) -- ~ s P ( r , ' " , r N , t ) , (4.40)

4.4. Smoluchowski Equation 185

and,

N

~ ( - - ) - E v~,. D,j. [~[v~.~](...)+ v~.(. . .)], i,j=l

(4.41)

where both the D~j 's and ~ are functions of all the position coordinates. Like the Fokker-Planck equation, the Smoluchowski equation can be used

to describe Brownian motion (on the diffusive time scale) of a single particle in an external field. Again, hydrodynamic interaction is absent in this case, and the potential energy r is now the potential due to the external field. The Smoluchowski equation now reduces to,

0 O---~P(r, t) - s176 s P(r , t ) , (4.42)

with r the position coordinate of the particle, and,

z2~(...) - D0 V~. [~[V~r V~( . . . ) ] , (4.43)

where,

1 kBT D o - /37 = 67rr/oa' (4.44)

is the Stokes-Einstein diffusion coefficient that we have already met in chapter 2 on diffusion of non-interacting particles. For a freely diffusing particle, where the extemal potential r is zero, this equation in shown in section 4.5 to confirm the results of chapter 2 for diffusion on the Brownian time scale.

Notice that on neglect of hydrodynamic interaction, according to eqs.(4.9) and (4.10), the microscopic friction matrix in super vector notation is a diagonal matrix,

-ri o . . . o o - A . . . o

T - . . . . , (4.45)

0 0 . . . 7I

where I is the 3 x 3-dimensional identity matrix and 0 the 3 x 3-dimensional zero matrix (with only zero's as entries). The microscopic diffusion matrix is

186 Chapter 4.

then also a diagonal matrix,

D o i 0 ... 0 D o i . . - 0

[ A A ~ \ L , - . . . . �9 t ~ . ' , , )

o o .. . D'oi

Diffusion on the Fokker-Planck and Brownian time scale of non-interacting Brownian particles is discussed in the following section.

4 . 5 D i f f u s i o n o f n o n - I n t e r a c t i n g P a r t i c l e s

An important correlation function, which is analysed for interacting particles in chapter 6 on diffusion, and which was already introduced in the previous chapter on light scattering, is the dynamic structure factor,

S,(k, t) - < exp{ik. (r(t - O) - r(t))} > . (4.47)

The subscript "s" here refers to the self dynamic structure factor, to make the distinction with the collective dynamic structure factor, although for dilute suspensions of non-interacting particles both are identical. There is a difference between the two dynamic structure factors only for more concentrated systems where interaction between the Brownian particles is of importance (see the discussion in subsections 3.8.2, 3.9.2 and chapter 6 on diffusion). In the following two subsections, the dynamic structure factor is calculated on the basis of eqs.(4.7,8). The conditional pdf's are calculated here on the basis of equations of motion for dilute and homogeneous suspensions in equilibrium without an external field. Notice that for the calculation of S,, the functions f and # in eqs.(4.7,8) are equal to,

f ( p , r ) - e x p { i k . r } ,

g(p,r ) - e x p { - i k , r} . (4.48)

The equations of motion for pdf's of non-interacting particles belong to a single class of differential equations �9 linear Fokker-Planck equations. The Smoluchowski equation for non-interacting particles in shear flow, which is considered later, also belongs to this class of equations. The following mathematical subsection deals with the solution of such linear Fokker-Planck

4.5. Free Diffusion 187

equations in their general form. The solutions of the relevant equations of motion in the present context follow from the expression for the general solution. The Smoluchowski equation, however, is a relatively simple equation of motion, which is solved without resort to the general solution. Those readers who are not interested in diffusion on the Fokker-Planck time scale, nor in diffusion on the Brownian time scale in a sheared system, may skip the next subsection and read subsection 4.5.2 on the Smoluchowski equation independently.

It is also shown in the following subsection how to derive equations of motion for certain ensemble averaged quantities directly from the equations of motion for pdf's, without first solving these explicitly. For example, expressions for the mean squared displacement can be obtained in this way, which were also analysed in chapter 2 on the basis of the Langevin equation.

4.5.1 Linear Fokker-Planck Equations

Consider the following differential equation for the pdf P of X,

O P (X t) - - V ~ - ( A . XP(X, t)) - V ~ - ( B . V~P(X, t)) (4.49) 0t '

with A and B matrices which are independent of X but may depend on time. The matrix B may be assumed symmetric without loss of generality, since

1 (B-+-B T) a, for arbitrary vectors a (the superscript "T" a - B . a - 5a. stands for "the transpose of"). For such matrices A and B, the differential equation (4.49) is referred to as a linear Fokker-Planck equation. In case these matrices are X-dependent, the resulting equation is referred to as non-linear. For interacting particles, the equations of motion which were derived in the previous sections are non-linear Fokker-Planck equations through their hydrodynamic and direct interaction terms. With the neglect of these interaction terms, the resulting equations of motion become linear.

Equations of motion for various moments can be obtained directly from the Fokker-Planck equation, without solving it first. Multiplying both sides with X or XX and integrating with respect to X gives, after performing some partial integrations (see exercise 4.1),

d d-7 < X(t) > - A. < X(t) > , (4.50)

d d~ < X(t)X(t) > - - 2 B + A. < X(t)X(t) > + < X(t)X(t) > . A T .

(4.51)

188 Chapter 4.

The equation of motion for the covariance matrix,

D - < (X( t ) - < X(t) > ) ( X ( t ) - < X(t) >) > , (4.52)

is thus,

d d~ D = - 2 B + A. D + D . A T . (4.53)

We seek a solution of eq.(4.49) of a Gaussian form,

1 I ( X - m) . M -1. ( X - m)} (4.54) P (X, t ) - (27r)~/2~/detM exp{-~

where M is an as yet unknown symmetric matrix and m an unknown vector, both of which may be time dependent.

That this Gaussian form is indeed a solution of the Fokker-Planck equation (4.49) can be verified by substitution. This is most easily done in terms of Fourier transforms. The Fourier transform of the Fokker-Planck equation (4.49) is,

O P ( k , t) - k . (A. VkP(k, t ) ) + k. B . k P(k, t) (4.55) Ot

with k the Fourier variable conjugate to X, and Vk the gradient operator with respect to k. The Fourier transform of the Gaussian form (4.54) is equal to (see subsection 1.3.4 in the introductory chapter),

1 P(k, t) - exp{- ik , m} e x p { - ~ k . M . k}. (4.56)

Substitution of this Fourier transform into the Fourier transformed Fokker- Planck equation (4.55) gives,

_ i k . d m lk . dM 1 [A M + M A T] k + k B k dt 2 - - -~- .k- - i k . A - m - ~ k . . . . . . . (4.57)

Since the vector k is an arbitrary vector, the linear and bilinear "coefficients" of k must be equal. Hence,

dm

dt dM dt

A. m , (4.58)

- 2 B + A- M + M . A T . (4.59)

4.5. Free D i f f u s i o n 189

These are precisely the equations of motion for < X(t) > and the covariance matrix D in (4.50) and (4.53), respectively. These two equations of motion are thus satisfied, and rn = < X > and M = D. This shows that the Gaussian form (4.54) is indeed a solution of the Fokker-Planck equation.

This particular solution should satisfy the initial condition,

P (X, t - O) - 5 ( X - Xo), (4.60)

with X0 - X(t - 0) a prescribed value of X at time t - 0. condition for the Fourier transform is thus,

The initial

P(k , t - O) - e x p { - i k . Xo}. (4.61)

That the above Gaussian form indeed satisfies this initial condition follows immediately from eq.(4.56), together with M(t - 0) - D( t - 0) - 0 and m( t - 0) - < X(t - 0) > - X0.

Thus, the Gaussian form in eq.(4.54) is the solution of the Fokker-Planck equation (4.49) subject to the initial condition (4.60). Moreover, the equations of motion (4.58,59) can be used to calculate both the mean m(t) - < X(t) > and the covariance matrix M (t) - D (t)

4.5.2 Diffusion on the Brownian Time Scale

The Smoluchowski equation for non-interacting particles is a relatively simple equation, which can be solved without having to resort to the preceding mathematical subsection.

Let us first calculate the structure factor from eq.(4.8). The pdf P ( X - r) in eq.(4.8) is the equilibrium pdf, which is equal to P~q (r) - 1/V, with V the volume of the system. The fortunate fact that allows the explicit evaluation of the integral in eq.(4.8) is that the function exp{ik, r} • peq is all eigenfunction of the Smoluchowski operator. The Smoluchowski equation (4.42,43) without the external potential reduces to,

0 P(r , t) - Do V~ P ( r t) Ot ' "

(4.62)

The initial condition here is,

P ( r , t ) - - r o ) , (4.63)

190 Chapter 4.

where ro - r(t - 0) is the initial value of the position coordinate of the Brownian particle. It is easily verified that,

Z~ (exp{ik-r} • P~q) - Do V~ (exp{ik. r} x P~q) = -Do k 2 (exp{ik. r} x P~q)

Since the operator exponential is formally defined by its Taylor expansion, this implies that,

exp { /~ t} (exp{ik. r} x P~q) - exp {-Do k2t} (exp{ik. r } x P~q)

According to eq.(4.8), the dynamic structure factor is thus simply equal to,

S,(k, t) - exp {-Do k2t} . (4.64)

Alternatively, the dynamic structure factor can be calculated from eq.(4.7), which reads for the present case,

f 1 / S~(k, t) - dro ~ exp{ik �9 ro} dr exp{- ik �9 r}P(r, t I ro, t - 0).

(4.65) The integral with respect to r is nothing but the Fourier transform of the conditional pdf with respect to r, which is easily calculated from the Fourier transformed Smoluchowski equation (4.62),

0~P(k, t) - -Do P(k, t) . (4.66) k 2

The initial condition for the Fourier transform follows from eq.(4.63),

P(k, t - 0) - f dr exp{- ik , r} 6 ( r - ro) - exp{- ik , ro}.

The solution of eq.(4.66) subject to this initial condition is,

P(k, t) - exp{- ik , ro} exp{-Do k2t} . (4.67)

Substitution of this result into eq.(4.65) immediately leads to the result (4.64) for the dynamic structure factor.

The solution of the Smoluchowski equation can be obtained from the expression (4.67) for its Fourier transform, by Fourier inversion,

1 f dk exp{ik- r} P(k, t) P(r, t l ro , t - 0 ) = (27r)3

1 [ I r - r 0 1 2 ] = (47rDot)3/2 exp - 4Dot , (4.68)

4.5. FreeDiffusion 191

which is precisely the result that we found on the basis of the Langevin equation in chapter 2 (see eq.(2.39)).

An expression for the mean squared displacement <1 ro - r(t) 12> can be found in several alternative ways. The first method is simply the integration of l ro - r 12 • the pdf in eq.(4.68). Secondly, the mean squared displacement can be found from the dynamic structure factor by expanding the defining equation (4.47) with respect to the wavevector (see also exercise 3.9),

1 k2 [2 S , (k , t ) - 1 - ~ < [ r ( t - 0 ) - r ( t ) > + . . . . (4.69)

Comparison with eq.(4.64) for S~ yields,

<1 r(t - 0) - r(t) 12> - 6 Do t , (4.70)

in accordance with the result obtained in chapter 2 (see eq.(2.21)). A third way to calculate the mean squared displacement is directly from the equation of motion for the pdf. Multiplying both sides of eq.(4.62) with r and r 2, respectively, and integration with respect to r yields (see exercise 4.1),

d dS < r ( t ) > - 0 , (4.71)

d d--t < r2(t) > - 6Do. (4.72)

Integration leads to the result in eq.(4.70).

4.5.3 Diffusion on the Fokker-Planck Time Scale

The Fokker-Plank equation (4.26,27) without an external potential reads,

aS P ( p ' r ' t ) - - ~ . V ~ P + T V v . ~ + f l - Vv P , (4.73)

and is subject to the initial condition,

P(p , r, t - 0) - 6(p - P o ) 6 ( r - ro), (4.74)

with Po and ro the initial values of the momentum and position coordinates of the Brownian particle, respectively.

The equilibrium pdf P ( X - (p, r)) is given by the Boltzmann exponential,

1 { / P ( o , r ) = V (27rMkBT) 3/2 exp - /3~-~ . (4.75)

192 Chapter 4.

A calculation of S, from eq.(4.8) on the Fokker-Planck time scale is much more difficult than on the Brownian time scale, because now we are not dealing with simple eigenfunctions. We evaluate S, on the basis of expression (4.7). The dynamic structure factor is equal, according to eq.(4.7), to the following integral,

&(k, t ) 1 (4.76) - / d r o exp{ik-ro} V (27rMkBT) 3/2

x epo e p P ( p , k , t I po, ro, t - 0),

where the k-dependence refers to the Fourier transformation with respect to r. Now let us define the following 6-m dimensional Fourier transform,

P(kp, k, t I p0, ro, t - 0) -

/ dp f dr exp{-ikp �9 p} exp{ - ik �9 r}P(p , r, t l po, ro, t - 0)

- f dp exp{-ikp �9 p ) P ( p , k, t [ po, ro, t - 0), (4.77)

where kp is the Fourier variable conjugate to p. This is nothing but the usual Fourier transform with respect to X - (p, r), where the Fourier variable is split into two 3-dimensional vectors kp and k. For the calculation of 5'8 we need this Fourier transform for kp = O. The Fourier transform can be calculated once the solution of the Fokker-Planck equation is known.

For the particular Fokker-Planck equation (4.73), where X - (p, r), the matrices A and B appearing in the more general Fokker-Planck equation (4.49) are given by,

1 i 0 ' (4.78)

and,

B ( t ) - ( - ~ I O ) 0 0 '

(4.79)

where 0 is the 3 x 3-dimensional zero matrix. The equation of motion (4.50) for the mean is thus,

d - ' - ~ - - ~ , ~ �9 (4.80)

4.5. FreeDiffusion 193

The solution of this equation is, 7 < p(t) > - Po e x p { - ~ t } ,

~ [ ~, ~ ] < r ( t ) > - r o + - P o 1-exp{- - rT t} . "7

Equation (4.51) for the average of the bilinear product yields,

d d---t < p( t )p( t )> = d d--t < r(t)p(t) > = d d~ < r(t)r(t) > =

27 M

< p(t)p(t) > -t-2fl-1"),I,

1 -y < p(t)p(t) > M

1

(4.81)

(4.82)

(4.83)

< r(t)p(t) > , (4.84)

[< r(t)p(t) > + < p(t)r(t) > ] . (4.85)

These equations of motion are easily solved,

M [ 2 7 ] < p(t)p(t) > - t--~- 1 - e x p { - ~ t } + popo exp{-~-~?t},(4.86)

< r(t)p(t) > - ropoexp{-~ t} + e x p { - ~ t } - 1

1 [ 23' 7 ] - PoPo- e x p { - ~ t } - e x p { - ~ t , (4.87) "Y

1 [ ~ ] < r(t)r(t) > - roro + (ropo + poro)~ 1 - e x p { - ~ t }

1 [oxp~ ~ 1] ~ + PoPo~-~ ~ t } -

2M "), 1 23' 1] [1 exp{ 7 + 17 ~ ( ~ t - ~ [exp{- ~ t } - - 2 - - ~ t }]) .(4.88)

These results were also obtained on the basis of the Langevin equation in chapter 2.

For the calculation of S, we need only to consider the expression,

< ( r ( t ) - < r(t) >) ( r ( t ) - < r(t) >) > - (4.89)

. .~':~2( 2M 7~t_: 1 [ e x p { - 2 7 ~ t } - l J - 2 1 1 - e x p { - S t } ] )

which follows from eqs.(4.88,82). The relevant Fourier transform in the expression (4.77) for the dynamic structure factor is thus equal to,

o ,, oro

194 Chapter 4.

(Ss-1)xlO ~ 0 .......... ~ I i

-0.5

-1-

-1.5

- 2 '

0 5 10 15 20 t[ns]

Figure 4.4: The initial decay of the serf dynamic structure factor on the Fokker-Planck time scale ( . . . ) and the Brownian time scale (~) . Typical values chosen for Dok 2 and 7 / M a r e 10 +4 s -1 arid 10 +8 s -1, respectively. The plot on the Brownian time scale should be considered as an extrapolation to small times, since the Brownian time scale is much larger than 20 ns.

{ o o )} { 1[ ' 7 ] } = exp{-ik, ro} exp - i k ' p o - 1 - e x p { - ~ t }

{ M k2 7 l[exp{ -23' 1] [1 { - ~ t } . xexp - ~ ( - ~ t - ~ ~ - t } - - 2 - exp "7 }])

Substitution of this result into eq.(4.76) and performing the Gaussian po- integration finally leads to the following relatively simple expression for the self dynamic structure factor that we set out to calculate,

S,(k, t) - exp - D o k 2 t + - - e x p { - ~ t } - i . 7

(4.90)

For times t >> M / 7 and Dok 2 << 7/M, this expression reduces, as it should, to eq.(4.64), which is valid on the Brownian time scale. The latter inequality here expresses the separation between the Brownian and Fokker-Planck time scales.

4.6. Smoluchowski Equation with Shear Flow 195

Figure 4.4 shows a plot of the initial decay of the dynamic structure factor on both the Brownian and the Fokker-Planck time scale. The initial slope of S, versus time is zero on the Fokker-Planck time scale �9 this is the "ballistic" regime, where the mean squared displacement is equal to <[ po/M [2> xt 2. The Brownian time scale is beyond the ballistic regime, so that times are al ways much larger than M/7. For those times the mean squared displacement is linear in time, which is the origin of the non-zero slope of S, versus time on the Brownian time scale.

4.6 The Smoluchowski Equation with Simple Shear Flow

Here we consider a system of Brownian particles which is subjected to simple shear flow. The suspension is thought of as being confined between two parallel fiat plates, which are moved in opposite directions with a certain velocity. In the absence of the Brownian particles this would induce a linearly varying fluid flow velocity for not too large relative velocities of the two plates. The coordinate system is chosen such that the fluid flow velocity u0 at a point r is given by (see also section 2.7),

uo(r) - F- r , (4.91)

with F the velocity gradient matrix,

O 1 0 / o o o ,

0 0 0 (4.92)

where -~ is the shear rate. This is a fluid flow in the z-direction with its gradient in the y-direction. A fluid flow described by eqs.(4.91,92) is called a simple shear flow. Brownian particles immersed in a fluid in simple shear flow are affected in their thermal motion by the fluid flow.

The nature of the hydrodynamic interaction is changed due to the shear flow. This is discussed in the following subsection in a qualitative manner. Quantitative results are derived in section 5.13 in the chapter on hydrodynamics.

The Smoluchowski equation changes, not only due to the different hydrodynamic interaction, but also as a consequence of the displacement of

196 Chapter 4.

Figure 4.5" The disturbance of the fluid flow in the neighbourhood of a Brownian particle B due to rotation of particle A, as a result o f the shear flow.

Brownian particles due to the fluid flow. The displacement of a Brownian particle depends on the local fluid flow velocity, which in turn depends on the position of that particle. The Smoluchowski equation in its most general form, including both hydrodynamic and direct interactions, is derived in subsection 4.6.2.

Even for non-interacting particles, Brownian motion is severely affected by shear flow. Diffusion in shear flow, for non-interacting Brownian particles, is analysed in subsection 4.6.3. The Smoluchowski equation for this case belongs to the class of linear Fokker-Planck equations discussed in subsection 4.5.1. The results from that subsection are used to calculate the solution of the Smoluchowski equation with shear flow.

4.6.1 Hydrodynamic Interaction in Shear Flow

The relation (4.9) between the force F/h which the fluid exerts on the i th Brownian particle and the velocities vj of all Brownian particles is incomplete for a sheared system. Even if the velocities of all the Brownian particles were equal to zero, the fluid would exert forces on the Brownian particles due to the fact that the fluid velocity is non-zero.

First of all, the velocity vj in eq.(4.9) should be taken relative to the local fluid flow velocity F . r j , that is, vj in eq.(4.9) should be replaced by v j - r . rj. This alone is not sufficient to fully describe the effect of shear flow. In addition, the local fluid flow around each Brownian particle is distorted by the presence of all the other Brownian particles. In particular, the fluid

4.6. Smoluchowsta" Equation with Shear Flow 197

flow velocity gradients cause the Brownian particles to rotate. Each rotating particle induces a fluid flow field which affects the other Brownian particles in their motion (see fig.4.5). The fluid flow in the neighbourhood of a given Brownian particle is thus equal to F . r plus a contribution of the fluid flow disturbance due to the presence of all other Brownian particles. Since the hydrodynamic equations are linear, this disturbance of the fluid flow is linear in F.

The total force that the fluid exerts on the i th Brownian particle can thus be written as,

N

F/h -- - - 2 "][~iJ( r l ' ' ' ' ' rN)" ( V j - F . r j ) + C i ( r l , . . . , rN)" F . (4.93) j= l

The disturbance matrices C~ describe the effect of the fluid flow disturbance as described above. They are matrices of indexrank 3, that is, each element of C~ is characterized by three indices. The double contraction " : " with respect to adjacent indices of Ci and F is thus a vector (see subsection 1.2.1 on notation in the introductory chapter).

In very dilute suspensions, where hydrodynamic interaction is absent, eq.(4.93) reduces to,

Fh -- --7 ( v i - F . ri) . (4.94)

The disturbance matrices C~ are equal to zero in the absence of hydrodynamic interactions.

4.6.2 The Smoluchowski Equat ion with Shear Flow

The derivation of the equation of motion for the pdf of the position coordinates of the Brownian particles on the Brownian time scale proceeds as in section 4.4. The only difference is that the hydrodynamic force, the first term on the right hand-side of eq.(4.30), must be replaced by the expression (4.93). Using eq.(4.37) for the Brownian force, we get,

- rN)- - r . r j + F j = l

-Vr,(I)(rl, "'- , rN) -- kBTV~ l n { P ( r l , - - . , rN, t)}. (4.95)

In order to express the momenta in terms of position coordinates, this equation must be rewritten in terms of the "super vector notation", as discussed in section

198 Chapter 4.

4.3 (eqs.(4.20-24)). The supervector notation must now be extended to include the extra terms. We define,

C s - ( e l �9 r , c 2 �9 r , . . . , c N �9 r ) , (4.96)

and,

F 0 . . . 0 0 F . . . 0

F , - . . . . . (4.97)

0 0 . . . F

The "super vector" velocity gradient matrix F , is a 3N x 3N-dimensional matrix.

Equation (4.95) reads in super vector notation,

0 - - T ( r ) . ( M - Fs" r ) + C ~ ( r ) - V ~ O ( r ) - k , TV~ ln{P( r , t)}. (4.98)

The velocities can now be expressed in terms of the position coordinates as,

dr P = F, �9 r + , ~ - 1 d7 = M ( r ) - [ - V ~ r kBTV~ ln{P(r , t)} + C~(r)] .

(4.99) This identifies the function H via r The general expression (4.4,5) for the equation of motion for the pdf becomes, in super vector notation,

O P( r , t) - /~s P ( r t) Ot ' '

(4.100)

where the Smoluchowski operator is given by,

z~(...) V~. D(r). [fl[V~r + V~(...)] - % . [r~.r (...) + C'~(r ) ( . . . ) ] , (4.101)

where, C'~ - f l D . C, . The microscopic diffusion matrix D is defined in eq.(4.34). In terms of the original position coordinates this equation reads,

O_p , t) ~ . sP( r , ,rN, t) , i)t ( r l , . . . rg , -- . . . (4.102)


and,

. . ) N

E V,.,. Dij" [ /~[V~r V~(- . . ) ] i,j=l

N

- E V,~. [F. rj (..-) + C}" r(...)], (4.103) j = l

where C~ - /~ ~n Djn �9 Cn. With the neglect of hydrodynamic interaction the Smoluchowski operator reduces to,

N N

Ls('" ") - Do E V~,. [~[V~,O](...)+ V, , ( . . . ) ] - E V~,. [r .r j (...)]. j=l j=l

(4.104) This Smoluchowski operator is the sum of two operators, one of which is proportional to the diffusion coefficient and another which is proportional to the shear rate. The operator which is proportional to the diffusion coefficient describes the tendency of the system to resist the effect of the shear flow. The larger the diffusion coefficient relative to the shear rate, the smaller the effect of the shear flow on the pdf. Faster diffusive motion more rapidly counter balances distortions due to the shear flow. In the Smoluchowski operator (4.103), which includes hydrodynamic interaction, there is in addition a mixed term, proportional to the product of the (microscopic) diffusion coefficient and the shear rate. The interplay between diffusion and shear distortion is discussed in some detail in the chapter on diffusion and in the chapter on critical phenomena.

4.6.3 Diffusion of non-Interacting Particles in Shear Flow

Let us now consider the effect of simple shear flow on the self dynamic structure factor (4.47) for a very dilute suspension. The diffusive motion of a Brownian particle is then described by the Smoluchowski equation (4.102,103) in shear flow, with the neglect of both direct and hydrodynamic interactions,

P(r , t) - Do V 2 P(r , t) - V~. [F. r P(r , t)] (4.105) Ot

This is a linear Fokker-Planck equation, meaning that this equation belongs to the class of equations which can be written in the form (4.49). The solution of eq.(4.105), subject to the initial condition,

P(r , t - 0) - 5 ( r - ro), (4.106)

200 Chapter 4.

is thus the Gaussian pdf (4.54). The equation of motion for the mean m and the covariance matrix M are given by eqs.(4.58,59), where the matrices A andB are,

A - r ~

B = - D 0 i . (4.107)

Hence,

d m

dt dM dt

r . m , (4.108)

2DoI + r . M + M . r T . (4.109)

The solutions of these equations, with the initial conditions m(t - 0) - ro and M(t - 0) - 0, read,

m(t) - exp{r t} , ro - ro + r - ro t (4.110)

- 2Do fo' dt' exp{r( t - t ' )}. exp{rT(t - t')}

1 ( r + r T) t + t 2 = 2Do t 1: + ~ 5r

M(t)

(4.111)

Here we used that r '~ = 0 for all n > 1 to rewrite the operator exponential exp{r t } as I + r t (see also the discussion in section 2.7). Notice that the above expression for the covariance matrix is identical to that obtained in chapter 2 on the basis of the Langevin equation.

The expression (4.7), with the functions f and g specified in eq.(4.48), yields the following expression for the self dynamic structure factor,

f 1 exp{ik, ro} f dr e x p { - i k , r}P( r , t) , s , ( k , t ) - dro Y

-- P(k,t)

where the Fourier transform of the pdf is equal to (see eq.(4.56)),

1 P(k, t) - e x p { - i k , m} exp{ - : -k . M . k}.

2 (4.112)

Using the expressions (4.110,111) for the mean and the covariance matrix thus yields,

S~(k, t) - 1 L dro exp{ - ik - r . r o t}

( 1 2 } exp -Dok2 t - ;yDok~ky t2 -g4 / Dok~t a . (4.113)

The integral with respect to the initial value ro depends on the precise geometry of the volume V. In the strict thermodynamic limit, where V tends to the entire ~a, this integral is a delta distribution ofkk~ x t, so that S,(k, t) - 0 for k~ ~ 0. In a light scattering experiment in which the self dynamic structure factor is measured, however, the volume V is the scattering volume, which is a finite volume. In that case the measured S, is strongly dependent on the geometry of the scattering volume. The intensity auto-correlation function (IACF) tends to zero within a time interval At of the order [,~k~ V1/3] -1 , w i t h V 1/3 the linear dimension of the scattering volume. A standard dynamic light scattering experiment is therefore insufficient to obtain meaningful information about diffusion in shear flow. This information is contained in the exponential function in the second line on the right hand-side of eq.(4.113). It is possible, however, to devise a two-detector dynamic light scattering experiment which eliminates the integral from the measured correlation function. In the two- detector experiment, the instantaneous output of detector A say, is correlated with the instantaneous output of detector B. The intensity correlation function is now the (normalized) intensity cross-correlation function (ICCF),

oS~(k A, k B, t) = < i(k A, t - O)i(k s, t) > / ( I (k A)/ (kS)) , (4.114)

where the superscripts A and B refer to the corresponding detectors. The arguments in section 3.6 on dynamic light scattering, which lead to eq.(3.76), are equally valid for the two-detector experiment. As an example, let us evaluate one of the ensemble averages occurring here,

< (E~(k A, t - 0) . fi~)(E~(k B, t) . fi~) >

,-~ < exp{ik A. r ( t - O) + ik B. r(t)} > .

"Cross terms" in which the particle number indices i and j refer to different particles, that is i ~ j , are zero for non-interacting particles and are omitted here. This ensemble average can be calculated just as above, using eq.(4.7),

202 Chapter 4.

except that the functions f and g are now equal to,

f ( r ) - exp{ik A. r}

g(r) - exp{ik B . r } .

Using this in eq.(4.7) together with eqs.(4.110-112) gives,

< exp{ik A. r ( t - 0) + ik B. r(t)} >

__ 1V fv dro exp{ikA �9 ro + ik B. (ro + r . r o t)}

1 2 ) ~2t3 • }

The other ensemble average occurring in the expression (3.76), in the present cross-correlation context, is identical to the above average, with k B replaced by - k B. The ICCF is thus found to be given by,

~l~(kA, kB, t) - 1 + [F(k A, k B, t) + F(k A, - k B, t)] (4.115)

• exp - 2 D o ( k S ) : t - 2"~Dokffk~ - 5-~ Oo(k~ ,

where we abbreviated,

F(k A, • B, t) - I V dro exp{ikA �9 ro 4 ik B. (ro + r . r o t)} I ~ .

(4.116) Now suppose that the wavevectors k a and k B are chosen such that, for a certain time t - t*,

kA ' ro + k B. (ro + r . r o t*) - o . (4.117)

The time dependence of the ICCF is now as follows. Both F-functions in eq.(4.115) tend to zero within a small time interval At ,~ [~/k~ V1/3] -a, due to their delta distribution like character. The function F (k A, - k B, t) remains equal to zero for all times (since ik a . ro - ik B. (ro + r . rot) ~ 0 for all t > 0), whereas the function F(k A, +k n, t) is zero until t ~ t*. For this time, according to its definition in eq.(4.116), F ( k a, + k n, t*) - 1. Actually, F ( k a, + k B, t) is sharply peaked around t = t*, with a width of the order [~/k~ V~/3] -~. According to the expression (4.115) for the ICCF, the top of this peak is equal to the numerical value of the exponential function of interest for

4.6. Smoluchowsld Equation with Shear Flow 203

^ [

gI

1 .06

1.04

1 .02

1 .00

Figure 4.6"

�9 '. . . . . I " "' I

- i f" i. I

5 . . . . I . . . . . i, I , 0 5 10 15

f [ m s ]

An experimental intensity cross-correlation function (ICCF). The upper set of data points is an enlargement of the lower set of data pionts. This figure is taken from Derksen (1991).

t = t*. To obtain experimentally numerical values for the exponential function at various times, measurements for various combinations of wavevectors must be performed. A measurement of a single ICCF as a function of time gives only information about the diffusive behaviour of the Brownian particle at one particular time t*. The full time dependence of the diffusive behaviour, as described by the exponential function in eq.(4.115), is now constructed from ICCF's obtained from experiments with various combinations of the two wavectors.

The above described time dependence is experimentally verified in fig.4.6. A sharp decrease at small times and an equally sharp peak at a particular instant of time. The occurrence of the sharp peak can be understood intuitively as follows. At time zero, the phase difference between the light scattered towards detector A and B is equal to (k a -F k B) �9 ro (see the discussion in section 3.2). The change of the phase of the light scattered towards detector/3 during a time t*, due to the shear flow, is equal to k B �9 F . ro t*. Adding this up to the phase difference at time t - 0, and demanding a net phase difference equal to zero, reproduces eq.(4.117). Thus, at the particular time t = t*, the Brownian particle is displaced by the shear flow over a distance corresponding to a zero

204 Chapter 4.

phase difference between the light scattered towards detector A at time t - 0 and towards detector B at time t = t*, giving rise to perfect correlation at that time. The measured correlation at time t* is non-perfect only due to the diffusive motion that occurred during the time interval t*.

There is an experimental difficulty concerning the normalization in eq.(4.114). The intensities I (k A) and I (k B) are proportional to the scattering volumes V A and Vf for the detectors A and B respectively. The ensemble average in the numerator of eq.(4.114), however, is proportional to the squared cross sectional volume] V a n V f 12. In writing eq.(4.115) for the ICCF, it is assumed that the ratio o f [ V a fq V~ 12 and V~ x Vf is equal to one. Evidently, in reality this ratio is smaller than one, and is different for each different choice of wavevectors. We shall not pursue this experimental detail here any further.

4.7 The Smoluchowski Equation with Sedimentation

When there is a mismatch of the mass density of the Brownian particles with that of the solvent, the Brownian particles attain a so-called sedimentation velocity due to the earth's gravitational field. Charged Brownian particles can also attain a certain mean velocity when subjected to an external electric field, the so-called electroforetic velocity. In this section, the effect of a constant external force on the equation of motion for the pdf of the position coordinates on the diffusive time scale is considered. The external force is assumed to act equally on all the Brownian particles. There are two things to be considered �9 the effect of a non-zero velocity of the fluid surrounding the particles (the so-called back flow) on hydrodynamic interaction, and the change of the equation of motion as a result of the additional external force. Hydrodynamic interaction and back flow are discussed in the following subsection. The equation of motion is considered in subsection 4.7.2.

4.7.1 Hydrodynamic Interaction with Sedimentation

In an experiment, the Brownian particles sediment in a closed container. Consider a flat cross sectional area of the container perpendicular to the sedimentation direction. The total volume of colloidal material that sediments through that area must be compensated by fluid flow in the opposite direction. Let qa denote the volume fraction of Brownian particles, which is the fraction of

4. Z Smoluchowski Equation with Sedimentation 205

Figure 4.7"

\ \ \ \

\

\ \ \ \ \ \

\ \ \ \

U s

F exf

\

\

\

\

\

\ \

The inhomogeneous back flow in a sedimenting suspension. On a local scale the back flow may be considered constant. The Smoluchowski equation applies to a subgroup of Brownian particles in the indicated region, where the back flow is almost constant.

the volume that is occupied by colloidal material. For a sedimentation velocity v,, the total volume of colloidal material that is displaced is compensated by an (average) fluid flow velocity u,, when,

u , ( 1 - q o ) + v,~o - 0 ,

since 1 - r is the fraction of the total volume that is occupied by the fluid. Hence,

u~ = ~ v~. (4.118) 1 - q o

The subscript "s" refers to "sedimentation". The fluid flow that compensates the volume flow of colloidal material is referred to as back flow. Since at the wall of the container the fluid flow velocity is zero (for so-called "stick boundary conditions"), the back flow may be inhomogeneous, that is, it may vary from position to position within the container. The above equation for the fluid back flow velocity is the back flow velocity averaged over a cross sectional area perpendicular to the sedimentation direction.

Here we discuss the case in which the back flow may be considered constant, independent of the relative position to the walls of the container. For a container with dimensions very large compared to the radius of the

206 Chapter 4.

Brownian particles, the back flow may be considered homogeneous on a local scale. The back flow is certainly inhomogeneous, irrespective of the size of the container. However, we analyse the sedimentation velocity in the chapter on sedimentation of a large subgroup of Brownian particles in a region within the container where the back flow is (to a good approximation) constant (see fig 4.7).

The Brownian particles can thus be considered to be immersed in a fluid with a homogeneous flow velocity u, as given in eq.(4.118). The hydrodynamic interactions in a suspension in which the fluid is homogeneously displaced are simply obtained by replacing the velocities in the expression (4.9) by the velocities relative to the fluid. There is no additional disturbance contribution as for the case of an inhomogeneous flow, like a simple shear flow. Hence, the force that the fluid exerts on the i th Brownian particle is given by,

N

F/h - - Y~ T i j ( r l , . . . , rN)" (vj -- u,) . (4.119) i,j=l

This equation can be used to obtain the Smoluchowski equation for a sedimenting suspension, in a similar manner as the original Smoluchowski equation (4.40,41) was derived in section 4.4.

4.7.2 The Smoluchowski Equation with Sedimentation

The derivation of the Smoluchowski equation is analogous to that in section 4.4. The only difference is that the hydrodynamic force in eq.(4.30), the first term on the right hand-side, is to be replaced by the above expression (4.119), and that there is an additional (external) force F ~t which is equal for all Brownian particles. On the Brownian time scale, the total force is zero. On the other hand, the total force on the i th Brownian particle is equal to the sum of the hydrodynamic force (4.119), the direct force - V~ ~, the Brownian force (4.37) and the external force. Hence,

N

o - _ z j = l

-kBTV,., l n{P( r l , . . . ,rN)} + F ~'t . (4.120)

As before, this expression is written in "super vector notation" in order to express the momentum coordinates in terms of the position coordinates. The

4.7. SmoluchowskiEquation withSedimentation 207

super vector notation was introduced in eqs.(4.20-24). Introducing further,

F ~ t _ ( F ~ t F ~ t . F ~ t 8 ~ ~ ~ ' " ~ ) ~

Nx

Nx

the expression (4.120) takes the form,

(P ) 0 - - T ( r ) �9 ~ - U, - V ~ O ( r ) - k B T V , ln{P(r, t)} + F~ ~t .

The velocities can now be expressed in terms of position coordinates as,

dr p dt M

= U , + T - l ( r ) . [ - V ~ e ~ ( r ) - k B T V ~ l n { P ( r , t ) } + F : ~ t ] . (4.121)

This identifies the function H via eq.(4.2), and the general expression (4.4,5) for the equation of motion for the pdf becomes, in super vector notation,

0 0---t P(r , t) - /~s P(r , t ) , (4.122)

where the Smoluchowski operator is given by,

V~. D( r ) . [/3 [V~O](. . .)+ V ~ ( . . . ) - 3F:~t( .. .)]

- V ~ . [U,( . . . ) ] . (4.123)

For later reference we reproduce here eq.(4.121) in terms of the original momentum and position coordinates,

Pi

M = u , + E Dij" [ - /3 [V~j ] - V~ In{P}] + D,j .flF ~t . (4.124)

j--1 j= l

The microscopic diffusion matrix D is defined in eq.(4.34) as kBT times the inverse of the microscopic friction matrix T. In terms of the original position coordinates the Smoluchowski equation reads,

O p ( r l , ' " , r N , t) -- / ~ s P ( r , ' " , r N , t ) , (4.125)

208 Chapter 4.

with,

. . ) N

V~,. Dij . [fl[V~j(I)](...)+ V ~ ( . . - ) - / 3 F ~ t ( .- .)] i,j=l

N - [u,(...)]

3=1

(4.126)

This Smoluchowski equation contains the solvent back flow velocity u,, which is related to the sedimentation velocity v, of the Brownian particles as given in eq.(4.118). The sedimentation velocity is also equal to the ensemble average of the velocities p~/M of each of the Brownian particles in the group of particles in the container where the local back flow velocity attains the particular value u~. In principle, to obtain the sedimentation velocity, one should solve the (stationary) Smoluchowski equation in terms of the back flow velocity, calculate then the mean velocity v, - < pi > /M from eq.(4.124), again in terms of the back flow velocity, and finally substitute eq.(4.118) to obtain a closed equation for the sedimentation velocity v,. This procedure is worked out in chapter 7 on sedimentation.

4.8 The Smoluchowski Equation for Rigid Rods

The equations of motion considered so far are valid for spherically symmetric Brownian particles. For such particles, rotational motion is not included in the stochastic variable X. For non-spherical Brownian particles, however, orientations of the particles must be included, since translational motion and rotational motion are now coupled. Clearly, the translational motion of a particle is affected by the orientation of neighbouring particles, and vice versa, in contrast to spherical particles. Due to the orientation dependence of the potential energy of an assembly of rods, the Brownian particles exert torques on each other. These torques, which depend both on the relative separations of particles and on their orientations, lead to rotational motion. Moreover, even for non-interacting rods, the translational motion is coupled to the orientation, since the translational friction coefficient depends on the orientation of the rod (see the discussion in subsection 2.8.2).

Here we consider cylindrically symmetric Brownian particles of which the orientation is characterized by a single unit vector fi, the direction of which is along the cylinder axis (see fig.2.5). On the Brownian time scale it is sufficient

4.8. Smoluchowski Equation for Rigid Rods 209

to consider only the positions and orientations. The stochastic variable to be considered here is thus the 6N-dimensional vector,

X - ( r l , r 2 , . . . , r N , i l l , 1~12,""", f iN) �9 (4.127)

The translational and the rotational velocities, on the Brownian time scale, are instantaneously relaxed to thermal equilibrium with the solvent (see chapter 2). As a consequence, the total force and torque on each Brownian particle is zero. The friction force and torque that the fluid exerts on each Brownian rod is thus balanced by interaction forces and torques. This fact can be used to derive the equation of motion in much the same way as for spherical particles. However, since the orientations are unit vectors, the relations (4.2-5), which were used for spherical particles, cannot be used here as they stand. We shall have to derive an alternative expression for the special case of rigid rod like particles.

Hydrodynamic interaction between rods is discussed on a qualitative level in the following subsection. In subsection 4.8.2 the Smoluchowski equation is derived, of which elementary consequences for non-interacting rods are discussed in subsection 4.8.3.

4.8.1 Hydrodynamic Interaction of Rods

The force as well as the torque that the fluid exerts on a rod depend on both the translational and angular velocities of all other rods. Due to, (i) fast propagation of fluid disturbances relative to the Brownian time scale, and (ii) the linearity of the hydrodynamic equations that describe the fluid flow (as discussed in section 4.2), there is a linear relationship between the forces F h and torques T/h which the fluid exerts on the i th Brownian rod on the one hand, and the translational velocities vj and angular velocities 12j on the other hand,

( Fhl ~

r} I TTT TTR 1 TRT Tnn ~1 h

( Vl ~

VN

k ~"~N )

, (4.128)

where the four 3N x 3N-dimensional microscopic friction matrices T depend on the positions and orientations of all the N rods. The supercripts T and

2 1 0 Chapter 4.

R refer to "translation" and "rotation". The calculation of these microscopic friction matrices is a difficult hydrodynamic problem. Not much is known about their explicit dependence on positions and orientations.

The angular velocities f~i and torques Ti h are relative to the center of mass of the i th rod, which is assumed hereafter to coincide with its geometrical center.

Without hydrodynamic interaction, the microscopic friction matrix reduces t o ,

I T TT T T R I _

"rRT T R R

CT TT 0 . . . 0 0 0 . . . 0 0 T TT . . . 0 0 0 - . . 0

TT 0 0 . . . "rNN 0 0 . . . 0

0 0 . . . 0 T ~ n 0 . . . 0

0 0 . . . 0 0 T nn �9 0 22 ""

RR \ 0 0 . . . 0 0 0 . . . T N N j (4.129)

Due to the linearity of the hydrodynamic equations, the translational friction matrices can be written as,

X~5 ~ - ~11~,~, + ~[i- ~,~,1, (4.130)

where 711 (7• is the friction coefficient for translational motion parallel (perpendicular) to the symmetry axis of the cylinder. Furthermore,

T nn I (4.131) ii "-- "~r

where % is the rotational friction coefficient. These forms for the friction matrices were already discussed in subsection 2.8.2 in chapter 2.

The Smoluchowski equation contains the inverse of the microscopic friction matrix, which is referred to as the microscopic diffusion matrix,

TRl a I oTT I T RT T RR DRT D n n


(DIT1 T . . . DTN T DT~ . . . DITNR ~

�9 �9 �9 , �9 �9

DTT r f rn Tn �9 . . D N N D N 1 . - . D N N

D ~ T "'" D n T1N DR11 n "'" D1RN R

k D ~ T . . . D ~ T D~v ~ . . . D ~ ,

�9 (4.132)

Notice that each of the 3 x 3-dimensional microscopic diffusion matrices D ij is a mix of an the 3 x 3-dimensional microscopic friction matrices.

Without hydrodynamic interaction, the "off-diagonal matrices" D~j, with i # j , are zero. According to eqs.(4.130,131) the "diagonal matrices" Dii are equal to,

D.T. T,, = Dllfiifii + D • fiifii] , (4.133)

DiR/n - D~i , (4.134)

where the parallel and perpendicular translational diffusion coefficients are equal to,

DI! - kBT/Tjl, (4.135) D• - kBT/7• , (4.136)

and the rotational diffusion coefficient is given by,

D~ - k B T / % . (4.137)

These diffusion coefficients were already introduced in chapter 2 in connection with the description of Brownian motion of non-interacting rods on the basis of the Langevin equation.

The fact that the translational microscopic friction matrices are orientation dependent for rod like particles, even in the absence of hydrodynamic interaction, couples the translational dynamics of a rod to its rotational motion. For spherical particles this is not the case, which circumstance allows for an analysis of translational motion without having to consider the rotational motion. The rotational motion of spheres, however, does show up in the calculation of the microscopic friction and diffusion matrices. This becomes particularly clear, considering the linear relation (4.128), which is also valid for spheres.

212 Chapter 4.

Contrary to rod like particles, the hydrodynamic torques Ti h are all zero for spheres on the Brownian time scale, since there are no other torques acting on a spherical particle. For spherical particles, the hydrodynamic torque is equal to the total torque, which is zero on the Brownian time scale, just as the total force (see also the discussion in section 5.11 in chapter 5). This can be used to express the hydrodynamic forces entirely in terms of translational velocities (see exercise 4.4), and shows explicitly that the microscopic friction and diffusion matrices in eqs.(4.9,34) include hydrodynamic interaction due to rotational motion of the spheres.

4.8.2 The Smoluchowski Equation for Rods

The vector X in eq.(4.127) is a 6N-dimensional vector which cannot attain arbitrary values in 6N-dimensional space, since the orientations fii are unit vectors, which lie on the unit spherical surface in ~a. Thus, the subspace of ~6N tO which X is confined is the product space,

~3Nx~xSx...x~, N x

where ,~ is the unit spherical surface in ~a. The "volume" W, which was introduced in section 4.1 on the derivation of the equation of motion for the pdf of X, is now the product of a volume W~ in ~aN and N surfaces ,~n, n = 1, 2 , . . . , N on the unit spherical surface in ~ a . see the sketch in fig.4.8a. The boundary of the set W is the product of a surface in ~3N and N closed curves on unit spherical surfaces.

The derivation of the Smoluchowski equation for rods is technically speaking a bit different than for spheres due to the fact that the two parts of X are elements of different spaces. The general idea of the derivation is the same as outlined in section 4.1. Let us go through the derivation here. It is convenient to introduce the position dependent part X, of X as,

X r - ( r l , r 2 , . . . , r N ) . (4.138)

The rate of change of the "number of points" in W due to the flux through the boundary of W~ is, just as for the spherical particles, proportional to the integral of dS, �9 (dX/dt) P, ranging over the boundary OW~ of W~. The rate of change of the number of points due to the rotational motion is a bit more complicated. First of all, this orientational contribution is proportional

4.8. Smoluchowsla" Equation for Rigid Rods 213

| ~//~Na xes X X .... X

d[n d-Sn

g

/ - A

I.ln

A

dl.x Uo \

Figure 4.8: (a) The "volume" W consists of a volume W~ in N aN for the position coordinates and N surfaces ,Sn , n - 1, 2 , . . . , N, on the unit spherical surface in ~a. (b) The boundary OW~ of W~ is a closed surface and the boundaries 0,~,~ are closed curves on the unit spherical surface, dS,~ is the infinitesimal scalar surface area on O,~n, and dl,~ is an infinitesimal vector length along the curve OSn with positive orientation. (c) The relevant component of the point current density through the boundary OSn is along the vector din x fi,~, which is perpendicular to the boundary of S,~ and tangential to the unit spherical surface.

214 Chapter 4.

to integrals ranging over the boundaries 0,~n of the N surfaces S,~ on the unit spherical surface. These boundaries are closed curves on the unit spherical surface (see fig.4.8b). Secondly, the integrand is equal to dl,~. fi,~ x (dfin/dt) P, with dl,~ an infinitesimal vector tangential to aS,~, with a positive orientation (see fig.4.8c). This can be seen as follows. First rewrite, dl,~. fi,~ x (dfi~/dt) P = dl~ x fi~ �9 (dfi~/dt)P. Now, din x fin is the vector with length [ din I (since dl,~ .1_ fin), perpendicular to the boundary OSn and directed outwards. Hence, din x fi,~ �9 (dfi,~/dr) P is the component of dfin/dt perpendicular to a,~,~, which is the component that must be integrated to obtain the rate of change of the number of points that leave the surface Sn.

Analogous to eq.(4.1), the equation for the rate of change of the number of points contained in W as the result of flow of points through its boundary is the sum of the rates of change due to the flux through OW~, 0S1,''" OSN,

fWr dXr fgl dSl "'" f~NdSn ~ P ( X , t ) -

- ~Wr dSr "~1 dSl f$2 dS2"" ~N dSN [dXrdt P(X, t)]

-- fWr dxr J~O,~l dll " ~, dS2 "'" f,~N d~N [~'~1 P(X,t)]

- L r dXr fdl d~'~l ,)~OS, d12 .. . . f~, dSN [f12 P(X,t)]

- L .

Here we used that the angular velocity fl~ of a long and thin rod is related to the orientation fii as follows (see section 2.8.2),

dtli f~i - fii x dt The integral ranging over 0W~ can be recast into an integral ranging over W~, just as in section 4.1, using Gauss's integral theorem. The integrals ranging over 0Sn can be recast into integrals ranging over ,~n using Stokes's integral theorem,


for any (well behaved) vector field F. Here, V~, is the gradient operator with respect to fin. Hence we obtain,

0 P(X, t) -

[ ( ) ] dX~ p(x , t) + E a, . V~, x ( a , p ( x , t)) . x V~ . . . . dt i=1

Since this equation is valid for an arbitrary set W, it follows that the integrands of the two above integrals are equal, precisely as in section 4.1, yielding,

] P ( X , t ) - - ~ V~,. T P(X, t ) + ( f i i x V a , ) . ( f l i P ( X , t ) ) ,

(4.139) where we used that fi,. Va, x (..-) - (fii x Va,)- (. . .) .

On the Brownian time scale the translational velocities dri/dt and angular velocities f~i are functions of the positions and orientations as a result of the balance of the hydrodynamic force and torque with the other forces and torques. That is, on the Brownian time scale the total force and torque on each particle are zero,

h I F B r 0 - Fj + F j + (4.140)

0 - T5 h + T J + T 5 B~, (4.141)

where the superscript I refers to direct interaction and Br to the Brownian contribution. Substitution of eqs.(4.128,132 ) for the hydrodynamic forces and torques gives,

N

vi - f l ~ [ D , T T - ( F ~ + F ~ ~ ) + D i T R . ( T 5'+TSmQ] , (4.142) j= l

N

" i -- /~ E [Di~ T" (F~ + F Br) + D~ R. (T 5' + TiB~)] . (4.143) j= l

As a last step the direct interaction and Brownian forces and torques must be expressed in terms of position and orientation coordinates.

The direct force is minus the gradient of the total potential energy (I) of the assembly of Brownian particles,

I Fj -- --Vrj (I)(rl , . . . , rN, 1~11,''', fiN). (4.144)

216 Chapter 4.

The direct torque is related to r as,

Tj I - -f i j x V~ (I). (4.145)

This expression is derived in exercise 4.5. The form of both the Brownian force and torque can now be found from

eqs.(4.142,143), in a similar manner as for spherical particles. For long times, the pdf is equal to the Boltzmann exponential ,-, exp{-flr }. The time derivative of the pdf in the equation of motion is then easily seen to be equal to zero, when the Brownian force and torque are related to the pdf as,

F f ~ - - k B T V~, ln{P}, (4.146) ~B~ _ -kBTf i j x V, b ln{P}. (4.147)

Substitution of these expressions for the Brownian force and torque, together with the expressions (4.144,145) for the direct force and torque, finally leads to the Smoluchowski equation for rigid rod like Brownian particles in its most general form,

0 0-t P ( r l , . . . , rN, s s t) -- ~S P ( r l , . . . , rm, d l , . . . , fiN, t ) ,

(4.148) with,

N

s - Z {V~,. Di TT. [~[V~(I)](..-)+ V~j(...)] (4.149) i,j=l

+V~,. Di T ' - [fl[fi~ x V~(I)I(-- . )+ fij x Ya~(...)]

+ a , • �9 %(---)] +fi, x V~,. D,~' . [fl[fij x V ~ r fij x V~(. . . ) ] } .

Due to its complexity this equation of motion is of very limited practical value. Moreover, there are no accurate expressions for the hydrodynamic interaction matrices available. In further chapters the Smoluchowski equation for rods will be analysed with the neglect of hydrodynamic interaction. In that case, only the microscopic diffusion matrices on the diagonal in the expression (4.132) are non-zero, which are given in eqs.(4.133,134).

It is convenient at this stage to define the rotation operator 7~i,

7~i(.- .) - fii x Va, ( . . . ) , (4.150)


and the average translational diffusion coefficient D and the difference of the two translational diffusion coefficients AD as,

- 31 t' 2D• , , D = kDll + (4.151)

AD = DI I - D• (4.152)

With these definitions, the Smoluchowski operator without hydrodynamic interaction reads,

N

/~s('" ") - ~ {/7) V~,. [r + V~,(...)1 (4.153) i=1

+ D~ 7~/. [/317~/01(-..)+ 7~,(...)]

+ AD V, , . [ f i / f i / - ~ i ] . [/3[V~,O](...)+ V~,( . . . ) ]} .

The last term in this Smoluchowski operator describes the coupling of translational and rotational motion as the result of the anisotropic microscopic translational friction.

The equation of motion for the pdf of the position and orientation of a rod in a very dilute suspension is,

0 0--t P(r , fi, t) - /~g P(r , fi, t ) , (4.154)

w i th /~ the Smoluchowski operator (4.153) without the interaction potential �9 , the form of which is given here explicitly for later reference,

Z~g(...) - /7) V ~ ( . . . ) + D~ 7~2( .. .)

[ li]. V~(. . . ) . (4.155) + A D V ~ . i f i 3

The squared rotation operator is defined as 75,.. 7~ - (6 x V~). ( f ix Va), in analogy with the Laplace operator V~ = V~ �9 V~.

The solution of the Smoluchowski equation (4.148,153) will be discussed up to leading order in concentration, as far as rotational correlations are concerned, in chapter 6 on diffusion. The Smoluchowski equation (4.154,155) is used in the same chapter to calculate the electric field auto-correlation function (EACF), as defined in the previous chapter on light scattering, for a system of non-interacting rods. In the following subsection, translational and rotational correlations in dilute dispersions are discussed to some extent. The results obtained here reproduce the results obtained on the basis of the Langevin equation, as obtained in chapter 2.

218 Chapter 4.

4.8.3 Diffusion of non-Interacting Rods

Consider the mean squared center of mass displacement of a freely diffusing rod like Brownian particle. The equation of motion for the dyadic < r(t)r(t) > is obtained by multiplying the Smoluchowski equation (4.154,155) with rr, and integrating with respect to r and ft. According to a theorem that is a direct consequence of Stokes's integral theorem (see exercise 1.5c in the introductory chapter),

# ( . . . ) - x - o ,

where ,~ is the unit spherical surface. Furthermore,

47r 1 47r

~ d S f i f i - - ~ - I , ~d~8:J: --~-I. It follows that the only remaining term is the first term on the right hand-side of the Smoluchowski operator (4.155),

d d---t < r ( t ) r ( t )> - b f dr ~ dS rrV~2P(r, fi, t)

= D / d r ~ d S P ( r , fi, t)V2~rr - 2/)i.

The last step here is verified in exercise 4.6. Similarly it is found that,

d d~ < r ( t ) > - O.

The solutions of these equations of motion, with the initial condition that r(t - O)=r(O), are,

< r(t)r(t) > - r (0)r (0)+ 2 D t i ,

< r ( t ) > - r(O).

It follows from these expressions that the mean squared displacement is given by,

< ( r ( t ) - r (O) ) ( r ( t ) - r(O)) > - 2Dt i . (4.156)

This result is identical to that for spherical particles, except that the translational diffusion coefficient is now the weigthed mean b as defined in eq.(4.151). The result (4.156) is in accordance with eq.(2.124), which was derived on the basis of the Langevin equation.

Let us now consider the time dependence of the orientation < fi(t) >, given that fi (t - 0) - fi(0). As for the translational mean squared displacement, the equation of motion for < fi(t) > is obtained by multiplying both sides of the Smoluchowski equation (4.154,155) with fi, and integrating with respect to r and ft. According to Gauss's integral theorem,

/drV . (...1 - 0,

so that the only remaining term is the second term on the right hand-side of the Smoluchowski operator (4.155),

d~ < fi(t) > - D~ dr dS fi~2 P(r, fi, t).

Now, from Stokes's integral theorem it follows that for any two (well behaved) functions f and g of fi,

J

= 0

and hence,

J.r dS f(fi)~g(fi) - - ~ dS g(fi)7~f(fi). (4.157)

Applying this result twice, we get,

f dr Jd dS fiT~ 2 P(r , fi, t ) - f dr Jd dS , ( r , fi, t)7~2 fi - - 2 < fi(t) > ,

where it is used that 7~2fi=-2fi (see exercise 4.6). The equation of motion we were after thus reads,

d d5 < f i ( t ) > - -2D~ < f i ( t ) > ,

the solution of which is,

< f i ( t ) > - exp{-2D~t} fi(0), (4.158)

in accordance with the Langevin equation result (2.141) or, equivalently, eq.(2.143).

In exercise 4.7 it is shown how to use the Smoluchowski equation to evaluate the time dependence of the depolarized scattered intensity at small scattering angles, after switching off a strong external field that fixes the orientation of the rods in a certain direction. Such an experiment can be used to determine the rotational diffusion coefficient.

220

Exercises

Exercises Chapter 4

4.1) Here we consider the derivation of equations of motion for ensemble averages directly from the equation of motion for the pdf.

(a) Suppose one wishes to derive an equation of motion for the ensemble average < f (X) >, for some function f. Multiply both sides of the linear Fokker-Planck equation (4.49) with f and integrate over X to show that,

d d~ < f (X) > - < [V~f(X)]. A . X > - < V~V~f(X) �9 B > .

Depending on the form of the function f , additional equations of motion for the ensemble averages on the right hand-side must be found to obtain a closed set of equations of motion.

Take the function f equal to X and XX, respectively, and derive the equations of motion (4.50,51).

(Hint" Use Gauss's integral theorem in m-dimensions,

f d X / ( X ) V ~ . ( . . .) - - f dX [V~f(X)]- (.. .). )

(b) Use the method as described in (a), with X = r, the position coordinate of a non-interacting Brownian particle, to derive the equations of motion (4.71,72) directly from the Smoluchowski equation (4.62).

4.2) The Brownian oscillator Two identical Brownian spheres are connected to each other with a spring.

The potential energy of the two particles with position coordinates rl and r2 1C I r~ - r212, where C is the spring constant. Define the is equal to �9 -

separation R = r l - r 2 between the two spheres and the center of mass r = 1 1 !(rx + r 2 ) Convince yourself that V~ - VR+ 7V~ and V, 2 - - V a + 7V

2 " r .

Use this to rewrite the Smoluchowski equation (4.40,41) for the two particles under consideration, with the neglect of hydrodynamic interaction, as,

c0 1 2 o-~P(R, r, t ) - Do {2,SCVn. ( R P ) + 2V~P + 5V,.P} .

Now try a solution of the form P (R, r, t) - P (R, t) P (r, t), and show that,

" P ( R , t ) - 2DoVn. [/~CRP(R,t) + VnP(R, t ) ] Ot

~ t P ( r t ) - 1DoV~P(r, t ) 2 "

The center of mass thus diffuses as a single sphere with a diffusion coefficient equal to half the Stokes-Einstein diffusion coefficient of the separate spheres of the Brownian oscillator. The Smoluchowski equation for the pdf of the separation R is a linear Fokker-Planck equation: Verify that this equation of motion is of the form (4.49) with A = -2Do/~CI and B = -2DoI. Solve the equations of motion (4.58) for m - < R > (t) and (4.59) for the covariance matrix M. Show that,

< R > (t) -

M ( t ) =

R(O) e x p { - 2 D o / 3 C t } ,

i /3C [1 - e x p { - 2 D o / 3 C t } ] .

The pdf P (R , t) now follows immediately from eq.(4.54). Verify that the expression for M for t ~ oo is in accordance with the equipartition theorem (see exercise 2.2).

4.3) Diffusion in an inhomogeneous solvent For very dilute suspensions, the diffusion coefficient is equal to Do in

eq.(4.44) only for a homogeneous solvent. Now suppose that the solvent is inhomogeneous in composition, so that the diffusion coefficient is different at each position, that is, the diffusion coefficient is a position coordinate dependent matrix, Do(r). Verify that the Smoluchowski equation for this case is,

_0 P ( r , t ) - V~. [Do(r). V~P(r t ) ] .

Ot

Show that the inhomogeneity of the fluid gives rise to an average drift velocity equal to,

d < r > - < V~. DoT(r) >

d-~

You can use the integration method as described in exercise 4.1.

4.4) (a) For spherical particles, the hydrodynamic torques are equal to the total

torque (provided no external field exerts a torque on the particles), which is zero on the Brownian time scale. Use this to show that eq.(4.128) yields


the following linear relationship between the hydrodynamic forces and the translational velocities,

[ + x Vl

�9 T R T ] �9 �9 . "1

V N

This is relation (4.9). This expression makes explicit the effects of rotations of the spheres on the translational hydrodynamic friction matrix.

(b) As spheres translate through a fluid they transfer energy to the fluid. Verify that the energy of dissipation is equal to - ~Y=I v j . F h, and is always positive. Show that this implies that D is positive definite, meaning that for any 3N-dimensional vector x # 0, x . D . x > 0.

4.5) The direct torque on a rod Suppose that a rod's orientation fi is changed by an infinitesimal amount

6ft. For a long and thin rod, the accompanied change in potential energy is,

(~(I) -- - - / V ~ dr f ( r ) . (r *fi),

where V ~ is the volume of the thin rod with its geometrical center at the origin. Furthermore, f(r) is the force per unit volume on a infinitesimal volume element of the rod at the position r relative to its center. We used here that the displacement of a volume element at r is equal to r 6ft. Verify each of the steps in the following sequence of equations,

- /vo d r f ( r ) . (r fi x (6fi x fi)) - - / v o d r f ( r ) . (r x (~fi x fi))

- ( 6 f i x fi). fvo dr ( f ( r ) x r) - (6fl x f l ) . T - 6fl. (fl x T ) .

Now, on the other hand,

6r - V a ~ . 6 6 .

Compare the two above equations to conclude that,

V ~ = f i x T .

For long and thin rods,

dr r x f ( r )~ , fi x Iv dr r f ( r ) o


so that T I ft. Use this to show that,

T - - 6 x V a ~ .

This is the expression for the torque on the jth rod in eq.(4.145). (Hint : For three arbitrary vectors a, b and c,

a x (b x c) - b ( a . c ) - c ( a . b) . )

4.6)* In this exercise we evaluate V2rr, 7~2fi and a . 7~fi, with a an arbitrary vector.

V2rr is a matrix of which the ij th- component is equal to V2rirj. Verify that V2r~rj - 26~j, with 6~j the Kronecker delta. Conclude that,

V~rr- 2t.

7~2fi is a vector with components,

7~2t21 7 2fl- ,

where fij is the jth that,

component of ft. Use the definition (4.150) of 7~ to show

l0 / t~ 3

--it2

Use this to verify that, ~ 2 U 1 - - --2~tl. Repeat this calculation for j - 2 and 3. Conclude that, R2fi - -2f t .

Let a be an arbitrary vector and define 7~fi as the matrix with components (7~fi)ij - 7~ifij. Show that,

a . ,,fi - a •

4.7) Small angle depolarized time resolved static light scattering by rods Consider a very dilute suspension of rigid rod like Brownian particles

which are strongly aligned in the z-direction by means of an external field. At time t - 0 the external field is turned off. The rods attain an isotropic

orientational pdf after a long time. The following light scattering experiment can be done to follow the rotational relaxation of the aligned rods. The polarization direction of the incident light is chosen in the z-direction, which is the alignment direction of the rods at time zero. The mean scattered intensity, with a polarization direction perpendicular to the z-direction, say in the x-direction, is measured at a small scattering angle as a function of time. The scattering angle is chosen such that �89 < 0.5 (k is the wavevector and L is the length of the rods). The ensemble averaged scattered intensity is given by (see eqs.(3.126,127)),

^2 ^2 R ~ .

1 kL < 0 5, and the "cross The jo-functions in eq.(3.127) are equal to 1 for 7 terms", with i ~ j , are zero for the dilute dispersion considered here. In eq.(3.127), fi, (rio) is the polarization direction of the detected (incident) light, which is along the x-axis (z-axis).

In this exercise, the time dependence of this depolarized small angle scattered intensity is calculated from the Smoluchowski equation (4.154,155), along similar lines followed in subsection 4.8.3 to calculate the time dependence of < 6(t) > in eq.(4.158).

In the following, the indices 1, 2 and 3 refer to the x, y- and z-direction, respectively.

First verify that (Vj is the jth component of Va, the gradient operator with respect to fi),

- { + + + + +

- 2 [?~2~3V2V3 -~- uI?~3VIV3 -~ uI?~2VIV2]

- 2 [?~i~71 -~- ~2~72 -~- ~3~73] } ( . . . ) .

Let f and g be arbitrary functions of ft. Apply the result (4.157) twice, to show that,

dS f ( u ) ~ 2 g(fi) - f dSg(u)7~2 f ( f i ) �9

Now multiply both sides of the Smoluchowski equation (4.154,155) with u3u 1 ^ 2 " 2 and fi~, and integrate to arrive at the following equations of motion,

d ^2^2 ^2 ^2 ^ )] d~ < U3Ux > - D,. -20 < U3U 1 > + 2 ( 1 - < u] > ~ ,

d [2 6 < d-'i" < '&~ > - D,.t - '5~>] .

Further Reading 225

Solve these equations to find the following time dependence of the small angle depolarized scattered intensity,

^2^2 1 1 e x p { - 6 D ~ t } - 4 exp{-20D~t} R ~-~< N3N 1 >-- ~ "~" 2i- ~ "

This can be used to determine the rotational diffusion coefficient. Rotational relaxation is discussed in more detail in subsection 6.10.2 in the chapter on diffusion.

An alternative way to determine both the (weighted mean) translational and the rotational diffusion coefficient is by conventional dynamic light scattering. This is discussed in the chapter on diffusion in subsection 6.10.1.

F u r t h e r R e a d i n g a n d r e f e r e n c e s

The book of van Kampen contains a detailed discussion on the "Use and abuse of the Langevin approach",

�9 N.G. van Kampen, Stochastic Processes in Physics and Chemistry, North Holland, Amsterdam, 1983.

More about the equivalence of Langevin equations and Fokker-Planck equations can be found in the above mentioned book of van Kampen and in,

�9 M. Lax, Rev. Mod. Phys., 38 (1966) 541. �9 C.W. Gardiner, Handbook of Stochastic Methods, Springer-Verlag, Am-

sterdam, 1983. �9 H. Risken, The Fokker-Planck Equation, Springer-Verlag, Berlin, 1984.

The original papers on the derivation of the Fokker-Planck and the Smolu- chowski equation from the Liouville equation for the pdf of the phase space coordinates of both the solvent molecules and the Brownian particles are,

�9 R.M. Mazo, J. Stat. Phys. 1 (1969) 89, 101, and 559. �9 J.M. Deutch, I.J. Oppenheim, J. Chem. Phys. 54 (1971) 3547. �9 T.J. Murphy, J.L. Aguirre, J. Chem. Phys. 57 (1972) 2098.

See also, �9 G. Wilemski, J. Stat. Phys. 14 (1976) 153. �9 W. Hess, R. Klein, Physica A 94 (1978) 71. �9 J.L. Skinner, P.G. Wolynes, Physica A 96 (1979) 561.

226 Further Reading

�9 U.M. Titulaer, Physica A 100 (1980) 251.

For the fluctuating hydrodynamics approach, see, �9 D. Bedeaux, E Mazur, Physica 76 (1974) 247. �9 B. Noetinger, Physica 163 (1990) 545.

Early discussions on the derivation of the Smoluchowski equation for flexible polymer chains, along similar lines as followed here are,

�9 J.G. Kirkwood, J. Chem. Phys. 29 (1958) 909 and J.J. Erpenbeck, J.G. Kirkwood, J. Chem. Phys. 38 (1963) 1023.

�9 R. Zwanzig, Adv. Chem. Phys. 15 (1969) 325. The book of Doi and Edwards contains a detailed account of the Fokker-Planck and Smoluchowski equation approach for polymers,

�9 M. Doi, S.E Edwards, The Theory of Polymer Dynamics, Clarendon Press, Oxford, 1986.

Conventional homodyne and heterodyne dynamic light scattering experiments on sheared systems are analysed in,

�9 B.J. Ackerson, N.A. Clark, J. Physique 42 (1981) 929. Two-detector dynamic light scattering experiments on a sheared suspension are discussed in,

�9 J.J. Derksen, Light Scattering Experiments on Brownian Motion in Shear Flow and in Colloidal Crystals, Thesis, TU Eindhoven, 1991.

Chapter 5

HYDRODYNAMICS

227

228 Chapter 5.

5.1 Introduction

On several occasions in previous chapters, the friction coefficient "7 of a single Brownian particle has been introduced as the ratio of (minus) the force that the fluid exerts on the particle and its velocity. The corresponding diffusion coefficient is given by the Stokes-Einstein relation Do - kBT/7. So far, we just quoted expressions for the friction coefficients in terms of the linear dimensions of the particles (see eq.(2.1) for a spherical particle and eqs.(2.92- 94) for rod like particles).

In case of interacting Brownian particles, the friction coefficient of each particle depends on the positions and velocities of the remaining Brownian particles" the fluid flow velocity induced by the motion of a Brownian particle affects others in their motion. Brownian particles thus exhibit hydrodynamic interaction. The friction coefficient T is now a matrix which depends on the positions of the Brownian particles, and the microscopic diffusion coefficients D that appear in the Smoluchowski equation follow from the Stokes-Einstein relation D - kBT Y -~, with T -~ the inverse matrix of T. The explicit evaluation of the position dependence of the microscopic diffusion matrices is a complicated hydrodynamic problem.

The present chapter is a treatise of hydrodynamics, aimed at the calculation of friction coefficients and hydrodynamic interaction matrices. Hydrodynamic interaction of spherical colloidal particles in an otherwise quiescent fluid, in a fluid in shearing motion and in a sedimenting suspension are considered. Friction of single long and thin rod like particles is also analysed.

Hydrodynamics is a phenomenological treatment of fluid motion, where processes on the molecular level are not considered. Therefore, only macroscopic quantities like the viscosity and the mass density of the fluid enter the equations of interest. The outcome of this hydrodynamic treatment is used in microscopic equations of motion for the Brownian particles, like the Smolu- chowski equation, which makes explicit reference to position coordinates of the Brownian particles. The large difference in relevant length and time scales between the fluid and the assembly of Brownian particles allows one to consider the fluid on a phenomenological level, without loosing the microscopics for the assembly of Brownian particles.

The mechanical state of the fluid is described by the local velocity u(r, t) at a position r in the fluid and at some time t, the pressure p(r, t) and the mass density p(r, t). All these fields are averages at time t over small volume elements located at the position r. These volume elements must be so small that

5.2. Continuity Equation 229

the mechanical state of the fluid hardly changes within the volume elements. At the same time, the volume elements should contain many fluid molecules, to be able to properly define such averages. In particular we wish to define the thermodynamic state of volume elements, which is possible when they contain a large amount of molecules, and when they are in internal equilibrium, that is, when there is local equilibrium. In this way the temperature field T(r, t) may be defined. The temperature dependence of, for example, the mass density is then described by thermodynamic relations. These thermodynamic relations are an important ingredient in a general theory of hydrodynamics. For our purpose, however, the temperature and mass density may be considered constant, both spatially and in time. Temperature variations due to viscous dissipation in the fluid are supposed to be negligible. At constant temperature, the only mechanism to change the mass density is to vary the pressure. For fluids, however, exceedingly large pressures are needed to change the density significantly, that is, fluids are quite incompressible. Brownian motion is not as vigorous to induce such extreme pressure differences. 1

Assuming constant temperature and mass density leaves just two variables which describe the state of the fluid" the fluid ftow velocity u(r, t) and the pressure p(r, t). Thermodynamic relations need not be considered in this case, simplifying things considerably.

5.2 The Continuity Equation

As was mentioned in the introduction, the density of the fluid may be considered constant, both spatially and in time. Such a constant density poses a restriction on the nature of the fluid flow, since now the number of fluid molecules within some given fixed volume W must be a constant in time, as otherwise the density inside that volume changes in time. The number of fluid molecules which are transported into this volume by the fluid flow through its boundary OW must be equal to the number flowing outwards through 014;.

Clearly, in the more general case of a spatially and timely varying mass density p(r, t), the rate of change of the density is related to the properties of the fluid flow velocity u(r, t). The rate of change of the mass of fluid contained in some arbitrary volume W, which mass is directly proportional

1The assumption of constant temperature and pressure is also a matter of time scales. The relaxation times for local temperature and pressure differences in the solvent are much faster than the Brownian time scale we are interested in here.

230 Chapter 5.

to the number of fluid molecules contained in W, is equal to the mass of fluid flowing through its boundary, in the direction perpendicular to OW. Formally,

t) - - ]o d S . dt w ' Here, dS is an infinitesimal vector directed outwards and normal to 01a2. The minus sign on the right hand-side is added, because the mass in ],V decreases when u is along the outward normal. The time derivative on the left hand-side can be taken inside the integral, while the integral on the right hand-side can be written as an integral over the volume ~V, using Gauss's integral theorem, yielding,

[0 ] ~ bTp(~,t) + v . {p(~, t)u(~, t)) - 0,

where ~' is the gradient operator with respect to r. Since the volume W is an arbitrary volume, the integrand must be equal to zero here. This can be seen by choosing for ~V a sphere centered at some position r, with a (infinitesimally) small radius. Within that small sphere the integrand in the above integral is (almost) constant, so that the integral reduces to the product of the volume of 14) and the value of the integrand at the point r. Hence,

a 0---t p(r, t) + V- {p(r, t )u(r , t)} - 0. (5.1)

This equation expresses conservation of mass, and is usually referred to as the continuity equation.

The above mentioned restriction on the fluid flow to ensure a constant mass density follows from the continuity equation by simply taking p time and position independent, that is,

v . u(~, t) - 0 . (5.2)

Being nothing more than the condition to ensure a constant mass density, this single equation is not sufficient to calculate the fluid flow velocity. It must be supplemented by Newton's equation of motion to obtain a closed set of equations.

Since generally the pressure in a fluid changes from point to point, a necessary condition for the validity of eq.(5.2) is that the density is independent of the pressure. To a good approximation this is indeed the case for most fluids. Suchs fluids are called incompressible. The continuity equation (5.2) is only valid for incompressible fluids and is sometimes referred to as the incompressibility equation.

5.3. Navier-Stokes Equation 231

5.3 The Navier-Stokes Equation

The Navier-Stokes equation is Newton's equation of motion for the fluid flow. Consider an infinitesimally small volume element, the volume of which is denoted as 5r. The position r of that volume element as a function of time is set by Newton's equation of motion. The momentum that is carried by the volume element is equal to p0 (Sr)u(r , t), so that Newton's equation of motion reads,

du( r , t ) = f , po (Sr) dt

where po is the constant mass density of the fluid, so that po (Sr) is the mass of the volume element, and f is the total force that is exerted on the volume element. Since in Newton's equations of motion r is the time dependent position coordinate of the volume element, and dr/dt - u is the velocity of the volume element, the above equation can be written as,

po (6r) r/0u(r, t)

Ot [ + u(r, t ) . Vu(r , t)] - f .

Here, V u is a dyadic product, that is, it is a matrix of which the ij th component is equal to V~uj, with V~ the differentiation with respect to ri, the i th

component of r (see subsection 1.2.1 on notation in the introductory chapter). The total force f on the volume element consists of two parts. First of all,

there may be external fields which exert forces on the fluid. These forces are denoted by (Sr) f~ t ( r ) , that is, fext is the external force on the fluid per unit volume. The second part arises from interactions of the volume element with the surrounding fluid.

The forces due to interactions with the surrounding fluid are formally expressed in terms of the stress matrix E( r , t), which is defined as follows. Consider an infinitesimally small surface area in the fluid, with surface area dS and a normal unit vector ft. The force per unit area exerted by the fluid located at the side of the surface area to which the unit normal is directed, on the fluid on the opposite side of the surface area, is equal to dS �9 E, with dS=fidS. This defines the stress matrix (see fig.5.1). The force of surrounding fluid on the volume element 5r is thus, per definition, equal to,

~a dS' �9 E ( r ' , t ) - f6 d r ' V ' . E ( r ' , t ) - (Sr) V . E ( r , t ) , 5 r r

232 Chapter 5.

ds

f - cIS. >-i,(r,i)

Figure 5.1" Definition o f the stress matrix ~ .

.......... '~..F

X"

Y

where 06r is the boundary of the volume element. We used Gauss's integral theorem to rewrite the surface integral as a volume integral. The last equation is valid due to the infinitesimal size 6r of the volume element at position r. The force fh on the volume element due to interaction with the surrounding fluid is thus given by,

fh(r, t) = (6r) V . ~ ( r , t ) . (5.3)

There are two contributions to the stress matrix �9 a contribution which is the result of pressure gradients and a contribution resulting from gradients in the fluid flow velocity.

Consider first the forces due to pressure gradients. Let us take the volume element 6r cubic, with sides of length 61. The pressure p is the static force per unit area, so that the force on a the volume element in the x-direction is equal tO,

(61) 2 x - -~61, y, z, t) - p(x + 61, y, z, t - - (61) 3 x -~z p(x, y, z, t),

where (6l) 2 is the area of the faces of the cube. The force on the volume element is thus - ( 6 r ) V p ( r , t). We therefore arrive at, V . N = - V p . The contribution of pressure gradients to the stress matrix is thus easily seen to be equal to,

s ( r , t ) - - p ( r , t ) i ,

5.3. Navier-Stokes Equation 233

with I the 3 x 3-dimensional unit matrix. This contribution to the stress matrix is referred to as the isotropic part of the stress matrix, since it is proportional to the unit matrix and therefore does not have a preferred spatial direction.

Next, consider the forces on the volume element due to gradients in the fluid flow velocity. When the fluid flow velocity is uniform, that is, when there are no gradients in the fluid flow velocity, the only forces on the volume element are external and pressure forces. There are friction forces in addition, only in case the volume element attains a velocity which differs from that of the surrounding fluid. The contribution to the stress matrix due to friction forces is therefore a function of spatial derivatives of the flow velocity, not of the velocity itself. This contribution to the stress matrix can be formally expanded in a power series with respect to the gradients in the fluid flow velocity. For not too large gradients (such that the fluid velocity is approximately constant over distances of many times the molecular dimension) the first term in such an expansion suffices to describe the friction forces. The contribution of gradients in the fluid flow velocity to the stress matrix is thus a linear combination of the derivatives Viuj(r , t), where Vi is the derivative with respect to the i th component of r, and uj(r, t) is the jth component of u(r , t).

There are also no friction forces when the fluid is in uniform rotation, in which case the flow velocity is equal to u = 12 x r, with 12 the angular velocity. Such a fluid flow corresponds to rotation of the vessel containing the fluid, relative to the observer. Linear combinations of the form,

Viuj(r , t) + Vjui(r , t ) , (5.4)

are easily verified to vanish in case u = f~ x r. The stress matrix is thus proportional to such linear combinations of gradients in the fluid velocity field.

For isotropic fluids, with no preferred spatial direction, the most general expression for the components ~ j of the stress matrix is therefore,

Y',ij - 770 Viuj + V j u i - ~ i j V . u ( r , t ) + ~o ~ijV. u . (5.5)

The terms ~ V.u( r , t) on the right hand-side are due to the linear combinations 2 V . u (r, t) is introduced to make the expression (5.4) with i = j . The t e r m - ~

between the curly brackets traceless (meaning that the sum of the diagonal elements of that contribution is zero). It could also have been absorbed in the last term on the right hand-side. The constants 770 and ~o, which are scalar quantities for isotropic fluids, are the shear viscosity and bulk viscosity

234 Chapter 5.

of the fluid, respectively. Notice that all terms ~., V- u(r, t) are zero for incompressible fluids. The contribution (5.5) to the stress matrix is commonly referred to as the deviatoric part of the stress matrix.

We thus find the following expression for the stress matrix for an isotropic fluid,

E(r, t) r/o {Vu(r , t ) + [Vu(r, t)] T 2 } - g I V . u(r, t)

+ {r V . u(r, t) - p(r, t)} i , (5.6)

where the superscript T stands for "the transpose of". Using the expression (5.6) for the stress matrix in eq.(5.3), and substitution

into Newton's equation of motion yields the Navier-Stokes equation,

Po 0u(r,t)

Ot + pou(r, t ) . Vu(r, t) -- r/o V2u(r, t) - Vp(r, t)

( 1 ) + r + g,~o v (v . u(r, t))+ f~'(~). (5.7)

For incompressible fluids, for which V- u(r, t) - 0, the Navier-Stokes equation reduces to,

Po Ou(~,t)

Ot + po u(r, t). Vu(r, t) - r/oV2u(r, t) - Vp(r, t) + f~t (r). (5.8)

Together with the continuity equation (5.2) for incompressible fluids this equation fully determines the fluid flow and pressure once the external force and boundary conditions for its solution are specified.

5.4 The Hydrodynamic Time Scale

In chapter 4, where fundamental equations of motion for probability density functions are considered, it is assumed that the realization of a fluid disturbance due to the motion of Brownian particles is instantaneous on the time scale under consideration (the Fokker-Planck or the Brownian time scale). That is, it is assumed there that the fluid flow and pressure disturbances, due to motion of Brownian particles, propagate with such a large velocity, that the flow and pressure can be thought of as being present throughout the fluid, without any time delay on the time scale under consideration. In that case the

5.4. Hydrodynamic Time Scale 235

hydrodynamic interaction matrices are determined by the instantaneous coordinates of the Brownian particles. Here we discuss the propagation velocity of disturbances, and compare the outcome with the Brownian and Fokker-Planck time scale.

There are two kinds of fluid disturbances to be distinguished" shear waves and pressure waves (also called sound waves). Shear waves are propagating tangentially sliding layers of fluid, and pressure waves are propagating pressure differences. The two types of disturbances are discussed in the following.

Shear Waves

A shear wave is induced by pulling a fiat plate with a certain velocity in a direction parallel to that plate. Consider a semi infinite quiescent fluid which is bounded by a flat plate of infinite extent (see fig 5.2a). The plate is located in the xy-plane. At time zero the plate's velocity is zero, and from that time on the plate is pulled along the x-axis with a certain non-zero velocity, v say. This motion of the plate induces motion of the fluid, consisting of sliding layers parallel to the plate. These shear waves propagate into the fluid in the positive z-direction. We solve the Navier-Stokes equation for incompressible fluids (5.8) for small velocities of the plate to obtain the propagation velocity of the shear waves into the fluid. A solution of the Navier-Stokes equation, subject to the appropriate boundary condition, can be found by setting the gradient of the pressure equal to zero. For small velocities of the plate, the Navier-Stokes equation may then be linearized with respect to the fluid flow velocity, yielding,

Ou(r,t) _ ~OV~u(r , t ) , z > O Ot Po

The solution of this partial differential equation, subject to the boundary condition u = v ~ atz - O, is ofthe form u(r, t) - u(z, t ) ~ , with ~ = ( 1 , O, 0). The problem thus reduces to solving the one-dimensional equation,

Ou( z , t ) 0 2

Ot po Oz 2

The initial condition is,

- - - - u ( z , t ) . (5.9)

> 0, t - 0 ) - 0 .

The boundary condition is,

- o , t ) - v .

(5.10)

(5.11)

236 Chapter 5.

~ ' - Z

2=3 i / / / / / / / / / / / / / i , / 7 / Y

O) / / / ...... / / / / / / / / / /

@ Figure 5.2: A fiat plate of infinite extent, located in the xy-plane, induces shear waves on displacement parallel to the xy-plane (a), and sound waves on displacement along the z-direction (b).

The solution of the problem (5.9-11) is constructed in exercise 5.3, with the following result,

u(z t ) = 2v fr162 dq exp{-q2} . ' (5.12)

The typical distance between two Brownian particles, in a moderately concentrated suspension, where hydrodynamic interaction is important, is a of the order 10 x a, say, with a a typical linear dimension of a Brownian particle. According to eq.(5.12), a shear wave traverses such a distance in a time interval of the order,

_(1 )2 Po TH = 4 , 1 0 a - - .

770 (5.13)

The time rH is the hydrodynamic time scale. The Brownian time scale rD on the other hand, is given by (see section 2.3 in chapter 2),

M 2 2 Pp (5.14) T D > > - - - -a ~ ,

"7 9 ~o

with M the mass, 7 the friction coefficient and pp the mass density of the Brownian particle. Since the mass density of the solvent and the Brownian particle are of the same order, the conclusion is that both time scales are of the same order of magnitude,

ro ~ rH. (5.15)

5.4. Hydrodynamic Time Scede 237

On the Brownian time scale, the propagation of shear wave disturbances, due to motion of Brownian particles, may therefore be thought of as being infinitely fast. The fluid flow may be considered as being present, without any time delay, in the entire fluid. Since rn is significantly larger than the Fokker- Planck time scale, however, the approximation of instantaneous realization of shear waves is questionable on the Fokker-Planck time scale.

Sound Waves

A pressure wave, or equivalently, a sound wave, is induced by moving the flat plate of infinite extent (which was considered in the above paragraph on shear waves) in the positive z-direction (see fig.5.2b). This upward velocity is assumed here to be so small, that the change 5p of the pressure and the fluid flow velocity u are small, so that the equations of motion can be linearized with respect to these changes. As will be seen shortly, the propagation velocity is infinite for strictly incompressible fluids. We therefore consider here the more general case of a compressible fluid. The change 5p of the density is also assumed to be small enough to allow for linearization. Furthermore, viscous effects are not essential for the calculation of the velocity of propagation of sound waves. Viscous effects damp the amplitude of sound waves, but do not affect their propagation velocity. Since we are only interested in the propagation velocity, viscous effects are neglected here, that is, the stress matrix (5.6) contains only the pressure contribution.

Due to the symmetry of the problem all functions are only z-dependent. Furthermore, the fluid flow is along the z-direction. We denote this velocity simply by u (z, t).

Suppose that the temperature of the fluid is uniform. The small change of the pressure is then related to the change of the density, as,

%(z t) - % 5p(z, t ) , ' Opo

where the derivative on the right hand-side is that of the equilibrium pressure (as a function of the temperature and the density) with respect to the density. Substitution of this expression into the linearized continuity equation (5.1) and the linearized Navier-Stokes equation (5.7), with r/o - 0 and ~0 - 0, gives,

0 0 u ( z , t ) - o 0-7 p(z, t) + o op o

t) + Opo Oz p(z' t) = o .

238 Chapter 5.

Differentiation of the first of these equations with respect to time, and substitution of the second equation into the resulting expression yields,

0 Op 0 2 ) Ot 2 Opo Oz 2 6p(z, t) - O . (5.16)

The solution of this equation is any function of the form,

6p(z, t) - 6 p ( z - v t ) , (5.17)

with,

v - ff-Po" (5.18)

This is a disturbance that propagates with a velocity v in the positive z- direction, without changing its shape. This is why eq.(5.16) is referred to as a wave equation. In reality, the shape of the disturbance changes due to viscous damping, which is neglected here. For strictly incompressible fluids the pressure becomes infinite on slightly increasing the density, so that v - c~. For real fluids Op/Opo is large (for water, 2.2 106 m2/s 2 and for an organic solvent like cyclohexane, 1.1 106 m2/s2), corresponding to a large propagation velocity (for water 1500 m/s and cyclohexane 1000 m/s) . The time that a sound wave requires to propagate over a typical distance of a few #m's is of the order 10 -9 s, which is in turn of the order M / 7 . This is smaller than the Brownian time scale, but larger than the Fokker-Planck time scale. Sound wave velocities are somewhat larger than propagation velocities of shear waves, so that the latter determine the hydrodynamic time scale.

The conclusion is that for interacting Brownian particles the approximation of instantaneous realization of fluid disturbances is correct on the Brownian time scale, but questionable on the Fokker-Planck time scale.

5.5 The Creeping Flow Equations

The different terms in the Navier-Stokes equation (5.8) can be very different in magnitude, depending on the hydrodynamic problem under consideration. In the present case we are interested in fluid flow around small sized objects (the colloidal particles). Let us estimate the magnitude of the various terms in the

5.5. Creeping Flow Equations 239

Navier-Stokes equation for this case. A typical value for the fluid flow velocity is the velocity v of the colloidal objects. The fluid flow velocity decreases from a value v, close to a Brownian particle, to a much smaller value, over a distance of the order of a typical linear dimension a of the particles (for spherical particles a is the radius, for a rotating rod a is the length of the rod). Hence, typically, [ V2u 1,~ v ia 2. Similarly, [ u . Vu [,~ v2/a. The rate of change of u is v divided by the time it takes the colloidal particle to loose its velocity due to friction with the fluid. This time interval is equal to a few times M / 7 , with M the mass of the colloidal particle and 7 its friction coefficient (see chapter 2). Introducing the rescaled variables,

U I - - U / V ,

r' - r /a ,

t' - t / ( M / 7 ) ,

transforms the Navier-Stokes equation (5.8) to,

~ v O U I

Po M Ot' P~ u' V 'u ' r/oVv,2 u, 1V, p + f~:~t

F . ~ . . . . . . a a 2 a

where V' is the gradient operator with respect to r'. Introducing further the dimensionless pressure and external force,

a pl _ ~ p ,

T]o V

a 2 f, ext = ~ f e z t

r/oV

transforms the Navier-Stokes equation further to,

a27 Ou' Po MTlo c3t'

F Re u ' . V'u ' - V'2u ' - V'p' + f ,~t

The dimensionless number Re is the so-called Reynolds number, which is equal to,

Re - po a v . (5.19) 7/0

By construction we have,

l u'.V'u'l lV'2u'J 1.

240 Chapter 5.

Hence, for very small values of the Reynolds number, the term ~, u �9 V u in the left hand-side in eq.(5.8) may be neglected. Furthermore, for spherical particles we have "7 - 67rr/oa so that poa27/M71o - 9po/2pp ,.~ 9/2, with pp the mass density of the Brownian particle. The prefactor of Ou'/at ' is thus approximately equal to 9/2. The time derivative should generally be kept as it stands, also for small Reynolds numbers. Now suppose, however, that one is interested in a description on the diffusive time scale TO >> M / 7 . For such times the time derivative Ou'/at ' is long zero, since u goes to zero as a result of friction during the time interval M/.y. One may then neglect the contribution to the time derivative which is due to relaxation of momentum of the Brownian particle as a result of friction with the solvent. The remaining time dependence of u on the Brownian time scale is due to the possible time dependence of the external force, which is assumed to vary significantly only over time intervals equal or larger than the Brownian time scale. The value of the corresponding derivative a u / 0 t can now be estimated as above �9 the only difference is that the time should not be rescaled with respect to the time M / 7 , but with respect to the Brownian time scale rD. We now have, t' - t/TD, U' -- U/V, and [au'/Ot' I~ 1. The transformed Navier-Stokes equation now reads,

9 Po M / 7 0 u ' 2 pp T D Ot'

+ Re u ' . V 'u ' - V '2u ' - V'p' + f ,~ t

and all derivatives of the fluid flow velocity u' are of the order 1. Since ro >> M/'y, the time derivative due to changes of the fluid flow velocity as a result of the timely varying external force may also be neglected.

For small Reynolds numbers and on the Brownian time scale, the Navier- Stokes equation (5.8) in the original unprimed quantities therefore simplifies to,

Vp(r, t) - 7/0V 2 u(r, t) - ff~t(r). (5.20)

This equation, together with the incompressibility equation (5.2), are the creeping flow equations. "Creeping" refers to the fact that the Reynolds number is small when the typical fluid flow velocity v is small.

A typical value for the velocity of a Brownian particle can be estimated 1 M < v 2 3 kB T (kB is Boltzmann's con- from the equipartition theorem, ~ > - ~

stant and T is the temperature). Estimating v .~, x/'< v 2 >, using a typical mass of 10 -~r kg for a spherical particle with a radius of 100 nm and the density and viscosity of water, the Reynolds number is found to be equal to 10 -2 .

5.6. The Osecn matrix 241

Hydrodynamic interaction matrices can thus be calculated on the basis of the creeping flow equations.

For small Reynolds numbers and on the Brownian time scale inertial effects of the fluid flow are unimportant, that is, the left hand-side of the Navier-Stokes equation (5.8) may be neglected. According to the creeping flow equations, the velocity of the fluid is then directly proportional to the external force on the fluid. Bacteria, which are of a colloidal size, thus experience the pre- Newtonian mechanics of Aristotle (on the Brownian time scale) : velocity is proportional to force. When the bacteria stops swimming, its velocity is zero instantaneously, or more precise, relaxes to zero within a very small time interval of the order M/7.

Notice that the inertial terms in the Navier-Stokes equation can be neglected only on the Brownian time scale. Hydrodynamic friction functions as calculated from the creeping flow equations (5.2,20) can therefore be used in the Smoluchowski equation but not in the Fokker-Planck equation. On the Fokker-Planck time scale only the term ,-~ u. Vu on the right hand-side of the Navier-Stokes equation can be omitted (for small Reynolds numbers), but the time derivative a u / a t must be kept. Hydrodynamic friction functions on the Fokker-Planck time scale should therefore be calculated from the equation,

P o ~ au(,,t)

0t = - V p ( r , t ) + ~oV 2 u(r, t ) + f ~ t ( r ) .

Hydrodynamic friction functions on the Fokker-Planck time scale are therefore time dependent. Such hydrodynamic friction functions are not considered here. From now on, we will restrict ourselves to the Brownian time scale.

5.6 The Oseen Matrix

An external force acting only in a single point r' on the fluid is mathematically described by a delta distribution,

f~ ' ( r ) - f o 6 ( r - r ' ) . (5.21)

The prefactor fo is the total force fdr ' f~t( r ' ) acting on the fluid. Since the creeping flow equations are linear, the fluid flow velocity at some point r in the fluid, due to the point force in r', is directly proportional to that point force. Hence,

u(r) - T(r- r'). fo.

242 Chapter 5.

The matrix T is the Oseen matrix. This matrix connectsthe point force at a point r ' to the resulting fluid flow velocity at a point r. That T is only a function of the difference coordinate r - r ' follows from translational invariance, or to put it in other words, from the fact that the choice of the position of the origin is of no significance. Similarly, the pressure at a point r is linearly related to the point force,

p(r) - g ( r - r ' ) . f0.

The vector g is referred to here as the pressure vector. Consider now an external force which is continuously distributed over the

entire fluid. Due to the linearity of the creeping flow equations, the fluid flow velocity at some point r is simply the superposition of the fluid flow velocities resulting from the forces acting in each point on the fluid,

- / dr' T(r - r ' ) - f ~ t ( r ' ) . (5.22) u ( r )

The same holds for the pressure,

p(r) - f dr' g ( r - r ' ) . f"~t(r'). (5.23)

In mathematical language, the Oseen matrix and the pressure vector are the Green's functions of the creeping flow equations for the fluid flow velocity and pressure, respectively. Once these Green's functions are known and the external force is specified, the resulting fluid velocity and pressure can be calculated via the evaluation of the above integrals. The calculation of the Green's functions is thus equivalent to solving the creeping flow equations, provided that the external forces are known.

Let us calculate the Oseen matrix and pressure vector. To this end, substitute eqs.(5.22,23) into the creeping flow equations (5.2,20). This leads tO,

f j dr' [V. f='(r') - 0,

/ dr' [ V g ( r - r') - r/oV2T(r - r') - I6 ( r - r ' ) ] . f~ ' ( r ' ) - 0 ,

where I is the 3 x 3-dimensional unit matrix. Since the external force is arbitrary, the expressions in the square brackets must be equal to zero, so that the Green's functions satisfy the following differential equations,

V . T ( r ) - 0 , (5.24)

V g ( r ) - r / o V 2 T ( r ) - i 6 ( r ) . (5.25)

5.6. The Oseen matrix 243

A single equation for the pressure vector is obtained by taking the divergence of the second equation, with the use of the first equation,

V 2g(r) - V'. iS(r) - V6(r ) .

Now using (see exercise 5.1),

1__ V2 _1 = - 6 ( r ) , (5.26) 47r r

it follows that,

1 V 1 - q- G ( r ) ,

g(r) - 47r r

where G is a vector for which V2G=0. It is shown in exercise 5.2 that, with the condition that G ~ 0 as r ~ c~, this implies that G - 0. Hence,

1 1 1 r g(r) - - - - V - =

4~r r 47r r 3" ( 5 . 2 7 )

The differential equation to be satisfied by the Green's function for the fluid flow velocity (the Oseen matrix), is found by substitution of eq.(5.27) into eq.(5.25), and using eq.(5.26),

V2[ 14-~r-1~- r/oT(r)] = [rr

.

An obvious choice for the term between the square brackets on the left hand- side of the above expression is of the form,

1 1 1 , 1 r r i - yoT(r) - ao~-~I + a~ r-- ~ r- ~ , 47rr

with O~0,1, n and m constants. These constants can indeed be chosen such that this Ansatz is the solution of the differential equation (with the boundary condition that T(r) --+ 0 as r ~ c~). A somewhat lenghty, but straightforward calculation yields, [ rr] 1 1 ~ + (528) - "

This concludes the determination of the Green's functions for the creeping flow equations. These functions, the Osccn matrix in particular, play a central role in the calculation of microscopic diffusion matrices.

In section 5.8, the microscopic diffusion matrices are calculated directly from the above expressions for the Green's function, in case the distance between the Brownian particles is large. This is the leading term in an expansion with respect to the inverse distance between the particles. Higher order terms in this expansion arc calculated in section 5.12.

244 Chapter 5.

5.7 Flow past a Sphere

For the calculation of hydrodynamic interaction matrices we shall need expressions for the fluid flow as a result of translation or rotation of a spherical Brownian particle. The following two subsections are devoted to the calculation of these fluid flow velocity fields.

Throughout this chapter we assume stick boundary conditions. That is, it is assumed that the velocity of the fluid at the surface of the Brownian particles is equal to the velocity of the corresponding surface element on the particles' surface. The fluid is thus assumed to "stick" onto the surface of the Brownian particles due to attractive interactions between the fluid and the core material of the Brownian particles. In that case the fluid flow velocity u(r) , for positions r on the surface of the Brownian particle, is related to the translational velocity v and the angular velocity [2 of that particle as,

u(r) - v + f ~ • r e 0 V , (5.29)

with rp the geometrical center of the spherical Brownian particle, its position coordinate, and O V its surface.

In the present case of a fluid containing Brownian particles in motion, the force field f~ t ( r ) in all previous equations represents the forces which elements on the surface of each of the Brownian particles exert on the fluid. These forces are concentrated on the surfaces of the Brownian particles. The expressions (5.22,23) for the fluid flow velocity and the pressure are now integrals ranging over the surface OV of the spherical Brownian particle,

u(r) - ~ v d S ' T ( r - r ' ) . f ( r ' ) , (5.30)

- ~v dS' g(r- r'). f(r'), p(r) (5.31)

where f(r ' ) is now the force per unit area that a surface element of the Brownian particle located at r ' exerts on the fluid.

There are two possible routes for the calculation of the fluid flow velocity. Via the differential creeping flow equations (5.2,20) or via the above integral Green's function representation (5.30,31) for the solution of the Creeping flow equations. Both routes are considered in the following.

The fluid flow velocity can be calculated from eq.(5.30) once the forces which the particles exert on the fluid are known. From the definition of the stress matrix, these forces are equal to E(r ' ) �9 fi, with fi the outward normal

5. 7. Flow past a Sphere 245

on a sphere. The stress matrix is in turn related to the fluid flow velocity and pressure as given in eq.(5.6). The Green's function representation (5.30,31) is thus an integral equation which is equivalent to the differential creeping flow equations. The advantage of the integral representation is, that one can substitute a guess for the forces, calculate the integral, and check whether the resulting expressions satisfy the boundary conditions of the problem. Such a procedure is feasible for a single sphere in an unbounded and otherwise quiescent fluid.

5.7.1 Flow past a Uniformly Translating Sphere

Consider a sphere with a constant velocity v in an unbounded and otherwise quiescent fluid. Without loss of generality we may take the center of the sphere at the origin.

The boundary condition at infinity for this problem is,

u(r) --. O, r --. oc . ( 5 . 3 2 )

The boundary condition on the surface of the sphere is the stick boundary condition (5.29), which, for the non-rotating sphere at the origi n, reads,

u(r) - v , r E O V ~ (5.33)

with OV ~ the spherical surface of radius a with its center at the origin. Let us first follow the route via the Green's function integral representation

(5.30) for a single sphere, by making a guess for the forces which the surface elements of the sphere exert on the fluid. The simplest choice is a constant, independent of the position r ~ on the surface of the sphere, and proportional to the velocity v of the sphere. That is, the force is proportional to the local fluid flow in the absence of the sphere,

r

f (r ' ) - 47ra 2 v , (5.34)

with c a constant, which must be chosen, if possible, such that the above mentioned boundary conditions are satisfied. Substitution of the Ansatz (5.34) into eq.(5.30) and using the expression (5.28) for the Oseen matrix, gives,

C l ~ o d S ' l [ ~ I + u(r) - 47ra 287r~70 v0 ] r - r ' I

(r- r')(r- r')] I r - r ' 12 �9 v . ( 5 . 3 5 )

246 Chapter 5.

The evaluation of the integral on the right hand-side is deferred to appendix A. The result can be made to satisfy the boundary conditions (5.32,33) with the choice, c = 67rr/oa. The Ansatz (5.34) is thus the correct one to obtain the solution. By simply replacing the position r by r - rp (with rp (t) ,-~ v t the position of the sphere) then yields the fluid flow due to translational motion of the sphere in an otherwise quiescent fluid,

u(r) { [ rp /rr /] 3_ a I + 12 (5.36) 4 [ r - r p I ] r - r v

1( a )3[i_3(r- ')(r-rp)]} +4 ]r -rp] r--rpi2 .V.

This expression can also be obtained directly from the creeping flow equations as follows. The continuity equation (5.2) is satisfied for fluid flow velocities of the form, u(r) - V x A(r) . Now suppose that the coordinate frame is inverted, that is, suppose that the problem is transformed to new coordinates (x, y, z)---} ( - x, - y, - z). Clearly, both u and v are then changed in sign. Since A is linear in v, it follows that A is the product of v with a vector that also changes its sign on inversion of the coordinates. Such a vector is V f ( r ) , with f a scalar function of r - ! r I. We thus arrive at the following form for the fluid flow velocity,

u(r) - V x ([Vf(r)] x v) - - v V 2 f ( r ) + (v . V)Vf ( r ) . (5.37)

Taking the curl V x from both sides of the eq.(5.20), with f~ t _ 0 for points inside the fluid, yields,

V • V2u(r) - V 2 IV • (V • A(r))] - V 2 [V(V. A ( r ) - V2A(r)] - 0 .

Since,

V . A(r) - V . (Vf(r) • v) - v . (V • Vf(r)) - 0,

the above differential equation for A reduces to, V2V2A(r) - 0. equation is satisfied whenever,

This

VV2V2f( r ) - 0 ,

since A - V f x v. A single integration gives, V2V2f(r)=constant. Since the fluid flow velocity tends to zero at infinity, and is related to second order

5. 7. Flow past a Sphere 247

derivatives of f , see eq.(5.37), fourth order derivatives of f are zero at infinity. The above cons tan t is thus equal to zero,

V2V2f(r) -- O.

Since outside the sphere, where r > 0, we have according to eq.(5.26) that V 21=0, which is also easily verified by direct differentiation. Thus, V2f is

of the form,

V2 f (r) Co ~ -t- C1 ~ r

with Co and C 1 constants. equation gives,

Since V2r=~, and X72r2=6, integration of this

f (r) c2 1 1 r2 --" --r "4- C3 + ~COT ~- ~C1

The constants c,, n - 0, 1,2 or 3, must now be determined such that the boundary conditions (5.32,33) are satisfied. Substitution of the above result for f into eq.(5.37) for the fluid flow velocity shows that the boundary conditions

3 1 3 are satisfied for co - - T a , C1 -- 0 and c2 - - ~ a , while the constant c3 is of no relevance, since the fluid flow velocity contains only derivatives of f . Replacing r by r - rp reproduces eq.(5.36).

The friction force F h that the fluid exerts on the sphere can in principle be calculated from the integral,

~6 r l F ~ : - d S ' E ( r ' ) . ~ . V o

A minus sign is added here, since F h is the force exerted by the fluid on the particle, while f is the force exerted by the particle on the fluid. The integral may be evaluated by substitution of eq.(5.36) into the expression (5.6) for the stress matrix, with V . u = 0. The pressure is found from Vp=r/oV2U, which follows from the creeping flow equation (5.20). This a lengthy calculation which can be avoided by recognizing that the choice for the force density (5.34) is a unique choice. That is, every other choice yields a different result for the fluid flow velocity. We found for the constant in eq.(5.34) the value c - 67rr/oa, so that one immediately obtains,

F h = - ~ v ~ dS' f(r ') - -67rr/oa v . (5.38)

This is Stokes's friction law for translational motion of a sphere.

248 Chapter 5.

5.7.2 Flow past a Uniformly Rotating Sphere

Consider a sphere with its center at the origin, rotating with a constant angular velocity f~. The boundary condition at infinity here is,

u(r) --, O , r - - , cx~. (5.39)

The stick boundary condition on the surface is,

u(r) - f l x r , r E O V ~ (5.40)

The simplest reasonable choice for the force which a surface element of the sphere exerts on the fluid, is a force that is proportional to the velocity of that surface element.

C f(r) - 47ra--- ~ 12 x r , (5.41)

with c an adjustable parameter, which should be chosen, if possible, to satisfy the boundary conditions (5.39,40). Substitution into eq.(5.30) yields,

c 1 dS' 1 i + r' ( x . u(r) - 4~ra 287r,o vo I r - r ' l I r - 12

(5.42) The integral is evaluated in appendix B. The result can be made to satisfy the boundary conditions (5.39,40) with the choice, c=127r~1oa. The following expression for the fluid due to a rotating sphere is then found,

u ( r ) - ( a ) a . x r . (5.43)

This flow represents sliding layers of fluid with an angular velocity equal to I't aa/r a, with r the radius of the spherical layer.

This result is obtained from the differential creeping flow equations (5.2,20) as follows. The fluid flow velocity may be expected to rotate along with the sphere, with an angular velocity that decreases with the distance to the sphere. Let f(r)f~ denote the angular velocity of the fluid at a distance r. The fluid flow velocity is then of the form,

u(r) - f ( r ) n x r = n x ( f ( r ) r ) .

Substitution into the continuity equation (5.2) yields,

V . [fl x (f(r)r)] - f~. [V x (f(r)r)] - O,

5. 7. F l o w past a Sphere 249

hence, V x ( f (r) r) - 0. This equation is satisfied when there is a function h(r) such that, f ( v ) r = V h ( r ) . The fluid flow velocity is thus of the form,

u(r) - ft • Vh( r ) . (5.44)

Taking the curl V x from both sides of the creeping flow equation (5.20) (with f~,t _ 0), and substitution of (5.44) into the resulting expression gives,

V x V2u(r) - V x [ft x VV2h(r)] - O.

This equation is satisfied when V2h(r)=0, which is the case for (see the discussion in the previous subsection),

h(r) ~ -~- C1 ~ r

with co and c~ constants, which should be chosen such that the boundary conditions (5.39,40) are satisfied. Substitution of this result into eq.(5.44) shows that the boundary conditions are satisfied for Co=--a 3, while Cl is not relevant, since the fluid flow velocity is proportional to the derivative of h. With this value of co, the result (5.43) for the fluid flow velocity is reproduced.

The rotational friction coefficient % is defined as the proportionality constant between the torque T h that the fluid exerts on the sphere and its angular velocity ft (see also section 2.8 on rotational motion in chapter 2),

T h = - % Ft .

Analogous to the calculation of the translational friction coefficient in the previous subsection, the rotational friction coefficient for a spherical particle can be calculated from the integral,

~5 rl T h = - d S ' r ' • E(r ' ) �9 ~S , V o

by substitution of eq.(5.43) into the expression (5.6) for the stress matrix. The pressure is found from Vp=~7oV2U, which follows from the creeping flow equation (5.20). Notice that a minus sign is added here, because T h is defined as the hydrodynamic torque which is exerted by the fluid on the sphere. This lengthy calculation can be avoided, by recognizing that the choice for the force density (5.41) is a unique choice. We found for the constant in eq.(5.41) the

250 Chapter 5.

value c - 127r~?oa, so that, with the use of r' x (f~ x r ' )=( r ' )2 f~- r ' r '- f~, one immediately obtains,

Th = -- ~vo dS' r' x f(r ') - - 871"r/oa3['~ . (5.45)

The rotational friction coefficient is thus equal to % - 87r~oa 3. This is Stokes's friction law for rotational motion of a sphere.

5.8 Leading Order Hydrodynamic Interaction

Before setting up a general procedure for the calculation of hydrodynamic interaction matrices, let us discuss a simple approximation which is almost an immediate consequence of the Green's function representation of the fluid flow velocity (5.22).

For the calculation of hydrodynamic interaction matrices for large separations between the Brownian particles, these particles can be considered as point-like. For such point-like particles rotations are of no importance, and the calculation becomes quite simple. This calculation is discussed in the present section. Hydrodynamic interaction of particles which are not very far apart is discussed in subsequent sections.

Remember that we are looking for expressions for the 3 x 3-dimensional microscopic diffusion matrices D ij, w h i c h by definition connect the total forces F/h, exerted by the fluid on the i th Brownian particle, to the velocities vj of the Brownian particles (see also eqs.(4.9,34)),

Vl D l l D I 2 "-"

v2 D21 D22 " '"

VN DN1 DN2 " '"

D I N F1 h D2N . F h

D N N FhN (5.46)

This expression is valid on the Brownian time scale and for small Reynolds numbers, as discussed in the sections 5.4,5. Coarsening to the Brownian time scale and for a small Reynolds numbers, a linear relation between velocities and forces is ensured. The instantaneous fluid disturbance approximation renders each of the time dependent quantities (velocities, forces and position coordinates) at equal times. The microscopic diffusion matrices are functions of the position coordinates of all N Brownian particles in the system.

5.8. Leading Order Interaction 251

As for the single sphere problem considered in the previous section, we assume stick boundary conditions for all N spheres. The fluid flow velocity u(r) for positions r on the surface of the i th Brownian particle is then related to the translational velocity vi and the angular velocity ~2i of that particle as,

u(r) - vi + f~i x ( r - r i) , r E 0V/, (5.47)

with r~ the geometrical center of the spherical i th Brownian particle, its position coordinate, and 0V/its surface.

The starting point for the calculation of the microscopic diffusion matrices is the Green's function representation (5.22) of the creeping flow equations. In the present situation, the external force fext is due to forces that surface elements of the Brownian spheres exert on the fluid, just as for the single sphere problems that were discussed in the previous section. For the multi sphere problem considered here, the integral in eq.(5.22) is now a sum of integrals ranging over the surfaces 0 ~ , j - 1 , . . . , N of the N spherical Brownian particles,

N P

u(r) - ~ ~0v~ dS' T(r - r ' ) . f j ( r ' ) , (5.48)

N P

p(r) E ~_ dS' g ( r - r ' ) . f j ( r ' ) , (5.49) j=l Jov~

where fj is the force per unit area that a surface element of Brownian particle j exerts on the fluid.

For stick boundary conditions, the two expressions (5.47) and (5.48) must coincide for positions r located on the surface of the i th Brownian particle. Hence,

N P

vi + f~i x ( r - ri) - j=IE ~ovjdS' T ( r - r ' ) . fj(r ') , r e 0Vi. (5.50)

Since this equation is valid for any position r on the surface 0Vi of particle i, both sides can be integrated over that surface. Due to symmetry, the rotational component on the left hand-side drops out, and we have,

1 ~o dSfio d S ' T ( r - r ' ) . f i ( r ' ) (5.51) v i = 4 7 r a 2 14

+47ra 21 ~ ~ y i d S ~ 0 v , dS, T ( r _ r , ) . f j ( r , ) . j#i

252 Chapter 5.

r v

z

Figure 5.3" Definition of the positions R and R' on the surface of Brownian particles relative to their position coordinates ri and rj, respectively.

It is shown in appendix A that,

~o dST(r-r') - i 2a v, -~o' for r' e 017/. (5.52)

The first term on the right hand-side of eq.(5.51) is thus equal to,

1 ~o d S ~ d S ' T ( r - r ' ) . f i ( r ' ) - - 1 Fh 47ra 2 ~ ~ 6rr/oa '

where the total force that the fluid exerts on the i th Brownian particle is equal to,

Fh(t) -- -- ~v~ dS' f / ( r ' ) . (5.53)

The double surface integrals in the second line on the right hand-side of eq.(5.51) can be approximated, in case the distance between the Brownian particles is large, as follows. First, the integrations are performed with respect to the translated coordinates R - r - r i and R ' - r ' - r j (see fig.5.3). Let 0 V ~ denote the spherical surface 0V/with its center at the origin. The integrals on the right hand-side of eq.(5.51) are written as,

1 { dS{ d S ' T ( R - R ' + r i - r j ) . f j ( R ' + r j ) . 47ra 2 j~yo j~y o

5.9. Fax6n's Theorems 253

Now suppose that the distance [ ri - rj [ between the Brownian particles i and j is much larger than I R - R' I < 2a. The Oseen matrix T ( R - R' + ri - rj) may then be replaced, to a good approximation, by T(ri - rj). With eq.(5.53) it then follows that,

1 dS dS ' T ( r - r ' ) . f j(r ' ) ~ - T ( r i - rj) �9 F j . 47ra 2 v~ v~

For these large separations between the Brownian particles, eq.(5.51) can thus be approximated as,

1 N h h vi = - ~ F i - ~ T(ri~,. r j ) . F j . (5.54)

67rr/oa jei

Comparison with the definition (5.46) of the microscopic diffusion matrices gives,

Dii - D o i , (5.55) 3 a

[i+ j <5.56) D i j - k B T T(ri - rj) - - 4 D o - - r i j , , r i j

where rij - r i - rj is the distance between the spheres i and j , and rij = rij / rij is a unit vector. Furthermore, Do - kBT/67rrloa " this expression for the Stokes-Einstein diffusion coefficient Do was already introduced in previous chapters.

Notice that our earlier result (5.38) for the translational friction coefficient of a single sphere, "7 = 67r~70a, is rederived here by integration of the Green's function representation of the creeping flow equations.

The above expressions, the Oseen approximation for the microscopic diffusion matrices, are valid for large distances between the Brownian particles, that is, for small values of a/vi i . These results are the leading terms in an expansion with respect to a/r i j . The next higher order terms are discussed in subsequent sections.

5.9 Fax6n's Theorems

Consider a fluid with a flow velocity field u0(r). Suppose a sphere is immersed in that fluid. Fax6n's theorems express the translational and rotational velocity that the sphere acquires in terms of Uo. These theorems can be used to calculate

254 Chapter 5.

the microscopic diffusion matrices" the fluid flow u0 in the neighbourhood of a given sphere is then the fluid flow velocity that is induced through the motion of other spheres.

Fax~n's theorems are derived from eq.(5.50) for a single sphere (N = 1), to which the homogeneous solution uo(r) of the creeping flow equations is added to the right hand-side,

vp + ftp • ( r - rp) - uo( r )+ ~ovdS' T ( r - r ' ) . f ( r ' ) , r E vOV, (5.57)

where OV is the surface of the sphere with its center at the position rp, vp its translational velocity and 12p its angular velocity. Furthermore, f is the force per unit area that a surface element of the sphere exerts on the fluid after immersion of the sphere in the fluid flow velocity field u0. Integration of eq.(5.57) over 0V, using the result (5.147) in appendix A for the integral of the Oseen matrix, gives,

1 h 1 ~0 dSuo(r). (5.58) vp = -67ryoaFp + 47ra 2 v

The fluid velocity field Uo is now Taylor expanded around the center of the sphere, r - rp,

1 ( r - r p ) ( r - r , ) " VpVpuo( rp )+ . . . , uo(r) - uo(rp) + ( r - r , ) �9 V p u o ( r , ) + :

(5.59) where Vp is the gradient operator with respect to rp. Due to the spherical symmetry of the surface O V, odd terms in the components of the vector ( r - rp) do not contribute to the integral in eq.(5.58). Substitution of the Taylor expansion into eq.(5.58) yields (for mathematical details, see exercise 5.7),

Vp 1 h

67rr/o------~Fp + uo(rp)+ 6a2Vp:uo(rp)

-t-V2pV2p [(...)uo(rp)-I-...-I-(-..)V2p .. . V2uo(rp) -I-...] .

The last term on the right hand-side is equal to zero. This can be seen from the creeping flow equations (5.2,20). Taking the divergence of eq.(5.20), noting that in the part of the fluid considered here ff~t ( r ' ) - 0, and using eq.(5.2), gives

2 V po=0, with po the pressure in the fluid without the sphere being immersed.

5.10. The Rodne-Prager matrix 255

Operating with the Laplace operator on eq.(5.20) thus gives, V2V2uo=0. The above expression thus reduces to Fax~n's theorem/'or translational motion,

1 h 6a2 2 v v = -67rr/oaFr, + uo(rp)+ Vpuo(rv) �9 (5.60)

Notice that in case uo ( r ) - 0, this reproduces Stokes's friction law (5.38). The rotational analogue of eq.(5.60) can be obtained similarly from eq.(5.57),

by multiplying both sides with ( r - rp) x and then integrating over the spherical surface OV (for mathematical details, see exercise 5.7),

3 a2~"~p _ 1 f

~ dS ( r - rp) x uo(r) (5.61) 47ra 2 _~v

1 dS ~ dS' (r - rp) x [T(r - r ' ) . f(r')] .

-1-47ra2 JOV JOV

Only the second term in the Taylor expansion (5.59) survives in the first term on the right hand-side. The first term in the Taylor expansion vanishes because of symmetry, while the third and higher order terms vanish because V2V2uo(r)=0. The first term on the right hand-side of eq.(5.61) is thus equal to (for mathematical details, see exercise 5.7),

1 f 1 ~_ a s ( r - r .) x uo(r) - ---a ~ V~ x uo( r . ) .

47ra 2 Joy 3 (5.62)

The second integral on the right hand-side of eq.(5.61) is related to integrals that were evaluated in appendix A and B, as is shown in appendix C. The result for that second term is proportional to the torque (see eq.(5.156)). Using this result finally leads to Faxdn's theorem for rotational motion,

1 1 f~v = -87rr/oa-----~ Tv a + ~Vp x uo(rv). (5.63)

Notice that this reproduces Stokes's friction law (5.45) in case uo(r) - 0.

5.10 One step further : the Rodne-Prager Matrix

One way to calculate the microscopic diffusion matrices, as a series expansion in the inverse distance between two Brownian particles, is by iteration. This method is known as the method of reflections. In the absence of hydrodynamic interaction, two particles (i and j say) have a translational velocity as given by

256 Chapter 5.

Stokes's law (5.38) �9 v i=- f lDoF h, and similar for particle j. The rotational velocity of a spherical particle in uniform translational motion in an otherwise quiescent fluid is zero. The fluid flow induced by the translational motion of particle i is given by eq.(5.36), with the velocity v equal to the above expression for v~. The effect of this flow field on the translational motion of particle j can be found from FaxOn's theorem (5.60), with Uo equal to the fluid flow field induced by particle i,

vj - -~Do ( F h + [1+ ~a2V~] M ( r j - r i ) . F h } ,

where,

3a [~ r_~] l ( a ) a [~ r~] - - + + - - 3 , M(r) 4 r 4 r

is the matrix appearing in the expression (5.36) for the fluid flow field induced by a uniformly translating sphere in an otherwise quiescent fluid. Comparing this expression with the definition (5.46) of the microscopic diffusion matrices, it is found that,

D , - DoI , (5.64)

- [1§

_ _ 1 a [I a i - o ~ o ] i C j , 3 o [ i + - ,

= Do -~ ro

with r 0 - ri - rj and ~ij - rij / r i j . This is an expression for the diffusion matrices that goes one term further than the leading order Oseen approximation which was discussed in section 5.8. The matrix on the right hand-side of eq.(5.65) is usually referred to as the Rodne-Prager matr/x.

This result is the first step in an iterative process. The next step would be the calculation of the flow field induced by particle j , the first order "reflected fluid flow field", and to use that field in Fax6n's theorem to obtain the translational velocity of particle i. This then leads to an expression for the diffusion matrices which is valid up to higher order in the inverse distance than the above Rodne- Prager approximation. This procedure can be repeated indefinitely and is known as the m e t h o d o f reflections. The method of reflections is discussed in detail in section 5.12.

5.11. Rotational Relaxation 257

5.11 Rotational Relaxation of Spheres

For a calculation which goes beyond the Rodne-Prager approximation that is discussed in the previous section, rotational motion of the spheres must be taken into account. In this section it is shown that, on the Brownian time scale, the torque exerted by the fluid on the spheres may be set equal to zero.

As discussed in chapter 2, the momentum coordinate of a spherical Brow- nian particle relaxes to equilibrium with the heath bath of solvent molecules on a time scale which is much smaller than the Brownian or diffusive time scale. As a result, the total force on each spherical Brownian particle is equal to zero on the Brownian time scale. In describing hydrodynamic interaction between Brownian particles, both translational and rotational motion are of importance, since both induce a fluid flow velocity that affects other particles in their motion. Due to the spherical geometry of the Brownian particles, the torque exerted by the fluid on each Brownian particle is also the total torque. Interactions, other than hydrodynamic interaction, do not give rise to torques due to spherical symmetry. In analogy with a total zero force, it is thus tempting to set the torque exerted by the fluid on each Brownian particle equal to zero on the Brownian time scale. This is justified when the relaxation time for rotational motion of a spherical particle is of the same order or smaller than the relaxation time for translational motion. The latter was found in chapter 2 to

2a2 be equal to M / 7 = ~ pp/rjo, with M the mass and O,= 67rr/oa the translational friction coefficient of the Brownian particle, a its radius, pp its mass density and 7/0 the viscosity of the fluid. Let us now determine the relaxation time for rotational motion of a sphere. Newton's equations for rotational motion were derived in section 2.8 (see eqs.(2.81,83,84)),

d J I d t - 7 - ,

J - I ~ . f t ,

I ~ - f v o d r p ( r ) [ v 2 i - r r ] .

The summation over molecules in eq.(2.84) is replaced here in the last line by an integral ranging over a spherical volume V ~ with its center at the origin, where p(r) is the mass density at the point r inside the spherical volume. Furthermore, J is the angular momentum, T is the torque, I ~ is the inertia matrix, and f~ is the angular velocity (these are the rotational analogues of translational momentum, force, mass and velocity, respectively). The inertia matrix is easily evaluated for a Brownian sphere with a homogeneous mass

258 Chapter 5.

density pp,

i~ 87r aS i = 15 PP "

Now consider a sphere with a certain angular velocity f~o at time zero. As was already discussed in section 2.8, the torque that the fluid exerts on the sphere is equal to -%f~(t ) at each time t, with % the rotational friction coefficient, which was calculated in subsection 5.7.2 �9 % =87rrloa 3. Using this in Newton's equations of motion, it is found that,

~ ( t ) - [2oexp{ 15r/0 } ppa2 t .

The rotational relaxation time is thus equal to ~a2pp/rlo. This relaxation time is of the same order as the relaxation time for translational motion

2 = M / 7 - "~a2pp/OO. The conclusion is, that both translational and angular momentum relax

to equilibrium with the solvent on the same time scale. As a consequence, not only the total force on a Brownian particle may be set equal to zero on the Brownian time scale (as discussed in section 2.6), but in addition the torque may be set equal to zero on that time scale. This is used in subsequent sections on hydrodynamic interaction to obtain a linear relation between the translational velocities and the hydrodynamic forces. In the previous section, where leading order hydrodynamic interaction was considered, rotations do not play a role. For the calculation of hydrodynamic interaction matrices for shorter distances between the particles, however, rotations must be taken into account.

5.12 The Method of Reflections

Consider two spheres, i and j, in an unbounded and otherwise quiescent fluid. In order to calculate the forces exerted by the fluid on these two spheres one should, in principle, calculate the fluid flow velocity field u(r) and the pressure field p(r) as a result of the motion of the two spheres. The forces are then obtained by integration of the stress matrix (5.6) over the surfaces of the spheres. The fluid flow velocity field satisfies stick boundary conditions on the surfaces of the two spheres,

u(r) = v i + f ~ i • for r E OVi ,

= vj + f~j x ( r - r j ) , for r E OV i . (5.66)

5.12. Method of Reflections 259

This boundary value problem is too complicated to solve in closed analytical form. Instead the problem is solved by iteration. There are two alternative ways of doing this" one can prescribe the velocities and calculate, by iteration, the hydrodynamic forces, or one can prescribe the forces and calculate the velocities. The former procedure leads to expressions for the inverse of the microscopic diffusion matrices (the microscopic friction matrices), while the latter procedure leads directly to the microscopic diffusion matrices. Since we are interested here in the microscopic diffusion matrices, as these are needed in the Smoluchowski equation, the latter procedure is followed here, saving the effort of a matrix inversion.

On the Smoluchowski time scale, the hydrodynamic torques may be taken equal to zero. Of course one may consider the purely hydrodynamic problem where the hydrodynamic torques are taken non-zero. This leads to a linear relationship between the translational and rotational velocities on the one hand, and forces and torques on the other hand. The microscopic diffusion matrices we are seeking are then found by setting the torques equal to zero. Here, we set the torques equal to zero right from the start of the calculation.

The flow field is calculated by iteration, resulting in a series expansion representation of the flow field u(r) in powers of a/rij, with rij the distance between the spheres. Thus, we write,

u ( r ) - u ( ~ u ( 1 ) ( r ) + u(2)(r)+ . . . , (5.67)

where each field u (n) (r) satisfies the creeping flow equations. The field u(~ is the fluid velocity field of sphere i, say, in the absence of sphere j. This field satisfies the boundary condition,

u(~ - v! ~ = -~DoF/h , for r e OVi,

where it is used that/3Do - 1/67r~oa. This is the fluid flow velocity field (5.36) of an isolated sphere in an otherwise quiescent fluid, with a translational velocity v! ~ and the corresponding Stokesian friction (5.38). This fluid flow velocity field influences particle j in its motion. The velocity of particle j follows directly from Fax6n's theorem (5.60),

V~ 1) -- h (0) 1 a2 2 u(O) - / 3 D o F j + u ( r j ) + ~ Vj (r j ) .

Up to this level, the Rodne-Prager result (5.64,65) for the microscopic diffusion matrices is obtained, as will be illustrated in subsection 5.12.3. The

260 Chapter 5.

rotational velocity of sphere j follows from Fax6n's theorem (5.63), with the hydrodynamic torque Tj h exterted by the fluid on sphere j set equal to zero,

f~l) _ 1Vj • u(~ . 2

The fluid velocity field u(1) (r) in the expansion (5.67) is the velocity field that is the result of immersing sphere j into the velocity field u(~ The "incident" field u(~ is said to be "reflected" by sphere j. The hydrodynamic problem

(1) to be considered now is a sphere (the sphere j) with translational velocity vj

and rotational velocity f~!l), which is immersed in a fluid flow field u(~ The resulting additional fluid flow velocity field, after immersion of sphere j , is the field u(1)(r) in the iterative expansion (5.67). The total flow field is thus u(~ + u(1)(r), which is equal to v~ 1) + f~l) x ( r - rj) on 8Vj for stick boundary conditions. The additional fluid flow field thus satisfies the following boundary condition,

u(1)(r) -- v~ ' ) - u(~ f~.') • ( r - r j ) , for r E 0 ~ .

The creeping flow equations are thus to be solved, subject to the above boundary condition. Once the reflected field u(1)(r) is calculated, the velocity of sphere i, in addition to the Stokesian velocity v~ ~ = -~DoF~, follows from Fax6n's theorem (5.60) with F h set equal to zero,

1 a 2 2 u(1) v! ~) - u( ' ) ( r , )+ g v , (r,) ,

while its rotational velocity follows from Fax6n's theorem (5.63) as,

~-~!2) = 1 ~7 i X U(1)(ri) 2

This yields an expression for the microscopic diffusion matrices which goes one step beyond the Rodne-Prager level. In the next iterative step, the field u(2)(r), resulting from the reflection of the field u(1)(r) by sphere i should be calculated. The additional flow field u(1)(r) + u(2)(r) is equal to the additional surface velocity v! 2) + f~!2) x (r - ri) on OVi,

u(~)(r) - v! ~) - u ( ' ) ( r ) + a ! 2) x ( r - r , ) , /o,- r ~ aV,.


The add/tiona/velocity of sphere j , in order to sustain the prescribed hydrodynamic force, is again found from Fax6n's theorem,

1 a2 2 U(2) v} 3) - u(2)(r j )§ g Vj ( r j ) .

Repeating this iterative procedure indefinitely solves the calculation of u(r) , assuming convergence of the resulting series expansion. The velocities are obtained from Fax6n's theorem as functions of the distance between the two spheres with increasing accuracy at each level of iteration,

(2) V!4) vi - vl ~ + + . . . ,

v ? v?) vj - vj + + + . . . . (5.68)

The angular velocity is given by a corresponding series expansion,

_ ,

~ j __ ~-~1)_[_ ~-~3)_1_ ~-~.5) ]_ "'" . (5.69)

Increasingly higher order terms in the series expansion of the microscopic diffusion matrices with respect to the inverse distance between the two spheres follow directly from the series representation (5.68) for the velocities.

We thus arrive at the following sequence of boundary conditions for the flow fields u('0, each of which satisfies the creeping flow equations,

u(~ -- v! ~ , for r E OVi, u( i ) ( r ) _ v( . i )_ u(O)(r ) + ~-~(.1) x ( r - r j ) , for r G OVj,

u(2)(r) v! 2) u(1)(r)+1-1! 2) • ( r - ri) , for r e OVi,

- v ( .3)-u(2)(r)+l"l (3) x ( r - r j ) , for r e OVj, u(3)(r) v!4) (3) r) fl! ') - r,) for r e OU u ( 4 ) ( r ) - - u ( + �9 • ,

(5.70)

where the angular velocities are obtained from,

f~l ~ = O,

fl~2~) _ 21 Vi • u ( 2 n - 1 ) ( r i ) , for n >_ 1 ,

(2~+1) _ 1 Vj • u(2~)(rj) for n > 0 J 2 ~ __

(5.71)

(5.72)

262 Chapter 5.

and the translational velocities are obtained from,

•(2.) i - -

r j

I a2,..,2 (2n-1)(ri)] , (5.73) -6.0/3DoF~'+(1-6,,o) u(2"-~)(ril+g v iu

h (2.) 1 a2 2 (2n)(rj) (5.74) -6,~oflDoFj + u (rj) + g Vj u .

Note that even indices relate to sphere i and odd indices to sphere j. The problem to be solved yet is the calculation of reflected fluid flow

velocity fields, that is, we have to find the flow field for which one of the boundary conditions in (5.70) on a spherical surface are specified. This problem is quite complicated and cannot be solved in closed analytical form. In each step in the iteration, the reflected field must be expressed in terms of a power series expansion with respect to a/rij, which is then truncated at the desired level.

5.12.1 Calculation of Reflected Flow Fields

General expressions for solutions of boundary value problems, like those for the reflected fields, can be obtained as series expansions with respect to gradients in the "incident fields", which specify the boundary conditions in eqs.(5.70). As we shall see in the next subsection, such gradient expansions lead to a power series expansion for the reflected fields with respect to the inverse distance between the particles. Each of the boundary value problems (5.70) is then decomposed into a set of simple boundary value problems, pertaining to each of the separate terms in the gradient expansion. Due to the linearity of the creeping flow equations, the solution of each of the original boundary value problems in eq.(5.70) is the sum of the solutions of these simple boundary value problems. The present subsection deals with the construction of reflected flow fields as a superposition of solutions of these simple boundary value problems. Subsequent subsections contain the explicit calculation of microscopic diffusion matrices employing the method discussed here.

Before discussing the simple boundary value problems, we introduce here some convenient notation conventions (see also subsection 1.2.1 on notation in the introductory chapter). First, the n-fold polyadic product of a vector a is written as,

a . . . a = a". (5.75) n X

5.12. M e t h o d o f Ref lect ions 263

This is a matrix of indexrank n with elements a i ~ a i 2 . . . a i , . The n-fold polyadic product of the gradient operator is written similarly simply as (V) n �9 the round brackets are used to indicate the polyadic nature of the product. For example, V 2 is the Laplace operator, while (V) 2 is the dyadic operator V V, which yields a matrix of indexrank 2 when operating on a scalar field. Secondly, the contraction symbol | is used to indicate contraction with respect to the maximum number of indices of either matrix occuring on both sides of the contraction symbol. For example, when A is a matrix of indexrank n, and B a matrix of indexrank m > n, then,

A | B - ~ Aj,,...j2 j, Bj~ j~...j,, j,+~...jm. (5.76) j l "" " in

Take notice of the ordering of indices. This contraction is thus a matrix of indexrank m - n. Similarly, when the matrix A on the left hand-side of the contraction symbol is of higher indexrank than the matrix B, then the number of left-over indices of the matrix A determines the indexrank of the resulting matrix. Thus,

A | jl ""'in

Aim... jn+ l J,,...J2 Ja BJl j2...jn

is again a matrix of indexrank m - n. This notation saves us the effort to write the summation explicitly. In manipulations which involve contraction symbols, however, one must carefully keep track of the order of indices.

Consider the following boundary value problem �9 find the velocity field u(r) that satisfies the creeping flow equations (5.2,20) together with the boundary condition,

u ( r ) - Uo( ) , f o r r e O V o , (5.77)

= 0 , for r ---, oo ,

where uo(r) is some known velocity field, which is referred to as the incident

f low field, and O V ~ is a spherical surface of radius a with its center at the origin. Each of the boundary value problems for u(n) (r) in eq.(5.70) is of this form when the coordinate flame is translated over the distance r~ for even n or over r 5 for odd n.

Let us first eliminate the pressure from the creeping flow equations. Taking the divergence of both sides ofeq.(5.20) for positions inside the fluid, where the external force is zero, and using eq.(5.2) yields V2p(r)=0. Operating with the

264 Chapter 5.

Lapace operator V 2 on both sides of eq.(5.20) then yields V2V2u(r)=O. The fluid flow velocity field thus satisfies the following two differential equations,

V . u ( r ) - 0, (5.78)

V2V2u(r) - O. (5.79)

The problem now is to solve these equations subject to the boundary condition (5.77).

The solution may be constructed by expansion with respect to gradients in the "incident flow field" uo(r) as follows,

00 1 (,+2) [(V')' u(r) - Z ~ U (r)| -u~ ( 5 . 8 0 )

/=0

where the index "0" on the square bracket is used to denote the value of the /-fold polyadic derivative at r' - O. The matrices U(m)(r) are of indexrank m, and are referred to as connectors, since they connect the known field uo(r) with the solution u(r) which we are seeking. The idea behind this expansion is as follows. As two Brownian particles are well separated, the flow field Uo in the neighbourhood of one particle, due to the motion of the other particle, is smooth, and only the first term(s) in the gradient expansion (5.80) need to be taken into account. For shorter distances between the two particles, higher order gradients in the flow field Uo become important. Thus, the number of terms included in the gradient expansion (5.80) determines the number of terms in the inverse distance power series expansion of the microscopic diffusion matrices that can be obtained with it.

The boundary value problem for each of the connectors is easily obtained from the Taylor expansion of the boundary condition (5.77) on OV ~

1 rt Uo( )]o uo(r) - ~ ~ | [(V')' r' . (5.81) /=0

Substitution of the gradient expansion (5.80) into the differential equations (5.78,79), and using the above Taylor expansion in the boundary condition (5.77) yields,

V . (U(t+2) (r))

V2V2 (U('+2)(r))

U( /+2) ( r )

= O,

{ J r I for r - e o o

for r e 0 V ~ .

(5.82)


TaMe 5.1 �9 Explicit expressions/or the connectors. These expressions for U(l+2)(r) are valid only when contracted with (V')luo(r'). The matrices H(~)(r) are defined as (V) ~ 1.

r

m!! - m ( m - 2 ) ( m - 4 ) . . . 5 . 3 . 1 .

U(2) ( r ) 4 ( r 2 - a2)H(2)(r) + aJ~H(~

.-- a 3 U(3 ) ( r ) -6-(r 2 - a 2)H(a)(r) - a3j~ H(1)(r)

u ( ' ) ( , )

U(5)(r)

a 5 a 3 h-~(r 2 - a2)H(4)(r) + ~-~(r: - a2)H(2)(r)i~

+ ~ i H ( 2 ) ( r ) + ~ i H ( ~

a 7 5) 9 a 5 r2 ~o• r2 - a2) H ( ( r ) - ~ ( -- a 2 ) H ( 3 ) ( r ) i

~ 9~ i H(~)(r) i -5~.,iH(3)(r) - 5,,-

U(6 ) ( r ) a 9 9 0 a 7 r 2 i2• 2 - a2)H(6)(r) + ~ ( - a2)H(4)(r)i~

+~I~9 ^ H(4) (r) + 9~ i H(2) (r) i 7 ~ -

U(7)(r)

U(S)(r)

a l l 1 4 a 9 5) i ~ , , ( r 2 - a2)H(7)(r) + ~o• 2 - a2)H( (r) i

a ~ 14a9 ~ H(3 ) ( r ) i ~ii IH(5)(r) + 9!! -

a 13 27a11 r 2 ~ ( r2 - a2)H(8)(r) + 12Xl l ! , ( - a2)H(~)(r) i

+ 1-~..'IH(6)(r) + "11!!

266 Chapter 5.

Once explicit expressions for the connectors as solutions to these "simple" boundary value problems are determined, the solution of the boundary value problem (5.77) follows directly from eq.(5.80) : simply replace the polyadic products r t in the Taylor expansion (5.81) of the known field uo(r) by the connector U (t+2) (r).

For (l + 2) < 8, the solutions are constructed in appendix D as linear combinations of the "basic" matrices H(m)(r)=(V) TM 1. Notice that in the

r

solution (5.80) we only need the connectors as a contraction with polyadic derivatives of u0(r') �9 many of the terms in the general expression for the connectors as the solutions of the boundary value problem (5.82) need not be considered, as uo(r) itself satisfies the creeping flow equations (5.78,79). For example, a term which is proportional to I I gives rise, on contraction with (V')4uo(r'), to a contribution which is proportional to VaVauo(r ' ) , which vanishes due to the creeping flow equation (5.79) for uo(r'). Such terms may be disregarded in solving the boundary value problem (5.82). The results of the calculations in appendix D are collected in table 5.1, where all terms which vanish on contraction with the corresponding polyadic derivative of uo(r') are omitted. Explicit expressions for the first five of the matrices H (m) (r) are also given in eq.(5.158) in appendix D.

Appendix D also contains the derivation of the following elementary properties of the basic matrices H (n) (r) =(V) n x_

V2 H(n)(r) = 0 V �9 H(~)(r) = 0

r. H(~+X)(r) = - ( n + 1)H(~)(r) V 2 (r2H('~)(r)) = - 2 ( 2 n - 1)H(n)(r)

v . :

(5.83)

The expressions in table 5.1 and the above elementary relations will be used in subsection 5.12.4 to obtain explicit expressions for the microscopic diffusion matrices.

5.12.2 Definition of the Mobility Functions

On the two particle level, expressions for the microscopic diffusion matrices are always linear combinations of the identity matrix I and the unit separation vector dyadic i'iji'ij, with i'i~ - rij/rij. These linear combinations can be re-

arranged as a sum of a matrix [I - i'iji'ij] and the dyadic product r~jrij. These


matrices are the projections perpendicular and parallel to ~ij, respectively. In rewriting the microscopic diffusion matrix as a linear combination of these two projections, the translational velocities of the spheres are decomposed in a component perpendicular and parallel to the line connecting the centers of two spheres. The general form of the diffusion matrices is thus written as,

D .

Di j

N

- Doi + Do ~ {A,(rij)i',ji'ij + B , ( r i j ) [ i - i'ij~,j]) , j = l , j r

= Do {A~(rii)riirii + B ~ ( r i j ) [ I - ~ i i ~ i i ] } , i ~ j . (5.84)

The summation in the expression for the "self' diffusion matrix D , accounts for the fact that a/1 particles in suspension reflect the field of the i th particle back to that particle. The scalar functions A, , B , , A~ and B~ are referred to as the mobility functions. These functions depend only on the scalar distance rij between the two spheres i and j. The mobility functions with i - j are sometimes called self-mobility functions, and those with i ~ j , distinct- or cross-mobility functions. The subscripts s and c refer to "self" and "cross", respectively. A Taylor expansion of the microscopic diffusion matrices is equivalent to the Taylor expansion of the four mobility functions. Rearranging the Rodne-Prager result (5.64,65) in the form (5.84), gives the leading order terms of these Taylor expansions,

A, - O ((alrij) 4) , B, - O ((alr~j) 4) ,

-- 2,,j ~ "q- 0 ( ( a / r i j ) ,

o 1(o) )4 _ 3 + ~ ~ q - O ( ( a / r i j ) B e - - 4 rij

(5.85)

In the following subsections we will calculate higher order contributions. The expressions (5.84) for the microscopic diffusion matrices are valid

for simultaneous interactions between two particles only. Contributions to these matrices, resulting from configurations where three or more spheres interact simultaneously, are more complicated functions of relative position coordinates. Three body interactions are considered in subsection 5.12.5.

5.12.3 The First Order Iteration

The field u(~ in the reflection expansion (5.68) is the fluid flow velocity of a single uniformly translating sphere (sphere i in this case) in an otherwise

268 Chapter 5.

quiescent fluid. This field was already calculated in subsection 5.7.1, and is given by eq.(5.36) with v = vi and r, = ri. Alternatively, this result may be obtained straightforwardly as an application of the method described in the previous subsection. Since the boundary condition in eq.(5.82) for the connectors is defined on the spherical surface OV ~ centered at the origin, we must translate our coordinate frame over the distance ri for the calculation of u(~ That is, the position coordinate is replaced by r + ri. The boundary condition (5.70) for the zeroth order field then reads,

u(~ - v! ~ = -/3 DoF/h , for r E O V ~ (5.86)

The Taylor expansion (5.81) of the "incident flow field" no(r) is now simply a constant equal to v! ~ The only remaining term in the gradient expansion (5.80) is therefore the first term 1 = 0. Hence,

u(~ + r,) - U(2)(r)| v! ~ .

The position dependence here is relative to the position coordinate of sphere i. Returning to the original coordinate frame, by replacing r by r - r~, gives,

u(~ - U(2) ( r - ri) | v! ~ . (5.87)

Substitution of the expression for the connector U(2)(r) given in table 5.1, and using the definition H( ' ) ( r ) - (V) m! reproduces the result given in

r )

eq.(5.36). The velocity v~ 1) in the reflection expansion (5.68) follows from Fax~n's

theorem, eq.(5.74) with n = 0. Using that V2H(2)(r)=0 and X72(r2H(2)(r))

=-6H(2)(r) (see eq.(5.83)) readily gives,

: _ 1 a 2 2 U(2)(rj/)].F/h} v51) /~ Do {F~ + [U(2)(rji)+ ~ Vii 3 a 1 a 3 ^ ,, ,, . = - / ~ D o { F ~ + [~, , [ i+ b i j i ' i j ]+~(7 /~) [ I -3 r i j ro ] ] �9 F~} (5"88)

This result reproduces the Rodne-Prager expression (5.64,65) for the microscopic diffusion matrices, and the corresponding expressions (5.85) for the mobility functions, as it should.

5.12.4 Higher Order Reflections

The next higher order term for the velocity of sphere i is v} 2). In order to calculate this additional velocity, the reflection of the fluid flow field u(~

5.12. Method o f Reflections 269

by sphere j towards sphere i must be calculated first, which reflected field was denoted as u(1)(r).

In order to calculate u(a)(r), we have to translate our coordinate frame over the distance rj, so that the boundary condition (5.70) for u(1)(r) is formulated on (9 V ~ and the solution can be constructed in terms of connectors, as described is subsection 5.12.1. That is, in all relevant equations we replace r by r + rj. The boundary condition (5.70) then reads,

u(1)(r + rj) - v ! 1 ) - u(~ + r j ) + ~'~'1) X r , for r E OV ~ �9 (5.89)

The rotational ve loc i ty ~..~.1) follows from eq.(5.72) with n - 0. Its explicit evaluation proceeds via the use of the following general relation for two vector fields a(rj) and b(rj) ,

a x (Vj x b) - Vj (a. b) - (a. V j ) b - ( b - V j ) a - b x (Vj x a ) .

With a(rj) - r, which is independent of rj, and b(rj) - u(~ it follows from eq.(5.72) with n = 0 that,

~-~1) X r -- ~ r (S) Viu (rj) -- ( ~ T j u ( ~ , (5.90)

where the superscript T stands for "the transpose of" the corresponding matrix. The Taylor expansion (5.81) of the above boundary condition (5.89) thus

reads,

(1) u(l)(r + rj) -- vj 1 [ ( O ) ( r j ) + ( V j u ( O ) ( r j ) ) T ] u(~ -- ~ r | Vju

- E rz (3 [(Vj)tu(~ �9 (5.91) /=2

(1) and u(~ have been calculated Note that both translational velocities v j in the previous iterative step, and are thus known functions of the forces. It is now apparent that the Taylor expansion of the boundary condition is actually a power series expansion with respect to the inverse distance between the two spheres. According to eq.(5.87), u(~ -- U(2)(rji) (S) v! ~ ~ a/r i j , so that each higher order term in the Taylor expansion corresponds to a higher order power in a/r i j . Each differentiation adds one order in a/r i j , so that the I th term in the Taylor expansion is of the o r d e r (a /r i j ) l+1 .

270 Chapter 5.

The first order field reflected by sphere j is simple obtained by replacing the polyadic products r I by the connector U tl+2) (r). Returning to the original coordinate frame, by replacing r by r - rj, thus yields,

U(I)(:I[ ") -- U(2)( I �9 - l ' j ) (~ [v~ 1 ) - u{~

- ~-~ ~.I ~ 1 U ( t + 2 ) ( r _ r j)(S) [ ( V j ) t u ( ~ . (5.92) /=2

We are now in a position to evaluate the additional force v~ 2), simply by substitution of the above flow field into Fax6n's theorem (see eq.(5.73) with n - 1). Clearly, this cannot be done rigorously, since then all the terms in the above sums should be evaluated. We restrict ourselves to expressions for the microscopic diffusion matrices which are accurate up to and including terms of order (a / r i j ) r. The summation in eq.(5.92), representing a power series expansion in a/rij, can then be truncated. As can be seen from table 5.1, or the general expression (5.159) given in appendix D, for n > 4 we have that U( '~)(r ) ,~I /P -3 as r ~ c~, while U(2)(r),,~l/r and U(3)(r) ,~l / r 2, since

H(m)(r), ,~l/r m+~ . Since the velocity v~ 2) in the expansion (5.68) is related to the value of u(X)(r) at r - ri, through Fax6n's theorem, the terms that need to be taken into account in eq.(5.92) can be determined without difficulty" no terms beyond 1 - 3 in the sum need be considered, and in these contributions a number of terms may be neglected in addition.

Notice that on substitution of the expression (5.92) for u(1)(r) into Fax6n's theorem (5.73), the differentiation is with respect to r only, after which r must be set equal to ri.

The mobility functions are evaluated explicitly with the use of table 5.1 and the properties (5.83) of the basic polyadic matrices H (m), together with explicit expressions for the polyadic matrices given in eq.(5.158) in appendix D. These calculations require a considerable effort and careful bookkeeping. The explicit expressions for the translational velocities v! 2) and u(1)(r) in terms of the hydrodynamic forces that one finds are the ingredients for the next higher order iteration.

To find the next higher order term v~ 3) for the force on sphere j , we must first find the flow field resulting from the reflection of u(1)(r) by sphere i. As before, our coordinate frame is first translated over the distance ri, so that the boundary condition (5.70) for ut2)(r) can be formulated on sphere i with its

center at the origin. Thus, r is replaced by r + ri. The term in the boundary condition (5.70) containing the angular velocity 1-1!2) is calculated precisely as before from eq.(5.71) with n - 1 (compare with eq.(5.90)),

f~2) x r : 1 [ ( , ) , ( r i ) ) T ] 7" | V,u (~,)- (V,u{'

Taylor expansion yields the following boundary condition for u(2)(r + ri) on OV ~ (compare with eq.(5.91)),

u{2)(r + ri) 1 [ {1> ri)(Viu<l)(ri))T] vl ~) - u{')(~,)- 7 ~ m V,u ( +

- ~ r I | [(Vi)/u(lI(ri)] . (5.93) /=2

Again, the solution to this boundary value problem is obtained by simply replacing r TM by the connector U(~+2)(r). Transforming back to the original coordinate frame, by replacing r by r - ri, gives,

u{~)(r) - U(2) ( r - r , ) | [v! 2 ) - u(1)(r,)]

_ 12 U ' 3 ' ( r - ri)(S)[Viu(X)(ri)+ (Viu(a'(ri)) T

- ~ u{'+~)(, . - ,.,)| [(v,)'u{')(,-,)]. 1=2

(5.94)

This is the equivalent of eq.(5.92) in the previous iterative step. Likewise, as in the previous iterative step, careful bookkeeping yields, after a considerable effort, explicit expressions for v~ .3) and u(2)(rj) in terms of the forces.

The iteration must be continued up to the level where only contributions of higher order than (a/r~j) r are found. It turns out that the iteration must be extended up to and including V~ 3) and v! 4) for sphere j and i, respectively. The final result for the mobility functions is,

3a Ac

2 r i j

+ 0 ((alrij)s),

+ 0 ((alr,j)s),

+ T r~S + o (/<,i,,,/1,

272 Chapter 5.

.8 A CtS B C~S

_2

Bc

Bs

Figure 5.4:

A S

2 3 % Exact numerical results for the moNlity functions (solid curves) and the approximation (5.95) (dashed curves). The exact results are taken from Batchelor (1976).

3 a 1 a = + -~ + 0 ((a/rij) 9) . (5.95) Be 4 rij

These results are valid approximations for sufficiently large distances between the spheres. More terms should be calculated in order to obtain results which are accurate also at smaller distances. Higher order coefficients have been calculated and tabulated by Cichocki, Felderhof and Schmitz (1988). Alternatively, the accuracy for smaller distances between the spheres may be improved by matching with exact asymptotic results for small separations between the spheres. The calculation of such small separation asymptotic results is referred to as lubrication theory (see for example Kim and Karilla (1991)). For such small separations there is a small gap between the two spheres filled with "lubricant" solvent, that is expelled from the gap when the two spheres approach. One may construct Pad6 approximants with the correct asymptotic behaviour both at larger distances (given in eq.(5.95)) and the small distances as predicted by lubrication theory.

In fig.5.4 the above results for the four mobility functions are compared with exact (numerical) results from Batchelor (1976). As can be seen from this figure, the approximation is not too bad, and we will use the results


(5.95) for the mobility functions to calculate transport coefficients is later chapters. We will always compare the outcome of these calculations with those where "exact" hydrodynamic interactions are employed. The use of the above approximation yields results which are not very much different from the "exact" results, as compared with common experimental errors. This is true for particles with a repulsive pair-interaction potential and also for mild attractive pair-interaction potentials. For stronger attractive pair-potentials, superimposed on the hard core repulsion, the values of the mobility functions at very small separations may be of major importance, in which cases one should be careful in using the approximation given in eq.(5.95). Results for transport coefficients obtained with the approximation (5.95) are most accurate for systems with a long ranged repulsive pair-potential, since in that case the average distance between the particles is relatively large.

5.12.5 Three Body Hydrodynamic Interaction

The preceding discussion on hydrodynamic interaction is restricted to two spheres. Results obtained so far can only be used to describe suspensions where the simultaneous hydrodynamic interaction of three or more Brownian particles is improbable as compared to pair-interactions. This is the case for dilute suspensions. We have to consider hydrodynamic interaction of three spheres simultaneously in order to predict the concentration dependence of transport coefficients up to somewhat larger concentrations. This three body problem is considered here within the framework of the method of reflections.

Consider three spheres, denumbered as i, m and j (see fig.5.5). The two spheres i and j interact hydrodynamically with each other via the intermediate sphere m : the sphere i creates a fluid flow field that reflects off sphere m, which in turn affects sphere j in its motion. This indirect contribution to the microscopic diffusion matrix is denoted by D!~ ), with i ~ j , where the super script "3" stands for "three body interaction". The sphere j reflects the flow field from sphere m back to sphere i, giving rise to an extra three body term to the self part of the microscopic diffusion matrix, which contribution is denoted byD!~ }.

The field that is reflected by sphere m is given by eq.(5.92), with the index j replaced by m,

274 Chapter 5.

d l)

_(3)

_13)

~/u(O) Figure 5.5: Three body hydrodynamic interaction.

i U(t+2) - ~ ~ ( r - r ~ ) | [(V~)tu(~ , 1=2

(5.96)

where u (~ is given in eq.(5.87) and v~) is given in eq.(5.88) with j replaced by m. For our purpose, the force F h in eq.(5.88) may be set equal to zero, since this term yields a two sphere contribution to the microscopic diffusion matrix, which we already considered in the previous subsections.

The force on sphere j due to the reflected field (5.96) from sphere m is simply obtained from Fax6n's theorem. Here we consider the leading order contributions in an expansion with respect to inverse distance between the three spheres. The second term in eq.(5.96)is of the order (a/rim) 2 (a/rj~) 2. All terms in the summation with l > 3 are of higher order in both inverse distances, while the first term and the term in the sum with 1 - 2 cancel. The cancellation of these two terms can be shown as follows. From Fax6n's theorem we have that v~) - u(~ ~al 2~7 m2 u(O)(r~). Furthermore, the leading order contribution of the connector U(4)(rj~) is, according to table 5.1, i~al arj m2 H(2)(rj,~)~+ ~1 a3IH(~ The unit matrix I appearing here gives rise to the Laplace operator V 2 upon double contraction with VV. Using these facts in the evaluation of the two terms, it is easily shown that they cancel.


Thus, in leading order, the only remaining term in eq.(5.96) is,

U(1) ( r ) - - 21 U(3) ( r - r~ ) | [V~u(~ (V~u'~ T ] . (5.97)

The additional velocity of sphere j is simply equal to u(a)(rj), since the Laplace operator in Fax6n's theorem contributes to a higher order term. With some effort this expression is evaluated explicitly with the help of table 5.1 and the expressions for the matrices H (m) (r) for m - 1, 3 given in appendix D. Interchanging the indices i and j leads to the following expression for the leading order three body contribution to the microscopic diffusion matrix with i C j ,

12 D ! ? - - - V o

r e = l , rn ~ i,.7

(1 - 3(i'im �9 ~'jm)2) ri~rj~. (5.98)

A summation over all intermediate spheres m is added here to account for all three body interactions in a suspension of N spheres that contribute to Dij. The next higher order terms are easily seen to be of the order (a/rim)P(a/rjm)q, with (p, q) - (2, 4) and (3, 3).

The three body matrix D!~ ) is calculated to leading order in precisely the same way. The flow field reflected by sphere j is given to leading order by,

u(2)(r) -- --~1 U(3) ( r - rj)(S)[~Tju(l '(rj)+ (~Tju(1)(rj)) T ] (5.99)

where the field u(1)(rj)is equal to D~ ). F~, with D~ ) given in eq.(5.98) with the indices i and j interchanged. The additional velocity of sphere i is simply equal to u (2) (ri), since the Laplace operator in Fax6n's theorem contributes to a higher order term. One finds with some effort,

DI ) - N N a a a

75 Do Y~ ~ ~ii~im 16 ~=~,~, ~=~,~, , , :

[1 - 3(kij " ~jm) 2 - 3(i',~. i'jm) 2 + 15(i'im. ~jm)2(~ij �9 ~jm) 2

-6( i ' i~ . fzj~)(~ij. ~,m)(~ij" f'j~)] �9 (5.100)

As before, we added summations over intermediate spheres. The next higher order terms are easily seen to be of the order (a/rij)P(a/rim)q(a/rjm) s, with (p, q, s) - (3, 2, 4) and (2, 2, 5).

276 Chapter 5.

These leading order expressions for the three body interaction terms allows for the approximate evaluation of transport coefficients up to concentrations of Brownian particles where the probability of three particle interactions becomes significant.

5 .13 H y d r o d y n a m i c In terac t ion in S h e a r F l o w

In this section we consider two spheres immersed in a fluid in linear shearing motion. That is, the fluid flow velocity field, without the two spheres being present, is given by,

uo(r) - I ' . r , (5.101)

where r is a constant matrix, independent of the position r in the fluid. A possible choice for this so-called velocity gradient matrix is,

O 1 0 / r - ~ o o o ,

0 0 0 (5.102)

representing a fluid flow along the x-direction, linearly increasing with position in the y-direction and independent of the z-coordinate. In subsection 4.6.1 we have conjectured the following form for the hydrodynamic force on a sphere i,

F , -

N -- E V i j ( r l , - . ' , rN)" (Vj -- r . r j) + C i ( r l , . . . , rN)" r . (5.103)

j=l

The microscopic friction matrices T ij were conjectured to be identical to those for spheres in an otherwise quiescent fluid, the inverse matrix of which is (proportional to) the microscopic diffusion matrix, which was considered in section 5.12. In any application of the Smoluchowski equation, the "inverse" relation is the relevant one, that is, we are interested here in the velocity in terms of the forces. The velocity of the i th sphere can be written as (see subsection 4.6.2 for details on the inversion of the above matrix equation),

N h t vi = - f l ~ Dij . Fj + r . r ~ + c~ �9 r ,

j=l (5.104)

5.13. Interaction in Shear Flow 277

where we introduced D - fl-~T -1, and the disturbance matrix C~ of indexrank 3 is the product of the microscopic diffusion matrix D and the matrix (2 in eq.(5.103). For a precise definition of C' in terms of D and C, one should transform to the "supervector notation" introduced in chapter 4 (see the equation in the text just below eqs.(4.101) and (4.103)). This precise definition is of no concern here. It is the general form of the velocities in eq.(5.104) which is of interest in the Smoluchowski equation (4.102,103). In the present section we show that the velocities are indeed of the form as conjectured in eq.(5.104), and an explicit expression for the disturbance matrices

' is derived, irrespective of its precise relation to the original "disturbance Ci matrices" C~.

Before doing so, let us first consider an isolated sphere immersed in a linear shear field.

5.13.1 Flow past a Sphere in Shear Flow

The translational and rotational velocity of a single, torque free sphere immersed in the linear shear field (5.101,102) follow immediately from the translational and rotational Fax6n's theorems (5.60,63),

Vp =

x -

-/3DoF ) + r . r , , (5.105)

1 ( r - r r ) 1 [V, x ( r . ( r - rv))] - - ( r - rp) | ~

where the superscript T stand for the transpose of the corresponding matrix. The fluid flow velocity field u(r) that exists after immersion of the sphere in the linear shear field can be calculated as the reflection of the "incident" linear shear field by the sphere, using the results of subsection 5.12.1. The flow velocity is written as, u(r) - uo(r) + Au(r), with Au(r) the reflected field. The stick boundary condition reads,

u(r) = u o ( r ) + A u ( r ) - v p + f l p x ( r - r p ) , for recOV~,

with 0Vp the surface of the particle. The boundary condition for the reflected contribution Au(r) is first reformulated on the spherical surface with its center at the origin, OV ~ by translation of the coordinate frame over the position coordinate rp of the sphere,

A u ( r + r p ) - - u o ( r + r p ) + v v T l 2 p • for r E OV ~ .

278 Chapter 5.

Substitution of the expressions (5.105) for the translational and rotational velocities, and eq.(5.101) for the incident field in the form u0(r) - r | F T, yields,

A u ( r + r p ) - -flDoF h- r (S) E , for r E O V o , (5.106)

where E is the symmetric part of F,

1 (r + r z) (5.107)

The above boundary condition is already in the form of a Taylor expansion, which apparently contains only a constant and a linear term in r. The reflected field thus follows immediately from what has been said in subsection 5.12.1, by replacing the polyadic products r t by the connectors U (t+2) (r). Returning to the original coordinate frame, by replacing r by r - rp, yields,

h U(3)(r r p ) | Au(r) - - U ( 2 ) ( r - rp)| f lnoFp - - (5.108)

Substitution of the explicit expressions for the two connectors as tabulated in table 5.1 finally yields the flow field that exists after immersion of the sphere (for brevity we denote here r - rp by R),

u(r) - F . r - ~ ( ~ ) - ( R ) ( R . E . R ) R - ( ~ ) E . R

{ 1 a } 3 a [ J + R R ] + ( R ) [I 315"R1 " (-/3D~ (5.109) + -

The first term on the right hand-side here is nothing but the incident field uo(r). The terms proportional to E represent the reflection of the linear shear field by the sphere, while the last term (the one proportional to F h) is the field due to the translational motion of the sphere relative to the local linear shear field. This last term is precisely the flow that is induced by a translating sphere in an otherwise quiescent fluid (compare with the result in eq.(5.36)).

5.13.2 Hydrodynamic Interaction of two Spheres in Shear Flow

The hydrodynamic interaction between two spheres in a linear shear field can be calculated precisely as for two spheres in an otherwise quiescent fluid with

5.13. Interaction in Shear Flow 279

the method of reflections, as described in section 5.12. The only difference is the presence of the extra terms in the zeroth order fluid flow velocity field u(~ This extra contribution (proportional to the shear rate) is most clearly revealed by comparing eq.(5.87) for the zeroth order field in case of an otherwise quiescent fluid (with v! ~ replaced by -/~DoFp h) and the above expression (5.108). The terms proportional to the shear rate on the right hand- side of eq.(5.108) are extra as compared to the corresponding field in case of an otherwise quiescent fluid. Since the creeping flow equations are linear, the two contributions give rise to a sum of two separate contributions for the translational velocities of hydrodynamically interacting spheres. The term

h proportional to the force Fp reproduces the microscopic diffusion matrices for spheres in an otherwise quiescent fluid, which were calculated in section 5.12. This proves the first part of our conjecture" the diffusion matrices Dij in eq. (5.104) are presicely the microscopic diffusion matrices for spheres in an otherwise quiescent fluid. The terms in eq.(5.109) for the fluid flow velocity around an isolated sphere in a linear shear field which are proportional to the shear rate give rise to the additional terms (the last two terms) in eq.(5.104). The method of reflections is used to prove the second part of our conjecture, related to the form of the extra last two terms on the right hand-side in eq.(5.104). In doing so, an explicit expression for the disturbance matrices C~ is established.

The method of reflections is applied with,

g _ r . r - f r - r , I n . ( 5 . 1 1 0 )

[( ( ) 5 a a r - ri r - ri ( r - ri) --2 I r _ r i I - i r - r i [ I r r ~ l ' E ' i r r~l '

which is the field (5.109) resulting from immersion of sphere i in a linear shear field, disregarding the term proportional to the hydrodynamic force. This field may be substituted in Fax6n's theorem (5.74) for n = 0, again disregarding the hydrodynamic force, to find the following expression for the velocity of sphere j on the Rodne-prager level,

v } l ) - r . rji + ()5

8 a E . r j i . E . - 5

(5.111) Up to the Rodne-Prager level we indeed find the form conjectured in eq.(5.104).

280 Chapter 5.

We will calculate the leading order term in the next reflection, along the same lines as in subsection 5.12.4 for spheres in an otherwise quiescent fluid. For the first order reflected field we can simple copy the general expression (5.92) from subsection 5.12.4,

u(')(~)

oo 1 (,+2) ), (0 ) ( ) ] - y ~ U (r-rj)| u rj .

/=2

(5.112)

The first term r . r on the right hand-side of the expression (5.110) for u(~

gives rise to a contribution for v~ 2) which is identical to the expression (5.111)

for v~ ~) with the indices i and j interchanged. The remaining terms in the eq.(5.112) for u(~ are of the order (a / r ) 3. These terms are used in the reflection expression (5.112) in leading order. Just as for the three-body interaction in leading order, the first and third term (the term with 1 - 2) cancel. The only remaining term is the second term involving the connector U (3). This leading contribution may be evaluated explicitly with a little effort, using the leading two terms for the connector U(3)(r) as given in table 5.1 and the explicit expressions for the matrices H(m)(r) for m = 1, 3 as given in eq.(5.158) in appendix D. In this way we arrive at the following expression for the disturbance matrix,

[ t 5 a 20 a 4 a t c , - - ~ + -5- ~,~,~r,~- 5 i~,j + (ir,~)

25 ~,~,~,j + o ((o/~,~1 ~) (5.113) 2 '

where we defined the indexrank 3 matrix, (Ir)~m, - 8i,~rm, with 8i,~ the Kronecker delta, and we used that, 1". r - (Ir) t �9 1" and F T. r -- (Ir) �9 1". We could have added a sum over all the intermediate spheres j here, to account for the fact that all the Brownian particles in the suspension reflect the field of sphere / back to that sphere.

In the chapter on critical phenomena, the divergence of the disturbance matrix is needed for the calculation of the critical behaviour .of the effective viscosity. Due to incompressibility, only the highest order term in the above

5.14. Interaction in Sedimenting Suspensions 281

expression contributes to the divergence. The result is,

75 s a b~j~j , Vi .C ' i - 2 , = , , ~ , , (5.114)

where the summation over all intermediate particles is written explicitly. This concludes our considerations on hydrodynamic interaction of two spheres in a linear shear field.

5.14 Hydrodynamic Interaction in Sedimenting Suspensions

Consider a suspension in a container in which the Brownian particles sediment with an average velocity v,, due to, for example, a gravitational force field. Since in the laboratory coordinate frame, which is fixed to the container, the total net flux of volume of colloidal material and solvent through a cross sectional area of the container is zero, there is a so-called back flow of solvent. This solvent back flow compensates the flux of volume due to sedimenting colloidal material. As discussed in subsection 4.7.1, the back flow may be considered uniform on a local scale. We may consider an assembly of many Brownian particles in a small subvolume in the container over which the back flow u, is approximately constant (see fig.4.7). We conjectured in section 4.7 on the derivation of the Smoluchowski equation for a sedimenting suspension, that hydrodynamic interaction is described by the microscopic diffusion matrices for spheres in an otherwise quiescent fluid, when the velocity of the sedimenting spheres is taken relative to the solvent backflow (see eq.(4.119)). This means that we can simply replace the velocities vj on the left hand-side of eq.(5.46) by v~ - vj - u~, and the microscopic diffusion matrices are identical to those for spheres in an otherwise quiescent fluid, which were considered in section 5.12. This conjecture is shown to be correct as follows.

Clearly, the field u*(r) - u ( r ) - u, , with u(r) the fluid flow around the sedimenting spheres, satisfies the creeping flow equations V . u*(r) -- 0 and V2V2u*(r) - 0, since u(r) itself satisfies these equations and u, is a constant. The stick boundary condition on the surfaces of the spheres in terms o f u* ( r ) reads,

u*(r) - v ; + x ( r - r j ) , r e aVj. (5.115)

282 Chapter 5.

The boundary condition at infinity reads u(r) - u,, or equivalently,

u*(r) - 0 , for r ~ o c .

Hence, the field u*(r) satisfies the same differential equations and boundary conditions as the fluid flow field of a system of moving spheres in an otherwise quiescent fluid, except that vj is to be replaced by v~. Moreover, Fax6n's theorem (5.60) for translational motion is valid for a sedimenting suspension, with the velocity vp of a sphere (for example sphere j) and the homogeneous flow field uo(r) replaced by their starred counterparts, that is, when vp is replaced by vv(r) - u~ and uo(r) by uo(r) - u~. It is essential here that u~ is a constant, independent of position, since vp on the left hand-side of FaxEn's theorem (5.60) is obtained after integration over the spherical surface of the particle. The rotational Fax6n's theorem (5.63) is not affected by the homogeneous back flow.

Thus, all relevant equations for the calculation of hydrodynamic interaction matrices remain unaffected by the back flow, except that all velocities are to be taken relative to the back flow velocity. Therefore, the entire analysis of hydrodynamic interaction between spheres in an otherwise quiescent fluid carries over to sedimenting spheres when all velocities are taken relative to Us .

This proves the conjecture that we made in subsection 4.7.1. One can simply use the expressions for the microscopic diffusion matrices which were obtained in section 5.12 to account for hydrodynamic interaction between sedimenting spheres.

5.15 Friction of Long and Thin Rods

We will think of a rod as a rigid string of connected spherical subunits, which 1 are referred to as beads (see fig.5.6). The radius of each bead is equal to 7 D,

with D the thickness of the rod. Each bead is labelled with an integer, ranging 1 1 from - T n to +Tn, with n + 1 - L/D the number of beads, where L is the

length of the rod. The rod contains an odd number of beads" for long and thin rods this choice is not a restriction. The reason for considering a bead model is that we developed knowlegde concerning fluid flow around spherical objects in previous sections, that may be exploited to study friction of such a string of spherical beads. It is possible, however, to calculate friction coefficients for ellipsoidal objects exactly, but we shall not consider such calculations here.

5.15. Friction of Long and Thin Rods 283

A

..f.U

o~

Figure 5.6: The bead model for a long and thin rod. L is the total length of the rod and D is the diameter of the beads. The orientation is given by the unit vector fi pointing along the long axis of the rod.

Starting point for our calculation of friction coefficients for rods in an o- therwise quiescent fluid is the translational Fax6n's theorem (5.60) for spheres, which can be applied to each separate bead. The translational velocity vj of t h e j t h bead is given by,

1 h 1 vj = -37rr/oDFj + uo(rj) + D2V~uo(rj) , (5.116)

�9 -'~ r l

Stokes friction o.f the bead Hydrodynamic interaction with other beads

1 where we used that the radius a of each bead is equal to 7D. The fluid flow velocity field uo(r) is the fluid flow that would exists in the absence of the jth bead. The last term in eq.(5.116) incorporates the friction of the jth bead due to hydrodynamic interaction with the other beads.

It is tempting to use the following expression for uo(r),

n / ,

u o ( r ) - ~ J~~-v~ dS' T(r - r ' ) . f~(r'). (5.1 17) i = - � 8 9 i r j

One should be careful to interpret the forces f~ : these forces are the forces that the surface elements of bead / would have exerted on the fluid, in the absence of bead j. These forces are not equal to the forces on bead / in case of the intact rod. The difference arises from the contribution to the total flow field as a result of the presence of bead j. For very long and thin rods, consisting of many beads, this difference may be neglected. There are only a

284 Chapter 5.

few neighbouring beads i of bead j for which the neglect is not allowed, but there are many more beads i, further away from bead j , for which the neglect is allowed. The relative error made in eq.(5.117) by taking the forces fi equal to the actual forces on each bead of the intact rod is small for long and thin rods.

Substitution of eq.(5.117) into Fax6n's theorem (5.116) now yields,

1

1 h ]o vj = -37rr/oDFj + ~ dS' 1 + D2V T(rj - r ' )- f i ( r ' ) . i = - � 8 9 i #3 Vi

(5.118) Furthermore, for the majority of beads i, the distance rj - r',~ rj - ri, the error being at most equal to the size of the beads. Moreover, rj - r~-( j - i)Dfi, with fi the orientation of the rod, which is the unit vector in the direction of the long axis of the rod (see fig.5.6). From the expression (5.28) for the Oseen matrix one finds, for r r O,

V2T(r) - 1 [ i -3}}] 47rr/or3

so that, for long and thin rods, eq.(5.118) can be approximated by,

1 1 h v j ,.~ - ~ F j - 3rr/oD ' 87rr/oD

1 �89

-8 ,oV E

[ 2 1 1 ] fail. ~ l i - j l 6 l i - j l 3 F~

i-- - ~ n , i y j

1 1 1 ] F h (5 119) l i - j l + 1---2 1 i - j [3 �9 �9

This is the equation from which translational and rotational friction coefficients are calculated in the next two subsections, for very long and thin rods for which "end effects" are negligible. A more accurate way to go about would be to invert the set of relations (5.119), in order to express the forces in terms of the velocities of the beads. The velocities are known when the motion of the rod is specified, so that the forces can then be calculated, from which expressions for the relevant friction coefficient follow immediately. This involves the inversion of a (n + 1) x (n + 1)-dimensional matrix, which can be done numerically with the help of a computer. Here we restrict ourselves to the derivation of limiting expressions for very long and thin rods, and compare with results of more accurate (numerical) calculations which, to some extent, include end effects.

5.15. Friction o f Long and Thin Rods 285

5.15.1 Translational Friction of a Rod

In case of a stationary translational velocity of the rod, the hydrodynamic forces F h on the beads are approximately equal for each bead. Only the beads near the ends of the rod experience differing forces : for very long and thin rods, the relative error is small when these "end effects" are neglected.

F h We may thus use that F) ~ g-gi-F h, where is the total force on the rod. Substitution of this approximation into eq.(5.119), summing both sides over all beads j, and noting that the translational velocity of all beads is equal to that of the center of the rod, v, yields,

- 1 v - ~.,,o~ {s..(~/D)~.~, + s . ( ~ / D ) [ i - < .<. ] )F~, (~ , :o)

where the following two functions are introduced,

f l I (L /D) - 1 -~

and,

1 1 [ 111 3 1 ~ ~ l i - j l - - 6 1 i - j [ a ' 8 n + l j=_�89 ,=-�89 ,~ ~

(5.121)

1 [1 1 1 1 8 n + l y~ E [ i - j l ~ 1 2 [ i - j I a " j=-}~ ,=-�89162

(5.122) These sums may be evaluated in leading order by replacing the summations by integrals, as discussed in appendix E. The results of these integrations are,

3 l n { L / D } , (5.123) f l I (L/D) = -~

3 f z ( i / D ) = - 4 1 n { i / D } . (5.124)

In case F h [[ fi, it follows from eq.(5.120) that,

V - - - - ~

1 3rrloL f l I (L/D) F h '

so that the corresponding friction coefficient is equal to,

27r~oL '711- l n { L / D } " (5.125)

286 Chapter 5.

In case F h _1_ fi, one obtains similarly,

4r~oL "/• = ln{L/D}" (5.126)

The limiting expressions (5.125,126) were already quoted in chapter 2 (see eqs.(2.93,94)). Notice that for the long and thin rods considered here, the perpendicular friction coefficient is twice as large as the parallel friction coefficient.

The translational diffusion coefficients DII and D• follow simply from the Stokes-Einstein relation, that is, Oil ' x - kBT/711 ,x. The explicit expressions for D and AD (see eqs.(4.151,152)) appearing in the Smoluchowski equation (4.154,155) are,

f ) - k B T [~flI(L/D)+~f• = kl~----~T ln{L/D } (5127) 37rr/oL 37rr/oL ' "

and,

A D - 37rr/oLkBT [flI(L/D ) _ f• = 47rr/oLkS----~T ln{L/D} . (5.128)

Broersma (1960) includes end effects for cylindrically shaped rods with ln{L/D} > 2 in an approximate way. His result is obtained from the limiting expression (5.126) by replacing the logarithm in the denomenator by In{ L / D } - v, with v - 0.12. The most simple expression that includes end effects in an approximate way is thus obtained by replacing the logarithm in eq.(5.126) by ln{O.89L/D}.

5.15.2 Rotational Friction of a Rod

The rotational friction coefficient % for a long and thin rod was defined in chapter 2 as (minus) the proportionality constant between the hydrodynamic t o rque 'T h that the fluid exerts on the rod and its rotational velocity f~ (see the discussion in subsection 2.8.2). The rotational velocity is assumed to be perpendicular to the orientation fi of the rod, that is, rotation around the long axis of the rod is neglected. The friction coefficient associated with rotation around the long axis is considered in exercise 5.10.

The velocity of bead i is equal to i Df~ • ft. The relative change of the velocity from one bead to the other is thus ,~ 1/i. For beads further away

5.15. Friction of Long and Thin Rods 287

from the center of the rod, one may thus consider the velocity of larger groups of neighbouring beads equal. Each bead in that group of neighbouring beads experiences the same friction force, which is proportional to the velocity of that group of beads. One may thus write the following expression for the friction force on a bead i,

F) - - C i D l2 x fi , (5.129)

where C is a yet unknown proportionality constant. This expression is not valid for beads close to the center of the rod, since there the relative change of the bead velocity is not small. The total torque on the rod, however, is determined by the forces on the beads further away from the center of the rod, since these forces are evidently larger than for beads closer to the center. Hence, for very long and thin rods, we may use the above expression for the forces on the beads in eq.(5.119), making a relative error that vanishes in the limit L / D ~ o0. Multiplying both sides of eq.(5.129) with rj • and summing over the bead index j yields the following expression for the hydrodynamic torque ,/--h o n the rod,

�89 3

Th = ~ r i • - - C D 2 ~ - - ~ ( L ) I t , (5.130) 1 i=-~n

where we used that k 2 ~j=l J -- ~k(k + 1)(2k + 1), which relation is easily proved by induction. The constant C is yet to be determined. This is done with the use of eq.(5.119), which leads to a second relation between the torque and the angular velocity. The constant C is then eliminated from the two equations, and resubstituted into eq.(5.130) to obtain the friction constant.

The second relation that is needed to determine the constant C is found from eq.(5.119), by multiplying both sides with rj x and summing over all beads j ,

1 Z 3D 2 1 7 'h + - - - - - g(L/D) Ft (5.131) 1-2 D 1 2 - - 3 7 r r / o D 87r~70 D

where the following function is introduced,

1 �89 1

g(L/D) - (n + 1) 3 ~ ~'~ a i=-�89 , u j---~n

i j ]i - j I + li - j ]a �9

(5.132)

288 Appendix A

This function is evaluated by replacing the summations by integrations, as discussed in appendix E, with the result,

1 ln{L/D} (5.133) g(L/D) - -~

Substitution of the expression (5.130) for the torque into eq.(5.131) results in the following expression for the constant C,

C _ _ 41r~7oD ~ 47r~oD 4 ln{L/D}" ln{L/D} + 5

The limiting expression for the friction coefficient then follows immediately from substitution of this expression for C into r

~-~?oL 3 7,. = 31n{L/D} " (5.134)

This result for very long and thin rods was already quoted in chapter 2 in eq.(2.92).

Broersma (1960) includes end effects for cylindrically shaped rods with ln{ L/D} > 2 in an approximate way. His result for % is obtained from the limiting expression by replacing the logarithm in the denominator in eq.(5.134) by ln{L/D} - u, with,

v - 0 . 8 8 - 7 I n { L / D } - 0 " 2 8 .

The most simple expression that includes end effects in an approximate way would be to replace ln{L/D} by ln{O.42L/D}, where u is taken equal to 0.88.

Appendix A

This appendix contains a number of mathematical expressions which are used in the main text of the present chapter. Results are obtained in the course of the evaluation of the integral J (r) of the Oseen matrix appearing in eqs.(5.35,52),

1

(5.135)

Appendix A 289

The integral (5.135) is calculated via the Fourier transform T(k) of the Oseen matrix,

1 T( r - r') - (27r)3 f dr T(k)exp{ ik . (r - r ' )} . (5.136)

The Fourier transform of the Oseen matrix follows from the Fourier transformed equations (5.24,25) (replace x7 by ik, as discussed in subsection 1.2.4 in the introductory chapter),

k . T(k) - 0 ,

ikg(k) + yok2T(k) - i .

Multiplying the second equation here with k. , and using the first equation, gives i k2g(k)=k. Hence,

k g(k) - - i k-- ~ .

Substitution of this result into the second of the above two equations, leads to the following expression for the Fourier transform of the Oseen matrix,

T(k) = 1 [ ~ _ k k ] ~7ok 2 --~ . (5.137)

Substitution into eq.(5.136) and subsequent substitution of the result into eq.(5.135) gives,

J(r) - ~5- v0 k-~ ~ - - ~ exp{ik. (r - r ' )} . (5.138)

sin{ka} / ,

r dS' e x p { - i k - r ' ) - 47ra 2 vo ka JO

, (5.139)

with a the radius of the spherical surface OV ~ Substitution into eq.(5.138) and transforming to spherical coordinates gives,

J ( r ) - 4a2 f dl~ [ i - 1~1~] fo ~ dk sin{ka} exp{ikl~, r} 71" ]r

(5.140)

The advantage of using Fourier transforms is that the (r - r')-dependence now enters as a product of two exponents. The integration with respect to r' is now easily done,

290 Appendix A

C

Imz

, . v

Rez

I - ~ �9 : : ~

Figure 5.7" The integration contours for the calculation of the integral in the last line in eq.(5.142) fork . r/a > -1, (a), andtr r/a < -1, (b).

where tr - k /k is the unit vector in the direction of k and f dl~ is the spherical angular integration ranging over the unit spherical surface in k-space. Let us now introduce the so-called principal value of an integral, which is defined as,

p f ( . . . ) - ! i ~ [ f - ~ ( . . . ) + / ~ 1 7 6 . (5.141)

The origin is thus removed from the integration range by taking the principal value of an integral. For integrands which are continuous at the origin, the integral is equal to its principal value. The k-integral in eq.(5.140) is now rewritten as follows (with z = ka),

fo ~176 dk sin{ka}ka exp{iklr r} - -21/,ooo dk sin{ka}ka exp{ikl~, r} (5.142)

= -~z 79 dk-~a exp{ik(a + k . r)} - 79 dk exp{ ik( -a + k . r)}

1 [ f?oo 1 k - r f_x~ 1 k . r ] = 4ia 79 dz-exp{iz(1 + ~ 1 } -- 79 d k - e x p { i z ( - 1 + 1} . Z a oo z a

Consider the first integral on the right hand-side in the last line here, for the case that k . r/a > -1 . In this case the integration range can be extended, without changing the outcome of the integration, to include the semi circle of infinite radius in the upper complex z-plane. The integral ranging over the

Appendix A 291

closed contour, as sketched in fig.5.7a, is equal to zero, since the integrand is analytic within the entire region enclosed by that contour (this procedure to calculate integrals is discussed in subsection 1.2.5 in the introductory chapter). From the definition of the principle value, eq.(5.141), is thus follows that (with z - e exp{iqp}),

1 { loxo( z,l+ r,} 79 dz exp i z(1 + ) = lim dz [f " oo z a elo , z a

{ ( = lim~loiL dqpexp ieexp{iqp} 1 + a - i T r , k . r / a > - l .

Here C, is half the circle with radius e at the origin in the upper complex z-plane (see fig.5.7a),

c< - {z I z - e exp{iq0} ; 0 _< qo <__ 7r} . (5.143)

For the complementary case that 1~. r /a < - 1 , the integration contour is closed in the lower complex z-plane, as sketched in fig.5.7b. In the same way it is found that,

f_,~ 1 { l~.r P d z - e x p i z ( l + oo z a

~)} - - i t , [<. r/a < - 1 .

The second integral in the last line on the right hand-side of eq.(5.142) is evaluated similarly, with the result,

79 dk-1 exp i z ( -1 - t - ) - iTr, k . r / a > 1, oo Z a

= - iTr , tr r /a < 1.

Collecting these results leads to the following expression for the integral in eq.(5.142),

L oo sin { ka }

dk ka

l(.r e x p { i k k . r } = r__ - 1 < < 1

2a ' a ' = 0 , o therwise . (5.144)

The integral J ( r ) in eq.(5.140) is now reduced to,

(5.145)

292 Appendix A

Figure 5.8: The integration range A S on the unit sphere in k-space. The angle a is set by the value o f a / r. For r - a, this integration range is the entire unit sphere.

COS

r

where AS is the following section on the unit sphere (~- - r / r ) ,

/xs - { t I -1 < l ( . r a} < 1 ~ . / - < - . (5.146)

r

This integration range is sketched in fig.5.8. Notice that, in case r E OV ~ that is, r - a, this integration range is the entire unit sphere. For this special case, the integral is easily calculated,

16 2a i J ( r ) - y r a ] : - SrOo 3~o ' r e OV ~ �9 (5.147)

This result is identical to eq.(5.52). In eq.(5.35), however, J ( r ) must be evaluated for r > a. For this more general case, it is convenient to rotate r onto the z-axis. Let the matrix A denote the rotation that maps r onto the z-axis (63=(0, 0, 1)),

A - r - r63 .

The inte^gral (5.145) is now rewritten in terms of the new integration variable ~: '=A �9 k. The new integration range is then the dashed area in fig.5.8, rotated into the xy-plane,

~ , o} - - - < k 3 < - , ()5.148

r r

where k~ is the z-component of the unit vector 1~'. The integral (5.145) can thus be rewritten as (A -~ is the inverse matrix of A, ~' and O' are the spherical

Appendix A 293

coordinates of k', and x'=cos { O'}),

J(r) 2a ~ , , dl~' [ I - (fill -1" l~')(A. -1. t(')] 2r [a /r

-- 2a fo dqJ J-~/," dx' [ I - ( A -1. I~')(A -1. 1~')] <5.149)

= ~ i - 2a d~' ' - o # d~' (A -1 �9 f(')(A -~ �9 ~').

The integral on the right hand-side in the last line here can be calculated by writing the matrix components explicitly,

jfo2r [a /r d~,' , -o# dz' (A -1 �9 I?)~(A -1 �9 l~')j

- E A 2 A j2 e~,' , - o # e~' k: k : . n~m~l

For reasons of symmetry, it is easily seen that this integral is zero for n 5r m. For n - m we have,

fo2,~ [a/,. [ 1 (ra_)3 ] d~ ' J-~/~ dz ' k" k'~ = 27r a _ r "3 , f o r n - - m - - l , 2 ,

= -~ --27r -r--3 +27r --r + , f o r n - - r n - 3 .

Hence,

2,~ [a/,. fo d~' a-a~,, dx' (A -1. l~t)(A -1. k') (5.150)

1 :3] : (A~IA~-q - A.'(21A~-q - A~IA~)27r J r - 3 (a)

+A~aA~3127r[ a-- ( a ) 3 ] r r

Now, for a rotation matrix the inverse is equal to its transpose, so that,

3 A~ 1A-~ + A~ 1A -1 j2 + A~ 1A y~ ~ A~ 1Asj - ~ij ,

294 Appendix B

with 5ij the Kronecker delta (Sij = 0 for i # j, 5~j - 1 for i - j). Furthermore, since A rotates r onto the z-axis,

r/r - A -1.e3 - (A11,A2 "1,A31) �9

The integral in (5.150) is thus equal to,

2r fa/r ~(I -1 f~X$ dl( kl( = fo dr d-a/r dx' (A. -1. )(A �9 k')

[a 1 ( a ) 3] rr [ a ( a ) 3] - i2~ - ~ + g 2 ~ - ; + .

(5.151)

Substitution into eq.(5.149) thus finally leads to the following expression for the integral we were after,

J(r) - 8rrlo ~ovo dS' T(r - r') (5.152)

1 3] a _ _ 3] rr

A p p e n d i x B

In this appendix the integral J (r) appearing in eq.(5.42) is evaluated,

~o dS' 1 [~ J(r) - v0 I r - r ' l + ( r - r')(r r')]

i r - - r ' ~ �9 (r' • ~ ) . (5.153)

Just as in the previous appendix, this integral can be rewritten as an integral of the Fourier transform (5.137) of the Oseen matrix as,

d(r) = ~-~ v0 k-~ ~ - ~ - e x p { i k . ( r - )} . ( x

The integration with respect to r' can be done as follows, using eq.(5.139),

~ovo dS' exp{- ik , r'}(r' x f~) = -if~ x Vk logo dS' exp{- ik , r'}

sin{ka}ka - -47ria2f~ x l~ d-~ sin{ka}ka = -4ria2f~ x Vk --

Appendix C 295

with V k the gradient operator with respect to k, and ~:=k/k. Performing a partial integration, the integral (5.153) can thus be rewritten as,

J(r) - 47ria 2 ~ d sin{ ka}

7r 2 f dl~ [ I - 1~1~]- (l~x f~)fo dk exp{ikl~-r}dk ka

[ :o ] 7r 2 ~2 x dl~ [~ 1 + i(l~. r) dk sin{ka} exp{ikl~, r} ka

with f dtc the spherical angular integration r~ging over the unit sphere in k- space. In the second line here, we used that kk. (k x f'/) =0, since, (k x f~)_l_l~. The k-integral on the right hand-side in the last line was already calculated in the previous appendix (see eq.(5.144)). Substitution of that result leads to,

J(r) - 7r 2 ~2x dl~k 1+i(1~ r) ~ , for - 1 < / ' -~<1

0 , otherwise

r / - r

The integration range AS is defined in eq.(5.146) and is depicted in fig.5.8. The last integral here was already calculated in the previous appendix (see eq.(5.151)). Substitution of that result into the above expression finally leads to,

J(r) - 8rr/o ~oyo dS' T ( r - r '). (r' x fl)

x 1 a) rr a a) = - 4 r a f ~ { [ I ( : 3 ( a ) + ~ 7 ( - r + ( a ) ] . r }

= -47ra~ f~ x r . (5.154)

Appendix C

Consider the second integral on the right hand-side of eq.(5.61),

j _ 4ra ~ov dS ~ov dS' ( r - r , ) x [ T ( r - r ' ) . f(r')l . (5.155)

The integral with respect to r can be expressed in terms of integrals that are calculated in appendix A and B as follows. First rewrite,

~av dS ( r - r,) x [ T ( r - r ') . f(r')] -

( r ' - rp) x ~ov dS T ( r - r ' ) . f ( r ' )+ ~ov dS ( r - r') x [ T ( r - r ' ) . f(r')].

296 Appendix D

Using the explicit form (5.28) of the Oseen matrix, the last integral is easily rewritten as,

Joy dS ( r - rv) • [ T ( r - r ' ) - f ( r ' ) ] -

( r ' - rv) x flog aS T ( r - r ' ) . f ( r ' ) + flog dS T ( r - r ' ) . [ ( r - r ' )x f(r ' ) ] .

Next transform to r" - r - r v, and rewrite the above equation as,

fioydS(r - rv)• [T(r - r ' ) . f(r')] - ( r ' - rv)x f iovdS"T(r ' ' - r ' + rv). f(r ' )

- ~oyodS"X(r ' ~ r ' + rv). [ ( r ' - rv) x f(r')] +~ovodS"W(r "- r ' + r ,) . [r"x f(r ' ) ] .

This expression needs be evaluated only for [ r' - rp l - a. The first two integrals on the right hand-side are evaluated in appendix A (see eq.(5.147)) while the last integral is evaluated in appendix B (see eq.(5.154)). Using these results we obtain,

~o dS(r- rv) • [ T ( r - r ' ) . f(r')] - a v 3~7o

( r ' - rv) x f ( r ' ) .

Substitution into eq.(5.155) then finally yields,

1 Tp h (5 156) g - - 127rr/o--------~ "

This term corresponds to the first term on the right hand-side of Faxdn's theorem (5.63) for rotational motion.

Appendix D

Before solving the boundary value problem (5.82) for the connectors U (n) (r), let us derive the properties (5.83) of the basic polyadic matrices,

H(~)(r) = V V . . . V 1 r ~ r

n •

These properties will be used here to derive the expressions for the connectors as listed in table 5.1.

Appendix D 297

Since V 2 ! - 0 for r ~ 0, as can be verified by performing the differen- t tiations, it follows immediately that V2H (n) - 0 and V . H ('~) - 0 for n >__ 1. These are the first two properties listed in eq.(5.83). The third property is proved by "moving r into the string of V-operators", as follows,

r . H (n+l)) ili2...in -- ~ r~VmVil m = l

1 Vi~- �9 �9 Vi. -

r

3 [ = E V m rmVia "'" V i . -- 3Vi i "'" V i . 1

m - - 1 r

3 [{ - Vi~ ~ V~

m--1

1 -3Vi~ . . . V i . -

r

3 [ - Vi~ ~ V~ r ~ V i 2 . . - V i . - 4 V i ~ . . . V i . 1

m = l r

3 rm �9 . . - v , , . . , v,~ E v~ - (n + 3)V,,.-. V,o 1

r r m- -1

=2#

(n + 1)Vi~ Vi. 1 (n + 1) (") = . . . . . : - H~...~ . (5.157) r

Next, using that ~r2H(n) - 0, and V H ('~) - H (n+l) by definition, one finds,

V 2 (r2H (~)) - 6H (~) + 4 r . H (n+l) .

From eq.(5.157) we thus obtain,

V 2 (r2H (~)) - - 2 ( 2 n - 1)H (") .

The last property in eq.(5.83) follows from the second and third property,

V . ( r 2 H (n)) - V r 2. H (") - 2 r . H ('') = - 2 n i l (n-l) .

This completes the proof of the properties listed in eq.(5.83). For explicit calculations of microscopic diffusion matrices, explicit ex-

pressions for the basic matrices are needed. Up to the level that is considered

298 Appendix D

in section 5.12, the first five basic matrices suffice. Straightforward differentiation yields,

H(O) = 1_ r

H ! 1 ) - r i /,3

H!~) 6ij rirj - r- S + 3 r---T-, (5.158)

= 3~iJrm + t)imrj + ~jmri _ 15rirjr_.__~m r.5 r7 '

= 3~J8~ + ~i~j~ + ~jm~i~ rirjrmrn r5 + 105 r9

- 1 5 ~ijrmr~ + $imrjr~ § ~jmrir~ + ~inrjrm + ~j~rirm -I- ~m~rirj

H!. 4) ~3mn

r7

Let us now consider the construction of the connectors, which are the solutions of the boundary value problem (5.82). We shall need explicit expressions for H ('~) | (V)~uo. These quantities may be obtained simply by first calculating I-I (n) and then contracting with (V)nuo. The explicit expressions for I-I ('~), however, become quite formidable for n > 5. It requires an enormous effort to calculate the desired contractions for n > 5 in this way. On performing the contraction, many terms yield identical contributions due to the symmetry of (V)'~Uo in its first n indices, and many terms vanish due to the creeping flow equation V2V2uo - 0. The easy way to obtain explicit expressions for H ('~) | (V)'~Uo, without having to calculate H (n) first, is as follows. Since (V)'~Uo is symmetric in its first n indices, we may interchange any of the last n indices in any term in the expression for I-I ('~). Many terms become equal by performing such interchanges of indices, which considerably simplifies the explicit expression for H (n). Secondly, since V2V2uo - 0, all

terms in H !n). which are proportional to a product of two or more Kronecker ~1 ""$n

delta's with differing indices (for example, (5il i3 (5i~ is ) may be disregarded" these terms vanish on contraction with (V)nUo. Keeping this in mind while differentiating 1- to obtain I-I ('~) and a little practice, readily leads to the results

T

listed in table 5.2. These results are needed in the sequel to derive explicit expressions for the connectors.

The first thing that comes to mind, is to represent the connectors by a linear combination of the basic matrices I-I ('~) and products of the basic matrices with the unit matrix. It is readily found that such linear combinations cannot be made to satisfy the boundary value problem (5.82). According to the properties

Appendix D 299

Table 5.2 �9 Explicit expressions for the contractions

H(") O (V)~uo.

revuo H(: )OVUo - ,~

H (2) | (V)2uo v2u~ + 3r2|176 1.3 r 5

H (3) Q (~7)3U0 9revv2uo r 5

5!v r~| �9 7 .7

H(4) Q (~7)4 Uo - 6 x 5! vr2~176 ,r + 7![ r'o(v)'uo,9 i

H (5) | (V)SUo 2 x 7!v r3| ~v2u~ 9!v r~| ~u~ - - . r 9 ' _ . r l l

H (6) 0 (V)6U0 - 3 x 9 vv r4~176 rs~176 .. ,:: + 11!! ,13

of the basic matrices listed in eq.(5.83), the creeping flow equations are also satisfied by combinations of the form r2I-I (n). Including such terms in a linear combination readily shows that the general form of the connectors is,

U(')(r) - cn(r 2 - a2)H(~)(r) + cn_2(r 2 - a2)H('~-2)(r)i i " ! ^ -1- c=_21H(n-2)(r) -F Cn_41H(~-4)(r)J:. (5.159)

Terms of the f o r m . . , i i need not be considered, since these give rise to terms �9 .. V2V~uo(r) - . . . 0, on contraction with (V)n-2uo(r) . The constants

' and ' cn, cn-2, cn_ 2 cn_ 4 can be chosen such that U ('~) (r) is the solution of the boundary value problem (5.82).

As an example, let us calculate U(r)(r). Form the properties of the basic matrices listed in eq.(5.83), it follows immediately that the above form satisfies the creeping flow equation V2V2U(r)(r) - O. Furthermore, since the basic

300 A p p e n d i x E

matrices tend to zero at infinity as H('~)(r) ,-~ 1 / r TM, it is easily seen that the above form for the connectors also tends to zero at infinity. To render the solution of the boundary value problem (5.82), the constants in eq.(5.159) must be chosen such that both U(Z)(r) - I r 5 on OV ~ and V . U(Z)(r) - 0. Since r is equal to a on OV ~ it follows that,

c~iH(S)(r) + c~iH(3)(r) i - Jr s , r e OV ~ .

This condition must be satisfied as a contraction with (V)Suo(r), so that the expressions in table 5.2 may be employed here, to find that,

Cst _ _ --all/9!l and ' - 14a9/9 v! �9 ~ C3 . . .

Next, the divergence of U(Z)(r) is easily obtained with the use of eq.(5.83),

V . U(Z)(r) - -14czH(6) ( r ) - 10c4H(4)(r)iF

all 1 4 a 9 H ( 4 ) ( r ) I - 0 9!! H(6)(r) + 9!!

This equation is satisfied for,

cr -- -aXl/(14 x 9!!) , and c5 - 14a9/(10 x 9!!).

Substitution of these constants into the expression (5.159) yields the expression for U(Z)(r) as listed in table 5.1.

Appendix E

Consider the function,

�89 � 8 9 1 1 ] f l I ( L / D ) - 1-~ 3 1 ~ ~ [ i - j l - -6 [ i - j [a " 8 n + l j=_~,~ ~ _ _ � 8 9

(5.160) For very large values of L I D - n + 1, the second term in the sum may be neglected in comparison to the first term, since the second term tends to zero at infinity much faster than the first term. The second term may be dealt with in the same manner as the first term is dealt with in the sequel. We leave it out here from the start since is does not contribute to the leading expression of fll for large L /D.

Appendix E 301

,,

j - t , j - 3 j -2 j-1 I I j j+l j4-2

Figure 5.9: The sum in eq.(5.161) equals the surface area of all rectangles, and the integral is the surface area under the solid curve.

Let us first evaluate the sum,

1 ~n 2

,=-l~. i~ [ i - j l

This sum equals the surface area of all the rectangles in fig 5.9. It can be 1 1 replaced by an integral, when the range ( - 7 n , gn) of the sum is large,

+ di . (5.161) ,=_ , , r aj+} j ] i - j l

The difference between the sum and the integral is the sum of the dashed surface areas in fig 5.9 (with their proper sign). For increasing L/D-ratios, this difference tends to a constant, while the sum itself goes to infinity. The relative error that is made by replacing the sum by an integral thus tends to zero as L / D tends to infinity. The leading terms in the above integral are,

1 l ( n + l ) _ j } 21n{j + ~(n + 1)} + 21n{~

This expression is substituted into eq.(5.160), where the sum over j is again replaced by an integral. Using the standard integral,

/ dz z TM ln{z} - z m+~ [in{z} m + l

1] + '

one ends up, to leading order in D/L, with the result given in eq.(5.123).


The sums which define ]'1 in eq.(5.122) and g in eq.(5.132) are evaluated in precisely the same manner, replacing summations by integrals.

E x e r c i s e s

5.1) * In this exercise we prove the following representation for the delta distribution,

1 1 - - v ~ - - ~ ( r - ro) 47r [ r - r o [

where the differentiation is with respect to r. Let f ( r ) be a smooth, but otherwise arbitrary function. Consider the integral,

1 f dr f ( r ) V 2

I ~ - r o l "

Verify that V 2 1 - 0 for any r except for r - ro, where the function [ r - r o l - -

i~!ol is not defined. The integration range in the above integral may thus be replaced by a spherical volume S, with an arbitrary small radius e centered around ro : outside that spherical volume the integrand is zero. For very small e, and provided that f ( r ) is a differentiable function, the integral may thus be rewritten as,

f dr f ( r )V2 1 fs 1 [ r r o [ = f(ro) d r V 2 - �9 I r - r o l "

Use Gauss's integral theorem, and translate the coordinate frame over the distance ro, to arrive at (OS ~ is the spherical surface with radius e at the origin),

fd~f(r)V ~ i ~o dS"'Vl [ r - r o [ - f(ro) so r "

Here, fi is the unit normal on the spherical surface, directed outwards. Verify 1 _ _ d 1 _ _ 1 Evaluate the surface integral, using spherical that f t . V 7 - a-77 - - ; r .

angular coordinates, to obtain,

f f/r/V [ r - ro [ -- -4~" f(ro) �9


This proves the delta distribution representation we were after.

5.2) * Consider the following boundary value problem,

V 2 f ( r ) - 0 , on ~3,

f ( r ) ~ 0 , for r--+oo.

Use Green's integral theorem (see subsection 1.2.2 in the introductory chapter) and the above properties of f ( r ) , to show that,

1 1 f dr' f(r')V'21 r - r ' I - f dr' I r - r ' ] V '2 f ( r ') - O.

Verify with the help of the representation of the delta distribution derived in the previous exercise, that,

- 47 r f ( r ) - 0 ;- f ( r ) - 0.

This proves that a function is identically equal to 0 when its Laplacian is equal to zero and the function itself is zero at infinity.

5.3) * In this exercise, the solution to the problem (5.9-11) is constructed. In chapter 4, a solution of the differential equation (5.9) was already determined in three dimensions (see eqs.(4.62,68)). In one dimension this solution reads (replace Do in eq.(4.68) by r;o/po),

~ po [ po(z-zo) 2] u~ o (z, t) - 47r~7o t exp - 4710t "

Although this is a solution of the differential equation (5.9), it does not satisfy the initial and boundary conditions (5.10,11) of the present problem. For example,

lim Uzo (z, t) - 6(z - zo), t l0

with 6 the 1-dimensional delta distribution. However, since the differential equation is linear, the following superposition is also a solution,

F u(z, t) - dzo f (zo) u~ o (z, t ) , o o

where f is an arbitrary function. This function can be chosen, such that the initial and boundary conditions of the problem are satisfied. Show that the choice f(zo) = 2 [1 - H(zo)] renders the solution of the problem (5.9- 11). Here, H (zo) is the Heaviside unit step function H (zo) - 0 for zo < 0, H(zo) - 1 for zo > 0. Show that this solution is identical to that in eq.(5.12).

5.4) The effective viscosity On a length scale that is large in comparison to the size of a Brownian

particle, a flowing suspension can be described as an "effective fluid" (see fig. 5.10). The Navier-Stokes equation applies also to suspensions, where the viscosity r/o is now replaced by the "effective viscosity" 77 ~ff of the suspension. This effective viscosity depends on the concentration of Brownian particles and the way they interact. In this exercise we calculate the effective viscosity up to first order in concentration. Interactions between the Brownian particles may be neglected at this level.

Assuming incompressibility of the core material of the Brownian particles, the effective viscosity determines the "effective stress matrix" just as 770 determines the stress matrix of a fluid in eq.(5.6) with V �9 u = 0,

E~Z(r, t) - r/~/y {VU(r , t) + (VU(r , t)) T} - P( r , t ) i .

Here, VU(r , t) and P(r , t) are the flow velocity gradient and pressure of the suspension at a position r at time t. These are averaged quantities over fictitious volume elements which contain many Brownian particles (see fig.5.10). The effective stress matrix is the corresponding volume average of the "microscopic stress matrix",

1 fvdr 'E(r~ r2 . . . , r N l r ' ) E ~ Z ( r ' t ) - V ' ' "

The position coordinate of the fictitious volume element V is r. The microscopic stress matrix depends on the positions of the spherical Brownian particles within the volume V. For N non-interacting Brownian particles there are N independent contributions to the total microscopic stress matrix, so that,

E~ff(r,t) = V dr' r,o(r')

N[jv Jr ] V dr' r,o(r') + dr' r.o(r') o \ V o


/

\

0

. . . . ~" - - ' -~-" 0 - ~ 0

- - - ~ �9

0

I I I I I I I I effective mi croscopic Figure 5.10: Figure on the left" the flowing suspension on a length scale large compared to the size of a Brownian particle. Figure on the right : A blow up of a fictitious volume element, showing the flow on a length scale smaller than the size of the Brownian particles. The dotted straight line indicates the flow velocity gradient pertaining to the effective flow.

In the last line, the integration range is split up into the volume occupied by the core of the Brownian particle V ~ and the remaining space V \ V ~ that is occupied by fluid. Without loss of generality, this sphere may be positioned at the origin. The index "0" on the stress matrix ~o(r ' ) is used to indicate the stress generated by just a single force free sphere in shear flow. In the fluid, outside the core of the sphere, the microscopic stress matrix is related to the fluid flow induced by the sphere, as given in eq.(5.6). We do not know, however, about stresses inside the core of the Brownian particle. Therefore, the integral over the core V ~ of the sphere is firstly rewritten as integrals ranging over space that is occupied by fluid.

Show that,

r~o(~') - v ' . ( S o ( e ) ~ ' ) - (v ' . So(e)) ~',

and use Gauss's integral theorem to arrive at,

-

where aV ~ is the surface of the sphere at the origin. V ' . ~o(r ' ) is the total force on a volume element at r', which is zero on the time scale on which


stresses in the core relax. This time scale is not larger than the Brownian time scale, so that the last integral here is zero.

Conclude that,

.'".r ,> ''o"> ". '+/.,.o'r o.r >]

Show similarly that,

N {V'uo(r ') + (V'uo 7/0 {VU(r, t)+ (VU(r, t)) T} - ,o v/v dr' (r')) T} "[ ] = V rio ~vo dS' {fi'uo(r') + uo(r')fi'} +/. \vo dr' Eo(r ') �9

The index "0" is used again to indicate that just a single particle is considered. Compare this with the above equation for the effective stress matrix, and

conclude that, apart from isotropic terms ,-~ I which do not contribute to the effective viscosity,

r,~z(~,t) - ~o {VU(r, t) + (VU(r, t)) ~} N

+ 'V ~vo dS' {(Eo(r ' ) . f i ' ) r ' - r/o [fi'uo(r') + uo(r')fi']} Q

Verify with the use of Gauss's integral theorem, that the integration range O V ~ may be replaced by an arbitrary surface which does not intersect O V ~ This is particularly handy for the explicit evaluation of the above integral : replace O V ~ by a spherical surface of infinite radius, so that all terms in the integrand which tend to zero at infinity faster than 1/r '2 may be omitted.

h Calculate the integral, using eq.(5.109) for u - Uo, with F v - O, since the single particle is force free, and with r = VU(r , t), since that is the local velocity gradient at the position of the fictitious volume element V. In the explicit evaluation of the integral you will need the following identity,

go dS' fi:h~n'vfi'q = --~a .

Verify that,

5], 77 ~z - 71o [1 + ~


where ~ - ~a3-~ is the fraction of volume that is occupied by the core material of the Brownian particles. This is Einstein's equation for the effective viscosity of a dilute suspension.

5.5) Oseen's approximation For point-like Brownian particles, the hydrodynamic force density in

eq.(5.22) is equal to,

N

h 6 ( r ' - r j ) . ff~t(r') - - ~ Fj j = l

Verify that this Ansatz reproduces the Oseen approximation (5.55,56) for the microscopic diffusion matrices.

5.6) Sedimentation of two spheres Two spheres in a fluid attain a certain steady state velocity due to a gravita-

tional force. The force is equal for both spheres. For small Reynolds numbers, the hydrodynamic force on each of the spheres is equal in magnitude, but opposite in sign to the gravitational force. Use the fact that the microscopic diffusion matrices are even functions of the separation vector between the two spheres, to show that the two spheres attain equal velocities v, which is related to the gravitational force F as,

V - ~ [D.(r i j )+ Dij(rij)]. F

= ~ [Dji(rij)+ Djj(rij)] . F .

Express the force in terms of the velocity. Use the Oseen approximation (5.55,56) for the microscopic diffusion matrices, and perform the matrix inversion to first order in a/rij. Show that,

[( 3~ F - 6ryoa ]: 1 4 rij 3 a rijrij]"

4 rij V .

Is there a direction of the relative separation vector/'~j where the friction coefficient is larger than 67ry0a ?

5.7) * In the derivation of the Fax6n's theorems, integrals of the form,

~ y dS (r - rp) ~ - ~oyo dS r ~,


are encountered. Here, aV ~ is the spherical surface of the Brownian particle with its center at the origin. These integrals are matrices of indexrank n. As an example, let us consider one of the elements (n~ + n2 + n3 - n),

dS~ Xnl y~2 Zn3 , V o

with x, y and z the cartesian components of r. Convince yourself that it follows from symmetry that this integral is zero

in case at least one of the numbers nl, n2 or n3 is an odd integer. Show from this, that in case n is an odd integer, the integral is zero, and that in case n is an even integer, and n > 4, the integral is proportional to the product of two (or more) unit matrices I. Verify that such products gives rise to a product of two Laplace operators on contraction with a polyadic product of gradient operators.

Eq.(5.62) is derived in a similar way. Notice that the outer product in the integral on the left hand-side of eq.(5.62) acts on uo(rp) in the Taylor expansion (5.59). Use the above arguments to obtain eq.(5.62).

In arriving at eq.(5.61), as a first step in the derivation of Fax6n's theorem for rotational motion, we used that,

4ra 21 ~yo dS r x [f~v x r] - ~2a2ftp.

Show that,

~o 47r dS~rirj = 5ij , vo 3

and r x [f~p x r] - r2f~ - r r . tip, in order to verify the above equation.

5.8) Hydrodynamic interaction of two unequal spheres Consider two spheres, i and j , with unequal radii ai and a j, respectively.

The first few terms in the reciprocal distance expansion of hydrodynamic interaction matrices are discussed here, starting from sphere i with a translational

1 h velocity vi - 6~o~ F i ' in an otherwise quiescent fluid. (a) Show that the Rodne-Prager matrix is now given by,

D i j - 6ryoaj rij -4 r~j


(b) The flow field of sphere i is reflected by sphere j. This first order reflected field is denoted as u(X)(r). Show that this field, to leading order, is equal to (see also the discussion above eq.(5.97)),

1 [ ] u(1)(r) - - ~ U ( 3 ) ( r - r j ) | V ju(~ (Vju(~ T aj

with,

rf]. u(~ - U(2,)(r- ri)(S) -67rr/oai

The indices a~,j on the connectors indicate which radius should be substituted for the radius a in the expressions in table 5.1. Use the expressions for the connectors given in table 5.1 together with eq.(5.158) for the basic matrices to show that, to leading order,

u(~ - 4 rij 67ryoai

This is nothing but the flow field induced by a point-like particle. Verify that,

Vju(~ + (Vju(~ T 1 1 [~_ 3~j~j] (r i / . Fh) . 47r~o ri 3

Use the expression for U(3)(r) in table 5.1 to leading order, and verify that,

D ~ = kB____~T 15 aia~ -- 67r~7oai 4 r4j rijl?ij "

In case a~ - aj, this reduces to the leading term for the mobility function A, in eq.(5.95).

5.9) Friction of a rod in shear flow For a rod in a fluid that is otherwise in shearing motion, the field u0 in

eq.(5.116) is the sum of the shear flow field F . r and the field induced by the remaining beads.

(a) Consider a rod with its center at the origin and with an angular velocity ft. Similar arguments as for a rotating rod in an otherwise quiescent fluid can be used to show that the force on a bead i is proportional to its velocity relative to the local shear flow velocity f~ x ri - F- ri. This relative velocity,


however, consists of a component parallel and perpendicular to the rods long axis. The proportionality constant between the force and the relative velocity may be different for both components. We therefore write,

r , ~ - -CII tiff. ( f t x r , - r . r , ) - C• [ I - tiff]. (ft x r i - r . ri)

Use this to show that the hydrodynamic torque on the rod is given by,

7 h = --y~ [f~ - ~ x r . f l ] .

A torque flee, non-Brownian rod in shear flow thus attains an angular velocity equal to fi x r . ft.

(b) Consider a rod in uniform translational motion with a velocity v. The force on bead i is again proportional to the relative velocity parallel and perpendicular to the rods long axis, with possibly different proportionality constants,

r , ~ - - c . , a a . ( v - r . r ) - Cz [ i - tiff]. (v - r . r) - - c . , a a . ( ~ - r , r~-iDF, f i ) - C • [I-tiff] . ( v - r . r~-iDF, fi),

where r~ is the position coordinate of the center of the rod. Calculate the constants CII,• and show that the total force on the rod is equal to,

r~ = -nil ~ " (v - r . r~) - n~ [ i - ~ ] . (v - r . r~).

(Hint" The term iDF. fl gives rise to sums over i~ [ i - j 1, which can be evaluated by replacing sums by integrals, as discussed in appendix E. These sums are then found to be of higher order in D / L than the sums stemming from the term v - F . r ~ , and can therefore be neglected to leading order. The physical interpretation of this mathematical result is obvious : the forces arising from the term,-, i D r . fi acting on the beads on one side of the center of the rod cancel with the forces on the beads on the opposite side of the center.)

5.10) Friction of a long and thin rod, rotating around its long axis. A rod rotates along its long axis, that is, the angular velocity fl is parallel to

the orientation fi of the rod. The positions of all beads thus remain unchanged, and each bead rotates with the same angular velocity.

Fur ther R e a d i n g 311

To obtain the friction coefficient for this rotational motion, Faxdn's theorem for rotational motion (5.63) can be used,

1 f ~ j - _ 7r ~ o D------ 5 Yj h +

Stokes .friction o] the bead

1 ~Vp x uo(rp)

�9 i l l

Hydrodynamic interaction with other beads

According to eq.(5.43) the fluid flow field due to a rotating sphere is zero for positions r ~, f~. The fluid flow field that a bead experiences due to the rotation of another bead is therefore small, and tends to zero for large distances between the two beads. This implies that for long and thin rods, hydrodynamic interaction between the beads may be neglected. Only the Stokes friction term on the right hand-side in the above equation is of importance.

When the small contribution from hydrodynamic interaction between the beads is neglected, the forces which surface elements of the beads exert on the fluid are tangential to the surface (see eq.(5.41)). Use this to show that the torque T h on the rod is equal to the sum of the torques Tj h of all beads, as if they were alone in an unbounded fluid,

�89 �89 T h -- ~ ~ dSrxf(r) - ~ Tih.

The position coordinate r is relative to the center of the rod. Use this result to obtain the following expression for the rotational friction

coefficient,

% - 7ryoLD 2 "

Compare this result with the friction coefficient (5.134) for rotational motion perpendicular to the orientation of the rod.


There are a number of books on hydrodynamics, with an emphasis on low Reynolds number flow past spheres, cylinders, etc.,

�9 J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics, Martinus Nijhoff Publishers, The Hague, 1983.

312 Further Reading

�9 S. Kim, S.J. Karilla, Microhydrodynamics, Principles and selected Ap- plications, Butterworth-Heinemann, Boston, 1991.

�9 G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge Univer- sity Press, 1967. The book of Kim and Karilla contains a chapter on lubrication theory. Relevant references concerning this subject can be found there.

A recommendable paper on "life at low Reynolds numbers" is, �9 E.M. Purcell, American J. of Phys. 45 (1977) 3.

Expansion of hydrodynamic interaction functions for two particles in a power series of the inverse distance are considered in,

�9 J.M. Burgers, Proc. Koninkl. Akad. Wetenschap. 43 (1940)425, 44 (1941) 1045.

�9 G.K. Batchelor, J.T. Green, J. Fluid Mech. 56 (1972) 375. �9 G.K. Batchelor, J. Fluid Mech. 74 (1976) 1. �9 B.U. Felderhof, Physica A 89 (1977) 373. �9 D.J. Jeffrey, Y. Onishi, J. Fluid Mech. 139 (1984) 261. �9 R. Schmitz, B.U. Felderhof, Physica A 116 (1982) 163. �9 R. Jones, R. Schmitz, Physica A 149 (1988) 373. �9 B. Cichocki, B.U. Felderhof, R. Schmitz, Physico Chem. Hyd. 10

(1988) 383. In later work, many hundreds of coefficients in the reciprocal distance expansion have been calculated.

The gradient expansion technique for the calculation of reflected flow fields, that is used in the present chapter (and, for example, also by Felderhof (1977)), has been put forward in,

�9 H. Brenner, Chem. Eng. Sci. 19 (1964) 703. This work has been used for the first time by,

�9 J.L. Aguirre, J.T. Murphy, J. Chem. Phys. 59 (1973) 1833, to obtain the very first terms in the reciprocal distance expansion.

An alternative to the method of reflections is the so-called method of induced forces, where the hydrodynamic forces on the surfaces of the particles are expanded in a multipole series. This approach is utilized to calculate the first few terms of the reciprocal distance expansion and the leading three body interaction terms in,

Further Reading 313

�9 E Mazur, W. van Saarloos, Physica A, 115 (1982) 21. Many particle hydrodynamic interaction is also considered in,

�9 K.E Freed, M. Muthukumar, J. Chem. Phys. 76 (1982) 6186. �9 M. Muthukumar, K.E Freed, J. Chem. Phys. 78 (1983) 511. �9 H.J.H. Clercx, EEJ.M. Schram, Physica A 174 (1991) 293, 325. �9 B. Cichocki, B.U. Felderhof, K. Hinsen, E. Wajnryb, J. B lawzdziewicz,

J. Chem. Phys. 100 (1994) 3780. �9 B. Cichocki, K. Hinsen, Phys. Fluids 7 (1995) 285.

Calculation of friction coefficients for rod like particles that go beyond the leading term for large L/D-ratios can be done by the so-called Oseen-Burgers method. The forces are then concentrated on a line, and are represented as a power series expansion in the position relative to the center of that line. The coefficients in this expansion are then found by minimizing the difference of the resulting flow field with stick boundary conditions on a cylindrical surface around the line of force, in an average sence. This method was first used by Burgers, and later refined by Broersma,

�9 J.M. Burgers, Ver. Koninkl. Ned. Akad. Wetenschap. 16 (1938) 113. �9 S. Broersma, J. Chem. Phys. 32 (1960) 1626, 32 (1960) 1632, 74 (1981)

6889. The effects of the precise shape of a slender body on its hydrodynamic friction coefficients is explored in,

�9 R.G. Cox, J. Fluid Mech. 44 (1970) 791. Friction coefficients of rods and flexible macromolecules are also considered in,

�9 J. Garcfa de la Torre, V.A. Bloomfield, Quarterly Rev. Biophys. 14 (1981) 1.

�9 M. M. Tirado, J. Garcia de la Torre, J. Chem. Phys. 71 (1979) 2581, 73 (1980) 1986.

Chapter 6

DIFFUSION

315

316 Chapter 6.

6.1 Introduction

In most cases, experimental data are macroscopic, ensemble averaged quantities. Properties of such macroscopic quantities find their origin in processes on the microscopic scale, where the motion of individual Brownian particles is resolved. The ultimate level of understanding macroscopic processes would be to start from equations of motion for the constituing particles and, by ensemble averaging, obtain the relevant equations for the macroscopic variable under consideration. For colloidal systems, the microscopic ingredients for calculating ensemble averaged quantities have been established in the previous two chapters. In chapter 4 the Smoluchowski equation is derived, which is an equation of motion for the probability density function of the position coordinates of the Brownian particles, and in chapter 5 explicit expressions for the microscopic diffusion matrices are obtained, which are needed as input for the Smoluchowski equation. The present chapter is concerned with the prediction of ensemble averaged diffusive behaviour in systems of interacting colloidal particles.

There are two types of diffusion processes to be distinguished : collective and self diffusion. Collective diffusion relates to the motion of many Brownian particles simultaneously, while self diffusion concerns the dynamics of a single Brownian particle, under the influence of interactions with surrounding Brownian particles. These two distinct diffusion processes are discussed on an intuitive level in the next two sections 6.2 and 6.3.

The interplay between shear flow effects and diffusion on the microstructure of systems at finite concentration is discussed on an intuitive level in section 6.4. The shear flow tends to distort the equilibrium structure, while diffusion tends to restore equilibrium. The relative importance of these two counter balancing processes determines the non-equilibrium steady state microstructure.

After the heuristic and introductory sections 6.2-4, quantitative results are derived from the Smoluchowski equation. We start with the evaluation of short-time diffusion coefficients up to second order in concentration in section 6.5, followed by the derivation of Fick's law for gradient diffusion in section 6.6, with an explicit evaluation of the gradient diffusion coefficient up to first order in concentration. The long-time self diffusion coefficient is calculated up to first order in concentration in section 6.7. The effect of a stationary shear flow on the static structure factor is considered in section 6.8.

The temporal evolution of the density and higher order probability density

6.2. Collective Diffusion 317

functions may depend on the history of the system, that is, may be coupled to states of the system at earlier times. To include such "memory effects", one can, in principle, consider the hierarchy of equations of motion for increasingly higher order probability density functions as obtained from the Smoluchowski equation. An alternative approach is to derive so-called memory equations from the Smoluchowski equation by means of projection operator techniques. Although these equations are as complicated as the hierarchy of equations mentioned above, there is in some cases an advantage in analysing such memory equations. The memory equation approach is the subject of section 6.9.

For rod like Brownian particles, rotational diffusion must be considered in addition to translational diffusion. The effect of rotational diffusion on the intensity auto-correlation function is considered in section 6.10 for non- interacting rods, as well as rotational relaxation to first order in concentration for rods with hard-core interaction.

6.2 Collective Diffusion

Imagine a colloidal system where the density of Brownian particles, at some instant in time, varies sinusoidally (such a sinusoidal density profile is some times referred to as a density wave). That is, at time t - 0 say, the macroscopic density p(r, t - 0) at position r is equal to,

p ( r , t - O) - f i+ p ( k , t - O) sin{k, r} , (6.1)

with p - N / V the average density of Brownian particles, and p(k, t - 0) the amplitude of the density wave. This density profile is sketched in fig.6.1. The wavevector k determines both the direction and the wavelength of the sinusoidal density variation. For changes of the position r perpendicular to k, the phase of the sine function does not change, so that the direction of k is in the "propagation direction" of the sinusoidal variation. A change Ar of the position r parallel to k leaves the sine function unchanged when I A r I - n x 27r/k, with n an arbitrary integer. Hence, the wavelength of the density variation is,

A - 27r/k. (6.2)

The sinusoidal density variation may be thought of as being the result of some fictitious external field. Now suppose that this field is turned off at

318 Chapter 6.

A

/

K

Figure 6.1" A density wave. The "propagation direction" is along the wavevector k and the wavelength is A - 2~r / k.

time t - 0. In a thermodynamically stable system, the amplitude of the density wave decreases with time due to the thermal motion of the Brownian particles (see fig.6.2). In the initial stage of the decay, the sinusoidal shape of the density wave will be retained. At a later stage, different wavevectors, or equivalently, different wavelengths come into play as a result of interactions between the Brownian particles. The strength of these interactions varies with the distance between the Brownian particles, leading to a distribution of relaxation times. Spatial inhomogeneities extending over varying distances relax to equilibrium with different relaxation times. The shape of the density variation is then no longer sinusoidal, but involves other "Fourier components" (other wavevectors) in addition.

The decay of such a sinusoidal density variation is a collective phenomenon, since many Brownian particles are displaced simultaneously. No- tice that p(k, t - 0) is the amplitude of the sinusoidal variation with wavevector k, which is just one of the many sinusoidal density variations that constitute an arbitrary spatially varying density. In addition to this particular wavevector, there are generally many more wavevectors contributing to the actual spatial variation of the density.

Let p(r, t) denote the space and time dependent macroscopic density, and J(r , t) the flux (or current density) of Brownian particles, which is the number of Brownian particles which move across a surface perpendicular to J per unit area and unit time. The continuity equation, which expresses conservation of

6.2. CollectiveDiffusion 319

Figure 6.2: The decay of a sinusoidal density profile. Initially the density profile will be more or less sinusoidal. At later times, however, the density profile is generally no longer purely sinusoidal, but involves many Fourier components.

the number of Brownian particles, reads,

0 0-Tp(r, t) - - v . J(r , t ) . (6.3)

The derivation of this equation is equivalent to the derivation in section 5.2 of the continuity equation (5.1) for fluid flow. The current density in the case of fluid flow is equal to p u, with p the number density of fluid molecules and u the fluid flow velocity. In the present case of diffusion, the flux is driven by gradients in the density of Brownian particles. For small gradients in the density, the flux is a linear function of these gradients. The flux at a certain position r may depend, through interactions with surrounding particles, on gradients at neighbouring positions. Furthermore, the flux at a certain time t may depend on states of the system at preceding times. The flux can thus formally be written as,

J(r, - - f d r ' f Z (r - r', t (6.4)

The integral kernel D(r, t) will be referred to simply as "the diffusion coefficient", which is 0 for t < O, since the temporal evolution of the density cannot depend on future profiles. To leading order in gradients in the density, and for otherwise translationally invariant systems, the diffusion coefficient is a function of the difference vector r - r' only. When the current density at a point r is fully determined by the instantaneous density gradient in that same point, so that there is no coupling with gradients in neighbouring points nor with preceding states of the system, the diffusion coefficient is proportional to a delta distribution in both position and time, that is,

320 C h a p t e r 6.

D(r - r', t - t ' ) - D(r, t ) 5 ( r - r')5(t - t'), so that J(r, t ) - - D ( r , t)Vp(r, t). In general, however, there is a coupling with gradients in the density at different positions, due to interactions between the Brownian particles, and the evolution at a certain instant of time may depend on states at earlier times.

Let us consider diffusion processes where "memory effects" are of no importance, that is, where the time dependence of the current density J is fully determined by the instantaneous density profile. In the absence of memory effects we have,

D(r - r ' , t - t ' ) - D(r - r ' , t ) 6 ( t - t ' ) . (6.5)

To avoid the unnecessary introduction of new symbols, the same symbol for the two diffusion coefficients on both sides of this equation is used. The time dependence of D(r - r', t) is now the result of a constantly changing density during relaxation of the initially purely sinusoidal density profile. This change of density with time affects the coupling between density gradients at different positions. Eq.(6.4) now reduces to,

J(r , t) - - f dr' D(r - r', t)V'p(r', t) . (6.6)

Substitution into eq.(6.3) and Fourier transformation with respect to position yields, with the use of the convolution theorem (see exercise 1.4c),

0 O---~p(k , t ) - - D ( k , t ) k 2 p ( k , t ) . (6.7)

The spatial Fourier transform of p(r, t) is defined as,

p(k, t) - f dr' p(r', t) e x p { - i k , r '}. (6.8)

The Fourier transformed diffusion coefficient D(k, t) is defined similarly. As discussed in subsection 1.2.4 in the introductory chapter, Fourier transformation is nothing but a decomposition in sinusoidal functions. The spatial Fourier transform p(k, t) is the amplitude of the sinusoidal component that contributes to p(r, t). The dynamics of such sinusoidally varying density profiles, which we referred to above, is thus fully described by the Fourier transform D(k, t) of the diffusion coefficient. The solution of eq.(6.7) is,

p(k, t) - p(k, t - O) exp{-D~(k, t ) k 2 t } , (6.9)


where the collective diffusion coefficient is defined as,

l D~(k, t) - 7 dt' D(k, t ' ) . (6.10)

The wavevector dependence of the collective diffusion coefficient does not involve the direction of the wavevector k when the system is isotropic, so that no preferred direction can be defined. For isotropic systems, the collective diffusion coefficient is a function of k -Ikl only.

The zero wavevector limit

For very small wavevectors (large wavelengths), the curvature of the sinusoidal density variation is negligible over distances equal to the range of interaction between the Brownian particles. The gradient of the density profile is then essentially a constant in regions containing many Brownian particles. The collective diffusion coefficient is then equal to the gradient diffusion coefficient, Dr , which describes transport of Brownian particles in a density profile with a constant gradient. Hence,

lim D~(k, t) - D r . (6.11) k---,0

The limit k ~ 0 should be taken with some care. In the strict limit that k becomes equal to 0, the term k2t in eq.(6.9) that multiplies the collective diffusion coefficient vanishes. This means that the corresponding density wave does not evolve in time. Physically this means that in the strict limit k ~ 0, gradients in the density disappear, and with it, the driving force for transport of Brownian particles. The limit in eq.(6.11) is therefore to be interpreted as : "take k so small, that gradients in the density may be considered constant over distances equal to the range of interaction between the Brownian particles".

In writing eq.(6.11) it is assumed that in the small wavevector limit the diffusion coefficient becomes time independent. The reason for this is as follows. The position dependence of the diffusion coefficient D(r - r', t) accounts for the effect of interactions of Brownian particles at r' with those at r. The effect of these interactions changes as the density profile changes its form in time, since the interactions then propagate from r to r' through a different "density landscape". That is, the time dependence of D(r - r', t) is due to the change of the form of the density profile with time. In case the gradient in the density is very smooth, however, it remains so for all times. Only very long wavelength density waves are present during the entire

322 Chapter 6.

r')

O - I" 0

' r = r I , i

!

RI '

Figure 6.3" The diffusion coefficient D(r - r', t) tends to zero over a distance of the order o f the range Rx o f interactions between Brownian particles. The figure shows a density variation which is smooth on the length scale Rx.

relaxation of smooth gradients. The form of the density profile therefore remains the same, and the time dependence of the diffusion coefficient is lost. Eq.(6.10) implies that the collective diffusion coefficient is time independent whenever the diffusion coefficient is time independent. We thus come to the following conjecture,

The collective diffusion coefficient is independent

o f t ime for small wavevectors . (6.12)

There is no rigorous proof of this statement. In the present chapter, this conjecture is verified up to first order in concentration (subsection 6.5.2 and section 6.6), and for weak pair-interaction potentials for arbitrary concentrations (section 6.9 on memory equations).

The diffusion coefficient D ( r - r ' , t) tends to zero over distances [ r - r ' I of the order of the range over which Brownian particles interact. For very smooth gradients of the density, we may therefore replace V'p(r', t) by Vp(r, t) in eq.(6.6) (see fig.6.3),

(6.13)


where the time dependence of the diffusion coefficient is omitted in view of the conjecture (6.12). According to eq.(6.10) we have D(k - O) - Dr . For this special case of very smooth gradients in the density, the continuity equation (6.3) reduces to,

0 0--t p(r, t) - Dv V2p(r, t ) . (6.14)

This is Fick's law. This equation of motion will be derived from the Smolu- chowski equation in section 6.6, resulting in an explicit expression for the gradient diffusion coefficient Dv in terms the interaction potential and the density ~ - N / V of Brownian particles.

Notice that the Smoluchowski equation (4.62) for non-interacting Brow- nian particles is of the form of Fick's law, except that the gradient diffusion coefficient is replaced by the Stokes-Einstein diffusion coefficient Do. For very dilute suspensions, where interactions are of no importance, the gradient coefficient thus becomes equal to the Stokes-Einstein diffusion coefficient.

Short-time and long-time collective diffusion

The initial decay of a purely sinusoidal density profile is described by the collective diffusion coefficient in eq.(6.10) at small times, which is referred to as the (wavevector dependent) short-time collective diffusion coefficient D~(k),

D ~ ( k ) - lim D ~ ( k , t ) - D ( k , t - 0 ) . t---,O

(6.15)

In practice, the short-time limit is reached for times which are of the order of a few times the Brownian time scale.

Late stage decay of the Fourier component of a density profile, that was originally purely sinusoidal with a particular wavelength A - 27r/k, is described by the long-time collective diffusion coefficient Dt~ ( k ),

Dry(k) - lim D ~ ( k , t ) . (6.16) t--~oo

It is difficult to assess the time at which the long-time limit is reached, if it reached at some finite time at all.

Notice that the conjecture (6.12) implies that the long- and short-time collective diffusion coefficients are equal at zero wavevector.

324 Chapter 6.

Light scattering

As we have seen in chapter 3, light scattering probes a single density wave, the wavelength of which is set by the scattering angle. Although many wavevectors contribute to the dynamics of density variations, light scattering probes only a single wavevector.

For spherical particles, the normalized density auto-correlation function is equal to the normalized electric field auto-correlation function (EACF) as measured with light scattering (see eq.(3.83)). This correlation function follows from eq.(6.9) as (see subsection 1.3.2 in the introductory chapter on correlation functions),

~E(k,t) =<p(k , t )p* (k , t - 0)> / < lp(k , t - 0) 2 > - e x p { - D ~ ( k , t ) k 2 t } , (6.17)

where the brackets < . . . > denote ensemble averaging over initial conditions. The above result can be reformulated in terms of the collective dynamic structure factor which was introduced in eq.(3.107),

1 N - - - ~ < exp{ik-(r , ( t - 0 ) - rj(t))} > . (6.18) S~(k, t) N i,j=--I

Comparison of the definition of t~E in eq.(3.83) and of S~ in eq.(3.107) yields,

S~(k, t ) /S (k ) - exp{-D~(k, t)k2t} , (6.19)

with S(k) the static structure factor, which can be measured in a static light scattering experiment. A dynamic light scattering experiment on a monodisperse system thus measures the collective diffusion coefficient for a wavevector that is set by the scattering angle, according to eq.(3.50).

6.3 Self Diffusion

Contrary to collective diffusion, which involves the transport of many particles simultaneously, induced by density gradients, self diffusion is related to the dynamics of a single particle in a system with a homogeneous density. The single particle under consideration is commonly referred to as the tracerparticle or the tagged particle, while the remaining Brownian particles are referred to as host particles.

6.3. Sel f Diffusion 325

The simplest quantity that characterizes the motion of a single Brownian particle is its mean squared displacement W(t) , defined as,

W(t) - <] r ( t ) - r(t - 0)12>, (6.20)

where r(t) is the position coordinate of the Brownian particle at time t. In chapter 2 on the diffusion of non-interacting Brownian particles, we have seen that for times v, ol,~nt << t << M/.y, with M the mass of the Brownian particle and 7 its friction coefficient, the mean squared displacement is equal to (see eq.(2.22)),

w ( t ) - v : ( t - o ) t , (6.21)

with v(t - 0) the initial velocity of the particle. For these very small times, the Brownian particle did not yet change its initial velocity v due to friction with the solvent. On the Brownian time scale however, where t >> M/7 , there have been many collisions of the Brownian particle with solvent molecules. This results in the typical linear dependence of W(t) on time (see eq.(2.21)),

W(t) - 6Dot , (6.22)

with Do - kBT/7 the Stokes-Einstein diffusion coefficient. For very small times, W(t ) ,~ t 2, while for larger times, W(t) ,.~ t. The cross-over between these two limiting forms occurs for times larger than the Fokker-Planck time scale, but smaller than the Brownian time scale. The time dependence of the mean squared displacement is sketched in fig.2.1.

Interaction of the tracer particle with surrounding Brownian particles clearly affects the time dependence of the mean squared displacement. The most obvious way to introduce the self diffusion coefficient for interacting systems, is to replace Do in eq.(6.22) formally by the serf diffusion coefficient D,. This diffusion coefficient may be time and wavevector dependent as a result of interactions with other Brownian particles. To make the connection with light scattering experiments, however, where the self dynamic structure factor S,(k , t) as defined in eq.(3.108) can be measured, the above definition of the self diffusion coefficient D, (k, t) is generalized as follows,

S,(k , t) - < exp{ik. (r(t - O ) - r(t))} > - e x p { - D ~ ( k , t ) k 2 t } , (6.23)

in analogy with its the collective counterpart (6.18,19). The self dynamic structure factor may be expanded in a Taylor series for small wavevectors (see

326 Chapter 6.

Figure 6.4: S e l f di f fusion o f a tracer particle through the energy landscape set up by the

host particles.

exercise 3.9),

S , ( k , t ) - 1 - 6 k2 <l r ( t - 0) - r(t)12 > + . . . . (6.24)

On the other hand, the defining relation of D, in eq.(6.23) may be expanded for small wavevectors as,

S~(k, t) - 1 - D , ( k - 0, t )k2t + . . . . (6.25)

Comparing the two Taylor expansions gives,

W ( t ) - 6 D , ( k - 0,t) t , (6.26)

which is the obvious generalization that we had in mind originally, in connection with eq.(6.22). Higher order terms in the Taylor expansions are related to higher order moments of the displacement of the tracer particle (see exercise 6.1 for the next higher order terms in the above Taylor expansions). The wavevector dependent self diffusion coefficient as defined in eq.(6.23) thus fully characterizes the dynamics of the position coordinate of the tracer particle. The zero wavevector self diffusion coefficient is related to the lowest order moment of the displacement of the tracer particle, that is, to the mean squared displacement, as given in eq.(6.26).

The above equations suggest the following experimental route for obtaining the time dependent mean squared displacement. According to eq.(6.23), a plot of In { S, (k, t) } / k 2 as a function of the wavevector for a given time may be extrapolated to k - 0 to obtain D~ (k - 0, t). Since S~ (k, t) is an even

6.3. S e l f Dif fusion 327

function in k, this can best be done by plotting versus k 2, which should yield a straight line for small enough wavevectors. The mean squared displacement then follows immediately from eq.(6.26).

Short-time and long-time self diffusion

On average, the tracer particle resides at positions where the "flee energy landscape", created through interactions with other Brownian particles, exhibits minima (see fig.6.4). Short-time diffusion of the tracer particle thus relates to its displacement out of such minima. The diffusive motion out of free energy minima is characterized by the short-time se l f diffusion coefficient D~(k),

D~(k) - lim D,(k, t ) . (6.27) t- - ,0

In the limit t -+ 0, the time is still understood to be larger than the Brownian time scale "rD ~> M/ 'y , so that the displacement is diffusive. The initial mean squared displacement is related to the zero wavector component of D~ (k), as described in eq.(6.26),

l imW(t) - 6D~(k - O) t . (6.28) t---,O

For later times, the tracer particle "climbes" free energy barriers, which changes the time dependence of the mean squared displacement. The self diffusion coefficient may then become time dependent. The mean squared displacement is then no longer a linear function of time. For very long times, however, where the tracer particle crossed many free energy barriers, one may expect that the mean squared displacement becomes a linear function of time again. The tracer particle then experienced many independent displacements, from one energy minimum to the other, which should result in diffusive behaviour again, in the sense that W ( t ) is directly proportional to t. The corresponding diffusion coefficient is the long-time diffusion coefficient DZ,(k),

lim Dl,(k, t) - Dl,(k) (6.29)

and,

lim W(t) - 6 DZ, (k - 0) t . (6.30) t---,oo

One may ask about the time at which the long-time limit is reached. This is the time that the tracer particle needs to cross many, say 100, energy barriers.

328 Chapter 6.

._1/2 , ! 6~

6Ds I 1 / , I

. . . . . . . . . . . . . [ _ . . . ,

t Figure 6.5: The mean squared displacement W(t) as a function of time. For very long times t >> 7i, W ( t ) becomes linear in time. This long-time limit is approached like ,~ t -1/2 within the so-called weak coupling approximation, as discussed in subsection 6.9.6. This result is indicated in the figure.

The energy landscape is not at all static, however. The host particles which create the free energy landscape, through their interaction with the tracer particle, are not fixed in space. They exhibit thermal motion, or equivalently, Brownian motion. The free energy landscape thus fluctuates with time on a time scale which is set by collective diffusion coefficient of the host particles. Suppose that the free energy landscape varies predominantly on a length scale Am - 2 r /k~ . The corresponding predominant wavevector k~ is the wavevector for which the static structure factor S(k) attains its maximum. The time scale ~-z on which this predominant structure exhibited many independent realizations is now estimated as,

rI >> 1/Dt~(km)k~, (6.31)

where the right hand-side is approximately the time that it takes a density wave of wavelength Am to fully relax. The time scale 7-i is called the interaction time scale. The long-time limit is reached when the tracer particle experienced many independent structural rearrangements of the free energy landscape due to collective Brownian motion of the host particles. This happens for times


t > ri. The actual displacement of the tracer particle need not be large to reach the long-time limit, since is does not have to cross energy barriers, but should just experience many independent realizations of that energy landscape.

For interacting Brownian particles there is an additional time scale as compared to non-interacting particles, the interaction time scale, which is related to structural rearrangements through collective diffusion. For purely repulsive interaction potentials, one may imagine that the tracer particle is hindered in its motion as time proceeds. For those cases, the long-time self diffusion coefficient is smaller than the short-time self diffusion coefficient. ~ The mean squared displacement as a function of time thus bends over to attain a smaller slope at long times. This is sketched in fig.6.5.

For non-interacting particles there is no such energy landscape, and there is no difference between long- and short-time self diffusion. Both the long- and short-time self diffusion coefficient are then equal to the Stokes-Einstein diffusion coefficient Do.

Once fig.6.5 is constructed experimentally, the long-time self diffusion coefficient can best be determined as the slope of W(t) versus 6t, instead of the quotient W(t) /6t . In the mathematical limit t ~ c~, both of these are the same. In practice this mathematical limit is never reached and the mathematical limit limt--,oo W(t) /6t is best determined as the experimental derivative dW(t)/d(6t) for large times.

6.4 D i f f u s i o n in S t a t i o n a r y S h e a r F l o w

The considerations in the previous sections are restricted to systems in equilibrium. What happens when a stationary shear flow is applied that brings the system out of equilibrium? Consider the fluid flow velocity field uo(r) - F. r, with I' the velocity gradient matrix, which is a constant matrix independent of the position r in the system. A shear flow in x-direction with its gradient in the y-direction corresponds to,

l 0 1 0 / o o o . 0 0 0

(6.32)

Here, ,~ is the shear rate, which measures the rate of change of the fluid flow velocity along the gradient direction. The shear flow disrupts the isotropic

1 In fact, DZs is smaller than D~ also for attractive interactions.

330 Chapter 6.

y

X

/ -

Figure 6.6" The competition between shear flow distortion and diffusion.

equilibrium microstructure, that is, the pair-correlation function and the static structure factor. A new anisotropic microstructure exists in the stationary state, which is the outcome of the competition between diffusion and shear effects. Diffusion, driven by shear flow induced microstructural gradients, tends to restore the equilibrium microstructure, while the shear flow tends to distort that structure (see fig.6.6). When diffusion is very fast (slow), the microstructure is little (severely) affected.

Let us try to estimate the relative importance of shear flow over diffusion. Consider a Brownian particle with a position coordinate rp relative to a second particle at the origin. The shear flow induced velocity of the Brownian particle, relative to the particle at the origin, is given by, vp-I r.rp [- ;~y~, with yp the y- component of the position coordinate. The time t, required for a displacement yp in the flow direction due to the shear flow is thus, t, = yp/vp - ;[-1. Diffusion tends to counter balance this relative displacement. It is not a simple task to estimate the time required for diffusion over the same distance in opposite direction, since the diffusion process is driven by the difference of the actual steady state microstructure under shear and the equilibrium static structure factor. A simple minded estimate for the diffusion time would be,

2 2 The factor 6 in tD -- yp/2Do, where we used eq.(6.22), with W ( t ) - yp. eq.(6.22) is replaced here by a factor 2, since we are considering here the mean squared displacement in one direction (the flow direction) only. The ratio of these two times gives an estimate for the amount of distortion, and is

6.5. Short-time Diffusion 331

commonly referred to as the Peclet number,

Pe = t___D = ~/y~ (6.33) t~ 2Do "

In the literature, the Peclet number is usually defined with yp replaced by the radius a of a Brownian particle or the range of their interaction potential. The Peclet number defined in that way, however, is not a correct estimate for the effect of shear flow on large scale microstructures. With increasing yp, the shear flow velocity becomes large, and diffusion is less effective in restoring the equilibrium structure. Hence, microstructures which are extented in the y-direction, or equivalently the gradient direction, are severely affected, even though "~a2/2Do may be small. There does not exist a single dimensionless number that characterizes the amount of distortion on all length scales.

The phenomenon that shear is always dominant over diffusion for structures which extend over large distances in the gradient direction leads to so-called singularly perturbed equations of motion for the pair-correlation function. No matter how small the shear rate is, there is always a region (where g is large) where the distortion is large. The mathematical consequence is that solutions of the Smoluchowski equation cannot be expanded in a Taylor series with respect to the shear rate. For large g, the solution of the Smoluchowski equation is a singular function of the shear rate. This feature is quantified in section 6.8.

6.5 Short-time Diffusion

Short-time diffusion coefficients are most easily evaluated with the use of the "operator exponential expression" (1.67) for correlation functions that was derived in subsection 1.3.2 in the introductory chapter. The stochastic variable X is now the 3N-dimensional vector r - (rl, r2 , . - . , rN), with rj the position coordinate of the jth Brownian particle. The correlation function of two aribitrary functions f and g of r is given by,

< f ( r ( t - 0)) g(r(t)) > = f dr g(r)exp{/~s t} [f(r) P(r)] , (6.34)

where P is the equilibrium probability density function (pdf) for an instantaneous value of r. The Smoluchowski operator is given in eq.(4.41), or alternatively in eq.(4.39) in terms of the "super vector notation" that was

332 Chapter 6.

introduced in section 4.3,

( . . . ) - D ( r ) . (6.35)

The gradient operator V~ is a 3N-dimensional gradient operator with respect to r, D(r) is the 3N x 3N-dimensional microscopic diffusion matrix and (I) is the total potential energy of the assembly of N Brownian particles. The pdf P(r ) in the expression (6.34) for the correlation function is directly proportional to the Boltzmann exponential, P(r) ,-, exp{-/~(I)(r)}.

For explicit calculations it is very handy to introduce the Hermitian conjugate s s of/~s, which operator is defined as,

f dr a(r)/~s b(r) - f dr [/~t s a(r)] b(r), (6.36)

for arbitrary functions a(r) and b(r). The action of the hermitian conjugated operator on the right hand-side of this definition is restricted to the function a(r), as indicated by the square brackets.

easily seen, by applying the above definition m times, that ( / ~ ) t = It is

(/~ts) m. It then follows from the definition (1.66) of the operator exponential and eq.(6.34), that (we abbreviate r(t - 0) = r(0)),

< f(r(O)) g(r(t)) > - f dr P(r)f(r)exp{/~ts t}g(r) . (6.37)

The advantage of this expression is that the operator now only acts on the single function g, and not on the product f x P of two functions. In exercise 6.2a it is shown, by means of partial integration, that,

Z~ts ( . . . ) - (V~ -/3[V~4)]). D( r ) . V~(. . . ) . (6.38)

The two functions f and g are different for self- and collective diffusion. Let us analyse the short-time self diffusion coefficient first.

6.5.1 Short-time Self Diffusion

The short-time self diffusion coefficient is defined in terms of the correlation function in eq.(6.23). The position coordinate r of the tracer particle is denoted here as rl, to distinguish it from the 3N-dimensional super vector r. The tracer

particle is thus the Brownian particle number 1. The correlation function in eq.(6.23) is obtained from the general expression (6.37), with the choice,

f ( r ) - e x p { i k . r l } ,

g(r) - e x p { - i k , r~}. (6.39)

Hence,

t)k2t} - fdrP(r)exp{ik �9 r, } exp{/~ts t} e x p { - i k �9 rl }. exp{-Ds(k ,

(6.40) Taylor expansion of both sides with respect to time, and equating the linear terms in time gives,

D , ( k , t - O)k 2 - D] (k )k 2 - - f dr P(r) exp{ik, rx}/~ts e x p { - i k , rx}

= - < exp{ik, rx}/~ts e x p { - i k , rx} >o, (6.41/

where the ensemble average < . . . >o with respect to the equilibrium pdf P is introduced,

<"" > o - fdrP(r ) ( . . . ) . (6.42)

One can now use that P is proportional to the Boltzmann exponential, implying that - f lP(r ) [V~r - V~P(r), to show by means of partial integrations that for any two arbitrary functions a(r) and b(r) (see exercise 6.2b),

< a(r)/~ts b(r) > o - - < [V~a(r)]. D ( r ) . [V~b(r)] >o �9 (6.43)

Combination of this identity with eq.(6.41), and using that,

V~ exp{:t=ik �9 r l } - - ( i / k , O, 0 , . . . , 0 ) exp{• r 1 },

(g-1)x

finally gives (with 1~ - k /k the unit vector in the direction of k),

D~ - < I~. D 1 , ( r ) - I~ >o �9 (6.44)

The 3 x 3-dimensional microscopic diffusion matrix Dll (r) is a function of all the coordinates r~ -.. rN. Notice that the short-time diffusion coefficient is independent of the wavevector k.

334 Chapter 6.

To obtain an explicit expression for D~ which is valid to first order in concentration, we can use the two-particle expression for D ~ (r) as derived in chapter 5 on hydrodynamics (see eqs.(5.84,95)),

N

Dl l ( r ) - Do i + ~ {A~(rxj)I'ljl'lj + Bs(rlj) [ i --" rljrlj j=2

} } , (6.45)

where the self-mobility functions are given by (see eq.(5.95)),

A~(rxj) = 154 a +2-11 a

B,(r l j ) - 17 a 16 "

(6.46)

These expressions are accurate up to order (a/rlj) 8, with a the radius of a Brownian particle. Since each term in the summation over particles in eq.(6.45) yields the same contribution, substitution of these expressions into eq.(6.44) gives N - 1 identical terms,

D : - D o (1+(N-1)fdrP(r)k.{A,(r12)~lz~l:§ } " k } .

(6.47) The pdf P( r ) is the only function in the integrand which depends on r3, �9 �9 �9 rN, SO that we can perform the integration with respect to these position coordinates to obtain the two-particle pdf,

/ / 1 dr3 . . , drNP(r) - P2(rl,r2) -- V 2 g ( r l , r2 ) , (6.48)

where the last equation defines the pair-correlation function g (see also subsection 1.3.1 in the introductory chapter). For the homogeneous and isotropic system under consideration, the pair-correlation function depends on r~ and r2 only through I rx - r2 I - rx2. We can therefore use that,

f drl f dr2 I'121"12 -- f0 ~ 47r ~ V dr12 r~2 , 3

to finally obtain (with the new integration variable x - r12/a),

D~ - Do 1 + r dx x2 g(ax) {A~(ax) + 2B,(ax)} . (6.49)

6.5. Short - t ime Dif fusion 335

The vo lume fraction ~ - ~ a 3 ~ is the fraction of the total volume that is occupied by the colloidal material, and # - ( N - 1 ) / V ~.. N / V is the number density of Brownian particles. To leading order in concentration, the pair-correlation function is simply the Boltzmann exponential of the pair- interaction potential V(r12), that is, g(r12) - exp{-flV(r12)}.

The "first order in volume fraction coefficient" for the short-time self diffusion coefficient is depending on the form of the pair-interaction potential through the pair-distribution function. One of the most simple pair-potentials is that of so-called hard-sphere systems. The pair-potential Uhs(r12) is then equal to zero for separations between the centers of two spheres larger than 2a, and is infinite when the cores of the Brownian particles overlap,

- 0 , for r 1 2 > 2 a ,

= oc , for r12 < 2 a . (6.50)

Hence, to leading order in concentration,

- 1 , for r 1 2 k 2 a ,

= 0 , for r 1 2 < 2 a . (6.51)

The self diffusion coeffient can now be written as,

D~ - D o { l + c ~ } , (6.52)

with, for hard-sphere interactions,

f OO a~ - dx x 2 {A~(ax ) + 2 B ~ ( a x ) } . (6.53)

Using the expressions (6.46) for the mobility functions it is a simple matter to calculate this integral. The result is,

111 a~ - 64 - - 1 . 7 3 4 - - . . (6.54)

The use of exact expressions for the mobility functions gives a~ - - 1 . 8 3 . . . , which differs about 5% from the above result.

Let us go one step further, and calculate the "second order in volume fraction coefficient" a~ in,

s sqo2 D : - }. (6.55)

336 Chapter 6.

There are two contributions to a~ that should be distinguished. One contribution comes from eq.(6.49) with the pair-correlation function expanded up to first order in the volume fraction. The other contribution comes from three- particle hydrodynamic interactions. These two contributions are denoted as

8 8 a2 (1) and a2 (2), respectively. Consider the former contribution. Specializing to hard-sphere interactions, the first order in volume fraction expansion of the pair-correlation function reads (see subsection 1.3.1 and exercise 1.12 in the introductory chapter),

- 1 , f or r x 2 > 4 a ,

= 1 + ~ 8 - 3r12 + a

= 0 , f o r r12 < 2a.

, f o r r12 E [2a,4a),

(6.56)

For ~ - 0, this expression reduces to the zeroth order expression (6.51). For

the calculation of a : (1) we need the difference between these two expressions, that is, we need the first order in ~ contribution, which we shall denote here as A g h s ( r l 2 ),

A g h s ( r l 2 ) -- 0 , f o r r12 ~_ 4a,

= 0 , for r 1 2 < 2 a .

, f o r r12 E [2a, 4a),

(6.57)

Replacing g in eq.(6.49) by this expression for Ag gives,

,(:) ~2 - dx x 2 8 - 3x + -~-~x 3 { A , ( a x ) + 2 B , ( a x ) }

2271 1467 = 256 + 128 ln{2} - - 0 . 9 2 6 . . - . (6.58)

8 For the second contribution a2 (2), we have to resort to the three-particle contribution to the microscopic diffusion matrix in eq.(5.100),

Di~ ) - 75 Do ~ Y~ a a a 16 rljrlm

j=2 m=2, m y j

[1 - 3(~1j . ~i~)2 - 3(~1~" rim): + 15(~1~ .rim^ )2(rlj . ri~)2

--6(l '1m" r j m ) ( r l j . I ' im)(I ' l j" rjm)] �9 (6.59)

I .@z3 ~3

Figure 6.7" The integrand in eq.(6.62) is a function oft:2, r13 and O2a only.

This expression can be substituted into eq.(6.44) for D~ to obtain a numerical

value for a2 . Each pair (j, m) in the above double summation yields an identical result upon averaging, so that we can set (j, m) - (2, 3), omit the double summation and multiply by the number of terms in the double sum, (N - I)(N - 2) ,,~ N 2. The integration with respect to r4, �9 �9 �9 rN can then be performed, yielding the three-panicle pdf,

/ / 1 d r y . . , drN P(r ) - P~(r:, r~, r~) - Y~ g~(r~, r~, r~). (6.6O)

The last equation defines the three-particle correlation function, which, for the special case of hard-sphere interactions and to leading order in concentration, is equal to zero when one or more of the cores of the assembly of three particles overlap, and is equal to 1 otherwise,

g3(r:,r2, r3) - 0

= 1

, for r:2 < 2a and~or r13 < 2a and~or r23 < 2a,

, otherwise. (6.61)

Furthermore, in the rotationally invariant system under consideration, D~ cannot depend on the direction of the wavevector. We can therefore average eq.(6.44) over directions of k. In appendix A it is shown that this averaging

l i . We thus find, amounts to the replacement of the dyadic product 1~1~ by

O~ 2 / / 48 47ra 3 : dr1 dr2 dr3g3(r: ,r2, r3)

• (h2" h3)

- -6(1"13" 1"23) ( r :2" I '13)(I '12 �9 I'23)] �9

338 Chapter 6.

o o

13o 0.6

0.~

0.2

A Z X \ ~

"**..

zx )- \ ~ ' - - ~ A \

A

. . . . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . . . ! . . . . . . . . .

0 0.1 0.2 0.3 o.s Figure 6.8: The short-time self-diffusion coefficient as a function of the volume fraction for hard-sphere colloids. The solid curve is eq.(6.63), the dashed curve is the linear approximation, where the ~2-term in eq.(6.63) is omitted. The symbols are experimental results from Pusey and van Megen (1983) (.), van Megen and Underwood (1989) (A), and Ottewill and Williams (1987) (+).

The value of the entire integrand is fixed once r~2 = r l - r 2 and r 1 3 - - r l - - r 3

are fixed. These coordinates determine the third relative distance appearing in the integrand : r23 = r2 - r3 = r13 - r12. The integrations with respect to r2 and r3 can be replaced by integrations with respect to r12 and r13, which corresponds to a simple shift of the origin. It follows that once these two (three- fold) integrals are performed, a constant, independent of rx results. The three integrals in the above expression can thus be replaced by, V x f dr12 f dr13. Next, the integrand is independent of the orientation and the position of the cluster of three particles. That is, once, for example, the scalar distances r12, r~3 and the angle 0~3 between r2 and ra are fixed, the value of the integrand is uniquely determined (see fig.6.7). Transforming to the spherical coordinates of r12 and r13 (with the z-axis for the r13-integration chosen along the direction of r12), the integration with respect to r12, r13 and 023 therefore leaves a constant, and the remaining integrations give simply a factor 87r 2. We thus arrive at the following expression (with x~2 - r l 2 / a and x~3 - r 1 3 / a ) ,

225 t~ (2) -- dx12 dx13 dO23 sin{O23} (6.62)

32

g3(rl r2 r3)( ~._~3 , , ( 12" 13) \ r23/

x [1 -- 3(1'12" I'23) 2 -- 3(1'13" 1"23) 2 + 15(I '13" I '23)2(I'12 �9 i'23) 2

--6(1"13" I'23)(1'12 " 1'13)(1"12 " I'23)] "- 1 .836 • 0 . 0 0 2 .

The numerical value of the integral, given in the last line, is obtained by numerical integration using S impson's quadrature with automatic stepw~dth determination. 2 It should be kept in mind that this numerical value is based on the leading order in the inverse distance expansion of the three body interaction matrix. There is as yet no consensus on the precise numerical value of c~.

The second order expansion (6.55) of the short-time self diffusion coeffi-

cient thus reads,

D: - Do (1 - 1.734r + 0.910qp 2 } . (6.63)

A more accurate value for the first order coefficient is -1 .83 . This theoretical prediction is compared with light scattering measurements in fig.6.8. The solid curve is eq.(6.63) and the dashed line is the linear in volume fraction approximation. The linear approximation does better over the entire volume fraction range than the second order approximation. Since at volume fractions of ,-~ 0.1 and higher, a linear volume fraction approximation is certainly invalid, higher order terms in the volume fraction must partially cancel.

The calculation of still higher order coefficients requires knowledge of higher order hydrodynamic interaction matrices and, in addition, expressions for higher order correlation functions.

6.5.2 Short-time Collective Diffusion

The short-time collective diffusion coefficient is defined in terms of the collective dynamic structure factor in eqs.(6.18,19). The collective dynamic structure factor is obtained from eq.(6.37), with the choice,

N f ( r ) - ) - - ~ e x p { i k . r i } ,

i=1

2Two features about the numerical evaluation of the three-fold integral are essential. First of all, the angular integration must have its first node at O2~ = 0 whenever I z~2 - zi~ [< 2, or else at arccos{(4 - z~ - x~3)/2 Zl2Xls}. The nodes must exactly fit into the angular integration range where g3 is non-zero. Secondly, the choice of the upper limits for the zig- and z~s-integration is a somewhat subtle matter. No matter how large x~2 and z~s are, there are always angles O23 such that x~z z - (a/r23) 3 is not small. The convergence of the integral stems from the effectively vanishing O2s-integration range once x~2 and x13 are large, since then a small change of O23 from its first node increases x2s significantly. This is also the reason why the angular integration stepwidth should be taken proportional to x~-~.

340 Chapter 6.

N

g ( r ) - ~ e x p { - i k . r j } . (6.64) j = l

Hence,

S(k) exp{-D~(k , t)k2t} = (6.65) N N

/ dr P(r) ~ exp{ik, ri} exp{/~ts t} E exp{-ik �9 rj}. i=1 j = l

Taylor expansion of both sides with respect to time, and equating the linear terms in time gives,

D~(k, t - 0)k 2 - D~(k)k 2 (6.66) 1 N N

= S(k) < ~ exp{ik, ri}/~t s E exp{-ik , rj} >o �9 i=1 j = l

The equilibrium ensemble average < ... >o is defined in eq.(6.42) �9 it is the ensemble average with respect to the equilibrium pdf P(r). Precisely as for self-diffusion this ensemble average can be written as (see exercise 6.2b),

[ ] [ N ] D~(k)k 2 - < V~ y~exp{ik, ri} �9 D(r) . V~ ~ e x p { - i k . rj} >o.

i=l j=l (6.67)

Using that,

N

V~ y~. exp{• = m--1

+i (k exp{ +ik. rl }, k exp{ +ik. r2 }, �9 �9 �9 k exp{ +ik. rN }),

(6.68)

then gives (with 1~ - k/k the unit vector in the direction of k),

H(k) D~(k) - Do S(k) ' (6.69)

with S(k) the static structure factor, and H(k) the hydrodynamic mobility function, which is equal to,

_ 1 N H(k) ~ y~ <(l~ .Di j ( r ) .

i,j=l Do l~) exp{ik. ( r , - rj)} >o �9 (6.70)

As for self diffusion, the short-time collective diffusion coefficient in eq.(6.69) can be expanded in a Taylor series with respect to the volume fraction,

D~(k) - Do {1 + a~(k)~ + a~(k)~ 2} . (6.71)

Contrary to the self diffusion case, the coefficients are now wavevector dependent. Let us calculate the coefficients for hard-sphere interactions. The static structure factor is calculated with the use of eq.(6.56) for the pair-correlation function,

s(k) sin{kr}

- 1 + 47r~ r.]a ~ dr r 2 ( g ( r ) - 1) kr

= 1 + V Sl(2ka) + V 2 S2(2ka), (6.72)

where,

S l ( X ) - - 24x f01 dz z sin{zx} - 24 [sin{x} - x cos{x}] X 3

(6.73)

a n d ,

- - - d z z s i n { z x } 8 - 6 z + z 3 . X

(6.74)

The integral in eq.(6.74) is easily evaluated explicitly by partial integration. Nothing is learned from this explicit (and long) expression, so that we do not display it.

Next, consider the volume fraction expansion of the hydrodynamic mobility function H(k). The "diagonal terms" in the double sum in its definition (6.70), those with i = j , yield precisely the expression for the short-time diffusion coefficient. From eq.(6.63) we can thus write,

H(k) 1 - 1.734~ + 0.910~ 2 (6.75) 1 N Di j ( r ) .

< (1~. 1~) exp{ik. ( r i - rj)} >o �9 { N ~ , j = l , ~ , ~ D o

As was mentioned in the previous subsection, a more accurate value for -1.734 is -1.83. Let us first consider the contribution from the two-particle microscopic diffusion matrices to the remaining "non-diagonal" terms. The three body hydrodynamic matrices also contribute to the ~2-coefficient �9 this contribution is calculated later on. The ensemble average in eq.(6.75) is

342 Chapter 6.

obtained from expression (6.56) for the pair-correlation function, and the two body microscopic diffusion matrix (5.84) for i # j ,

Dij(r) - Do {Zc(rij)rijrij + Bc(rij) [I- rijrij] } , (6.76)

together with the expressions (5.95) for the cross-mobility functions,

3 a a 75 a - - - + T ' A~(rij ) 2 rij rij

3 a 1 a - + ~ . (6.77) Bc(rij) 4 rij

These expressions are accurate up to order (a/rij) 9. For identical Brownian particles, each of the terms in the summation contributes equally, so that the sum may be replaced by the ensemble average of just one pair of particles (for example i - 1 and j - 2), multiplied by twice the number of pairs of particles = N ( N - 1 ) ~ N 2. We thus obtain (with r - rl2),

H(k) - 1 - 1.734r + 0.910qp 2

(6.78)

+ - ~ N t c [ f d r g h , ( r ) e x p { i k . r } { Z ~ ( r ) ~ + B ~ ( r ) . [i - i'i'] }] k �9 .

In the evaluation of this expression a divergent integral is encountered, corresponding to the linear terms a/rij in the mobility functions. The integral which is problematic is equal to,

where,

[/ C/r)] I -- tr dr gh,(r) exp{ ik - r} /~Oo " ~ '

T ( r ) - flDo -~ r

is the Oseen matrix (see eq.(5.28)). The integrand tends to zero at infinity like ,,~ r -1, which is too slow for convergence for any k. Let us rewrite this integral as,

I =

/~Do


The first integral on the right hand-side is convergent, since gh, (r) - 1 is zero at infinity. The second integral is just the Fourier transform of the Oseen matrix. This Fourier transform is ,~ [ I - kk] (see eq.(5.137) in appendix A of chapter 5), so that the innerproduct of the Oseen contribution with k is equal to 0. Therefore, the divergent integral does not contribute to the short-time collective diffusion coefficient. Hence,

H(k) - 1 - 1.734r + 0.910qp 2

+ ~ 1 ~ . dr (gh~(r)- 1) exp{ik, r}a--r [I + i'i'] �9 [~ (6.79)

+ ~1:. [/drgh,(r)exp{ik.r} {A*~(r)~+B:(r)[]:-~] } ] . 1~.

The starred mobility functions A*~(r) and B:(r) are the mobility functions as given in eq.(6.77) with the Oseen contribution subtracted : the Oseen contribution is contained in the first integral on the right hand-side of the above expression. Up to the level of approximation of our calculations in chapter 5, we have, according to eq.(6.77),

3 75 A*~(r) - _ (a) +__~. (a)

l(a) ( r ) - 2 r (6.80)

Since for rotationally invariant systems H(k) is independent of the orientation of the wavevector, one may average the above expression with respect to the direction of the wavevector. As shown in appendix A, this amounts to the replacement,

kl~exp{ik, r} --, hl(kr)i + h2(kr)~, (6.81)

with,

1 [ s i n { x } - x c o s { x } ] , hi(x)- (6.82)

and,

1 [3xcos{x}- (3 x 2)sin{x}] . h2(x ) - x--- ~ (6.83)

344 Chapter 6.

Notice that for x ~ 0, h x (x) ~ 1/3, while h2 (x) ~ - x 2 / 15. The substitution (6.81) transforms eq.(6.79) into, 3

n(k) - 1 - 1.734~o + 0.910~o 2 (6.84)

+6r/5 fo ~ dr r2(gh~(r) -- 1)ar [2hl(kr) + h2(kr)]

+4~',5 dr 2 �9 r ghs(r) {hl(kr)[a:(r)+ 2B:(r)] + h2(kr)a~(r)}.

Notice that the (a/r)3-terms in the combination A~ + 2B~ cancel. Substitution of the expression (5.56) for the pair-correlation function and eq.(6.80) for the starred mobility functions gives,

H(k) -- l+qp {-1.734 + Hl(2ka)}+tp 2 {0.910 + H~l)(2ka) + H~2)(2ka)}, (6.85)

where,

and,

~0 1

Hi(x) - - 18 dz z [2hl(zx) + h2(zx)]

+ 3 f ~ dz [75z-Sh1(zx)-(z-X-~ 6475 - 5 h2(zx)] , (6.86)

HO)(x) - 3 dz -~z- h~ - -6---~z h2(zx) 8 - 6 z + ~ z 3

+ 18 dz z [2h l (zz )+ h2(zz)] 8 - 6z + ~z a . (6.87)

As for the structure factor, the integrals can be evaluated explicitly with some effort, but we do not display the long resulting expressions here, since nothing is learned from them.

The additional contribution H~ 2) (2ka) to the second order in volume fraction coefficient arises from the three body contribution to the microscopic diffusion matrices in eq.(6.75). The leading order in the reciprocal distance expansion of the three body microscopic diffusion matrix is given in eq.(5.98) (with i = 1 and j - 2),

(6.88) Di 3 )= Do E - - a a m----3

-

aThe integral f o drr2ghs(r)h2(kr)A*~(r) is discontinuous at k - 0. The integral is 0 for k - 0 (since h2(0) - 0), but non-zero for k ~ 0. Whenever H(k - 0) appears, what is meant is its limiting value for k ~ 0.


This expression can be substituted into eq.(6.70) for H(k). This leads, however, to a non-convergent integral. Precisely as in the case of the two-partic!e contribution in eq.(6.78), there are non-convergent terms which are _L k. These terms do not contribute to H(k). Since the expression (6.88) is the leading term in the inverse distance expansion, which is simply obtained from Fax6n's theorem (5.60) by substitution of the fluid flow field reflected by the intermediate particle, and the fluid is assumed incompressible, the divergence of the above expression for D ~ ) with respect to r~ is zero. This can also be verified by direct differentiation. Fourier transformation thus yields,

Is (~ (~ ' (1 - 3(I'13" I'23) 2) i'131'23 exp{k, r12}]" l~ - 0.

Subtraction of this equation from the expression that is found by simply substituting eq.(6.88) into eq.(6.70) for H(k), and using the averaging procedure (6.81) over directions of the wavevector, gives,

1 i f i (a)'(a)' H~2)(2ka) - W dr1 dr2 dr3 {g3(ri,r2, r3) - g(r2, r3)} ~

• (1 -

This integral is convergent for any value of the wavevector, contrary to the integral where the pair-correlation function g(r:, r3) is not subtracted from the three-particle correlation function.

Exactly the same reasoning to arrive at eq.(6.62) for the three body term for self diffusion, finally leads to,

H~2)(2ka) = 13516 f2 ~ dx12 L c~ dx13 L r d023 sin{023} (6.89)

x {g3(rl,r2, r3)-g(r2,r3)} (Xl---~2) 2 x2. (1 -3 ( ih . " i'2.)') [ Xl2)'J-( ~'12" I'13)(i'12. l'23)h2(2ka~-~)] X (1"13 �9 l'23)h1(2~a 2

For hard-sphere interactions this function of 2ka can be evaluated by numerical integration using Simpson's quadrature. The numerical integration is a bit tricky" the x12-integral converges slowly in an oscillatory fashion.

Collecting results we find the following expression for the first and second order volume fraction coefficients of the short-time collective diffusion

346 Chapter 6.

40~ |

3 0 -

20 $2

10

0 $1

- l O . . . . I . . . . . . . . . I ......... 2o- |

10-

O_ H1 " " "'

-I01 c21

-20 ......... I ......... I ......... I ......... I .........

5 �9

o

-5:

-10

-15 ....... I ......... I ......... I ......... I ......... 0 2 4 6 8 10

2ko

Figure 6.9" The fUllCtioI'Js S1, $2, (a),H1, /-/2(1) and 11(22), (b), defined in eqs.(6.73,74), (86,87) and (6.89), respectively, versus 2ka for hard-sphere colloids. Also plotted in (c) are the first and second order in ~ coefficients in eqs.(6.90,91).


coefficient for hard-sphere suspensions,

c~(k) = -1.734 + H~(2ka)- S,(2ka), (6.90)

o ~ ( k ) - 0 . 9 1 0 --J- H~l)(2ka) -f- H~2)(2ka) - S2(2ka)

+S~(2ka) - [Hl(2ka) - 1.734] S1 (2ka). (6.91)

Here, the volume fraction dependence of 1/S(k) is Taylor expanded up to second order. The functions $1, $2, HI, H~ 1) and H~ 2} are given in eqs.(6.73,74), (6.86,87) and (6.89), respectively. These functions are plotted in fig.6.9 versus 2ka, together with the two coefficients a~ and a~. Note the functional similarity of the H- and S-functions.

For the zero wavevector limit we find that, up to order ~2,

D:(k - 0)-D -Do 1-6441 +67 2 = {1+,.559 -14.S 1_8~+34~2

(6.92) Using more accurate two-body mobility functions gives a first order coefficient of 1.45, which differs about 7% from the above result 1.559. The above numerical value o f - 14.8 is numerically accurate up to -4-0.2. Notice that the second order in volume fraction coefficient is quite large. The second order term is as large as the leading order term for qp ~ 0.1, so that the range of validity of the second order expansion is quite limited (probably to volume fractions less than about 0.05). The higher order coefficients are so large that a Taylor series expansion in the density is probably not very realistic. Many higher order terms must be included to obtain a result that is accurate up to some appreciable volume fraction. The expansion (6.92) is compared to dynamic light scattering results on a hard-sphere like suspension in fig.6.10. The solid line is the second order prediction in eq.(6.92), which is indeed seen to coincide with the experimental data over a very small volume fraction range. The dashed line is eq.(6.92) where only the linear term in ~ is kept. As for self diffusion, this supposedly less accurate expression is in very good agreement with the experimental data. The significant higher order terms partially cancel, leading to an almost perfect but fortuitous agreement with the linear order in volume fraction result.

An alternative derivation of the leading concentration dependence of Fick's gradient diffusion coefficient Dv is given in the next section, and indeed agrees with the above expression. This then confirms the conjecture (6.12) up to first order in volume fraction for hard-sphere interactions.

348 Chapter 6.

1.2

s 0c 0o

1.1

I I I I I ,, 7

7 o -- o7O o --

o7 o

o 7 oj

-- 0 J --

J

J -- 0 ~

S,,~,, , ,!, . . . . . . . . I . . . . . , ..... I . . . . . . . . . I . . . . . . . . . J,,, . . . . . . . 0 0.02 0 0 4 0 .06 ~, 0.0B 0.10

Figure 6.10: Comparison of eq.(6.92) for the concentration dependence of the short-time collective zero wavector diffusion coefficient (solid curve) with experiments on a hard-sphere like dispersion. Data are from van Kops-Werkhoven and Fijnaut (1981). The dashed line is eq.(6.92) to//near order in V0.

I I I I

1

H

0.6 -

0.2 . . . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . I . . . . . . . . .

0 2 /+ 6 2ka 10 Figure 6.11" The prediction (6.85) for H(k) (solid curve) compared to experimental data for a suspension of charged colloidal particles. The volume fraction and radius o f the particles is resca/ed to "effective va/ues", to account for the interactions due to the charge on the particles, by the requirement that the maximum of the theoretical curve coincides with the experiments. Data are taken from Philipse and Vrij (1988).


Since Sx, $2, HI,//2(1) and 11(22) are zero for infinite values of their argument, the long wavelength limit of the short-time collective diffusion coefficient is found to be equal to the short-time self diffusion coefficient,

D ~ ( k ~ ~ ) - D~ - 1 -1 .734~+0 .9107 , 2 . (6.93)

For large wavevectors, D~ is equal to D~ because all the cross-terms (those with i # j) become equal to zero, due to the rapidly oscillating imaginary exponential exp{ik �9 (r~ - rj)}. Contrary to the zero wavevector limit, the second order coefficient is small for large wavevectors. The second order term becomes less important with increasing wavevectors.

The second order in volume fraction contribution to H (k) is smaller than for the collective diffusion coefficient. This is due to the very large second order coefficient for the static structure factor. Moreover, since the second order contribution becomes smaller at larger wavevectors, a comparison of the expansion (6.85) of H(k) with experimental data as a function of the wavevector is feasible. The experimental determination of H(k) requires both dynamic light scattering and static light scattering measurements. A comparison with experiments is made in fig.6.11. The experimental data shown here are for a charged colloidal system, with a Debye length which is about 1/3 of the hard-core diameter of the particles. In comparing with our theoretical result for hard-spheres, an "effective volume fraction" and an "effective hard-core diameter" are fixed by fitting the position and height of the maximum in H(k) to the theoretical expression (the actual volume fraction is 0.101, compared to the effective volume fraction of 0.15, and the actual hard-core radius is 83 nm, compared to the effective radius of 118 nm). To within experimental errors, the agreement is quite satisfactory, although the volume fraction of 0.15 used here is probably beyond the range of validity of an O(~ 2) approximation.

6.5.3 Concluding Remarks on Short-time Diffusion

A striking difference between the general expressions (6.44) for the short- time self diffusion coefficient and (6.69) for the short-time collective diffusion coefficient is the factor 1 / S (k). This difference can be understood intuitively as follows. At the short times under consideration here, on average, a tracer particle moves out of free energy minima, as was discussed in section 6.3. For such displacements only hydrodynamic interaction of the tracer particle with the surrounding host particles is of importance. This is why the expression

350 Chapter 6.

(6.44) for the short-time self diffusion coefficient contains only hydrodynamic functions. Direct interactions of the tracer particles with the host particles is implicit in the ensemble average through the pdf, and reflects the modification of hydrodynamic interaction as the configuration of host particles changes. The situation is entirely different for short time collective diffusion. There, direct interactions are of importance, which is reflected in the appearence of the static structure factor in eq.(6.69).

For zero wavevectors we have, according to eq.(6.69),

1 dII(~) D~ - H(0) , (6.94)

67rr/oa d~

where II is the osmotic pressure of the suspension. In the next chapter on sedimentation, we shall see that the derivative of the osmotic pressure with respect to the density ~ may be interpreted as a "driving force" for gradient diffusion. The remaining factor on the right hand-side of eq.(6.94) is usually referred to as a "mobility" for short-time collective diffusion. Notice that with the neglect of hydrodynamic interaction, H(k) - 1, so that the mobility reduces to 1/67rTloa. The hydrodynamic mobility function H(k) incorporates the effect of hydrodynamic interaction on the total mobility. Notice that this mobility function is always smaller than 1 for zero wavevectors, but that at finite wavevectors H(k) may be larger than 1. Hydrodynamic interaction always slows down collective diffusion at long wavelengths, but may enhance diffusion at finite wavelengths.

On the pair level, and for hard-spheres, self diffusion is seen to be slowed down by interactions, while collective diffusion for long wavelengths is enhanced. This is intuitively appealing, since the displacement of a tracer particle is hindered due to repulsive interactions, while macroscopic inhomogeneities are restored faster when Brownian particles repel each other. Attractive forces are expected to decrease the collective diffusion coefficient (see exercise 6.4).

As will be shown in the chapters on critical phenoma and demixing kinetics, attractions can lead to a considerable decrease of the collective diffusion coefficient. For relatively strong attractions, the collective diffusion coefficient may even become negative. This implies that there is "uphill diffusion", that is, particles diffuse from regions of lower concentration to regions of larger concentration due to the attractive forces between them, giving rise to growth of inhomogeneities in time. This is an instability which leads to demixing of the system into two phases, each with a different concentration.

6.6. Gradient Diffusion 351

6.6 Gradient Diffusion

Consider a density gradient that is very smooth on the length scale of the range of interaction between the Brownian particles. We wish to derive an equation of motion for the density in case of smooth inhomogeneities from the Smoluchowski equation (4.40,41),

N

Ot - y~ V~,. Dij . + ] , (6.95) i , j= l

where P - P( r l , r 2 , . . . , rN, t) is the probability density function (pdf) of the position coordinates rj of the N Brownian particles, and Dij is the position coordinate dependent microscopic diffusion matrix, for which explicit expressions are derived in chapter 5.

The equation of motion for the macroscopic density p(rl, t) is obtained from the Smoluchowski equation by integration with respect to the position coordinates r2, ra, �9 �9 �9 rN, using the following relation between the N-particle pdf P ( r l , . . . , rN, t) and the density,

p ( r , , t ) - NPl(r , , t ) - N f dr2fdr3...f drNP(r1,r2,...,rN, t).(6.96)

This relation is discussed in subsection 1.3.3 in the introductory chapter. The problem that arises is that the microscopic diffusion matrices de-

pend on the position coordinates of all the Brownian particles in the system under consideration. Upon integration, this leaves integrals with respect to r2 , . - - , rN, involving products of Dij with P(r~, r2 , - . . , rN, t), which cannot be reduced further. We restrict ourselves here to concentrations which are so small, that events where more than two Brownian particles interact simultaneously hardly occur. For such small concentrations, the two-particle expressions for the microscopic diffusion matrices in eq.(5.84,95) may be used, which functions depend only on the difference of two particle positions. This leaves integrals involving only the lowest order pdf's, which are amenable to explicit evaluation.

The mobility functions on the pair level are given in eqs.(6.45,46) and (6.76,77). It is convenient to rewrite the self microscopic diffusion matrix as,

N

Dii - Doi + AD,( r i j ) . (6.97)

352 Chapter 6.

An explicit expression for the matrix AD, follows from eqs.(6.45,46). This matrix depends only on the relative separations r~j - r~ - rj of two Brownian particles.

Furthermore, to make any progress, it is necessary to assume that the total potential energy �9 of the assembly of N Brownian particles is pair-wise additive, that is,

N (~(rl, r2~"-~rN) -- Z

i , j = a i < j

V(rij) . (6.98)

For spherically symmetric Brownian particles, the pair-interaction potential V is a function of the absolute distance rij - [ ri - rj [ between two particles. For many systems, the approximation of pair-wise additivity of potential interactions is a very good approximation (for monodispers hard-sphere systems this is even exact). Hydrodynamic interactions, on the other hand, are certainly not pair-wise additive, as is evident from the expressions for the three body interaction matrices that were derived in chapter 5 (see eqs.(5.98,100)).

For identical Brownian particles each term in the summation yields upon integration an identical result, where i and j - 1 are special, since we do not integrate with respect to rl. Integration of the Smoluchowski equation (6.95) thus gives, with rl replaced by r,

1 0 N Ot p(r' t) -V.fdr2fdr3...fdrND~.[VP+~PVr (6.99)

+(N-1)V . f dr2f drz.., f drND12.[V~2P+flPV,2r ,

where V is the gradient operator with respect to r. The problem is to reduce the integrals to expressions containing the density as the single unknown variable. A number of integrals must be evaluated. Let us consider, as an example, one of these integrals. The analysis of the remaining integrals proceeds along similar lines. Substitution of the expressions (6.97,98) into the second term in the first integral on the right hand-side of the integrated Smoluchowski equation gives rise to the following integral,

N

~V. f dr2. . . f drN ~ AD~(I r - r i I) I=2

N

j=2


There are N - 1 terms with I - j , which all yield the same result for identical Brownian particles. We can therefore choose l - j - 2, and the integration with respect to r3, r 4 , ' " , rN is only over the pdf, resulting in the pdf P2, which is defined as,

P2(r, r 2 , t ) - f dr3"" f drNP(r, r2, r3,'",rN,t). (6.100)

There are (N - 1 ) (N - 2) terms with I # j , which again yield identical results upon integration. We choose 1 - 2 and j - 3. The integration with respect to r4, r s , . - . , rN is only over the pdf, resulting in the pdf P3, which is defined as,

P3(r, r2, r3, t ) - fdr4""fdrNP(r, r2,r3, r4,"',rN,t). (6.101)

The above integral thus reduces to,

I - ( N - 1)flY. f dr2 A D , ( r - r2)P2(r, r2, t ) - V V ( I r - r2 I)

+(N- 1)(N-2)flV.fdr2fdr3AD,(r-r2)P3(r, r2, r3, t ) . VV(] r - - r3 1).

The pair- and three-particle correlation function g and ga for an inhomogeneous suspension are defined as (see also subsection 1.3.1 in the introductory chapter),

P2(r, r2, t) =

P3(r, r2, r3, t) =

1 N2 p(r, t)p(r2, t)g(r, r2, t ) , (6.102)

1 N3 p(r, t)p(r2, t)p(r3, t)ga(r, r2, r3, t ) . (6.103)

These correlation functions account for the interactions between the Brownian particles, and are simply equal to 1 for non-interacting particles. On the pair level considered here, the pair-correlation function is equal to the Boltzmann exponential of the pair-interaction potential,

g(r, r2, t) - g(~ r - r2 I) - exp{-f lV(I r - r2 I}, (6.104)

which is time independent. For larger concentrations the time dependence of the correlation functions is of importance. In general, these correlation functions are depending on the history of the system, and give rise to the memory effects that were mentioned in the introduction and section 6.2. In

354 Chapter 6.

principle, the time dependent correlation functions can be found from the Smoluchowski equation. This is a very complicated matter, which needs no consideration on the pair level.

As the next step towards the derivation of the equation of motion for the density, the inhomogeneities in density are assumed to be small. Thus, we rewrite the density as,

p(r, t) = fi + Ap(r, t ) , (6.105)

with A p a small deviation from the mean density p - N/V. The integrals may be linearized with respect to this small deviation. Furthermore, on the pair level, the integrals may also linearized with respect to the mean density. The integral involving the three-particle pdf turns out to be of higher than linear order in either/~ or Ap, and is therefore omitted. After linearization the integral reduces to,

I = iv-1 f N2 fiflV, dr2 AD~(r - r2) Ap(r2) g(O)([ r - r2 I)" VV(I r - r2 I).

One integral is omitted here" this is an integral over an odd function in r - r2, which integral is zero. Notice that AD~, g(O) and V are all even functions, so that their spatial derivatives are odd functions.

The density profiles considered here are smooth on the length scale of the range of interactions between the Brownian particles. Since Ap in the above integral is multiplied by V V, the density may be expanded to leading order in a Taylor expansion as follows,

Ap(r2, t) -- Ap(r, t) + (r2 - r ) . VAp(r , t ) . (6.106)

Upon substitution into the integral, only the second term survives, since the first term gives rise to an integral over an odd function, which is zero. Hence,

I __ N - 1 N----i-fiflVVAp(r, t) " f dr2 (r - r 2 )AD, ( r - r2)

x g(~ r - r2 I)" XTV(I r - r2 I).

The next step is to transform to the new integration variable r' - r - r2, using that V'V(r') - ~'dV(r')/dr', and integrating over the directions of r', using that f d~'~'~' - ~I , finally gives,

I = N - 1 4 r V2 fo r' dV(r') g 2 3 ~ ~ Ap(r, t) r dr' aA~(r')g(~ dr'


The remaining integrals in eq.(6.99) are treated similarly. The final result is,

0 O---~p(r , t) = Do {1 + c~v qo} V2p(r, t ) , (6.107)

with the first order in volume fraction coefficient being equal to,

O l v - - fo ~176 dV(ax) - / 3 d z x 3g(~ dx

+ fo ~176 dx x 2 {A~(ax) + 2B~(ax)} g(~

[ - dx x 3 f~(ax)g(~ + A~(ax) dx '

(6.108)

where,

f~(ax) - x2 d ~ A~(ax)-B~(ax)) -~z x 2 +4 A~(ax)-Br x dB~(ax) + .(6.109)

dx

The mobility functions are defined in subsection 5.12.2, and explicit expressions are given in eqs.(6.46,77).

Eq.(6.107) is Fick's law (6.14), with an explicit expression for the gradient diffusion coefficient Dv to first order in concentration.

The above integrals are easily evaluated for hard-sphere interactions, with the use of the following relations,

dg(~ - ~ ( r - 2a) dr

dV(r) g(O) (r) __ __fl-i dr dr

- f l - l ~ ( r - 2 a ) , (6.110)

where 6 is the 1-dimensional delta distribution. The first integral on the right hand-side of eq.(6.108) is easily calculated

with the help of the above relations, and is found be equal to 8. The second integral is also easily evaluated using our approximate expressions (6.46) for the self-mobility functions. Its numerical value is -111/64 - -1 .734.- - . The function f~(ax) is found from the expression (6.77) for the cross-mobility

3r5 _-s The third integral now turns out to be equal functions to be equal to --T-x . r5 -4 .707 . . . Hence, t o - 5 + 25---~ =

av - 1.559. (6.111)

3 5 6 Chapter 6.

Using more accurate hydrodynamic interaction functions yields a value of 1.45.

To first order in volume fraction this is identical to our earlier result (6.92) for the zero wavevector and short-time collective diffusion coefficient. In fact, each of the three integrals in eq.(6.108) is equal to one of the three separate terms in the zero wavevector limit of a~ in eq.(6.90). The structure factor Sl(2ka - 0) is equal to - 8 , and the hydrodynamic function Hl(2ka - O) is equal to -4 .707 . - . . The conjecture (6.12) is thus verified to first order in volume fraction for the special case of hard-sphere interactions.

6.7 Long-time Self Diffusion

In this section, the long-time self diffusion coefficient is calculated for hard- sphere suspensions up to first order in volume fraction. In the following subsection, the method for such a calculation is outlined. It is argued that the long-time self diffusion coefficient can be found from an Einstein relation, where the friction coefficient is that of the tracer particle. This is the proportionality constant between an external force acting on the tracer particle (not on the host particles) and its resulting velocity. That friction coefficient is modified by interactions with the host particles, and is an ensemble averaged quantity with respect to a pdf which is distorted due to the external force on the tracer particle. Subsection 6.7.2 contains the evaluation of that pdf as the solution of the Smoluchowski equation. Finally, the long-time diffusion coefficient is calculated in subsection 6.7.3.

6.7.1 The Effective Friction Coefficient

In chapter 2 on diffusion of non-interacting Brownian particles, we have seen that the diffusion coefficient Do is related through the Einstein relation Do - kBT/7 with the friction coefficient 7 of the Brownian particle with the solvent. The mean-squared displacement of a Brownian particle (without an external force) is thus related to the stationary velocity that the particle attains when subjected to an external force. Now suppose that the Brownian particle interacts with neighbouring Brownian particles. The pure solvent is thus replaced by a dispersion, and the friction coefficient is now an "effective friction coefficient" 7 ~ff, the numerical value of which is affected by the interactions of the tracer particle with the host particles. It is tempting to

6. 7. Long-time Self Diffusion 357

assume an Einstein relation between the long-time self diffusion coefficient and the effective friction coefficient, that is,

DI - k s T / ~ Z . (6.112)

That this is indeed a valid relation can be seen from the Langevin equation approach as described in chapter 2. In the Langevin equation (2.2,3) for the position and momentum coordinate of the tracer particle, the friction coefficient is now replaced by the effective friction coefficient, and the fluctuating force is now the "effective force", which is due to interactions with both the fluid molecules and the Brownian host particles. The analysis given in chapter 2 to derive eq.(2.21) for the mean squared displacement can now be carried over to the effective Langevin equation, provided that the time scale is taken much larger than the time scale of fluctuations of the position coordinates of the host particles. The effective fluctuating force is delta correlated in time (see eq.(2.5)) only on this larger time scale. The analysis of chapter 2 can now be copied to arrive at eq.(2.21), where the friction coefficient is equal to the effective friction coefficient. Comparison with the definition (6.30) of the long-time self diffusion coefficient immediately leads to eq.(6.112). The time scale on which the effective Langevin equation with a delta correlated effective fluctuating force is valid, is the interaction time scale Tt that was discussed in section 6.3 (see eq.(6.31)).

The problem is thus to calculate the stationary average velocity < vt > of the tracer particle due to an external force F ~t. The brackets < . . . > denote ensemble averaging over fluctuations of the actual velocity due to interactions with the host Brownian particles. We have seen in chapter 5 that the velocity of the tracer particle (particle number 1 say) is related linearly to the hydrodynamic forces F~ on all Brownian particles in the suspension,

N h vt = - f l ~ D l j . F j .

j = l

On the other hand, the total force on each of the particles is zero on the Brownian time scale. The hydrodynamic force is just one of the various forces that a Brownian particle experiences. In addition to the hydrodynamic force, there is the direct interaction force - V j ~ , with �9 the total potential energy of the assembly of Brownian particles, and the Brownian force - k B T V j ln{P}, with P the probability density function (pdf) of the position coordinates. The tracer particle is the only Brownian particle that is subject to the external force

358 Chapter 6.

F ~t. Since the total forces are equal to zero, the hydrodynamic forces are equal to minus the sum of the remaining forces. Hence (diij is the Kronecker delta),

N < vt > - /~ Z < Dlj" [ F ~ ' 6 a j - Vj(I)- k , TV j ln{P}] > .

j=l

For identical Brownian particles this expression reduces to,

< Vt > - fl < D l l > " F~:t+ < v[ > + < vt B~ > , (6.113)

where the direct interaction velocity < v[ > is the contribution to the velocity due to direct interactions,

< v[ > - - f l < Dl1" Vl (I) "~-(N- 1)D12. V2(b > , (6.114)

and the Brownian velocity < vt B~ > is the contribution to the velocity due to Brownian motion,

< vt B~ > - - < D~,. Va ln{P} + ( N - 1)Dx2. V2 ln{P} > . (6.115)

The ensemble averages are with respect to a pdf P, which is affected by the external force that acts on the tracer particle (see fig.6.12). The probability of finding a host particle just in front of the translating tracer particle is expected to be larger than in its wake. The first problem to be solved is the evaluation of this distorted pdf. This is done in the next subsection for hard-sphere suspensions, up to leading order in interactions. In subsection 6.7.3, each of the ensemble averages in eq.(6.113) is evaluated, and with it, the proportionality constant between the velocity < vt > and the external force. The resulting expression for the long-time self diffusion coefficient, up to first order in volume fraction, then follows immediately from eq.(6.112).

0 0 0 0 0 o .......... o o

0 0 0 o 0

F ext

Figure 6.12: The deformation of the pair-correlation function around the tracer particle due to its translational motion.

6. 7. Long-time Self Diffusion 359

6.7.2 The Distorted PDF

The ensemble averages discussed in the previous section are with respect to a pdf where an external force F ~*t acts on the tracer particle, while the host particles are force free. To leading order in concentration we may consider the case where there is only one host particle, that is, the suspension is so dilute that the tracer particle interacts just with a single host particle at each instant in time. The pdf that we need is the stationary solution of the Smoluchowski equation (4.40,41) with N - 2, that is, there are two particles, the tracer and the host particle. The potential energy �9 of such a system of two particles, where a force acts only on the tracer particle (particle 1 say), is equal to V ( r 1 2 ) - r l �9 F ext, where V is the pair-interaction potential. The stationary Smoluchowski equation (4.40,41) thus reads,

0 -- 1 " [Dll-{ Trlg fJg rlg-flg ezt} 2I- D12"{ 7r2 g P Tr2g}]

The pdf P here is the two-particle pdf P - P(r~2). The above differential equation is now transformed to the relative position r~ - r2, which is abbreviated here simply as r. Since V' = V,~ = -V~ 2, it is easily found that,

0 = V - ( D , ~ - D2~). 2VP + 2 /~PVV- /3PF ~*t] , (6.116)

where we used that D 2 2 "- D l l and D~2 = D 2 1 . Our interest here is in small external forces, for which the velocity that the tracer particle attains varies linearly with that force. To obtain such a linear relationship we linearize with respect to the external force. For zero external force, the solution of the Smoluchowski equation is proportional to the Boltzmann exponential p(o) ,~ e x p { - ~ V } . We shall seek a linearized solution of the form,

P ( r ) = P(~ + ~aL(r)~. F~*t] . (6.117)

The factor ~a is introduced here to render L(r) dimensionless. The mathe- mathical problem is to find the function L(r) for which this expression is the solution of eq.(6.116). To this end, eq.(6.116) must be reduced to a differential equation for L(r). To achieve this, eq.(6.117) is substituted into the differential equation (6.116), the resulting equation is linearized with respect to the external force, and the microscopic diffusion matrices are expressed in terms of the mobility functions (see subsection 5.12.2) in order to perform

360 Chapter 6.

the differentiations explicitly. With a little effort the following differential equation for L(r) is found (use that i'. V(...) - d(. . .)/dr, for a function (...) of r),

p(O)(r ) [s(r)r2 d ( l d L(r) ) d L(r) + q(r) ~- p ( r ) ~ ~r r dr r dr r

( d ) [ L ( r ) 1 ] d L(r) ~ + P(~ s(r) rdr r r 2a

L(r)r p(r)]2a

(6.118)

where the functions p(r), q(r) and s(r) are respectively defined as,

p(r) - r2d&(A,-B,-A~+B~)r 2 A , - B , - A ~ + B ~ d + 4 + (B, - B~)

= 7 --- 1125 ( r ) 41598 (a ) 6 + - T - r 3 7 5 (a ) r] ,

- 5 + rp(r) + 3A, - 3A~ + 2B, - 2B~

= - - + 2 - + - - r 4

- 1 + A , - A ~

1 3a (a ) 3 15 (ra-) 4211 (a) 6r 754 (a ) r = . . . . ~_ _ _ _ + _ _

2 r r

q(~)

(6.119)

, (6.120)

(6.121)

Here, we substituted the expression (5.95) for the mobility functions. For the special case of hard-sphere interactions, the solution of the differential equation (6.116) is constructed in appendix B. The solution reads,

L(r) = 108 0 0

o ( ) - -0.824 + 0 (a/r) s . (6.122) r

The ensemble averages in eq.(6.113), which determine the long-time self diffusion coefficient, can now be evaluated with the above expression (6.117,122) for the pdf. This is done in the next subsection.

6.7.3 Evaluation of the Long-time Self Diffusion Coefficient

Now the explicit expression for the distorted pdf is known, the ensemble averages in eq.(6.113) can be evaluated explicitly.

6. Z Long-time Self Diffusion 361

1 I I

-07o 0.6

04 t 0 2 1 - - . ~ ,

t . . . . . . . , I . . . . . . . I , . , . . . . . . I . . . . . I . . . . . ~ ' ~ 0 01 02 03 5 ~ 0.5

Figure 6.13" Dynamic light scattering and FRAP data compared to the theoretical prediction (6.130). The dynamic light scattering data are taken from van Megen and Underwood (1989) (.), and the FRAP data are taken from van Blaaderen et al. (1992) (o) and Imhof and Dhont (1995) (A , .).

To first order in the external force, the first term on the right hand-side of eq.(6.113) is an average with respect to the pdf p(o), which is unaffected by the external force. The averages of the off-diagonal elements of D l l are zero, because these are odd functions of the cartesian components of the interparticle separation. Only the diagonal elements survive the ensemble averaging with respect to p(o). The first term is therefore nothing but the short-time self diffusion coefficient in eq.(6.44) (multiplied with 3F~=t), which was found to be equal to,

/7 < Dll > " F e=t ~ e=t - 3D, F - 3Do { 1 - 1.734~} F ~t (6.123)

Next, consider the direct interaction velocity. Substitution of the expressions (5.95) for the diffusion matrices, and assuming a pair-wise additive potential energy (6.98), gives, for identical Brownian particles,

q- {As(r12)-Ac(r,2)} i'1~i'124-{Bs(r12)-Bc(r,,)} [i-f12f,2]] .V,V(r12),

where terms ~,, (N - 1)(N - 2) are omitted, since these terms are of order ~2. In the derivation of this result, it is used that l~72V(r12 ) - -~71V(r12) . The

362 Chapter 6.

average with respect to the unaffected pdf p(0) is zero, since the integrand is an odd function of r~2. Only the additional contribution ,-~ F ~t to the pdf in eq.(6.117) survives the integration. Using that V~ V(r~2) - b12dV(r1~)/dr12, and renaming r = r12, yields,

f dV( ) < v[ > - -a132Do# drg(r)L(r) dr [1 + A , ( r ) - F

where g(r) is the pair-correlation function. Integration with respect to the directions of r, using that f di-/-~ - ~ I , and using the delta distribution relation (6.110) finally leads to,

< v[ > - ~Do 4L(2a)[1 + A,(2a) - A~(2a)] ~F ~t = -flDo 0.127 ~F ~t. (6.124)

The numerical value here is obtained from the explicit expressions (6.95) for the mobility functions and the expression (6.122) for L(r).

The Brownian velocity is evaluated as follows. To leading order in interactions, the N-particle correlation function gN, defined as,

1 P(~ -- V N g N ( r l , ' ' ' r N ) , (6.125)

is a product of pair-correlation functions,

g N ( r l , ' ' ' , rN) -- 1"I g(ri, r j ) . i < j

(6.126)

Substitution of these expressions into eq.(6.115) for the Brownian velocity, using that V2P - - V I P , disregarding terms ,-~ (N - 1)(N - 2), and performing a partial integration with the use of Gauss's integral theorem yields,

< v~ r >=/3Doq~[~dzx2L(ax) a p(ax) F ~t = -f lDo 0.250 qoF ~t .

(6.127) The function p is defined in eq.(6.119).

Collecting results, we thus find that,

(6.128)

Hence,

7~ff = kBT 1 1)o 1 - 2.111~o § 0 (~2) �9

(6.129)

6.8. Diffusion in Stationary Shear Flow 363

The first order in concentration dependence of the long-time self diffusion coefficient now follows from eq.(6.112) as,

Dt~ - Do ( 1 - 2.111qp + O (qp2)) . (6.130)

Using more accurate expressions for the mobility functions, the first order coefficient is found to be equal to -2.10. The calculation is considerably simplified when hydrodynamic interaction is neglected. Each of the separate contributions to the velocity of the tracer particle is substantially different from the above results, but the net result is remarkably close to the correct value. One finds that the first order coefficient is then equal to - 2 (see exercise 6.6).

A comparison of eq.(6.130) with dynamic light scattering and FRAP data is given in fig.6.13. As can be seen, the agreement is very good up to large volume fractions. The agreement at larger volume fractions is probably fortuitous.

6.8 Diffusion in Stationary Shear Flow

The intention of this section is to survey the interplay between convective motion due to shear flow and diffusive motion. As discussed in section 6.4, there is no single dimensionless number that characterizes the effect of convective shearing motion relative to diffusion for all length scales. The relative velocity of two Brownian particles due to the shear flow is large for large separations. Shear flow effects are therefore always dominant for large interparticle separations, even for small shear rates. Mathematically, this gives rise to equations of motion which are singularly perturbed by the shear flow. As a result, the pair-correlation function and the static structure factor cannot be expanded in a Taylor series of the shear rate. That is, the static structure factor is a so-called non-analytic function, or equivalently, a singular function of the shear rate.

These intuitive ideas are quantified on the basis of the Smoluchowski equation to leading order in concentration. For low concentrations, and disregarding hydrodynamic interaction, the stationary Smoluchowski equation for the shear rate dependent pair-correlation function g(r I "~) reads,

(9 (r I~) 0 2DoV {~[VV(r)]g(r I~/) + Vg(r I~/)} ~7 {F rg(r I~/)} ~-~g - _ , _ . . .

(6.131)

364 Chapter 6.

This is the Smoluchowski equation that is derived is subsection 4.6.2 (see eq.(4.102,104)) for N = 2, transformed to the spatial separation r = rl - r2 between the two Brownian particles. This is the most simple equation that still contains the essential features of shear induced structural distortion of a fluid like system.

The singular nature of the distortion that was discussed before is apparent from the structure of this equation of motion. The last term in the above equation is the term which perturbs the Smoluchowski equation as a result of the shear flow. For large separations r, this term is large relative to the remaining terms, even for small shear rates. The effect of shearing motion is always dominant for large separations, so that for these large distances the solution of eq.(6.131) cannot be expanded in a Taylor series in the shear rate,

(6.132) where 9 ~q (r) is the equilibrium pair-correlation function (without shear). The pair-correlation function is therefore said to be a "singular", or equivalently, a "non-analytic" function of the shear rate for small shear rates. An expansion in a power series of the shear rate cannot be used in the Smoluchowski equation (6.131) to obtain an approximate solution for small shear rates.

A perturbation of an equation that leads to a solution which can be Taylor expanded with respect to a small parameter that quantifies the magnitude of the perturbation is commonly referred to as a regular perturbation. The solution is then called a regular, or equivalently, an analytic function of that small parameter. The Smoluchowski equation is a singularly perturbed equation, the solution of which is a singular, or equivalently, a non-analytic function of the small parameter, which will be specified shortly. The mathematical theory dealing with this class of singularly perturbed equations is referred to as singular perturbation theory. Of particular interest here is what is usually referred to as boundary layer theory. For those readers who are not familiar with boundary layer theory, exercise 6.7 is added to get a taste of the essential features. More about this subject can be found in Bender and Orszag (1978), Nayfeh (1981) and Hinch (1991).

An experimental verification of predictions that follow from eq.(6.131) can be achieved by means of light scattering, where the static structure factor is measured, which is related to the Fourier transform of the pair-correlation


function as,

- 1 + # f dr' [g(r' [ "~)- 1] exp{ik, r '} , (6.133) S(k 1"~)

with/~ - N/V the Brownian particle number density. Therefore, instead of solving the equation (6.131) for the pair-correlation function, the Fourier transformed equation is considered here, which reads (see subsection 1.2.4 in the introductory chapter),

0 S(kl;r) - 2Dok ~ S(klZl) 1 + ~ kBT - ~ k l ~ - (6.134)

2Do f + kBT(27r) 3k. dk' k 'V(k ' )[S(k - k'l+) - 1],

where the choice (6.32) for the velocity gradient matrix r is used. V(k) is the Fourier transform of the pair-interaction potential V(r), and kj is the jth component of the wavevector k. For "~ - 0, the solution of eq.(6.131) is of course the equilibrium pair-correlation function g~q (r), and the solution of eq.(6.134) is the corresponding structure factor S~q(k). Subtraction of the zero shear equation from the full equation (6.134) leads to,

0 k2 ' : , /k l~-~S(k[+) - 2Do {S(klS')- s~q(k)}

2Do + kBT(27r) 3 k. f dk' k'V(k')[S(k - k'l+) - S'q(I k - k' 1)1 I

The singular behaviour of the pair-correlation function at large distances leads to singular behaviour of the structure factor at small wavevectors. Actually, the above equation is in a standard form of a singularly perturbed equation, where the highest order derivative is multiplied by the small parameter (the shear rate in the present case).

To leading order in interactions the relevant length scale is the range Rv of the pair-interaction potential. The above equation is therefore written in a dimensionless form, by expressing the wavevector in units of Rv. Let us introduce the dimensionless wavevector,

K = k x Rv. (6.135)

The above equation in dimensionless form reads,

Pe~ oK2 S [ - K 2 (6.136)

366 Chapter 6.

+ 1 ksT(2~r)3K. [S - [ - S~q(IK- ,

where the so-called Peclet number is defined as,

p e O = ;~R~,. (6.137) 2Do

This is precisely the dimensionless number that was introduced in section 6.4 eq.(6.33), with the relative y-coordinate yp between two Brownian particles replaced by the range of the pair-potential Rv.

In the following subsection, the first term in the asymptotic expansion of the solution of the Smoluchowski equation (6.136) for small Peclet numbers is constructed, using boundary layer theory. Although the outline is more or less self-contained, you may prefer to go through exercise 6.7 first.

6.8.1 Asymptotic Solution of the Smoluchowski Equation

The term on the left hand-side in eq.(6.136) is approximately of the same order of magnitude as the first term on the right hand-side, when K 2 ~ P e ~ For K >> ~/Pe ~ the perturbing term (the left hand-side of eq.(6.136)) is small in comparison to the remaining terms. For these large dimensionless wavevectors the structure factor is a regular function of P e ~ For K << V'P e ~ on the other hand, the perturbing term is large, and the structure factor is a singular function of P e ~ The region in K-space where the structure factor is a singular function of P e ~ is usually referred to as the inner region or the (mathematical) boundary layer, while the region where the structure factor is regular is referred to as the outer region. The width of the boundary layer is thus ,-, x/Pe ~ (see also exercise 6.7). The strategy to solve singularly perturbed equations like eq.(6.136), is to separately construct the solution in the inner and the outer region (the so-called inner and outer solution), and match the two by choosing appropriate values for integration constants.

The inner solution: K < x/Pe ~

To obtain the inner solution, that is, the solution in the inner region, the dimensionless wavevector is rescaled with respect to some (possibly fractional) power of P e ~ in such a way, that the rescaled equation becomes regular. Let


us therefore introduce the rescaled wavevector,

q - K/(peO) ~' " (6.138)

Rewriting the Smoluchowski equation (6.136) in terms of this rescaled wavevector yields,

1- -2v L S (q [peO) _ q2 {S (q [Pe ~ - S ~q (q(Pe~ (6.139) ( P e o) ql Oq2

+ kBT(27r)a(Pe~ q . fdq , q,V(q,(peO)~)[S(q - q ' l P e ~ - S~q([ q - q' [ (Pe~ ] ,

where we have used the same symbol for the non-equilibrium structure factor as a function of the rescaled wavevector q as for the original structure factor as a function of K. For u - 1/2, the derivative on the left hand-side of this rescaled equation is no longer multiplied by the small parameter Pe ~ and thereby looses its singular nature. The width of the boundary layer is thus rescaled to unity. The rescaled equation is now regular in v/Pe ~ by construction, so that,

~ - So (q ]Pe ~ + (Pe~ 1/2 S1 (q ]Pe O) "~- Pe 0 S2 (q ]Pe ~

+ (Pe~ a/2 Sa ( q l P e ~ + ' " . (6.140)

The Pe~ of the expansion coefficients Sj is due to the residual Pe~ of the rescaled pair-potential and the equilibrium structure factor in eq.(6.139). These functions are not Taylor expanded with respect to Pe ~ but are kept as they stand (see also the remark at the end of exercise 6.7).

Here we consider only the leading term So in the so-called singular perturbation expansion (6.140). Substitution of the expansion (6.140) into eq.(6.139) and equating terms with equal powers of ~ P e ~ leads to a set of equations for the expansion coefficients Sj. Provided that (Pe~176 --+ 0 for Pe ~ ~ O, the leading order equation is,

0 ( . o) - e {So (. o) - (6.141)

This equation is solved in appendix C under the restriction that,

lim So(k 1-~) - Seq(k). (6.142) P e ~ . - ,0

368 Chapter 6.

For Pe ~ > 0, the solution reads, in terms of the original dimensionless wavevector K,

IK~+K~ } 1 K2 K~ + g A S ( K I Pe~ - S(K [ Pe~ - S~q(K) = peoi Q exp K, Pe ~

x f,.:oo dQ(K~+Q'+K~){S "q (r + Q 2 + K ~ ) - S'q(K)} { 1.} Q Iq + sQ + I(~ (6.143)

x exp K1 Pe ~ "

The + ( - ) in the upper integration limit is to be used for positive (negative) values of K1. Notice that this expression does not contain undetermined integration constants, which are usually needed to match the inner to the outer solution. This is due to the condition (6.142). This expression must therefore coincide with the outer solution for wavevectors in the outer region. Also notice the dependence of the distortion (6.143) on the reciprocal value of Pe ~ indicating its singular behaviour. That the condition (6.142) is satisfied follows from the delta distribution representation (6.254) given in appendix C.

The outer solution : K > ~/Pe ~

The shear term in eq.(6.136) is small in comparison to the remaining terms when K >> ~/Pe ~ For these wavevectors, the shear induced perturbation is regular, so that the solution may be expanded in a power series in P e ~

S (K I Pe ~ - S'q(K) + Pe ~ Sx(K) + (Pc~ 2 S2(K) + - . . . (6.144)

Substitution into eq.(6.136) and equating terms of equal powers in Pe ~ yields the following equation for the linear coefficient,

S~(K) - //i S~q(It) - If2kBT(2r~3K., , dK 'K 'V(K ' )S~(K-K ' ) .

(6.145) This is an integral equation for Sx (K), which may be solved by iteration. The first iterated solution is simply,

1 K~ 0 KaK2 d S~ q(K). (6146) S ~ ( K ) - K2 cOK2 S~q(K) = K 3 dK

The second iterated solution is obtained by adding to the above first iterated result the integral on the right hand-side of eq.(6.145), with Sx taken equal


to the first iterated solution. This contribution is relatively small, due to the almost anti-symmetric integrand, and we shall be satisfied here with the first order iteration. Hence,

peoKIK2 , 4 AS ( K I P e ~ - S ( K I ~ ) - S ~ q ( K ) - Pe~ ~--2-S~q(h')

K 3 dK (6.147)

Notice that the zero wavevector limit of this expression does not exist �9 zero wavevector limits of this expression depend on the path in K-space along which the origin is approached. For example, taking K1 - K2 --+ 0 and Ii'3 - 0 gives, S~ ~ dS~q(K)/d(K2)lK=o, which is a non-zero quantity (remember that S~q(K) is an even function of K, so that dS~q(K)/dK is zero at K - 0, but dS~q(h')/d(l(2) is non-zero). On the other hand, along the path K1 = 0 = Ka and K2 ~ 0, the limit is zero. There is no ambiguity here, since the expression (6.147) is only valid for non-zero wavevectors K > x//Se ~ in the outer region.

Match of inner and outer solution and structure of the boundary layer

Since the above determined inner and outer solution do not contain any adjustable integration constants so as to match both for K ~ x/Pe ~ the inner solution (6.143) must reduce to the outer solution (6.147) for K > v/Pe~ In order to show this, it is convenient to rewrite eq.(6.143) by introducing the new integration variable X - Q - 1(2,

A S ( K ] P e ~ = 1 • ( K 2 + +

• {S ~q ( 4 K 2 + X 2 + 2 X K 2 ) - S~q(K)}

x K? + + + s + 2XK ) X + 2XK x exp K1 P e ~ 3K1 P e ~ "

When either one or both of the conditions,

1K~ -}- K3 2 K~ ! K12+~ !>>1 ] ]>>1 peoK1 ' 3PeOK1

are satisfied, only very small values of X contribute to the integral, since then the exponential functions tend to zero already for small values of X. When

370 Chapter 6.

in addition the wavevector and shear rate are such that for all X's which contribute significantly to the integral, the following conditions are satisfied,

X ( X 2 + 2 X K 2 ) I + l K2X2 3Pe~ 3Pe~

I<<l ,and X2 << 21XK2 [ ,

the inner solution becomes equal to,

1 4-0o

Pe~ fo dX (K2 + 2XK2)

x (S ~q (~/K2 + 2 X K 2 ) - S~q(K)}exp - K 1 P e ~ "

Expanding the equilibrium structure factor at K 2 + 2XK2 around X - 0 to leading order yields,

A S ( K i p e o ) _ 1 dS~q(K) fo'r" { X K 2} P e ~ K1K2 K dK dX X exp K 1 P e ~ "

The integral is standard, and the result is found to reproduce the outer solution (6.147). The inner solution thus indeed coincides with the outer solution in the outer region.

The above inequalities, which must be satisfied in order that the inner solution reduces to the outer solution, actually define the boundary layer, or equivalently, the inner region in K-space. These inequalities are a more precise definition of the structure of the inner region than our earlier simple estimate K < V'Pe ~ Clearly, the structure of the boundary layer is quite complicated.

There is one feature that should be noted about the structure of the boundary layer. In the inner solution (6.143), the Peclet number only appears as a product with the component K~ of the wavevector K along the flow direction. In view of the obvious condition (6.142) we therefore have that,

/ _ \

lim A S [ K I P e ~ - 0. K1 ---,0 k , /

(6.148)

This is also trivially true for the outer solution (6.147). There is thus no shear flow induced distortion perpendicular to the flow direction, where K1 - 0. The inner and outer solution therefore coincide for any shear rate when K1 - 0, and the extent of the boundary is empty in these directions in K-space. The boundary layer is quite asymmetric.


Figure 6.14" The relative structure factor distortion A S~ S ~q as measured by light scattering for a charged colloidal system (a), and as calculated from eq.(6.143) with the Percus-Yevick equilibrium structure factor for an effective hard-sphere suspension with qa = 0.45 (b). The central area in the left figure is blocked out by a beamstop. The experimental configuration used here is depicted in figure c. The sample is located between two horizontal glass plates, o f which the upper plate rotates. The scattered intensity is collected on a fiat screen, which in turn is imaged onto a camera. The vectors v and e are used to indicate the orientation of figures a and b relative to c. For more experimental details see Yan and Dhont (1993).

372 Chapter 6.

Since the inner solution (6.143) reduces to the outer solution in the outer region, this expression is valid throughout K-space.

An experiment

Although the result (6.143) for the structure factor distortion is based on a simplified Smoluchowski equation, where only leading order direct interactions are taken into account and hydrodynamic interaction is neglected, generic features are probably correctly predicted. In fig.6.14 a comparison between experimental light scattering measurements on a charged colloidal system and eq.(6.143) is made. Plotted is the relative structure factor distortion AS/S ~q. For the equilibrium structure factor, that is needed as an input to calculate the structure factor distortion, the Percus-Yevick structure factor for hard-spheres is used, with an effective hard-core diameter and volume fraction which are determined by scaling the actual crystallization concentration to that of a monodisperse hard-sphere system. The qualitative agreement is striking. There is no quantitative agreement in the sense that the actual magnitude of the theoretical relative structure factor distortion does not agree with the experimental result. This is not surprising, since for the large concentration of the colloidal system used here, both higher order direct interactions and hydrodynamic interaction are certainly significant.

6.9 Memory Equations

In the preceding sections we considered either short-time diffusion processes or stationary states, for which memory effects are of no significance. In general, however, memory effects must be included. How memory effects come into play can be understood by considering a Brownian particle that at some instant moves in a certain direction. Through direct and hydrodynamic interactions, other particles are affected in their motion, which in turn affect other particles. These "disturbances" propagate through the suspension and may return to the particle under consideration. These disturbances take some time to return, and render the motion of the particle under consideration to depend on its motion at earlier times. These memory effects are most clearly revealed by so-called memory equations for correlation functions, which are derived from the Smoluchowski equation. Memory effects are made explicit in these alternative equations of motion through the so-called memory function. These memory functions are very complicated correlation functions. The

6.9. Memory Equations 373

difficulty with this approach is, that an exact equation of motion is derived, but the single memory function that contains all the physics can only be evaluated in an approximate way for special cases, and a kind of "working hypothesis" must be employed to get to more general results. A more physically appealing way to go about would be to make justifiable approximations in each step in a derivation. Nevertheless, the memory equation method is valuable, not only for special cases, but also to gain insight in memory effects in general.

6.9.1 Slow and Fast Variables

Let a(r ] X(t)) denote a function of the phase space coordinates X, a so- called a phase function, or equivalently, a microscopic or stochastic variable. The ensemble average of such a microscopic variable is the corresponding macroscopic variable. An example of such a phase function is the microscopic density,

N

p(r IX(t)) - ~ 6 ( r - r j ( t ) ) . j = l

(6.149)

On previous occasions, we sometimes denoted this phase function simply as p(r, t). In the present context it is more convenient to denote the dependence on the phase variable explicitly, as operators will be encountered which act on that variable. As shown in subsection 1.3.3 in the introductory chapter, the ensemble average of the sum delta distributions in eq.(6.149) is precisely the macroscopic density. In this example the phase variable X is the set of position coordinates of the particles in the system under consideration �9 X -- ( r l , ' ' ' , r N ) .

Suppose that a microscopic variable a(r I X(t)) is a conserved variable. This means that in a given volume W, the "amount of a", fw dr a(r I X(t)), changes in time only by flow of a through the boundary 014; of W. There are thus no sources or sinks where a is created or annihilated. An example of such a conserved variable is the number density of particles of a certain species, provided that there are no chemical reactions going on in which that particular species participates. Let j~(r [ X(t)) denote the current density of a. Only the component of j parallel to the (outward) normal fi on 014; contributes to the change of the amount of a in 14;. By definition we thus have,

d - dS.jo( I X( t ) ) , dr a(r IX(t)) w

374 Chapter 6.

where dS - dS fi, with dS an infinitesimal surface area on OW. Since the volume W is arbitrary, and can be chosen infinitesimally small, it follows from Gauss's integral theorem that (compare with the derivation of the continuity equation in section 5.2),

0 O~ a(r J X(t)) = -V.j~(r IX(t)).

Fourier transformation with respect to r thus gives 0 ~a(k , t) ,-, k (replace V by ik). In fact, the current density is driven, on average, by gradients in a (and possibly gradients of other variables), so that the temporal evolution is ,,~ k 2. The Fourier transform of conserved variables are thus slowly varying variables for small wavevectors k. The reason for such slow dynamics for small wavevectors is that particles must be displaced over large distances. There is a natural division of variables in slow and fast variables. The non- conserved variables are fast variables, also for small wavevectors, since for these variables there is an additional contribution to ~ I X(t)), the Fourier transform of which remains finite in the zero wavevector limit.

In a closed system, without chemical reactions going on, the conserved variables are the number densities, the momentum density and the energy density. This is different for the subsystem of Brownian particles in a fluid. The Brownian particles exchange both momentum and energy with the fluid, so that the momentum and energy density associated with the Brownian particles is not conserved. The only conserved variable for the subsystem of Brownian particles is their number density. Being the only slow variable, we specialize in the following to the number density p(r I X(t)) of the Brownian particles.

6.9.2 The Memory Equation

The slow temporal evolution of p(k I X(t)) for small wavevectors can be exploited to derive an equation of motion for its auto-correlation function, which is the memory equation referred to earlier. The density auto-correlation function is equal to (see subsection 1.3.3 in the introductory chapter),

1 p. Ni , j~? S~(k,t)=-~<p(klX(t)) (k I X(0)) > - exp{ik.(ri(O)-rj(t))}>

= f dX p(klX)exp{~st } [p*(k ] X)P(X)] (6.150)

= f dXP(X)p*(kIX)exp{~tst}p(k I X) ,

where P is the equilibrium pdf for an instantaneous value o fX - (rx, �9 �9 �9 rN), and/~t s is the Hermitian conjugate of the Smoluchowski operator/~s (see exercise 6.2 and section 6.5),

Z~ts( .. .) - (Vx - fl [Vx(I)]). D ( X ) . V x ( . . . ) . (6.151)

Instead of the supervector r - (r~, �9 �9 �9 rN) used earlier, this vector is denoted here by X, to distinguish it from the position coordinate r. According to eq.(6.150), the time dependence of the density correlation function is determined by the time dependence of the phase function,

c(k IX It) - e x p { / ~ f s t } p(k IX). (6.152)

The time dependence of this phase function is understood to be due to the time t appearing in the operator exponent" in eq.(6.150) the phase variable is a mere integration variable, of which the time dependence is irrelevant. The equation of motion for the density correlation function is obtained from the equation of motion for c, which simply follows from its definition (6.152) by differentiation,

O c(k I X I t) - z2fs c(k I X I t) - exp{/~tst}z~t s p(k I X). Ot

(6.153)

The temporal evolution of c is now "decomposed" into a part that is coupled to the slow variable (the density) and the rest that is coupled to the remaining fast variables. To this end, the following projection operator onto the density 75(k) is introduced,

< ( ' " ) l l p (k I X) > 75(k)( '") - < p(k I X)llp(k I X) > p(k I X). (6.154)

Here, the inner product < h IIg > of two arbitrary phase functions h and g is introduced for convenience of notation,

< h llg > - < h g* > o - f dXP(X)h(X)g*(X). (6.155)

The ensemble average < . . . >0 is with respect to the equilibrium pdf P(X) . What the operator in eq.(6.154) does, is to project a phase function onto the density, or equivalently, 75(k)( .. .) is the component of the phase function ( . . . ) that is "parallel to the density". A phase function ( . . . ) is said to be "orthogonal to the density" when the inner product < ( . . . )IIp(k I X) > is

376 Chapter 6.

0, and hence 9(k) ( . . - ) - 0. In this way, the space that is spanned by all conceivable phase functions is split into two subspaces �9 the subspace parallel and perpendicular to the density: The component of a phase function (.-.) that is parallel to the density is P(k)( . . . ) , which varies slowly with time (at least for small wavevectors), while the orthogonal component evolves rapidly in time.

Once a phase function is projected onto the density, a subsequent projection should not change the function, that is,

75(k)75(k)(...) - 75(k)(.. .) . (6.156)

This property is easily verified from the definition (6.154). The operator Q(k) - 2 " - 75(k) is the projection perpendicular to the density, onto the subspace of fast variables (2" is the identity operator which leaves a function unchanged). It is easily verified that,

Q ( k ) Q ( k ) ( . . . ) - Q(k) ( . . . ) , (6.157)

Q(k)75(k)(...) - 75(k)Q(k)(...) - 0. (6.158)

Any phase function (...) can be written as the sum of 7~(k) (. . .) and Q( k)( . . . ), where the first term is in the subspace of the slow variables (the density in our case) and the latter term is in the subspace of fast variables. Both projection operaters are Hermitian, that is, for two arbitrary phase function h and g,

< hl175(k)g > - < 75(k)hilg > , (6.159)

and similarly for Q(k). This is easily verified from the definition of the projection operators.

Let us now decompose the phase function in the equation of motion (6.153) appearing under the operator exponent in its fast and slow component,

0 c(klX It) - exp{/~tst}/3(k)/~t s p(k I X) + exp{/~tst}Q(k)/~ t p(k I X) Ot

(6.160) The first term on the right hand-side is proportional to the c itself,

exp{/~tst}75(k)/~t s p ( k l X ) - f~(k)c(k I X I t) ,

where the following wavevector dependent function is introduced,

f~(k) = < s p(k [ X)llp(k [ X) > (6.161) "

The second term on the right hand-side of eq.(6.160) is rewritten with the use of the following identity (see exercise 6.8),

exp{/~tst} - exp{Q(k)/~tst} (6.162)

fot dt' exp{/~ts(t - t')}75(k)/~ts exp{Q(k)/~tst'} �9 +

The last term in eq.(6.160) is thus equal to,

exp{Z~tst}Q(k)Z~t sp(k IX) - f (k lXl t ) (6.163)

fotdt ' exp{Z~ts(t-- t') }75(k)/~ts f (k ] X I t) , +

where the following phase function is introduced,

f ( k i X It) - exp{Q(k)Z~tst}Q(k)Z~tsp(k I X). (6.164)

Notice that the combination appearing in the integral under the operator exponential is directly proportional to the density. Using that for two arbitrary phase functions h and g (see exercise 6.2c),

< Z~h IIg > - < h IIZ~ g > ,

and that Q(k) f(klXlt) - f(klXlt), it is found that,

75(k)/~ts f (k [ X [ t) - p(klX)

(6.165)

p(klX)

p(klX)

< z~ts f(k i X I t)IIp(k I X) > < p(k i X)IIp(k I X) >

< f(k ! X I t)llZ~ts p(k I X) > < p(k I X)IIp(k I X) >

< f (k I X I t)il Q(k)/~ts p(k I X) > < p(k I X)IIp(k I X) >

p(klX) M(k, t),

where the following function is defined,

M(k, t) - < f (k l X l t)llf(k l X l t - 0) > . (6.166) < p(k I X)IIp(k I X) >

With these formal mathematical manipulations, the equation of motion (6.160) can now be rewritten in the following appealing form,

~ fOt 0 c(klXlt) a(k)c(kiXl t )+ dt'M(k t - t ' ) c ( k l X l t ' ) + f ( k l X l t ) . Ot '

(6.167)

378 Chapter 6.

It is evident why the function M is commonly referred to as the memory function. The function f is in the space of fast variables, since Q(k) f - f . Due to the similar structure of eq.(6.167) and the Langevin equation discussed in chapter 2, and the fast temporal behaviour of f , this phase function is usually referred to as a fluctuating "force". The function f~ has the dimension s -~, and is therefore referred to as the frequency function. Since, according to eq.(6.150), S~ = < clip >, and f is perpendicular to p, the equation of motion for the density auto-correlation function follows immediately from eq.(6.167), by multiplying both sides with p and ensemble averaging with respect to the equilibrium pdf,

0 f0 O---t S~(k, t) - f~(k) S~(k, t) + dt' M ( k , t - t') S~(k, t') . (6.168)

This is the memory equation for the density auto-correlation function. Since the memory function in eq.(6.166) is proportional to the auto-

correlation function of the rapidly varying fluctuating force, M is expected to decay to zero over a time interval on which S~ hardly changes. This suggests the following approximation of the memory equation,

__0 s (k, t) - s , (k , t) Ot

where the effective frequency is equal to,

f0 ~176 f~r - gt(k) + dr' M ( k , t') .

The density auto-correlation function is thus predicted to be a single exponential function of time. However, this is not what is observed experimentally. The error that is made in the above approximation is, that the density is not the only slow variable, but in addition, phase functions that are equal to products of two, three --. Fourier transformed densities are also slow variables. The fluctuating force f in eq.(6.164) therefore contains slow components, which are parallel to these products. As a consequence the memory function does not go to zero in a time interval on which S~ remains virtually constant. All products of conserved variables should be added to the space of slow variables, and the projection operators should project onto that extended space. The projection operators are then matrix operators. Extending the above analysis to the multi-dimensional space of slow variables, including products of conserved variables, is the starting point of what is referred to as mode-mode

coupling theory. Resulting memory functions are now very complex quantities, which can generally only be calculated when making ad hoc mathematical simplifications.

An alternative to the extension of the space of slow variables in order to include products of conserved variables, is Mori's fractional expansion. The idea here is to derive an additional memory equation for the fluctuating force appearing in the above memory equation (6.167). The projection operator in this subsequent derivation is then onto an additional slow variable, which is constructed such that it is perpendicular to the density. The new fluctuating force is then perpendicular to the two slow variables (the density and the additional slow variable). This procedure can be extended up to a level that is believed to be sufficient to virtually exhaust the space of slow variables.

The treatment of mode-mode coupling theory and Mori's expansion are beyond the scope of this book, and we will analyse the memory equation as derived above, without assuming a fast decaying memory function.

A similar memory equation as for the density auto-correlation function can be derived for the self correlation function S, defined in eq.(6.23). The system of Brownian particles is now a mixture of a single tracer Brownian particle and a concentrated species of host particles. There are now two conserved variables : the number density of both the tracer and host particles. The subspace of slow variables is now spanned by these two number densities and all their products.

Repeating step by step the above analysis, where the projection is now onto the density,

pa(k [ X(t)) - e x p { - i k �9 r 1 ( t)}, (6.169)

of the tracer particle, with X - ra the position coordinate of that particle, gives the following memory equation for S,,

(9-[ S,(k, t) - f~(k) S~(k, t) + dt' M,(k , t - t') S,(k, t') . (6.170)

The self frequency function is defined as,

a , (k ) - < z2*s m(k [ X)IlPa(k [ X) > , (6.171)

and the self memory function is defined as,

M,(k, t) - < L (k I X I t ) l lL(k I X I t - 0) > , (6.172)

380 Chapter 6.

with the self fluctuating force equal to,

f , ( k l X t) - exp{Q(k)Z~tst}Q(k)Z~t s p ~ ( k [ X ) . (6.173)

The frequency- and memory function are subscribed with an "a" to indicate that they relate to self diffusion. The analogous functions in eq.(6.168) for the density auto-correlation function are therefore also referred to as the collective frequency and memory function.

The self memory function is not a rapidly varying function of time for two reasons" just as for the collective memory function, the products of conserved variables are also slow, and, in addition, the density of the host particles is slow, which is not taken into account in the self projection operator.

6.9.3 The Frequency Functions

The time-integral in the memory equations may be neglected for short times. The memory equations should then reproduce the short-time expressions for the collective and self dynamic structure factor of section 6.5.

Consider collective diffusion first. The solution of the memory equation (6.168) for short-times is,

S~(k, t) - S(k) exp{f~(k)t}. (6.174)

Comparing this expression with the definition (6.19) of the collective diffusion coefficient (in the absence of memory effects) immediately gives,

1 O~(k) k ~ - -f~(k) - - N S(k) < z~ts P(K I X)llp(k I X) > ' (6.175)

where it is used that < pllp > - NS(k) . This is exactly the expression (6.66) that was derived earlier in subsection 6.5.2. Hence,

a(k) - -Do H(k) k2 S(k) " (6.176)

The solution of the memory equation (6.170) for the self dynamic structure factor for short-times is similarly,

S~(k, t) - exp{f/~(k)t}. (6.177)

From the definition (6.23) of the self diffusion coefficient it follows that,

O:(k) k 2 - -f~,(k) - - < Lt s pa(k I X)llP~(k I X) >, (6.178)

which expression is identical to eq.(6.41) derived in subsection 6.5.1, so that,

~ s ( k ) - - - - o k2. (6.179)

Memory effects need to be taken into account for longer times. An alternative approach, that implicitly includes memory effects, was developed in section 6.7 to calculate the long-time self diffusion coefficient. In general, expressions for memory functions are needed in order to include memory effects.

6.9.4 An Alternative Expression for the Memory Functions

The time dependence of the fluctuating force f is modified by the projection operator Q(k), which multiplies the Hermitian conjugate Smoluchowski operator in the operator exponential in the definition (6.164,173) of f. This complicates the evaluation of the memory functions (6.166,172), and it is desirable to have an alternative expression in which the Hermitian Smoluchowski operator is not modified by this projection operator.

Such an alternative expression can be derived for the Laplace transform of the memory function, which is defined as,

M(k, z) - dt M(k, t ) e x p { - i z t } . (6.180)

The variable z is the Laplace variable, conjugate to t. When the Laplace transform is known, the transformation may in principle be inverted to obtain the memory function M(k, t). Substitution of the definition (6.164) of the fluctuating force f into eq.(6.166) for the memory function and Laplace transformation yields,

< exp{ - i z t + Q(k)Z~fst)f(k I X I 0) Ilf(k IX 10) > M(k , z ) - dt < p(klX)lip(k IX) >

1 1 < f ( k l X l o ) l l f ( k l X l O )>, (6.181)

<p(klX) llp(klX)> iz-Q(k)Z~ts

where 1 / ( i z - Q(k)Z~ts) is the inverse operator of i z - Q(k)Z~ts (see exercise 6.9 for mathematical details). The operator 1/(iz - 75(k)Z~) is usually referred o t t as the modified resolvent operator, while 1/(iz - s is referred to as

the resolvent operator. We wish to express the modified resolvent operator entirely in terms of the resolvent operator, and use that result to obtain an

382 Chapter 6.

expression for the memory function in terms of the resolvent operator only. First of all, it is easily verified that,

1 1 _ 1 75(k)Z~t s 1 i z - Q(k)~ts - i z - E.t s - i z - ~ t s i z - Q(k)/~ts "

For brevity we shall use the following short-hand notation for the resolvent and modified resolvent operator, respectively,

A - 1 ~ i - i ' t i z - ~ i z - Q(klZ~

The above identity thus reads,/~ - A - A/5(k)/~ts/~. Repeated application of this identity yields,

-A-A

. a - [A- , ,) ,)

a - , a . . . . ]

[,~=~o (-1>" (7~(k>/~ts.j) ,'] 75(k)/~ts ' . (6.182>

Since by definition,

75(k)Z~tsJf(klX10) - p(k[X) < Z~ts.Af(k I X I 0) II p(k I X) >

< p(klX)lip(kiN) >

it follows that,

[,=~o ( - I ) " (75(k)/~ts.A) "] 75(k)/~ts~f(klX10)

o o n

= < Z~ts.af(klXl0lllp(klX) > y~. (-1)" (73(k)/~ts.A) p(k]X) < p(k IX)11 p(klX) > .=o

oo (< Z~tsAp(k I X)lip(kiN) > p(klX) < Z~tsjf(k I X I 0)lip(KIN) > ~-~ (-1)" (kiN)lip(kiN) > < p(k I X)II p(k I X) > ~=o p(klX). < #(kiN)lip(kiN) > + < Z~tsAp(k l X)lip(kiN) >

In the last line the geometrical series is resummed. Use of this expression in eq.(6.182), and subsequent substitution into eq.(6.181) for the Laplace

transform of the memory function finally gives the altemative expression we were after,

M(k,z) = 1 [ 1

< f(k IX [ O)II f (k lX [ O) > < ,o(k[ X)[Ip(k I X) > i z - ~ . t s

< s > < .1., p(klX)llf(klXlO)> i z - ~ S

t . J

< p(k I x)IIp(k I X) > + < Z~*s _~, p(k I X)IIp(k I X) > "

(6.183)

This expression contains the Smoluchowski operator without being multiplied by the projection operator Q(k), in contrast with the original expression (6.181).

The alternative expression for the self memory function Mr,(k, z) is obtained from the above expression by simply replacing f by f, and p by p~. Note that < plllpl > - 1, while < pllp > - NS(k), with S(k) the static structure factor.

6.9.5 The Weak Coupling Approximation

An explicit evaluation of the memory functions in eqs.(6.166,172) for collective and self diffusion, respectively, is feasible for weak direct interactions, with the neglect of hydrodynamic interaction. The memory function may be expanded to first order in the pair-potential V for these weak potentials. The resulting expression for the memory function is referred to as its weak coupling approximation. This approximation is considered in the present subsection without hydrodynamic interaction. The microscopic diffusion matrix D is then a diagonal matrix, with the diagonal elements equal to the Stokes-Einstein diffusion coefficient Do.

First consider the memory function for collective diffusion. As a first step in the evaluation of the memory function, an explicit equation for the fluctuating force must be derived from its definition (6.164). From the expressions (6.161,176) for the collective frequency function, it immediately follows that,

f(k ! X [ O) - Q(k)Z~tsp(k I X ) - Z~tsp(k [ X ) - <Z~tsp(k I X)I[,o(k [ X)> <p(k I X)II P(k [ X)>

k2 = /~tsp(klX ) + D o S ~ p ( k l X ) ,

p(klX)

384 Chapter 6.

where H(k) is set equal to unity, since we do not include hydrodynamic interaction. From the explicit expression (6 151) for/~t it now follows that,

[1 ] f ( k J X [ 0 ) - flDoik, y~ [V j r rj}+Dok 2 S(k) 1 p(klX).

j=l (6.184)

Since for a zero potential energy ~, the static structure factor is equal to 1, the fluctuating force f is of first order in ~. The leading contribution to the collective memory function is therefore of second order in r Up to that order, the Hermitian conjugate/~t s of the Smoluchowski operator in the first term between the square brackets in eq.(6.183) and in the numerator of the second term can be taken equal to DoV~ (this is/~t s with �9 - 0). Both terms in the numerator of the second term between the square brackets are then equal to 0, since,

1 1 <s < - - - - ~ f ( k l X l 0 ) l l s ) >

-k2Do = iz + k2Oo < f ( k l X l 0 ) l l p ( k l X ) > = 0,

because f _1_ p, and,

1 1 < ~ p ( k I X)II f (k I X I 0) > - < p(k I X)II f (k I X I 0) > - 0, iz-f..ts iz+k2Do

The feature that makes these explicit evaluations feasible is that p is an eigenfunction of the free diffusion operator DoV~c. Up to O(~2), only the first term between the square brackets in eq.(6.183) survives,

1 1 M(k,z) - NS(k) < iz-OoV2x f ( k l X l 0 ) l l f ( k l X l 0 ) > + " o ( r '' .

The expression (6.184) for the fluctuating force f can now be used here to obtain, for identical Brownian particles,

D2k4 [ 1 ]2 M(k,z) = iz + Dok 2 S(k) 1 (6.185/

+ D2 k2fl2S(k-----~ [r162 i z - DoV}I [V1r exp{-ik �9 rl} II [V,r exp{-ik" r~} >

+(N-l) O~ k2f~2S(k)[r [~71 r exp{-ik �9 rl} I1 [~72(I )] exp{- ik �9 r2} >.

This expression can be evaluated further, assuming a pair-wise additive potential energy (see eq.(6.98)), and by the introducing the Fourier transform of the pair-potential,

Vi@ - (2rr)a ~ dk k V(k) exp{ ik - ( r j - rn)}. (6.186) n = l , r t ~ j

The action of the resolvent operator is now easily evaluated. Substitution of eq.(6.186) into eq.(6.185) yields,

D~ k4 [ 1 ]2 M(k,z) = iz+Dok 2 S(k) 1 (6.187)

D2ok2f12 N N I" I' (k~. 1r k:)V(k~)V(k2) ~,, ~,, ]dk'jdk2 iz + Do[l kl_k l 2 +k~]

,

x < exp{ikl . (rl - r . ) - ik=-( r l - rm}) >o

D~k2fl2 k ~ ~ / d k l f dk2 -(kl" 1~)(k2" k)V(kl)V(k2) + ( N - l ) (2~r)SS(k) " = , , - , ~ , . . = , ~ 2- zz q7 Do il kll ~2~r i?T/~l 5]

x < exp{iki. (ri - rn) - in2" (r2 - rm) + i k - ( r : - rl)} >o �9

To zeroth order in interactions, the ensemble averages here can be calculated with the use of constant pdf's, independent of position coordinates. The ensemble averages of the exponential functions then reduce to delta distributions. Consider for example the average of the exponential function in the second term on the right-hand side of eq.(6.187). For m = n, and m, n 7t 1, we have,

< exp{ik , - (r~ - r . ) - ik2-(rx - r .} ) >o 1

= V2 / dr1 / dr~ exp{i(kl - k 2 ) - ( r l - r . )}

1 [ dr e x p { i ( k t - k2 ) - r} - = p a ~ 6(kl - k2).

For m r n, and m, n r 1, on the other hand, we have,

< exp{ikx- ( r a - r,~) - i k 2 - ( r l - rm}) >o

- 1 / d r fdr , fdrmexp{i(kl k 2 ) . r l } e x p { ikl rn}exp{ik2 rm} - - V 3 1 . . . . i

= V ----g-6(kx)a(k2) V dr exp{i(kl - k 2 ) - r } ,

--~k I ,k 2

386 Chapter 6.

where 6ka ,k2 - - 1 for kl - k2, and 6 k l ,k2 - - 0 for kl ~ k2. Since the integrand in the second term is proportional to k l and k2, the latter average does not contribute. The first average yields, upon substitution into the second term in eq(6.187),

"Second term" = Do2k2/~ 2

(27r)aS(k) ~ f dk' (k ' . k)2V2(k')

iz + Do[I k ' - k 12 +k '2 ]" (6.188)

For the last integral on the right hand-side of eq.(6.187), the following different combinations of n and m must be distinguished �9 (n - 2, m - 1, # 2), ( n - 2 , m r 1 ,2) , (n r 1,2, m - 1 , r r 1, 2, m r 1, 2, n) ,and (n r 1, 2, m - n). Only the first and last combinations are not proportional to either or both 6(kl) and 6(k2), and are therefore the only combinations which contribute to the memory function. The average of the exponential

(2~r)a 6(kl + k2 - k) for functions in the last term in eq.(6.187) is equal to v (2~)6 6(kx - k2)g(k1 - k) for the last combination. the first combination, and v~

Explicit evaluation of the third term in eq(6.187) is now easy, leading to,

]2 D~k4fl 2 #2 V2(k) Do 2 k 4 l 1 - M(k,z) = iz + Dok 2 S(k) S(k) iz + Dok 2

D~ k2~2 ~ / . . . . dk , ( [k - k'] l~)(k' l~)V(Ik' kl)V(k') + (k' l~)2V2(k'). + iz + Do [ I k ' - kl 2 + k'2 ]

Since to leading order in the pair-interaction potential the static structure factor is equal to 1 - fl ~ V ( k ) , the first two terms on the right hand-side

cancel. Hence, it is finally found that,

Do 2k2f12 : [ a �9 k'- k ' - �9 U(k , z ) = (2r)aS(k) t 'J"k'([k-k'l fr t r kl)V(k')+(k ' fc)2V2(k') iz + Do [ [ k ' - k 12 -q-k '2 ] (6.189)

This concludes the explicit evaluation of the collective memory function in the weak coupling approximation. Notice that in the zero wavevector limit (in the sense discussed in section 6.2) the collective memory function vanishes faster than k 2, in accord with the conjecture (6.12).

The evaluation of the self memory function proceeds along similar lines. The leading order expansion of the self fluctuating force is (remember that < pl Ilpx > = 1),

L(k I X I 0) - 13Doik. [Vl(I )] exp{ - ik , r l} . (6.190)

Precisely as for collective diffusion, the complicated second term on the right hand-side of eq.(6.183) for the self memory function (replace p by pi, and f by ]'8) does not contribute to leading order in the potential energy. Hence, to leading order,

M , ( k , z ) - <

= D~o k 2~2~r162 .<

1 i z - s L ( k I x I o)IIL(k ! x I o) >

1 iz DoV 2 e x p { - i k , rl}[~7x~] II e x p { - i k , rl}[~Tx~] > .

- - X

This term is precisely the second term on the right hand-side of eq.(6.185) for the collective memory function (apart from the static structure factor). From eq.(6.188) we thus immediately obtain the following expression for the self memory function,

M,(k, z) - D2~ f (k ' . (r (27r)3 ~ dk' iz + Do[I k ' - k 12 +ka ] " (6.191)

Notice that, contrary to the collective memory function, the zero wavevector limit of the self memory function does not vanish faster than k 2.

For long times the memory equation (6.170) can be written as,

0 07 s , (k , t) - [a , (k) + M,(k, z - o)] S,(k, t ) . (6.192)

This follows from the fact that, for large times, in the integral in eq.(6.170), S, (k, t') is essentially equal to S, (k, t) over the range where the self memory function tends to zero. The long-time and zero wavevector limit of the self diffusion coefficient, which is defined in eq.(6.23), is thus equal to,

Dl,, - - l i m 1 [as(k)+M~(k z-O)] k---*O k'2

- Do 1 (27r)3 (6.193)

In the second equation use is made of the expression (6.179) for the self frequency function, which reduces simply to -Dok 2 when hydrodynamic interaction is neglected. Since the above expression is valid for weak interaction potentials, and hydrodynamic interaction is neglected, a comparison with experiments on systems with long ranged pair-potentials is the only sensible

388 Chapter 6.

thing one can do. For charged Brownian particles the screened Coulomb pair- potential is an obvious choice for the further explicit evaluation of eq.(6.193). In that case, V(r) - A e x p { - ~ r } / r forr > 2a, with n-~ the so-called screening or Debye length. One easily finds that (use, f o dx x 2 / (x 2 + 1)2 _ 7r/4),

D r " - D~ [ 1 - (~V~176 ' ~ a (6.194)

where Vo - V(r - 2a) - A exp{2~a}/2a is the value of the pair-potential at contact, that is, for r - 2a. This expression makes sense for large screening lengths, say ~a ~ 0.1 or smaller. The present approximation breaks down for larger concentrations, where both hydrodynamic interaction and hard- core interaction become more important. Notice that, although this is not an approximation to leading order in concentration, but rather to leading order in the strength of the pair-interaction potential, Dt~ is predicted to vary linearly with the volume fraction ~;. Moreover, for a given contact potential V0, the decrease of DZ, with concentration depends on the screening length ~-1 as exp{-4~a } / ~a. This quantity may be varied by varying the salt concentration of the solvent.

The result (6.194) predicts that Dt~ < Do - D~ both for repulsive and attractive interactions, since the amplitude A of the pair-potential enters as A 2 .

Although there are some experimental results on long-time self diffusion of charged colloids (see H/irtl et a1.(1991)), these are too scarce to test the prediction (6.194).

6.9.6 Long-Time Tails

In the previous subsection we obtained an expression for the zero wavevector and long-time self diffusion coefficient in the weak coupling approximation. Let us go one step further, and ask for the time dependence of the zero wavevector self diffusion coefficient or, equivalently, for the time dependence of the mean squared displacement, for long times. That is, we ask for the way in which the true long-time limit is reached.

To obtain the asymptotic long-time dependence of D~(k - 0, t), first of all the definition (5.23) of the self dynamic structure factor is substituted into the memory equation (6.170),

a~(k) 0 D,(k , t) ~ k~ ~- t - ~ D , ( k , t)

- fo tclt'M'(k't-t')[:2 exp{-D~ (k, t')k2t' + D~(k, t)k2t} .

Now take the limit k ~ 0 from both sides to obtain,

D,(0 t) + lim f},(k) 0 rio t M,(k, t') k--~o k - - T - + t--~D,(O t) - - l i m dt' k2 . (6.195) ~ k--,O

This is a differential equation for D,(0, t), which you are ask to solve in exercise 6.10, with the result,

o.,0,) ' ( k-~o k - - Y - + dt' 1 - k2 . (6.196)

An interchange of the order of time-integration has been performed here, similar to that in exercise 2.1. According to eq.(6.26), the zero wavevector limit of the self diffusion coefficient is related to the mean squared displacement of the tracer particle as, D,(0, t) -<1 r l ( t ) - rl(0) 12> /6t, with rl the position coordinate of the tracer particle. The mean squared displacement can be expressed in terms of the velocity auto-correlation function < vx(t).vx (0) > of the tracer particle, using that r l( t) - ra (0 ) + fo dt'va(t') (here, v I is the translational velocity of the tracer particle). Since in an equilibrium system < Vl(t -+- T). VI(T) > is independent of r, we have,

1 lo t fot t") D,(O, t) = 6-7 dt' dr" <vx(t'). Va( >

1 fo' f_~-'" = - - dt' d ( t ' - tit) ( v l ( t / - tH)- V l ( 0 ) > . 6t t,,

The integral with respect to t ~ - t" is now written as a sum of two integrals, ranging from - t " to 0 and 0 to t - t", respectively. For both integrals an interchange of the order of integration is performed, similar to that in exercise 2.1, to find that (with t' - t" renamed as t'),

D~(0, t) - -~ dt' 1 - 7 < Vl (t/) " Vl (0) > �9 (6.197)

Comparison to eq.(6.196) yields the following integral relation between the velocity auto-correlation function and the self memory function,

lim " ' Jo (1 k---~O k 2 = -- dt' -- 7 -3 < v l ( t ' ) " V 1 (0) > -'[- lim k--~O

M~(k,t') ] k2

(6.198)

390 Chapter 6.

0 o 0 0 o o Ooo _ 09 ~ 1 7 6

o oOy o o 0 o 0 0 ~ 0 0 ~ 0 0

Q Figure 6.15" @ Visualization of interaction of the tracerparticle with its "cage" ofhostparticles which leads to the algebraic decay of the self memory function for long times. (a) is the initial state, (b) depicts the reversal of the initial velocity due to cage interaction.

Differentiation of this equation with respect to t twice results in a relation between the velocity correlation function and the memory function for t > 0,

1 -3 < v l ( t ) , vi(0) > + k-olim M~(k,k 2 t ) = O.

Substituting this back into eq.(6.198) shows that the left hand-side of this equation is a delta distribution at time t - 0. Hence, for t >_ 0,

1 3 < v l ( t ) . Vl(0) > + lim M,(k, t) f~,(k) - k--,o k 2 - - 2 5 ( t ) ~ i ~ k2 . (6.199)

Let us now evaluate the velocity auto-correlation from this exact expression in the weak coupling approximation, with the neglect of hydrodynamic interaction. Instead of taking the long-time limit, we set out to evaluate the memory function as a function of time for long times. To this end, first notice that the eq.(6.191) is the Laplace transform of,

M,(k,t) -D2~ fdk'(k' fc)~v ~ k It} 15 �9 (k ' ) exp{-Do [1 k ' - 12 +k '2

For long times, only small values of k' contribute to the integral, so that,

M,(k t ) - D2~ 2 , (2rr)a ~ V2(k - 0) /dk ' (k ' . l i ) exp {-Do [I k ' - k 1= +k'2 ] t} .

The integral is evaluated in appendix D, with the result,

M,(k, t) = D2o/3=fiV=( k _ O)Tra/2t_s/2 exp{_lDok2t} k 2 2(27r) 3

x [l k2t(2Do)-a/2 + (2Do)-5/2] . (6.200)

With eq.(6.199) and the explicit expression (6.179) for the self frequency function without hydrodynamic interaction, it is thus found that,

3Do2/32t5 V2 )-5/2 < v ( t ) . v(0) > - 6Do6(t ) - 2(27r) 3 (k - O)7r3/2(2Dot . (6.201)

The interpretation of this result is as follows. The memory equations derived in this section are projections of the Smoluchowski equation, which is an equation of motion that is valid on the Brownian time scale. On that time scale the momenta of Brownian particles are always in thermal equilibrium with the solvent. The contribution to the velocity auto-correlation function due to relaxation of the velocity of the tracer particle with the heath bath of solvent molecules is therefore proportional to a delta distribution in time. This is the origin of the first term on the right hand-side of eq.(6.201). The second term is due to interactions with host particles. Notice that this term is always negative, so that the velocity of the tracer particle is reversed at later times relative to its initial velocity. This can be visualized as interaction of the tracer particle with the "cage" of surrounding host particles, as depicted in fig.6.15. The initial velocity (fig.6.15a) is reversed at later times due to recoil of the tracer particle by the deformed cage of host particles (fig.6.15b). The algebraic decay of the velocity correlation function at long times is commonly referred to as a long-time tail, to distinguish that decay from fast exponential decay with time.

Such a long-time tail is also present (although very small in amplitude) for the velocity auto-correlation function of a single Brownian particle in an unbounded fluid. The interactions are now with solvent molecules instead of the host Brownian particles. The fluid flow returns at a later time to the Brownian particle, giving rise to a long-time tail. The Langevin equation for a single Brownian particle that describes these long-time tails contains a memory term, and is usually referred to as a retarded Langevin equation. These memory effects are neglected in chapter 2.

In which way do the self diffusion coefficient and mean squared displacement attain their asymptotic long-time limit? To answer this question, we can differentiate eq.(6.195), and subsequently integrate from some large time t to c~, to obtain,

0__ D t D~)/32PV2(k _ O)Tr3/2(2Do)-5/2 t-3/2 D,(O, t )+ t Ot ~(0, t) - D, + 3(2~r) 3

where eq.(6.200) is used in the zero wavevector limit, and we assumed that l imt. .~ t o D, (0, t) - O. This assumption will turn out to be self-consistent

392 Chapter 6.

with the result obtained with it. The above equation for D~ (0, t) can be solved with the method described in exercise 6.10, to obtain,

Do~2pv2(k _ O)Tr3/2(2Dot)-3/2 D.(0, t) - Dl~ - 3(27r) 3 (6.202)

From eq.(6.26) we thus immediately obtain the following asymptotic time dependence of the mean squared displacement of the tracer particle,

<1 r l ( t ) - r l (0)I > _ 6Dl, t /~2~ V 2 /2 47rv~ (k - 0)(47rD0t) -~ . (6.203)

The linear true long-time dependence of the mean squared displacement is thus predicted to be approached algebraically like ,-~ - t -~/2, as indicated in fig.6.5. The amplitude of the long-time tail is small, and it is a difficult matter to obtain reliable experimental values for the exponent which describes the algebraic approach of the mean squared displacement to its true long-time behaviour.

6.10 Diffusion of Rigid Rods

All of the preceding sections in this chapter are about spherical Brownian particles. The present section is about diffusion of rigid rod like Brownian particles, for which both translational and rotational Brownian motion must be considered. Two subjects are considered here �9 the intensity auto-correlation function for an isotropic system of non-interacting rods (isotropic means that each orientation of a rod is equally likely), and rotational relaxation to first order in concentration. Hydrodynamic interaction is not considered, since not much is known about the hydrodynamic interaction functions for rods.

6.10.1 The Intensity Auto-Correlation Function (IACF)

For non-interacting spherical particles, the field auto-correlation function (EACF) has the simple form ,.~ exp{-Dok2t}. The IACF, which is the function that is measured in a DLS experiment, is related to the EACF through the Siegert relation (3.78). Rotational Brownian motion of spherical particles does not affect the scattered intensity, so that only the translational diffusion coefficient Do appears in the EACE For rod like Brownian particles this is different. When a rod is rotated, without being translated, the interference

6.10. Diffusion of Rigid Rods 393

of the electric field strengths scattered by different volume elements in the rod changes, and so does the scattered intensity. Fluctuation of the scattered intensity thus contains both translational and rotational components for rod like Brownian particles.

In section 3.10 in the chapter on light scattering, the following expression for the normalized EACF was found (see eqs.(3.132,129) and (3.121)),

[TE(k, t)-- Zi,~=, <Jo(�89 exp{ik" (r i (O)-r j( t ))} >

N l Lk fij) exp{ik (ri r j )}> Zi,j=, <Jo( 1Lk" fii)jo(7 " " -

It is assumed here that the relative difference A c/~ in the dielectric constants parallel and perpendicular to the rods long axis is small. The function jo(x) is equal to sin{x }/x, and fi is the unit vector along the long axis of the rod, which specifies its orientation.

In very dilute isotropic suspensions, where the rods effectively do not interact, the "cross terms" with i # j are zero, because exp{ i k . r i } is equally likely negative and positive. The above formula for the normalized EACF then reduces to,

1 (1 ,) ~E(k,t) -- P(k) <jo ~Lk . fi(0 jo Lk. fi(t exp{ik . ( r (0)- r ( t ) )} >,

(6.204) where P(k) is the average scattered intensity normalized to unity at zero wavevector, the so-called form factor,

47r J dfi jg (6.205)

The explicit evaluation of the form factor is the subject of exercise 3.12a. Since for the present case the stochastic variable is X - (r, fi), the correlation function (6.204) is equal to (see eq.(1.62) in the introductory chapter),

~E(k,t) = 4r V P(k) (1 ) (1 )

xjo ~Lk.fi(O) jo ~Lk.fi(t) exp{ik.(r(O)-r(t))}P(r, fi, tlro, fio, t -O),

where it is used that the equilibrium pdf for (r, fi) is equal to P - 1/(47rV) in an isotropic system of non-interacting rods. The integral }' dfi is over the

394 Chapter 6.

entire unit sphere. The conditional pdf P(r, fi, t[ro, rio, t - 0) is the solution P(r , fi, t) of the Smoluchowski equation (4.154,155),

O P ( r , fi, t) - s P(r, fi t) Ot ' '

(6.207)

where/~} the Smoluchowski operator for non-interacting rods,

- bV~(...)+ D , . ~ 2 ( - . .)

+ A D V ~ . [ f i f i - 3 i ] .V~( . . . ) , (6.208)

together with the initial condition,

P(r , fi, t - O) - 6 ( r - ro)6(fi - rio). (6.209)

The rotation operator is defined as,

# ( . . . ) - , a x (6.210)

There are now three different diffusion coefficients, related to rotational diffusion (D~) and translational diffusion parallel (DII) and perpendicular (D• to the rods long axis. In the Smoluchowski operator (6.208), D is the weighted average of the two translational diffusion coefficients and AD is the difference between the two (see eqs.(4.151,152)). According to the hydrodynamic calculations in section 5.15 for very long and thin rods, DII and D• differ by a factor of 2 (see eqs.(5.125,126)), and D and AD are given by eqs.(5.127,128) in terms of the length L and thickness D of the rods. Since the friction coefficients 711, • - kBT/DII , • for translational motion parallel and perpendicular to the rods long axis are different, the instantaneous friction of a rod depends on its velocity relative to its orientation. The last term ,,~ AD in the Smoluchowski operator (6.208) describes this coupling between translational and rotational diffusion. We shall first calculate the EACF with the neglect of this coupling term. In the second part of this subsection, the EACF is calculated for short times, including coupling between translation and rotation.

The conditional pdf in eq.(6.206) depends on r and r0 only through their difference r - r0. The r- and ro-integrations are therefore transformed to integrations with respect to r and r-- r0. The first integration simply yields the

volume V of the system, and the latter integration gives the Fourier transform of the conditional pdf. Hence,

1 j j ( 1 ) ( 1 ) ~ E ( k , t ) - 4 r P ( k ) dfi dfiojo ~Lk-5o jo ~Lk-f i P(k, fi, fio, t),

(6.211) w h e r e ,

- [ d(r - ro)P(r, fi, t [ ro, rio, t) exp{- ik �9 (r - ro)}, P(k, fi, rio, t) J

(6.212) is the Fourier transform of the conditional pdf. The equation of motion for the Fourier transformed conditional pdf, without the translation-rotation coupling term, is obtained by Fourier transformation of eq.(6.207,208), with AD - 0 (replace V~ by ik, as discussed in subsection 1.2.4 in the introductory chapter),

r ~ Y-:-P(k, fi rio, t) - [ - D k 2 + D, "I~ 2 ] P(k, fi, rio, t). Ot

(6.213)

The initial condition follows by Fourier transformation of eq.(6.209),

P(k, fi, rio, t - 0) - 5(fi - rio). (6.214)

The solution is most easily constructed in terms of spherical harmonics (Ap- pendix E is a short reminder of the most important properties of these special functions). The property that is specially useful here is that the spherical harmonics Ytm(fi), l = 0 , 1 , 2 , - . . , - l < rn <_ l, are eigenfunctions of the squared rotation operator,

7~ 2 Ylm(fi) - - l(1 + 1)Yt~(fi). (6.215)

The Fourier transform of the conditional pdf is expanded in a spherical harmonics series,

oo l

P(k. ~, ~o, t) - E E .,~(k. ~o. t) Yy(~) . 1=0 m = - I

(6.216)

Substitution of this expansion into the equation of motion (6.213), and equating coefficients, yields,

0 . ,~(k, ao. t) - I -ok:

0--~ - D~l(1 + 1)] arm(k, rio, t).

396 Chapter 6.

Hence,

a te(k, rio, t) - exp{-Dk2t} exp{-D~l(1 + 1)t} azm(k, rio, t - 0).

From the initial condition (6.214) and the closure relation (6.264) in appendix E it follows that,

a o . t - 0 ) -

where the superscript �9 denotes complex conjugation. We thus find the following series expansion representation of the Fourier transform of the conditional pdf without translation-rotation coupling,

oo 1

P (k , fi, rio, t) - exp{-Dk2t} ~ ~ exp{-D~l(l + 1)t}Ylm*(fio)ytm(fi). / = 0 m=-l

(6.217) The EACF now follows from eq.(6.211) as,

(20

[TE(k, t) -- exp{-Dk2t} ~ exp{-D~l(l + 1)t} St(kL), (6.218) / = 0

where the coefficients are equal to,

l 1Lk fi) [ 2 Zm:-z 1~ dfi Yzm(fi) jo(7 �9 Sz(kL) dfi j~)(�89 fi) " (6.219)

Since the direction of the wavevector k in the integrals ranging over the entire unit sphere is irrelevant, k may be choosen along the z-axis, so that k . fi - k cos{O}, with O the angle that fi makes with the z-axis. The only ~-dependence in the numerator stems from the spherical harmonic (~ is the second spherical angular coordinate of fi). Since Yt ~ ,,~ exp{im~} (see appendix E), the integration with respect to ~ in the above expression yields a 0, so that only the terms with m - 0 survive the integration in the

. ! ~ p . lth numerator. Furthermore, Yt ~ (fi) - V 4# ", (x), with Pt the order Legendre polynomial, which is an odd function of x - cos{ O } for odd values of l, so that St is non-zero only for even values of I. The above expression (6.219) for the coefficients St thus reduces to,

41 +1 [fll dx P21(x) jo (}g]gx)] 2

S21(kL) -- 2 fZldXi2(1Lkx~ ' ~ .--.-- ~ li, j k ~ /

S2t+i(kL) - O, (6.220)

6.10. Diffusion of Rigid Rods

I I I ' 0.8- - So , ,

- S 2

O.l,

- I ! ! i i i i i i ! i ' i ! 1 i i 1 i | | ! ! ! | i

0 5 kL Figure 6.16" The expansion coefficients S2t as functions of k L.

15

397

and the expansion (6.218) of the EACF reduces to,

exp{-/)k2t} [So(kL) + S:(kL) exp{-6D~t} (6.221)

+S4(ki) exp{-20D~t) + S6(kL)exp{-42D~t} + - . . ] .

The coefficients S2z can be evaluated numerically as functions of kL, using the explicit expressions of the Legendre polynomials given in appendix E. The result is plotted in fig.6.16. As can be seen from this figure, the above expansion may be truncated after the S2-term when k L < 10. In those cases the EACF is the sum of two exponentials, exp{-Dk2t} and exp{-[[)k2-J[ - 6D~] t}. Notice that, according to fig.6.16 and the expression (6.221) for the EACF, a significant contribution from rotational diffusion is observed only when kL > 5. Rotation of a rod does not lead to a significant change of the intensity for smaller values of k L.

The above analysis is based on the Smoluchowski equation for a non- interacting rod, where the term ~ AD, which accounts for coupling between translation and rotation, is neglected. We now analyse the effect of coupling of translation and rotation in the short-time limit.

398 Chapter 6.

The effect of translational and rotational coupling

An analysis as given above with the inclusion of the coupling term is a complicated matter. Fortunately, the relevance of the coupling term can be analysed, quite straightforwardly, in the short-time limit. To this end we employ the operator exponential expression (1.67) for correlation functions that is derived in the introductory chapter, in the same way as for the derivation of short-time results for a system of interacting spheres in section 6.5. Since the stochastic variable here is X - (r, fi), and the pdf P(X) - 1/47rV, we have,

t) = 1 (1Lk fi)exp{ik 4 VP(k) f dr / dfJjo �9 .r}

X exp{/~t} [ J0(2Lk-f l )exp{- ik .

= 1+ 1 (1Lk fl)exp 47rVP(k) f dr / dfijo �9 {ik. r}

• [jo (~Lk. f l )exp{- ik , r } ]+ "O(t2) '' .

In the second equation, the operator exponential is Taylor expanded up to first order in time. Only the coupling term in the Smoluchowski operator (the term ,.~ AD) needs be considered here. Substitution of that term into the above expression leads to (with x - cos { O }),

f d r / d f i Jo(2Lk . f i ) exp{ik .r }

[ 3I] [ J ( 1 ) ] x A D t V , . tiff- .V~ o ~Lk.f i exp{- ik- r} +"O(t2) ''

1 ( 1 ) [ ~] = -27r AD k2t f-1 dx j~ -~Lkx x 2 - + "O(t2) '' .

Exponentiation of the resulting short-time expression gives rise to an additional factor exp{-C(kL) AD k2t} to the EACF,

O (k,t) exp{- [C(kL)AD+D] k2t} [So(kL) + S2(kL)exp{-6D,.t} (6.222)

+S4(kL) exp{-Z0D~t} + S6(kL)exp{-42D~t} +. . . ] .

0.3

0.1

Figure 6.17"

, , , I ,~, '~, , , , , , 5 kL 15

The coupling function C, which characterizes the relevance of coupling of translational and rotational diffusion, as a function of k L.

The coupling function C(kL) is equal to,

1 ,

This function can be evaluated numerically and is plotted in fig.6.17. Since the decay time of the EACF due to translation is typically 1/ /)k 2, and AD is not much different from D (see eqs.(5.127,128)), the additional contribution due to the coupling of translation and rotation may be neglected for kL < 5, as can be seen_ from fig.6.17 (there is a 10% contribution from coupling at times where D k2t ,~ 1). Since we found earlier that a significant contribution from rotational diffusion is only found for kL > 5, the conclusion is that coupling between translation and rotation is of importance as soon as rotational diffusion itself is of importance. Rotational diffusion and the coupling with translational diffusion are both important (for kL > 5) or essentially absent (for kL < 5).

There are thus three regions in wavevector space to be distinguished. For k L < 5 only translational diffusion is observed, while for 5 < k L < 10 both translational and rotational diffusion are observed, including the coupling between translation and rotation, and where the EACF (6.221) may be truncated after the S2-term. For k L > 10, higher order rotational terms in the expansion (6.221) become relevant in addition (for 10 < kL < 15, the only significant higher order contribution is the S4-term).

400 Chapter 6.

The EACF is a sum of two exponential functions of time for k L < 10. The difference with the case where coupling is neglected, is that Dk2t is now replaced by [C(kL)AD + D]k2t. The EACF is now a sum of two exponential functions in time, with a frequency F1 - [C(kL)AD +/)]k 2, and a frequency F2 - [C(kL)AD + D]k 2 + 6D~. The amplitude of the latter exponential function relative to the first (which ratio is equal to $2 ( k L) / So ( k L) ) is significantly different from zero only for k L > 5. The experimental procedure to measure both D and D~ is thus as follows. First perform measurements of the IACF for wavevectors kL < 5, and fit the IACF via the Siegert relation (3.82) with gE -- exp{-F t} . In this wavevector range, F - Dk 2, since both rotation and coupling are insignificant. Determine the average translational diffusion coefficient D from the slope of a plot of F versus k 2. Then subtract the two frequencies, F2 - F~ = 6D~, as obtained from measurements in the wavevector range 5 < kL < 10. This difference should be wavevector independent, and gives a value for the rotational diffusion coefficient D~.

6.10.2 Rotational Relaxation

Consider a system of perfectly aligned rod like Brownian particles. The alignment is achieved by means of an external field. Suppose that this external field is turned off at time t -- 0 say, so that the system returns to its equilibrium isotropic state, where each orientation of a rod is equally likely. This process of rotational relaxation can be monitored by means of time resolved small angle depolarized static light scattering, as discussed in chapter 3 (a sketch of the experiment is given in fig.6.18). In case the polarization direction of the incident light (rio) is perpendicular to that of the detected scattered light (fi,), the scattered intensity is given by eq.(3.127). For small scattering angles, such

l k L scattering amplitudes j0 (1Lk �9 fi) are essentially equal that < 0.5, the to 1. Apart from a constant prefactor, the time dependent scattered intensity R(t) is then equal to,

1 N R(t) - ~ ~ < (fi , . fii)(fi,, fij)(fio" fii)(fio" f i j )exp{ik. ( r i - rj)} > .

i,j=l (6.224)

The time dependence is entirely due to that of the pdf, with respect to which the ensemble average is calculated. The polarization direction of the incident light is chosen parallel to the alignment direction rio at time t - 0, so that,

R(t - 0) - 0. (6.225)

7

Figure 6.18" The optical train for a small angle depolarized light scattering experiment. 191 and P2 are polarizers, S is the sample and D the detector. The arrows o f the polarizers indicate their polarization direction. The polarization direction o f the incident light rio is along the z-direction, and of the detected light essentially along the x-direction.

In the present subsection, R(t) is calculated to first order in the concentration for rods with hard-core interaction. Hydrodynamic interaction will be neglected.

The scattered intensity is evaluated in exercise 4.7 for non-interacting rods, by solving equations of motion for ensemble averages which are derived from the Smoluchowski equation. Here we derive the result obtained in that exercise in an alternative way, using spherical harmonics. This approach is employed later in this subsection to analyse the effects of interactions to first order in concentration.

For non-interacting rods the "cross terms" with i r j do not contribute. The scattered intensity is then equal to,

-'2 -,2 R(t) - , (6.226)

where fi~ ( ~ ) is the component of fi of a rod in the z- (x-) direction. We used here that rio is in the z-direction, and fi~ is in the x-direction. The pdf P(fi, t) satifies the Smoluchowski equation (4.154,155), integrated with respect to r (since P (fi, t) - f dr P (r, fi, t)). According to Gauss' s integral theorem, the only term that survives the r-integration is the purely rotational term in the Smoluchowski operator,

O p(fi t) - D~7~2P(fi t) (6.227) 0t ' ' "

The initial condition is that fi is along the z-direction,

P ( , a , t - o ) - - (6.228)

402 Chapter 6.

As in the previous subsection, the pdf is most easily constructed as a spherical harmonics expansion (see appendix E for a summary on spherical harmonics),

co l

P(fi, t) - ~ ~ c~,~(t)Yz m(fi). / = 0 m=-I

Substitution into the equation of motion (6.227), and using that the spherical harmonics Yt m are eigenfunctions of 7},. 2 with eigenvalues - l ( l + 1), yields,

arm(t) - exp{-D~l(l + 1)t} ~tm(t - 0).

From the initial condition (6.228) and the closure relation (6.264) in appendix E, it follows that,

- o ) - Yt '(fi0).

We thus find the following expression for the pdf,

oo l

P(fi, t ) - ~ ~ exp{-D~l(l+ 1)t}Ytm*(fio)Ytm(fi). 1=0 m=-l

(6.229)

This expression allows for the evaluation of the scattered intensity (6.226),

oo l * ~ ^2 ^2 R(t) - y~ ~ exp{-D~l(l + 1)t}Yt m (rio) j dfi u~ u~ Ytm(fi).

1 = 0 m=-I

In spherical coordinates we have, ^ 2 2 u~ u~ - cos 2 { qo } cos 2 { O } sin 2 { 19 }. Since Yl '~ ,~ exp{imcp}, the integrals with respect to qo are only non-zero for

,/21+1 lth m - 0. Furthermore, Yt ~ (fi) - v 4~ Pt (x), with Pt the order Legendre polynomial, which is an odd function of x - cos{O} for odd values of l. Hence, only the integrals for even values of I are non-zero. The above expression for the scattered intensity thus reduces to,

OO

R(t) - ~ exp{-D~2n(2n + 1)t} 4n + 1 4

n--O

P2,~(1) f_lldx ( x 2 - x 4) P2,~(x).

After substitution of the identity x 2 - x 4 = s P4 + ~P2 + ~Po which 35

identity follows from the explicit expressions for Legendre polynomials in eq.(6.256) in appendix E, use can be made of the orthogonality relation (6.255) in appendix E for Legendre polynomials, to finally obtain,

0.1

R

0.06

0.04

0.02

0 0.2 0.4 0.6 D~ t 1.0 Figure 6.19: The scattered intensity R as a function of D~t for non-interacting rods.

1 1 4 exp{-20D~t} (6.230) R(t) - -i5 + -~ exp{-6D~t} - ~

This reproduces the result of exercise 4.7, where a different method of solution was employed.

The scattered intensity is plotted in fig.6.19 as a function of D~t. The scattered intensity goes through a maximum before the rods relax to the fully isotropic state. At the time the maximum scattered intensity is observed, there are more rods having an orientation that gives rise to an optimum depolarization of scattered light than in the isotropic state. You are asked in exercise 6.11 to show that the depolarized scattered intensity of a rod is maximum when its orientation makes an angle of 45 ~ with both polarization directions rio and ft,. Hence, before the isotropic state is attained, their is a transient state where many rods have an orientation of about 45 ~ with both polarization directions.

Let us now consider the effect of direct interactions between the rods to first order in concentration, where only pair interactions are of importance. The scattered intensity (6.224) now consists of two terms,

R(t) ^2 ^2 ( N - 1 . . . . < ~l~z~ > + ) < ~ , ~ ~ z l ~ z ~ exp{ik. ( r l - r~)} >

J dill UxlUzl 2 ^2 P ( t l l , t ) + ( N - 1 ) f d r l f d r 2 J d f i l J dfi2

x ~ ~ 2 ~ z 1 ~ 2 exp{ik" ( r ~ - r 2 ) } P ( r l , r2, fi~, fi2, t ) ,

(6.231)

404 Chapter 6.

where ~ i is the x-coordinate of fii, and similarly for the z-coordinate. There are two pdf's to be calculated, P(fi, t) and P(r~, r2, ill, fi2, t). The Smolu- chowski equation of the latter pdf is very complicated, and not amenable to further analysis. The thing that saves us from the analysis of that very complicated equation of motion, is a separation of the time scales for orientational and positional relaxation. During reorientation of the rods, their position coordinates adapt relatively fast to the new orientational configuration. Let lp be a typical relative displacement of the rods that is needed to equilibrate the positional correlations. The time ~-p required for equilibration of positions is then approximately equal to l~/2D. On the other hand, a significant change of the orientations requires a time To ,.~ 1/D,.. From the expressions (5.127,134), the ratio of these two time scales is found to be equal to,

~ - . (6.232) To 2

Since lp is at most equal to L and the most relevant values, for interacting rods, are probably much smaller than L, this ratio of time scales is small. This means that on the time scale rp, the position coordinates are always in equilibrium during orientational relaxation. Now let P(r~, r2 [ ill, fi2 [ t) denote the conditional pdf for the positions of two rods at time t, given their orientations. If one is willing to accept the separation in orientational and translational time scales, this pdf is always the equilibrium pdf. Up to leading order in concentration, on the pair level, this pdf is equal to the Boltzmann exponential,

P( r l , r2 I 1~11,1~12 It) - exp{-flV(rl - r2, Ul, tl2)} V f dr exp{-~V(r, fix, fi2)}

exp{-flV(rl - r2, I~II, 112)} V f dr [exp{-flV(r, ill, fi2)} - I] + V 2'

with V the pair-potential. The integral in the denominator is of the order 4,~ p3 -ff-~v, where Rv is the range of the pair-interaction potential V(rl - r2, ill, fl2). For very large volumes V of the system, the above pdf thus reduces to,

P ( r l , r2 ]fix,fi2 It) = 1

V2 exp{- f lv ( r l - r2, ill, fi2)}.

The pdf P(ra, r2, fll, fi2, t) - P( r l , r21 ill, fi2 It) x P(fil , fi2, t) now simpli-


ties to,

1 P(r l , r2, 1:11,112, t) -- V2 exp{-/3V(rl - r2, fi~, fi2)} • P(fix, fi2, t).

(6.233) Substitution of this expression into eq.(6.231) for the scattered intensity yields,

R(t) f ,,2 ,,2 p(fi t) (6.234) dfii uxl Uzl 1

+/~ f dill f dfi2 ~iz2~2Z2zl~2 h(k, 1~11, 1~i2)P(fia, fi2, t),

where,

h(k, ill, fi2, t) - f dr exp{-flV(r, Il l , 1:12)} exp{ik, r} (6.235)

= f dr [exp{-flV(r, fi~, fi2)} - 1] exp{ik �9 r}.

In the last equation here, it is assumed that k V ~/a >> 1, so that the integral of exp{/k �9 r} over the scattering volume V is essentially zero (see also the discussion at the end of section 3.5 in the chapter on light scattering). The instantaneous adjustment of positional correlations simplifies the calculation in the sense that only purely orientational pdf's need be evaluated.

Notice that for the calculation of R(t) to first order in concentration, the pdf P(fil, t) must be evaluated up to first order in concentration, while the pdf P(fil, 02, t) needs be known only to zeroth order, since the second term in eq.(6.234) is already multiplied by the density/~. To zeroth order, P(flx, 02, t) simply factorizes as P~ t) x P~ t), where the superscript "0" referres to non-interacting rods. The scattered intensity is therefore fully determined by the single particle pdf's, up to first order in concentration.

The equation of motion for P(fi 1, t)

The equation of motion for P(fi~, t) is obtained by integrating the Smolu- chowski equation (4.148,153) over all the position coordinates and the o- dentations f i2 , " ' , fiN. The only terms which survive these integrations are, according to Gauss's and Stokes's integral theorems,

9

406 Chapter 6.

The potential energy r is now assumed pair-wise additive,

N

(I ) ( r l ' ' ' ' rN ' l~ l l ' ' ' ' l~ lN) -- Z i , j = l , i < j

V (ri - rj, fii, fij).

For identical rods, the above equation of motion then reduces to,

(6.236)

__ "2 69 P(Ul , t ) - D ~ P ( f i ~ , t) Ot

h - ( N - i ) D r f l T ~ l . f d r l / d r 2 J d f i 2 [ 7 ~ l V ( r l - r 2 , I~11, 1~12)] P( r l , r2, 1~11, tl2, t) �9

Substitution of the form (6.233) for the pdf under the integral, and using the factorization discussed above, finally yields, 4

f 0---t P( f i l , t ) - D ~ 7 ~ P ( f i l , t ) - f i D ~ ~ l P ~ �9 dfi2Tl(fil, fi2)P~ t),

(6.237) where,

/~ r~l(l~ll, 1~12) -- - - f i f dr [~1V(r, 01, 02)] exp{-/3V(r, 01, fi2)}, (6.238)

is the torque on rod 1, averaged over the position coordinates of the remaining rods with respect to the Boltzmann exponential (see exercise 4.5). This is the kind of averaged torque that one expects to appear, as a result of the assumption that position coordinates are in equilibrium at each instant of time. The integral on the right hand-side of the equation of motion (6.237) is proportional to the torque on rod 1, averaged over the orientations of another rod. Eq.(6.237) is the equation of motion for a single rod, with the addition of an "external torque", which is due to interaction with other rods.

Before the equation of motion for P(fil, t ) can actually be solved, the as yet unknown torque Tz (ill, fi2) must be specified. In addition, in order to obtain an explicit expression for the scattered intensity, the function h(k, ill, fi2) in

4In equilibrium, where eq.(6.233) is not an approximation, this equation, together with eq.(6.240) for the average torque in case of hard-core interaction, leads to,

Constant - ln{P(fi)} % 2DL 2 f i f dd' [ f~ x d'[ P(d').

This is Onsager's equation which can be used to predict the isotropic-nematic phase transition for long rods (see Onsager (1942,1949)).

6.10. Diffusion of Rigid Rods

Figure 6.20: The excluded volume for two rods. The oblique coordinates 11, 12 and la are parallel to fi 1, fi2 and perpendicular to both these orientations, respectively.

/ L2~"/

' 1 I

I ' �9 L." I " I / . - i

P ' ' J / / /

/ / . / J

407

eq.(6.235) should be evaluated. In doing so, the pair-interaction potential V(rl - r2, fix, fi2) must be specified. This is done in the following paragraph for a hard-core interaction, that is, V is 0 when the hard-cores of two rods do not overlap, while V is infinite when they do overlap.

Evaluation of h (k, fix, fi2) and TI (Ul, u2)

The evaluation of the h-function proceeds as follows. For hard-core interactions, the function exp{-/~V} - 1 in the integrand in the defining equation (6.235) for h is zero in case the cores of the two rods do not overlap, and is equal to - 1 in case of overlap. The integration range is therefore equal to the volume that is traced by the center of a rod when this rod is translated relative to a second fixed rod such that the cores always overlap. This so- called "excluded volume" is depicted in fig.6.20. The integration can best be performed with respect to a new coordinate frame, with one axis ll parallel to i l l , o n e axis 12 parallel to fi2, and the third 13 perpendicular to both fix and fi2. The Jacobian of the transformation from rectangular cartesian coordinates to this oblique coordinate frame is I sin{7} I-I 61 • I, where 7 is the angle between fil and fi2. Hence,

j_~ - d l3exp{ik . 13(fia x fi2)} h ( k , i l l , 1~i2) - - - - [ i l l X 1:i 2 [ D

x f�89 Y_�89 dl2 exp{ik �9 3_~Ldl~ exp{ik./11~11 }

= - 2 D L : I f'l • f'2 [ j o (Dk . (fi~ • f i2) ) jo~-Lk, f i~ ) jo~-Lk , fi2).

In writing the expression (6.231) for the scattered intensity, the scattering

408 Chapter 6.

(1Lk . fax,:) of the rods are set equal to unity, which is allowed amplitudes jo J

for wavevectors for which �89 < 0.5. In this wavevector range, according to the above expression, the h-function is wavevector independent, and is equal tO,

h(k, ill, fi2) - h(k - O, t l l , t l 2) - - - 2 D L 2 I fil x fi2 [ . (6.239)

There are problems with the coordinate transformation when the two orientations 1~i 1 and fi2 are parallel. According to the above formula, the h-function would be 0. This is evidently not correct. However, the excluded volume for parallel rods is equal to 7rD2L and differs by a factor of [sin{-y}l L_. from the

' r / 2 D

excluded volume for the case considered above. Hence, for large L/D-ratios, the range of angles 7 where eq.(6.239) is a bad approximation is extremely small. Although in our experiment we start with perfectly aligned rods, it takes a very short time of reorientation to reach the situation where eq.(6.239) is correct.

The average torque r r 1 is simply related to the function h, since,

T1(fix, fi2) -- -- f dr 7~iV(r, l~ll, I~12) ] exp{-f~V(r, fix, fi2)}

= fl-17~ f dr [exp{-/3V(r, ill, I~12)} -- I]

= / ~ - 1 ~ 1 h ( k - 0, 1~11,1~12) .

Substitution of the eq.(6.239) for h gives,

T1 (IAI1,1~12) = -2/3-XDL27~ [fix • fi2 [

fil x fi2 (6.240) = 2f l - lDL2(f i l" u2) ] tl'-I • fi'~l "

The last equation is obtained by straightforward differentiation (which is most easily performed with the use of lfil • fi21 1 - (fi . fi2)2). This expression can be used in the equation of motion (6.237) to calculate the single-particle pdf. According to eq.(6.240), the translationally averaged torque for rods which interact via their hard-cores only, is zero when fil _1_ fi2. This is easily understood on the basis of fig.6.21. A little thought shows that side- side contacts as depicted in fig.6.2 l a do not contribute to the average torque. There is always another similar side-side contact resulting in a opposite torque. Only side-tip contacts as depicted in fig.6.2 l b give a non-zero contribution to the average torque that rod 2 exerts on rod 1. It is easily seen that such


Figure 6.21" For hard rods the force on rod 1 acts only at the point of contact with rod 2. Side-side contacts (a) do not contribute to the average torque. Only side-tip contacts (b) give a non-zero contribution to the translationally averaged torque on rod 1.

side-tip contacts result in a torque that is in the direction of fi~ • fi2. For the same reason, the torque is also zero when fix I1 f12. This does not follow from eq.(6.240), however, since that expression is not valid when the two orientations are parallel.

Solution of the equation of motion for P (ill, t)

In order to solve eq.(6.237), let us first of all calculate the integral on the right hand-side. In evaluating integrals, instead of using the expression (6.240) for the average torque, it is more appropriate to expand the entire integral in a spherical harmonics series. Hence,

p

f lT~P~ fi2)P~ - ~ ~-~%q(t)Ypq(fal),(6.241) p=O q = - - p

with,

( t ) - �9 po(, x, t ) t ) .

(6.242) This expression for the 7-coefficients is a very complicated one, and can be evaluated only with great effort. In the following paragraph these coefficients are evaluated in a so-called mean field approximation. As will turn out shortly, we shall need explicit expressions only for the first few coefficients. The

410 Chapter 6.

solution of eq.(6.237) is again represented by a spherical harmonics series,

oo l P(l~ll, t) - E E ~ m( I~11)"

1=0 m=-I

Substitution of the expansion (6.241) into the equation of motion (6.237) and equating coefficients of spherical harmonics yields the following differential equation for the coefficients ate,

0 O---t arm(t) - -D~l(1 + 1)c~tm(t)+/SD~7,~(t).

The solution is easily determined, and the pdf is found to be equal to,

l ~/21+ 1 P( t l l , t ) - E E --v 47r Ylm*(fi~

/=0 m=-l

[ Jo' ] x exp{ -D~l ( l+l ) t } + #D d t ' ' [ t m ( t ' ) e x p { - O r l ( l + l ) ( t - t ' ) } .

(6.243)

Since the 7-coefficients are, in principle, known functions of time, this is an explicit solution of the equation of motion, which can be used to evaluate the scattered intensity.

Mean field approximation for the 7-coefficients

The explicit evaluation of the coefficients %q in eq.(6.242) is a cumber- some exercise. The following physically appealing approximation simplifies their evaluation considerably.

The integral with respect to fi2 in the expression (6.242) for the 7- coefficients, is the torque on rod 1, averaged over the orientations of a second rod. This averaged torque is now approximated by the torque on rod I with all other rods having their average orientation < fi2(t) >0, where < . . . >o denotes averaging with respect to the pdf p0. In reality there is a spread in the orientation of the remaining rods around this average. These variations in orientation are neglected, and the torque on rod I is taken equal to the torque that it would experience when all remaining rods would have their mean orientation. The force produced by such uniformly aligned rods is a "mean force field", and the approximation is referred to as a mean tield approximation. Formally the mean field approximation reads,

/ dfi2 T(a~, fi2)P~ t) ~ T(fi~, < fi2(t) >o). (6.24.4)

The average orientation has been calculated in chapters 2 and 4 (see eq.(2.141) and eq.(4.158)), and follows alternatively from the spherical harmonics expansion (6.229) together with,

1 ~ [Y1-1(i~12)_ Yl1(1~12)]

1~i2 _ 1 / ~ [Y1-1(1~!2).~_ Yl1(1~12)] . . 4/~-yy VS- (a2)

Orthogonality of spherical harmonics leads to,

< fi2(t) >o - f dfi2 fi2P~ t) - exp{-2D, t} e3, (6.245)

with ea - (0, 0, 1) the unit vector along the z-direction. Replacing fi2 in eq.(6.240) for the torque by this average thus yields,

f fi~ x ~3 du2 T(fil,fi2)P~ ~ 2fl-lDL2Uzllfi-~__ x ~--~1 exp{-2Dr t} .

(6.246) Substitution of this approximation into the defining expression (6.242) for the 3,-coefficients yields,

%q ,~ - 2 D L 2 exp{-2D~t} d dua ~za P~ t) [ fi---~__ • (6.247)

where a partial integration has been performed (see eq.(4.157)). To make further progress, the action of the rotation operater on the spherical harmonic under the integral must be explicitated. As will turn out in the following paragraph, the only 7-coefficients that are relevant for the scattered intensity are "/00, 0'20 and %0. For our purpose it is therefore sufficient to consider

�9 / ~ , 1 5 Pv (x cos { O }), for - 0, 2 and 4. For - 0 this ~1 ypO (1~11 ) _ V 4~- t~'l - 1 P p is zero, since Po - 1. Hence,

700 = 0. (6.248)

Straightforward differentiation, with the use of the explicit expressions for the Legendre polynomials given in eq.(6.256) in appendix E, yields,

~1/92(x) -- ~z31 -- "-~-~zl I~11 >( e3.

412 Chapter 6.

Substitution of these expressions into eq.(6.247), using that [ fil • e3 [ = ~/1 ^ 2 and applying the mean field recipe once agian, it is found that, ~tz 1 ,

720 "~ -6DL2 i 5 'exp{-2D~ t} f dl~I1P~ t)Ztzl ^2 ~/i " " 2 ~ -- /tzl

5" ^ 2 "~ -6DL2 -~r exp{-2D~t} < ~tzl >20 ~ 1 - - < ~tzl >0

5" - - 6 D L 2 ~--~ exp{-6D~t}~/1 - e x p { - 4 D ~ t } , (6.249)

and,

( 4 9 [ ~ 15~2 ] ~/1 ^2 "74 ~ 2DL 2 9 exp{_2D~t} f df1po(fi1, t) ~t41 ~ z l 0 - - - - ?-tzl

,~_2DL 2 ~9 exp{-2D~t} < Uzl >4 - - T < Uzl >20 ~/1-- < fizX >20

- _ 2 D L 2 9---exp{-lOD~t}--~exp{_6D~t}]~/1-exp{-4D~t}.

(6.250)

This concludes the calculation of the 7-coefficients in mean field approximation which are needed to evaluate the scattered intensity.

Surely, the mean field approximation is not a very accurate one, and further results should be considered as semi-quantitatively.

Evaluation of the scattered intensity

Using the pdf (6.243) and the expression (6.239) for the h-function to evaluate the scattered intensity (6.234) gives,

R(t) - 1-"51 [ I+pD~ fo t dt'70o(t')] 1 t

[ex, - 2#DL 2 J dO1J dfi2 z2~,fi~2Z2~lfi~2 I fi, • fi21 pO(fi~, t)pO(fi2, t).

The first three terms on the right hand-side are found in precisely the same way as the result for non-interacting rods. The coefficients "~0o, 720 and "740 are calculated in the previous paragraph in mean field approximation so that the only remaining task is the evaluation of the integrals in the last term in the right hand-side of the above equation. Within a mean field approximation, however, this last term vanishes, since < ~1,~2 >0 = 0. Since the above expression is valid to within such a mean field approximation, the last term may therefore be discarded.

- ~D2L~ the The final result for the scattered intensity is thus (with ~ hard-core volume fraction)

L 1 1 4 exp{-20D~t} + ~F(D~t) (6.251) R(t) - -~ + ~-~ exp{-6D~t} - ~-~ ~ �9

The orientational relaxation function is given by,

8 ~ 5 ~/1 exp{--4x F(z) - 7~ ~ e x p { - 6 Z ) ~ o ~ dx - ) (6.252)

10i /oZ [ exp , -F--Tr ~ exp{-20z} dx exp{X0x}- g exp{14x} - - 4 x .

This function is plotted in fig.6.22, together with the scattered intensity for various values of Lop. There are a few features to be noted here. First of all, the volume fraction in eq.(6.251) is multiplied by the large number L/D. Contrary to spherical particles, interactions are of importance also for low volume fractions. Indeed, the volume fraction need not be large to assure that two rods interact at certain orientations. Since Lqp ,,~ DL 2 is the volume of an imaginary disk of diameter L and thickness D (which disk is spanned by rotating a rod around its center perpendicular to its long axis) it is actually the volume fraction of such imaginary disks that is a measure for the significance of interactions. Secondly, the orientational relaxation function F is not only zero at time 0, but so is its first order derivative. Hence, for small times, F(D~t) ,,~ (D~t) 2, implying that in the initial stages of relaxation, interactions do not play a role. This is more generally true and is due to the fact that when the rods are perfectly aligned, the translationally averaged torque on a rod due to remaining rods vanishes. In the initial stages of orientational relaxation, the only torque on a rod is due to interaction with solvent molecules, not with other rods. The initial slope of R(t) versus t is therefore concentration independent and equal to that for non-interacting rods. However, as can be

0.02

Chapter 6.

F

0.00

-0.01

- 0 . 0 2 0.0 0.2 0.4 0.6 0,. t 1.0

414

o.~2 ] 1 1

R @

O.04

0.00 o.o o.2 D~t o.6

Figure 6.22" (a) The orientational relaxation function F in eq.(6.252) and (b) the scattered intensity R(t) in eq.(6.251), for several values of ~ , as functions of D~t.

Appendix A 415

seen from the plot in fig.6.22a, the time interval where interactions are not important is extremely small, so that it will be difficult to verify this prediction experimentally (for ~ - 1, the relative contribution of the interaction term is 1.5%, 4.5% and 14% for D~t - 10 -4, 10 -3 and 10-2, respectively). Thirdly, as can be seen from fig.6.22b, repulsive interactions tend to enhance orientational relaxation. The maximum scattered depolarized intensity, where an optimum number of rods have an orientation of about 45 ~ relative to the polarization directions rio and fi,, is achieved at earlier times, and the subsequent decay to the fully isotropic state is faster. For the larger concentrations the intensity goes through a minimum value as a function of time at the later stages of relaxation. This may be an artifact of the mean field approximation and/or the value of ~ may be too large, so that higher order terms in the concentration should be taken into account. Notice also that the number of rods having an orientation of about 45 ~ with both polarization directions is increased due to interactions, since the scattered intensity at its maximum is larger for larger concentrations. The enhancement of orientional relaxation due to hard-core interactions is confirmed by the time dependence of < fi(t) >, which you are asked to evaluate in exercise 6.12.

The scattered intensity is a completely different function of time when both polarization directions of the polarizers in fig.6.18 are rotated over 45 ~ In that case the scattered intensity is maximum at time t - 0, since then all the rods have an orientation that leads to maximum depolarization. The intensity then decreases monotonically with time. This time dependence may be calculated in exactly the same way as in the preceding.

Appendix A m

The orientational average f (k) of some function f (k) over the directions of k is defined as,

y ( k ) - 1

47r f d~: f ( k ~: ) .

The integral here ranges over the unit sphere in k-space, that is, over the spherical angular coordinates of the wavevector.

In case f - k l k 2 , with kj the j t h component of k, the orientational average is zero, since this is an odd function on the unit sphere. Furthermore,

416 Appendix B

the average of f - k~ is independent of j , and is equal to,

~ 3 ~ m lfo2'~ fo '~ lk2 4~" dqok dOk k 2 sin{Ok} cos2{Ok} - 5

where Ok and qok are the spherical angular coordinates of k. The orientational average of the dyadic product kk is thus equal to,

kk - g i .

This result is used in subsection 6.5.1 for the evaluation of the short-time self diffusion coefficient.

Next consider the orientational average,

1 fdk~:kexp{ ikk , r } - - - - - - 1~1~ exp{ik �9 r } = 1

47rk 2 f df exp{ikfc, r}.

The gradient operators with respect to r acting on the exponential "brings two times ik down". The reason for writing the second equation is that the integral appearing there is easily calculated, to obtain,

1 V~v~sin{kr} l<lcexp{ik �9 r} - k2 kr

Now using that V~ - r !~ ~ ( . . . ) , leads to the result quoted on the right hand-side of eq.(6.81), together with the expressions given in eqs.(6.82,83).

Appendix B

For hard-sphere suspensions and to leading order in concentration, the solution L(r) of eq.(6.118) is most naturally constructed as a power series expansion with respect to a/r. Since the functions p, q and s are given by such expansions, valid up to order (a/r) 8, the expansion of L must be truncated also at that order,

L(r) - E a~ + 0 ((a/r)8) . (6.253) n-~ l

Terms corresponding to n < 0 do not occur, since L(r) must vanish for r ~ c~. For hard-sphere interactions and to leading order in the concentration,

A p p e n d i x B 417

the pdf p(o) is a constant for r > 2a. For these distances the above expression for L can be substituted into the differential equation (6.118), which then reduces to a polynomial in a / r . The constant coefficients that multiply each power of a / r must be zero. A little bookkeeping yields the following relations for the expansion coeffcients of L,

( - 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - 2 4 0 0 0 0 2 4 0 - 1 2 10 0 0 0

- 1 5 9 0 45 18 0 0 2 0 _7_5 16 0 - 3 6 28 0 2

\ --ff-'281 0 ---'~135 25 0 1052 4 0 )

C~2

C~3

�9 O~ 4

C~5

0~6

( 0 0

15/4 o

-159/16 37 /s

Notice that the coefficient c~2 is undetermined. The remaining coefficients are easily expressed in terms of c~2, since the matrix contains only zero entries at the upper right of the diagonal. One finds,

C~ 1 - - 0

15 9 =

9 27

41 19

501 249 c~ = 224 + 1--~ c~2'

1953 1017 C~7 = I ~ C~2 �9

512 256

The as yet undetermined coefficient a2 is calculated as follows. The derivative of P(~ in the differential equation (6.118) is, to leading order in concentration, proportional to the delta distribution 5(r - 2a) (see eq.(6.110)). The differential equation is now integrated from r - 2a - e to r - 2a + c, with c a vanishingly small number. The only term that survives after integration and taking the limit e I 0, is the term proportional to the delta distribution. Hence,

d L ( r ) I - 0 for r - 2 a . s ( r ) r dr r r 2a '

Substitution of eq.(6.253) for L together with the above expressions for the coefficients yields a single equation for c~2. It is found that c~2 = -1 .1676 . - - .

418 Appendix C

This concludes the calculation of the solution of the differential equation (6.118). The result is explicitly written in eq.(6.122).

Appendix C

Solving the differential equation (6.141) under the restriction (6.139) requires the following representation of the 1-dimensional delta distribution"

Let f ( x ) denote a function on ~, with i f (x) - df ( x ) / dx lim~._,oo f ( x ) - o0, then,

> O, and

~ ( x - Xo) - H(x - xo) lim elo i f (x)e exp { - f ( x ) - f (x~ ' (6.254)

where H (x) - 0 for x < 0 and H (x) - 1 for x > O, the so-called Heaviside unit step-function.

The proof of this statement is as follows. We have to show that, for an analytic function g(x),

I - l i m f ~ ~ 1 7 6 o dxg(x) if(x)e e x p { - f ( x ) - f ( x ~ - g ( x o ) . e

The first step is to integrate with respect to y - f(x). Since f ' (x) > 0 this is a proper coordinate transformation, and since f(oo) - c~, the new upper integration limit is c~. Hence, with f-1 the inverse of f ,

I - l i m -1 [oo ~,o e Jr(,0)

dy g(f- l (y)) exp {-- y - f ( xo )~ ~

J s

For very small values of e, the exponential function tends to zero for values of y which are close to f(xo). The only values of y which contribute to the integral are very close to f(xo). In the limit of vanishing e, g(x) in the integrand may be set equal to g ( / - l ( y _ f(xo))) - g(xo),

I - g(xo)lim-1 /:r ~o e J](~o)

dy exp ( - y - f(xo) }

The remaining integral is standard, and the result confirms the representation (6.254).

Appendix C 419

The differential equation (6.141) is solved by variation of constants. First consider the so-called homogeneous equation, where S ~q is omitted,

c3 peo) q2 peo) qi b--~q So(q I - So(q[ .

Straightforward integration yields,

So(. l , , ~ - C(ql, q3) exp{Lql foqZdx (q~q_x2~_q~)} { : : )}

ql 3 "

Here, C is an integration constant, which is in general a function of ql and q3 since we integrated with respect to q2. The idea of the method of variation of constants is to make C a function of q2 as well, in such a way that the full equation (6.141) is satisfied. Substitution of the above expression into the differential equation, with C understood to be a function of q2, yields a differential equation for C, which is easily integrated, to obtain,

~o(. I ~ ~

-- dx (q21 + x2+ q23)S ~q x 2 ~/Pe ~ if q, (~/q21 + + q23• ql {--( ql ) x ( 1 q, o )} • exp q2 q~+~q~+q2 ___ q2+ x 2+q32

3 I ~" "

This expression is finite for all q's when the integration constant C' is 0 and the unspecified lower integration limit is - ~ in case ql < 0, and + ~ in case ql > 0. The resulting expression is then,

1/4-00 (k/q ) S ~ 1 7 6 - ~11 2 dx ( q ~ + x 2 + q ~ ) S ~' 1 2 + x 2 + q ~ x x / P e ~

{ _ ( 1 q 2 ) _ x ( 1 ) } x cxp q2 q~ + ql + ~ _ qa ~ + ~ 2 ql 3 ql ~ + q3 �9

Returning to the original dimensionless wavevector (see eq.(6.138)) gives,

So(~ I ~ ~ - .:/,.,/~, '~ ('<~§ + ~ ) ~" § +

. e x , I 1 1t Pe~

420 Appendix D

The condition (6.142) is now verified with the use of the delta distribution rep-

resentation (6.254). With x - Q, xo - K2, f (Q) - ~ (K? + ~ + and c - P e ~ the right hand-side of the above expression is indeed seen to become equal to S ~q (K) in the limit where Pe ~ vanishes. Subtraction of S~q(K) from both sides, and using the delta distribution representation (6.254) leads to eq.(6.143).

Appendix D

The integral that must be evaluated to arrive at the result (6.201) for the velocity auto-correlation function is,

' -- f d k t ( k ' , k) 2 e x p { - D o [[ k ' - k 12 + k t 2 ] t } .

This integral is independent of the orientation of k, so that we may chose k along the z-direction. The integral is then given by,

I __. exp{-Dok2t} f dk~ exp{-2Doki2t} f dk~ exp{-2Dok;2t}

/ , • dk'3 2k3 2 exp{-2Do[k3 2 - k k;l t},

with k~ is the jth-component of k'. Using that f_~ dx e x p { - a x 2 } - ~/Tr/a,

the first two integrals are both found to be equal to ~/Tr/2Dot. The third lk+k~ as, integral is rewritten, with z - - g ,

f dk'32k'32 exp{-2Do[k'3 2 - k k~3]t} - -

exp{-~Dok2t} dz z 2 +-~ + zk exp{-2Doz2t}

The third term between the square brackets does not contribute, since the corresponding integrand is an odd function of z. Using the earlier mentioned standard integral and f_~ dx x 2 e x p { - a x 2 ) = ~ ~/Tr/a, finally leads to,

1 ] I - -~exp{--~Dok2t}Tr312t -5/2 k2t(2Do) -3/2 + (2Do) -5/2 .

This result leads to the expression (6.200) for the self memory function.

Appendix E 421

Appendix E

It is not feasible to give a self-contained survey of the theory of special functions (which is a mathematical discipline in itself) in an appendix. In this appendix we just summarize the equations which are relevant for the calculations in section 6.10. A treatment of the theory of special functions can be found in, for example, Arfken (1970) and Jackson (1975). Spherical harmonics are discussed in many books on quantum mechanics, since the rotation operator is proportional to the quantum mechanical angular momentum operator.

Spherical harmonics are constructed from Legendre polynomials. These are polynomials Pt(x) of degree l, defined on x E [-1, 1], which are orthogo- hal, in the sense that,

l 2 dx Pt(x)PI,(X) - ~5l~, , (6.255)

a 2 / + 1

with t~t l, - 1 for 1 - l', and al t, - 0 for I ~ l', the Kronecker delta. Starting with Po(X) - 1, all other Legendre polynomials may be constructed from the above orthogonality relation. The first few Legendre polynomials are,

Po(x)- 1, -

l [ 5 x a - 3 x ] , P3(x) - -~

1 [35x4- 30x2+ 3] P 4 ( x ) - g

1 P s ( x ) - g

(6.256)

Higher order polynomials may also be obtained from lower order functions with the use of the recurrence relation,

2 / + 1 1 Pt+a (x) - l + 1 x Pt(x) l + 1 Pl-x (x) . (6.257)

422 Append ix E

Alternatively, Legendre polynomials may be calculated from Rodrique 's for-

mula,

1 d I )l P~(x) - 2t/! dx z (z 2 - 1 .

For even l, Pt(x) is an even function of z, for odd l it is an odd function. Furthermore, Pt(1) - 1 and P t ( - 1) - ( - 1)t.

From these Legendre polynomials, the so-called associated Legendre func-

tions are constructed as,

d m - -

dx m

It may seem that m should be non-negative. However, substitution of Ro- drique's formula for Pt shows that we may have negative m, not smaller than -I . Furthermore, it is clear that P y - 0 for m > I. Hence, the values that m is allowed to have are, - l < m < 1.

Spherical harmonics Yl ~ are now defined as,

Yl'~(O qo) - ( - 1 ) m ~ 21 + 1 ( 1 - m)l ' 47r (l + m)l Pt~(c~176 (6.258)

Here, O and qo are spherical angular coordinates (0 _< O _< 7r and 0 _< qp _< 27r). These angles specify a unit vector fi with x- y- and z-components equal to,

fi~ = sin(O} cos{~,},

fly - s in{O}s in{~} ,

- cos{O}, (6.259)

so that Yt m (O, qo) is also written simply as Ytm (fi). Since Pt ~=~ (x) - Pt (x),

it follows from the above definition that Yt ~=~ (fi) - v 4~ Pt (x - cos { O }).

_ ./2t+1 From From Pl (z - 1) - 1, we thus obtain, Yl ~=~ (fi - (0, 0, 1)) v 4~ �9 the above listed equations, explicit expressions for spherical harmonics may be calculated. The first few are,

1 voo( )-

y l - x ( f i ) - ~ 8 ~ s i n { O ) e x p { - i q o ) ,

Appendix E 423

ra0 (a) -

~(a) =

r : ~ ( a ) -

y2-1 (s -

r~0(a) -

~(a) -

r~(a ) -

~/3 ~ c o ~ { O } ,

- ~ 8 ~ sin{O} exp{i~'},

3 ~sin2{O}exp{-2iqo},

~ / 5 sin{O}cos{O}exp{-i~,} 3 ~

( c o : { O } - ,

-3 ~ s i n { O } cos{O} exp{iq:},

3 ~sin2{O}exp{2iq:},

(6.260)

Spherical harmonics satisfy the orthogonality relation,

~ r : * ( a ) r~, m'(a) - ~,,, ~ ~ , . (6.261)

This can be verified from the orthogality relation for Legendre polynomials. The set of spherical harmonics is a complete set of functions. This means that a function f(fi) of the two spherical angular coordinates may be written as,

o o l

f ( f i ) - Y~ ~ ftm Ytm(fi). (6.262) / = 0 m = - l

From the orthogonality relation (6.261), it follows immediately that the expansion coefficients are equal to,

ftm - / dO' f(fi')Ytm*(fi'). (6.263)

Substitution of this latter formule back into eq.(6.262) yields,

o o l

f(~) - ~ ~ /'~' f(~ l = O m = - l "

')rtm'(~') r : ( ~ , ) .

Hence, by definition,

oo l

~(fi - fi') - ~ ~ Ytm*(fi ') Ylm(fi). (6.264) /=0 m = - I

Such relations are known as closure relations. The thing that makes spherical harmonics important, at least for our purpose, is that they are eigenfunctions of the squared rotation operator, with eigenvalues - l(l + 1). Relations for operating with a single rotation operator on spherical harmonics can be obtained with so-called/adder operator techniques. For our limited use of spherical harmonics these relations are not needed, and we shall not summarize these here.

Exercises

6.1) Non-Gaussian behaviour of displacements In section 6.3, the zero wavevector self diffusion coefficient was shown to

be related to the mean squared displacement (see eq.(6.26)). The next higher order wavevector dependence of D~ (k, t) is discussed in this exercise.

According to the definition (6.23),

S,(k, t) - f dAr P(Ar , t ) e x p { - i k . Ar} ,

where Ar -- r(t) - r(t - 0) is the displacement of a Brownian particle during a time interval t, and P(Ar , t) is the pdf for such a displacement. For an isotropic system, this pdf depends on the magnitude [Ar[ of the displacement only, not on its direction. The integral over the directions can thus be done (see eq.(5.139) in appendix A of chapter 5). Perform a Taylor expansion of the sine function of k [Ar [ in the resulting expression up to "O(k6) '' to show

1X 3 1 5 t h a t ( s i n { x } - x - ~ + x + . . . ) , iT6

1 k2 [2 1 k4 4> S~(k,t) = 1 - ~ <JAr >+1--~ <[Ar[ + . . . .

The brackets < . . . > denote averaging with respect to the pdf P(Ar , t). Exponentiate this expansion to obtain, up to "O(k6) '',

{ 1 2 l k4 [3 <[Ar[4 > 2 2 ] } . S~(k, t) - - exp - ~ k <1Ar[2> +3-~ - 5 <[ Ar[ >

I k 2 2 Show that the equation S, - exp{- ~ <[ Ar I > } is exact for Gaussian displacements Ar (you may verify this by Fourier transformation of the isotropic Gaussian pdf for Ar). The value of the k4-term between the square brackets is therefore a measure for the non-Gaussian behaviour of the position coordinate.

Extend the discussion in section 6.3 on the experimental procedure to obtain the zero wavevector self diffusion coefficient by light scattering, to include the measurement of the non-Gaussian k4-term. For short and long times, that is for t << TX and t >> rt respectively, the particle displacement is expected to be Gaussian, while for t ~ r~ the non-Gaussian contribution to the self structure factor should be maximum. For charged colloidal systems and for liquid argon, relatively small non-Gaussian contributions are found (Gaylor et al. (1981), van Megen et al. (1986), Rahman (1964)) while for hard-sphere dispersions there seem to be relatively large non-Gaussian contributions (van Veluwen and Lekkerkerker (1988)).

6 . 2 ) *

(a) Use Gauss's integral theorem and disregard surface integrals ranging over surfaces located at infinity to derive the expression (6.38) for ~t s, which is defined by eq.(6.36). Use that the microscopic diffusion matrix D is symmetric.

(b) Use Gauss's integral theorem to show that, for arbitrary functions a(r) and b(r),

f dr P(r)a(r)V~- D(r) . V~b(r) =

- f dr [a(r)V~P(r) + P(r)V~a(r)]. D(r ) . V~b(r).

Substitute P(r) ,~ exp{-/3@(r)} to verify eqs.(6.43,67). (c) The operator/~t s is the Hermitian conjugate of the Smoluchowski opera-

tor s with respect to the so-called unweighted inner product fdXh(X)g* (X) of two phase functions h and g (in subsection 6.9.1 we used the more general notation X for the stochastic variable, which is the super vector r for calculations on the Brownian time scale). Eq.(6.165) states that this conjugate operator is Hermitian with respect to the weighted inner product (weighted with the pdf P ( X - r - ( r l , - . . , rN)) ~ exp{- /~( r )}) . Show this by means of partial integrations using Gauss's integral theorem.

6.3) Cumulant expansion In section 6.5 on short-time diffusion, the linear term in time in the expres-

sion (6.37) for the correlation function of two phase functions f ( r ) and g(r) was evaluated for self and collective diffusion. Expanding up to the second order term in time gives,

< f ( r (0 ) )g ( r ( t ) ) >0 = f dr P(r)f(r)g(r) + t f dr P(r)f(r)~.tsg(r)

1 t2 / dr P(r)f(r)s + "O(t3) '' +

12 --< f ( r ) g ( r ) >o +t < f(r)/~tsg(r) >o + ~ t < f(r)/~ts2g(r) >o +"O(t3) '' ,

where < . . . >0 denotes ensemble averaging with respect to the equilibrium pdf P(r) . Exponentiate this expression, and show that,

< f ( r ( 0 ) ) g ( r ( t ) ) > o

< f ( r ) g ( r ) > o 1 t2 } ,, = exp K~t + -~K2 +"O(t a) ,

where the so-called first and second cumulant are respectively given by,

< f(r)/~tsg(r)>o

< f ( r ) g ( r ) > o

K2 = < f(r)/~ts2g(r)>o _ K~. < f ( r ) g ( r ) > o

This cumulant expansion can of course be extended to include higher order terms in time. The first cumulant is considered in section 6.5 for the particular choices (6.39) and (6.64) of the functions f and g, for self and collective diffusion, respectively. In this exercise we calculate the second cumulants for self and collective diffusion without hydrodynamic interaction.

(a) An identity, known as the Yvon identity, that can be used for the evaluation of second cumulants, reads,

< a(r)Vi(I)(r) > o - fl-1 < Via(r) >o,

for an arbitrary phase function a(r). Proof this identity with the help of Gauss's integral theorem and the fact that P(r) ,-~ exp{-/3(I)(r)}.

(b) For self diffusion, the functions f and g are given in eq.(6.39). With- out hydrodynamic interaction, eq.(6.44) for the first cumulant reduces to to

K1 - -Dok 2. Neglecting hydrodynamic interaction, assuming identical host particles and a pair-wise additive potential energy, show that the second cumulant for self diffusion is given by,

K2 - 3D~k2fdr VxV,O(r) >o

= ZD~ d,g~,(,)T, N V~,(,) .

The function ght(r) is the pair-correlation function and Vht (r) the pair-interaction potential for a host particle and the tracer particle. The Stokes-Einstein diffusion coefficient Do is that of the tracer particle.

(c) For collective diffusion, the functions f and g are given in eq.(6.64). Without hydrodynamic interaction, eq.(6.69) for the first cumulant reduces to K1 - -Dok2/S(k). Under the same assumptions as in (b), show that the second cumulant for collective diffusion is given by,

K2 + K~ - D2o k2 S(k) [k2 + f l ( N - 1)kl~ "< VIVlV(rl2)

+ [VlV2V(rl2)] e x p { - i k - ( r l - r2)} >o]

D~k2 [ k 2 + / ~ f dr g (r ) [1-exp{- ik , r}] (1~. V)2V(r)] . s(k)

Now use that (f di" are integrals over the unit sphere),

f de (~. e): =

f d~ e x p { - i k . r } -

fd i " exp{ - ik , r}(l<, f.)2 _

471"

3 ' sin{kr}

4 r ~ kr '

2kr cos{kr} - ((kr) 2 - 2)sin{kr} 47r (kr) 3

to reduce the above expression for the second cumulant to,

K2 -- Daok2[ 4r fo ~ s~(k) k~ (s(k)- 11 + - f s (k )~ d~g(r)

x ~ ~ (k,)~

_3r2 d2 U(r)dr 2 3kr cos{kr} + (kr) 3((kr) 2 - 3)sin{kr} }]

Notice that the derivatives of the pair-potential do not exist for hard-core interactions. It might be that the cumulant expansion does not exist for hard- sphere systems, or that the higher order time dependence should be calculated via an alternative route.

(d) Show that the first cumulants for self and collective diffusion on the Fokker-Planck time scale are zero. The relevant operator is now the Fokker- Planck operator (4.19). As a first step you should calculate the Hermitian conjugate operator/~tFp with respect to the unweighted inner product, analogous to the calculation in exercise 6.2a for the Smoluchowski operator.

This result shows that it is essential to know the time scale on which an experiment is performed. On the Smoluchowski time scale, the intensity auto- correlation function has a finite slope at time t = 0, while on the Fokker-Planck time scale the slope is zero.

(Hint" In (b) and (c) you can use that,

< f ( r ) / ~ t s 2 g ( r ) > o - < [/~ts f(r)] [/~t s g(r)] > o .

This statement is proved in exercise 6.2c.)

6.4) Gradient diffusion Without hydrodynamic interaction, the first order in volume fraction coef-

ficient av in eq.(6.108) for the gradient diffusion coefficient reduces to (with x - r /2a) ,

fo cr d V ( a x ) c~v - - fl dx x 3 g(~ ( ax ) dx

Now suppose that in addition to hard-core interaction, there is a pair-potential V+(r ) (for r >_ 2a). Use the relation (6.110), which is valid for the hard-core part of the pair-correlation function, to show that,

a v -- 8 h- f2~ x 3 d e x p { - ~ V + ( a x ) } dz

Let the additional potential be equal to a square well potential,

V+(r) - 0 , f o r

= - c , for

= 0 , for

0 < r < 2 a ,

2a < r < 2a + A ,

r > _ 2 a §


is the depth of the square well and A its width. The depth e is positive for attractive potentials and negative for repulsive potentials. The derivative in the integral for av is now equal to,

dexp{- f lV +(r)} = (exp{fle} - 1) [5(r - 2a) - 6(r - 2a - A)] . dr

Use this to show that,

[ ] av - 8 - (exp{fle} - l ) ( 2 + - - - 8 . a

Is diffusion enhanced or slowed down due to attractive interactions? Interpret this result.

The combination 1 + av r can be made negative for strong attractions. This implies that the gradient diffusion coefficient is negative, so that smooth gradients in the density increase their amplitude in time. In that case, the system does not relax to the homogeneous state, but rather develops inhomogeneities. This is the initial stage a phase separation. Up to first order in concentration, a negative diffusion coefficient is nothing more than a formal result. The first order in volume fraction contribution is now larger in magnitude than the zeroth order term, so that higher order terms in concentration can not be neglected. Nevertheless, this calculation may serve to illustrate the mechanism of (spinodal) phase separation. Chapter 9 is devoted to the kinetics of such phase transitions.

6.5) A n ef fect ive m e d i u m approach

In subsection 6.7.1, the effective friction coefficient 7 ~:: is introduced as the friction coefficient that a single particle experiences due to both friction with the solvent and interaction with the remaining Brownian particles. It is tempting to consider the suspension as an "effective medium" for the tracer particle. The composite system of fluid and host particles is then formally replaced by a fluid with the properties of the suspension of host particles.

The viscosity of this effective medium is equal to,

77 ~:: - 7/o[1 -t-

to first order in concentration (see exercise 5.4). The effective friction coefficient, to within the effective medium interpretation, is equal to,

7 ~:: - 67rrl~:fa - 6:ryoa [1 + ~

Compare this with the result (6.129), and conclude that the effective medium interpretation is in error (the correct value for the first order in volume fraction coefficient is 2.11, while the effective medium value is 5/2).

In fact, the value 5/2 is independent on the kind of pair-interaction potential at hand. Go through exercise 5.4 to convince yourself that the effects of interactions on the viscosity show up in the second and higher order in volume fraction coefficients. In contrast, the value of the linear in volume fraction coefficient for 7 ~ff is strongly dependent on the kind of pair-interaction potential under consideration (the value 2.11 is valid for hard-core interactions). This shows that, at least to leading order in concentration, an effective medium approach is in error.

6.6) Long-time self diffusion without hydrodynamic interaction

The evaluation of the leading order concentration dependence of the long- time self diffusion coefficient is considerably simplified when hydrodynamic interaction is neglected. Repeat the analysis of section 6.7 to show that,

(a): fl < DII > "F ex t - t 3 D o F e x t �9

(b)" L ( r ) - - 2 ( a / r ) 2 .

(c)" < v[ > - -/~Do 2 ~;F ~t .

(d)" < v, s~ > - 0 . Conclude that D~, - Do { 1 - 2r + O (r Although each of the separate contributions to D~ are very much different from the results which are obtained with the inclusion of hydrodynamic interaction, the end result is quite close to the exact result -2.10 for the first order coefficient. The effect of hydrodynamic interaction on the distortion of the pdf and on the short-time self diffusion coefficient almost counter balance each other.

6.7) * Boundary layer theory

This is an exercise for those readers who are not familiar with boundary layer theory. A simple singularly perturbed differential equation is analysed in order to illustrate the method. More about singular perturbation theory, boundary layer theory in particular, can be found in Bender and Orszag (1978), Nayfeh (1981) and Hinch(1991).

Consider the following differential equation for y - y(x I e) on x E [0, c~), with e a small number,

d - 1 +


with the boundary condition,

- 01 ) - 0 .

In a naive approach one might try to expand the solution in a power series of the small parameter, that is, one assumes that the solution is a regular function (or equivalently, an analytic function) of e,

y ( x l s ) -- go(X)"~- s s ) -~- . . . ,

The boundary condition implies that y,~(x - 0) - 0 for all n. Verify by substitution of this regular expansion into the differential equation, and equating coefficients of each power of e, that,

yo(x) - 1 + x 2 . (6.265)

This solution does not satisfy the boundary condition yo(x - 0) - 0. The conclusion is that y is a non-analytic function of e, at least in some neighbourhood of x = 0. The point is, that the differential equation with e simply put equal to zero lacks the freedom to adjust integration constants such as to match the boundary condition. Such a lack of freedom always occurs when the highest order derivative in a differential equation is multiplied by the small parameter, since the order of the differential equation is then reduced when the small parameter is set equal to zero. The further conclusion is, that in some neighbourhood of x - O, e d y / d x is not small in comparison to the other terms in the differential equation, since it evidently can not be neglected. This implies that the derivative d y / d x is of the order 1/c, which is a large number for small c. The solution of the differential equation thus changes very rapidly around x = 0 in order to adjust to its prescribed value at x = 0. That region is called the boundary layer or the inner region. The remaining set of x-values is the outer region. The approximate solution (6.265) is only valid in the outer region, where d y / d x is not very large, so that c d y / d x is indeed small in comparison to the remaining terms in the differential equation.

The idea to find an asymptotic approximation in the inner region is to introduce a new variable z - x / e ~, with v chosen such, that in the new differential equation the highest order derivative is no longer multiplied with the small parameter, and thereby looses its singular nature. This new variable is referred to as the boundary layer variable. Verify that with v - 1 the rescaled differential equation reads (use the same symbol for y as a function

121 o I I . . . . . . . . . . . . -

r

0.8 . . . . . i - - _

0.4

0 0.1 0.2 X 0.3 Figure 6.23: The inner solution in eq.(6.266) (V ~) and outer solution in eq.(6.265) (V~ together with the exact solution (solid line), for c = 0.02.

of z as for the function of x),

d dzY(ZlC) + y(z I,) - 1 + J z 2 .

Since now the highest order derivative is no longer multiplied by the small parameter, the solution may be expanded in a power series of e,

y ( z l s ) -- yO(Z) + s + s + - ' " .

The boundary condition is yn (z - 0) - 0 for all n. Show that,

yo(z) - 1 - e x p { - z } .

Returning to the original x-variable, we thus have,

yo(x l e) - 1 - exp{ -x /e} . (6.266)

The range of validity of this solution is z - e z < 0.1, say, since the term e 2z 2

was omitted from the differential equation for yo(z). This function changes rapidly in the small interval x E [0, e), so as to match with its boundary condition. This small interval is the boundary layer. At larger values of x,


outside the boundary layer, the derivative of the solution is not so large, so that the solution (6.265) is a good approximation (this happens for x >> e).

We now have two approximate solutions which are valid in two separate regions in [0, oz), namely, for x E [0, ,-~ 1 ) and for x >> e (x > 5e say, where e x p { - x / e } ~ 0). The approximate solutions in these regions are referred to as the inner solution and the outer solution, respectively. The two solutions are both good approximations in the region x E (Se, 1-!6), the so-called matching region. For somewhat larger values of e, higher order terms in the above expansions must be determined to enlarge the matching region for the two asymptotic approximations.

The above analysis is given without any knowledge about the exact solution. Verify that the exact solution is,

y(x) - 1 - e x p { - x / e } - 2~ 2 exp{-x /e} + x 2 - 2xe + 2e 2 .

Show that this expression reduces to (6.266) in the inner region and to (6.265) in the outer region. For e - 0.02, the inner and outer solution are plotted in fig.6.23, together with the exact solution.

A more accurate inner solution is obtained when in the equation for yo(z) the right hand-side of the differential equation (1 + e2z 2) is not approximated by its leading term (1 + e2 z 2 ,,~ 1), but is kept as it stands. In the simple example considered here, however, the differential equation for yo(z) is then precisely the exact differential equation. In section 6.8, such a procedure corresponds to keeping the Pe~ of S ~q (q v/P e ~ instead of expanding up to the leading term, which is S ~q (0). This makes the coefficients Sn in the expansion (6.140) for the inner solution Pe~ This procedure renders the leading inner solution So valid also in the outer region.

6.8) * The operator identity (6.162) is derived as follows. First define the operator,

z ~ ( t ) - exp{/~tst}-exp{Q(k)/~tst } .

Differentiate this definition with respect to t, and show that,

0--0--/~(t) - s ) + 75(k)s exp{ Q(k)s t } Ot

First solve the homogeneous equation (the above equation with the second term on the right hand-side omitted), to find,

A(t) - C exp{~tst},


where C is an integration constant. Now let C be a function of time, in such a way that the solution satisfies the full differential equation. A differential equation for C is obtained by substitution of the homogeneous solution into the full (inhomogeneous) differential equation. This equation is easily solved. Show that/k(t) is equal to the integral on right hand-side of eq.(6.162).

6 . 9 ) * The resolvent operator The resolvent operator R(z) of an operator 60 is defined as the inverse of

the operator i z - (9 (iz is to be read as i zZ , with Z the identity operator). In subsection 6.9.4 it is used that the resolvent operator equals the Laplace transform of exp{ Ot },

- f0 T l i rn dt exp { - i z t + (gt } .

Expand the operator exponential in its defining Taylor series and integrate term by term to show that,

fo ~ dt exp{- iz t + (gt } oo 1 ( _ i z + ~ ) ~ T ~ + 1

- - ~-~ (n + i)' = (--iZ -~- O) -1 [--~ -'~ exp{(--iz + O)T}] .

The operator exponential vanishes for r --, oo. Verify that the resolvent operator T~(z) is indeed equal to the Laplace transform of exp{Ot}.

6.10) * The differential equation (6.195) reads,

O----D (0, t) - g( t ) D~(O, t ) + t Ot ~

with,

[ /0' g(t ) - - k-~olim ft ) + dt, ~ k 2 .

First solve the homogeneous equation, where g is set equal to zero, and show that,

D~(0,t) - C / t .

This solution contains an undetermined integration constant C. Make this integration constant time dependent, in such a way that the full differential

equation for D~(0, t) is satisfied. In solving the differential equation for this time dependent integration constant, notice that the short time self diffusion coefficient D~ (0, 0) is related to the self frequency function, as given in eq.(6.178,179). Show that,

1 fot dt" g(t") D,(0, t) - ~-

Substitute the above expression for g, and perform an interchange of order of integration, as was also done in exercise 2.1, to obtain eq.(6.196) for D,(0, t).

6.11) Depolarization of light by scattering The scattering amplitude B of an optically homogeneous, thin and long

rod is proportional to (see subsection 3.10.2 in chapter 3 on light scattering),

Consider orientations fi of the rod in the xz-plane spanned by the polarization directions rio and fi, of the incident and scattered radiation (see the sketch of the experimental set up in fig.6.18). Verify that the depolarized scattered intensity is proportional to,

rio" B . fi, ,-~ Ae sin{O} cos{O},

with O the angle between fi and the z-axis. Maximize this expression with respect to O.

(The answer is O = 45 ~ Rods with such an orientation contribute most to the depolarized scattered intensity).

6.12) Orientational relaxation Consider an assembly of interacting rods which are oriented along the

z-axis at time t - 0. The average time dependent orientation of a single rod (rod 1, say) is equal to,

> - d,a t ) .

The orientation I~i 1 c a n be expressed in terms of spherical harmonics (see the analogous expression for fi2 of rod 2 below eq.(6.244)). We need not consider the x- and y-component of < fil (t) >, since these are zero by symmetry of

{a~_,(t)}

0.6 - D So 0 0.5 -

0.4

0.2

. . . . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . . 0 0.2 0.4 0.6 D~ t 1.0

Figure 6.24" The z-component of (0 l(t)> as a function of D~t for several values of ~qo.

the problem under consideration (performing the calculation for these components, you will encounter integrals with respect to ~ which vanish). Use the expression (6.243) for the pdf P(fix, t) together with the orthogonality relation (6.261) for spherical harmonics, to show that,

[ /o ] < ~ lx ( t )> - e x p { - 2 D ~ t } + ~D~ dt"y~o(t')exp{-2D~(t - t')} ~ ,

with ea - (0, 0, 1) the unit vector along the z-direction. Evaluate ~,~ 0 in mean field approximation for rods with hard-core interactions to obtain,

3 ..... k/1 exp{-4D~t} 71o - - 2 D L 2 ~ exp{-4D~t} -

Conclude that,

< tl l ( t ) > -- e xp{ - -2Or t } + -~ ~ G(Or t ) e3,

with,

G ( z ) - - - ~ ~- exp{-2z} d x e x p { - 2 x } ~ / 1 - e x p { - 4 x } .

F u r t h e r R e a d i n g 437

This function is negative for z - D~t > 0, so that the orientational relaxation is faster due to hard-core interactions. The z-component of the orientation is

z: plotted in fig.6.24 for various values of ~ .


The data in fig.6.8 on short-time self diffusion of hard-sphere colloids are taken from,

�9 P.N. Pusey, W. van Megen, J. Phys. (Paris) 44 (1983) 285. �9 W. van Megen, S.M. Underwood, J. Chem. Phys. 91 (1989) 552. �9 R.H. Ottewill, N.St.J. Williams, Nature 325 (1987) 232.

The data in fig.6.10 on short-time collective diffusion of a hard-sphere colloid are taken from,

�9 M.M. Kops-Werkhoven, H.M. Fijnaut, J. Chem. Phys. 74 (1981) 1618. The data in fig.6.11 on the hydrodynamic mobility function are taken from,

�9 A.P. Philipse, A. Vrij, J. Chem. Phys. 88 (1988)6459. The data in fig.6.13 on long-time self diffusion are taken from,

�9 W. van Megen, S.M. Underwood, J. Chem. Phys. 91 (1989) 552. �9 A. van B laaderen, J. Peetermans, G. Maret, J.K.G. Dhont, J. Chem.

Phys. 96 (1992) 4591. �9 A. Imhof, J.K.G. Dhont, Phys. Rev. E 52 (1995) 6344.

Dynamic light scattering measurements on diffusion are also described in, �9 P.N. Pusey, J. Phys. A: Math. Gen. 11 (1978) 119. �9 M.M. Kops-Werkhoven, C. Pathmamanoharan, A. Vrij, H.M. Fijnaut, J.

Chem. Phys. 77 (1982) 5913. �9 M.M. Kops-Werkhoven, H.M. Fijnaut, J. Chem. Phys. 77 (1982) 2242. �9 W. van Megen, R.H. Ottewill, S.M. Owens, P.N. Pusey, J. Chem. Phys.

82 (1985) 508. Depolarized light scattering by optical anisotropic spheres can be used to study "self motion" of particles in concentrated suspensions (fluid or crystalline). See,

�9 R. Piazza, V. Degiorgio, Phys. Rev. Lett. 67 (1991) 3868.

Batchelor has been the first to rigorously calculate the first order concentration

438 Further Reading

dependence of various diffusion coefficients. See, �9 G.K. Batchelor, J. Fluid Mech. 52 (1972) 245, 74 (1976) 1,131 (1983)

155 (a corrigendum to this paper is in the J. Fluid Mech. 137 (1983) 467). Calculations of this kind with a different mathematical flavour can be found in,

�9 B.U. Felderhof, J. Phys. A 11 (1978) 929. �9 B.U. Felderhof, R.B. Jones, Faraday Discuss. Chem. Soc. 76 (1983)

179. �9 B. Cichocki, B.U. Felderhof, J. Chem. Phys. 89 (1988) 1049, 94 (1991)

556. Self diffusion is also discussed in,

�9 M. Venkatesan, C.S. Hirtzel, R. Rajagopalan, J. Chem. Phys. 82 (1985) 5685.

�9 T. Ohtsuki, Physica A 110 (1982) 606. Diffusion in binary mixtures is discussed in,

�9 R.B. Jones, Physica A 97 (1979) 113. An extensive overview of literature concerning diffusion of spherical Brow- nian particles and an outline of theoretical approaches concerning that subject can be found in,

�9 P.N. Pusey, R.J.A. Tough, J. Phys. A 15 (1982) 1291, Faraday Discuss. Chem. Soc. 76 (1983) 123.

�9 R.J.A. Tough, P.N. Pusey, H.N.W. Lekkerkerker, C. van den Broeck, Mol. Phys. 59 (1986)595.

�9 J.M. Rallison, E.J. Hinch, J. Fluid Mech. 167 (1986) 131. �9 R.B. Jones, P.N. Pusey, Annu. Rev. Chem. 42 (1991) 137.

The effect of three body hydrodynamic interaction on diffusive and rheological properties are discussed in,

�9 C.W.J. Beenakker, P. Mazur, Physica A 126 (1984) 349. �9 P. Mazur, Far. Discuss. Chem. Soc. 83 (1987) paper 3. �9 A.J.C. Ladd, J. Chem. Phys. 88 (1988) 5051. �9 H.J.H. Clercx, The Dependence of Transport Coefficients of Suspen-

sions on Quasitatic and Retarded Hydrodynamic Interactions, Thesis, TU Eindhoven, The Netherlands.

The experimental result in fig.6.14 is taken from, �9 Y.D. Yan, J.K.G. Dhont, Physica A 198 (1993) 78.

Further Reading 439

This reference also contains an overview of the experimental work and computer simulations that have been done one the effect of shear flow on colloids, with an equilibrium fluid-like or crystalline structure. An extensive overview on sheared colloids with a crystalline equilibrium structure is,

�9 B.J. Ackerson, J. Rheol. 34 (1990) 553. The theoretical approach in section 6.8 to describe the shear induced deformation of the static structure factor in dilute suspensions is largely taken from,

�9 J.K.G. Dhont, J. Fluid Mech. 204 (1989) 421. The numerical solution of the two particle Smoluchowski equation for hard spheres in shear flow is discussed in,

�9 J. Blawzdziewicz, G. Szamel, Phys. Rev. E 48 (1993) 4632. Our expression (6.143) for the distortion of the structure factor is quite similar (but not exactly equal to) a result derived by Ronis on the basis of a "fluctuating diffusion equation", in,

�9 D. Ronis, Phys. Rev. A, 29 (1984) 1453, Phys. Rev. Lett. 52 (1984) 473. An alternative approach towards the calculation of the structure factor distortion, starting from an equation of motion with a single wavevector independent relaxation time, can be found in,

�9 J.E Schwarzl, S. Hess, Phys. Rev. 33 (1986) 4277. In fact, our equation of motion (6.141) is quite similar to the equation proposed in the above paper, except that the corresponding relaxation time in eq.(6.141) is wavevector dependent (,-~ k-2). For the calculation of the shear viscosity in the zero shear limit, it is sufficient to calculate the linear response distortion (Sa (K) in eq.(6.144)). Since the width of the boundary layer vanishes in the zero shear limit, it is sufficient to use the linear response result in integrals which represent the effective viscosity. The (numerical) evaluation of the linear response result for hard-core interactions, including hydrodynamic interaction, can be found in,

�9 G.K. Batchelor, J. Fluid Mech. 83 (1977) 97. �9 W.B. Russel, A.E Gast, J. Chem. Phys. 84 (1986) 1815. �9 N.J. Wagner, W.B. Russel, Physica A 155 (1989)475. �9 N.J. Wagner, R. Klein, Coll. Polym. Sci. 269 (1991) 295.

The main part of section 6.9 on memory functions is taken from, �9 B.J. Ackerson, J. Chem. Phys. 64 (1976) 242, 69 (1978) 684.

440 Further Reading

�9 W. Dieterich, I. Peschel, Physica A 95 (1979) 208. The weak coupling approximation (6.194) for the long-time self diffusion coefficient for a long ranged screened Coulomb pair-interaction potentials was first derived in,

�9 J.A. Marqusee, J.M. Deutch, J. Chem. Phys. 73 (1980) 5396. FRAP experiments on colloids with a very long ranged pair-interaction potential are described in,

�9 W. H~rtl, H. Versmold, X. Zhang-Heider, Ber. Bunsenges. Phys. Chem. 95 (1991) 1105. The Smoluchowski equation for two particles with hard-core interaction and without hydrodynamic interaction can be solved exactly. To leading order in concentration, memory effects may be analysed on the basis of this exact solution. The exact solution is derived in,

�9 S. Hanna, W. Hess, R. Klein, Physica A 111 (1982) 181. �9 B.J. Ackerson, L. Fleishman, J. Chem. Phys. 76 (1982) 2675.

Experiments on the long-time tail of the mean squared displacement can be found in,

�9 G.L. Paul, P.N. Pusey, J. Phys. A : Math. Gen. 14 (1981) 3301. �9 M.H. Kao, A.G. Yodh, D.J. Pine, Phys. Rev. Lett. 70 (1993) 242.

These papers also contain many references to theoretical work on long-time tails.

A treatment of the mathematical theory of special functions can be found in, �9 G. Arfken, Mathematical Methods for Physicists, Academic Press, Lon-

don, 1970. A more applied treatment can be found in,

�9 J.D. Jackson, Classical Electrodynamics, John Wiley & Sons, New York, 1975.

Particularly clear texts on singular perturbation theory, including many exam- pies, are,

�9 C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scien- tists and Engineers, McGraw-Hill, New York, 1978.

�9 A.H. Nayfeh, Introduction to Perturbation Techniques, John Wiley & Sons, New York, 1981.

�9 E.J. Hinch, Perturbation Methods, Cambridge Press, Cambridge, 1991.

Molecular dynamics simulation results for liquid argon on the non-Gaussian

Further Reading 441

behaviour of particle displacements can be found in, �9 A. Rahman, Phys. Rev. A 136 (1964)405,

and on charged colloids in, �9 K. Gaylor, I. Snook, W. van Megen, J. Chem. Phys. 75 (1981) 1682.

Experiments on the non-Gaussian behaviour of charged and hard-sphere colloids, respectively, are described in,

�9 W. van Megen, S.M. Underwood, I. Snook, J. Chem. Phys. 85 (1986) 4065.

�9 A. van Veluwen, H.N.W. Lekkerkerker, Phys. Rev. A 38 (1988) 3758.

Experimental work on the validity of the "effective medium approach" discussed in exercise 6.5, up to large concentrations, can be found in,

�9 A. Imhof, A. van Blaaderen, J. Mellema, J.K.G. Dhont, J. Chem. Phys. 100 (1994) 2170.

The book of Doi and Edwards contains three chapters on the dynamics of rigid rods. For larger concentrations, Smoluchowski equations with "effective diffusion coefficients" are used to evaluate the dynamics for interacting rods,

�9 M. Doi, S.F. Edwards, The Theory of Polymer Dynamics, Clarendon Press, Oxford, 1986. The book of Berne and Pecora contains a treatise of rotational diffusion and dynamic light scattering,

�9 B.J. Berne, R. Pecora, Dynamic Light Scattering, John Wiley, New York, 1976. See also,

�9 S.R. Arag6n, R. Pecora, J. Chem. Phys. 82 (1985) 5346. The original papers where the isotropic nematic phase transition in systems of long and thin hard rods is presented are,

�9 L. Onsager, Phys. Rev. 62 (1942) 558. �9 L. Onsager, Ann. N.Y. Acad. Sci. 51 (1949) 627.

Chapter 7

SEDIMENTATION

443

444 Chapter 7.

7.1 Introduction

Sedimentation is the phenomenon that Brownian particles attain a certain velocity under the action of an external field. This translational velocity is referred to as the sedimentation or settling velocity. The most common example of an external field is the earth's gravitational field. For small particles, the sedimentation velocity in the earth's gravitational field is very small, and sedimentation can only be observed by artificially increasing the gravitational field by means of centrifugation. The sedimentation velocity evidently depends on the mass and size of the Brownian particles, so that a measurement of the sedimentation velocity may be used for characterization. The difference in sedimentation velocity for particles of different mass and size may also be exploited to separate different species of Brownian particles.

For larger concentrations the sedimentation velocity is affected by interactions. This can be used to characterize the pair-interaction potential via the measurement of settling velocities. In section 7.2, the concentration dependence for hard-sphere interactions and long ranged repulsive interactions is discussed, and a qualitatively different settling behaviour is found for these two systems (sedimentation of sticky spheres and superparamagnetic particles is discussed in exercises 7.2-4). Both hydrodynamic and direct interactions are essential ingredients for predicting settling velocities of interacting Brownian particles. The major problem in the theory of sedimentation of interacting Brownian particles is the occurrence of divergent ensemble averages, which arise due to slow decrease of expressions for hydrodynamic interaction functions with increasing interparticle distance. The hydrodynamic influence of the walls of the confining container must be taken into account to resolve these convergence problems.

One of the important points is the existence of backflow, which is due to the above mentioned hydrodynamic influence of the walls of the container. In the laboratory coordinate frame, the volume flux of colloidal material through a cross sectional surface area perpendicular to the sedimentation velocity is always compensated by fluid flowing in opposite direction. The total volume flux must be zero. As discussed in subsection 4.7.1, the backflow may be considered homogeneous over distances small compared to the size of the sample container and at the same time large compared to the average distance between Brownian particles. In the statistical mechanical treatment of sedimention we analyse a large subgroup of Brownian particles within the container for which the backflow may be regarded uniform (see also fig.4.7).


On that local scale, the zero net volume flux condition requires that (see eq.(4.118)),

qp u, = v , , (7.1)

1 - ~

where v, is the sedimentation velocity, u~ is the local backflow velocity, and is the fraction of the total volume that is occupied by colloidal material, the

volume fraction of Brownian particles. Clearly, backflow tends to decrease sedimentation velocities, more so at larger volume fractions. Although the fluid backflow may be considered constant on a local scale, allowing statistical mechanical analysis for a uniform backflow, it certainly varies significantly from point to point over distances comparable to the size of the container. More about this non-uniformity of the fluid backflow can be found in section 7.3.

When sedimentation is allowed to proceed over an extended period of time, the so-called diffusion-sedimentation equilibrium is established. Con- centration gradients then exist, such that sedimentation is counter balanced by gradient driven diffusion. This type of equilibrium in an external field is discussed in section 7.4, and section 7.5 is concerned with the dynamics of sediment formation from an initially homogeneous suspension.

Sedimentation at infinite dilution

Consider a very dilute suspension, in which the average distance between Brownian particles is so large that they do not interact with each other, not by direct interactions nor hydrodynamically. In the stationary state, each Brownian particle attains a velocity (on average), such that the corresponding friction force with the solvent precisely compensates the external force F ~t that acts on that Brownian particle. The friction force for a single spherical Brownian particle is equal to 67rr/oav, (see chapter 5), with 7/0 the viscosity of the solvent and a the radius of the sphere. Hence,

1 0 ~ezt v, - - - - - - - . (7.2) 6~'~oa

The superscript "0" is added to the sedimentation velocity to indicate that this is the velocity at infinite dilution, where interactions are absent. Suppose that the external field is due to the earth's gravitational field. The external force is then proportional to the mass of the Brownian particle, corrected for

4 4 6 Chap ter Z

buoyancy, and is easily seen to be given by,

47r 3 F ~ t - g --~-a (pp - p f ) , (7.3)

with pp and p f the specific mass of the colloidal material and the fluid, respectively, and 9 - I g 1- 9.8 m/s 2 the earth's acceleration. Using this in eq.(7.2), we find that,

2a 2 0 v~ - g -~-~o (pp - p f ) " (7.4)

The sedimentation velocity thus varies with the radius a of the Brownian particles as ,,~ a 2. Larger particles (with identical specific mass) sediment faster than smaller particles. When particles are typically smaller than about a = 10 - 100 n m , sedimentation velocities in the earth's gravitational field are very small, and sedimentation experiments must be performed by centrifugation. The above formula still apllies, except that the acceleration 9 is now the centrifugal acceleration w 2 l, with w the angular velocity of the centrifuge and I the distance of the container from the center of rotation. The ratio of the sedimentation velocity and the accelaration of the external field,

S - Iv, ] / w21 , (7.5)

is independent of the acceleration, and is commonly referred to as the sedi- m e n t a t i o n coef f ic ient .

7.2 Sedimentation Velocity of Interacting Spheres

In this section we consider a monodisperse suspension of spherical Brownian particles. The sedimentation velocity as a function of concentration is established for two different systems" spheres with hard-core interaction and with very long ranged repulsive interaction. Qualitatively different sedimentation behaviour for these two systems is predicted, in accord with experiment.

A formal expression for the sedimentation velocity follows immediately from eq.(4.124) by ensemble averaging the velocity vi of a Brownian particle i,

N v, - < vi > - u, + y~ < Dij" -13 [V, .~ ] - V,.j In{P}] >

j=l

(7.6) \ ] j--1

7.2. Sedimentation of Spheres 447

Explicit expressions for the microscopic diffusion matrices Dij, which describe hydrodynamic interaction between the Brownian particles, are derived in chapter 5 on hydrodynamics, including three body contributions. The total potential energy �9 of the assembly of N Brownian particles will be specified later, when explicit expressions are derived for particular pair-interaction potentials.

There are two terms on the right hand-side of eq.(7.6) to be distinguished. The middle term is the average velocity that each Brownian particle attains due to the force - [V~] - kBTV In{P}, which is zero when the pdf attains its equilibrium form ,~ exp{- f l~ }. This term is only non-zero when the pdf is distorted due to the external force. Such a distortion gives rise to a force on the particles which tends to drive the system back to equilibrium. The last term is simply the average velocity that each particle would attain when it experiences only the external force F ~t. The sedimentation velocity is thus simply equal to u, + ~jS1 ( flDij �9 Fj >, with Fj the sum of the two above mentioned forces.

The brackets < . . . > denote ensemble averaging with respect to the probability density function (pdf) P of the position coordinates of the Brownian particles. The first problem to be considered is therefore the evaluation of that pdf, which is the subject of the following subsection. In subsection 7.2.2, an explicit expression for the sedimentation velocity valid up to linear order in concentration is established. Subsections 7.2.3 and 7.2.4 are concerned with the explicit calculation of the concentration dependence of the sedimentation velocity for spheres with hard-core interaction and very long ranged repulsive interaction, respectively.

7 .2 .1 Probability Density Functions (pdf's) for Sedimenting Suspensions

The probability density function (pdf) of the position coordinates of the Brow- nian particles, with respect to which ensemble averaged stationary sedimentation velocities must be calculated, is the solution of the stationary Smolu- chowski equation (4.125,126),

0 N o-7 P(ra , rN, t) - 0 = - E P )

i=1 N

= ~ V ~ . D~j. [~[V~.r + V~jP] (7.7) i,j=l

448 Chapter 7.

N N

- y~ V , , . Dij" [flF~*tP] - Y~ V~,. [u,P] . i,j=l j = l

The stationary solution P of this equation is translationally invariant when the number density of Brownian particles is position independent. P is then independent of the choice of the origin of the coordinate frame, so that it can be written as a function of differences of the position coordinates,

P - P(r2 - r l , r 3 - r2, r4 -- r 3 , ' " , r N -- r N - 1 ) �9

Since now, V ~ P - [V,~_~ - V~+~j]P for j # 1, N (with rij - ri - rj), and V,~ P = -V,2~ P, V~NP - V~ N N-a P ' we have,

N N - 1

E - + E j=l j=2

- V~+~ j] P + V, NN_IP - - O.

The last term in the Smoluchowski equation (the term ~ u,) is thus equal to 0. This means that a mere translation of the system as a whole with a constant velocity does not affect the pdf.

Let us now consider the term,-~ F ~t in the Smoluchowski equation. On the pair-level, where only two particles interact simultaneously, the microscopic diffusion matrices D ij are functions of r i - - r j only. Hence, with N - 2,

�9 [ ] - (r21 V~, Dij" f l F ~ P - D l l D1 2 ~- D21 -]- D 2 - - - r~ l ) i,j=l =~0

q- n " D I 1 -q- V r2 " D 2 2 q- V r l " D 1 2 + Vr2 " D21 �9 - - 0

=0 =0

since Dl l = D n and D~2 - D21. Therefore, the term ,-~ F ~t in the Smolu- chowski equation is equal to 0 also. Thus, for concentrations where events of simultaneous interactions between three and more particles are insignificant, the Smoluchowski equation reduces to that for the equilibrium situation. Sedimentation does not affect the equilibrium pdf for a dilute monodisperse suspension, where only two body hydrodynamic interaction needs be considered.

For larger concentrations, where three particles may interact simultaneously, the term ~ F ~t in the Smoluchowski equation (7.7) has an additional


contribution ,-~ D!~ ), arising from three body hydrodynamic interaction (see subsection 5.12.5). The additional contribution is,

3 Vr~

i,j=l

�9 D ! ~ ) �9 F ~ * t p =

[~7rl "e~ 3)-~ ~Trl" D~ 3) "~- ~7rl " D(3)13

+ V~ �9 D ~ ) + V ~ . D ~ ) + V~ " D(3)23

+ Vr3" D(~)+ Vr3" D ~ ) + V,-3" D(~)] " F ~ t P

+ [-Di ) - ) + Di )+ ) +

To leading order in the inverse distance expansion of the microscopic diffusion matrices, the divergences here are all zero due to incompressibility of the solvent (this may also be verified by direct differentiation of the explicit expressions (5.98,100) for these leading order approximations). From these explicit expressions it is also seen that the last two terms on the right hand-side do not vanish (for example, D~2 ) - D~ ) T ~ D~), where "T" stands for "the transpose of"). The conclusion is thus that beyond the pair-level pdf's are affected by the external force due to hydrodynamic interaction. This is due to the fact that three or more particles attain different velocities as a result of their simultaneous hydrodynamic interaction, depending on their relative positions. In contrast, two particles attain equal velocities when no other particles interfere hydrodynamically. When three or more body hydrodynamic interaction is relevant, the pdf with respect to which ensemble averages are to be calculated differs from the equilibrium pdf.

For particles with a very long ranged repulsive pair-interaction potential, the relative distance between the particles is large compared to the size of their hard-cores, even for concentrations where higher order direct interactions are of importance. For such systems, hydrodynamic interaction of more than two particles simultaneously is insignificant. In that case the pdf is the equilibrium pdf, also for concentrations where higher order direct interactions are important. Sedimentation in such a system is discussed in subsection 7.2.4.

There are two circumstances where the pdf differs from the equilibrium pdf, even on the pair-level. When the Brownian particles experience different external forces, the above arguments fail. This is the case, for example, when the suspension is subjected to a homogeneous external field and the Brownian

450 Chapter 7.

particles do not have the same size. Such polydispersity effects are not considered here. Secondly, the pdf is not translationally invariant when the number density of Brownian particles varies with position, such as in a sedimentation- diffusion equilibrium and during sediment formation, which are discussed in sections 7.4 and 7.5, respectively.

7.2.2 The Sedimentation Velocity of Spheres

In this section, an expression for the sedimentation velocity of spheres up to linear order in qo is derived (~ is the volume fraction of colloidal material). To that order, only two body hydrodynamic interaction needs be considered, so that the pdf may be taken equal to the equilibrium pdf ,-~ exp{- f l~ }, as discussed in subsection 7.2.1. The middle term in eq.(7.6) for the sedimentation velocity is zero in this case. The last term in that formal expression, however, is divergent, due to the Rodne-Prager contribution to the microscopic diffusion matrices. As discussed in section 5.10 in the chapter on hydrodynamics, the Rodne-Prager contribution constitutes the first two terms in the expansion of the microscopic diffusion matrices with respect to the reciprocal distance between the particles (see eqs.(5.64,65)). This leading contribution to Dij is found with the total neglect of reflections of the fluid flow fields generated by the translating spheres, as if these spheres were alone in an unbounded fluid. The reflection contributions are calculated in section 5.12, and are found to vary asymptotically for large distances like ,-~ 1/r 4 (see eqs.(5.84,95)). They do not give rise to convergence problems. The Rodne-Prager contribution, on the other hand, varies asymptotically like ,-~ l / r , and tends to zero too slow to assure convergence. The divergence does not occur when the finite extent of the container is taken into account. The walls of the container do have an effect on the overall fluid flow in such a way that ensemble averages are finite, as they should be (the effects of the walls of the container on the overall fluid flow is considered in more detail in section 7.3). One way to cope with these divergences, without explicitly considering the walls of the container, is to subtract ensemble averages which show the same kind of spurious divergence, but for which, from physical reasoning, their finite value is know a priori. In this way divergent ensemble averages are identified with a priori known finite valued quantities, and wall effects are corrected for.

Let us decompose the sedimentation velocity as given in eq.(7.6) into two parts : the Rodne-Prager contribution, which is the divergent contribution, and

the remaining finite terms,

o Vs -- Us-I-V s-l- < R l j > "~F ~xt+ < A D a / > -/3F ezt , j=2 j=l

(7.8)

0 ~ext where we have chosen i - 1, and where v, - /67r~oa is the sedimentation velocity without interactions. The matrices R~j comprise the Rodne-Prager contribution as given in eq.(5.64,65). For j - 1, eq.(5.64) for the Rodne- Prager contribution gives rise to the term v ~ while for j # 1, { ( )3 }

__ 1 Z [ i - 3f"lj~'lj] , j r 1, 3 a [i + f"ljf"lj] + 7 Rl j - Do -4 rlj rlj (7.9)

w i t h r l j - rl - r j , and I ' l j - - rljlrlj. Furthermore, AD~j is D l j with the Rodne-Prager contribution subtracted, and Do - kBT/67ryoa is the Stokes- Einstein diffusion coefficient. Explicit expressions for the reflection contribution AD/j are derived in section 5.12 in the chapter on hydrodynamics (see eqs.(5.84,95)),

N ADl l - Do Z ( A A s ( r l j ) r l j r l j -it- A B s ( r l j ) [ i - l"lj:f"lj] } , (7.10)

j=2 ADl j - D O (AA~(rij)~ij~ij + A B ~ ( r i j ) [ i - r l j r l j ] } , j r 1,(7.11)

where the mobility functions with the Rodne-Prager contribution subtracted are given by,

AA,(r , j ) - 154 (r-~j)

i (a) A B . ( r l j ) - 16 rlj

4 11 a

+T + o ((o/,,,)~),

+ o ((o/,, ,),), ( )7 a AA~(ris) - ~ + O ((air , j )9) ,

aBc(r l j ) - O ((al.,j)"). (7.12)

The divergences due to the Rodne-Prager contributions can be resolved as follows. Let u ( r l r a , r 2 , . . - , r s ) denote the velocity at a point r in the suspension. For positions r in the fluid, this velocity is the fluid flow velocity

4 5 2 C h a p t e r 7.

generated by the N sedimenting spheres, while for positions inside the core of a Brownian particle i, this is the velocity of that particle. In the laboratory reference frame, the average velocity at any point r in the suspension is equal to zero, that is, the net flux of volume is zero. Formally, the net zero volume flux condition in the laboratory reference frame reads,

- 

= f d r l . . . f d r g u ( r [ r l , . . . , r g ) P ( r l , . . . , r N ) . (7.13)

This ensemble average is actually divergent. The fluid flow field is the sum of fields generated by the spheres as if they were alone in an unbounded fluid, plus reflection contributions. The former contribution varies like ,~ 1/I r - rj [, leading to divergent contributions, for exactly the same reason that makes the sedimentation velocity (7.8) divergent.

Now let U ( r ] rx,. �9 �9 ,rN) denote the velocity that a sphere with its center at r attains, given that all the remaining N spheres have positions r x , . . . , rn . The sedimentation velocity can now be expressed as,

v , - f dr~ . . . f d r N U ( r l r l , . . . , r N ) P ( r x , . . . , r N I r ) , (7.14)

where,

P ( r l , . . . , rN I r) - P ( r x , - . . , rN, r ) / P ( r ) , (7.15)

is the conditional pdf for rx,. �9 �9 rN, given that there is a sphere at position r. As we will see shortly, the divergent Rodne-Prager terms (partly) cancel, when the zero net volume flux condition (7.13) is subtracted from eq.(7.8) for the sedimentation velocity. The integrals in both eqs.(7.8) and (7.13) are divergent, but their difference is well defined (except for a single conditionally convergent term, to which special attention will be given later on). The sedimentation velocity is therefore written as,

vs rl u ,rl ,r, (7.16)

The velocities u and U are now written as a sum of two contributions �9 the contribution that gives rise to divergences, and the remaining convergent reflection contributions. The reflection contributions to u and U are denoted

7.2. Sed imen ta t ion of Spheres 453

simply by A~, and Au, respectively. We can thus write,

N u(r [ r l , . . ' , rN) -- E u o ( r - r j ) + Au, for r in t he f l u i d ,

j= l

= v~ , f o r r in a core , (7.17) N

0 U ( r I r l , . . . , r N ) -- v s -~- E V o ( r - r j ) --~ A, U . (7.18)

j= l

The field uo(r - r j ) is the fluid flow velocity at r due to translational motion of sphere j , as if that sphere were alone in an unbounded fluid with a uniform backflow u~. According to eq.(5.36) this field is equal to (with r' - r - r j ) ,

{ 3 a [ 4 ~ r-75-] r ' r ' ] 41 ( a ) 3 [ ~ r ' r ' ] } uo(r') - I + + I - 375-1 �9 (v, - u , ) . (7.19)

The reflection contributions to the total fluid flow velocity is contained in A~,. Similarly, Uo(r - rj) is the velocity that a sphere with its center at r attains due to hydrodynamic interaction with sphere j up to the Rodne-Prager level, that is, with the neglect of reflection contributions. From the translational Fax6n's theorem (5.60), it follows that,

la2 2 (r rj) (7.20) U o ( r - r j ) - u o ( r - r j ) + ~ V~Uo - .

The first term on the right hand-side of Fax5n's theorem (5.60), with F h = , o in the expression (7.18) for U. Substitution - F ~t gives rise to the term v,

of eqs.(7.17,18,20) into eq.(7.16), and noting that -~ov. - u. + O(r gives the following expression for the sedimentation velocity,

o V' V" v~ - u~ + v~ + + + W + 0(~2) , (7.21)

with, f

V' - ~ / d r [ g ( r ) - l ] u o ( r ) ~ (7.22) J r >a

1 2t~ f 2 ( r ) , (7.23) V" = ~a drg(r) V~uo

which are the contributions without reflections, and with W the contribution due to reflections, which is the well behaved last term on the right hand-side of eq.(7.8),

w

454 Chapter Z

These expressions are most easily obtained after multiplying the right hand- side of eq.(7.16) with,

1 1 v f d r ( " ' ) - ~ [ f y t ~ i d d r ( . . . ) + f~o~ dr(." ") ] �9

Also note that the conditional pdf is zero when r is inside a core of a Brownian particle.

Substitution of the expression (7.19) for Uo into eq.(7.22) for V' , using 0 that v , - u, -- 1-qal v, - v, + O(qo), which follows from eq.(7.1), and

performing integration with respect to the spherical angular coordinates yields, with x - r/a,1

o f~ dx x [g(ax) 1] . (7.25) V ' - 3~v , >1

Similarly, substitution of the expressions (7.10-12) for the microscopic diffusion matrices into the expression (7.24), using that F ~t - 6~'r/oav, ~ + 0(~0), and integration with respect to the spherical angular coordinates yields,

f0 ~ W -- qpv ~ dx x2g(ax) {AA~(ax )+AAc(ax )+2AB, (ax )+2AB~(ax ) } .

(7.26)

In the above equations we introduced the pair-correlation function g(r) which is defined as (see also subsection 1.3.1 in the introductory chapter),

g([ r 1 -- r 2 J) -- W 2 / d r 3 / d r 4 . . . / d r N P(r l , r2, r3, r4 , - . . , rN). (7.27)

It is a "renormalized" pdf, such that g(r) ~ 1 for r ~ c~. Since the probability that hard-cores of Brownian particles overlap is zero, the pair- correlation function is zero for r < 2a. To leading order in concentration, the pair-correlation function is equal to the Boltzmann exponential of the pair-interaction potential V (r),

g(r) - e x p { - f l V ( r ) } . (7.28)

Notice that it is assumed here that the pair-interaction potential is spherically symmetric. The spherical angular integrations cannot be performed so easily

1Use that, fdi~ I - 47r~I and fd~fHc - ~-~[, with f di" is the integration with respect to the spherical angular coordinates, that is, the integration ranging over the unit spherical surface.


when the pair-interaction potential is anisotropic, such as for example for spheres with an embedded magnetic dipole moment in an external magnetic field. In exercise 7.4 you are asked to evaluate the sedimentation velocity for such a system.

Both V' and W are perfectly well defined, but V" in eq.(7.23) is a conditionally convergent integral in the sense that when integration with respect the spherical angular coordinates is performed first, the value of the integral is found to be equal to zero, while if integration with respect to the magnitude r of r is performed first, the integral does not exist, since the integrand varies asymptotically like r 2 x 1/r a (the factor r 2 originates from the Jacobian for the transformation to spherical coordinates). We will have to repeate the procedure of subtracting a similar divergent integral with a well defined physical meaning in order to remove the terms giving rise to convergence problems.

The total force at a given point in the suspension, averaged over the positions of all the spheres, is simply the gradient of the ensemble averaged pressure. There is a pressure drop due to the gravitational force field to which shear forces do not contribute on average. The local shear force per unit volume at some point r, given the positions of the N spheres, is equal to ~7~. 32aev(r I r l , . . . ' rN), with ~d,v the stress matrix without its isotropic part, that is, with the pressure contribution subtracted. This stress matrix is commonly referred to as the deviatoric stress matrix. Hence,

0 -- f drl..-f drN [~7r" ~dev(rlra,... ,rN)] /Z~(r,,--. rN). (7.29)

In the incompressible fluid the deviatoric stress matrix is given in eq.(5.6) with the omission of the pressure term,

I - { V , u ( , I - I ,

where the superscript "T" stands for "the transpose of". Taking the divergence from both sides, and using incompressibility (that is, ~7~. u - 0, see eq.(5.2)), yields, again for points r in the fluid,

V~. E] d~(r [ r l , " . , rN) -- r/oV 2 u(r J r1 , ' " , rN).

Substitution of the decomposition (7.17) for the fluid flow field and omitting the reflection contribution (which is already accounted for in the contribution W), eq.(7.29) yields,

456 Chapter 7.

where X]o d*v is the deviatoric part of the stress matrix in the core of a Brownian particle with its center at the origin. Gauss's integral theorem may be used to convert the last integral in the above equation to an integral ranging over the spherical surface OV ~ of a Brownian particle at the origin,

O - p [ f dr r /oV~uo(r )+Jo dSfd*'(r)] > a V o '

where fdev _ ~dev. ~ is the part of the force per unit area that a surface element of the core exerts on the fluid that is related to the deviatoric part of the stress matrix. This is not the total force, since the pressure forces are omitted. You are asked in exercise 7.1 to show that, for an isolated sphere in an unbounded incompressible fluid, the above integral over the deviatoric surface force is

0 equal to-47r~7oa(v,- u~). Since v, - u~ - v~ + O(~), the above equation can now be written, to leading order in concentration, as,

0 = /~ [ f dr r/oV~uo(r)-47rr/oav~ . > a

Subtraction of this result from the expression (7.23) eliminates the convergence problems and leads to,

= - f~ dr 1 z o la2~ [g(r) - 1] V~ uo(r) + , ~ v , . (7.30) V" 6 > a

The Laplacian of Uo is found from eq.(7.19) to be equal to,

u o ( r ) - - 3 a [ i _ 3rr ] 2 .

Integration with respect to the spherical angular coordinates in eq.(7.30) (see the footnote to eq.(7.25)) shows that the integral vanishes. Hence,

1 V " - ~qov ~ . (7.31)

This concludes the analysis of the divergence of the expression (7.23) for V".

Let us summarize the results obtained above for clarity and later reference. The sedimentation velocity is given in eq.(7.21), valid up to first order in concentration,

o V' V" v, - ( 1 - ~ ) v , + + + W + O ( ~ ~) (7.32)

7.2. Sed imen ta t ion o f Spheres 457

0 ), which follows from eq.(7.1). where it is used that u, - - ~ v, § O(qo 2 Furthermore, V', V" and W are given in eqs.(7.25,31,26), respectively (with

- ~ /~ ) ,

/ ,

v ' - 3 ~ o v ~ dx x [g(ax) - 1] , (7.33) J x >1

1 V / t ~ _ 0 - 2 ~ v , , (7.34)

~ x2 g ( a x ) { A A ~ ( a x ) + A A ~ ( a x ) + 2 A B ~ ( a x ) + 2 A B ~ ( a x ) } . W - ~ov~

(7.35)

The combination u, + V' + V " accounts for backflow effects and hydrodynamic interaction up to the Rodne-Prager level ("near-field hydrodynamic interaction"), while W accounts for hydrodynamic interaction beyond the Rodne-Prager level ("far-field hydrodynamic interaction").

In the following two subsections, the sedimentation velocity is evaluated explicitly for two special cases : for Brownian particles with hard-core interactions, and for particles with a strong and long ranged repulsive pair-potential.

7 . 2 . 3 Sedimentation of Spheres with Hard-Core Interaction

Clearly, the sedimentation velocity of interacting spheres depends on the kind of pair-interacting potential via the pair-correlation function g. The pair- correlation function is the Boltzmann exfponential (7.28) of the pair-interaction potential. In the present subsection, hard-sphere interactions are considered where the pair-potential Vh, is infinite when two cores of Brownian particles overlap, and is zero otherwise,

Vh~(r) -- 0 , f or r > 2 a ,

oo , for r < 2 a . (7.36)

The pair-correlation function ghs for hard-sphere interaction follows simply from the Boltzmann exponential (7.28),

gh, (r ) -- 1 , f o r r >_ 2 a ,

0 , for r < 2a. (7.37)

458 Chapter 7.

0.6

0.4

0.2

I

2,,

I I I

o_ _%0

0 0.1 0.2 0.3 2 0.5 Figure 7.1"

0 A plot o f R - Iv~ [ / [v~ I versus qa. The solid line is the prediction (7.40). The data are taken from �9 Buscall et a.1. (1982) (A), and Kops-Werldmven and Fijnaut (1982) (o). The dashed curve is according to eq.(7.90).

The contribution V' in eq.(7.33) is easily evaluated for this pair-correlation function, with the result,

9 V' = --2r .~ (7.38)

Substitution of the expressions (7.12) for the mobility functions into eq.(7.35) readily leads to,

o W = -1.441qpv, . (7.39)

The sedimentation velocity is thus found to be equal to,

o [1 6.441 + 0(r . V s - - V s - - qO (7.40)

More accurate expressions for the mobility functions, including higher order terms in a / r, yield a numerical value for the first order concentration correction o f -6 .55 instead of-6.441.

The prediction (7.40) for the sedimentation velocity is compared to experiments on hard-sphere like suspensions in fig.7.1. As can be seen, there is agreement up to volume fractions of about 0.05. Higher order interactions become important for larger volume fractions.

7.2. Sedimentat ion o f Spheres 459

7.2.4 Sedimentation of Spheres with very Long Ranged Repulsive Pair-Interactions

Suppose now that in addition to the hard-core interaction there is a very long-ranged repulsive pair-interaction. Typically this is the case for charged Brownian particles in a de-ionized solvent. The additional interaction potential is a screened Coulomb potential,

V(r) - A e x p { - x r } , for r >_ 2a . (7.41) r

The reciprocal of the parameter x is a measure for the range of the pair- interaction potential, and is referred to as the screening or Debye length (see also section 1.1 and exercise 1.9 in the introductory chapter). This is an accurate expression only for larger interparticle separations. For small interparticle separations this equation does not apply. However, since the pair-correlation function (7.28) is small for such short distances, an accurate expression for the pair-potential in this range is not essential to obtain accurate estimates for the sedimentation velocity.

For these very long ranged repulsive interactions, the Brownian particles tend to keep a maximum distance. The structure is a more or less ordered structure, where particles reside on "lattice sites", although in the fluid phase the thermally activated excursions around these sites is considerable. For these very long ranged potentials, a first order in volume fraction expansion as for hard-spheres does not make sense, since many particles interact simultaneously already at small volume fractions. In order to correctly predict the sedimentation behaviour of these "ordered" systems, we need to go beyond the simple Boltzmann exponential expression (7.28) for the pair-correlation function, which assumes simultaneous direct interactions of only two particles. The pair-correlation function 9(r) is now sharply peaked around the maximum possible interparticle distance. This interparticle spacing is related to the volume fraction as,

- a Ce qp-l/Z, (7.42)

where the dimensionless proportionality constant C'e depends on the particular structure of the "lattice" of the "ordered" structure.

The simplest approximation to the sharply peaked pair-correlation function would be a delta distribution centered at the peak position, left from which g - 0, and right from which g = 1 (see fig.7.2). Formally, such an approximation

460 Chapter 7.

Figure 7.2: The thin solid curve is a sketch of the pair-correlation function of a charged colloid with a large screening length ~-1 in comparison to the hard-core radius a, and the thick line represents the simple approximation (7.43).

reads,

e q.(7.z+3)

~J

@? r

r - a

g(r) - H(r - ~)[1 + a Cg 6(r - ~)] , (7.43)

where H(x) is the Heaviside unit step function (H - 0 for x < 0, and H - 1 for x > 0), and Cg is a dimensionless proportionality constant.

Since the distance between the Brownian particles is large, hydrodynamic interaction involving three or more Brownian particles is insignificant in comparison to two body interaction. The three body microscopic diffusion matrices D!~ ) w i t h / # j and DI~ ) vary asymptotically for large distances as (a/r)4 and (a / r ) r, respectively (see eqs.(5.98,100)). We assume here that such terms are negligible. This means that only the Rodne-Prager contribution to the mobility functions is of importance, so that AA~,~ and AB~,~ are negligible, and hence, W ~ 0.

Since we assume here that three and more body hydrodynamic interaction is insignificant, due to the large separations between the Brownian particles, the pdf is the equilibrium pdf, as discussed in subsection 7.2.1. Higher order hydrodynamic interaction causes the pdf to deviate from its equilibrium form.

It is now easily seen that the only remaining contribution is V' in eq.(7.32), which is equal to,

Vs o V I - v , + +

= v ~ [1 + 47ra/5 { - / < , . < ; r r + aC a / > / r r 6 ( r - ~ ) } + 0(~)]

- v, xC~ + 3C~Cg + O(~o) . (7.44)

"Ordering" is thus predicted to lead to a qo ~/a dependence of the sedimentation velocity.

7.3. Non-uniform Backflow 461

tn(1-R)

Figure 7.3"

-2

0

w

-2 -6 tn

0.6

0.2

0 002 h~ 0.0t~ 3"

_ o ln{~} (a). The straight line A plot o f In{1 - R} with R I v~ I/[v~ l, versus has a slope o f 1/3. A plot o f R versus ~ is given in (b). The dashed straight

3 2~1/3 line is the hard-sphere result (7.40), and the solid line is R - 1 - 7C e with Ce - 1.36. The colloidal system consists of silica particles in de-ionized ethanol. Data are taken from Thies-Weesie et al. (1995).

Experimental sedimentation data on charged colloids are shown in fig.7.3, both on a log-log scale and a linear scale. The initial slope of In { 1 - R}, with

0 R a short hand notation for I v, I / Iv~ I, versus ln{~} in fig.7.3a is indeed found to be equal to 1/3 to within experimental errors. In fig.7.3b, the same data are plotted on a linear scale, showing the enormous difference between the sedimentation behaviour of hard-spheres in eq.(7.40) (the dashed straight line) and of charged spheres with a long ranged pair-potential. Adding salt to the dispersion results in a decreasing screening length ~-1, thereby reducing the range of the pair-interaction potential. A smooth transition from the q;1/a_ behaviour to a q;1-behaviour of the sedimentation velocity is observed when adding salt. In the intermediate regime all kinds of exponents between 1/3 and 1 may be observed (see Thies-Weesie et al. (1995)).

7.3 Non-uniform Backflow

On a length scale that is large compared to the average distance between the Brownian particles and small compared to the width of the container, the solvent backflow that compensates the volume flux of colloidal material may be considered uniform on average. In the previous section, where explicit

462 Chapter 7.

expressions for the sedimentation velocity of interacting spheres are derived, the average backflow velocity u, is assumed to be position independent. In the present section, the non-uniformity of the backflow on the length scale of the container is analysed in an approximate manner.

The backflow velocity is non-uniform if, and only if, the ensemble averaged volume flow < u( r I r~,. �9 �9 rN > is non-uniform. The volume flow velocity is the velocity of a volume element in the suspension, ensemble averaged over the position coordinates of the Brownian particles, irrespective of whether that volume element is in the fluid or inside the core of a Brownian particle. In the sequel we shall consider the ensemble averaged volume flow rather than the backflow. The analysis of the previous sections applies on a local scale, where sedimentation velocities were calculated relative to this slowly spatially varying volume flow velocity.

The aim here is to establish non-uniformity of backflow rather than to derive equations which are very accurate and quantitatively correct. Only the lowest order hydrodynamically induced force moments will be taken into account, simplifying things considerably (hydrodynamic interaction is thereby described on the Rodne-Prager level). Within this approximation we derive an "effective" creeping flow equation for the ensemble averaged volume flow velocity. That effective creeping flow equation is subsequently solved for a parallel plate geometry, and the validity of assuming uniform backflow on a local scale is discussed.

The effective creeping flow equations

Subtraction of the volume flow velocity < u(r [ r~ , . . . , rN) > from the sedimentation velocity yields the sedimentation velocity relative to the coordinate frame in which the volume flow in zero : this relative sedimentation velocity is the velocity v, that is calculated in section 7.2. In reality, this ensemble averaged volume flow is position dependent. In order to calculate this position dependence, an effective creeping flow equation for the volume flow can be derived, which can be solved for simple geometries of the container which encloses the suspension.

The creeping flow equations for the fluid flow velocity reads (see the chapter on hydrodynamics, eqs.(5.2,20)),

V~p(r J r1 , . . . , rN) - - r/0V~u(r J r1 , . . . , rN) - - f ( r [ r l , . . . , rN), (7.45)

V~. u ( r l r ~ , . . . , rN) = 0, (7.46)

where V~ is the gradient operator with respect to the position r in the fluid, p

7.3. Non-uniform Backttow 463

is the pressure and f is the total force per unit volume that is exerted on the fluid at the position r. The above equations may be extended to apply also for positions inside the cores of the Brownian particles by suitably defining extensions of the fields u and p to within the cores. We do not specify these extensions explicitly, since we will not need them. The above creeping flow equations are simply averaged over the position coordinates of the Brownian particles, irrespective of whether r is inside a core or not, assuming that we suitably defined these appropriate extensions.

The force on the fluid is concentrated on the surfaces of the Brownian particles and on the walls of the container. Disregarding spatial variations of the hydrodynamic forces over the surfaces of the Brownian particles, the hydrodynamic force is equal to,

~ext N f ( r I r ~ , . . . , rg ) -- 47re 2 Z t~(I r -- rj]--a), (7.47)

j=l

with ~ the 1-dimensional delta distribution. Let P ( r ) and U(r ) denote the ensemble averaged pressure and volume

velocity,

-- / d r1 . . . / d rNP(r l , . . . , rN)p ( r [ r l , . . . , rN) , (7.48) P ( r )

and similarly for the velocity. For identical Brownian particles, ensemble averaging of eq.(7.45), using eq.(7.47), yields,

Fext V~P(r) - yoV~U(r) - N 4ra------ ~ < ~(1 r - rp l - a ) > ,

where rp is the position coordinate of an arbitrary Brownian particle. To lowest order in concentration, the ensemble average can be calculated with respect to the pdf for rp which is equal to 1/V for positions further away from the container wall than the radius a of the spheres (V is the volume of the container), and equal to zero otherwise. The interaction between the Brownian particles and the wall is thus assumed a hard-core interaction. The above effective creeping flow equation then reduces to,

F~ t fd - - - d r , t~(I r - r , I - a ) V~P(r) ~?0V~U(r) P 4 ra 2 (rp)>a

~ext

-- fi47ra 2 Jo~-v~ dSp H(d(rp) - a ) ,

464 Chapter Z

Figure 7.4: The integration range with respect to the particle coordinate r v. The minimum va/ue of the spherical angular coordinate 0 is equal to em= arccos{(d(rp) -- a)/a}.

2 a

f /

;? 0,

dtr~)>a

where we introduced the smallest distance d(rv) between r v and the wall of the container, H(z) is the Heaviside function (H - 1 for x < 0, H - 1 for x > 0), and 0 E is the spherical surface with radius a with its center at r. The Heaviside function in the surface integral of the above expression limits the range of integration to the region where d(rv) > a. The surface integral is easily evaluated with the help of fig.7.4. The following explicit effective creeping flow equations are thus found,

F ~ t [ (d(r ) 1)] (7.49) V , P ( r ) - ~oV~U(r) - q~ ~ a a 1 + H(2a-d(r)) 2--~-- '

V , . V(r) - 0. (7.50)

The last of these equations follows trivially from eq.(7.46). These effective creeping flow equations are identical to those for an ordinary fluid on which an external force,

f~ t ( r ) r'~' [1 + H(2a -d(r))(d(r) (7.51)

per unit volume acts. This is a constant in the bulk of the suspension (where d(rv) > 2a), but varies with position in the neighbourhood of the walls, as a consequence of deficiency of colloidal material near the walls. The external

11 and force is sketched in fig.7.5 for the case of two parallel plates at z - - ~ z = -t-�89 l, extending to infinity in the x- and y-directions.

The boundary condition for the effective flow velocity U follows simply by averaging the stick boundary condition u(r I r~,-. �9 rN) -- 0 for r on the walls O W of the container,

U(r) - 0 , for r E O W . (7.52)

7.3. Non-uni form Backf low 465

Figure 7.5"

J

I

K -~c

F ::ext

I f~,,t 1 g

i

! I

q -F I~2L

The external force in eq.(7.51) for the parallel plate geometry. The two plates are located at z - 4-�89 and extend to infinity in the x- and y-directions.

Solution of the effective creeping flow equations

The effective creeping flow equations are most readily solved for a geometry consisting of two parallel plates which extend to infinity in the x- and y-directions, and which are some finite distance I apart in the z-direction (see fig.7.5). From the symmetry of this problem it follows that the only non-zero component of the velocity is in the y-direction, which component is only depending on the z-coordinate. This component of the velocity is denoted as Uy(z). The incompressibility equation (7.50) is trivially satisfied. The components of the creeping flow equation (7.49) read,

0 P - 0,

Ox 0 2 F ~ t [ (d(r)

0 p _ 770 (z) - - - - - 1 + H ( 2 a - d ( r ) ) O y -ff ~z 2 Uy - qa ~ a 3 2 a

0 oZ P - O.

_1)]

Differentiation of the second equation with respect to z and using the last equation yields,

0 a 1 1 - ~ o - ~ V ~ ( z ) = 0 for - ~l + 2~ < z < - I - 2a

J F~t[ for 1 1 = ~P2a~a 3 ' - ~ l < z < - ~ l + 2 a ,

466 Chapter Z

I F ~ I for 1 1 = - q O 2 a 3 a3 ' ~ l - 2 a < z < ~l. (7.53)

It follows from these equations that within the bulk of the suspension Uy is a polynomial in z of order 2, while near the walls Uy is a polynomial of order 3. The appropriate length scale for the z-coordinate is 2a near the walls and 1 in the bulk. Since Uy is symmetric in z, the solution to the above equations is thus written as,

V~(z) (1) ~-/-Izl

Uo+Ux 2 2a

1 for - -~l < z

1 1 , for - - ~ l + 2 a < z < - ~ l - 2 a , (7.54)

~ + g 2 ( 2 l - l z [ ) 2 (l/-'zl) 3 2a +U3 7 2a ' (7.55)

1 1 1 z < - - ~ l + 2 a and - ~ l - 2 a < z < -~l.

The 6 as yet unknown coefficients must be determined through boundary conditions and continuity requirements. The boundary condition (7.52) implies that,

~l) - o V ~ ( z - +-~ (7.56)

The volume flux integrated across the gap between the two plates is equal to zero in the laboratory frame. This means that,

�89 gz U~(z) - o . �89 (7.57)

Furthermore, it follows from the creeping flow equation that the velocity and its l l + 2 a a n d z - 1 first two derivatives are continuous at z - - 7 71 - 2a. These

three continuity conditions and the conditions (7.56,57), together with the differential equation (7.53), lead to 6 equations for the 6 unknown coefficients in the representation (7.54,55) of the velocity. This set of linear algebraic equations is solved with a little effort. The solution is most naturally expressed in terms of the dimensionless distance Z - z / l and the small parameter e - 4a / l , which number measures size of the Brownian particles relative to the relevant linear dimension of the container. The solution of the creeping flow equations (7.49,50) for the parallel plate geometry is thus found to be equal to,

3 ) 18(1_ 1)z2] 758, ,v0, (1

7.3. Non-uniform Backflow 467

3 I I I J

2 ~'= " 0 1 '

1 i .5

U, 0 I

-1

- 2 . . . . . . . . . I . . . . . . . . . ! . . . . . . . . . I . . . . . . . . . I ......... -0.5 -0.3 -0.1 0.1 Z 0.3 0.5

Figure 7.6" 0 The volume velocity scaled on ~ [v~ [, for a parallel plate geometry, as given

in eq.(7.58,59) for e - 4a / l - O. 1 and 0.5 as a function o f Z - z / l .

1 1 for ~ { - 1 - b e } _ Z ~ ~ { 1 - e } ,

0 9 l - e + e 2 Iv, I - ~ ~ ( 1 - 2 1 z i ) (7.59)

9 ( 12 13 ) (1_21Z] ) 2 3 ] + 5 I - ~ + g - 5 ( 1 - 2 1 Z 1 ) 3 ,

1 1 1 for 21 _ < Z < 7 { - l + e } and ~ { 1 - e } < Z < - _ 2"

This solution is plotted in fig.7.6 for e = 4a/l equal to 0.1 and 0.5. The volume flow velocity is proportional to the volume fraction and the sedimentation velocity of a single non-interacting sphere. Clearly, an increase of both quantities leads to an increase of the back flow velocity. There is a small region of width ,~ e near the walls where there is strong variation of the flow velocity with position. In the bulk of the container, on the contrary, the velocity profile is smooth and parabolically shaped. For very large containers in comparison to the size of the Brownian particles (small e), the flow velocity

o does not tend to zero, but remains parabolic with a value for Uy/~ l v, [ equal

to 5 - 18 - - 3 near the walls and + 3/2 in the middle of the container. Non-uniform back flow is therefore always present, also for large containers.

468 Chapter 7.

In the previous section, sedimentation velocities were calculated relative to the volume fixed coordinate frame. The measured sedimentation velocity in the laboratory reference frame is therefore equal to the sedimentation velocity as calculated in the previous section plus the volume velocity as plottea in rig.7.6. Locat sedimentation velocities depend on the position in the conta/ner where the measurement is performed. A larger sedimentation velocity will be measured in the bulk of the container than near the walls. When these sedimentation velocities are averaged with respect to the position in the container, the resulting velocity is precisely equal to that calculated in the previous section, since the average volume velocity is zero (see eq.(7.57)). Common experimental methods, where the time dependence of position averaged optical properties are probed, measure precisely such a position averaged sedimentation velocity. The existence of non-uniform backflow therefore does not preclude experimental tests of the theoretical predictions made in the previous section. This justifies the comparisons in figs.7.1,3 of experimental and theoretical sedimentation velocities.

7.4 The Sedimentation-Diffusion Equilibrium

Consider a closed container as depicted in fig.7.7. Sedimentation occurs in the initially homogeneous suspension for large enough Brownian particles, the sedimentation velocity of which is considered in previous sections. Brownian particles collect at the bottom of the container due to sedimentation, where the density increases and a sediment is formed. After some time an equilibrium barometric height distribution is attained, where the sedimentation flux of colloidal particles is counter balanced by diffusion, which diffusion flux is driven by concentration gradients.

An essential difference with the sedimentation problem considered in previous sections is that there is no solvent backflow, since the net volume flux of colloidal material is zero, being the sum of the counter balancing sedimentation and diffusion fluxes. The relevant Smoluchowski equation which applies to the present situation is therefore eq.(7.7) with the solvent backflow velocity u, equal to 0,

N

i,j=l (7.60)

The solution of this equation is simply proportional to the Boltzmann expo-

7.4. Sedimentation-Diffusion 469

Figure 7.7" The sedimentation-diffusion equilibrium. Each black dot represents a Brownian particle.

nential,

�9 �9

�9 �9

�9 �9 �9 �9

, �9 �9

i 0 �9 ~ O~ : �9 ~ O ' I o o o o � 9 �9 � 9 el I - - - - O w �9 O O o U o I l " e ~ o o �9 o . % O o o o..oo o e o ol IOqW _BOWOIP- O e ~ o O , , O g l I o ~ 1 7 6 �9 e- e g . o e o o , e e e l

J J Ioo o m O W o . e o OO �9 ~.e I OOo$ �9 �9 o - e �9 �9 eooOe_on -~d i ~ - �9 �9 ~'. . . . . . . -~,'1 �9 �9 �9 o o O o Og

e Ol.~..t..~.~_%~ �9 o - e �9 - 0 e._~._j - ~ o

(7.61)

This pdf renders the term between the square brackets in the equation of motion (7.60) equal to O. Since the microscopic diffusion matrices multiply that term, hydrodynamic interaction is irrelevant for the equilibrium pdf, contrary to the pdf for the sedimentation problem considered in previous sections.

For non-interacting Brownian particles, where ~ = 0, the pdf (7.61) reduces to a product of single particle pdf's P(rj) , - exp{flF e=t. r d }, each of which is proportional to the macroscopic density p(rj),

M*g. r} p(r) ~ exp{flF ~ ' . r} - exp ~ ~ . (7.62)

In the last equation it is used that in the earth's gravitational field, the external force is given by eq.(7.3) (with M* - ~a3(pv - py) the mass of a Brownian particle corrected for buoyancy). The density varies exponentially with the distance from the bottom of the container in the direction of the external field. For interacting Brownian particles this barometric height distribution is modified by direct interactions.

7 .4 .1 B a r o m e t r i c H e i g h t D i s t r i b u t i o n for I n t e r a c t i n g

P a r t i c l e s

To obtain the extension of the barometric height distribution (7.62) to include interactions, the equation of motion for the one-particle pdf P~ (r) - f dr2. . , f dru P(r, r2 , - . . , ru) must be derived. This equation is actually

470 Chapter 7.

the equation of motion for the macroscopic density, since 191 - p /N (see also subsection 1.3.1 in the introductory chapter). The pdf P(r , r2 , . - . , rN) in eq.(7.61) satisfies the differential equation,

0 - fl[V~(I)]P + V ~ P - f lF~tP. (7.63)

This is nothing but the expression between the square brackets in the Smolu- chowski equation (7.60) with j taken equal to 1 and renaming r~ - r. The total potential energy (I) of the assembly of Brownian particles is now assumed pair-wise additive, that is, (I) is assumed to be a sum of pair-potentials V,

N

(I)(rl,... ,rN) -- ~ V(rij). (7.64) i , j = l i < j

Using the definition of the pair-correlation function g,

f dr3.. , f drN P(r, r', r 3 , . . . , rN) -- Pl(r)Pl(r ' )g(r , r ' ) , (7.65)

and assuming identical Brownian particles, integration of eq.(7.63) yields,

0 - - k s T V ~ ln{p(r)}-f dr' [V~V(Ir - r' I)] P(r')g(r,r')+ F~t. (7.66)

There are three contributions to be distinguished here. The first term on the right hand-side is the Brownian force at position r (see eq.(4.37)), the second term is the average direct force that particles exert on a particle at r, while the last term is the external force. These three forces cancel in equilibrium. Notice that for non-interacting particles, where V - 0, this equation reproduces the barometric height distribution (7.62).

To make further progress, the pair-correlation function g must be expressed in terms of the density. This can be done by making use of the fact that the pair-force V~V(I r - r' I) effectively limits the range of the r'-integration to a spherical region around r with a radius equal to the range Rv of the pair-potential. For small gradients in the density, such that the density varies linearly over distances of the order Rv, the following expression for the pair- correlation function may be used (with R - r - r'),

- 1 dg~q(RIp p(r)) x [V~p(r)] .R (7.67) g(r, r') - g q(RIp - p(r)) + dp -2 "

Here, g~q(RIp - p(r)) is the pair-correlation function for an equilibrium system with homogeneous density p = p(r). The closure relation (7.67) is

7.4. Sedimentation-Diffusion 471

nothing but a Taylor expansion of g around the local density at r (notice that 1 the density between the two positions r and r' is 7[V~p(r)]. (r - r') larger

than p(r)). Higher order terms are small when the density profile is smooth over distances of the order Rv. For such smooth density profiles, the density p(r') in the integral in eq.(7.66) may likewise be Taylor expanded as,

p(r') - p ( r ) + [V~p(r)]. R . (7.68)

Substitution of eqs.(7.67,68) into eq.(7.66) and performing the spherical angular integrations 2 yields,

4Ir 0 = -k=TV~ln{p(r)} + F~=t+ T [V~p(r)]

Xfo~176 1 dg~q(Rldp-p(r)) ] dR g~q(R[p - p ( r ) ) + ~p(r)

Since the local osmotic pressure H(p(r)) is equal to,

g(p(r))- kBTp(r)- 27rp 2 fo ~ R 3dV(R) -~ (r) dR d ~ g ~q(RIp- p(r)) , (7.69)

the above equation can also be written simply as,

0 - F ~ t - [V~ ln{p(r)}] dlI(p(r)) dp(r) " (7.70)

For density profiles which vary spatially in a non-linear fashion over distances comparable to the range of the pair-interaction potential, higher order derivatives of the density must be included in the second term on the right hand-side of this equation. The second term on the right hand-side is proportional to the diffusion flux induced by concentration gradients, while the first term is proportional to the sedimentation flux. In equilibrium the two fluxes counter balance each other.

In general eq.(7.70) is a non-linear equation in the density, since the osmotic pressure is a non-linear function of the density. A model for the density dependent osmotic pressure for a homogeneous equilibrium suspension must be specified in order to predict density profiles on the basis of eq.(7.70). In section 7.5, where the dynamics of sediment formation is considered, a numerical example of a barometric height distribution will be given.

2Use that VrV(R) - dV(R) 15 % and that the spherical angular integration over the unit dR

sphere of the dyadic product l~l~ is equal to ~ I , with l~ - R / R and I the unit matrix.

472 Chapter 7.

7.4.2 Why does the Osmotic Pressure enter Eq.(7.70) ?

For a molecular system, the expression on the right hand-side of eq.(7.69) is equal to the mechanical pressure (where the pair-correlation function now relates to interactions between the molecules). In the present case of a colloidal system we identified that expression with the osmotic pressure. The origin of such an identification is as follows. Consider two Brownian particles which are displaced relative to each other over a very small distance. The interaction potential is equal to the reversible work needed to accomplish that relative displacement. This work is a sum of two terms. First of all, there is the work related to the direct interaction between the two Brownian particles. Secondly, the solvent adjusts its structure so as to minimize its free energy in the field imposed by the two Brownian particles. This change of free energy adds to the work required for displacement of the two Brownian particles. The direct interaction of two Brownian particles is thus actually a free energy interaction as far as the second contribution is concerned. This free energy contribution to the pair-correlation function in the right hand-side of eq.(7.69) renders the osmotic pressure instead of the mechanical pressure. This can be shown thermodynamically by considering the driving force that a Brownian particle experiences in an inhomogeneous system without an external field. According to eq.(7.70) that driving force must be equal to,

F - - [V~ln{p(r)}] dlI(p(r)) _ 1 - - p(r---~V~II(p(r)). (7.71) dp(r)

Let us rederive this result on the basis of thermodynamic arguments. In equilibrium the chemical potential is a constant, independent of position. The driving force for diffusion is therefore equal to gradients in the chemical potential. Let/~,(r) and #B(r) denote the local chemical potential per solvent molecule and per Brownian particle, respectively. The force on a solvent and a Brownian particle are thus respectively equal to,

F, = -V~#~( r ) , (7.72)

FB = - - V r P B ( r ) , (7.73)

where a minus sign is added since the diffusion current is directed towards regions of lower chemical potential, so as to minimize the free energy. The two chemical potentials are not independent quantities �9 they are related by the Gibbs-Duhem relation (at constant mechanical pressure and temperature),

p(r)V~#B(r) + p,(r)V~#,(r) = 0 , (7.74)

7.5. Dynamics of Sediment Formation 473

with p,(r) the local number density of solvent molecules. Since the volume fraction of solvent plus Brownian particles adds up to unity, the two number densities are related as,

v~p~(r) + vBp(r) - 1, (7.75)

where v, and vB are the volume of a solvent molecule and a Brownian particle, respectively. What we are interested in here, is the velocity of the Brownian particles relative to the fluid. This relative velocity may be obtained by noting that any force per unit volume, acting on the solvent and the Brownian particles alike, does not produce a relative velocity, and furthermore, that the force per unit volume on the fluid is equal to F,/v,. Subtraction of this force from the actual forces on the fluid renders the fluid force free, resulting in a zero velocity of the solvent. The velocity of the Brownian particles relative to the fluid is thus obtained from the force per Brownian particle which is equal to,

VB F - FB F , . (7.76)

Vs

Use of eqs.(7.72-75) now yields,

1 F = V~#~(r). (7.77) p(r)v

Since the local osmotic pressure is equal to - ( # , ( r ) - #~ with #o the chemical potential of the pure solvent, this equation reproduces our earlier result in eq.(7.71), and confirms the identification of the right hand-side in eq.(7.69) with the osmotic pressure.

7.5 The Dynamics of Sediment Formation

In the previous section we have considered the density profile after sedimentation-diffusion equilibrium is reached. In the present section we discuss the transient density profiles which exist during sediment formation, starting from a homogeneous suspension.

When there is no equilibrium, there is a mismatch between the two forces in eq.(7.70). The non-zero current density J ( r , t) of colloidal material can then be written as,

J ( r , t) - velocity x p(r, t) - mobility x total force x p(r, t)

= M(p(r , t ) ) x [F~t - [V , ln{p(r,t)}]dII(p(r't))]xp(rdp(r) , t), (7.78)

474 Chapter Z

where M is a density dependent "mobility", that is, a "reciprocal friction coefficient". This current density can be decomposed into two contributions. The diffusion contribution is written as,

Jail(r, t) - -D(p ( r , t))V~p(r, t ) , (7.79)

where an "effective diffusion coefficient" D is introduced,

dn(p) D(p) - M(p) . (7.80)

dp

This is a generalized Einstein relation in the sense that it generalizes the Einstein relation Do - kBT/7 for non-interacting particles, which is derived in chapter 2 (see eq.(2.37)). The second contribution to the total current density of colloidal material is due to the gravitational force, giving rise to sedimentation,

J,~d(r, t) -- M(p(r, t)) p(r, t) F ~t . (7.81)

The equation of motion for the density now reads,

0 b7 p(r, t) -

- V ~ . J(r , t) (7.82)

[ dII(p(r, t))] V~- M(p(r, t)) -p ( r , t)F ~*t + [V~p(r, t)] dp(r) "

Without an external field this diffusion equation reduces to,

0 o-7 p(~' t) - v~ . [D(p(r, t))V~p(r, t ) ] , (7.83)

where eq.(7.80) for the effective diffusion coefficient is used. For very weak inhomogeneities, D(p(r, t)) ,,~ D(/5), with/5 - N / V the average density. In that case the above equation of motion becomes,

0 0-7 p(~' t) - D(~)v~p(r, t ) . (7.84)

The effective diffusion coefficient D(~) is therefore precisely the gradient diffusion coefficient that is introduced in section 6.2 on collective diffusion (see eq.(6.14)). In that section we indeed discussed systems with weak inhomogeneities, such that diffusion coefficients may be considered as if the system

7.5. Dynamics o f Sediment Formation 475

were isotropic. The effective diffusion coefficient D(~) can thus be measured, for example, by means of light scattering as the gradient diffusion coefficient in a homogeneous system.

The mobility M that is introduced in eq.(7.78) is by definition the proportionality constant between the sedimentation velocity and the total force on a Brownian particle (velocity - M ( p ) x total force). This is precisely the proportionality constant that we considered in previous sections for small concentrations. Since for non-interacting systems the Stokes-Einstein relation (7.80) reduces to Do - k B T / 7 - kBT/67rrloa, and the sedimentation velocity

o force/67rrloa, eq.(7.80) can also be written as, is then given by v, -

D(~) = I v, I 1 dlI(Z). (7.85) Do IvOl kBT d~

For hard-sphere suspensions the first order concentration dependence of the gradient diffusion coefficient is (see eqs.(6.107,111)),

D(fi) - Do {1 + 1.559~} , (7.86)

where qp is the volume fraction of Brownian particles. The numerical constant 1.559.-. is found with the use of the approximate expressions for the microscopic diffusion matrices derived in chapter 5 on hydrodynamics. Using more accurate expressions for these matrices yields a numerical value of 1.45-... Furthermore, for hard-sphere interaction and up to first order in concentration, we have,

dn(~) d~

= k B T { I + 8 ~ } . (7.87)

Substitution of these results into eq.(7.85) yields the first order concentration dependence of the sedimentation velocity,

0 Iv~ l - Iv, I[1 - 6.441 qp] , (7.88)

in accordance with the result found in subsection 7.2.3 in eq.(7.40).

The diffusion equation (7.82) describes the dynamics of sediment formation, provided that the density is smooth over distances of the order of the range of the pair-interaction potential. In fact, the time evolution of any smooth non-equilibrium initial density profile is described by eq.(7.82), like

476 Chapter Z

for example the expansion of a compact sediment. This equation may be solved (numerically) once an equilibrium equation of state is specified, that is, once the osmotic pressure as a function of the density for a homogeneous equilibrium system is specified, and once the concentration dependence of the mobility is known. Semi-emperical expressions for these density dependent quantities can be obtained by fitting to experimental data. More about such calculations and experimental observations of density profiles can be found in the references given at the end of the present chapter. Instead of going into these accurate calculations in detail, we discuss a numerical calculation of sediment formation in a hard-sphere suspension, which is simple but approximate.

A simple numerical example of sediment formation

Consider a homogeneous hard-sphere suspension that settles under gravity along the z-direction, where the top of the container is at z = L while the bottom is at z - 0. The intermediate density profiles, before sedimentation- diffusion equilibrium is reached, can be obtained from the diffusion equation (7.82) once the equation of state and the mobility are specified. The equation of state that can be used is the Carnahan-Starling equation of state, which reads,

II(p) p k B T 1 + qo + qO 2 - qO 3 - . (7.89) (I --So) a

This is an accurate equation of state up to volume fractions q0 of about 0.5. The [1 - 6 441 ~] to first order in concentration mobility M(#) is equal to 6,~o~

(see eq.(7.40)). To extend this formula to higher order in concentration, one could fit the data in fig.7.1 with a suitable empirical formula. In turns out that the fit formula,

1 M(~) - (1 -qo) 6 , (7.90)

67r~?oa

provides a reasonable fit to the data, as can be seen from fig.7.1 (the dashed line), and approximately reproduces the exact result (7.40) to first order in concentration. Substitution of eqs.(7.89,90) into the diffusion equation (7.82), and transforming to dimensionless quantities yields, in case the external field is


directed in the negative z-direction, from the top to the bottom of the container,

0 (9 { [ lq-4qo-+-4qp2--4qo3+qo4 0 ] } 0-r qo(Z, 7") - 6Z (1 - q;)6 qo Pe" + (1 - qo) 4 0Z qa "

(7.91) Here, the following dimensionless variables are introduced" Z - z / L is the z-coordinate in units of the height L of the container, r - t D o l l 2 is the time in units of the time required for free diffusion over the height of the container, and,

p e , _ _ L I I (7.92) k B T '

is the so-called sedimentation Peclet number, which is the ratio of the potential energy required to displace a Brownian particle over the height of the container against the extemal field, and the average kinetic energy. In the earth's gravitational field, typical values for the sedimentation Peclet number are 105 - 6.106 (for spheres with a diameter ranging from 500 - 2000 n m with a density of 1.8 g / m l in water in a container with a height of 10 cm).

Initially the volume fraction is homogeneous and equal to ~o, say,

- o ) - , z ( o , . (7.93)

Furthermore, the flux at the bottom and the top of the container must be zero,

1 +4qp+4qo2-4qo3+qo 4 (9 0 -- T Pe~+ (1 - qo) 4 OZ q;' for Z - 0 and 1. (7.94)

In order to satisfy this zero flux condition at the bottom and top of the container, the density there at time r = 0 is different from ~0. The initial condition (7.93) holds in the interior of the container, while the concentration at the bottom and top are fixed by the zero flux condition (7.94).

In the initial stages, the density develops inhomogeneities only at the bottom and the top of the container, which gradually spread out to the interior.

The non-linear diffusion equation (7.91), subject to the initial condition (7.93) and the boundary condition (7.94), must be integrated numerically. It is a too difficult equation to be solved analytically. To ensure numerical stability for acceptable stepsizes (in the sense that computation times are acceptable), implicit procedures must be used to numerically integrate the partial differential equation (7.91), due to the very large values of the sedimentation

478 Chapter 7.

0 . 8

J

9

I I I

F e X t

0 . 4 _ , - -

- I, _ �9 ~ ,

_ ~,Z

o.o ......... , . . . . ,

0.0 0.2 0.4 0.6 Z 1.0

Figure 7 . 8 :

The volume fraction versus the distance from the bottom of the container, in units o f its height, for Pe ~ - 500. Each curve is a profile at a certain instant of time" 1) T - 1.8410 -5, 2) 7.5810 -4, 3) 1.5810 -3, 4) 2.4010 -3, 5) 3.21 10 -3, 6) 4.03 10 -3 , 7)4.9310 -3 , 8)6.7510 -3 , 9) 1.3310 -2 . Data points are generated by means of a simple explicit numerical integration procedure. The accuracy is ~,, 0.5% (except forcurve 9 where the accuracy is ,.~ 5%). The solid line is the barametric height distribution as obtained by numerical integration of the equilibrium condition (7. 70).


Peclet number. In fig.7.8 numerical results are given for a relatively small sedimentation Peclet number of 500, where a simple explicit numerical procedure is used. As can be seen from that figure, after a short time a clear fluid layer developes at the top of the container, while particles accumulate at the bottom. An "interface" at the top of the container is then seen to move downwards with a constant velocity until it meets the sediment that is formed at the bottom. From then on the sediment compacts relatively slowly until sedimentation-diffusion equilibrium is reached. For more realistic sedimentation Peclet numbers, much larger than 500, the interfaces are sharper and the sediment volume fraction is larger. Apart from these two differences, density profiles in fig.7.8 are quite similar to those for much larger sedimentation Peclet numbers (see Auzerias et al. (1988) for a detailed account and many examples).

There are a few things to be said about the above example. First of all, the Carnahan-Starling equation of state (7.89) is not a good approximation for very large volume fractions (say ~ > 0.5). From our numerical results in fig.7.8, however, the volume fraction in the sediment is very large, and a better equation of state should be used there (in fact, the concentration of the sediment in fig.7.8 is unphysical). Secondly, when sedimentation is slow, crystallization can occur at the bottom of the container during sedimentation. One should use an equation of state that allows for the inclusion of such a phase transition during sedimentation. When sedimentation is fast in comparison with the time it takes the system to form crystallites, the sediment remains in a non-crystalline non-equilibrium state, where the spheres are not able to find their crystalline equilibrium positions due to structural hindrance. After sedimentation is completed, a crystalline layer on top of the sediment may be formed, where the structural hindrance is much less pronounced than in the lower parts of the sediment.

The sedimentation velocity revisited

The velocity of the interface between the clear fluid and the part of the dispersion wt-~:re the volume fraction equals the initial volume fraction is the velocity which is experimenlly identified as the sedimentation velocity. The experimental sedimentation velocity is most commonly obtained as the velocity of the inflection point in the interface, where 02~/OZ 2 - O. In the analysis of section 7.2, the sedimentation velocity was identified as the

480 Chapter 7.

velocity of a Brownian particle in the bulk of the suspension, away from the interface. One may rightfully ask whether the interface velocity is equal to the velocity of Brownian particles in the bulk. In comparing the first order in volume fraction predictions with experimental results in figs.7.1 and 7.3 both velocities are implicitly assumed equal. That the two sedimentation velocities are virtually equal can be seen as follows. Suppose that the two velocities were unequal. In case the interface velocity were smaller than the velocity of Brownian particles in the homogeneous part of the suspension, below the interface, a deficiency of Brownian particles below the interface would be the result, leading to a broadening of the interface, in contradiction with the numerical results in fig.7.8. In case the interface velocity were larger, an increase of the concentration just below the interface would be the result, to values larger than the initial concentration, again in contradiction with the numerical results in fig.7.8. The sedimentation velocity that is experimentally obtained as the velocity of the interface is thus equal to the sedimentation velocity of Brownian particles in the homogeneous suspension, up to errors which are determined by the width of the interface.

More quantitatively this may also be seen from the non-linear diffusion equation (7.91) (or more generally from eq.(7.82)). Neglecting shape changes of the interface, the density around the interface may be written as, q~(Z, r) = T(Z - Zi(r)) , where ZI(r) is the as yet unknown position of the interface. Substitution into the diffusion equation (7.91), and subsequent integration across the interface, leads to,

q O o ~ dr = - (1 - ~o) 6 ~oPe ~ ,

or, in the original coordinates,

dzi(t) dt

- M(p0) I ~,~t I �9 (7.95)

The velocity dzt(t ) / dt of the interface is thus equal to the sedimentation velocity of particles in a homogeneous suspension with concentration po, provided that there are no significant changes of the interracial profile during sedimentation. Concentration gradients do not affect the velocity of the interface at the top of the container.

Exercises

7.1) * The deviatoric part of the force that the fluid exerts per unit area on the surface of a translating sphere in an unbounded incompressible fluid is equal to (see eq.(5.6)),

fd~(r) -- Yo { V u o ( r ) + [Vuo(r)]T} �9 ~,

with Uo given in eq.(7.19). Show by differentiation that,

fa~v (r) - 3,o [ i - (v,- u,). 2a

Integrate this expression over the the spherical surface OV ~ of the Brownian particle with its center at the origin to obtain,

~ovo dS fa~(r) - -47rr/oa(v~ - u~).

This is used to arrive at the expression (7.30) for the contribution V " to the sedimentation velocity.

7.2) Sedimentation o f "sticky spheres" Consider spherical particles with a hard-core repulsion and an additional

attractive square well potential,

v+(, -) - 0 , for O < _ r < 2 a ,

= - e , for 2 a < r < 2 a + A ,

= 0 , for r > _ 2 a + A .

e is the depth of the square well and A its width. Show that,

~ 6.441qD + f(e, A)~-{- O(q;2)] , V s - - V s - -

with,

3 2 15 ( 2 + (exp{~e} - 1) -7.441 + ~ (2 + + -~-

27 (,,2 + +

24 16


Now let the depth of the square well tend to infinity and at the same time let the width tend to zero, such that,

a - l i m (exp{/~e}-l) 2+ ~ ----r (x )

A ~ O

- 8 ] - 1 2 lim (exp{fle}-l) A (; ----~ C ~ a

A ---, 0

is a fixed constant. This is the sticky sphere limit, where the surfaces of the spheres are "covered with glue". The parameter a is a measure for the stickiness of the spheres. Expand the term between the square brackets in the above expression for f(e, A) to linear order in A/a, and perform the sticky sphere limit to show that,

0 v, = v, [1 + {-6.441 + 0.488c~} ~p + O(~2)] .

Using more accurate mobility functions, which include higher order terms in the reciprocal distance expansion, the combination -6.441 + 0.488c~ is found to be equal to -6.55 + 0.44c~. Explain the enhancement of the sedimentation velocity due to attractive interaction (the result of exercise 5.6 may help).

7.3) Sedimentation of superparamagnetic panicles Consider Brownian particles of which the cores carry a magnetic moment.

In general, the anisotropy of the magnetic interaction results in a non-zero torque on the core, which is mediated via the magnetic dipole. This invalidates expressions for the hydrodynamic interaction functions that were derived in chapter 5. There it was assumed that the cores of the Brownian particles are torque free. However, when the volume of the magnetic material is small, its magnetic dipole can rotate independently of the material, that is, without an accompanied rotation of the core. Such a magnetic core is called superparamagnetic, and the orientational relaxation is referred to as Ndel relaxation. In these cases the cores of the Brownian particles remain torque free, despite the anisotropic interactions.

This is of course different when rotation of the magnetic moment requires rotation of the core of a Brownian particle. In that case the orientations must be treated as additional stochastic variables, just as for rod like Brownian particles. For those systems the calculation of the mobility functions in chapter 5 must be repeated without the assumption of zero torques but with given fixed torques unequal to zero.

In the present exercise we consider the sedimentation velocity of Brownian particles carrying a superparamagnetic core. To leading order in the density,


the pair-correlation function is equal to,

g(r, I~11,1~12) -- ghs(r) exp{- f lV( r , 1~11, u2)},

where gh~ is the hard-sphere pair-correlation function (see eq.(7.37)), and V is the pair-interaction potential of two magnetic dipoles m~ - m fi~ and m2 - mh2, with m the magnitude of the dipoles and h~,2 the orientation of the dipoles of particles 1 and 2,

m ~ ~o ~ . ~: - 3(e. ~)(e. ~:) V(r, hi, h2) - 47r r 3 '

for r > 2a, with ~ - r / r and #o the magnetic permeability of vacuum. Verify that the expressions (7.32-35) remain valid, except that the pair-correlation function is now equal to the orientational average,

1 Jd fax fd fa2g(r , fal fa2) g ( ~ ) - (4~)~ ' "

The integrations with respect to orientations range over the unit spherical surface. The angular integrations here do not allow for analytical evaluation. These integrations should be done numerically. Suppose, however, that the magnetic interactions are weak, so that the pair-correlation function can be Taylor expanded as,

j [ ] g(~) - g~'(~) (4,~): ] da~ da: ~ -ZV(r , a~, a~)+ Z:V~(r, al, a~) �9

Show that for these weak interactions,

g(r) - gh~(r) [1 + f12 m4#o 2 1 ]

48~r 2 ~~ "

Use this expression in eqs.(7.33-35) and verify that,

v~ - v, o 1 + -6.441 + 0.966 • 327ra3 qp+O qp2 .

V', V" and W contribute 1, 0 and -0.034 to the numerical coefficient 0.966, respectively. Explain why the sedimentation velocity increases due to the magnetic dipolar interactions, similar to what was found for sticky spheres in the previous exercise.


15 : I I f 1 0 -

5

~ ~ J "i 0 1 2 z :

Figure 7.9: The magnetic contribution to the first order in volume fraction coefficient for the sedimentation velocity, f , versus z - ~m2#o/327ra 3. The dashed line is the weak magnetic interaction approximation.

For stronger magnetic interactions, where ~m2#o/327ra a > 1, the first order in volume fraction coefficient must be obtained by numerical integration, a The function f of 3m2#o/327ra s in,

v, - v ~ [1 + {-6.441 + f(~m2#o132raS)} ~ + 0 (~2)] ,

as obtained by numerical integration, is plotted in fig.7.9. The weak interaction result discussed above (the dashed curve) is a reasonable approximation up to flm2#o132ra 3 ~ 2. Furthermore, the higher order terms in the Tay- lor expansion of the hydrodynamic interaction functions with respect to the reciprocal distance hardly contribute.

7.4) Supcrparamagnetic particles in an external magnetic field In this exercise the sedimentation of superparamagnetic spheres in an exter-

nal homogeneous magnetic field is discussed (the definition of superparamagnetic magnetic particles is given in the previous exercise). The homogeneous

aFor numerical integration, the pair-correlation function is most conveniently written as

ghs(r) X G( pm2P~ ar

32,~.~ (e) 3) with, k

i [2~" 1 1 (r

magnetic field does not exert a force on the particles but only a torque, and therefore tends to align the magnetic dipoles. In case of a strong magnetic field the dipoles are perfectly aligned so that the pair-correlation function is equal to,

/~m~#0 1 - 3~z 2} g(r) - gh~(r)exp -- 47r r 3 '

with ghs the hard-sphere pair-correlation function (7.37) and ~ the z-component of the unit separation vector ~- - r / r . The direction of the magnetic field is chosen in the z-direction (not necessarely parallel to the gravitational field, which may have a different direction). The above form for the pair-correlation function is obtained from the pair-potential given in the previous exercise with both magnetic moments chosen along the z-direction. Contrary to the case without an external field, the pair-correlation function is anisotropic, that is, it depends on the direction of r. The spherical angular integrations with respect to r in the expressions for V', V" and W therefore also range over the pair-correlation function. The relevant expression for V' is eq.(7.22),

V' - /~f~ dr [g (r ) - l ]uo (r ) , >a

and for V" eq.(7.30),

1 V tt - - - a

6 f~ 1 zp dr [ g ( r ) - 1] V~uo(r )+ ~ v ~ .

>a

Show from eq.(7.24) that the relevant expression for W is,

W _ _ f dr g(r ){[AA,(r ) + AA~(r) - A B e ( r ) - ABe(r)] t t J

+ [ABe(r)+ ABe(r)] I } . / ~ F ~t .

These expressions can be evaluated analytically for weak magnetic interactions, whet:- ~m2#o/327ra 3 < 1. The pair-correlation function is then approximatel3 equal to,

g(r) - ghs(r) [ l - t i m 2 # ~ 1 - 3 ~ ] 47r r 3 "


Verify the following mathematical identities (f d~" denotes integration over the unit spherical surface),

- o ,

f d~" ( 1 - 3~)H- = 167r 15

1/2 0 0 / 0 1/2 0 . 0 0 -1

Use these identities to show that the sedimentation velocity, for weak magnetic interactions, is given by,

o I / 1'2 ~ - 0 1/2 v 8 { 1 - 6.441~} "] - 1.868 ~ ~ - ~ ~ 0 0 ~ / 1 o o v. -1

V', V" and W contribute - 33/15, 1/5 and 0.132 to the numerical factor -1.868, respectively. The use of more accurate hydrodynamic interaction functions hardly changes this numerical coefficient. Show from the above result that for arbitrary directions 13 of the external magnetic field the sedimentation velocity is given by,

{ /3m2/~~ } ~ flm2#o (B.vO) 1~+ O (~p2) v~ - 1 - 6.441qa- 0.934 ~ 3-~r~ v, + 2.802 qp 3--~~

Notice that the sedimentation velocity is generally not parallel to the gravitational field (which is parallel to v~ This is due to the anisotropy of the pair-correlation function in combination with hydrodynamic interaction (see exercise 5.6, which shows that two particles in an unbounded fluid generally sediment in a direction different from that of the external force as a result of hydrodynamic interaction). The sedimentation velocity is parallel to the gravitional field only when the magnetic field is either parallel or perpendicular to the gravitational field. The difference in sedimentation velocity for these two special situations is 2.802 ~ (flm2#o/327ra3)v ~

In view of the previous exercise, the above weak magnetic interaction result is probably a reasonable approximation up to flm2#o/327ra a ~ 2. As in the previous exercise, results for stronger magnetic interactions can be obtained by numerical integration.


7.5) Relation between the hydrodynamic mobility function and sedimentation

In this exercise an alternative formula for the first order concentration dependence of the sedimentation velocity for spheres is derived. As will turn out, the first order in volume fraction coefficient is related to the hydrodynamic mobility function that was introduced in subsection 6.5.2 in the chapter on diffusion (see eq.(6.75)).

Consider a sedimentation-diffusion equilibrium where the sedimentation Peclet number is so small that the concentration and its gradients are small everywhere, so that a first order in concentration consideration suffices. Ac- cording to eq.(7.70) we have,

V~p(r)- PF~' dII/dZ

= s(k o) F

where in the second equation it is used that S(k ~ O) - kBT/(dlI/d~). Just below eq.(7.84) it is argued that the diffusion coefficient in eq.(7.79) is the gradient diffusion coefficient Dr . Use eq.(7.81) together with the above expression to show that for counter balancing diffusion and sedimentation fluxes,

ksT Dv(p(r)) - M(p(r)) S(k ~ 0)"

Now note that it follows from the conjecture (6.12) in the chapter on diffusion that Dv = Dt~(k ~ 0) = D~(k ~ 0). Use eq.(6.69) for the short-time collective diffusion coefficient to verify that that,

M(p(r)) - H(k ~ O)

where H(k) is the hydrodynamic mobility function defined in eq.(6.75). Verify that the hard-sphere result for H(k ~ O) in the numerator of eq.(6.92) as obtained in subsection 6.5.2 reproduces the sedimentation result in eq.(7.40).

7.6) Do rods align during sedimentation ? Consider a single non-interacting rod that uniformly sediments in an other-

wise quiescent fluid. Convince yourself that friction with the solvent does not give rise to a torque on the rod, so that there is no preferred direction for the rod. Hence, at infinite dilution, rods do not align during sedimentation.

Show from eqs.(5.120,123,124), with F h = - F ~t, that the orientationally averaged sedimentation velocity of a long and thin rod is equal to,

j 1 o 1 dfi v = ln{L /D} F ~t < v~ > - 47r 3rr/oL

When the orientation of the rod is parallel to the gravitational field, the sedimentation velocity is equal to F ~t ln{ L / D }/27r~oL, while for a perpendicular orientation the velocity is F ~t ln{L/D}/4rrloL. Note that the difference with the orientationally averaged sedimentation velocity is never larger than about 50%. In experiments on Tobacco Mosaic Virus at finite concentrations, variations of the sedimentation velocity with the applied field due to alignment of about 10% are observed (see Hearst and Vinograd (1961)). At infinite dilution no alignment effects are observed.

7.7) Use that II - - ( # , ( r ) - / t ~ and the Gibbs-Duhem relation in the form p(r)d#B(r) + p,(r)d#,(r) - 0, together with the relevant equations in subsection 7.4.2, to show that the gradient diffusion coefficient is equal to,

D(/~) - M(/~) 1 - ~ dln{/5} "

7.8) Sketch the transient density profiles when the mobility M (qo) increases with qo upto some volume fraction qom and then decreases again, with qom larger than the initial homogeneous volume fraction qoo. Why is there no sharp interface formed in the upper part of the container?

7.9) Instead of a homogeneous initial density profile we consider here the evolution of the density starting with a situation where all particles are concentrated in a very thin layer located at a height Zo, say. The concentration within that layer is assumed constant. Mathematically, such a situation is described by the initial condition,

v ( z , t - o) - Co 6 ( z - zo ) ,

where Co is equal to the thickness of the layer multiplied by the volume fraction in that layer, and 5 the 1-dimensional delta distribution. For non-interacting particles, where II - pkBT and M(~) - 1/67r~oa, the equation of motion (7.82) for the density reads,

0 0 [ , F~t 0 (z,t)] , z > O O---t ~o( z , t) - Do -~z ~o( z t ) 3 I I + -~z ~o


The zero flux condition at the bottom of the container is,

0 Oz~(Z , t) + D I F~tl ~( z , t) - 0 , for z - O .

Show that the function u(z, t), defined as,

ivOl ~(z, t) - u(z, t) exp - 2Do (z - Zo) 0 2 } Ivslt ,

4Do

satisfies the Smoluchowski equation of a free particle without an external field,

0 0 2 O--~u(z, t) - Do-~z2 U(Z , t) ,

with the initial and boundary condition,

~ ( z , t - o ) - 1 o ~(z,t) + I o D o G 5 v , [ u(z, t) -

Co ~(z - zo) ,

O , for z = O .

The solution of the Smoluchowski equation of a free particle without an external field, Po, subject to the intitial condition Po (z, t - 0) - CoS(z - zo) is (see chapters 2 and 4),

co {/z_zo 2} Po(z, t) - (47rDot)l/2 exp - 4Dot "

Verify by partial integration that u(z, t) can be expressed in terms of Po as follows,

~(z,t) P o ( z - zo, t ) + Po(z + zo, t)

+lV'Do[ o d z ' P o ( z + z ' , t ) exp 2 D o ( Z ' - Z ~ ,

and conclude that,

Co [ { (z-zo) 2} { ( z + z o ) 2}] q o ( z , t ) - (47r~ot)1/2 e x p - 4-D~ + e x p - "4~9o-t •

exp Iv~ Iv~ l Ivs ~ '1 ~ - 2 D o , z - Zo) . . . . . + exp - dx e x p { - x 4Do 7r1/2Do --~o Z~ +*~176 ''

~/4Dot

2}.

490 Further Reading

10

Co Zo - 1 ~ 3

t+ \ 7"10-2 3"10-2 / ~

/ ~"2

_

o 0.4 zA ~ 1.2 Figure 7.10: Density profiles for non-interacting particles, initially concentrated in a thin layer at z - zo. Plotted is qo(z, t)/Cozo versus z/zo for various values o f Dot/Z2o, which are indicated in the figure.

0 ~ F e x t The value of iv ~ [ zo / Do zo I ]/kB T is chosen equal to 10.

o For Dot/zg and ]v~ I t/Do << 1, this density profile is equal to the pdf of the free particle (in one dimension) in an external field (see exercise 2.4), while for t ~ c~ this solution reproduces the barometric height distribution (7.62). The transition from the approximately Gaussian peak with a downward velocity

o [v~ [ to the barometric height distribution is clearly seen in fig.7.10, where the above solution is plotted for various times.

F u r t h e r R e a d i n g and R e f e r e n c e s

Batchelor was the first to account correctly for the divergent terms that occur in the calculation of sedimentation velocities in,

�9 G.K. Batchelor, J. Fluid Mech. 52 (1972) 245. A short discussion on the history of the sedimentation problem is also contained in this paper. Batchelor improved on earlier work by,

�9 J.M. Burgers, Proc. Kon. Nederlandse Akad. Wet. 44 (1942), 1045,

Further Reading 491

1177, 45 9, 126. �9 C.W. Pyun, M. Fixman, J. Chem. Phys. 41 (1964) 937.

The value of the first order in volume fraction coefficient for hard-spheres that was found in the two above papers is -6.88 and -7.16, respectively, not very different from the correct value -6.55. The differences are due to a partly incorrect account of the divergent terms, and the use of approximate expressions for the hydrodynamic interaction functions. A few years after Batchelor's 1972-paper, alternative routes to dispose of the divergent terms were found in,

�9 R.W. O'Brien, J. Fluid Mech. 91 (1979) 17. �9 E.J. Hinch, J. Fluid Mech. 83 (1977) 695.

Sedimentation in polydisperse systems is discussed in, �9 G.K. Batchelor, J. Fluid Mech. 119 (1982) 379. �9 G.K. Batchelor, C.-S. Wen, J. Fluid Mech. 124 (1982) 495.

with a corrigendum to these papers in the J. Fluid Mech. 137 (1983) 467.

A recommendable paper on several aspects of sedimentation is, �9 E.J. Hinch (E. Guyon et al. eds.), Disorder and Mixing, Kluwer Aca-

demic Publishers, 1988, page 153.

The experimental data in fig.7.1 are taken from, �9 R. Buscall, J.W. Goodwin, R.H. Ottewill, T.E Tadros, J. Colloid Int. Sci.

85 (1982) 78. �9 M.M. Kops-Werkhoven, H.M. Fijnaut, J. Chem. Phys. 77 (1982) 2242.

The experimental data in fig.7.3 are taken from, �9 D.M.E. Thies-Weesie, A.P. Philipse, G. N~igele, B. Mandl, R. Klein, J.

Coll. Int. Sci. 176 (1995)43.

Accurate (numerical) results for the pair-correlation function are known for hard-core particles with an additional screened Coulomb potential as given in eq.(7.41). These results can be used to evaluate the sedimentation velocity explicitly, without making the approximation (7.43) for the pair-correlation function. Details of such calculations can be found in,

�9 G. N~igele, B. Steiniger, U. Genz, R. Klein, Physica Scripta T55 (1994) 119.

�9 G. N~igele, B. Mandl, R. Klein, Progr. Coll. Polym. Sci. 98 (1995) 117.

492 Further Reading

Sedimentation measurements on charged systems in de-ionized water are discussed in,

�9 T. Okubo, J. Phys. Chem. 98 (1994) 1472.

The sedimentation velocity of a random array and ordered arrays of ather- mal, non-Brownian particles varies like qo 1/~ and qa ~/a, respectively. Such sedimentation phenomena are discussed in,

�9 H.C. Brinkman, Appl. Sci. Res. A 1 (1947) 27. �9 H. Hasimoto, J. Fluid Mech. 5 (1959) 317. �9 S. Childress, J. Chem. Phys. 56 (1972) 2527. �9 P.G. Saffman, Stud. Appl. Math. 52 (1973) 115. �9 E.J. Hinch, J. Fluid Mech. 83 (1977) 695.

A.A. Zick, G.M. Homsy, J. Fluid Mech. 115 (1982) 13. A.S. Sangani, A. Acrivos, Int. J. Mult. Flow 8 (1982) 343.

Detailed accounts on the influence of the walls of the container on the sedimentation velocity can be found in,

�9 C.W.J. Beenakker, P. Mazur, Phys. Fluids 28 (1985) 767, 28 (1985) 3203.

�9 P. Nozi~res, Physica A 147 (1987) 219. �9 U. Geigenmtiller, P. Mazur, J. Stat. Phys. 53 (1988) 137. �9 B.U. Felderhof, Physica A 153 (1988) 217. �9 B. Noetinger, Physica A 157 (1989) 1139.

The contents of the section on the backflow velocity is largely taken from Geigenmiiller and Mazur (1988).

A classic paper on sedimentation-diffusion is, �9 G.J. Kynch, Trans. Far. Soc. 48 (1952) 166.

For elaborate theoretical treatments on sedimentation-diffusion and many experimental results, the following papers may be consulted,

�9 K.E. Davis, W.B. Russel, Adv. Ceram. 21 (1987) 573, Ceramics Trans. 1B (1988) 673, Phys. Fluids A 1 (1989) 82.

�9 F.M. Auzerias, R. Jackson, W.B. Russel, J. Fluid Mech. 195 (1988) 437. �9 S. Emmett, S.D. Lubetkin, B. Vincent, Coll. Surf. 42 (1989) 139. �9 K.E. Davis, W.B. Russel, W.J. Glantschnig, Science 245 (1989) 507. �9 S.D. Lubetldn, D.J. Wedlock, C.E Edser, Coll. Surf. 44 (1990) 139. �9 K.E. Davis, W.B. Russel, W.J. Glantschnig, J. Chem. Soc., Far. Trans.

87 (1991) 411.

Further Reading 493

�9 J.L. Barrat, T. Biben, J.P. Hansen, J. Phys., Condens. Matter 4 (1992) L l l .

�9 J.S. van Duijneveldt, J.K.G. Dhont, H.N.W. Lekkerkerker, J. Chem. Phys. 99 (1993) 6941.

The identification of the sedimentation mobility with the hydrodynamic mobility function as described in exercise 7.5 is discussed in,

�9 W.B. Russel, A.B. Glendinning, J. Chem. Phys. 74 (1981) 948.

Experimental results on the alignment of rod like Brownian particles during sedimentation are given in,

�9 J.E. Hearst, J. Vinograd, Arch. B iochem. B iophys. 92 (1961) 206. Some theoretical considerations are given in,

�9 J.M. Peterson, J. Chem. Phys. 40 (1964) 2680.

Sedimentation is sometimes seen to give rise to layered structures. See for example,

�9 D. Siano, J. Coll. Int. Sci. 68 (1979) 111. Siano suggested that a spinodal demixing mechanism could be at work to give rise to these stratified structures. A different mechanism has been suggested by,

�9 W. van Saarloos, D.A. Hyse, Europhysics Lett. 11 (1990) 107. This paper may also be consulted for more references on this phenomenon.

Chapter 8

CRITICAL PHENOMENA

495

496 Chapter 8.

In this chapter the origin of a diverging correlation length is discussed, and some of the resulting so-called critical phenomena are considered. The correlation length is roughly the distance over which two particles interact. The range of this effective interaction can be much larger than the range of the pair-interaction potential" two particles may interact with each other via other particles (see also the discussion on the pair-correlation function in the introductory chapter, subsection 1.3.1). These cooperative intervening interaction effects become very pronounced close to the spinodal, and in particular, close to a critical point.

This chapter is restricted to mean-field considerations, and renormalization and mode-mode coupling theory are not addressed. A general discussion on critical exponents and their inter-relations is also not given here. For such specialized topics references can be found at the end of this chapter in the section Further Reading and References. After an introductory section, the long ranged behaviour of the pair-correlation function is analysed in two different ways : by the Omstein-Zernike approach and the Smoluchowski equation approach. The advantage of the Smoluchowski equation approach is that effects of external fields, such as shear flow, may be included. The anomalous effects of shear flow on the pair-correlation function near the critical point are discussed in section 8.3. The generalization of the Omstein-Zernike static structure factor to include shear flow is derived in that section. The implications for the shear rate and temperature dependence of the turbidity are discussed in section 8.4. Section 8.5 is concerned with the dramatic effects of increasing correlation lengths on collective diffusion. Finally, the anomalous behaviour of the effective shear viscosity is the subject of section 8.6.

8.1 Introduction

A gas-liquid critical point may occur in colloidal systems where the pair- interaction potential has an attractive component. An example of such a system are silica cores coated with stearyl alcohol chains dissolved in benzene. Since benzene is a marginal solvent for stearyl alcohol, the alcohol chain brushes on the surfaces of two colloidal particles rather dissolve in each other than in the solvent. This results in a very short ranged attractive interaction super imposed onto the hard-core repulsion (this potential is sketched in fig. 1.1d, in chapter 1). The quality of the solvent is diminished as the temperature is lowered, resulting in a stronger attraction, and giving rise to thermodynamic instability


Figure 8.1" Experimental phase diagram of stearyl silica in benzene. The vertical axis is the temperature and the horizontal axis is the concentration in terms of the fraction of the total volume that is occupied by the silica cores of the Brownian particles. The points relate to various expe- rimenta/techniques to determine the phase lines. This phase diagram is taken from Verduin and Dhont (1995).

19

18

0 ~ I-"

17

- -s tab le - - CRITICAL POINT

\

- g e t - - . 8,,oo

\ \ \

\ \

\ \

-F\ \

\ \

\

\ \

\

\\

16 0 0.1 0.2 to 0.3 0.4

7"

and in particular to a gas-liquid critical point. An experimental phase diagram of such a system is given in fig.8.1. At larger temperatures, benzene is a good solvent for stearyl alcohol, resulting in a stable dispersion. At lower temperatures there is a binodal and a spinodal. These lines mark the transition from stable to meta-stable and from meta-stable to unstable, respectively. The mechanism that leads to instability when lowering the temperature to below the spinodal is discussed in detail in the next chapter. Disregarding the gel- line in fig.8.1 for the moment, a homogeneous system with a temperature and concentration located between the binodal and spinodal (for example the point A in fig.8.1) phase separates into two liquid phases (the points B and C in fig.8.1). The phase separation mechanism here is referred to as condensation or nucleation. A homogeneous system with a temperature and concentration below the spinodal (for example point A' in fig.8.1) will also separate into two liquid phases (again the points B and C in fig.8.1). The phase separation mechanism is now quite different from condensation, and is referred to as spinodal decomposition. More about these phase separation mechanisms can be found in the next chapter. Contrary to the phase diagram of simple molecular systems, a gel-line is found for colloidal systems. Below

498 Chapter 8.

the gel-line the system is mechanically unstable, and a more or less rigid and very long lived, space filling network of mutually connected colloidal particles is formed. The gel is a solid in the sense that it sustains a finite yield stress : the meniscus of the gel does not flow when tilting the cuvette. The more familiar standard phase diagram is that for simple molecular systems, where the gel-line is absent, and the binodal and spinodal extend to higher concentrations as sketched by the dashed lines in fig.8.1. Since for larger volume fractions the binodal and spinodal are located below the gel-line, the actual phase separation of the silica dispersion is not into two liquid phases with volume fractions B and C, but into a fluid with volume fraction B and a gel with volume fraction D, as depicted in fig.8.1.

There is a special point where the system changes from being stable to unstable, without first becoming meta-stable. This is the point where the binodal and spinodal meet, and is known as the critical point. This point is indicated in fig.8.1 by the vertical arrow. For some not yet understood reason, if there is any, the gel-line intersects the critical point.

A system is unstable when the osmotic pressure II decreases with increasing number density/~ = N/V, and (meta-)stable when II increases. That is,

dII d/5 < 0 " :- thermodynamically unstable. (8.1)

In the (meta-) stable region of the phase diagram this derivative is positive, while below the spinodal, in the unstable region of the phase diagram, it is negative. A kinetic derivation of this well known thermodynamic result will be given in the next chapter. This stability criterion can be understood thermodynamically as follows. Define ~ = 1//~, the reciprocal concentration. Now note that,

dH ~2dII ~2 d (OZ) _2d2(m/N) dp tN '

with N the fixed total number of Brownian particles in the system and A the Helmholtz free energy of the system. A negative (positive) value of dII/d~ thus implies a reciprocal density dependence of the Helmholtz free energy (per colloidal particle) as sketched in fig.8.2a (fig.8.2b). Consider a system with a concentration ~ - Co in the homogeneous state. Now suppose that an instantaneous realization of the fluctuating density mimics the separation of the system into two parts, each with a different concentration, c_ and c+, as

A -ff

i \ , , / i i \ 72" i I ', I

~, V .. ~, '4/

Figure 8.2: The local density dependence of the Helmholtz free energy per particle A / N in case dII / d~ < O, (a), and in case dII / d~ > O, (b). The points �9 are the free energies of the system when decomposed into two fluids with concentrations c_ and c+.

sketched in fig.8.2a,b. The Helmholtz free energy per colloidal particle of this "demixed" system is obtained from the intersection of the straight line connecting the two points (0_ - 1/c_, A(~_)/N_) and (0+ - 1/c+, A(~+)/N+) and the vertical line at the original reciprocal concentration 0o - 1/co of the homogeneous system. ~ Since the systems tends towards a state where the Helmholtz energy attains a minimum value, it is clear from fig.8.2a that when dII/d~ < 0, any demixing fluctuation, no matter how small in amplitude, decreases the Helmholtz free energy, so that the demixed state does not return to the homogeneous state but evolves in time towards more complete demixing. On the other hand, when dII/dp > 0, demixing (with small amplitudes) increases the Helmholtz free energy, so that the system returns to the homogeneous state. In the meta-stable region of the phase diagram, phase

1This can be understood as follows. From N+ ~+ + N_ ~_ - N00 and N+ + N_ - N (with N• the number of colloidal particles in the "phase" with concentration c+), it follows that N+ - N(~0 - ~_ )/(~+ - ~_ ) and N_ - N(~0 - ~+)/(~_ - ~+). Since the Helmholtz free energy is extensive, the free energy Aaem of the demixed system is equal to A(O_ ) + A(~+). Hence,

Aa~m ~o - ~+ A ( ~ _ ) ~o - ~ - A(~)+ ) [ _

N ~ _ - f ) + N_ v + - ~ _ N+

This is a linear function of % that connects the points (~_, A(~_)) and (~+, A(~+)). This proves the above statement concerning the free energy per particle in the demixed system.

500 Chapter 8.

Z n

n

m

I I I I I I I

\ oo Oo

~ o

O o ~ o

%oo ~ ^ 0~176

oOVO0oo ' - ' U O 0 0

~ 4 4 ~ + + . ~ o % 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 _ .e,11~§ . . . . . . . O o o o _ o o ^ o o o o

. . . . . �9 M.I..I..I...II..I.§247 ...., v 0 0 0 ~ I " § 2 4 7 § § "1" .I. § § § ..i ~

I - - -~--:;-~:~--:-5"S-~'-2:2::-~T:"000~00"000~00000~"

0 1 2 K,,, 10,2 [m_~,-] Figure 8.3" The scattered intensity as a function of the squared scattering wavevector k at various temperatures just above the critical point �9 18.05 o C (o), 18.10 0 C (o), 18.14~ (+), 18.66~ (.). The critical temperature is 17.95~ Data are taken from Verduin and Dhont (1995).

separation may occur, despite the fact that dII/d~ > 0, but is induced by large amplitude demixing fluctuations. This will be discussed in more detail in the next chapter.

The spinodal is the set of number densities and temperatures where the system becomes unstable, that is, where dII/d~ - O. Since the osmotic pressure is a function of the temperature and the concentration, this is an implicit relation between these two quantities, which defines the spinodal.

The static structure factor S (k) for zero wavevectors k is related to dII/d/5 as,

1 s ( k o) - (8.2)

7/

with fl - 1 /kBT (kB is Boltzmann's constant and T is the temperature), so that the intensity of scattered light at small scattering angles diverges as the spinodal is approached. Fig.8.3 illustrates the divergence of the scattered intensity of light at small scattering angles on approach of the critical point by lowering the temperature of the stearyl silica/benzene dispersion along the vertical line in fig.8.1 just above the critical point. On the other hand, the

8.2. Long Ranged Interactions 501

static structure factor at zero wavevector is equal to,

s(k --, o) - + p f dr (8.3)

where h ( r ) - g ( r ) - 1 is the total-correlation function, and g(r) is the pair- correlation function. The integrated total-correlation function therefore begins to diverges on approach of the spinodal, or in particular, on approach of the critical point. This implies that the total-correlation function goes to 0 for large distances between two colloidal particles more and more slowly on approach of the spinodal, implying in turn that the range of effective interactions becomes very large. On and below the spinodal each Brownian particle interacts, or equivalently is correlated, with all other Brownian particles in the system. This long ranged behaviour of the total-correlation function is further analysed in the following section.

The diverging range of effective interactions, which is roughly measured by the so-called correlation length, gives rise to "critical" (or "anomalous") behaviour of various quantities, such as the static structure factor (which diverges at zero wavevector as discussed above), the effects of an externally imposed shear flow (sections 8.3 and 8.4), the collective diffusion coefficient (section 8.5) and the shear viscosity (section 8.6).

8.2 Long Ranged Interactions

As discussed in the previous section, experiments indicate the existence of very long ranged effective interactions between colloidal particles in the neighbourhood of the spinodal, resulting in a non-zero pair-correlation function for large separations. The following two subsections are devoted to the calculation of the asymptotic behaviour of the pair-correlation function for such large distances. Two approaches are discussed : the Ornstein-Zernike approach and an approach that is based on the S moluchowski equation which is derived in chapter 4. In both cases the colloidal particles are assumed spherically symmetric.

8.2.1 The Ornstein-Zernike Approach

Close to the critical point the correlation length becomes very large and ultimately diverges due to interactions mediated via intervening particles. Inter- actions "propagate" from one particle to the other via the remaining particles,

502 Chapter 8.

as sketched in fig.8.4. In case of colloidal particles with a short ranged attractive pair-interaction potential (to which the phase diagram in fig.8.1 pertains), at low concentrations and close to the spinodal, there must exist temporary, very open clusters with an extent of the order of the correlation length.

The total-correlation function h(r) - 9 ( r ) - 1 is identically equal to 0 without any interactions (here, g(r) is the pair-correlation function that was introduced in subsection 1.3.1 in chapter 1). With interactions, h(r) is non- zero, up to the distance r between two particles where effective interactions, or equivalently, correlations, are lost. The total-correlation function thus measures the strength of effective interactions, or equivalently, the amount of correlation, between the position coordinates of two particles a distance r apart. At very low concentrations, where at most two colloidal particles interact simultaneously, the total-correlation function ho(r) is equal to (see eq.(1.55) in chapter 1),

ho(r) - e x p { - ~ V ( r ) } - 1, (8.4)

with/~ - 1/kBT (kB is Boltzmann's constant and T the temperatUre), and V(r) the pair-interaction potential. This total-correlation function is referred to here as the bare total-correlation function, since it does not include indirect interactions via other particles. For larger concentrations, the total-correlation function is similarly related to the effective interaction potential V ~:: (r), which includes the intervening effects of other particles (see eq.(1.59) in chapter 1),

h(r) - exp{-f lV ~H(r)} - 1. (8.5)

The total-correlation function can be expressed, approximately, in terms of the bare total-correlation function by the following reasoning, due to Ornstein and

Figure 8.4" Propagation of interactions from particle i to 2, via intermediate particles.

o Rv


dq /

o/Xo o Figure 8.5" Propagation of correlations via a single intermediate volume element dra (a) and two intermediate volume elements dr3 and dr4 (b).

Zemike (1914). Bare correlations, without intervening particles being present, are thought of to propagate from one particle to the other, via intermediate particles. Consider two particles which are affected in their interaction due to the presence of particles in a single (infinitesimally) small volume element of volume dra located at position ra, as sketched in fig.8.5a. The most simple Ansatz for the first correction to the bare total-correlation function for intermediate interactions is probably drap(ra)ho(ra - ra) h0(r3 - r2) (to allow for inhomgeneities, the total-correlation function is written as a function of r instead of r - I r ]). Here, p(ra) is the number density of colloidal particles at position ra, so that drap(ra) is the number of particles in the volume element dra. This Ansatz satisfies the condition that, when either bare correlations between particle 1 or particle 2 and the particles in dra are absent, there is no extra correlation between particles 1 and 2 due to the presence of particles in dra. The contribution of all other volume elements is now obtained by adding the intervening effects over all positions, that is,

-- 1"2) -- ho(r l - r 2 ) [ - f dr3 p ( r 3 ) h o ( F 1 - r3)ho(F 3 - r2) . h(rl

The next higher order in concentration correction is due to correlations which are subsequently mediated via two volume elements at r3 and r4 as sketched in fig.8.5b. This correction is similarly equal to,

/ dr3 / dr4 p ( r 3 ) p ( r 4 ) h o ( r i - r3)ho(r3 - r4)ho(r4 - r2).

Continuing in this way one obtains (with r 0 = ri - rj),

ho(rx2) + f dr3 p(ra)ho(r13)ho(r32) h(r12)

504 Chapter 8.

+ f dr3 f dr4 p(r3)p(r4)ho(r13)ho(r34)ho(r42)

+ fdr3fdr4fdrsp(r3)p(r4)p(rs)ho(r13)ho(r34)ho(r45)ho(r52) +. . .

= ho(r12)+ fdr3 p(r3)ho(r13) x [ho(r32)+ f dr4 p(r4)ho(r34)ho(r42)

+ fdr4fdrsp(r4)p(rs)ho(r34)ho(r45)ho(r52)+ ""].

The expression between the square brackets in the last equation is nothing but h(ra2), so that the above equation reduces to an integral equation for the total-correlation function,

h(ra - r2) - ho(rl -- r2) ~- / d r 3 p(r3)ho(rx - r3)h(r3 - r2).

This is the Ornstein-Zernike equation. Since the bare total-correlation function is known once the pair-interaction potential is specified, the Orstein-Zemike equation can be used to calculate the "dressed" total-correlation function h(r).

The notion of propagating bare correlations used to derive the Omstein- Zemike equation is a bit too simplified. Accurate expressions for the total- correlation function must be obtained from the Omstein-Zemike equation by replacing the bare total-correlation function by another correlation function, which is referred to as the direct-correlation function for obvious reasons. Expressions for that correlation function must then be found independently, where the Ornstein-Zemike equation is to be considered as defining that function. Notice that for relative separations r < 2a, where hard-cores of radius a of two colloidal particles overlap, both h(r) and ho(r) are equal to - 1. The integral term is not equal to 0 for such small separations, showing that the above Omstein-Zemike equation is not correct for these small separations. The bare total-correlation function is a reasonable approximation for the direct- correlation function for r > 2a, but is quite different for smaller distances where r < 2a.

The feature that the bare total-correlation function and the direct-correlation function have in common, is that they are short ranged, that is, they are both zero beyond a distance that is comparable to the range of the pair-interaction potential. Such a short range of the direct-correlation function is to be expected by way of construction of the Omstein-Zernike equation. In fact, from the divergence of the volume integral of the total-correlation function, it can be shown quite easily that the volume integral of the direct-correlation

r

c(

R v "- ~ h(I r - r ' l )

/ v V

r')

? I

r=r'

~r'

Figure 8.6: The short ranged direct-correlation function c ( [ r l - - ral) - c(r') and the total correlation function h(I ra - r2 I) - h(I r - r ' I), as functions ofr' forr >> Rv. The dashed line is a sketch of the first three terms in the Taylor expansion (8.7).

function remains finite (see exercise 8.1). The short ranged character of the direct-correlation function is the only feature that we shall use here.

The more appropriate form of the Ornstein-Zernike equation is,

h( r , - r2) - c(r , - r~) + f dr3 p(r3)c(rl - r3)h(r3 - r2) , (8.6)

where the bare total-correlation function ho(r) is replaced by the direct- correlation function c(r).

Asymptotic solution of the Ornstein-Zernike equation

Close to the spinodal, and in particular close to the gas-liquid critical point, the long-ranged behaviour of the pair-correlation function can be obtained from the Ornstein-Zemike equation (8.6), making use of the short rangedness of the direct-correlation function. Consider distances r - 1 r~ - r2 [>> Rv, with Rv the range of the pair-interaction potential. Since the direct-correlation function is short ranged, the "dressed" total-correlation function under the integral in eq.(8.6) may be Taylor expanded, since I r~ - r 3 1 < Rv, as indicated in fig.8.6,

h ( r 3 - r2) - h(r3 - ra + r l - r2) - h ( r 3 - r~ + r) (8.7) 1

= h(r) + (r3 - r~) . V~h(r) + ~ ( r 3 - r l ) ( r3 - r l ) " V ~ V ~ h ( r ) + . . . ,

506 Chapter 8.

where V~ is the gradient operator with respect to r - r l - r2. For rotationally and translationally invariant systems, where the density is a constant, p(r3) -

- N / V , h(r) - h(r) and c(r) - c(r), the Omstein-Zernike equation (8.6)

reads (with r ' - r l - r 3 ) ,

h(,) r + z dr ' c ( r / )

1 - fi [V~h(r)] . f dr'c(r')r' + 5~ [V~V~h(r)] �9 f dr'c(r')r'r' + . . . .

Integration with respect to spherical angular coordinates yields,

dr' c(r') -

f dr' c( r ' ) r ' -

f dr' c ( r ' ) r ' r ' =

with I the unit matrix. Hence,

47r dr' c(r')r '2

~0 ~176 47r dr' c(r')r '4 i 3

r 2 h(r) - c(r) + coh(r) + 2V~h(r) + . . . , (8.8)

where we abbreviated,

j~o CX) Co - 47r ~ dr' c(r')r '2 , (8.9)

27r _ fo ~ = - 2 p (8.10)

For large separations, the direct correlation function is 0, since this is a short ranged function, and the remaining terms represented by . . . in eq.(8.8) are vanishingly small. The Omstein-Zernike equation thus reduces to a simple

differential equation,

c2 V~ 2h(r) . (8.11) h(r) - 1 - co

The solution is, 2

h(r) - ( A R v ) e x p { - r / ~ } r

, for r >> R v , (8.12)

2In subsection 1.2.5 in chapter 1, a similar differential equation is solved by means of Fourier transformation (see eq.(1.28)), where the short-ranged direct-correlation function plays the role of the delta distribution. You may also verify by substitution that the expression (8.12) is the solution of eq.(8.11).


where A is an as yet unknown dimensionless integration constant and,

c2 (8.13) - 1 - Co '

is a temperature and density dependent parameter with the dimension of length, the so-called correlation length, which measures the range of effective interactions, or in other words, the distance over which the total-correlation function tends to 0.

The correlation length can be expressed in terms of the osmotic compres- sibility, noting that Fourier transformation of the Omstein-Zemike equation (8.6) yields, with the use of the convolution theorem (see exercise 1.4c),

1 S(k) - l + ~ h(k) -

1 - p c ( k ) "

From eq.(8.2) it thus follows that,

dII 1 - / 5 c(k --. 0) - ~ d/5 "

On the other hand we have,

c(k 0) - # f dr' c(r') - co.

Substitution of the above equations into eq.(8.13) yields,

~ c~ (8.14)

Since on approach of the spinodal, dII/d~ ~ O, the correlation length diverges when c2 is well behaved. This confirms the interpretation of experimental observations as discussed in the introduction.

An expression for the static structure factor S(k) in terms of the correlation length can be obtained by Fourier transformation of the total-correlation function as given in eq.(8.12),

S(k) -- l+#/dr h(r) e x p { - i k �9 r}

= 1 + 47rp fa ~~ dr h(r) sin{kr}

r 2 kr

1 = 1 + 4r /5 (ARv) ~-2 + k 2"

508 Chapter 8.

The expression (8.12) for the total-correlation function that is used here is only valid for r >> Rv, so that the above expression for the static structure factor is valid only for wavevectors k << 2~r/Rv. According to eq.(8.2), the integration constant A follows from the above expression as,

} A R v = 47rfi~2 /~ - 1 . (8.15)

The static structure factor is thus equal to,

s ( k ) -

[/~dH~__~] -1 _~_ (]g~)2 C21 ~2 _[_ (k~)2

1 + (k~) ~ - 1 + (k~) 2

where in the second equation the expression (8.14) for the correlation length is used. An order of magnitude estimate of r is obtained from its definition (8.10), using that the range of the direct-correlation function is of the order Rv. With the use of eqs.(8.9) and (8.13) it follows that,

- - - [ , -

and hence,

+ ( R ~ / r ~ ~ R~,,

because Rv/~ << 1. Since the above expression for the static structure factor is valid only for k << 27r/Rv, it follows that, to a good approximation,

c~1~ 2 S(k) - for k << 27r/Rv (8.16)

1 + (k~) ~ '

The results (8.12,16) for the total-correlation function and the static structure factor are commonly referred to as the Ornstein-Zernike total-correlation function and static structure factor, respectively.

8.2.2 Smoluchowski Equation Approach

The above results can also be obtained from the stationary version of the Smoluchowski equation (4.40,41), which reads,

0 -- ~ [V n ~ ( r l , . . . , r N ) ] P ( r l , . . . , r N ) + v , , P ( r x , . . . , r N ) , (8.17)

where V~a is the gradient operator with respect to the position coordinate r~, (I) is the total potential energy of the assembly of N Brownian particles, and P is the pdf of the position coordinates. An equation for the pair- correlation function g(r, r ') can be obtained from the Smoluchowski equation by integration with respect to r3, �9 �9 �9 rN, since, by definition,

/ d r 3 . . , f drN P ( r , r', r 3 , . . . , rN) -- /91 (r)Pa (r ' )g(r , r ' ) , (8.18)

where P1 (r) - 1/V for a homogeneous system (see subsection 1.3.1 in the introductory chapter). Assuming pair-wise additive direct interactions, that is, assuming that the total potential energy (I) can be written as a sum of pair-potentials,

N

( I ) ( r l , - ' ' , rN) -- E V(rij), (8.19) i , j = l i < 3

with rij -1 r~ - rj l, the integration with respect to r 3 , . . . , rN is easily performed for identical Brownian particles to obtain (with r - r l - r2 and r t -- r l -- r3) ,

where,

o - (8.20)

g3( r , r ' ) f

F,~d(r) - -fi J dr' [V,,U(r')] 9 ~ i ' (8.21)

is the indirect force of particle 2 on particle 1, which is the contribution to the total force that is mediated via intervening particles. The three-particle correlation function g3(r, r ') - g3(r~ - r2, ra - r3) is defined as,

g 3 ( r l -- r2, rl - r3) - V 3 J d r 4 . . " J drN P ( r l , r2, r3, r 4 , . - . , r N ) . (8.22)

For the translationally invariant system under consideration here, the three- particle correlation function depends on the position coordinates r~, r : and ra only via their differences r - r~ - r2 and r ~ - rx - r3. In order to obtain a closed equation for the pair-correlation function, the three-particle correlation function must be expressed in terms of pair-correlation functions. The most simple Ansatz for such a closure relation is to assume that correlations are pair-wise independent, which formally means that,

g3(r, r ') - g(r) g(r') g(] r - r ' ]). (8.23)

510 Chapter 8.

Q ....

Rv Q s

I I

/ I I i I I I

1 1 1 ' 1/ I ' I 1 1 1 , t II Q

~1 I I.._ _

r>>R -->

Figure 8.7" A sketch of an arrangement of the particles 1, 2 and 3 for which a closure relation is needed. The dashed line is the total-correlation function relative to the position coordinate of particle 2.

This closure relation is known as the Kirkwood superposition approximation. What is neglected here is part of the influence of a third particle on the correlation between two other particles. For the particular situation we are interested in here, the superposition approximation can be improved to some extent, by accounting for the effect that the presence of particle 2 has on the correlation between particles 1 and 3. The crucial point here is, that in the integral (8.21) that defines the indirect force, the distance r' - r~ - ra is always smaller than Rv since for r' > Rv the pair-force X7~,V(r') is zero. On the other hand the distance between particles 1 and 2 is much larger than Rv, since these are the large distances for which we are seeking a solution of eq.(8.20). A closure relation is therefore needed only for special configurations where particles 1 and 3 are close together, while particles 1 and 2 are far apart. Such an arrangement is sketched in fig.8.7. The effect of the presence of the distant particle 2 is to enhance the number density in the neighbourhood of the neighbouring particles 1 and 3 to ~g(R), where R is the distance between particle 2 and the particles 1 and 3 (see eq.(1.58) in the introductory chapter). The most obvious choice for R is the distance from the point inbetween particles 1 and 3, and the position of particle 2, that is,

1 1 I R - [ ~(rl + ra) - r2 [-[ r - ~r I. The effect of the distant particle on the correlation between the two neighbouring particles is accounted for by simply replacing #(r') in the superposition approximation (8.23) by the same pair-

l r t 1 t correlation function at the enhanced density/3#([ r - ~ [) - / ~ + ~h(I r - ~r [). We are interested here in the asymptotic solution ofeq.(8.20) for large distances


1 r' r, where h([ r - 5 I) is small, since h(r) ---, 0 as r ---, c~. The enhancement of the density around the two neighbouring particles can therefore be considered small, so that the pair-correlation function may be Taylor expanded up to leading order,

g ( r ' ) = dg(r') ~r ' l) (8.24) d----~ Ph( l r - .

The correlation functions on the right hand-side are understood to relate to the number density/5. Substitution of this result into the superposition approximation (8.23) yields an improved superposition approximation,

g3(r, r') - g(r) g([ r - r' [) {g(r ') + dg(r') l r , } d-----~ f i h ( l r - ~ 1) �9 (8.25)

What is still neglected in this closure relation is the effect that particle 1 has on the correlation between particles 2 and 3, and of particle 3 on the correlation between particles 1 and 2.

Substitution of the closure relation (8.25) into the expression (8.21) for the indirect force, and subsequent subsitution into the Smoluchowski equation (8.20) yields the following equation for the pair-correlation function,

V~g(r) + fig(r)[VrV(r)+ fi f dr' [V~,V(r')] (8.26)

xg(Ir-r'l){g(r')+ dg(r') 1 r' }] d----~Ph(I r - ~ I) �9

For precisely the same reasons that allowed for the Taylor expansion (8.7) in the Omstein-Zernike approach, both correlation functions g(l r - r' 1) and

1 t h ( I r - 7r I) in the integrand can be Taylor expanded around r' - O. In the present case it is the pair-force V~,V(r') that is zero for r' > Rv, while both correlation functions are smooth functions of r' when r >> Rv (see in this respect also fig.8.6). The Taylor expansions read,

g ( ] r - r ' l)

h ( I r - ~r ' I)

1 ! ! - g(r) - r ' . V~g(r )+ ~ r r �9 V~V~g(r)

- t-r 'r 'r ' | V~V~V~g(r )+ . - . (8.27) 6

1 r, l r , r, - h ( r ) - �9 g

1 - - - r ' r ' r ' (5) V~V~V~h(r) + . . . (8.28)

48

512 Chapter 8.

Furthermore, only linear terms in h(r) must be retained for the calculation of the asymptotic solution for large distances, since h(r) ~ 0 when r ~ e~. Noting that g(r) = h(r) + 1, substitution of the Taylor expansions (8.27,28) into eq.(8.26) and keeping only linear terms in h(r) yields,

- V , h ( r ) + fl {h(r) + 1} [V,V(r)]

+/~ fl [V,h(r)] �9 dr' [V,,V(r')] r' g(r') + -~ fi - dfi

/ { 1 - r ~ [v,v,v,h(r)] o dr' [V,,V(r')I r'r'r' ~g(~') + ~ ~

(8.29)

dg(,') } d~ "

Since V~,V(r') is an odd function of r' and g(r') is an even function, integrals like,

f dr' [V,,V(r')]g(r') , fdr'[V,,V(r')]g(r')r'r', are zero. Terms which are proportional to such integrals of odd functions are omitted in eq.(8.29). The angular integrations in eq.(8.29) can be performed after subsitution of V~,V(r') - ~' dV(r ' ) /dr ' , with ~' - r ' /r ' the unit vector along r', and using that,

f d~'/" i-' - 47r [ (8 30) 3 '

f d~' r~ ' rj ' ~k ' ~z ^' = 4~15 [~k~ + ~k~j, + ~,~jkl, (8.31)

where the integration ranges over the unit spherical surface and where 6~j is the Kronecker delta (6q - 0 for i ~ j , and 6ij - 1 for i - j). For r >> Rv, where V~V(r) - O, eq.(8.29) thus yields,

dII 2 ] 0 - V~ f l v h ( r l - f l r ' V ~ h ( r ) , (8.32)

where,

2 r ~2 fo ~176 r, 3 dV ( r' ) II - # kBT - T dr' dr' g( r ' ) , (8.33)

/o { 1 ~ = 27r _ ,5 dV(r') g ( r ' ) + ~ . (8.34) 1"5 p dr' r dr' -8 dp

The quantities denoted here as 1I and E are short hand notations for the expressions on the right hand-sides which are found from the Smoluchowski equation (8.29) after performing the spherical angular integrations. The expression on the right hand-side of eq.(8.33) is precisely the osmotic pressure, which is denoted as II. The differential equation (8.32) is satisfied when,

E h(r) - V~h(r) . (8.35)

dII/d~

Since 1-c0 - fl dlI/dp, this is precisely the differential equation (8.11) that we found from the Ornstein-Zernike approach, when the following identification is made,

f i e - c2. (8.36)

This is of course not an exact relation, because the closure relation that was used to arrive at eq.(8.35) is not exact. With this identification, the asymptotic behaviour of the total-correlation function in eq.(8.12) together with the expression (8.14) for the correlation length are reproduced by the present Smoluchowski equation approach. The small wavevector behaviour of the static structure factor in eq.(8.16) is also recovered independently from the present approach, provided that one can show that ~ E ~ R~,. You are asked to show this in exercise 8.2.

There is an important feature to be noted about the differential equation (8.32). Very close to the critical point, fldII/d~ is very small, so that the first term between the square brackets in eq.(8.32) is not large in comparison to the higher order terms in h(r), which are neglected. This invalidates the linearization of the Smoluchowski equation with respect to the total-correlation function very close to the critical point. To describe critical phenomena extremely close to the critical point, higher order terms in h(r) must be included in eq.(8.32).

8.2.3 A Static Light Scattering Experiment

The Ornstein-Zemike static structure factor (8.16) can be used to measure the correlation length. Since for k << 27r/Rv the form factor of the Brownian particles is equal to 1, the scattered intensity I is directly proportional to the static structure factor with a wavevector independent proportionality constant C (see eq.(3.66) in the chapter on light scattering)" 1 - C~ (~-2 + k2). A plot

514 Chapter 8.

I I I

0 1 K', , IO ~' [m-2] 4

Figure 8.8:

3.5

~3.0

0 7 2.5

I I I @

2.0 I I I - 2 . 0 - 1 . 5 ~Otog(T_T~)-o.5 o.o

(a) The reciprocal scattered intensity versus k 2 for various temperatures (from top to bottom �9 18.66~ 18.56~ 18.45~ 18.34~ 18.24~ 18.14~ 18.10~ 18.05~ 17.98~ The critical temperature is 17.95~ (b) The correlation length versus temperature on a double logarithmic scale. Data are taken from Verduin and Dhont (1995).

of the reciprocal intensity against k 2 is therefore a straight line with slope C -1

and an intercept at zero wavevector equal to C -1~2. The square root of the ratio of the intercept and the slope renders the correlation length. No absolute light scattered intensities are needed to measure correlation lengths. An example is given in fig.8.8, which data are on the same silica dispersion for which the phase diagram is given in fig.8.1. Fig.8.8a shows reciprocal intensities versus k 2 for various temperatures and fig.8.8b is a double logarithmic plot of the correlation lengths (determined via the above described procedure) versus T - T~, with T~ - 17.95 ~ the critical temperature. The slope of the reciprocal intensity versus k 2 is seen to be a weak function of the temperature, showing that c2 = / 3 E is a well behaved function at the critical point. The solid line in fig.8.8b is a best linear fit, resulting in,

- (190 + 10 nm) x ( T - T~) -~176176 . (8.37)

The diameter of the silica particles is 80 nm, which is of the order of the prefactor of 190 rim. The value 0.522 4- 0.023 for the so-called critical exponent of the correlation length is in accordance with its theoretical mean- field value of 1/2.

8.3. Shear Flow Effects 515

8.3 The Ornstein-Zernike Static Structure Factor with Shear Flow

In the chapter on diffusion, sections 6.4 and 6.8, it is shown that the pair- correlation function and the static structure factor are singularly distorted by a linear shear flow velocity at large separations and small wavevectors, respectively. The reason for large effects on correlations by weak shear flows is the arbitrary large shear induced relative velocity that two Brownian particles attain for larger distances between those particles. Diffusion cannot restore shear induced changes of correlations at larger distances, no matter how weak the shear flow is. The relative velocity of the Brownian particles induced by the shear flow is always larger than the restoring diffusion velocities when the distance between the particles is large. Since correlations become very long ranged close to the critical point, this singular effect of shear flow becomes more and more pronounced on approach of the critical point. The Ornstein- Zernike static structure factor is therefore severely affected by weak shear flows, more so closer to the critical point. The singular distortion of the microstructure of systems close to the gas-liquid critical point due to shear flow is analysed in the present section.

The externally imposed linear fluid flow velocity uo(r) at position r is equal to,

uo(r) - F . r , (8.38)

with F the velocity gradient matrix. For a flow in the x-direction and linearly increasing in the y-direction, F is equal to, /010/

F - , ~ 0 0 0 . (8.39) 0 0 0

The shear rate ;y measures the gradient of the flow velocity in the y-direction. The stationary equation of motion for the pair-correlation function is ob-

tained in exactly the same manner as in the previous subsection 8.2.2 �9 the Smoluchowski equation (4.102,104), where hydrodynamic interaction is neglected, is integrated with respect to r3, �9 �9 �9 rN, assuming pair-wise additivity of the total potential energy (see eq.(8.19)). Recalling the definitions (8.18,22) it is found, precisely as in the previous subsection, that,

0 = 2DoV~ �9 {V~9(r [+) +/39(r ];y)[V~V(r) - r i . a ( r ];y)] }

516 Chapter 8.

j r . r g(r I;f)] , (8.40)

where Do is the Stokes-Einstein diffusion coefficient and,

F,,d(r I;Y) -- --P f dr' [V~,V(r')] g3(r, r ' l+) (8.41) g(r I+) '

is the shear rate dependent indirect force. Eq.(8.40) is (the divergence of) eq.(8.20) with an extra term that describes the influence of the shear flow. The shear rate dependence of the correlation functions is denoted here explicitly. The pair-correlation function is anisotropic, that is, it is a function of the vector r and not only of the absolute distance r - I r l as for the unsheared system.

Mathematically, the singular nature of the perturbing shear flow is due to the large numerical value of the term that describes the effect of the shear flow (the last term in eq.(8.40)) for large separations r, relative to the remaining terms which tend to zero for large separations. Even though ~ may be small, the perturbing term is large for large separations.

The same closure relation (8.25) that was used for the unsheared system is employed here, except that the correlation functions are now shear rate dependent. Moreover, for the calculation of the asymptotic behaviour of the pair-correlation function at large distances, the same procedure as for the unsheared system that is used in subsection 8.2.2 can be employed here �9 use of the Taylor expansions (8.27,28) and linearization with respect to the total correlation function h(r l~ ) - g(r I,~) - I yields (compare to eq.(8.29)),

0 - 2DoV~ �9 {V~h(r 1"7) +/3 {h(r I~) + 1} [V~U(r)]

{ 21 dg(r'l~/)}d/5 + #/3 [V~h(rl;r)]. f dr' [V~,V(r')] r' g(r 'l;/) + fi

- { 1 dg(r'l;Y)}} - p [V V.V,h(rl+)]| r ' r ' r ' ~g(r' I ;Y)+ ~-g/~ dp

- V ~ . [ F . r h(r I';/)] �9 (8.42)

The spherical angular integrations cannot be performed without knowledge about the anisotropic r'-dependence of the pair-correlation function g(r' I "~). A crucial point here is, that in the integrals, g(r' I ~/) is always multiplied with the pair-force V~,V(r'), which is zero for r ~ > Rv. The shear rate dependence of the integrals is therefore related to the distortion of the pair- correlation function for short distances. The shear induced distortion of the

0 Figure 8.9: A sketch of the pair-correlation function near contact for attractive spheres. g+ is the contact value, and the slope of the dashed line is equal to the slope dg / drl+ at contact.

( 9

\


2a . . . . r

pair-correlation function for such short distances is much less pronounced than the distortion for larger distances, because the perturbing term (the last term in eq.(8.40)) is larger for larger distances. Distortions for large distances are significant for shear rates where distortions for short distances are still insignificant. The order of magnitude of the combination F . r g(r [ "~) for small distances is "~ r g+, with g+ the contact value of the pair-distribution function (see fig.8.9). The order of magnitude of the first term between the curly brackets in eq.(8.40) is dg/drl+, the slope of the pair-correlation function at contact of the hard-cores (see fig 8.9). Since in equilibrium, without shear flow, the terms between the curly brackets in eq.(8.40) cancel, the perturbing shear term is small in comparison to the each of the terms between the curly brackets when ,~ r g+ << 2Do [dg/dri+ [. Hence,

Re ~ << Rv 1 dln{g) dr

[ =~ g(rl,:y)~g~q(r) , for r<__Rv, (8.43)

where g ~q (r) is the equilibrium pair-correlation function, without shear flow, and P e ~ is the bare Peclet number, defined as,

p e o = 2Do " (8.44)

The right hand-side in the inequality in (8.43) can be large for the systems with attractive pair-interaction potentials under consideration here, since the contact value of the pair-correlation function is large and the pair-correlation function decreases rapidly with increasing distance (see fig.8.9). Therefore, for not too large bare Peclet numbers, the shear rate dependence of the pair-correlation

518 Chapter 8.

function in the integrals in eq.(8.42) may be neglected. The spherical angular integrations can now be performed with the help of eqs.(8.30,31), precisely as in subsection 8.2.2, to obtain,

dII r

0 - 2OoV~t /~- - ~ h(r I~/) - / 3 Z V ~ h ( r l + ) . ~ - V ~ . [ r . r h ( r l + ) ] , (8.45)

where II and E are given in eqs.(8.33,34), with the pair-correlation function equal to the equilibrium pair-correlation function, that is, the pair-correlation function of the quiescent system, without shear flow. Both dII/d/~ and E are thus the same quantities as encountered in subsection 8.2.2 where a quiescent system without shear flow is considered. The differential equation (8.45) is the generalization of the Smoluchowski equation (8.32) which includes the effects of shear flow.

For short distances r' < Rv, the pair-correlation function g(r ' I -~) is a regular function of the shear rate, since the perturbing term is now small for small shear rates in comparison to the remaining terms in the Smoluchowski equation (8.40) (for a more detailed discussion, see section 6.8 in the chapter on diffusion). This means that g(r' I'~) can be Taylor expanded in a power series of the shear rate for r' < Rv. For not too large shear rates, the integrals in eq.(8.42) are therefore linear functions of the shear rate, or more precisely, linear functions of Pe ~ so that there are additional linear terms in the Smoluchowski equation (8.45). These terms may be neglected when the inequality in (8.43) is satisfied.

The differential equation (8.45) can be solved by Fourier transformation to obtain the shear rate dependent Ornstein-Zernike static structure factor. Just as for an unsheared system, the static structure factor is defined as,

1 N S(kl~) - ~ Y~ < exp{ik. (ri - rj)} > - 1 +/5 h ( k ] ~ ) ,

i,j=l (8.46)

"~kl OS(kl~) = 2D~Z(k)k 2 { S ( k l ~ ) - s~q(k)) Ok2

(8.47)

with h (k 1"7) the Fourier transform of the shear rate dependent total-correlation function h(r I'~) - g(r I'~) - 1. This is the quantity that is measured in a static light scattering experiment. Notice that the static structure factor is not just a function of k - I k I, but of the vector k. The scattered intensity of a sheared system is thus no longer isotropic, and depends on the direction of the scattering wavevector. You are asked in exercise 8.3 to show that Fourier transformation of eq.(8.45) yields,

where kj is the jth component of k and where the wavevector dependent effective diffusion coefficient is equal to,

+ k2~ . (8.48)

The equilibrium static structure factor S ~q (k) is the Ornstein-Zemike static structure factor (8.16) without shear flow, with c~ = fie (see eq.(8.36)),

1 ~2 s~q(k) = fie 1 + (k() 2" (8.49)

These equations are correct for small wavevectors k << 27r/Rv, due to trun- cation of the Taylor expansions (8.27,28).

Above the spinodal, in the (meta-) stable region of the phase diagram, where dII/d~ > 0, the effective diffusion coefficient is positive for all wavevectors. 3 In the unstable part of the phase diagram, however, dII/dp < O, so that D~SS(k)is negative for wavevectors k < - - - - ~ - ] ( - d I I / d ~ ) / E (see also exercise 8.4). In the next chapter it will be shown that this implies that sinusoidal density variations corresponding to such small wavevectors will increase in time, eventually leading to phase separation. In the (meta-) stable region in the phase diagram, all sinusoidal density variations (with small enough amplitude) decay towards the homogeneous state. The discussion in the present chapter is restricted to phenomena in homogeneous systems in the (meta-) stable region of the phase diagram, where the effective diffusion coefficient is always positive.

The differential equation (8.47) is solved in the appendix A, with the result,

1pe [._~~176 (]k~ X2+k2a) AS(kI~/) - S ( k l a / ) - S ~ q ( k ) - k x Jk2 dxQ + 1

x [seq(]k~+ x2+ k~)-seq(k)] exp{ kl Pe (P(k) l ' a= ' -P(k))} ' (8"50)

where the functions Q and P are equal to,

Q(k} - (k Ru} = [1+(k r =] , {8.51}

P(k) fo k= " .

aThis is true provided that E is positive. Since the static structure factor is positive by definition, it follows from eq.(8.49) that this is indeed the case.

520 Chapter 8.

The + ( - ) sign in the upper integration limit in eq.(8.50) is to be used for positive (negative) values of kl Pe. The dressed Peclet number Pe that is introduced here is equal to,

(~ dII~-1 ;~ R~r 1 A/~2 P e - -d-~p] P e ~ 2 D r O) = f l E / n ~ 2Do' (8.53)

where the bare Peclet number Pe ~ is defined in eq.(8.44). The amount of distortion of long ranged correlations is measured by this dressed Peclet number, while the bare Peclet number measures the amount of distortion of short ranged correlations. The numerical value of the dressed Peclet number is much larger than the bare Peclet number, since ~ dlI /d~ is small close to the spinodal. This confirms the reasoning that led us to neglect the shear rate dependence of the integrals in the Smoluchowski equation (8.42).

Notice that the dressed Peclet number is roughly obtained from the bare Peclet number by replacing the range of the pair-interaction potential R v by the correlation length ~ of the quiescent system.

Scaling

The expression (8.50) for the static structure factor looks quite complicated. It can be substantially simplified by scaling the wavevector to the correlation length. Let us therefore introduce the dimensionless wavevector,

K = k~ . (8.54)

Define the relative static structure factor distortion ~ as,

- S ( K I ; 7 ) - S ~ q ( K ) . (8.55) S~q(K) - 1

Scaling the wavevectors in eqs.(8.49-52) to the correlation length, and substitution of the expression (8.49) for the static structure factor, yields a relatively simple expression for the relative distortion, namely,

~ ( K I A ) - )~KIJK~ ', -- ~ Ki '

where,

F(KIA)

.3. Shear Flow Effects 521

~igure 8.10: dinus the relative static structure factor distortion (8.55) as a function o f K~ u~d K~ with Ka - 0 (left column) and as a function of K1 and Ka with K2 - 0 right column). The values o f A increases from top to bottom as indicated. qumbers indicate the maximum and minimum values o f ~. The scales on the "Q, K2 and Ka axis are indicated. For example, in the left lower figure, K1 anges from - 35 to +35.

522 Chapter 8.

1 1 (X 5 + ~ (X 3 - K2 3) (1 + 2K 2 - 2K2 2) +

(8.57)

-h':), and where A is a dimensionless number, equal to,

1 - " (8.58)

Besides being a more simple expression than eqs.(8.49-52), there is a fundamental feature about these new expressions, namely, that the both the shear rate dependence and the temperature dependence (through the correlation length) are now entirely described in terms of the single dimensionless number A. Identical numerical values of A give rise to the same relative distortion ~, considered as a function of the scaled wavevector K. A single numerical value of A relates to many different shear rates and temperatures. Notice, however, that the scaled wavevector is also temperature dependent. This scaling behaviour of the static structure factor has profound implications for the shear rate and temperature dependence of for example the turbidity, flow induced dichroism and viscosity.

Since �9 = 0 for zero shear rates where A = 0, and A occurs in eq.(8.56) only as a product with K~, there is no distortion in directions perpendicular to the flow direction,

lim A S ( K I ; y ) - 0 . (8.59) K1 ~ 0

The relative static structure factor distortion �9 is plotted in fig.8.10 as a function of (K1, K2, 0) and (K~, 0, K3) for various values of A. First of all it is seen that for A < 1 there is hardly any distortion, while there is severe distortion for A > 1. The transition from "weak" to "strong" shear flow thus occurs at A ,~ 1,

A < 1 =~ weak shear flow, } A > 1 =~ strong shear flow. (8.60)

According to eq.(8.58), A ,-~ ,~ ~4, so that, on approach of the critical point, smaller and smaller shear rates are sufficient to give rise to significant distortions. In other words, at a constant shear rate, distortions increase on approach of the critical point. This is a result of the unlimited increase of the correlation length (.


Figure 8.11" The static struc~,ure factor as a function of K~ and K2 with K3 = 0 (upper figures) and of K1 and K3 with K2 = 0 (lower figures), for A = 10 and 100. The most left figure is the equilibrium Omstein-Zernike static structure factor. A value of 1/100 is chosen forthe quantity (Rv/~)2( f lE/R~) . The mostright figure is an experimental scattering pattern (with K2 = 0).

As can be seen from fig.8.10, the relative distortion (8.55) is positive in directions where KI = -K~ and Ka = 0. This means that a more pronounced microstructure ~s induced by the shear flow in these directions. Such an enhancement of microstructure can be understood intuitively by decomposing the simple shear flow into an extensional flow and a rotational flow, as depicted in fig.2.3 �9 the extensional flow drives colloidal particles towards each other in the directions where x - - y .

Correlation lengths of the sheared system

A plot of the static structure factor S(K I';/) itself instead of its relative distortion @ is given in fig.8.11. This figure illustrates that, for A > 1, the static structure factor is severely affected in directions where the component of the wavevector along the flow direction is non-zero, that is, when K1 ~ 0, and remains intact in directions where K1 - 0. In a light scattering experiment this results in a bright stripe of scattered light, which is indeed observed for colliodal systems (see the most right figure in fig.8.11) as well as for near

524 Chapter 8.

critical binary fluids (see for example Beysens and Gbadamassi (1981)). The experimental result in fig.8.11 is for a mixture of polydimethylsiloxane and stearyl coated silica particles in cyclohexane close to its critical point.

That the microstructure is unaffected by the shear flow in directions where K1 = 0 follows from eq.(8.59). That result, however, is obtained with the neglect of the shear rate dependence of the short ranged behaviour of the pair- correlation function in the integrals in the Smoluchowski equation (8.42). As discussed before, there are linear terms in P e ~ for not too large values of P e ~ (such that the inequality (8.43) is almost satisfied) which should be added to dI I /d~ and E in the Smoluchowski equation (8.45). Eq.(8.59) therefore holds up to linear terms in Pe ~ and the correlation length ~0 in directions where K~ - 0 is a regular function of Pe ~ that is,

- + r + + . . . , (8.61)

where the expansion coefficients ~(n) are of order unity. Whether the correlation length increases or decreases due to shear flow is determined by the sign of the coefficient ((1). The calculation of ((1) requires an analysis of the static structure factor distortion at large wavevectors, or equivalently, of the pair-correlation function at short distances.

In directions where Kx r 0, such an expansion certainly fails. In those directions a very small bare Peclet number gives rise to a large distortion close to the critical point, where A is large, also for very small shear rates. The correlation length of the sheared suspension is now a non-analytic (or equivalently, a singular) function of Pe ~ As discussed above, there is structure induced in the direction where KI = -1(2 and Ka = 0. The "size" of the induced structures is of the order of 27r/kin, where km is the magnitude of the wavevector where the maxima in �9 occur. It is apparent from fig.8.11 that the sheared static structure factor decreases first for very small wavevectors relative to the equilibrium static structure factor, before becoming larger at some finite wavevector. Hence, no very long ranged correlations are induced, and the correlation length always decreases due to shear flow, also in the direction where structure is induced (except may be in directions where K1 - 0, as discussed above).

8.4. Turbidity 525

8.4 The Temperature and Shear Rate Dependence of the Turbidity

The most simple experimental quantity that measures changes in microstructural properties is the turbidity r. The definition of the turbidity and the derivation of an expression for this quantity in terms of the static structure factor is derived in the following paragraph. The subsequent paragraph discusses scaling properties of the turbidity as derived from the shear flow distorted static structure factor in the previous section. Finally, the predicted scaling behaviour is tested against experiments on the stearyl silica/benzene suspension of which the phase behaviour was discussed in section 8.1, and of which the phase diagram is given in fig.8.1.

The definition and an expression for the turbidity

Consider an experiment where the intensity of a laser beam, directed along the z-direction, is measured before and after passing through a suspension. These intensities differ by an amount equal to the total scattered intensity, provided no absorption of light occurs. Conservation of energy requires that,

A [I(z) - I ( z + dz)] - - A dz dI (z ) Js dS I~(0, ) (8.62) -~z - R qo ,

with A the cross-sectional area of the laser beam, l(z) the intensity at the point z as measured relative to the point where the laser beam enters the suspension (see fig.8.12), dz is an infinitesimally small increment of that distance, and 1, is the intensity that is scattered by the infinitesimally small scattering volume V~ = A dz in the suspension located between z and dz. The spherical angular dependence of the scattered intensity is denoted explicitly. The integral ranges over a spherical surface SR with an arbitrary large radius R. The scattered electric field strength at points on SR is decomposed into two perpendicular polarization directions,

fi~o - ( - sin{~}, cos{qo}, O) ,

and,

fie - (cos{O} cos{~o}, cos{O} sin{~o},- s in{O}),

where O and qr are the spherical angular coordinates. According to eqs.(3.66,56), the intensity scattered from the volume element at z as sketched in fig.8.12 is

526 Chapter 8.

d7

I I I I i I II II II I I

...... [ I

Z : L taser beam cuvette

A

Figure 8.12: The turbidity measurement. The laserbeam is polarized in the x-direction and propagates along the z-direction, entering the cuvette at z = O. The cross sectional area of the beam is A and the length of the cuvette is I.

equal to,

h(o, ~,) - I ( z ) A dz R2 C~. P(k) S(k I ;r) f(O, v ) , (8.63)

with C~- a constant equal to,

k~) ,-p - es , C~- (47r12PVp21 ef 12 (8.54)

and (with rio = (1, 0, 0) the polarization direction of the laser beam),

f(O, ~;) - ( a o . ao) ~ + ( ~ . ao) ~ = sin2{qa} + cos2{cp} cos2{O}. (8.65)

The wavevector k in eq.(8.63) is equal to,

k - k o - k , - -ko (sin{O} cos{qp}, sin{O} sin{qp}, cos{O} - 1), (8.66)

where ko = (0, 0, 1) and k~ are the incident and scattered wavevector, respectively. Substitution of eq.(8.63) into eq.(8.62) yields the following differential equation for the intensity of the laser beam,

dI(z) = --r I ( z ) , (8.67)

dz

8.4. Turbidity 527

with r the turbidity of the suspension, which is equal to,

j~o 2r fo r 7 - C, dqp dO sin{O} P(k) S(k[-~) f (O, qo), (8.68)

where it is used that,

JsR(..-) - R 2 fo2'~ dcp rondo sin{O} ( . - . ) .

The solution of eq.(8.67) is simply,

It - lo e x p { - r 1}, (8.69)

with It the intensity of the laser beam that passed through the cuvette of length l, and with lo the incident intensity. This is the famous Lambert-Beer law when the loss of intensity were due to absorption, in which case the turbidity should be replaced by the extinction coefficient. Here we assumed no absorption, so that the loss of intensity is entirely due to scattering.

The turbidity can be measured with the use of eq.(8.69), simply by measuring It relative to the incident intensity 10. On the other hand, the turbidity can be calculated from eq.(8.68), once the wavevector dependence of the form factor and the static structure factor is known. Turbidity measurements can thus be employed to study the shear rate and temperature dependence of the static structure factor for systems close to the critical point as calculated in the previous section.

A scaling relation for the turbidity

The change of the turbidity on applying a shear flow relates to the change of the static structure factor according to eq.(8.68) as,

j~o 2~r j~o r r('~) - r *q - C, d~ dO sin{O} P ( k ) / X S ( k l q ) f(O, ~p), (8.70)

where r('~) is the turbidity of the sheared system and r ~q of the unsheared, quiescent system, and A S ( k I ~/) - S ( k I ~) - S~q(k). For small values of the bare Peclet number, such that the inequality in (8.43) is satisfied, the distortion of the static structure factor for larger wavevectors where k > 27r/Rv is negligible. The change in the turbidity is then related to the distortion of the static structure factor for small wavevectors, which is calculated in the previous section.

528 Chapter 8.

The integration with respect to 19 can be recast into an integration with respect to the dimensionless scaled wavevector in eq.(8.54). Using that sin{O/2} - ~/(1 -cos{O}) /2 , it is found from eq.(8.66) for k that, k -

2ko sin{O/2} (see also exercise 3.5). Hence, dk - kok/1 - k2/4k2o dO, and

cos{O} - 1 - k2/2k2o . Furthermore, sin{O} - 2 sin{O/2}~/1 - sin2{O/2},

so that, sin{O} - (k/ko)~/1 - k2/4ki '. Transforming from O-integration to k-integration in the expression (8.70) for the turbidity thus yields,

_

CT /2r [2ko k2 ~ dqOao dk k P(k) zXS(kl;Y)

[ { x sin 2{qo}+cos 2{qo} 1 - ~00

(8.71)

1

where eq.(8.65) for f(O, q;) has been used. For the small wavevectors under consideration here, the form factor may

be taken equal to 1. In addition, the spherical coordinate O may be assumed small enough to Taylor expand the wavevector (8.66) to linear order,

k ~ -ko 19 (cos{T}, sin{T}, 0) ~ - k (cos{T}, sin{qo}, 0) . (8.72)

Let us now denote the relative distortion �9 in eq.(8.55), with the scaled wavevector equal to K - k( - - K (cos { ~ }, sin{ qo }, 0), as �9 t, that is,

~t(K, ~IA) - ~ (K - - K (cos{~}, sin{~}, 0)]A). (8.73)

The change of the turbidity in eq.(8.71) can now be rewritten as,

_

C~~o2~ [2K0 ~o d~ao dK K Ot(If, qO ]')) [s~q(K) - 1] (8.74)

[ { ( K ) 2 1 ( K ) 4 } ] x sin2{cp} +cos2{cp} 1 - /To + ~- ~ ,

where Ko = ko~. As a last step in the derivation of a scaling relation for the turbidity, the correlation length is assumed large enough in comparison to the wavelength of the light to set the upper integration limit in the above expression equal to oo, and to neglect the terms ~,, (K/Ko) 2. This can be done when the integrand is essentially zero for K > K, and (K/K0)2 < 1/10 say, hence, ~ > AK/2. Typical values for K are found by numerical integration

8.4. Turb id i ty 529

to be equal to 2 - 6 in the range A - 10 - 1500. The correlation length should therefore be of the order of the wavelength of the light or larger. For such large correlation lengths, the dependence on the dimensionless number Ko is lost, and the change of the turbidity is completely determined by the numerical value of A. With the use of eq.(8.49) for the equilibrium static structure factor, the above expression for the change of the turbidity can be written in the scaling form we were after,

C~ 1 v(;y) - v *q - ( k o R v ) 2 f l E / R ~ , T(A), (8.75)

where the turbidi ty scal ing func t ion T ( A ) is equal to,

L2= f0 ~ K ~t(K,~l~/) (8.76) - d K K : +-----S

The experimental implication of this relation is as follows. For two experiments at two different shear rates and temperatures, such that the numerical value of A is equal for both experiments, the same turbidity change should be measured. In other words, when the shear rate dependence of the change of the turbidity at various temperatures is plotted as a function of A, these data should collaps onto a single curve. That "master curve" is the turbidity scaling function (8.7 6).

Experimental data on the stearyl silicafoenzene system that is discussed in the introduction are plotted in fig.8.13a. This figure shows the shear rate dependence of the turbidity for various temperatures. As can be seen, on approach of the critical point, a larger effect of shear flow is measured for the same shear rate. Close to the critical point, very small shear rates are sufficient to diminish the turbidity substantially. This is formally due to the large value of A ,,~ ~ (4 for small shear rates, as a result of the large correlation lengths ~ close to the critical point. Physically these larger effects on approach of the critical point are due to the fact that smaller shear rates are sufficient to affect correlations that extend over larger distances. As can be seen from fig.8.13b, the experimental data collaps onto a single curve when plotted as a function of A, and moreover, the data follow the theoretical prediction (8.75,76) quite closely. Relating the measured change T('~) -- r ~q of the turbidity to the scaling function T(A), and the product -~ ~4 to A (where the correlation length for each temperature is calculated from eq.(8.37)), involves unknown proportionality constants. In constructing fig.8.13b from fig.8.13a, these two proportionality constants were used as "fitting parameters". There is some

530 Chapter 8.

0 r-sL-n_

1=7" QJ

6" -2000

4+000

Q I

, I ,

~0

T (*C)

1841

18 35

18 30

18 25

1821

18 18

18 10

18 06

-- 18 01 I

o 8o 12o o

Figure 8.13"

o I I I

~x

-6

- 8 . . . . I . . . . I . . . . I . 5oo ~ 1ooo ISOO

(a) The turbidity as a function of the shear rate for various temperatures. The system here is the stearyl silica/benzene system that is discussed in the introduction. The critical temperature of the suspension is 17.95 ~ (b) The same data as in (a), but now plotted as T(A) ,,~ T(;y) - r ~q versus A. The solid line is the turbidity scaling function in eq.(8.76) obtained by numerical integration. Data are taken from Verduin and Dhont (1995).

discrepancy between the proportionality constant relating ,~ ~4 to A and its estimated value. This may be due to our neglect of hydrodynamics and the approximations involved in the closure relation (8.25) that was employed (see Verduin and Dhont (1995) for more details).

8.5 Collective Diffusion

Besides the long wavelength microstructure, also the diffusive behaviour of the Brownian particles changes drastically on approach of the critical point. This section is concerned with the anomalous behaviour of the short-time collective diffusive coefficient. The short-time self diffusion coefficient, on the contrary, is well behaved near the critical point, as shown in exercise 8.6.

In subsection 6.5.2 in the chapter on diffusion the following expression for the short-time collective diffusion coefficient is derived,

D (k) - Do H(k) S(k) '

(8.77)

8.5. CollectiveDiffusion 531

where the hydrodynamic mobility function H (k) is an ensemble average of hydrodynamic interaction matrices D ij, which are referred to as the microscopic diffusion matrices,

N D~j 1 ~ < (1~. �9 1~) exp{ik . ( r i - r j ) ) >o , (8.78)

H(k) - -N i,j=l Do

with l~ - k/k and where < . . . >o denotes ensemble averaging with respect to the equilibrium pdf.

With the neglect of hydrodynamic interaction, in which case H(k) - 1, it follows from eqs.(8.16,14,36) that DeS is equal to the effective diffusion coefficient in eq.(8.48),

- + E . (8.79)

Close to the spinodal, and in particular close to critical point, where/3 dII/db << 1, the short-time diffusion coefficient is much smaller than the diffusion coefficient without interactions, the latter of which is equal to the Stokes-Einstein diffusion coefficient Do. The decrease of the collective diffusion coefficient on approach of the critical point is commonly referred to as critical slowing down, and is observed experimentally both in molecular and colloidal systems.

Hydrodynamic interaction gives rise to an additional term for the short- time collective diffusion coefficient. As we are concerned here with the effects of long ranged correlations on diffusion, it is sufficient to use the leading term in the Taylor expansion of the microscopic diffusion matrices with respect to the reciprocal distance between the Brownian particles. These leading order expressions are derived in the chapter on hydrodynamics (see section 5.8 and exercise 5.5, and for the first few higher order terms see eqs.(5.84,95)),

o0 + o } 8.80,

D~j - Do 67rr/oaT(r) + O , i r j , (8.81)

where T( r ) is the so-called Oseen matrix (see section 5.6 in the chapter on hydrodynamics),

T( r ) = 1 1 [ i + H ' ] (8.82) 87rr/o r

532 Chapter 8.

with ~ = r/r. The first term between the curly brackets in eq.(8.80) is the only term that survives when hydrodynamic interaction is neglected, and yields the above result (8.79) for the short-time collective diffusion coefficient. The Oseen contribution to the "off-diagional" microscopic diffusion matrix in eq.(8.81) gives the leading order correction to D~ as a result of hydrodynamic interaction. Let AD~ denote the corresponding additional contribution to D~. It follows from eq.(8.78) for the hydrodynamic mobility function that, for identical Brownian particles (with r - ri - rj),

[j ] AD~(k) = S(k) ~r drg(r) T(r) exp{ik, r} �9 l~

67rr/oaD0 s(k)

+

zf . [/dr h ( r ) T ( r ) e x p { i k , r}] . l~

S(k) p [~" dr T(r) exp{ik, r} �9 1~.

The total-correlation function h(r) is equal to g(r) - 1. The integral in the first equation is divergent, since the integrand tends to zero at infinity too slowly. The first integral in the second equation, however, is convergent, since h(r) --. 0 for r ~ ~ . The divergent contribution is contained in the last integral in the above expression. On taking the inner product with 1~, however, the divergent integral is seen not to contribute, since the Fourier transform of the Oseen matrix is ,~ [ I - 1~1~] (see eq.(5.137)in appendix A of chapter 5). Substitution of eq.(8.12) for the total-correlation function, and using expression (8.82) for the Oseen matrix now yields,

if ] AD~(k) - S(k) ~ ~ " dr h( r )T( r ) exp{ik, r} �9 1~ (8.83)

3Doa_ [ f exp{-r /~} 1 []+i'~] exp{ik r}] k = 4 S(k) p (ARv)(r >dr r -r " " "

The lower integration limit is taken equal to r - ~ with ~ the smallest distance at which the total-correlation function is well represented by the Omstein- Zernike form (8.12). This lower limit is a few times the range Rv of the pair-interaction potential. Since the above expression is independent of the direction of the wavevector k, its direction may be taken along the z-axis, so that the above expression reduces to (with ~ di- the integral ranging over the unit spherical surface, and x - kr),

~ i -k- ) 3Doe [ Je oo q~ [ ( z ) 2 ] r aD~(k) = p ( A R v ) _ dr exp{-r/~} r di" 1+ exp{ikz}

3 D o a oo 1 (z) [ (z)2] { z} 4S(k) p (ARv)2r fe dr exp{-r/~} f-1 d 1 + r exp ikr r

3Doa 4r fkOOdxexp{_x/(k~)} [sin{x } d 2 sin{x)] = 4S(k)t~ (dRv)--ff ~ x dx 2 x "

Since the validity of the above expressions is limited to the wavevector range where kRv << 1, and since g is a few times Rv and Rv << (, it is easily seen that the lower limit in the last integral may be set equal to 0. Using that f o dx e x p { - x / a ) sin {x }/x - arctan {a}, and performing two partial integrations yields,

3Ooa AD~(k) - 4 S(k) fi (ARv)-~ 1

Substitution of eq.(8.16,36) for the static structure factor, of eq.(8.15) for ARv, with fldlI/d# << 1, and using eq.(8.14) for the correlation length, finally yields,

- Do( /a (1 + 2) + AD;

-- Do (flE/a 2) (1 + (k~) 2) + ~ F(k~) , (8.84)

with F the Kawasaki function,

F ( z ) - ~31+z 3z 2 [z+ (z 2 - 1 ) arctan{z}] . (8.85)

This function is plotted in fig.8.14. Two limiting cases are of interest here. For wavevectors where k( << 1, the above expression reduces to (use that 1 3 1 5)~ arctan{z) ~ z - 5z + gz

D~ - Do a 3 )2 4) ] . ( l + ( k ~ ) 2 ) + ~ ( 1 + ~ ( k ~ +O((k~) )

Since flE/a 2 is of the order unity and ( >> a close to the critical point, the Kawasaki contribution is dominant, and the above expression reduces to,

a ( 3 ) ) D~ - Do~ l + g ( k ~ ) 2 = 6~o~ l + g ( k ~ ) 2 , k~<<l . (8.86)

534 Chapter 8.

2.5

F(z)

2.0

1.5

Figure 8.14: 1.0

I I ! 1 i

0 0.5 1.0 1.5 z 2.0

The Kawasaki function (8.85).

The second limiting expression of interest is for somewhat larger wavevectors where k( >> 1. Using that arctan{z} - 7r/2 - 1/z + O(1/za), now yields,

[ (~)2 2) a(~_ ~ D~ - D0 (/3E/a 2) (1 + (k~) + ~- k~ + O

Since this expression is valid in the small wavevector range where k << 27r/Rv, and Rv is of the same order of magnitude as the radius a of the Brownian particles, it follows that ka << 1, so that the Kawasaki contribution is again dominant, and the above expression reduces to,

a ( ~ 3) kBT 37r D~ - Do~- k( + ~ - 67rr/o( -if k~r k~ >> 1. (8.87)

For wavevectors for which k~ ~ 1, the Kawasaki contribution is also dominant close to the critical point, since then ( >> a, so that eq.(8.84) can be approximated in the entire wavevector range where k << 27r/Rv as,

a kBT D~ ,,~ Do -~ FCk~) - 67rr/o,~ F(k~). (8.88)

The wavevector dependence of the short-time collective diffusion coefficient thus changes from a linear function of the wavevector (for k~ >> 1) to a constant (for k~ << 1). This behaviour is also apparent from the plot of the Kawasaki function in fig.8.14.

It should be noted that higher order hydrodynamic interaction is neglected, since only the Oseen contribution is considered. The corresponding higher order contributions to the hydrodynamic mobility function H(k) in eq.(8.78)

8.6. Shear Viscosity 535

remain finite at the critical point. These so-called "background" contributions must be dealt with when comparing experimental data with the Kawasaki prediction. This is not a trivial matter. Furthermore, there has been some debate on the viscosity that should be used in the Stokes-Einstein diffusion coefficient Do. For colloids this is clearly simply the solvent viscosity, but for molecular systems things are less clear (see Kawasaki (1970) and Kawasaki and Shih-Min Lo (1972)). Experiments seem to verify the above predictions for the collective diffusion coefficient (see for example, Lao and Chu (1975) and Meier et al. (1992)).

Further away from the critical point, where ~ < 10 a say, the contribution of the short-time collective diffusion coefficient without hydrodynamic interaction can be important, depending on the numerical value of ~E/a 2. Note also that formal extrapolation of experimental diffusion coefficients in the range k~ >> 1 to k~ - 0 yields Do (flE/R~)(a/~) 2, which relates to the small contribution that one obtains with the neglect of hydrodynamic interaction.

In the previous two sections 8.3 and 8.4, where the effect of shear flow on correlations is considered, hydrodynamic interaction is neglected. In that case the short-time collective and effective diffusion coefficient are equal (see eqs.(8.48) and (8.79)). One might conclude that the neglect of hydrodynamic interaction in the previous two sections is not justified, since we found above that the leading order effect of hydrodynamic interaction on the short-time collective diffusion coefficient, represented by the Kawasaki function, is usually dominant. However, not only is the expression for the contribution of hydrodynamic interaction to D ~ff in eq.(8.48) a different one than for D~, but also the effect of shear flow is to severely diminish the range of the total- correlation function in most directions. Integrals like in eq.(8.83) are therefore much smaller than for the sheared case considered in the previous sections, and as a result, the corresponding Kawasaki contribution to the effective diffusion coefficient in eq.(8.48) is much smaller than in the present case.

8.6 Anomalous Behaviour of the Shear Viscosity

The range of correlations is large close to the critical point and ultimately diverges. This implies that close to the critical point many Brownian particles interact simultaneously, and at the critical point each Brownian particle interacts with aH other Brownian particles in the system. This is the mecha-

536 Chapter 8.

nism that leads to very large and ultimately infinite forces that are required to induce relative displacements of Brownian particles, corresponding to a large and ultimately diverging shear viscosity. It is known that the divergence of the (zero frequency) shear viscosity for molecular systems is extremely weak, and probably only occurs on very close approach of the critical point, beyond the mean-field region. Hydrodynamic interaction, absent in molecular systems, is of major importance for the viscous behaviour of suspensions and leads to a much stronger divergence of the shear viscosity. The aim of the present section is to predict the divergence of the (zero frequency) shear viscosity of colloidal systems. In addition, the very pronounced shear thinning behaviour close to the critical point is considered.

The first problem is to derive a microscopic expression for the shear viscosity, that is, an expression that relates the shear viscosity to an ensemble average of functions of the position coordinates of the Brownian particles. The next step is to evaluate the ensemble average with the use of results from section 8.3 on the shear rate dependence of the static structure factor.

8.6.1 Microscopic Expression for the Effective Shear Viscosity

Let ~r be the rate at which energy is dissipated per unit volume. Suppose a simple shear flow with velocity gradient matrix (8.39) is induced by applying a force F on a fiat plate. The constant velocity of that plate relative to a second stationary plate is "~l, with I the distance between the two plates (with the suspension inbetween). The rate of energy dissipation is ~lF. The force F and the shear rate "~ are related, by definition, through the viscosity r/as, F / A - ~77, with A the surface area of a plate. Hence, U - ,~lF/1A - ~7~ 2. On the other hand, the dissipated energy is given in terms of the hydrodynamic forces F h that the fluid exerts on the Brownian particles i = 1, 2 . - - N , and the extra velocity AV~ that each particle attains as a result of the applied shear field,

1 N = V ~ < AVi . F h > , (8.89)

i=1

with V the volume of the system, and < . . . > denoting ensemble averaging. Hence,

1 N r/ -- .~2 V ~ < AVi . F h > . (8.90)

i=1


The shear induced velocity of a Brownian particle i is the local velocity of the suspension, 1" �9 ri, with ri the position coordinate of the i th Brownian particle, plus a contribution due to the disturbance of the local fluid flow by the other Brownian particles. The incident flow field 17'. r is scattered by each of the Brownian particles, thereby affecting the motion of the other Brownian particles. This contribution is denoted as C~ �9 1". Hence,

AV~ - F . ri + C~(rl, r 2 , . . . , rN) " 1". (8.91)

! The disturbance matrices Cj of indexrank 3 are complicated functions of all the position coordinates of the Brownian particles. Leading order expressions are derived in section 5.13 in the chapter on hydrodynamics. For the calculation of the anomalous behaviour of the shear viscosity, these leading order expressions suffice. According to eq.(5.113), the disturbance matrix is then a sum of matrices C depending on just two position coordinates (rij - ri - rj),

N C~ - ~ C(r i j ) . (8.92)

3=1 j r

For the evaluation of the effective viscosity we will need the explicit leading order expression for the divergence of the vector C �9 F, which was evaluated in section 5.13 (see eq.(5.114)),

75 a ( C ( r , j ) �9 r ) - y r . (8.93)

where ~ij - rij /r i j .

On the Smoluchowski time scale, the total force on each Brownian particle is zero, so that the hydrodynamic forces F ) are equal to minus the sum of the direct force,

I F j -- - - V j ( I ) ( r l , r 2 , " ' , r N ) , (8.94)

and the Brownian force,

(8.95)

with Vj the gradient operator with respect to rj, and PN the pdf of the position coordinates. In equilibrium (,~ - 0) these two forces add up to zero, yielding the Boltzmann pdf PN "~ e x p { - ~ / k B T } . In a sheared system the external force induces an unbalance between the two forces, so that PN is no longer

538 Chapter 8.

equal to the Boltzmann exponential. This effect of the shear flow on the pair- correlation function g - V 2 f dr3. .- f drN PN was analysed in section 8.3. The ensemble average in eq.(8.90) is to be taken with respect to the shear rate dependent pdf.

Substitution of eqs.(8.91,94,95) into eq.(8.90) gives,

1 N - ~/~v ~ < ( r . r , + c',- r ) . (v,~ + kBTV, ln PN) > .

i=1 (8.96)

There are further contributions to the viscosity which stem from direct interaction of solvent molecules with the Brownian particles, the hydrodynamic viscosity, and from interaction between solvent molecules. These contributions will not be considered here. The direct interactions between the Brownian particles become long ranged upon approach of the critical point, while the other direct interactions remain short ranged and do therefore not contribute to the anomalous behaviour of the effective viscosity. Actually, in much the same way as eq.(8.96) will be analysed in the present section, the anomalous part of the hydrodynamic viscosity can be evaluated, with a totally negligibly small result, confirming that the short ranged direct interactions between the solvent molecules and the Brownian particles do not contribute. The interactions between the Brownian particles become long ranged, and only these give rise to the anomalous behaviour of the viscosity.

The sum of the various contributions to the viscosity that are well behaved at the critical point are referred to as the background viscosity. The background viscosity is the contribution stemming from short ranged interactions, and changes smoothly right up to the critical point.

8.6.2 Evaluation of the Effective Viscosity

The microscopic expression (8.96) for the viscosity is written, for convenience, as a sum of four terms,

- ~$ + ~? + ~ + ~ , (8.97)

with,

1 N

~ v ~ < (c',. r ) . v , r > , i--1

r/c sr =

1 N "? = ~ v Z < (r. ri). V,r > ,

i--1 1 N

#~v ~ < (c~. r) . k~T v, 1. PN >, i=l

1 N r/Br = ~/2V E < ( r . r i) . k B T V i l n P N > , (8.98)

i=1

where the superscripts Br and �9 refer to the Brownian and direct force terms respectively, and the subscripts C and r to the terms involving C~ and 1"- ri.

Most of the terms here are regular functions of the bare Peclet number Pe ~ which do not contribute to the anomalous behaviour of the viscosity but constitute contributions to the background viscosity.

Let us consider each of the contributions to the effective viscosity in eq.(8.98) separately.

The contribution ~7~

Substitution of eq.(8.92) for C~, assuming a pair-wise additive potential energy and identical colloidal particles yields,

~2 r/~ = ~-2 f d R g ( R ] ' ~ ) ( C ( R ) �9 r ) . VRV(R)

t~ 3 + ~ f dr f dR g~(rt, r I#)[V~V(~)]. (C(R) �9 r ) .

The first integral on the right hand-side probes the shear rate dependence of the short ranged r-dependence of the pair-correlation function, since it is multiplied by V~V(r). As we have seen in section 8.3 (see in particular eq.(8.43)), the pair-correlation function g(r [ ~) is a regular function of Pe ~ for r <_ Rv , with Rv the range of the pair-interaction potential. The first integral is therefore a regular function of Pe ~ and does not contribute to the anomalous behaviour of the viscosity. The second integral may be evaluated as follows. In order to separate the anomalous part form the background contribution, the total-correlation function is decomposed in a long ranged and a short ranged contribution, ht and h, respectively,

g(r 1#) - 1 + ht(r I+) + h~(r I+). (8.99)

Formally, the long ranged part is defined as the asymptotic solution of the Smoluchowski equation for large distances as found in section 8.3. The

540 Chapter 8.

remainder is the short ranged part. What is important is that the short ranged part is a regular function of Pe ~ since by definition h, is zero for distances larger than a few times Rv. The anomalous contributions to the viscosity are due to the long ranged contribution of the correlation functions. Much the same procedure that was used in section 8.3 can be applied here to evaluate the integral.

First of all, linearization with respect to the long ranged contributions ht is allowed since the total-correlation function goes to zero at infinity. After substitution of the decomposition (8.99) into the closure relation (8.25), with r replaced by R and r' by r, such a linearization leads to,

r/c ~ - ~-{ dr dR [1 + h i ( R - r I+)+ h t (R]#)+ h,(R ]-~)h,(R- r I+)

+h~(R - r I'~) + h,(Rl~/)ht(R - r I'~) + h , (R I ~/)h,( R - r I~/)]

( d g ( r , ' ~ ) { 1 1 }) x g(rl;~)+ dp # h t ( R - 2 r l ; y ) + h ' ( R - 2 rl;~) [V~V(r)] . (C(R).r) .

The underlined terms only probe the short ranged distortion of the correlation functions, and therefore do not contribute to the anomalous part of the viscosity. For example, the first underlined term ,-, h , (R - r l'~) is only non-zero for I R - r [ smaller than a few times Rv. Since the factor V~V(r) limits the integration range of r to r < Rv, this implies that the integration range of R is limited to a few times Rv.

Secondly, since r < Rv, the correlation functions h i ( R - r l'~) and h l ( R - ~r I ~/) are smooth functions of r for large distances R. These correlation functions may therefore be Taylor expanded to first order in gradients,

hl(R - r 17) 1

ht(R - ~r I~)

- ht(Rl~/) - r . Vnht(Rl~/), 1

- ht(Rl~) - ~ r . VRh,(RI~).

Substitution of these expansions and a further linearization with respect to ht yields,

d# r h~(Rl '~)- ~r . VRh,(R[5) [V~V(r)]- (C(R)- r ) .


The underlined terms do not contribute upon integration, since the corresponding integrand is an odd function of either r or R (note that both V~V(r) and C(R) are odd functions). Finally, g(r I ;r) may be replaced by the equilibrium pair-correlation function up to O (Pe~ and the spherical angular integrations with respect to r can be performed, just as in section 8.2 (see eq.(8.30)), to yield,

]/ ~7r - ,~--~ -~p - k s r dR (C(R) �9 r ) . Vnh(Rl;y) , (8.100)

where II is the osmotic pressure of the quiescent suspension (see eq.(8.33)). Since the hydrodynamic interaction matrix C goes to zero in an algebraic fashion, as can be seen from eq.(5.113) for the disturbance matrix, the above integral probes the long ranged behaviour of the total-correlation function, and may therefore contribute to the anomalous behaviour of the effective viscosity.

The contribution ~

Using that the pair-interaction potential and the pair-correlation function are even functions, and assuming again identical Brownian particles, it is found that,

f drtg(Rl#) (r. R)-VRV(R). (8.101)

Only the short ranged behaviour of g is probed here, since g is multiplied in the integrand by VRV(R). Hence, r/~ is regular in Pe ~ and does not contribute to the anomalous behaviour.

The contribution r/c B~

In order to evaluate this contribution, we use the superposition approximation on the N-particle level, that is, PN is approximated as,

1 N (ri rj[+). (8.102) PN - - V N II(iCj)= 1 g -

This approximation becomes exact on the pair level and probably describes the essential features of higher order interactions in an approximate way. This approximation implies that,

N

V1 ln{PN} - y~ Vl lng(rl -- rj [;y). j=2

542 Chapter 8.

Substitution of this expression together with eq.(8.92) into eq.(8.98) for r/~ ~ readily leads to,

f ~ 4/--- ~ kBT dR (C(R) �9 r ) . Vng(R];y) (8.103)

z~ f da f dr I~)g q- -~ kBT g(R (r - R I+) V~g(r I#)" (C(R) �9 r ) .

The first term on the right hand-side cancels against a term in r/r in eq.(8.100). The second term may be evaluated by decomposing each of the pair-correlation functions in its short and long ranged part as in eq.(8.99). The integrand in the second integral in eq.(8.103) is thus written as,

{1 + ht(R I ~/)+ h , (R [~)} {1 + ht(r- RI~/)+ h , ( r - R I ~/)}

• {V~ht(r I;~) + V~h~(r I;Y)}.

Products of the short ranged parts give rise to a regular contribution to the viscosity and may be disregarded. Furthermore, odd functions of r may be disregarded since these yield a zero result upon integration. Linearization of the above product with respect to the long ranged parts then leaves the following terms to be analysed,

h t ( r - a l;y ) V~h,(r I'~) h~(r - R I+) h , (R I +) V~h,(r [~/) ht(R I~) h,(r - R I~) V~h,(r I'~)

h~(r - R[~) V~ht(r I'~) h , (R I #) h,(r - R I+) V~ht(r I~/)

(a) (b) . (~) (d) (~).

In term (a), ht(r- R I#) may be Taylor expanded around r - 0, since h,(r I if) is short ranged. Noting that C(R) is an odd function of R, this term yields the following contribution to the viscosity,

~3 (a) - - k s T -~ f dr rV~h,(r [;y) �9 f dR ( c ( a ) r ) �9 Vnht(R I S/) .

The second, third and last term, (b), (c) and (e), are only non-zero when both I rl and I R I are less or at most a few times Rv, and are therefore regular terms in Pe ~ In term (d), ht(r I ~) may be Taylor expanded around r - R to first order in gradients. Finally, h,(r 17) may be replaced by the equilibrium


short ranged part h~q(r) of the total-correlation function up to 0 (Pe~ This leads to the following contribution to the viscosity,

Z3 (d) - kBT ~ f dr f dR h,(r - R[a / ) (C(R) �9 r ) . VRht(RI'~)

Z3 = kBZ }7 / dr' h.(r'l~/) / dR (C(R) �9 r). VRh~(RI;~).

Putting things together we arrive at the following expression for the anomalous contribution ~?c s~ to the shear viscosity,

yg" - kBr -~ [1 - C,] f dR (C(R) �9 1"). V R h ( R I S ) , (8.104)

where,

C. = -4,, fo~176 dr r ' [ h:q - 31 rh'~q(r)]dr J + O ( P e . ~ . (8.105)

Being related to the short ranged part of the total correlation function, C, is a well behaved function at the critical point.

T h e c o n t r i b u t i o n r/~ ~

For identical Brownian particles, eq.(8.98) for 778 ~ is easily reduced to,

1 [ = 2 +2 kBT JR>a dR ( r . R ) . VRh(RI'~)-

An application of Gauss's integral theorem leads to a surface integral over the spherical surface with a radius d, the hard-core diameter of the colloidal particles, since V n . ( r . R) - 0. This integral probes the distortion of the total-correlation function at distance equal to d, and therefore contributes only to the background viscosity. 4

There are two terms that possibly lead to anomalous behaviour : the terms in eqs.(8.100) and (8.104). Summing these terms yields,

~1 - kBTT. [

4The effect of hydrodynamic interaction on the shear rate dependence of the total- correlation function makes the integral non-absolutely convergent (see Batchelor (1977) and Wagner (1989)). Since we neglected hydrodynamic interaction as far as the distortion of the pair-correlation function is concerned, this problem does not occur here.

544 Chapter 8.

Since fldII/dp ~ 0 on appoach of the critical point, while C, remains finite, the term ,-~ fldII/d~ may be neglected. The relevant expression for the calculation of the anomalous behaviour of the shear viscosity is therefore,

10 2 - ks T fa>d d R ( h ( R l - ~ ) - h~q(R))VR �9 (C(R) �9 r ) , (8.106)

where Gauss's integral theorem is used (the surface integral at I R l - d is omitted, being regular in P e~ and where it is used that,

fR dRh q(R) V R " ( C ( R ) " r ) - O ' >d

which follows from eq.(8.93) by spherical angular integration. The Fourier transform of the total-correlation function is related to the static structure factor as S - 1 + t~h, so that the Fourier transform of h(R I'Y) - h~q(R) is equal to AS(kl-~)/fi - ( S ( k ] - ~ ) - S~q(k))/~. An explicit expression for AS(k [ ~/) is derived in section 8.3. We therefore rewrite eq.(8.106) with the help of Parseval's theorem (see exercise 1.4b) as,

1 kBTP f 7/ - 87ra ~-~C~ dk A S ( k l S l / ( k ) , (8.107)

with,

I(k) - f >da Ivy. (c(a). r)] exp{-ik. R}

57r aa k . r . k (kd) 2 = 4 k 2 f ( k d ) , (8.108)

where the cut-off function f is equal to,

f ( x ) - [ ( S x S - l O x 3 - 1 2 0 x ) c o s z + (5x4-30x2+120)s inz] / (16x 5)

5 f ~ sin z 16x dz z " (8.109)

That the integral I(k) is indeed equal to the expressions (8.108,109) is shown in appendix B. The function f is called a cut-off function because it limits the integration range in the integral in eq.(8.107) for the viscosity to small wavevectors. As can be seen from fig.8.15, where f is plotted, the cut- off function effectively limits the integration range to wavevectors kd < 4, while the major contribution is from wavevectors kd < 2. This is indeed


Figure 8.15: The cut-off function (8.109).

f(x)

0.5

0 ~ "

l

I . . . . . . . . . I . . . . . . . . . I . . . . : 2 t~ 6

X

the wavevector range for which the expression for the shear flow distorted static structure factor as derived in section 8.3 is valid. If the cut-off function would have had a longer range, extending to wavevectors for which kd > 6, corresponding to wavelengths of the order d ,~ Rv and larger, we would have been forced to introduce in an ad hoc manner an upper limit for the wavevector integration range in eq.(8.107). Fortunately the introduction of such an uncontrolable parameter is not necessary.

A scaling relation for the non-Newtonian shear viscosity

A scaling relation for the viscosity can be obtained, using eqs.(8.107-109), by transformin~ the k-integration to K - k (-integration. The shear viscosity can now be wr :en as,

71o - 2~. 7 ~n -~v (~EIR~')-7/4(pe~ C. N(A, ~-ld), (8.110)

where ~ - 4__ ~ is the volume fraction of Brownian particles and 770 is the shear viscosi~ " ~f the solvent. The viscosity scaling function N is a function of the two di: ~sionless numbers A, which is defined in eq.(8.58), and ( - l d . This functior an integral over the relative static structure factor distortion that is introd~ d in eq.(8.55),

N( J t ,~ l) - 1 f q#(Kl~) �9 A3/4 dKK~ K2 1 + 1{ 2 f ( K ( - l d ) " (8.111)

To arrive at �9 ~ expression, eqs.(8.16,36) for the equilibrium structurefactor are used, tog,, :~er with the Stokes-Einstein relation Do - kBT/67rrloa.

546 Chapter 8.

From the experimental point of view it is more convenient to plot the viscosity as a function of the shear rate, that is, as a function of the bare Peclet number, rather than A. To this end the scaling function N* is defined as,

N*(Pe*,~-~d) - ( P e * ) - 1 / 4 N(A,~- ld ) , (8.112)

where the alternative bare Peclet number P e* is directly proportional to the bare Peclet number,

P e * - A (~-1d)4 - Pe~ (8.113)

Remember that the dimensionless number/3E/R~, is well behaved right up to the critical point. Since the numerical value of f in /R~ is not known apriori, the shear rate is expressed here in terms of the alternative bare Peclet number Pe* rather than in terms of Pe ~ Eq.(8.110) for the viscosity is thus rewritten as,

,7o - ( C~ N*(Pe*, (8.1 14)

The viscosity scaling function N* can be calculated by numerical integration, after substitution of the explicit expression (8.56,57), and is plotted in fig.8.16a as a function of the two dimensionless numbers Pe* and ~-x d. As can be seen from this figure, at a fixed temperature, corresponding to a fixed correlation length, the viscosity hardly changes on increasing the shear rate for small shear rates. For these small shear rates, the suspension is said to behave as a Newtoni~ fluid, meaning that the viscosity is independent of the shear rate. The range of shear rates where the suspension behaves as a Newtonian fluid is referred to as a Newtonian plateau. The thick line in fig.8.16a indicates the extent of the Newtonian plateau, which is seen to diminish as the critical point is approached. Closer to the critical point longer ranged correlations exist, so that smaller shear rates are sufficient to significantly distort the microstructure. The viscosity decreases with increasing shear rates beyond the Newtonian plateau, where the microstructure is increasingly distorted. This phenomenon is commonly referred to as shear thinning.

As can be seen from fig.8.16a, the zero shear viscosity diverges as the critical point is approached. The zero shear viscosity is plotted against the correlation length on a double logarithmic scale in fig.8.16b. The dashed line is a straight line with slope -1 . Clearly,

77(';/~ O) ... N*(Pe* --. O) ~ ~/d, (8.115) 770

8.6. Shear Viscosity

"" a l l tn N" ] ' - ~ ~----~ 0

64- \,..~

I | -

2

10-3 ~ 10-,s o

>

547

Figure 8.16: (a) The scaling function N* as defined in eqs.(8.111,112), which is directly proportional to the viscosity, as a function of ~-ld and the alternative bare Peclet number Pe*. The thick line indicates the extent of the Newtonian plateau. (b) The logarithm of the zero shear viscosity scaling function plotted against the logarithm of ~ -~ d. The dashed line is a straight line with slope -1 .

for ~ >_ 3 d, say. The viscosity is thus predicted to diverge in the same manner as the correlation length. The so-called critical exponent of the shear viscosity is thus equal to that of the correlation length, which is 1/2 in the mean-field region (see eq.(8.37)). This is a much stronger divergence than for molecular systems, where the critical exponent is known to be approximately equal to 0.06 (see Sengers (1985) and Nieuwoudt and Sengers (1989)). The difference between colloidal and molecular systems is that particles interact hydrodynamically. With the neglect of hydrodynamic interaction, the disturbance matrix is zero, and we would have found no anomalous behaviour at all. The strong divergence of the viscosity of colloidal systems is entirely due to hydrodynamic interaction. There is probably no anomalous behaviour of the shear viscosity of molecular systems in the mean-field region. So far there are no experimental results on colloidal systems available that allow for a test of the prediction in eq.(8.115).

548 Appendix A

Appendix A

The differential equation (8.47) is quite similar to the differential equation (6.141) in chapter 6 on diffusion, which is solved in appendix C of the same chapter. We will again need the representation (6.257) of the delta distribution which is proved in appendix C of chapter 6" let f(x) denote a function in ~, with f '(x) - df(x)/dx > 0, and lim~_.,oo f(x) - c~, then,

8(x - xo) - H(x - xo) lim,10 f'(x)e exp { - f(x) -e f(xo) } , (8.116)

where H(x) - 0 for x < 0 and H(x) - 1 for x > 0, the so-called Heaviside unit step-function.

The differential equation (8.47) is solved by variation of constants. First consider the so-called homogeneous equation, where S ~q is omitted,

A/k 1 0S(kl~r)

Ok2 = 2D ~z(k)k 2 S(kl ; r ) �9

Straightforward integration yields,

{ § ) S(kl;r) - C(kl, k3)exp ~ + k 2 (kl 2 + x

Here, C is an integration constant which is in general a function of kl and k3 since we integrate with respect to k2. Using eq.(8.48) for the effective diffusion coefficient and eqs.(8.14,36) for the correlation length, this equation reduces to,

1 S ( k l ~ / ) - C(kl,k3)exp klPe P(k)} ,

where the function P(k) and the dressed Peclet number are given by eqs.(8.52) and (8.53), respectively. The idea of the method of variation of constants is to make C a function of k2 as well, in such a way that the full equation (8.47) is satisfied. Substitution of the above expression into the differential equation, with C understood to be a function of k2, yields a differential equation for C which is easily integrated to obtain,

S(kl ) 1 1 /k2

klPe

Appendix B 549

where Q(k) - 2Pe D~YY(k)k2/'~ is given in eq.(8.51). This expression is finite for all k's when the integration constant C' is 0 and the unspecified lower integration limit i s - c ~ in case k~ < 0 and +c~ in case kl > 0. With e - Pe and f ( x ) - -4-P(k)lk2=~/ka (+ when kl > 0 and - when kl < 0) in the representation (8.116) for the delta distribution, the above expression (with C' - 0) is easily seen to become equal to S~q(k) for Pe ~ 0, as it should.

Subtraction of S ~q (k) from both sides, and using the delta distribution representation (8.116) leads to eq.(8.50) for the static structure factor distortion.

Appendix B

In this appendix we evaluate the integral,

- fn dR [Vn. ( C ( R ) . F ) ] e x p { - i k . I(k) >d R}.

Substitution of eq.(8.93) for the divergence of the hydrodynamic function leads to,

I(k) = 72a6 r./ l : t , dR R -4 J dR RIt, e x p { - i k . RR} (8.117) >d

where the integral ~; dl~(...) with respect to the spherical angular coordinates ranges over the entire unit spherical surface. This integral is equal to (see also eq.(5.139) in appendix A of chapter 5),

J d l~ l~ l~exp{- ik , nl~} = R2 VkVk J d R e x p { - i k . RI~}

4rr sin{kR} = - R 2VkVk kR '

with Vk the gradient operator with respect to k. Now using that Vkg(k) = k:dg(k)/dk, with k - k /k , for a differentiable function g of k - I k I, yields,

s in{kR}_~R 2 1 d sin{kR} 1 d [ 1 d sin{kR}] VkVk kR - kRd(kR) kR + kkR4kRd(kR) kRd(kR) kR "

Substitution of this result into eq.(8.117), and using that F �9 I - 0, yields eq.(8.108),

I(k) - 57ra3k. r . k (kd) ~ - - 4 k 2 f (kd) ,


where (with z = kR),

[1 sin,z ] f ( x ) - 1 5 x d z - ~ - ~ z z d z z "

Two partial integrations gives,

- 15x [ / - c~ f (x) t X 5

2sin{x} X 6

+ 15fOOdz sin{z}] Z7

(8.118)

This function may seem divergent at x = 0 at first sight. However, each of the divergent contributions from the three separate terms here cancel. This is most easily seen by rewriting the integral by means of successive partial integrations,

fx ~176 dz sin { z } 1 dz sin{ z } d z- 6 _ i sin{ x } + dz z~ - - g Yzz - ~ ~ g z~

_ 1 sin{x} 1 f O~dz cos{z} d z- 5 _ _

- ~ ~ 30 ~z "'" ( 1 1 1 )

= sin{x} 6x 6 120x 4 ] 720x-------- ~

( 1 1 1 ) 1 f~176 } + cos{x} 30x5 360x a t- 720x 720 z "

Substitution of this expression for the integral into eq.(8.118) for the function f yields eq.(8.109). The value ofthis function for x = 0 may now be evaluated by Taylor expansion of the sine and cosine functions, and is equal to 1.

E x e r c i s e s

8.1) Short-ranged character of the direct-correlation function Use the convolution theorem (see exercise 1.4c) to show that Fourier

transformation of the Omstein-Zemike equation (8.6) for a homogeneous system leads to,

Zh(k) pc(k) = 1 + ph(k) "


Conclude that #c(k ~ 0) - 1 at the critical point, since S(k --, O) - 1 + ph(k ---+ 0) ~ cr on appraoch of the critical point. This reflects the short-ranged character of the direct-correlation function.

8.2) Order of magnitude estimate of/3E Show from eq.(8.33) that at the critical point, where dH/d~ - O,

/0 { 1 47r '3dV(r') g(r')+ fi = kBT 3. p dr'r dr' -2 d~

Consider pair-interaction potentials, where a short-ranged attractive part is superimposed onto a hard-core repulsion (see for example fig.l.ld in the introductory chapter). For such potentials, dV(r')/dr' is zero everywhere except for distances r' around r' ~ Rv. Verify that the above expression can therefore approximately be written as,

3 ~ dr' R---~v dr' 9(r') + -~ dp "~ k . T .

Disregard the difference between the factor 1/2 that multiplies dg(r')/d~ in this expression and the corresponding factor 1/8 in eq.(8.34) for E, to show that this implies,

1 f lE/R~ ~ 1--0"

This estimate is actually an estimate for the expression (8.34) for E with the factor 1/8 replaced by 1/2 and is therefore a rather crude estimate.

8.3) * In this exercise, Fourier transformation of the Smoluchowski equation (8.45) is shown to result in the equation of motion eq.(8.47) for the static structure factor. In writing eq.(8.45) we have omitted a term ,,~ V~V(r), since we are after the asymptotic solution of the Smoluchowski equation for r >> Rv. However, Fourier transformation involves integration with respect to all r's, so that we must keep this short ranged term. Let C(r) denote the corresponding short ranged term in eq.(8.42) that was neglected in eq.(8.45), and write, instead of eq.(8.45),

dII 0 = 2DoV~{/3--d-- ~- h(r ]+) - fie V~h(r ]~)} - V~. { r . r h(r ]+)} + C(r).


Fourier transformation of this equation without the shear flow term is easy (simply replace V~ by ik as discussed in subsection 1.2.4 of the introductory chapter),

dII 0 - -2Dok2{fl-d-fip_ +/~Ek 2} h~q(k) + C(k).

Let us now consider the Fourier transform of the shear flow term. Verify each of the following steps (Vk is the gradient operator with respect to k),

-fdr {r.rh(r 17)} exp{- ik . r} - -ik.fdr {r.rh(r 1'7)} exp{- ik . r}

= (kV,). rrfdr h(r [,~) exp{- ik , r} - (kVk)" rrh(k I ;y).

The Fourier transform of the Smoluchowski equation, including the shear term is thus,

dII 0 = -2Dok2{/~-d-- ~ + ~Ek 2} h(k]~/)+ (kVk)" rTh(kl /) + C(k).

Subtract the corresponding equation without shear flow to eliminate C(k), substitute the form (8.39) for the velocity gradient matrix, and use that the static structure factor is equal to S = 1 + fih, to arrive at the equation of motion (8.47).

8.4) Spinodal decomposition The present chapter relates to (meta-) stable systems, where dII/d~ > O.

For negative values of dII/dp the time derivative in the Smoluchowski equation must be retained, because the system then decomposes into two phases, so that the pair-correlation function changes with time. Verify that the Smoluchowski equation (8.45) now reads,

O h 2 h(r, tl~/) DE 2h( tl'Y)} V~ { r r ,tl'~)}, Ot (r t I,:),) - 2DoVe{/3 dH ' - d - f i - V ~ r , - �9 �9 h(r

where the time dependence of the total-correlation function is denoted explicitly. Consider the unsheared system, where the last term on the right hand-side is absent. Fourier transform with respect to r and show that,

h(k, t) - h(k, t - 0) exp { - 2 D ~z(k)k2t},

where the effective diffusion coefficient is given in eq.(8.48). Show that density waves with k < ~/-7-~dn/E are unstable, and that the total-correlation

function grows most rapidly at the wavevector k = , /_drI /2E �9 V d~ "

The above equation describes the time evolution of the total-correlation function in the initial stages of the phase separation. To describe later stages, linearization with respect to h of the Smoluchowski equation (8.40,41) is no longer allowed, since then h is not small, as it increases exponentially in time during the initial stage.

The equation of motion may also be solved with the inclusion of the shear flow term. Such equations of motion are discussed in detail in the next chapter.

8.5) The turbidity o f an unsheared system (a) For an unsheared system in equilibrium, the static structure factor

in eq.(8.68) for the turbidity is a function of k -1 k [ only. Perform the r to arrive at,

T = C~ 7r 9/o~dO sin{O} (1-I-cos2{O}) P(k)seq(k).

Now suppose that the Brownian particles are so small that P(k) ~ 1 over the entire scattering angle range (this is the case when koa < 0.5, say). Suppose furthermore that the system is far away from the spinodal, such that S ~q (k) ,,~ S ~q (k - O) over the entire scattering angle range. Show that in that case,

87r kBT T ~q - C~ --~ dII /d~"

This equation offers the possibility to characterize the pair-interaction potential for small particles by means of turbidity measurements, since according to eq.(8.33), with g(r') - exp{-/~V(r') }, the first order concentration expansion of the osmotic pressure reads,

jfo (~ 27r/52 r, a d exp {-/~V(r')}

II - ~kBT + - ~ kBT dr' dr'

Integrals of this kind are considered for example in exercise 6.4 for hard- spheres with an additional square well attraction. Evaluate the osmotic pressure to first order in concentration for such an attractive square well pair- potential in terms of its depth e and width A. Now let A ~ 0 and e --, e~,

such that,

a - 12 lim (exp{15e}-l) A ~; ----+ OO a

A ~ 0

remains finite. This is the sticky sphere limit introduced in exercise 7.2. Show with the help of results for the derivate of g with respect to the distance as obtained in exercise 6.4 that,

Notice that C~ ,-~ qo, so that the turbidity increases linearly with concentration for small volume fractions. This equation applies to the colloidal system consisting of silica particles coated with stearyl alcohol chains and dissolved in benzene, of which the phase diagram is given in fig.8.1, and for which a number of experimental data were shown in the present chapter. Turbidity measurements on dilute samples can thus be employed to characterize the pair-interaction potential of these particles through the single parameter a.

(b) Consider now a system of small particles close to the critical point. Show that for this case,

1 c ~ . f~ [ ~ o 2 - (k/ko) ~ + ~(k/ko)' 1 C~ . C(2kor - - ~ d k k =

~~ = k~o ~ ~o 1 + (k~)~ 2 k?, ~ '

where,

L R d 2 - 4x + 4x 2 4 + 2 z: 4 + 4 z 2 + 2 z 4 a ( ~ ) - z ~ ~ l + z ~ - z~ + z, l n { l + ~ ) "

Since/3E is well behaved near the critical point, this expression offers the possibility to determine the temperature dependence of the correlation length.

8.6) Se l f diffusion near the critical point The short-time self diffusion coefficient is given in eq.(6.49) in terms of the

self-mobility functions in eq.(6.46), which describe the effect of hydrodynamic interaction. Use the leading order term of the self-mobility functions in eq.(6.46) together with eqs.(8.12,15) for the total-correlation function with f ldI I /d~ << 1, to show that,

[ 5 1 ] D~ - Do 1 - 1 . 7 3 4 ~ - T f l F ~ / a 2 H ( a l ~ ) '

Further Reading 555

with,

exp{-z X} H ( z ) -

373

Show that at the critical point, H(a/~) - 1/8, so that the the short-time self diffusion coefficient remains finite at the critical point, contrary to the short-time collective diffusion coefficient.


There is an enormous body of literature on critical phenomena. Beside some classic papers and books, only a limited number of references on subjects relating to the contents of the present chapter are given here. Accounts on critical phenomena, some of which include the phenomenological Ginsburg-Landau theory, renormalization and mode-mode coupling theory, about which nothing has been said in the present chapter, are,

�9 H.E. Stanley, Introduction to Phase Transitions and Critical Phenoma, Oxford Science Publications, New York, 1971.

�9 S.K. Ma, Modem Theory of Critical Phenomena, Benjamin, London, 1976.

�9 P.C. Hohenberg, B.I.Halperin, Pev. Mod. Phys. 49 (1977) 435. �9 P. Pfeuty, G. Toulouse, Introduction to the Renormalization Group and

to Critical Phenomena, John Wiley & Sons, Chichester, 1978. �9 J.D. Gunton (C.P. Euz, ed.), Dynamical Critical Phenomena and Related

Topics, Springer Verlag, Berlin, 1979. �9 L.D. Landau, E.M. Lifshitz, Statistical Physics, Part 1, Pergamon Press,

Oxford, 1980. �9 J.J. B inney, N.J. Dowrick, A.J. Fisher, M.E.J. Newman, The Theory of

Critical Phenomena, Oxford Science Publications, Oxford, 1993. There is an extensive overview of literature on critical phenomena up to about 1970 in the serie of books,

�9 C. Domb, M.S. Green, J.L. Lebowitz (eds.), Phase Transitions and Critical Phenomena, Academic Press, London, 1972. The original paper of Omstein and Zemike is,

�9 L.S. Ornstein, F. Zemike, Proc. Sect. Sci. Med. Akad. Wet. 17 (1914) 793.

556 Further Reading

Further developments on the Omstein-Zernike theory are discussed in, �9 M.E. Fisher, J. Math. Phys. 5 (1964) 944. �9 A. Mtinster (R.E. Burgess ed.), Fluctuation Phenomena in Solids, Aca-

demic Press, New York, 1966.

Static light scattering and turbidity experiments on critical fluids and macro- molecular systems, where the Omstein-Zernike static structure factor plays an important role, are described in,

�9 P. Debye, B. Chu, H. Kaufmann, J. Chem. Phys. 36 (1962) 3378. �9 V.G. Puglielli, N.C. Ford, Phys. Rev. Lett. 25 (1970) 143. �9 Th. G. Scholte, J. Pol. Sci. 39 (1972) 281. �9 P. Schurtenberger, R.A. Chamberlin, G.M. Thurston, J.A. Thomson,

G.B. Benedek, Phys. Rev. Lett. 63 (1989) 2064. Very close to the critical point, the total-correlation function attains the modified Omstein-Zernike form h ~ e x p { - r / ~ } / r l+v/2, where, for a 3- dimensional fluid, 77 is a constant ~ 0.05. See,

�9 M.E. Fisher, J. Math. Phys. 5 (1964) 944, Rept. Prog. Phys. 30 (1967) 615.

Amongst other things, a mode-mode coupling treatment of collective diffusion in molecular systems is given by Kawasaki in,

�9 K. Kawasaki, Annals of Physics 61 (1970) 1. In this treatment for molecular systems, the viscosity of the solvent in the expressions in section 8.5 appearing via the Stokes-Einstein diffusion coefficient is to be replaced by an effective viscosity. A self-consistent theory which eliminates the ambiguity related to this effective viscosity was published a few years later in,

�9 K. Kawasaki, Shih-Min Lo, Phys. Rev. Lett. 29 (1972) 48. The treatment in section 8.5 for colloidal systems leaves no doubt that in the mean-field region the viscosity to be used is simply the viscosity of the solvent, not some effective viscosity. The experimental verification of the Kawasaki contribution to the collective diffusion coefficient is not easy, preliminary due to the fact that higher order hydrodynamic interaction contributions are not taken into account in the theory, which constitute, what is referred to as "the background contribution". The subtraction of this background contribution is not a trivial matter. Two experimental papers on collective diffusion close to the critical point are,

�9 Q.H. Lao, B. Chu, J. Chem. Phys. 62 (1975) 2039.

Further Reading 557

�9 G. Meier, B. Momper, E.W. Fischer, J. Chem. Phys. 97 (1992) 5884.

The phase diagram in fig.8.1 is taken from, �9 H. Verduin, J.K.G. Dhont, J. Coll. Int. Sci. 172 (1995)425.

The closure relation (8.25) for the three-particle correlation function near the critical point was first proposed in,

�9 M. Fixman, J. Chem. Phys. 33 (1960) 1357. The content of sections 8.3,4 is taken from,

�9 J.K.G. Dhont, H. Verduin, J. Chem. Phys. 101 (1994) 6193. This paper also contains an analysis of flow induced dichroism near the critical point and relaxation of turbidity and dichroism after cessation of an externally imposed shear flow. A renormalization group theoretical approach to describe static correlations in sheared molecular systems close to their critical point can be found in,

�9 A. Onuki, K. Kawasaki, Annals of Physics 121 (1979) 456. �9 A. Onuki, K. Yamazaki, K. Kawasaki, Annals of Physics 131 (1981)

217. The last paper contains an extensive analysis of experiments on sheared critical fluids. Such experiments are described in,

�9 D. Beysens, M. Gbadamassi, L. Boyer, Phys. Rev. Lett. 43 (1979) 1253. �9 D. Beysens, M. Gbadamassi, J. Phys. Lett. 40 (1979) L565. �9 D. Beysens, NATO Adv. Study Inst. Ser., Ser B 73 (1981) 411. �9 D. Beysens, M. Gbadamassi, Phys. Rev. A 22 (1980) 2250. �9 D. Beysens, M. Gbadamassi, Phys. Rev. Lett. 47 (1981) 846. �9 Y.C. Chou, W.I. Goldburg, Phys. Rev. Lett. 47 (1981) 1155.

Turbidity experiments on a sheared critical colloidal system of stearyl silica particles in benzene are described in,

�9 H. Verduin, J.K.G. Dhont, Phys. Rev. E 52 (1995) 1811. The data in figs.8.3,8,13 are taken from this paper.

One of the first serious attempts to predict the critical behaviour of the shear viscosity of molecular systems is from Fixman,

�9 M. Fixman, J. Chem. Phys. 36 (1962) 310. �9 W. Botch, M. Fixman, J. Chem. Phys. 36 (1962) 3100. �9 M. Fixman, J. Chem. Phys. 47 (1967) 2808. �9 R. Sallavanti, M. Fixman, J. Chem. Phys. 48 (1968) 5326.

558 Further Reading

The above cited 1970 paper of Kawasaki contains a mode-mode coupling calculation of the viscosity, and the conclusion, made with some reservation, is that the viscosity is well behaved at the critical point. In two subsequent papers an exponential divergence is predicted,

�9 T. Ohta, Prog. Theor. Phys. 54 (1975) 1566. �9 T. Ohta, K. Kawasaki, Prog. Theor. Phys. 55 (1976) 1384.

On the basis of the same mode-mode coupling approach, it is concluded in, �9 D.W. Oxtoby, W.M. Gelbart, J. Chem. Phys. 61 (1974) 2957,

that the viscosity diverges logarithmically. Renormalization group calculations, including the non-Newtonian behaviour of the viscosity, can be found in,

�9 A. Onuki, K. Kawasaki, Phys. Lett. A 75 (1980) 485. �9 A. Onuki, Physica A 140 (1986) 204.

All the above references relate to molecular systems. The Smoluchowski approach for colloidal systems as described in section 8.6 is taken from,

�9 J.K.G. Dhont, J. Chem. Phys. 103 (1995) 7072. This paper contains some references on experimental work on shear viscosity of various kinds of systems near their critical point. Non of these, however, allow for a test of the prediction (8.115). Overviews of experiments on molecular systems to determine the critical exponent for the viscosity can be found in,

�9 J.V. Sengers, Int. J. Thermophysics 6 (1985) 203. �9 J.C. Nieuwoudt, J.V. Sengers, J. Chem. Phys. 90 (1989) 457.

Chapter 9

PHASE SEPARATION KINETICS

559

560 Chapter 9.

In all previous chapters we considered the dynamics of stable systems. The present chapter is concerned with phase separation kinetics of an initially homogeneous system into two fluid-like phases, each with a different concentration of colloidal particles. These two phases are also referred to as the "gas" (the phase of lower concentration) and the "liquid" (the phase of higher concentration). The phase transition is commonly referred to as the gas-liquid transition. As will be seen in this chapter, the pair-interaction potential for a monodisperse system must have an attractive component for such a transition to occur. 1 For initially homogeneous meta-stable or unstable systems, the macroscopic density develops inhomogeneities, eventually leading to a state where two phases with different concentrations or microstructures coexist. The time dependence of the macroscopic density is therefore the interesting quantity here. The emphasis in this chapter is on phase separation kinetics from the unstable state rather than from the meta-stable state.

Before entering into quantitative theories, qualitative considerations are given in the introduction" the distinction between phase separation from the meta-stable and unstable state is described, and the different stages that can be distinghuised during decomposition from an unstable state are discussed. Next, the classic phenomenological Cahn-Hilliard theory for decomposition from the unstable state and the Smoluchowski equation approach are discussed. The Smoluchowski equation approach allows for the analysis of the effect of shear flow on the evolution of the macroscopic density �9 this is the subject of section 9.3. These two sections are concerned with the initial stage of phase separation, where inhomogeneities just began to develop. An experimental technique to investigate phase separation kinetics is small angle time resolved static light scattering, where the static structure factor is measured. The connection between existing inhomogeneities of the macroscopic density and the static structure factor is made in section 9.4. The quantitative analysis

1A gas-liquid transition can also occur in mixtures where all pair-interaction potentials are purely repulsive, as for example in a suspension of hard-sphere colloids with added polymer. There is now an "effective" attraction between the colloidal particles that is induced by the polymers. When two colloidal particles are close together, polymer is expelled from the gap between the two particles, leading to an excess osmotic pressure of the polymer that drives the colloidal particles together. These effective attractions can be so large that they give rise to gas-liquid phase separation. In addition, there are phase transitions also in monodisperse systems where no attractive component of the pair-interaction potential is present" for example, crystallization can occur in monodisperse hard-sphere colloids. This is a phase transition from a meta-stable state. Such transitions will be addressed in this chapter only on a qualitative level.

of phase separation kinetics becomes much more complicated during the later stages of the phase separation, where inhomogeneities are well developed. Non-linear equations of motion for the static structure factor and scaling relations, pertaining to later stages, are discussed in section 9.5. The theoretical findings are compared with experimental results in section 9.6.

9.1 Introduction

Phase separation can occur for systems in the meta-stable part of the phase diagram and in the unstable part. The line in the phase diagram that separates the stable region from the meta-stable region is the binoda/, and the line that separates the meta-stable region from the unstable region is the spinodal. The standard phase diagram is sketched in fig.9.1a, and an experimental phase diagram is given in fig.8.1 in the previous chapter. The point where the spinodal and binodal meet is the critical point. On lowering the temperature at the critical concentration, the system changes from being stable to unstable without first becoming meta-stable. In addition to the spinodal and binodal there is a gel-line in the experimental phase diagram 8.1 where the system locks into a non-equilibrium state where strings of mutually connected colloidal particles exist.

As is pointed out in the introduction of the previous chapter, a system is thermodynamically unstable if, and only if, dII/d~ < 0 (see eq.(8.1)), with II the osmotic pressure and ~ - N / V the number density of colloidal particles. A typical plot of the osmotic pressure at a given temperature versus the reciprocal concentration ~ - 1//~ is given in fig.9, lb (exercise 9.1 contains a derivation of the typical density dependence as sketched in fig.9.1 for a so-called van der Waals fluid). The thick part of the curve is the region where dII/d~ < 0, that is, the homogeneous system with a (reciprocal) concentration in that range is unstable. The two points �9 are two points on the spinodal for the given temperature, where dII/d~ - O. The thermodynamic reason for phase separation of an initially homogeneous system from the unstable state is discussed in the introduction in chapter 8 �9 a negative value of dII/d/~ implies that the Helmholtz free energy A depends on the reciprocal concentration as depicted in fig.8.2a, so that any demixing fluctuation, however small in amplitude, leads to a decrease of the free energy. 2 To understand how

2A microscopic picture of the mechanism leading to demixing from the unstable state is discussed in subsection 9.2.2.

562 Chapter 9.

it is possible that phase separation also occurs for homogeneous systems in the meta-stable region (indicated by the thin solid line in fig.9.1b,c), where dII/d~ > 0, the reciprocal concentration dependence of the Helmholtz free energy A must be considered over the entire concentration range. The reciprocal concentration dependence of the free energy follows from that of the osmotic pressure through the relation II - -OA/OV IN - - d ( A / N ) / d ~ , and is sketched in fig.9.1c. Although demixing fluctuations of small amplitude lead to an increase of the free energy (such as the demixing in two phases indicated by the points b and c), demixing fluctuations with a large amplitude can lead to a decrease of free energy (such as the demixing in two phases indicated by the points B and C). The free energy of the demixed systems is obtained by the point of intersection of the straight line connecting the reciprocal concentrations of the two phases b and c or B and C, and the vertical line at the reciprocal concentration of the homogeneous system" these points are indicated by a o. Hence, phase separation from the meta-stable state is initiated by fluctuations of sufficiently large amplitude, contrary to phase separation from the unstable state, where any fluctuation, no matter how sma/1 in amplitude, leads to a lowering of the free energy. The probability for a large amplitude fluctuation is small, so that phase separation occurs only after some time, the so-called induction time. The induction time is roughly inversely proportional to the probability for a demixing fluctuation to occur that leads to a lower free energy. There is a larger probability for a fluctuation resulting in a relatively large change of the concentration in a small portion of the system than for such a change in a large portion of the system. Phase separation from the meta-stable state is therefore initiated by the fluctuation induced formation of sma/1 entities of significantly different concentration than the initial concentration. These entities are referred to as nuclei. The phase separation mechanism from the meta-stable region is referred to as nucleation or condensation. Phase separation from the unstable state is referred to as spinodal decomposition, and occurs without any time delay, since density fluctuations of small amplitude are always present. A little thought shows that the system attains the smallest possible free energy, after phase separation is completed, when the (reciprocal) concentrations of the two coexisting phases are equal to those obtained from a double tangent construction, as depicted in fig.9, l c. Note that an equal slope of A / N versus ~ implies equal osmotic pressures in the two coexisting phases. The concentrations of the two coexisting phases are two points on the binodal, and are also marked by a �9 in fig.9.1. It is easy to see that there are no demixing fluctuations that lead to a lowering of


T

l / \ \ \ \

\ \

\

\

I

~~mm

I \ \ \ \ \ \

Figure 9.1" (a) The standard phase diagr~n (the temperature T versus (reciprocal) concentration ~ = 1/~). (b) A typical reciprocal density dependence of the osmotic pressure II and (c) of the Helr~oltz free energy. The fllick solid line represents concentrations where the system in unstable, the thin solid line where the system in meta-stable, and the dashed line where the system is stable. The pionts �9 mark the spinodal and binodal points. A demixing fluctuation leading to two phases with reciprocal concentrations b and c leads to an increase of the free energy, while a fluctuation leading to two phases with reciprocal concentrations B and C leads to a lowering of the free energy. The free energies of the dem/xed state are indicated by a o.

564 Chapter 9.

the free energy of a homogeneous system with concentrations in the range where the free energy is represented by the dashed curve in fig.9.1 : for such concentrations the system is stable.

It should be noted that the Helmholtz free energy plotted in fig.9.1c is the free energy without the contribution of gradients in the density. For an inhomogeneous system there are also contributions to the free energy due to concentration gradients. As soon as the system becomes inhomogeneous, for example after a demixing fluctuation occurred, the free energy is larger than calculated on the basis of fig.9.1c. This is why fluctuations which induce small gradients in the density are preferred fluctuations for initiating phase separation.

Consider a homogeneous system that is unstable. In practice such a system may be prepared by suddenly cooling the system from a temperature in the stable region in the phase diagram to a temperature in the unstable part. Such a process is commonly referred to as a quench. Right after a quench the system starts to develop long ranged correlations. These correlations develop up to a point where they render the system unstable, from which time on the macroscopic density develops inhomogeneities. The evolution of the density is sketched in fig.9.2. As will be seen shortly, right after the quench, in the initial stages of the phase separation, one particular sinusoidal density fluctuation is amplified most rapidly. 3 This is due to the fact that the growth rate of sinusoidal density fluctuations with wavelength A is proportional to A-2, while the driving force for phase separation diminishes with decreasing wavelength. The proportionality of the growth rate with A -2 merely expresses the fact it takes more time to displace particles over larger distances. 4 The driving force for growth decreases with decreasing wavelength A, because the creation of short wavelength density variations corresponds to large gradients of the density, which are not preferred, since these lead to a large increase of the free energy. There is therefore one particular optimum wavelength for which the corresponding density waves grow most rapidly. This is depicted in fig.9.2a. In the initial stage of the phase separation, both the change 6p of the density and gradients of the density are small. The initial stage is also referred to as

aSinusoidally varying density profiles are also referred to as density waves. 4In previous chapters we have seen that sinusoidal density variations relax roughly like

exp{-Dk2t}, with D a diffusion coefficient and k the wavevector, which is related to the wavelength A as k = 27r/A. The factor k 2 in the exponent relates to the A -2 dependence referred to here.

i

P+

P_

. , . - - .!~?. ; : : - : -~:

2 : - "" �9 . : . ' . ; . ,

,~,<.,,,,,,~,>,,,,~,~ s.~,,,~,< . . . . . . . i v T - . . . . : " . . ~ V ' 7

P_

P+ S T A G E

m

, - ~ ~,: , , ~ . i ".-..'.-.~. �9 - ..

�9 �9 . '..'~ . . . . 12~:-: ." ';.:." ,',,.~",:- . . - . . . . , ~ i , ~ # . . . : : . : . : ~ r

! > ; 9 .....

I , . :,, .., :.~:.,;, ', ~ . . . . . . , �9 . . . . . . . ~ - ":,~.~.-:'--~,,;',',',.~. ;./

�9 '.. ", ~. ": " : ' , r 2 ' . ' . / "" :. " ,,~'L~; ,'. ~l" .',,~ , " , : " ; "L [ ~,., / . ': . - . . -..',,.:. . " . . . , .i.. ~ : , . - . ~ l . , . , ' , : : . , . , , , . " - ' ]

r,+/

_ / P_

1 I

L J

I / ."':.~". ~:.~.~,:,~,~'.'.," . [ :~;.i.;'#...'.: .~',:. ~- y.,"

r

Figure 9.2" A sketch of the time development of the density after a quench in the unstable part of the phase diagram. Time increases from top to bottom. The left column of figures is a sketch of the density versus position, while the right column depicts the corresponding morphology of density variations in the system itself. The concentrations ~+ and ~_ are the binodal concentrations.

566 Chapter 9.

the linear regime, since equations of motion for the density may be linearized with respect to 5p. Then there is the so-called intermediate stage, where 6p is not small so that linearization is no longer allowed. Gradients of the density are still small, as in the initial stage, due to the long wavelengths that demix. This stage is depicted in fig.9.2b. Subsequently, the decomposition reaches the transition stage where the lower and larger binodal concentrations (/5_ and ~+, respectively) are attained in various parts of the system, as sketched in fig.9.2c. At this stage, sharp interfaces between the regions with concentrations close to/~_ and p+ exist. Inhomogeneities are now large, and higher order terms in gradients of the density come into play. In the late stage of the phase separation the interfaces develop" concentration gradients sharpen and the interfacial curvatures change to ultimately establish coexistence (see fig 9.2d). We thus arrive at the following classification of the different stages during decomposition,

Initial stage �9 5p/fi is small, gradients are small ("diffuse interfaces"),

Intermediate stage �9 6p/~ is not small, gradients are small (" diffuse interfaces"),

Transition stage 6p/~ i8 large, gradients are not small ("sharp interfaces"),

Final stage �9 ~p/p is large, gradients are large ("very sharp interfaces").

Equations of motion for the density in the initial and intermediate stage can therefore be expanded to leading order with respect to gradients of the density, while the leading non-linear contribution in 6p/~ must be included in the intermediate stage. The first higher order terms in an expansion with respect to gradients of the density, which must be included in the transition stage, are referred to here as describing the dynamics of sharp interfaces, while even higher order terms describe the dynamics of very sharp interfaces in the final stage. These very sharp interfaces have a width of the order of a few particle diameters, except in case of quenches very close to the critical point, where the equilibrium interfacial thickness may be large.

As density waves grow independent of their spatial direction, the morphology of regions with lower and higher concentration is an interconnected

9.2. Initial Decomposition Kinetics 567

labyrinth structure. Restructuring of interfaces in the final stage may lead to a different morphology when coexsistence is reached. In an experiment performed under the influence of the earth's gravitational field, in the final state two bulk liquids are in coexistence rather than an interconnected state.

9.2 Initial Spinodal Decomposition Kinetics

A homogeneous system that is quenched from the stable state into the unstable region of the phase diagram, by a sudden change of the temperature, develops inhomogeneities after the development of correlations that render the system unstable. Let ~ - N/V denote the number density of colloidal particles in the homogeneous state, before decomposition occured, and let p(r, t) denote the macroscopic number density as a function of the position r in the system at time t after the system became unstable and started to demix. Define the change of the macroscopic density 6p(r, t) relative to that in the homogeneous state as,

p(r, t) - /5 + 6p(r, t ) . (9.1)

In the initial stage of the phase separation we have,

5p(r,t) << 1, (9.2)

allowing linearization of equations of motion for the macroscopic density with respect to the change 6p of the density. In the present section we describe the initial stage of phase separation on the basis of the Cahn-Hilliard theory, which is a phenomenological theory based on thermodynamic arguments, and on the basis of the Smoluchowski equation (4.40,41) as derived in chapter 4, with the neglect of hydrodynamic interaction.

9.2.1 The Cahn-Hilliard Theory

In a thermodynamic type of approach, demixing can be described as transport of colloidal particles between volume elements which are internally in equilibrium. These volume elements are supposed to be so small that the wavelengths of density variations which are unstable are very much larger than the linear dimension of the volume elements. This is only possible when large wavelength density variations are unstable �9 this will indeed turn out

568 Chapter 9.

to be the case, as could have been anticipated on what thas been said in the introduction. On the other hand these volume elements are supposed to be so large that they contain many colloidal particles and that the range of correlations between particles is small in comparison to its linear dimensions, in order to make a thermodynamic description of each volume element feasible. Furthermore, each volume element is supposed to be in thermal equilibrium at each instant during the initial stage of the phase separation. This can be achieved only when the rate of demixing is small in comparison to the relaxation time of density variations with a wavelength that fits many times into a volume element. That this is indeed the case is due to (i) the fact that it takes more time to displace colloidal particles over larger distances (for demixing) than over smaller distances (for internal equilibration) and (ii) the fact that the diffusion coefficient which describes the large wavelength demixing is very much smaller than the diffusion coefficient pertaining to relaxation of small wavelength density variations, as will be shown shortly. The latter is reminiscent to critical slowing down, discussed in the previous chapter. The system is thus supposed to be in local equilibrium. The idea of fast relaxing small wavelength density fluctuations and slowly growing large wavelength density variations is depicted in fig.9.3.

If one is willing to accept the above assumptions, the Helmholtz free energy Av(p(r, t)) per unit volume of a volume element at position r, with a homogeneous density p(r, t), is well defined and exhibits the double minimum form as a function of the local reciprocal density as sketched in fig.9, lc. Such a double minimum form can be modelled by a fourth order polynomial in the change 6p(r, t) of the local density relative to ~ - N / V as follows,

Av(p(r,t)) Av(p) + a16p(r, t) + -~a2(6p(r, t)

1 1 , )4 + t) . (9.3)

The coefficient a4 is positive, since for small and large densities the free energy is large and positive. Notice that this free energy is the free energy of a volume element in which fluctuations of the density with wavelengths larger than its own linear dimension are absent, since these simply do not "fit into" the volume. This is the free energy per unit volume of an infinitely large system (with a homogeneous density equal to/~ + 6p ) in which the large wavelength density fluctuations are constrained by means of an external field. In a statistical mechanical calculation of the double minimum form of the free


Figure 9.3"

J f

Small volume dements are in internal equilibrium due to fast relaxation of small wavelength density fluctuations, while the large wavelength density variations grow slowly.

energy, one should thus include an external field to account for the absence of density variations with wavelengths that are very much larger than the range of correlations between the colloidal particles.

The Helmholtz free energy of the total system is not simply equal to the sum of the free energies (9.3). There is also a contribution from the diffuse interfaces separating volume elements with different densities, that is, there is also a contribution arising from gradients in the density. Since we are only interested in the dynamics of the long wavelength density variations, leading to small gradients in the density, we may formally expand this contribution with respect to gradients in the density, and keep only the leading term. Since the free energy is invariant against inversion of the coordinate frame, that leading term is proportional to fdr [~75p(r, t)[ 2, where the integral ranges over the volume of the entire system under consideration. The proportionality constant

1 is denoted here as 7~, where n is referred to as the Cahn-Hilliard square gradient coefficient. This constant is assumed to be positive �9 for negative values the formation of gradients would decrease the free energy, leading to a different kind of instability, where the short wavelengths, corresponding to large gradients, are also unstable. The total Helmholtz free energy is therefore the sum (read : integral) of the "bulk contributions" in eq.(9.3) and the above discussed "diffuse interface contribution",

A[p(r,t)] f { 1 1 )3 A(#) + dr al6p(r, t) + -~a2(6p(r, t)) 2 + -~a3(6p(r, t)

570 Chapter 9.

1 1 } + -4a4(Sp(r, t ) ) 4 + ~ [ V S p ( r , t)12 . (9.4)

The square brackets in A[p(r, t)]) denote functional dependence" the numerical value of the free energy now depends on an entire function of position, not just on a single numerical value of the density as for a homogeneous system. An operator that maps functions onto real space is called a functional.

Transport of colloidal particles does not occur when the chemical potential it(r, t) of the volume elements located at r is a constant. The number density flux j (r, t ) of colloidal particles is therefore driven by gradients in the chemical potential. When these gradients are not too large, the number density flux is simply proportional to the gradient,

j ( r , t ) - - D V t t ( r , t ) , (9.5)

where D is a phenomenological transport coefficient. The chemical potential is in turn related to the functional dehvative of the free energy with respect to the density (see appendix A for an introductory discussion on functional differentiation and the derivation of this result),

5A[p] #(r, t) - @(r, t) al + a2~fl(r, t)-+- a3(Sp(r, t )) 2

+a4(Sp(r, t)) 3 - ~V25p(r, t) . (9.6)

Now, conservation of the number of colloidal particles requires that, 5

0 0-7 p(r, t) - - V - j ( r , t ) . (9.7)

Using eqs(9.4-6) in this equation of motion, linearization with respect to 5p(r, t) and Fourier transformation with respect to r (replacing V by ik as discussed in subsection 1.2.4 in the introductory chapter) finally yields,

(9 k2 k2 o-7 p(k,t) - - v + ] (9.8)

The solution of this equation is simply,

6p(k, t) - 5p(k, t - 0) exp{-D #s (k) k 2 t} , (9.9)

5You may repeate the analysis in section 5.2 on the continuity equation in the chapter on hydrodynamics to derive this equation.

9.2. Initial Decomposition Kinetics

Figure 9.4" A sketch of the growth rate of sinusoidal density variations as a function of their wavevector. The dashed curve is for a deep quench, the solid line for a shallow quench.

DEEP QUENCH

/

/

/ /

/

\ \

!

/

S H A L L O W Q U E N C H k

\

\

\ kc \

\

\

\

571

where the effective diffusion coefficient is defined as,

D~ff(k) - D [a2 + Ir 2] �9 (9.10)

The initial Fourier transform 5p(k, t = 0) is the particular realization of the macroscopic density after correlations developed that renders the system unstable, before phase separation occurred.

From eq.(9.9) it is clear that density waves with a wavelength A - 27r/k are unstable when D~]Y(k) < 0. For these wavevectors k, the amplitude 6p(k, t) of the corresponding density wave grows exponentially with time. Since t~ > 0, the effective diffusion coefficient can become negative for certain wavevectors only when a2 < 0. From the definition (9.3) it follows that (differentiations are at constant N, and # is the chemical potential of the colloidal particles),

d2(A/V) d (aA/ON) d# 1 dII a2 = d,~2 = d/~ = ~/p /~ d p ' ( 9 . 1 1 )

so that negative values of the effective diffusion coefficient corresponds, according to eq.(8.1), to a thermodynamically unstable system, as it should. For density variations with a wavelength A = 27r/k for which D ~if (k) < 0, diffusion occurs from regions of low density to larger density. This phenomenon is often referred to as upM'll diffusion.

The "growth rate" of a sinusoidal density variation is equal to - D ~fi (k) k 2, and is sketched in fig.9.4. The wavevector km of the most rapidly growing density wave is easily found by straighforward differentiation,

a2 k m - i ~ " (9.12)

572 Chapter 9.

The so-called critical wavevector k~ is the wavevector beyond which density waves are stable. That is, for any k > k~, D ~yf (k) > 0. The critical wavevector is easily found to be equal to,

k~ -- ~/ a_2 _ v/~k~. (9.13) V

Density variations with small wavevectors decompose slowly because it takes longer times to transport colloidal particles over large distances. Density variations with larger wavevectors decompose slowly because the driving force for uphill diffusion diminishes, as a result of the fact that less free energy is gained when larger density gradients are created.

Note that a deeper quench, where -a2 is relatively large, results in a larger value for the most rapidly decomposing wavevector kin.

9.2.2 Smoluchowski Equation Approach

The description given in the previous subsection is based on thermodynamic arguments. A microscopic derivation of the Cahn-Hilliard result (9.9,10) can be given on the basis of the Smoluchowski equation (4.40,41). The Smoluchowski equation is the equation of motion for the probability density function (pdf) P - P(r l , r2 , . . . , rN, t) of the position coordinates rj, j - 1, 2 , . . . , N, of all N colloidal particles in the system, and reads, with the neglect of hydrodynamic interaction,

0 N 0-'7 P - Do ~ V~j. [ f l [V~] P + V~P] , (9.14)

j = l

where Do is the Stokes-Einstein diffusion coefficient,/~ = 1/kBT (with kB Boltzmann's constant and T the temperature), and ~ - ~ ( r l , r ~ , . �9 � 9 rN) the potential energy of the assembly of colloidal particles. Since,

/ f 1 dr2.-- drN P(r l , r2 , . . . , rN, t) -- Px(rl,t) - ~ p ( r l , t ) , (9.15)

with P~ a reduced pdf (see subsection 1.3.1 in the introductory chapter), an equation of motion for the macroscopic density can be obtained from the Smoluchowski equation (9.14) by integration with respect to all the position coordinates, except for r~. In order to integrate the Smoluchowski equation, a pair-wise additive interaction potential is assumed, that is (with rij - I r ~ - rj I),

N

�9 (rl, r2 , - . . , r s ) -- Y~ V(rij), (9.16) i , j = l , i<:i


with V the pair-interaction potential. Further introducing the pair-correlation function g as (see also subsection 1.3.1 in the introductory chapter),

P2(ra, r2, t) - f d r 3 . . . f d r 4 P ( r l , r 2 , r3, . . . ,rN, t)

- PI (ra, t) P1(1"2, t) g(r~, r2, t ) , (9.17)

the integrated Smoluchowski equation reads, for identical Brownian particles (with ra renamed as r and r2 as r'),

0 O--t p(r, t) - Do [V2p(r, t) (9.18)

+ /3V. p(r, t) f dr' [VV(I r - r' [)] p(r', t) 9(r, r', t)] ,

where V is the gradient operator with respect to r. There are two terms to be distinguished on the right hand-side : the first term between the square brackets describes the effect of Brownian motion, while the second term represents the effects of direct interactions. The combination,

t) - - f dr' [VV(I r - r' I)] p(r', t) g(r, r', t ) , Flnt(r, (9.19)

is the direct force on a colloidal particle at r due to particles in a volume element with position r', averaged with respect to the position of the latter. We will come back to the role of these two contributions in rendering the system unstable in the next subsection.

Consider the initial stage of the phase separation, where the change 6p of the macroscopic density, as defined in eq.(9.1), is small. Let ,Sg denote the accompanied change of the pair-correlation function,

9(r, r', t) - 9o(1 r - r' 1) + 59(r, r', t ) . (9.20)

Here, 9o is the pair-correlation function after the quench, before phase separation occurred. Linearization of the Smoluchowski equation (9.18) with respect to these changes yields,

o_ 6p(r, t) - Do [ V26p(r t ) + 3~V . f dr' [VV(I r - r' ])]

Ot x (6p(r' , t)go(lr- r ' l) + p6g(r,r',t))] . (9.21)

To obtain a closed equation for 6p, the change 6g of the pair-correlation function must be expressed in terms of 5p. Such a closure relation may

574 Chapter 9.

f f f f

J

Figure 9.5" The statistical local equilibrium assumption implies fast relaxation of short wavelength density fluctuations in comparison to the slowly demixing large wavelength density variations, rendering the pair-correlation function locally equal to the equilibrium pair-correlation function.

be obtained as follows. The important feature is that the pair-correlation function in the integral in the Smoluchowski equation (9.21) is multiplied by the pair-force VV(I r - r' [), so that a closure relation is only needed for small distances I r - r ' l < _ Rv, with Rv the range of the pair-interaction potential. Correlations over such small distances establish much faster than the demixing rates of the very long unstable wavelengths, simply because it takes more time to displace colloidal particles over larger distances. On a coarsened time scale that is much larger than relaxation times of density fluctuations with wavevectors k >__ 27r/Rv, but which still resolves the phase separation process, the pair-correlation function in the integral in the Smoluchowski equation may therefore be replaced by the equilibrium pair-correlation function. This is the statistical equivalent of the thermodynamic local equilibrium assumption made in the Cahn-Hilliard approach as described in the previous subsection. The statistical local equilibrium assumption is illustrated in fig.9.5. The equilibrium pair-correlation function is to be evaluated at the instantaneous macroscopic density inbetween the positions r and r'. Hence, to first order in 6p, and for I r - r ' l < Rv,

6g(r, r', t) - 5g~q(] r - r' I) ) d9 ~(I r - r' I) ~+~,

dp O( ,t),

( 0 . 2 2 )

and,

go( I r - r'l) - g~q(lr- r ' l ) , (9.23)

where 9 ~q is the equilibrium pair-correlation function for a homogeneous system with density ~ and the temperature after the quench. The two relations (9.22,23) are certainly wrong for distances I r - r ' l comparable to the wavelengths of the unstable density variations. For such distances the system is far out of equilibrium. The validity of the relations (9.22,23) is limited to small distances, where [r - r ' [< Rv. Substitution of eqs.(9.22,23) into the Smoluchowski equation (9.21), renaming R = r - r', yields,

0 O---t 6p(r, t) Do [ V26p(r, t) +/~ffV . f dR [VnV(R)] (9.24)

( )] • g~q (R) 6p(r - R, t) + fi dfi 5p(r - �89 t) ,

with XTR the gradient operator with respect to R. This equation of motion can now be Fourier transformed to yield (for mathematical details, see exercise 9.2),

0_ 5p(k t) - - D ~fl (k) k 2 5p(k, t) at '

(9.25)

where the effective diffusion coefficient is given by,

D~ff(k) Do [1 + 27rflfi f o ~ d R R 3 dV(R)dR

x ( 2 g ~ q ( R ) j ( k R ) + p d g ~ d ~ R ) j ( l k R ) ) ] .

The j-function is equal to,

(9.26)

j (x) - x cos{x} - sin{x} x3 . (9.27)

The equation of motion (9.25) is formally identical to the Cahn-Hilliard equation of motion (9.8), and its solution is given by eq.(9.9). The effective diffusion coefficient (9.26) may seem different from the Cahn-Hilliard diffusion coefficient (9.10) on first sight. However, since in the integrand in eq.(9.26) the factor d V ( R ) / d R limits the integration range effectively to values R < Rv,

576 Chapter 9.

and the wavevectors of interest are those for which k Rv << 1, the j-functions may be expanded up to quadratic order of their argument. Taylor expansion of the sine and cosine functions in eq.(9.27) gives, j (x) - - 1/3+ x 2/30+O(x 4), so that the diffusion coefficient (9.26) is equal to,

D ~//(k) - Doff ~ + 5 3 k 2 , (9.28)

up to terms of order D0(kRv) 4, where,

27r ~2 fo ~176 dR R 3 dV(R) II - ~ kBT - -~ dR ~ g ~ q ( R ) , (9.29)

and,

E _._ 27r fo o', dV(R) ( 1 15 p dR R 5 dR g~q(R) + -~

dg~q (R)) d~ " (9.30)

The expression in eq.(9.29) for II is precisely the osmotic pressure. Compari- son of eq.(9.28) for the effective diffusion coefficient with the Cahn-Hilliard expression (9.10) identifies,

Da2 - Do~ an ' (9.31)

D x - D o f l E . f Using that dII/d# - fid#/dp, and a2 - d#/d# (see eq.(9.11)), with # the chemical potential of the colloidal particles, the first of these equation reduces tO,

D/Do - ~ ~ . (9.32)

This identifies the transport coefficient D in the Cahn-Hilliard theory. The second of the above equations, together with the expression (9.30) for E, identifies the Cahn-Hilliard square gradient coefficient fr in terms of the microscopic quantities V(R) and g~q(R). Needless to say that this identification is approximate, since the closure relation used here is approximate. 6 Note that

6Note that eq.(9.28) is precisely the expression for the effective diffusion coefficient that we found in the previous chapter on critical phenomena" see eq.(8.48). In eq.(8.36) we made

2 7 r - oo the identification fie - c2 - --~-p fo dr' c(r') r ' 4, with c(r ') the direct-correlation function.


eq.(9.12) for the wavevector of the most rapidly growing sinusoidal density component can be written as,

k ~ - - - ~ / 2 E . (9.33)

A deeper quench, where -/3dII/dp is relatively large, results in larger value for kin.

9.2.3 Some Final Remarks on Initial Decomposition Kinetics

There are a few subtleties involved in the approximations made in the two approaches.

First of all, when a system is quenched close to the spinodal, the linearized theory fails. The reason for this is that close to the spinodal, where fldII/d~ is small, the linear contribution Doff (dII/d~)6p(r, t) is no longer dominant over higher order contributions in 6p(r, t), even in the initial stage of the phase separation. For such shallow quenches, non-linear terms in the equation of motion for the density are important in the early stage of the phase separation.

A second point is the identification of the pair-correlation function g0 with the equilibrium pair-correlation function at time t - 0 when the system became unstable but before significant phase separation occurred. This can only be done to within errors of the order 6p(r,t - 0)/~. The equations of motion are therefore only appropriate when 6p(r, t) >> 6p(r, t = 0). It may happen, depending on the kind of system under consideration, that this requirement is already outside the range of density changes where linearization is allowed. The amplitudes of equilibrium fluctuations of the density are then so large that linearization is never allowed. For such systems and quenches there is no linear demixing regime.

Thirdly, there are rapidly and stable fluctuations of the macroscopic density with a wavelength which is comparable to the linear dimensions of the volume

Comparing this with the above identification one ends up with,

27r fo ~ fo ~ dV(R) ( 1 dgeq(R) ) ---if- ~ dR c(R) R 4 - 2r --~ ~ dRR 5 dR geq( R) + "8 ~ d~ "

This is an approximate relation, since both the Cahn-Hilliard and Smoluchowski equation approaches are approximate. The correspondence is however quite reasonable (see Dhont et al. (1992)).

578 Chapter 9.

elements in the Cahn-Hilliard approach. These fluctuations cause the average density within the volume elements to fluctuate. The equation of motion (9. 7) should therefore be supplemented with a rapidly fluctuating term. The theory that deals with such an additional term is commonly referred to as the Cahn-Hilliard-Cook theory. In the Smoluchowski equation approach there is no reason to add such a rapidly fluctuating term, so that one may conclude that for colloidal systems the contribution of such a fluctuating term is negligible.

It should be noted that eq.(9.5) for the number density flux of colloidal particles in the Cahn-Hilliard approach assumes that no coupling with other volume elements occurs. To account for non-local effects, one might genera- lize eq.(9.5) to,

j(r, t) - - f dr' D(r - r ' )V'#(r ' , t ) . (9.34)

For the special case that D(r - r') - D 5(r - r'), with 5 the 3-dimensional delta distribution, the local equation (9.5) is recovered. The Cahn-Hilliard analysis can be repeated with such a more general non-local diffusion flux, performing a partial integration, omitting the surface integral, and using the convolution theorem (see exercise 1.4c), to obtain the equation of motion (9.9) with,

D~H(k) - D(k) [a2 + g k2] . (9.35)

Little can be said about the wavevector dependence of the effective diffusion coefficient without a microscopic model. The Smoluchowski equation approach shows that with the neglect of hydrodynamic interaction, the local diffusion assumption (9.5) is correct. The approximate treatment of hydrodynamic interaction in subsection 9.5.2 suggests that hydrodynamic interaction does not give rise to non-local diffusion in the linear regime. For colloidal systems it is therefore expected that the wavevector dependence of the effective diffusion coefficient is simply a linear function of k 2.

It is important to realize that the Cahn-Hilliard type of approach applies to the initial and intermediate stages only. Only the leading term in an expansion of the free energy with respect to density gradients is taken into account. In the transition and late stage, where large density gradients exist, higher order terms in such an expansion are certainly important.


Figure 9.6:

FB .

~ r

The direct and Brownian force on a colloidal particle, indicated by , , in a inhomogeneous system. For an attractive direct force, its direction is towards the region with larger concentration, as sketched here.

The mechanism that renders a system unstable

To understand on a microscopic level why a system can become thermodynamically unstable, let us rewrite the Smoluchowski equation (9.18) as,

0_ (Sp(r, ~) - - M ~ . p(r, t) [FS~(r, ~) + FZnt(r, t)]

cot (9.36)

where F I~t is the direct force (9.19) and,

FU~(r, t) - - k u T V ln{p(r, t )} , (9.37)

is the Brownian force on a colloidal particle at the position r. Furthermore, M - fl Do is a "mobility". Now consider a colloidal particle at r in an inhomogeneous environment, as sketched in fig.9.6. The inhomogeneous macroscopic density may be thought of as an instantaneous realization of the fluctuating density. A little thought shows that the Brownian force is always directed towards the region with lower concentration, as depicted in fig.9.6. Now suppose that the pair-interaction potential is purely attractive. The direct force F I~t is then directed in the opposite direction, towards the region with a larger density, since in that region there are more neighbouring colloidal particles attracting the particle under consideration : this can also be seen formally from eq.(9.19) for the direct force, using a purely attractive pair-interaction potential. On lowering the temperature, the Brownian force diminishes, since that force is directly proportional to the temperature. The

580 Chapter 9.

direct force, however, increases in magnitude, due to the fact that the pair- correlation function becomes more pronounced (to leading order in the density this follows from the expression g - exp { - V/kB T}, where V < 0 for an attractive pair-potential). At the temperature where IF tnt I>1 F I, the net force on the colloidal particle is directed towards the region with a larger density. This is the mechanism that is responsible for uphill diffusion, and leads to phase separation. The pair-potential is never purely attractive for real systems since there is always a hard-core repulsion, and there is a competition between the repulsive and attractive components of the direct force. For large densities one can imagine that hard-core repulsion becomes dominant, leading to stabilization. This causes the spinodal to shift to smaller temperatures on increasing the density at sufficiently large concentrations. At smaller concentrations the attractive force can be dominant, leading to an increase of the spinodal temperature on increasing the concentration.

Clearly, the pair-interaction potential must have an attractive component to give rise to a gas-liquid spinodal. For temperature independent pair-potentials the critical point is an upper criticalpoint, meaning that the critical temperature is larger than spinodal temperatures. When the pair-potential is temperature dependent, the system may have a lower critical point, in which case the standard phase diagram in fig.9.1 a is upside-down.

9.3 Initial Spinodal Decomposition of Sheared Suspensions

In this section we analyse the effect of shear flow on the evolution of the macroscopic density in the initial stage, where linearization with respect to changes of the density with time is allowed. Being based on thermodynamic reasoning, the Cahn-Hilliard approach is not easily extended to include effects of shear flow. As yet, it is not known how to extend thermodynamics to include shear flow, if possible at all. Within the Smoluchowski approach, however, it is rather straightforward to include a shear flow.

The Smoluchowski equation (4.102,104) for a sheared system, with the neglect of hydrodynamic interaction reads,

0 N N

0--7 P - Do Y~ V, ; . [/3[V,,~ 1P + V~P] - y~ V,~. [ r . r j P]. (9.38) j=l j=l

9.3. Decomposition of Sheared Suspensions 581

P is again the probability density function (pdf) of the position coordinates of all colloidal particles. The velocity gradient matrix r defines the externally imposed shear flow. We will use,

0 1 0 / o o o ,

0 0 0 (9.39)

with -~ the shear rate. This matrix correponds to a flow velocity r . r in the x-direction, with its gradient in the y-direction. The x-, y- and z-direction are referred to as the Bow, gradient and vorticity direction, respectively.

The analysis of the Smoluchowski equation (9.38) is much the same as for the unsheared case considered in subsection 9.2.2. There is one essential difference with the unsheared case, however, concerning the closure relation for the pair-correlation function. Short wavelength density fluctuations relax fast, as for the unsheared system, rendering the short ranged behaviour of the pair-correlation function equal to the pair-correlation function gstat of a stable homogeneous sheared system (the superscript "stat" refers to "stationary"). This is not the equilibrium pair-correlation function, since the shear flow may affect short ranged correlations. For a zero shear rate g,t=t becomes equal to the equilibrium pair-correlation function. For the sheared system the closure relation (9.22) is replaced by,

dgstat ( r ~ ~g(r,r,,t],~ ) _ r - I~/)8.(~__+_r: tl~) for Ir r' d/~ r~ 2 , , - l< Rv. (9.40)

Also, the pair-correlation function at time t - 0, after the quench when the system became unstable before significant phase separation occurred, is equal to g~tat. Notice that the shear flow renders g,tat anisotropic, that is, it is a function of the vector r - r', not just of its length ]r --- r' I.

The analysis of subsection 9.2.2 can be copied to the present case, except that all pdf's are shear rate dependent, to obtain the following equation of motion for the Fourier transform of the macroscopic density, (0 0)

~-~ - ,~ kx ~ t~p(k, t[-~) - - D ( k 1,~) k 2 ap(k, t[,~), (9.41)

where kj is the jth component of k and where D(k[,~) is a diffusion coefficient, equal to,

1 dV(R) D(k]'~) - Do 1 - -~~ f dR R dR (~r " R)2 (9.42)

582 Chapter 9.

x 2g"t~t(R],:/) sin{k R} + # ___ k . R d/~ �89 '

where 1r - k / k and R - R / R are unit vectors. The shear rate dependence of the density is denoted explicitly.

The unstable density waves have a wavelength that is much larger than the range Rv of the pair-interaction potential. The effect of the shear flow is therefore much more pronounced for the demixing density variations than for the short ranged part of the pair-correlation function. In fact, as we have seen in section 8.3 of the previous chapter (see especially eq.(8.43)), the shear rate dependence of the pair-correlation function for distances of the order Rv and smaller may be neglected when the bare Peclet number Pe ~ - ;/R~, ~2Do is not too large. This dimensionless number measures the distortion of structures with linear dimensions of at most Re. The distortion of the large scale structures with linear dimensions A formed during spinodal decomposition is measured, roughly, by the Peclet number ;/A2/2Do. Since A >> Rv, severe effects of shear flow on the decomposition kinetics are observed even for small bare Peclet numbers. We may therefore replace g~t~t(R[;~) by the equilibrium pair-correlation function g~q(R), up to terms of O(Pe~ The spherical angular integrations in eq.(9.42) can now be performed (see exercise 9.3 for mathematical details) to obtain, not surprisingly, the effective diffusion coefficient (9.26) for the unsheared system,

D(k['~) - D~ll(k) , up to Do • O(Pe ~ . (9.43)

For larger shear rates, where P e ~ is not small, spinodal decomposition is affected by the distortion of short-ranged correlations. In the sequel these short ranged distortions are neglected.

The solution of the Smoluchowski equation (9.41), with D(k [,~) replaced by D~H(k) in eq.(9.26), reads,

~p(k, t ['~) -- 6p(k -- (k l , k2 + ;#,k3),t -- 01'~) e x p ( - D e l l ( k , t [ ; r ) k 2 t } , (9.44)

where the time and shear rate dependent effective diffusion coefficient is equal tO,

D~Z(k'tl~/)- ~klt 2 dxDell k~q-x2q-k~ k2q-x2"bk2 k2 . (9.45)

For shear rates for which Pe ~ is large, D~Yf(k) must be replaced here by D(k I,~).

Notice that there is a time dependence in the exponential prefactor in eq.(9.44). Hence, besides the exponential function, also the wavevector dependence of the initial density variation contributes to the evolution of the density.

The integral (9.45) for the effective diffusion coefficient can be done explicitly, with a little effort, after substitution of the small wavevector expansion (9.28) for D ~ff (k), to yield,

[ { 12 dII K1 K2-~t + ~ 1( 1 ('~t D e f f ( k , t l ;y ) - Do fl -~p 1 + K2

{ ( K2 § 2KI K2;# + 2 K2(ZTt)2 ) + (flE/R~) (h'~ + K~) 1 + K2

§

(9.46)

1 4 }] K~ + 2K~K~;Tt + 2K?h'~(;#) 2 + K~K2(;#) a + gli~ ('~t) 4 1(2

Here, K = k Rv is a dimensionless wavevector. The sheared system is unstable when there is a wavevector for which

D ~ff (k, t - 01;Y) < 0. From eq.(9.45), however, it follows that,

D ~ Z ( k , t - 0['~) - D~YY(k), (9.47)

so that a sheared system is unstable if, and only if, the unsheared system is unstable. The spinodal is therefore not shifted by applying a shear flow. This is true only within the approximation (9.43), where the shear rate dependence of D(k I'~) is omitted. This omission is correct up to O(Pe~ The conclusion is thus that the spinodal is shifted only slightly for not too large values of this bare Peclet number. Since large effects on the demixing kinetics should be observed already for small values of P e ~ meaningful experiments can be performed where the shift of the spinodal may be neglected.

Since the time always appears in eq.(9.45) as a product with kl, it follows that in directions where kx - 0 there is no effect of shear flow,

= (9.48) D~Z(k ' t 1#)1~=o _-o"

Density variations in the (y, z)-plane, that is the gradient-vorticity plane where kz - 0, are therefore not affected by shear flow.

Apart from the exponential prefactor in eq.(9.44), the growth rate of density variations is equal to - D ~fl (k, t l,~) k 2. This is an anisotropic function, that

584 Chapter 9.

Figure 9.7: The anisotropic growth rates -D4f (k , t [ -~)K 2, with K - k Rv a dimensionless wavevector, for various times ;rt (see eq.(9.46)). The ratio of the two dimensionless numbers fldII/d~ and f lE /Rv is taken equal to -1/10 here. Negative values for the growth rate, corresponding to stable fluctuations, are not shown. The left column of figures is for K3 = 0 the right column for K2 - O. The vertical scales are the same for all figures. For the two lower figures the wavevector scale in the K1-direction is 10 times as small as compared to the other plots.


i . . . . . . . . . . . . . ~ . : . . . . = . . . . ...

y -.~. :

. . . ~ . :

Z . . . . . " - .

Figure 9.8" A density wave in the plane where K1 - 0 (a) leads tO planar regions of sma/ler and larger concentration of infinite extent (b), indicated by the dotted and dashed areas, respectively. In reality, these regions have a finite extent, equal to the wavelength A - 27r /kl for wavevectors kl at which the growth rates are large, while there exists a interconnected labyrinth structure in the planes where K1 - O, due to the fact that sinusoida/density variations with an arbitrary direction within these planes are unstable (c).

is, a function of the vector k, not just of its length k -1 k I. Moreover, the anisotropy changes as time proceeds. A plot of the anisotropic growth rates at various values of ,~t is given in fig.9.7. The spherical symmetrical growth rates become ellipsoidal like for small times (see the figures for ,~t - 1 and 2). In the velocity-gradient plane, where K3 = 0, the ellipsoid makes an angle with the K~ and K2 axes, while in the velocity-vorticity plane, where K2 = 0, the long axis of the ellipsoid is along the line where K~ = 0. The angle of the major axis of the ellipsoidal distortion in the (K~, K2)-plane with the K2-axis is seen to decrease for larger values of ,~t (note that in the lower two figures in fig.9.7 the Kl-scale is expanded by a factor of 10 relative to the other figures). For somewhat larger values of ~t, the growth rates along the major and minor axes of the ellipsoid in the (K1, K2)-plane diminish, while in the (K1, Ka)-plane the growth rate along the minor axis diminishes. According to eq.(9.48), growth rates in directions where K1 = 0 are not affected by shear flow, so that the corresponding cross sections of all the figures in fig.9.7 are identical. For larger values of ,~t the only remaining unstable modes are those where K1 is small (note that the bottom figures in fig.9.7 the Kl-scale is 10 times as small as compared to the other figures). The structure of a

586 Chapter 9.

suspension where only density variations for which K1 -~ 0 are unstable is one where regions of different concentrations are formed which extend in the flow direction. An example of such a unstable density wave and the corresponding microstructure is sketched in fig.9.8a, with the correponding density variations sketched in fig.9.8b. The extent of these elongated structures is equal to the wavelengths in the Kl-direction where the growth rates are large, as indicated in fig.9.8c. The three dimensional interconnected labyrinth structure that exists in a decomposing system without shear flow, as discussed in the introduction, now reduces to a similar kind of two dimensional structure in planes perpendicular to the flow direction, that is, in planes where K1 - 0.

This is schematically depicted in fig.9.8c. The ellipsoidal like deformation of growth rates for values of ~/t smaller than approximately 5, say, is indeed observed experimentally in molecular systems by means of light scattering. The quasi two dimensional growth for larger values of ~/t has also been seen in computer simulations and experiments on polymer systems. These results will be discussed in more detail in section 9.6, where experimental results are compared with theoretical expectations.

9.4 Small Angle Light Scattering by Demixing Suspensions

In the foregoing, predictions are made concerning the time development of the macroscopic density. These predictions can be tested experimentally by means of light scattering. In this section the relation between scattered intensities and inhomogeneities of the macroscopic density is discussed.

According to eqs.(3.55,61) in the chapter on light scattering, the intensity I scattered by a suspension of identical spherical colloidal particles is equal to,

1 N ~ 1 < exp{ik. (ri - rj) } > . (9.49) I - C* P(k) ~, , j=

Here, C* is a wavevector independent constant,

c*- y, Io - r2 (4r) 2 #(ft," fi0 I B ( k - 0) l , (9.50)

with V, the scattering volume, that is, the volume from which scattered intensity is collected, lo is the incident intensity, r the distance between the

9.4. Light Scattering 587

scattering volume and the detector, ko - 2r/A with A the wavelength of the light in the solvent, and fi. (rio) the polarization direction of the detected (incident) light. Furthermore,

B ( k - o ) - I - ' s I , (9.51) el

where Vp is the volume of a colloidal sphere, ~p the volume averaged dielectric constant of the colloidal particles and ef the dielectric constant of the solvent. The constant C* is a proportionality constant which is not affected by the demixing process. The form factor in eq.(9.49) is equal to,

P(k) - If~162 kr fo dr r 2 ~(~)-~1 ' (9.52)

with a the radius of the colloidal spheres and e(r) the dielectric constant in a colloidal particle at a distance r from its center. The form factor describes the interference of electric field strengths scattered by volume elements within a single colloidal particle. The form factor is practically equal to 1 for wavevectors ka < 1. For our purpose the form factor may therefore be omitted from eq.(9.49). The last factor in eq.(9.49) is the interesting quantity here. This factor describes interference of electric field strengths scattered by distinct colloidal particles. The ensemble averaging < . . . > is with respect to the position coordinates.

In chapter 3 on light scattering, the ensemble average in eq.(9.49) is analysed for homogeneous suspensions. In the present case, however, the suspension is inhomogeneous, and that analysis must be extended to include these inhomogeneities. The ensemble average will be denoted here, as for homogeneous suspensions, by S(k, t), the static structure factor, which is time dependent due to the ongoing phase separation. For identical colloidal particles we have,

1 N - ~ ~ < exp{ik. ( r i - rj)} > (9.53) S(k, t) -__ N id=__

1) f dr1 f dr2 P(r l , r2, t) exp{ik. (rl - r2)}. 1 + (iv-

Using eq.(9.17) for the probability density function (pdf) P(r l , r~, t) of the two position coordinates, together with eq.(9.15) that connects the single

588 Chapter 9.

particle pdf with the macroscopic density, and introducing the total-correlation function h = g - 1, it is found that, for N >> 1,

1 / / S(k,t) - 1 + ~ dr1 dr2 p ( r l , t ) p ( r 2 , t) (9.54)

x [1 + h(r,, r2,t)] exp{ik. ( r x - r2)}.

Now, for large distances Ira - r2 1, the total-correlation function is small, since h ~ 0 as I rx - r2 I-- ' cr The total-correlation function must therefore be retained only for distances equal to a few times the range Rv of the pair-interaction potential, r The remaining long ranged correlations give a small contribution and may be neglected. Let h~ denote this short ranged contribution to the total-correlation function, s By definition, h,(rl, r2, t) - 0 for distances i rx - r21 larger than a few times Rv. The exponential function is almost constant on the length scale of a few times Rv for wavevectors with k Rv << 1, so that,

1 f dr l f dr2 p(rl,t)p(r2,t)h~(rl,r2,t) exp{ik. ( r l - r2)} N

.~ drl p(rl, t)fdr2p(r2,t)h,(ri,r2,t) (9.55)

We thus arrive at the following expression for the scattered intensity,

S(k,t)

= C * S ( k , t ) , /

- A(t) + ~ <15p(k, t)[2>i,~it , for kRv << 1, (9.56)

where A(t) is a wavevector independent "base line" equal to,

A(t) - 1 + f dr 1 fl(rl, t) f dr 2 p(r2, t) h s ( r , , r2, t). (9.57)

In eq.(9.56) it is assumed that the scattering volume is so large, that it contained many independent realizations of the initial macroscopic density before phase

7This is only true for inhomogeneous systems. For homogeneous systems, the I in 1 + h in eq.(9.54) gives rise to a delta distribution in k, as discussed in detail in section 3.5 in the chapter on light scattering, which does not contribute at finite wavevectors.

8Formally h8 could be defined as the h - hz, with ht the long ranged part of the total- correlation function, which is the solution of the Smoluchowski equation when linearized with respect to h. Such a decomposition in a short and long ranged contribution is discussed in more detail in the previous chapter.

9.4. Light Scattering 589

"DEMIXIN5 PEAK"| ~; ~_..~'MO LECU LAR PEAKS"

. . . . . . . . .

Figure 9.9: k A sketch of the scattered intensity by a system that started to decompose. The thick solid line is equal to C* <1 6p(k, t)[2>i~t, which is the small wavevector contribution due to developing inhomogeneities, while the thin solid line represents the "molecular contribution".

separation occurred" the average < ..- >init is the ensemble average with respect to initial realizations of the density. Furthermore, p(k, t) is replaced by 6p(k, t) - p(k, t) - p" the difference between p(k, t) and 6p(k, t) is proportional to the delta distribution in k, which is zero for non-zero wavevectors.

Note that the static structure factor is anisotropic for systems under shear, that is, it is a function of the vector k, not just of its magnitude k - [ k [, contrary to uns~eared systems.

These equa:ions relate the scattered intensity to inhomogeneities of the macroscopic density. A few approximations are made to arrive at these expressions, and in reality the "base line" A(t) is weakly wavevector dependent due to long ranged correlations and due to slight variations of the exponential function in eq,(9.55). This weak wavevector dependence is insignificant for sufficiently developed inhomogeneities.

In section 9.2 on initial decomposition kinetics we have seen that small wavevector "xnf ~'~omogeneities develop, and as a result, the intensity of light scattered at th~ :orresponding small scattering angles is enhanced. The situation is sketche , in fig.9.9. For wavevectors kRv << 1 a pronounced intensity peak grows it: ~me. The main "molecular static structure factor peak" that is also present i~ stable systems is located at kRv ,~ 27r.

In the transition and late stage of the demixing process, where sharp interfaces come into play, the relevant scattered intensities extend up to much

590 Chapter 9.

larger wavevectors, since the spatial variations of inhomogeneities are now no longer smooth on the length scale of a few times Rv. The exponential function exp{ik �9 (rl - r2)} can no longer be set equal to 1 in eq.(9.55) for these larger wavevectors. The above formula for the scattered intensity is only valid at most up to wavevectors k Rv < 1/2, say, and does not describe structural changes on length scales comparable to the thickness of very sharp interfaces (except may be close to the critical point where the interfacial thickness is large). Light scattering by interfaces with a thickness of the order Rv is not described by eqs.(9.56,57). Formation of sharp interfaces and restructering do, however, affect the evolution of large scale structures, and for example do affect the time dependence of the scattered intensity at small scattering angles. The above equations for the scattered intensity can therefore be used to study interface formation and restructuring, albeit in an indirect way. Scattered intensities at larger wavevectors reflect in a more direct way the dynamics of interface formation and restructuring. For these larger wavevectors, short ranged correlations contribute to the wavevector dependence of the scattered intensity through the term on the left hand-side of eq.(9.55). Short ranged correlations must now be taking into account explicitly. Scattering by interfaces is a complicated problem that will not be considered here (see, however, exercise 9.5).

9.5 Demixing Kinetics in the Intermediate Stage

As we have seen in section 9.2 on the initial stage of spinodal decomposition (without shear flow), the size of density inhomogeneities does not change with time. The wavevector km in eq.(9.13) or (9.33) of the most rapidly growing density wave is independent of time. Mathematically speaking this is the consequence of linearization of the equation of motion of the density with respect to its changes relative to the average density/~ - N/V. Beyond the initial stage of the phase separation, linearization of the equation of motion is no longer allowed. Non-linear terms must be taken into account to describe demixing in the intermediate stage. What the initial stage and the intermediate stage have in common, however, is that the density varies smoothly on the length scale of the order of the range Rv of the pair-interaction potential. For the transition and late stage, the evolution of large scale structures couples to the dynamics of small scale structures (the interfaces with a thickness of a few times Rv), corresponding to large density gradients. In these stages, the

9.5. The Intermediate Stage 591

evolution of large scale structures can only be predicted when non-linear terms involving higher order gradients of the density are also taken into account. In the intermediate stage such a coupling of large wavevector dynamics with small wavevector dynamics is not present, simply because small scale structures are not yet present.

In the present section, the evolution of the static structure factor in the intermediate stage is analysed (without shear flow). To begin with, the evolution of density waves with the neglect of hydrodynamic interaction is considered in the subsequent subsection. The role of hydrodynamic interaction is discussed in subsection 9.5.2. Examples of numerical solutions of the non-linear equation of motion are given in subsection 9.5.3. When decomposition has evolved to an extent that a dominant length scale can be identified, the static structure factor exhibits so-called dynamic scaling behaviour. This is the subject of subsection 9.5.4. Scaling functions are obtained from numerical solutions of the equation of motion for the structure factor, which are compared to experimental results in section 9.6.

9.5.1 Decomposition Kinetics without Hydrodynamic Interaction

Let us first rederive the linearized equation of motion (9.25,28) discussed in subsection 9.2.2 in an alternative way that allows for the inclusion of non- linear terms. Instead of Fourier transforming the Smoluchowski equation eq.(9.24) with respect to r, one may alternatively Taylor expand the changes

1 6p(r - R, t) and 6p(r - 7R, t) ofthe density around R - 0 in case the density is smooth on the length scale Rv, since the factor VRV(R) in the integrand effectively limits the integration range to R < Rv. Substitution of the Taylor expansions,

6p(r -- R, t) 6p(r, t) - R . V6p(r, t) (9.58) 1 1

+ R R . VV~p(r, t) - gl~l~R" VVVp(r, t) + . - . ,

and,

~p(r 1 -- ~n, t) 1

8p(r, t) - ~ R . X76p(r, t) (9.59)

+ m~- vv~p(r, t) - ~ R m ~ " VVVp(r, t) + . . . ,

592 Chapter 9.

into eq.(9.24) and keeping only linear terms in 5p(r, t) yields,

0 t) - Do [ V25p(r , t ) - /3~V. f dR [VnV(R)]

x R .VSp( r , t ) g~q(R)+ ~fi -dp (9.60)

1 dg~q(R_____~))}] + R R R ' V V V S p ( r , t ) ( 6 g ~ q ( R ) + --~# d~ "

Since VRV(R) is an odd function of R and g~q(R) an even function, integrals like,

f dR [VRV(R)]g~q(R) f dR [VRV(R)]g~q(R)RR,

are zero. Terms which are proportional to such integrals of odd functions are omitted in eq.(9.60). The spherical angular integrations can be performed after substitution of VRV(R) - RdV(R)/dR, with R - R/R the unit vector along R, and using that,

f d R R R - 47r~ (9.61) 3 '

f dR RiRjRkRt = 4Z [6ij6kl "~- 6ik6jl "~- 6il6jk] (9.62) 15

where the integration ranges over the unit spherical surface and where 6~j is the Kronecker delta (Sij - 0 for i ~ j, and 5ij - 1 for i - j). The equation of motion now reduces to,

0 [dII V25p(r ' t ) - E V2V25p(r, t ) ] , o-7 p(r, t) - Do/ V (9.63)

with II and E given in eqs.(9.29,30), respectively. Fourier transformation reproduces eq.(9.25,28), since upon Fourier transformation V is to be replaced by ik, as discussed in subsection 1.2.4 in the introductory chapter.

The above procedure can be applied to include higher order terms in 6p(r, t). We are interested here in an equation of motion for the static structure factor. Inhomogeneities are quite well developed in the intermediate stage, so that we can neglect the "base line" A(t) in eq.(9.56). As will be seen in the experimental section 9.6, the relative contribution of this baseline is indeed

small. The relevant equation for the static structurefactor in the intermediate stage is therefore,

1 S(k, t) - -~ <15p(k, t)12>init . (9.64)

For unsheared isotropic suspensions, the static structure factor is a function of k - Ik l only. Hence,

0 S ( k , t ) . 1 O f / O-t NOt dr dr' < 6p(r, t) 5p(r', t) >i,it exp{ik. (r - r')}

1// = 2 -~ dr dr' < 6p(r', t) 05p(r, t) Ot >~n~t e x p { i k . ( r - r ' ) } . (9.65)

The last equation follows from,

f f 05p(r,t) dr dr' < 5p(r',t) Ot >init exp{ik. ( r - r')} =

f f 0@(r', t) dr' dr < p(r.t) Ot >,. . e . .p{ ik- ( r - r')}.

which in turn follows from inversion invariance of the ensemble averages, meaning that these do not change under the transformation r ~ - r and r' ~ - r ' .

The equation of motion (9.18) is now substituted into eq.(9.65) and subsequently expanded with respect to 5p(r, t) and 5p(r', t), as discussed in the first part of this section, but now including higher order terms. We will assume here that 6p(r, t) for a fixed position and time is approximately a Gaussian variable. This is certainly wrong in the transition and late stage, where the probability density function (pdf) of the density is peaked around two concentrations, which ultimately become equal to the two binodal concentrations. In the initial and intermediate stage such a splitting of the pdf is assumed not to occur, and the pdf is approximately "bell-shaped" like a Gaussian variable. When one is willing to accept the Gaussian character of the macroscopic density, averages < . . . >i~it of odd products of changes in the density are zero, while averages of products of four density changes can be written as a sum of products containing only two density changes (see subsection 1.3.4 on Gaussian variables in the introductory chapter, in particular Wick's theorem (1.81)). Hence, in the expansion of the integrand in eq.(9.65) with respect to 6p, only even products need be considered, and averages of products of four density changes can be reduced to products of two densities with the

594 Chapter 9.

help of Wick's theorem. Furthermore, as discussed above, in the intermediate stage there is no need to take higher order spatial derivatives then fourth order into account. Extending the Taylor expansion (9.22) to third order (with g ~ - g~ r - r' I)),

@(r , r', t) - d#~q ~Sp(~_.+_r?_ t) -4 1 d2 g eq ~2 , 2d,~2

~ ~ p 2 ( r _ ~ t) 4 ~, 2

1 d3g eq (r_.+.~ 6 dp 3 ~503 t) ~, 2 ~

yields, after a considerable effort (see appendix B for mathematical details),

: ] 0~ ' + 2k2

[ d3II d2E - Do/3k2S(k, t) [-~p3 + dp2

+2Vo/~k2S(k, t) [E~ < 5p(r, t)V25p(r, t) >i~it +E~ <1V@(r, t)[2>init],

where,

(9.66)

k 2] < 5p2(r, t) >i~it

E O . _ _

and,

47r15 fo ~dR R5 dV(R)dR (5_8 dg~qdp(R) 5 d2g~q(R) + -~fi d#2 + 5t52 ~ ) , (9.67)

E. 47r fo~dRRsdV(R) (5 d2g~q(R) 12d3g ~q) = 1---5 dR -~ d~ 2 + ~ d~ 3 . (9.68)

Notice that averages like < 5p2(r, t) >init are independent of position, but are still time dependent. In fact, these averages can be expressed in terms of integrals over the static structure factor as follows.

Evaluat ion of the ensemble averages in terms of the static structure factor

First consider the average < 5p2(r, t) >in,. Integration of the static structure factor (9.64) with respect to k, for isotropic systems, yields,

dk S(k, t) - -~ d r k < 6p(r, t) 6p(r', t) >i,it exp{ik. (r - r')}

1// = (2r) 31 fdr < >init ~P2(r,t) ,

where it is used that f dk exp{ik. (r - r')} = (27r)35(r - r'), with 3(r - r') the 3-dimensional delta distribution (see subsection 1.2.3 of the introductory chapter and exercise 1.3a). Since there is no prefered position in the system on average, the ensemble average with respect to initial conditions is independent of position. It is thus found that,

1 / 1 /o < 6p2(r, t) >,,~it- (27r)3 p dk S(k, t) - ~ fi ~176 k2S(k, t) . (9.69)

The average < 6p(r, t)V~Sp(r, t) >i~it is calculated as follows. Using Green's second integral theorem, with the neglect of surface integrals (see eq.(1.7) with X = r), yields similarly,

f d k k 2 S(k, t) - -1 f d r f d r ' f d k < ,p(r, t ) ,p ( r ' , t)>~nit ~7~2 exp{ik" ( r - r ' ) ) N

_- -1 j<~r i.r,/.k < [V~,p(r, t)] ,p(r' t)>,,~,t exp{ik. (r-r')} N '

_ _ - 1 f dr < [V~Sp(r, t)] So(r, t) >,,~,t N

Since the ensemble average is position independent it follows that,

< 5p(r, t)V~Sp(r, t) >,,~,t - 1 fo ~176 k4 27r2 fi dk S(k, t) . (9.70)

The neglect of surface integrals in Green's integral theorem means that the influence of the boundaries of the container of the system on the decomposition process is not considered. Similarly,

/ d k k2 S(k, t) - N / d r / d r ' / d k < Sp(r, t) Sp(r', t) >~,~it Xz~. X7~,exp{ik.(r-r')}

= N f d r / d r ' f d k < [~7~5p(r, t)].[XT~,bp(r', t)] >i,it exp{ik. ( r - r ' )}

= N er < t)]

so that,

1 f o ~ d k 4 <1 ~7~Sp(r, t)12>i~it= ~-ir~ k S(k, t) - - < 5p(r, t)V~Sp(r, t) >i,~it.

(9.71) It is important to note that the static structure factor that is integrated with respect to the wavevector in the above equations, is only that part of the

596 Chapter 9.

static structure factor that relates to the demixing process, and is given in eq.(9.64). The integration therefore does not extend to infinity, but really goes up to some finite wavevector of the order of a few times km,, where the demixing peak of the static structure factor attains its maximum value. The "molecular contribution" to the static structure factor (the thin solid line in fig.9.9) is understood not to be included in any of the above equations. In an experiment, the integrals over the static structure factor in the above equations can be obtained by numerically integrating the intensity peak at small scattering angles that emerges during demixing.

The explicit non-linear equation of motion for the static structure factor is now obtained from eq.(9.66) by subsitution of eqs.(9.69,70,71), to yield,

_o s(k t) = Ot '

- 2 Do ~k2S(k , t) -~p + Ek 2

] f ~ - Do k S(k t) + , ~fiz2 dk 'k '2S(k ' , t)

- 2 Doflk2S(k t) N ~ p dk' k '4S(k ' t) (9.72) 271-2 ~

with E ~176 = E ~ - E ~ Keeping only the first term on the right hand-side in the above equation of motion reproduces the linear theory result (to see this, multiply both sides of eq.(9.25) by @*(k, t) and average with respect to initial conditions).

Simplification of the equation of motion

Not all terms on the fight hand-side of the equation of motion (9.72) are equally important. Neglect of the irrelevant terms simplifies the equation of motion considerably and reduces the number of independent parameters.

The wavevector dependent contribution ,~ E k 2 in the very first term on the right hand-side of eq.(9.72) is essential, even thought the wavevectors of interest are small. This is due to the fact that near the spinodal dlI /d# is small and negative. The wavevector dependent contribution ,-, d2E/dfi 2 k 2 to the second term, however, is not essential, since d3II/d~ 3 is not small, except possibly for quenches close to the critical point. For the small wavevectors under consideration one may neglect the contribution ,-, d2E/dp 2 k 2 in the second term on the right hand-side in eq.(9.72). Physically this means that the local density dependence of the contribution of gradients in the density to

the Helmholtz free energy is neglected, that is, the density dependence of the Cahn-Hilliard square gradient coefficient is neglected.

Furthermore, the dimensionless numbers/3p2d3II/dp 3 and f l~2E~176 are probably not of a different order of magnitude. The ratio of the third and second term on the right hand-side of eq.(9.72) is thus of the order,

third term second term

= 0 dk' k'2(k ' Rv)2S(k ', t) / dk' k '2S(k ', t) .

This ratio is small since k 'Rv << 1, so that the third term may be neglected against the second term.

The equation of motion (9.72) thus reduces to,

O S(k t) = - 2 Do flk2S(k, t) [d-~~ ] 0--t ' + Ek2 (9.73)

d3II P fo dk ' k'2S(k ' t) - Dof lk2S(k , t ) dp 3 27r2 , .

This equation contains the relevant features of phase separation in the intermediate stage, for quenches away from the critical point.

Shift of km (t) and k~ (t) with time

In subsection 9.2.2 it was found that in the initial stage the wavevector of the most rapidly growing density wave is independent of time, and coincides with the maximum of the static structure factor (see eq.(9.33)). This is no longer true beyond the initial stage. The wavevector of the relatively most rapidly growing sinusoidal density variation is easily found from eq.(9.73),

foSk'k' s(k ' d/5 3 47r 2 , �9 (9.74)

Since daII/dp a > O, and fodk ' k '2S(k ' t) - 2~2- < 62p(r t)>i~it evidently increases with time, km (t) shifts to smaller wavevectors as time proceeds. This means that the regions of lower and higher density increase their size, due to decreasing growths rates of the density in regions where the density is small or large, in which regions the binodal concentrations are approached. This ultimately leads to the formation of sharp interfaces.

The wavevector where the static structure factor peaks, which is denoted as km,(t), does not coincide with kin(t) beyond the linear stage. Since the

598 Chapter 9.

maximum of the structure factor shifts to lower wavevectors we must have that k~ (t) < kin, (t) beyond the initial stage.

The critical wavevector k~(t), beyond which density waves are stable, is easily seen to be equal to,

(9.75)

just as in the initial stage. Note that according to eq.(9.69) and the Gaussian character of the density,

eq.(9.74) can also be written as,

dn(Z + 6p(r, t)) kin(t) = < d~ >i,it / 2E , (9.76)

which expression is to be taken seriously up to second order in 6p(r, t). This expression reduces to eq.(9.33) for km during the initial stage where 6p(r, t) is small compared to p.

The dimensionless equation of motion

For numerical purposes and to reduce the number of parameters, the equation of motion (9.73) is rewritten in dimensionless form. First, using eq.(9.74) for kr, (t), it is found that eq.(9.73) can be written as,

(k) Oot S(k, t) - 4Do/3Ek4(t) kr,(t) [ 1( k 1 - -~ k~(t) S(k, t). (9.77)

Let us now introduce the dimensionless wavevector K and time r,

K - k/k~,o, (9.78)

dII 2 r = - 2 D o / 3 - ~ kin. o t , (9.79)

where km,o = km(t = 0) is the wavevector of the most rapidly growing density wave during the initial stage, which is given in eq.(9.33), a The dimensionless variable r is the time in units of the time that a particle with an effective diffusion coefficient -Do fldlI/dp requires for diffusion over a

aThe assumption here is that the integral fodk ' k'2S(k ', t - O) is of no significance.


- 1 distance ,,~ k~,o. The equation of motion (9.77) in the desired dimensionless form reads,

[ 1] 0--~ S ( K , T) - km,O K 2 - - -~K 4 S ( K , T) . (9.80)

The ratio km (r) / kin,0 is similarly written in dimensionless form, using eq.(9.74), as,

/o k~,o - 1 - C d K ' K ' 2 S ( K ', r) , (9.81)

with,

6 ' - d'H/a ' 2E 2Z 47r 2

> O, (9.82)

and K ' - k ' /k~,o. The number of parameters is thus reduced to the single dimensionless constant C.

9.5.2 Contribution of Hydrodynamic Interaction

In the above description of spinodal decomposition kinetics we have neglected hydrodynamic interaction. In the present subsection the effect of hydrodynamic interaction is considered in an approximate way. It is not feasible to tackle this problem by simply starting with the S moluchowski equation with the inclusion of hydrodynamic interaction. On integrating the Smoluchowski equation to obtain an equation of motion for the macroscopic density, integrals containing three particle correlation functions are encountered. Moreover, these integrals probe the long ranged non-equilibrium part of the correlation functions. A sensible closure relation then requires a separate analysis of the Smoluchowski equation for the three particle correlation function. These equations are extremely complicated and not amenable to further analysis.

Instead of considering the very complicated equation of motion for the three particle correlation function, the following reasoning allows for an approximate evaluation of the effect of hydrodynamic interaction. Consider a subdivision of the entire system into small volume elements, as was done in subsection 9.2.1 on the Cahn-Hilliard theory. The linear dimensions of the volume elements are small in comparison to the unstable wavelengths but

600 Chapter 9.

/ /

Figure 9.10:

\,,/ . - " /

". i .

The distinction between hydrodynamic interaction of particles within a single volume element and the long ranged interaction between distinct volume elements.

should contain many colloidal particles. There are now two contributions from hydrodynamic interaction to be distinguished : hydrodynamic interaction between colloidal particles within a volume element and long ranged hydrodynamic interaction between different volume elements.

The short ranged hydrodynamic interaction between particles within single volume elements is simply accounted for by replacing the Stokes-Einstein diffusion coefficient Do in the equation of motion (9.73) by a "renormalized diffusion coefficient", which is denoted by D(o ~n). This expresses the change of the mobility of the colloidal particles within a volume element due to their mutual hydrodynamic interaction. The renormalized diffusion coefficient is virtually wavevector independent for the small wavevectors of interest here.

The long ranged hydrodynamic interaction of colloidal particles in distinct volume elements may be treated as follows. The additional velocity that particles in a certain volume element attain is equal to the solvent velocity u(r, t) that is induced by the motion of the colloidal particles in the other volume elements, with r the position of the volume element under consideration. That solvent velocity is in turn related to the forces F h that the fluid exerts on each colloidal particle, as (see eq.(5.22) in chapter 5 on hydrodynamics, with f~t(r ' ) - - p(r', t) Fh(r ', t)),

u(r, t) - - f dr' T( r - r ' ) . p(r', t )Fh(r ', t ) , (9.83)

where the Oseen matrix is given by (see eq.(5.28)),

[ rr] 1 1 i + W ( r ) - s ,7o ; z �9 (9.84)


The expression (9.83) may be considered as the continuous version of eq.(5.46), where eqs.(5.55,56) for the leading order microscopic diffusion coefficients are used : the variable r now plays the role of the particle number index. On the Brownian time scale there is a balance of the hydrodynamic, Brownian and direct forces. The Brownian force is equal to -kBTV'ln{p(r', t)}, (see also eq.(9.37)) while the direct force is given in eq.(9.19). Hence,

Fh(r ', t) -- kBTV'ln{p(r', t)} +fdr" [V'V(I r ' - r " I)] p(r", t)g(r ' , r", t).

(9.85) The additional contribution to the equation of motion for the macroscopic density now follows by substitution of eqs.(9.83,85) into the continuity equation (see eq.(5.1) in chapter 5 on hydrodynamics),

O6p(r, t) = - V . [p(r, t) u(r, t)] - ksT [V6p(r, t ) ] - f d r ' T ( r - r ' ) �9

Ot Ihyd~o

{V 'p ( r ' , t )+ ~p(r',t) f dr"[V'V(lr'-r"l)]p(r",t)g(r',r",t)} , (9.86)

where it is used that V . T(r) - 0. The subscript "hydro" refers to the additional contribution due to hydrodynamic interaction.

The additional contribution to the equation of motion for the static structure factor now follows from eq.(9.65). Using the same closure relation for the pair-correlation function g as before (see the expression just above eq.(9.66)), and expanding up to fourth order in ~p's, yields, with some effort,

fo ~ k' 2) [2k + ( + l n ] ~ _ k , ] ] . (9.87) • k' k' k k - k'

The somewhat complicated mathematical manipulations needed to arrive at this result are given in appendix C.

As before, the wavevector integral extends up to ~, k~(t), that is, only the structure factor relating to the existence of density inhomogeneities due to the ongoing phase separation is integrated" the molecular contribution to the structure factor is not included (see also the discussion around fig.9.9).

Introducing the dimensionless wavevector K and time r, see eqs.(9.78,79), finally leads to the following additional term to the equation of motion for the

602 Chapter 9.

static structure factor,

OS(K,r) Or L

O O

- C' K 4 S(K, r) dK' f ( K ' / K ) S(K' r) [ h y d r o ~ ~

(9.88)

where C' is depending on the quench parameters,

C I _._ _ _ -----. _ _ 3 fi (k,~,oa) Do L~176 ~q dg~(R).~ (9.89) 40 z ~ dR I,~ g (n)+~ d~ ) '

and the function f is equal to,

f(z) = z (i- z') [2 1 - Zl] " z+(l+z2)lnil+z (9.90)

The constant C' is most likely positive, due to the large positive values of g'q (R) and dg -q (R)/dp at contact. The dimensionless time r is given in eq.(9.79), except that Do is replaced by the renormalized diffusion coefficient

D(o'0. Note that hydrodynamic interaction does not contribute to linear terms. To

within the approximations made here, initial decomposition kinetics is only affected through a renormalization of the Stokes-Einstein diffusion coefficient. Furthermore, hydrodynamic interaction does not contribute to the K2-terms in the equation of motion. The zero wavevector limit of demixing rates is therefore unaffected, so that the location of the spinodal does not depend on hydrodynamic interaction, as it should.

The additional contribution (9.88) couples the rate of change of the static structure factor at a certain wavevector to values of the structure factor at other wavevectors. Such a coupling of dynamics of different density waves also occurs in the equation of motion without hydrodynamic interaction, via the integral in eqs.(9.80,81). The difference with coupling caused by hydrodynamic interaction is that the latter is non-local (in k-space).

9.5.3 Solution of the Equation of Motion

The equation of motion for the static structure factor with the inclusion of hydrodynamic interaction is the sum of eqs.(9.80) and (9.88),

[ jo Or S(K, 7") - h "2 S(K, r) 1 - C dK' K'2S(h "', 7)

-~ s+~ + s<:~ c' i " dK' f(s<:'IS<:) S(s~' r)l (9.91) . ~ �9

Jo J


The dimensionless time is defined in eq.(9.79), with Do replaced by the renormalized D(o ~'~).

The equation of motion (9.91) is easily solved numerically, where the wave vector integration extends up to the non-zero wave vector where the actual structure factor becomes equal to the initial structure factor.

Figs.9.1 la and b show numerical solutions with C - 0.01, and with C' = 0.001 and C' - 0.1, respectively. The inserts are a blow up for earlier times. Since C' measures the contribution of hydrodynamic interaction, the effects of hydrodynamic interaction are more pronounced for the latter solution given in fig.9.1 lb. Clearly, hydrodynamic interaction tends to displace the maximum wavevector km,(t) more rapidly to smaller wavevectors and increases the relative growth rates at later times.

A remarkable feature of the equation of motion is that, except for the very early times, the structure factorpeaks are insensitive to the precise initial wavevector dependence S(K, T -- O) Of the static structure factor. For the numerical solutions given in fig.9.11, the initial structure factor is simply taken equal to I for all wavevectors.

As can be seen from this figure, the maximum in the structure factor shifts to smaller wavevectors. The wavevector kin(t) of the most rapidly growing density wave is therefore smaller than the wavevector km,(t) where the structure factor peaks. The critical wavevector k~(t ) is quite close to k ~ (t), resulting in a decrease of the structure factor just beyond its maximum.

Without hydrodynamic interaction, 1 - C f ~ dK' K ' 2 S ( K ', 7) is equal to (km(T)/k,~,o) 2 (see eq.(9.81)). With the inclusion of hydrodynamic interaction, however, this is no longer true. The wavevector km (T) of the relatively most rapidly growing density wave is now obtained from eq.(9.91) by differentiation,

(9.92) 2

,, , i �9

= 0 without hydrodynamic interaction

km,o C' dK' f ( I i ' / K ) - ~ K' dK' ,, ,, r

contribution oS hydrodynamic interaction

where a partial integration has been performed. The critical wavevector k~ (r), beyond which density waves are stable, is now given by,

604 Chapter 9.

ooo , S k m,

- idk, k,2S

i O S 4--

S ~ i i i i i I i I I ! ~ ~ ~ ! ! ! �9 r

i i I i i i I

V 1 0 1 1.5 I \

k / k m ~ 100 "

2000

(~ : c,"t'-: S/S(k-kms ) A' t -= 1 -~ . . . . . . .

,~,, ,T/, ~ , - K I " I

0 ' " ' ~ ' ' " ' " ' ~ 1.5 K k/k.,s Figure 9.11" (a,b) The numerical solution of the equation of motion (9.91) for C - 0.01, and C" - 0.001 and C" - 0.1, respectively. The initia/structure factor S (K, t - O) is simply taken equal to 1. The inserts show the structure factor for early times. (c) A test o f the scaling relation (9.96). (d) A test o f dynamic similarity scaling. In figs. c and d, all curves in figs. a and b where S > 100 are included. The solid (clashed) lines correspond to fig.a (b).

1 - c dK' K'~S(K ' , , ) - ~ k~,O (9.93) J

= 0 without hydrodynamic interaction

+ 2 k~,O e g ' I ( K ' / K ) S ( K ' , , ) - O. i

contribution o.f hydrodynamic interaction

The simple relation k~(t) - Vc2 kin(t) no longer holds when hydrodynamic interaction is taken into account.

9.5.4 Scaling of the Static Structure Factor

The solution of the equation of motion (9.91) in the intermediate stage shows scaling behaviour due to the dominance of a single length scale. This dominant length scale is related to the sharp maximum of the structure factor in figs.9.11 a,b, and is equal to,

L(t) - 2 ~ / k ~ . ( t ) , (9.94)

where kin, (t) is the wavevector at which the static structure factor peaks. The dominance of such a single length scale implies that distances can only be measured in units of that single length scale, so that,

< 5p(r, t)5p(r', t)>i~it _ F ( Ir - r ' l ) < ~p2(r,t) >init _ L(t) , " (9.95)

The normalizing denominator on the left hand-side fixes the value of the scaling function F(x) to unity at x = 0 for all times. It follows that (with x - I r - r ' i /L ( t ) ) ,

1 s(k, t) = ~ <1 ~p(k, t)I~>,~,,

_ I f dr f dr' < 5p(r, t)5p(r', t)>init exp{ik (r r')} N 1 j / ( )

= N <~p2(r't)>i'~it dr dr 'F I t - r'l exp{ik- ( r - r')} L(t)

sin{klr-r'l} = - - <Sp2(r,t)>i~it I r - r ' l i r - r ' [ 2 F

L(t) k [ r - r ' I

_- 47r_ L3(t) < 5p2(r, t) >i~it fo ~ dx x F(x) sin{k L(t) x} p kL( t )

606 Chapter 9.

In the fourth equation the integration with respect to the spherical angular coordinates of r - r ' have been performed (see also eq.(5.139) in appendix A of chapter 5, where a = 1 is the radius of the spherical surface). From equation (9.69), which expresses the position independent average < 6p2(r, t) >init in terms of an integral over the static structure factor, it now follows that,

S(k, t) L-3(t) 2 fo ~176 dz x F(z) sin{k L(t) x} f o dk' k'2S(k ', t) = -~ k L(t) " (9.96)

The right hand-side of this dynamic scaling relation is a function of k L(t) ~, k / k ~ ( t ) only. Therefore, plots of the quantity on the left hand-side of eq.(9.96) versus k/k~, (t) for various times must collaps onto a single curve.

Notice that it follows from the scaling equation (9.96), together with eq.(9.94) for the dominant length scale, that plots of S(k, t)/S(km~(t), t) versus k/k~,( t) for various times should also collaps onto a single curve. This is verified in fig.9.1 ld. This scaling means that the structure factor peaks have the same form, and differ only in the location of their maxima. One might call this scaling dynamic similarity scaling.

It should be noted that the scaling functions in figs.9.1 lc,d are in principle depending on the initial state of the density and the values of the parameters C and C' in the equation of motion eq.(9.91), which in turn depend on the quench depth and possibly on the particular manner the quench is realized. However, it is found from numerical calculations that there is remarkably little variation of the scaling functions on varying the initial ensemble average <[ 8p(k, t - 0) [2>init and the values of the parameters C and C'. This properly of the equation of motion (9.91) makes the dynamic scaling functions universal in the sense that they are independent of initial conditions and quench characteristics. 1~ To within numerical accuracy the scaling functions as given in figs.9.1 l c,d apply for any physically reasonable choice of these quantities. Ix Although the evolution of the static structure factor as sketched in fig.9.1 l a is very different from that in fig.9.1 l b, the scaling forms are identical to within numerical accuracy" the solid lines in figs.9.1 lc,d refer to

10provided that the quench is deep enough. The equations of motion derived here are valid for quenches not too close to the spinodal where j~ dlI/dp and ~2daII /d~a are not very small.

11"Physically reasonable" is any choice where kms(t)/km,o smoothly evolves from 1 to smaller values. That is, any choice of S(K, r = 0), C and C ~ for which the non-linear terms in the equation of motion (9.91) are insignificant at zero time are termed "physically reasonable". Non-linear terms should thus become important solely due to the growth of the static structure factor.

9.6. Comparison to Experiments 607

the system in fig.9.11 a, the dashed lines to the system in fig.9.1 lb. Scaling is always approximate since there is not a truly dominant length scale. As can be seen from figs.9.1 l c,d, scaling becomes more accurate as time proceeds, and ultimately all curves converge to the thick solid curve in these figures. These thick solid lines are the dynamic sca/ing functions. The universality of these scaling functions admit a direct experimental verification of the ideas developed in this section. 12

9.6 Experiments on Spinodal Decomposition

During the linear regime of spinodal decomposition eq.(9.9) predicts a time independent location of the wavevector k,~,0 of the most rapidly growing density wave. Moreover, plots of In { S(k, t) ) / k2t versus k 2 should be time independent straight lines with a slope equal to D n = DoflE and an intercept Da2 - DofldII/d~. No such linear k2-dependence of these so-called Cahn- Hilliard plots is observed when non-local diffusion occurs (see eq.(9.35)). Such non-local diffusion is not to be expected for colloidal systems as discussed in subsection 9.2.3. Sometimes these characteristics of the initial stage are indeed observed, but in most experiments they are not observed. Beside the reasons discussed in subsection 9.2.3 for not observing a linear initial decomposition, it may well be that in some experiments the decomposition is so fast that a first meaningful measurement can be performed only beyond the initial stage. The scattering peak emerging at small wavevectors is always observed, together with the displacement of its maximum to smaller scattering angles due to non-linear coupling.

Spinodal decomposition of sheared systems can be studied by means of light scattering, just as for unsheared systems. According to eqs.(9.44) and (9.56), with the neglect of the wavevector independent baseline, the intensity

X2The dynamic similarity scaling function in fig.9.1 ld is almost perfectly described by the simple function,

s(k,t) s ( k =

{ 1) 3} - exp -30 (~'~m~

608 Chapter 9.

scattered by a decomposing sheared system is proportional to,

1 S(k, t[-~) = ~ <[ ap(k, t['~)[2>

= <l t~p(k - ( k l , k 2 + ;~t, k3) , t - - 0 1 " ~ ) 1 2 > exp { - 2 D *H (k, t] ~)k2t}.

Apart from the exponential prefactor, which in principle also contributes to the time and wavevector dependence of the scattered intensity, the scattering patterns should resemble the time and wavevector dependence of the anositropic growth rate - D ~f/(k, t I;r)k 2 as depicted in fig.9.7. So far no scattering experiments of this kind have been performed for colloidal systems. Experiments on binary fluids are reported by Chan et al. (1988,1991), Perrot et al. (1989) and Baumberger et al. (1991). Scattering patterns taken from Baumberger et al. (1991) are given in fig.9.12 (see also Chan et al. (1988)). There is a striking resemblance between these experimental results and our predictions in fig.9.7 �9 in the (kl, k2)-plane the ellipsoidal scattering pattern is rotated relative to both axis, contrary to the patterns in the (kl, ka)-plane, where the major axis of the ellipsoid is oriented parallel to the k3-axis. Furthermore, the predicted decrease of the angle between the major axis of the ellipsoidal scattering pattern in the (kl, k~)-plane and the k2-axis at later times is observed. This reorientation of the ellipsoidal scattering pattern in the (k~, k2)-plane at larger values of ~t towards alignment along the k2-axis ultimately leads to quasi two dimensional growth.

The prediction (9.48) that shear has (almost) no effect in directions where kx = 0 is also found experimentally. Perrot et al. (1989) state that "--. in the direction perpendicular to the flow and to the shear, the characteristic length is nearly insensitive to the shear and is identical to that obtained without shear flow", and Chan et al. (1991) state that "in the direction perpendicular to the flow and the shear, shear seems to have little effect on the growth". In fact, the effect of shear in these directions is expected to be of order Pe ~

Also mentioned by Perrot et al. (1989) and Baumberger et al. (1991) is the diminishing intensity along the major axis of the ellipsoid in the (kl, k2)- plane. This feature seems to be in accord with the theoretical predictions in fig.9.7.

As can be seen from eq.(9.45) for the effective diffusion coefficient, growth rates scale with ,;/t. Such a scaling is observed by Baumberger et al. (1991), who state that "-.. varying ,~ at constant t is similar to varying t at constant

As we have seen in section 9.3, the only unstable density waves for large

9.6. Comparison tO Experiments 609

k2 kl

- 0.9 2.1 /+.2 6 .3

Figure 9.12: Scattering patterns of a sheared demixing binary fluid (isobutyric acid and water) in the (kl, k2)-plane (top figures) and the (kl, ka)-plane (bottom figures). These figures are taken from Baumberger et al. (1991).

shear rates are those where the component k~ of the wavevector along the flow direction is small, leading to a kind of two dimensional growth (see fig.9.8). Such two dimensional growth has been observed in experiments on polymer systems (Hashimoto et al. (1995)) and binary fluids (Perrot et al. (1989)). The title of the latter reference referres explicitly to this phenomenon : "Spinodal Decomposition under Shear" Towards a Two-Dimensional Growth ?". Such a "Dimensional Reduction in Phase-Separating Critical Fluids under Shear Flow" was first predicted theoretically by Imaeda and Kawasaki (1985).

The extension of the theory on the initial stage in section 9.3 to include leading non-linear terms in the intermediate stage, as was done in section 9.5 for unsheared systems, is probably feasible, but is not yet explored. One of the still open questions here is whether stationary states of sheared unstable systems can exist. Extending the linear theory as mentioned above may give answers to this fundamental question.

Let us now turn to the decomposition kinetics in the intermediate stage. Figs. 9.13a and 9.14a show experimental scattering curves of a spinodally decomposing microemulsion system and a binary polymer melt, respectively. These experimental curves are much alike the theoretical curves in fig.9.11 a,b. In particular the shift of the maximum of the structure factor peak towards smaller wavevectors is indeed observed. Moreover, dynamic similarity scaling is seen to apply in figs.9.13c,14c, and is in reasonable agreement with the theoretically predicted scaling function (the dashed curve in these figures). There is some deviation for the larger wavevectors. These deviations are

610 Chapter 9.

3-1-

1

S . . . . I ' '

~., , . ] - 1 2

- ( k - k "

- 0 . 5

0

I ' ; ' ' ' ' ' ' ' 1 ' ' ' ' ' ' ' ' ' 1

Figure 9.13: (a) Scatteringpeaks during decomposition of a AOT/water/decane microemulsion. (b) A test of the dynamic scaling relation (9.96). (c) Dynamic similarity scaling. The dashed curve is the theoretical prediction (see fig.9.11d). The arrows indicate changes as time proceeds. Experimental data are taken from Mallamace et al. (1995).

i I , 1 , , i , 1 | [ , ! i ~ ! ~ ~ l

K

~~~-

~ ~

~ ~g

.~~~

~ ~m

_~ ~

~-

..

~~

" ~

~

K ~

" ~.

~"

~"

i=i

~-"

~~.

,~~

- �9

~

~~

~

.~.~

~

~-

~~

,~

t~

.

~~

~ ~

~

" ~

"

~ ~

'h

' ~

~ ~i

~

3

e~

i=..

i ~

..==.

i

.===

i

612 Appendix A

due to scattering of sharp interfaces which begin to form. Mathematically, these contributions are neglected in our theory through the neglect of terms of O(K 6) in the equation of motion (9,91). These sharp interface contributions to the experimental intensities yield the experimental value of the integral f dk' k'2S(k',t) much too large, since these large wavevectors contribute relatively most. This is the reason why the scaling relation (9.96) is not verified by experiments, as shown in figs.9.13b,14b. Due to the already developing sharp interfaces the above mentioned integral is grossly overestimated, giving rise to experimental scaling functions with a diminishing amplitude (the arrows in fig.9.13,14 indicate trends with increasing time).

The sharp scattering peaks that are measured during the initial and intermediate stages of spinodal decomposition are broadened by multiple scattering to an extent that depends on the transmission coefficient. An iterative method to correct low angle scattering data for multiple scattering is described in appendix D. The curves in figs.9.13,14 are not corrected for multiple scattering since transmission data are not given in the corresponding references.

A well known empirical scaling relation for the static structure factor is due to Furukawa (1985). This scaling function is much broader than the scaling function that we found for the intermediate stage, and applies probably only in the transition and final stages. One of the features of sharp (and very sharp) interfaces is a decay of the static structure factor at larger wavevectors like ,,~ k -4 (see exercise 9.5). This so-called Porod behaviour is one of the ingredients for constructing the Furukawa scaling function. Such behaviour is absent in the intermediate stage where sharp interfaces are yet to be formed.

Appendix A

Functional differentiation is used in subsection 9.2.1 on the Cahn-Hilliard theory to derive an equation of motion for the density on the basis of thermodynamic arguments. The mathematical notion of functional differentiation is introduced in the present appendix, and in addition it is shown that the chemical potential in an inhomogeneous system is related to the functional derivative of the Helmholtz free energy.

A function is a prescription to associate a real (or complex) number to each element in the vector space ~ . Likewise, a functional is a prescription to associate a real (or complex) number to a function which belongs to some

Appendix A 613

linear function space 9 r . In the sequel we shall not specify the function space .7 r explicitly, but simply assume that the functions have properties, such as continuous differentiability, necessary to justify certain mathematical steps. A simple example of a functional is (with p an integer),

F[g(r)] - f dr 'gP(r ') .

For each function g defined on ~a, the integral yields generally a different real number. The functional F thus maps functions onto real numbers. One may ask for the change 5F of the functional F as a result of a small change 5g(r) of the function g. The answer is easy �9 simply Taylor expand (g + 5g) p = gP + p gP-~ 5g + 0(5g2), yielding, up to 0(5g2),

5F - Fig(r) + dig(r)] - Fig(r)] - f dr'pgP- l(r ') 5g(r ') .

This is the first term in "the Taylor expansion of the functional" F. The function p gP-~ is referred to as the first order functional derivative of F, which is more generally denoted as 5F[g]/Sg(r), and is simply a function of r. The concept of functional differentiation is easily extended to more complicated functionals of the form,

Fig(r)] - f dr' G(g(r ')) ,

with G a differentiable function on ~. Proceeding as above one immediately finds that,

5F[g] 5F - f dr' 5g(r ')5g(r ') ,

where the first order functional derivative is now equal to,

5F[g] dG(g(r)) @(r) eg( )

Things become a little bit more complicated when the functional involves integration over spatial derivatives. Consider for example the functional,

F[g(r)] - f dr' IVg(r')l:

614 Appendix A

The functional (9.4) that is encountered in the Cahn-Hilliard theory contains such a contribution. To first order in t~g it is found that,

5F - f dr '2 [V'g(r')]. [V'Sg(r')].

In order to find the first order functional derivative, the differentiation must be removed from 5g. This can be done by applying Gauss's integral theorem, omitting the surface integral at infinity (the function space .T is thus supposed to consist of functions which vanish at infinity fast enough to be able to omit such surface integrals). It is thus found that,

5F - - f dr '2 [V'2 g(r')]Sg(r') .

The first order derivative of this functional is thus equal to,

5F[g] = - 2 V2g(r) .

5g(r)

You should now be able to reproduce eq.(9.6) for the functional derivative of the functional (9.4).

Many of the applications of functional differentiation relate to the calculation of a function for which a functional attains its maximum or minimum value. Since the variation 5g in the above first order Taylor expansions is arbitrary, and can therefore be chosen either positive or negative, it is easily seen that a functional attains its maximum or minimum when the first order functional derivative is zero. This is the functional analogue of calculating the argument where a function attains extreme values by setting its first order derivative equal to zero. As a very elementary example you may verify that the minimum distance between two points (a, y(a)) and (b, y(b)) in ~2 is a

straight line, by minimizing the length f~ dx r + (dy(x)/dx)2 with respect to y - y(x). The first order functional derivative is found by partial integration, using that 5y(x - a) - 0 - 6y(x - b),

5F[y] dy(x) /dx

5y(x) ~/1 + (dy(x) /dx) 2

This functional derivative is equal to 0 only if dy(x) /dx - constant, which means that the functional attains its maximum or minimum Value in case the

Appendix B 615

two points are connected by a straight line. In order to determine whether the extremum is a maximum or minimum (or not an extremum at all), the Taylor expansion must be extended up to second order (when the second order derivative is also zero, even higher order derivatives must be calculated). The sign of the so-called second order functional derivative determines whether the extremum is a maximum or minimum for the functional. We do not go into these higher order expansions here, since we do not need them for our purpose.

In eq.(9.6) we identified the local chemical potential #(r, t) with the first order functional derivative 5A[p]/Sp(r, t) of the Helmholtz free energy functional A. This can be understood on the basis of the thermodynamic relation,

5A - - I I S V - SST + #6N ,

with II the osmotic pressure and S the entropy. Here we are interested in changes of the free energy due to the change of the number of particles N, with fixed volume V and temperature T. For an inhomogeneous system, the total volume is subdivided into small volume elements, with fixed volumes V = dr' and fixed temperature, as was done in subsection 9.2.1 on the Cahn-Hilliard theory. A change 5A of the free energy due to changes 5N = 5dN(r ') of the number of particles dN(r ') in volume elements with positions r', corresponding to a change 5p(r') - 5dN(r ' ) /dr ' of the density, is thus equal to,

5 A - / # ( r ' ) S d N ( r ' ) - / d r ' # ( r ' ) S p ( r ' ) ,

where #(r') is not simply the chemical potential of a homogeneous system, but contains additional contributions relating to densities of neighbouring volume elements to account for inhomogeneity, as discussed in subsection 9.2.1. From this equation we see that by definition,

5Alp] / z ( r ) - (Sp(r) "

This is used in eq.(9.6) to express the chemical potential in terms of the density and its spatial derivatives.

Appendix B

The derivation of the equation of motion eq.(9.66) requires a considerable effort. The mathematical treatment of one of the terms encountered in the

616 Appendix B

derivation is discussed in this appendix. Other terms are treated similarly. One of the typical terms which are encountered is,

I - / d r / d r ' < t~p2(r, t)[V~Sp(r, t)] ~p(r', t) >i,~it exp{ik. (r - r')}

- V ~ , , / d r / d r ' < ~Sp2(r,t)~Sp(r",t)~Sp(r',t) >init exp{ik. ( r - r'))l~,,=r

In the last line, r" is to be taken equal to r after the differentiation is performed. An application of Wick's theorem (1.81) yields,

< 6p2(r. t) 6p(r". t) 6p(r'. t) >ini, - < 6p2(r. t ) > , . , < 6p(e'. t)~p(r', t)>,~,,

+ 2 < 6p(~. t) ~p(~'. t) >,.,, < 6p(~. t) ~p(r t) >,~,,.

The first term on the right hand-side contributes,

(9.97)

< ~p=(r, t) >,.,, V~,, fd=fdr' <,Sp(r",t)6p(r',t)>i~i, exp{ik. ( r - r')}lr,,=,

= < ~p~(~.t)>.~,. f d r f d ~ ' < [V~,(~. tl] ~,(r'. t)>,o,, exp{ik. ( r - r')}

= - k 2 < ~p2(r, t) >init/dr/dr' < ~p(r, t) t~p(r', t) >i~it exp{ik- (r -- r')}

= - k 2 < 6p2(r, t) >i,it N S(k, t).

In the third line, Green's second integral theorem (1.7) is used, with the omission of surface integrals. The second term on the right hand-side of eq.(9.97) contributes,

2V~,,/dr/dr'< ~p(r, t)t~p(r', t)>ini, < ~p(r, t)~Sp(r", t)>i,,t exp{ik. (r-r'))ir,,=r

= 2 < 6p(r, t)V2$p(r, t)>i,itfdrfdr' <~Sp(r,t)~Sp(r',t)>init exp{ik.(r-r ')}

= 2 < 6p(r, t) V26p(r, t) >i~it N S(k, t).

The term under consideration here is thus equal to,

I - N S(k, t) [ -k 2 < ~p2(r, t )>, , i t + 2 < 8p(r, t)V~t~p(r, t) >i,it].

The averages with respect to initial conditions are independent of positon, since there is no preferred position on average. They are, however, time dependent.

Averages like < ~p(r, t )V~V~p(r , t) >i,~it are zero, since each component of the vector V~V~6p(r, t) is equally likely to be positive and negative, independent of the local value of 6p(r, t).

Appendix C 617

Appendix C

To illustrate the mathematical manupilations needed to obtain the contribution of hydrodynamic interaction to the equation of motion for the static structure factor from eq.(9.86), let us consider one of the terms that must be evaluated,

2 ' -- -~fifdrfdr'fdr"'Tij(r-r')exp{ik.(r"'-r)}

x fdR[Vn, V(r)] _ r'", r' 1 �9 d-----j--- <SP( t)[V,,Sp(r,t)lSp( -R, t )Sp(r ' - - jR, t)>init.

Here, summation over repeated indices is understood. Substitution of the Taylor expansions (9.58,59), disregarding odd functions of R and ensemble averages of an odd number of density changes @, and performing spherical angular integrations with respect to R according to eqs.(9.61,62) leads to,

I - a2fifdrfdr'fdr"'Ti,(r-r')exp{ik.(r"'-r)}

x < 5p(r'", t)[V.,Sp(r, t)] {-1-6 5p(r', t)[V.j V"Sp(r' , t)]

3 V' 3 } + ~ 5p(r')[V.j 25p(r', t)] - ~ [V..Sp(r', t)] [V..V.jSp(r', t)] >,~it,

with a - -i-g4~ f o dR R 5 dv(n)dn dg'q{n)d~ �9 The last term between the curly brackets in the above equation does not contribute, as can be seen by partial integration with respect to r', using Gauss's integral theorem with the neglect of surface integrals, and using that V ' . T(r - r') - 0 �9 partial integration shows that the integral is equal to minus itself, and is therefore zero. Now using Wick's theorem (1.81), the above expression can be rewritten as,

3 X fifdrfdr, Tij( r r' fdr'"exp{k (r"' I - - ~ a ~ - ) e x p { i k - ( r ' - r ) } i �9 - r ' )}

x {< 6p(r'", t)6p(r', t)>init< [V.,6p(r, t)] [V.jV'26p(r ', t)] >i.it

+ < 6p(r'", t) [%; V'26p(r ', t)] >~.,< [%,6p(r, t)] 6p(r', t)] >~.,}.

Since < 5p(r'". t ) @ ( r ' , t) >i..t is a function of r'" - r' only, we have that,

f dr'" exp{ik. ( r " ' - r')} < 5p(r"', t)6p(r', t) > i n i t =

i f d r ' f dr'" exp{ik. ( r " ' - r')} < 5p(r'", t)6p(r' t) >i~it- P S(k, t)

V ~ "

618 Appendix D

Performing partial integrations, it is similarly found that,

f dr'"exp{ik.(r"'-r')} < 5p(r"', t)[V~;V'25p(r ', t)] >ini,= -/5 k2ik S(k, t) .

Using these expressions in the above formula for the integral I yields,

I = 3 1:2 S(k t) f dr f dr' ( k - T ( r - r ' ) . k ) e x p { i k . ( r ' - r ) }

x { < 6p(r, t)[V'26p(r ', t)] >,.,t + k 2 < 6p(r, t)6p(r', t) >,,~,t}.

Substitution of the Fourier inversion formula,

T ( r - r') -

1 (27r)3 f dk' T(k') exp{ik'. ( r - r ')},

and performing a partial integration with respect to r' yields,

3 1 /~2 f S ( t ) ( k T ( k k') k)[k ' : k 2] I = ga(2rr)a S(k,t) dk' k', . . . . ,

where the integration variable has been changed to k - k'. The integration with respect to the spherical angular coordinates of k' can be performed explicitly, using that T(k) = ~ 1 [~ _ ~kk] (see eq.(5.137) in appendix A of chapter 5). Since the spherical angular integral is independent of the direction of k, that direction can be chosen along the z-axis. In this way one obtains (with

- cos{O'}),

f~ 1 - x 2 dl~' ( k . T ( k - k ' ) . k ) = 27r k2 k, 2 dx r/o 1 (k 2 + k '2 - 2kk'x) 2

[ ] _ 7r 1 k' k,2) k - k' - ,okk' 2k + ( k 2 + l n l k + k , I .

All other contributions turn out to be proportional to the same integral, leading to the expressions (9.88-90) for the additional contribution of hydrodynamic interaction to the equation of motion for the static structure factor.

A p p e n d i x D

In the intermediate stage pronounced inhomogeneities exist, which scatter a considerable fraction of the incident light. When scattered intensities are

Appendix D 619

hO

_.-. 0 ~ . 0 ~ " ~

~ ~ , ~ . . k s

�9 �9

Figure 9.15: First and higher order scattering events that contribute to the experimental intensity corresponding to the scattering wavevector k,.

large, multiple scattering events can certainly not be neglected. There is a certain probability that a photon that is scattered once will be scattered again, leading to so-called double scattering. An additional scattering of that photon gives rise to triple scattering, etc. etc.. These higher order scattering events are schematically depicted in fig.9.15. The experimental scattered intensities are the sum of intensities due to single, double, t r ip le . . , scattering events. That is,

I(ko - k~) - I~(ko - k,) + I2(ko - k,) + I3(ko - k,) + . . . , (9.98)

where ko and k, are the incident and scattered wavevector, respectively. The wavevector k used in previous sections is simply equal to ko - k,. The relation (9.56) between the scattered intensity and the static structure factor is valid only when higher order scattering events can be neglected. In order to compare data with theoretical predictions for the static structure factor, experimental intensities must be corrected for multiple scattering. This can be done as follows. Let a be the fraction of incident light that is scattered once. The fraction of that total scattered intensity that is scattered in the direction

1r of the scattered wavevector is equal to S ( k o - k ~ ) / ~ dlr where the integral extends over all directions, that is over the entire unit spherical surface. Notice that this integral is a constant, independent of ko. We thus find that,

Ii(ko - k~) - a Io S(ko - k , ) . (9.99) dl~" S(ko - k")"

In secondary scattering events, the same fraction a of 11(ko-k',) for a certain scattering wavevector k', is scattered again. The fraction of that light that

620 Appendix D

is scattered into the direction k~ is equal to S ( k ; - k ~ ) / ~ ; dk~S(k'~-k~) - S (k '~ - k~)/5f d l r k"). The total double scattered intensity is now

^ l

obtained by adding all contributions for different directions k.. Hence,

I~(ko - k , ) - a 5~ dl~'~ 11(ko - k'~)S(k; - k,)

3: dl~" S(ko - k")

The n th order scattered intensity is similarly related to the ( n - 1)th order scattered intensity as,

I ~ ( k o - k,) - Ot ~ dl(18 ln-1 (ko - k : )S(k : - k,)

& " S(ko - k 7)

Substitution into eq.(9.98) yields,

- a ~; dl~', I(ko - k',)S(k', - k~) (9.100) I ( k o - k ~ ) - a l o S ( k o k,) + dl~ S(ko - k") ~ dl~" S(ko - k")

In analogy with eq.(9.99), the experimental static structure factor S ~p is defined as,

I(ko - k~) - a Io S~*P(ko - k~)

dl~" S~*P(ko - k")

Defining the relative static structure factor S~ as,

S~(ko - k~) - S ( k o - k~)

dl~" S(ko - k~)

Ii(ko - k,)

5f dl~ 11(ko - k~) , ( 9 . 1 0 1 )

and similarly for ,q~P

.q_,~P(ko - k,) - S=~(ko - k , )

a t 7 S ~ ' ( k o - k',')

I(ko - k,)

dl~ I(ko - k~) , (9.102)

eq.(9.100) reduces to,

S ~ ( k o - k , ) - S ~ ' ( k o - k , ) - u /dl~'~ SeX,(ko_ k~s)Sr(k~ s - k~). (9.103)

The fraction a of the light that is scattered is equal to,

a - l - T , (9.104)

Appendix D 621

where T is the transmission coefficient, which is the fraction of the light that in not scattered. Transmission is an experimentally quite easily accessible quantity, so that the above equation may be regarded as an integral equation for S~, where both a and S~ ~p are known.

The experimental relative static structure factor is easily obtained by numerical integration of experimental data, so that the static structure factor that is relevant for comparison with theory can be obtained by solving the above integral equation with respect to S~(k0 - k,). This can be done by iteration. First calculate the integral on the right hand-side with S~ - S~ ~p to obtain a first estimate S! x) for the static structure factor. 13 Then calculate the integral with ST - S! ~) to obtain a second, better estimate S! 2). Repeat this up to a level where subsequent estimates do not differ to within some desired accuracy. This then yields the (relative) static structure factor S~ - limn--,oo S~ n) for which theoretical predictions are made in previous sections. ~4

There are a few approximations involved in the derivation of the integral equation. First of all, changes of polarization directions upon scattering are neglected. Since we consider scattering in forward directions, these changes are small, and may be safely neglected. Secondly, we added intensities instead of electric field strengths, thereby neglecting interference. It is assumed here that multiple scattering occurs between volume elements which contain many colloidal particles, so that the scattered intensity from each volume element can be described as if it where macroscopically large. The intensity scattered by each volume element is then proportional to the static structure factor, which is indeed assumed in the derivation given above, and phase relations of electric field strengths of light scattered by different volume elements is lost. This is probably a reasonable approximation for the present situation, where large scale inhomogeneities exist. Thirdly, there is in principle a dependence

laFor numerical purposes, the integral is most conveniently written as,

Jo" Jo" dk, .S, rXP(ko - k',)Sr(k', - k , ) - dv/ dO' sin{O'} .S,~P(2ko sin{O'/2})

x S~ (k0 X/2 [1 - sin{O,} sin{O'} cos{~r - cos{O,} cos{(9'}]),

where 08 is the scattering angle, which is related to the scattering vector as k - I k0 - k8 I= 2k0 sin{O~/2}.

14Th e rate of convergence of the iterative scheme is greatly enhanced when in each iterative

step instead of S~ (n), the average (S(,. n-x) + S~ n))/2 is substituted for Sr to calculate the

integral. In the very first iterative step one then uses S~ ~p / 2 instead of.q..~P for Sr to calculate the integral.


of multiple scattering contributions on the geometry of the scattering volume, since part of the scattered intensity by volume elements at the edge of the scattering volume will leave the suspension and will not be scattered again. For large scattering volumes, with a relatively small surface area, this geometry dependence is insignificant.

Exerc i ses

9.1) Stability and decomposition kinetics of a van der Waals fluid A van der Waals fluid is defined as a one-component fluid (or a suspen-

sion of monodisperse colloidal particles) with a hard-core repulsion and an additional attractive pair-interaction potential w of infinite range. Subdivide the entire system into little volume elements as was done in subsection 9.2.1 on the Cahn-Hilliard theory. These volume elements are now so small that the additional pair-interaction potential is a constant over distances equal to the linear dimensions of the volume elements, but at the same time so large that they contain many particles. Such a long ranged pair-interaction potential is not realistic, but it allows for an analysis of thermodynamic behaviour and phase separation kinetics. Despite the unrealistic nature of the pair-interaction potential, the equation of state of a van der Waals fluid exhibits all features that one expects for gasses/fluids. The equation of state is analysed in (a), thermodynamic stability is considered in (b) and decomposition kinetics in (c).

Let us first derive an expression for the free energy of a van der Waals system (this derivation is taken from van Kampen (1964)). The canonical configurational partition function is equal to,

Q N - - 1 f d r l . . . / d r g e x p { - f l ~ ( r l . . . r s ) ) N! ' '

a f f {1 N! arNx(ra,..-,rN)exp N }

n,m=l

where the so-called "characteristic function" X is 0 when two or more hard- cores overlap and 1 otherwise. The characteristic function enters through the hard-core part of the interaction potential ~, which is infinite when two


or more hard-cores overlap and 0 otherwise. Let Nj denote the number of particles in the jth volume element. The partition sum is now rewritten in terms of a sum of all possible realizations { Nj } of these so-called occupation numbers. Since the additional pair-potential w is supposed to be constant within the volume elements, the partition function can be written as (n is the number of volume elements),

1 N! I # r l ' . "#rN~ . . . . . # r N + I - N ~ ' " " # r N Q N - N!{~N~}I-IjNj. ~ ~, ,, .- .,

N1 in V 1 Nn in Vn {1 } x x ( r l , ' - ' , r N ) exp --~fl~. . wijNiN:i .

Here, wij is the long ranged pair-potential evaluated at the distance between the volume elements i and j. Each of the integrals pertaining to a single volume element renders the average volume available to a single particle, taking into account that part of the total volume is excluded due to the presence of the other particles. This free volume is approximately equal to A - NjS, with A the volume of a volume element and (5 being a measure for the core size of the particles. Hence,

1 N~ QN - N! {N~j) I]j Nj'. rIj (A - Nj6) N~

This result can also be written as,

{1 } exp - -~ fl Z wij Ni Nj .

z,3

QN - ~ e x p { - f l ~ ( N 1 , - - . , N n ) } , {N~}

with,

1 ~'~wijN, Nj ~I/(N1,""", gn) - - -kBT~_,(Nj ln{ A - g j 6 } - gj ln{Nj} + Nj)+-~ .. J ',~

(9.105) Stirling's approximation ln{Nj!} - Nj ln{Nj } - Nj is used here. The canonical partition function is related to the Helmholtz free energy A as A - - kB T In{ QN }. For large N's, �9 is sharply peaked around its minimum value, and positive and large otherwise. There is therefore a dominant term in the above sum that defines the partition function, pertaining to the occupation numbers where tI, attains its minimum value. Hence,

A - ,IJ(N~,. . . ,Am), (9.106)


where the occupation numbers are those for which ~ attains its minimum value. 15

(a) Assume that the density is homogeneous, that is, assume that,

A Nj - N V , for all j ,

where V is volume of the entire system under consideration. Show from eqs.(9.105,106) that the free energy is now equal to (note that n - V / A ) ,

A--kBT( ln{V-N'} ) 1 N + N - ~ W o V '

where,

WO = I A2 V ~. . wij =

z~3

1 _ 47rL ~ r2 V f ~ > d r f ~ > d r ~ w ( ] r - r ' l ) - dr w(r) .

Since w is defined only outside the hard-cores, the integration ranges do not include distances smaller than the diameter d of the cores. Notice that for an attractive additional pair-potential w the parameter Wo is positive. Now use that the osmotic pressure is equal to II - -OA/OVIN.r to show that (with

- N / V ) ,

H fikB T 1 fi2 1 - p$ - -~w~ .

This is the van der Waals equation o f state. Verify that for positive wo and low enough temperatures, the qualitative features sketched in figs.9, l a-c are

15Notice that the minimization of �9 is constrained by the condition that the total number of particles in the canonical ensemble is a constant, that is,

N - ~_~Nj - constant. J

The actual function that one should minimize is therefore,

* t ( N 1 , ' " , N n ) - ~ ( N 1 , " ' , N n ) - A E NJ ' J

where A is a Lagrange multiplier, which can be determined after minimization. In this way van Kampen (1964) constructs, quite elegantly, the two-phase equilibrium states. We do not go into this matter here.

confirmed by this result. Use that (5 equals four times the core volume of a particle : this is the simplest approximation for (5, being half the volume that is mutually excluded for a pair of particles. The van der Waals equation of state is only qualitatively correct due to the approximate nature of the treatment of the free volume and the unrealistic assumption of infinitely long ranged attractive pair-interactions.

(b) Show from the stability criterion (8.1) that the homogeneous state with density p is unstable when,

flWo 1 (5 > /~ (1 -/Sdi) 2" (9.107/

Verify that the the minimum value for the function 1/x(1 - x) 2 is 27/4 which is attained for x - 1/3. Conclude that there is no unstable homogeneous state when/3wo/~ < 27/4, and that the critical temperature is given by T~ - • TM

- - 27 ' kB6" (c) Equations (9.105,106) allow for the construction of the Helmholtz free

energy functional of the density for an inhomogeneous state. To this end, the summations over volume elements in eq.(9.105) are to be replaced by volume integrals. This can be done as follows. Instead of working with number densities, it is more convenient here to work with a quantity that is proportional to the volume fraction of colloidal particles,

- N j < 5 1 A .

When $ is taken equal to four times the core volume of a particle, this is four times the volume fraction in the jth volume element. According to eqs.(9.105,106), the free energy can be written in terms of this concentration parameter as,

A= kBTA ( [ { 1 - ~ j } ] ) l ( ~ ) 2 - - 7 - - Z ~pj In ~ + In{(5} + 1 + 7 E WijqOiqOj.

3 s ,3

The summations can be identified as integrals as follows,

E A ( ' " ) J - / d r (.. �9 ) ( r ) ,

J

Verify that the free energy can now be written as,

A[v(r)] - kBT , + + 1]) [,n{ 1 r' . + ~-~ i dr i dr' w(I r - I)qo(r')~(r)


The functional dependence of A on ~(r) is denoted as usual by the square brackets. Show by functional differentiation that (when you are not familiar with functional differentiation, you may consult appendix A),

[lnIX 'r'l ] ~q0(r) - (5 r + ln{6} + 1 - 1 - qo(r)

1 f r' q - ~ dr 'w( I r - I)qo(r').

The chemical potential is equal to #(r) - 6A[p] 6A[~] 6p(r) = 6~(r) X 6. Verify that the particle current density is equal to,

kBTD Old j ( r ) - - D V # ( r ) - - r V~(r ) - r ' [Vw([ r - r ' I)]r

Apply Gauss's integral theorem to arrive at the following equation of motion,

__0 Ot ~(r, t) = -kBTD8 1 4~(r, t! +. 3f2.(r, t) t) 12 qo2(r,t) (1 - r i ' IVy(r,

kBTD5 f + ~(r, t) ( 1 - ~ ( r , t))2 V2cP( r, t) + D dr' w([ r - r' [)V'2~(r ', t ) ,

Where the time dependence of q0 is now denoted explicitly. Linearize with respect to 6~p(r, t) = ~p(r, t) - ~, with ~ = ~ , and show that,

0 kBTD~ f 0--'t 6~(r, t) - q~(1 -q5)2 V26~( r, t) + D dr' w(I r - r' I)V'26qo(r ', t ) .

Fourier transform this equation of motion with respect to the position coordinate r with the help of the convolution theorem (see exercise 1.4c) to show that,

6p(k, t) - ~p(k, t - 0)exp { - D ~ff(k) k 2 t } ,

where the effective diffusion coefficient is equal to,

D~Z(k) - kBTD6 f~ q5(1 - q5) 2 + D dr e x p { - i k �9 r}w(r) .

~, > d �9 Y -~(k)

Expand the Fourier transform w(k) up to "O(k2) '' , to show that,

D~Z(k) = kBTD5 - D k2w ].

The parameter w0 is defined in exercise (b), while,

W 2 - -

This is the standard form of the Cahn-Hilliard diffusion coefficient. Verify that D~ff(k - 0) < 0 whenever the instability criterion in eq.(9.107) is satisfied, as it should. Use that D/Do - fl~ (see eq.(9.32)) and the van der Waals equation of state in (a) to show that D~f.t(k - O) - DofldII/dp, in accordance with our general expression (9.28) for the effective diffusion coefficient. Derive an expression for E (see eq.(9.28)) in terms of the interaction parameter w2, and verify that E > 0 for an attractive long ranged pair-interaction potential w.

9.2) * Fourier transformation of eq.(9.24) with respect to r yields integrals of the type,

I(k) - ik. f dr / dR [Vnv(n)]f(R)Sp(r - aR, t )exp{- ik �9 r ) ,

where a is either 1 or 1/2. Verify each of the following mathematical steps which lead to an expression for the integral in terms of the Fourier transform 5p(k, t),

I(k) = ik. f dr f dR[V nv( n)] f ( R)~p(r-aR, t) exp{-ik. (r-aR) } exp{-iak.R}

= ikfdR[VnV(R)lf(n)exp{-iak.R}fd(r-aR)6p(r-aR , t) exp{-ik. ( r -aR)}

= $p(k, t)ik.fdR[VnY(R)]f(R)exp{-iak.R}.

Now use that VnV(R) - RdV(R)/dR, with R - R/R, and verify that (Vk is the gradient operator with respect to k),

/ dR [VRV(R)]f(R)exp{-ik �9 R} ik .

- i k . fo ~r dR R 2 dY(R) ~I~ f (R) f dR It, e x p { - i a k . I~R}

v/R/ 1 / dI~ f (n) Vk dR e x p { - i a k . RR} - i a R

- ik. fo ~ dR R 2 dY(R)_d_R f(R) 1 Vk 4r sin{akR} - i a R akR


dV(R) 1 sin{akR} = ik . ]oo dR R 2 f (R ) d----R i a------R V k 47r -- a k R

= ik . fo ~176 dR R 2 dV(R) 1 a2R2 d-----~ f (R) - i a R 47r k j ( a k R ) .

In the third equation it is used that,

sin{akR} I "

dl~ exp{:t:iak. R} - 47r a k R J

(9.108)

This mathematical identity is derived in appendix A of chapter 5 (see eq.(5.139)). The j-function is defined in eq.(9.27). Conclude that,

I(k) - -(Sp(k, t)47rak 2 ~ ] r162 dR R 3 dV(R) f (R ) j ( a k R ) . dR JO

Use this result to verify eqs.(9.25,26).

9.3) * To obtain eq.(9.43) for the diffusion coefficient defined in eq.(9.42), integrals of the kind,

sin{k. ,

I - f f d R ( l : r 2 ~r R}

must be evaluated, where 5~dl~ is the integration with respect to spherical angular coordinates ranging over the unit sphere. Show that this integral is equal to,

I - - - - m 2 (KR) 2 0 a dR [ exp{iak. R) + e x p { - i a k . R) ]],~=~ ,

where a is to be set equal to 1 after the differentiation is performed. Use eq.(9.108) to show that,

I - - 4 r j ( k R ) ,

with the j-function defined in eq.(9.27). Verify eq.(9.43).

9.4) Stability and demixing of confined suspensions In this chapter we have considered systems of infinite extent, where density

waves of infinite wavelength become unstable on the spinodal. Suppose now that the suspension in contained in a cube with sides of length L. The maximum


wavelength of density waves is now L, corresponding to wavevectors 2r/L. Suppose that the container is still large enough to neglect the influence of the walls of the container. Show that the spinodal is now given by,

d,~ = -E~ .

At a given density the spinodal temperature is thus lower than for a system of infinite extent.

Consider a rectangular geometry with two small equal sides of length l and a large length L : L >> I. Argue that upon cooling, density waves with wavevectors along the long side will become unstable first. The demixing process will then have a one-dimensional character.

In a realistic description of the shift of the spinodal due to a confining geometry, the effects of the walls on the microstructure of the suspension should be taken into account, which is not a simple matter.

9.5) Porod's law Porod's law states that sharp and very sharp interfaces give rise to a scat-

tered intensity that varies like ,-~ k -4 for large wavevectors. Let us describe the interfaces as the (infinitely sharp) boundaries between an optically homogeneous assembly of spheres, polydisperse in size, and a homogeneous solvent. According to eqs.(3.199,100) the scattered intensity of such an assembly of spheres is proportional to,

I(k) ~, foo~176 da Po(a) [ka cos{ka}(ka) 3- sin{ka}]

where Po is the probability density function for the radius a of the spheres. Verify that for large wavevectors,

j~0 ~176 I(k) ,~ k-' da Po(a) cos2{ka}.

For large wavevectors, cos{ ka} has many oscillations as a function of a over intervals where the pdf Po(a) remains virtually constant. Convince yourself that for such large wavevectors,

./o de Po(a) fO ~ 2{ka} ~ da Po(a) sin2{ka}.

630 Further Reading

Show from this that it follows that,

I(k) ~ k -41fo r [cos 2 1 k - ' -~ da Po(a) {ka} + sin2{ka}] - ~ .

A much more sophisticated treatment of scattering by interfaces can be found in Tomita (1984,1986).


A few of the original papers on the Cahn-Hilliard theory are, �9 J.W. Cahn, J.E. Hilliard, J. Chem. Phys. 28 (1958) 258, 31 (1959) 688. �9 M. Hillert, Acta Metallica 9 (1961) 525. �9 J.W. Cahn, Acta Metallica 9 (1961) 795. �9 J.W. Cahn, J. Chem. Phys. 42 (1965) 93. �9 J.W. Cahn, Trans. Metall. Soc. Aime 242 (1968) 166. �9 H.E Cook, Acta Metallica 18 (1970) 297. �9 J.E. Hilliard (ed. H.J. Aronson), in Phase Transformations, American

society for metals, Metals Park OH, 1970, chapter 12. The 1958 paper of Cahn and Hilliard is concerned with the contribution of gradients in the density to the free energy. Extensions of the Cahn-Hilliard theory, including computer simulations, are,

�9 J.S. Langer, Annals of Physics 65 (1971) 53. �9 J.S. Langer, M. Bar-on, Annals of Physics 78 (1973) 421. �9 J.S. Langer, M. Bar-on, H.D. Miller, Phys. Rev. A 11 (1975) 1417. �9 K. Kawasaki, Prog. Theor. Phys. 57 (1977) 826. �9 K. Kawasaki, T. Ohta, Prog. Theor. Phys. 59 (1978) 362, 59 (1978)

1406. �9 R. Evans, M.M. Telo da Gama, Mol. Phys. 38 (1979) 687. �9 K. Binder, J. Chem. Phys. 79 (1983) 6387. �9 K. Binder, Coll. Pol. Sci. 265 (1987) 273. �9 C. Billotet, K. Binder, Z. Phys. B 32 (1979) 195. �9 G.E Mazenko, Phys. Rev. B 42 (1990) 4487. �9 A. Sariban, K. Binder, Macromolecules 24 (1991) 578. �9 P. Fratzl, J.L. Lebowitz, O. Penrose, J. Amar, Phys. Rev. B 44 (1991)

4794.

Further Reading 631

�9 A. Shinozaki, Y. Oono, Phys. Rev. lett. 66 (1991) 173. �9 J.A. Alexander, S. Chen, D.W. Grunau, Phys. Rev. B 48 (1993) 634. �9 T. Koga, K. Kawasaki, Physica A 196 (1993) 389.

In the 1975 paper of Langer, Bar-on and Miller, an expression for the time dependence of km (t) is found for molecular systems that is similar to eq.(9.74). They also derive the identification in eq.(9.69). A few of the above papers start from equations of motion for the density, and solve these (numerically), including the late stage. It turns out that this is not realistic. Scaling behaviour is predicted in a more reliable way from heuristic considerations about the driving mechanisms during the transition and late stage. See,

�9 K. Binder, D. Stauffer, Phys. Rev. Lett. 33 (1974) 1006. �9 E.D. Siggia, Phys. Rev. A 20 (1979) 595.

Nonlocal diffusion, discussed in subsection 9.2.3 and section 9.6, is also considered in,

�9 P. Pincus, J. Chem. Phys. 75 (1981) 1996. �9 K. Binder, J. Chem. Phys. 79 (1983) 6387.

This work is on polymer systems.

The effect of sharp interfaces on scattering properties are discussed in, �9 G. Porod (eds. O. Glatter, O. Kratky), Small Angle X-ray Scattering,

Academic Press, London, 1982, page 30. �9 H. Tomita, Prog. Theor. Phys. 72 (1984) 656, 75 (1986) 482.

A S moluchowski equation approach to spinodal decomposition for rigid rod like Brownian particles, where correlations are neglected (that is, where the pair-correlation function is taken equal to 1), can be found in,

�9 T. Shimada, M. Doi, K. Okano, J. Chem. Phys. 88 (1988) 7181. The Smoluchowski approach as discussed in subsection 9.2.2 is taken from,

�9 J.K.G. Dhont, A.EH. Duyndam, B.J. Ackerson, Physica A 189 (1992) 503.

�9 J.K.G. Dhont, A.F.H. Duyndam, B.J. Ackerson, Langmuir $ (1992) 2907.

Theory on the effect of shear flow on decomposition kinetics can be found in, �9 T. Imaeda, A. Onuki, K. Kawasaki, Prog. Theor. Phys. 71 (1984) 16. �9 T. Imaeda, K. Kawasaki, Prog. Theor. Phys. 73 (1985) 559. �9 A. Onuki, Physica A 140 (1986) 204. �9 J.K.G. Dhont, A.EH. Duyndam, Physica A 189 (1992) 532.

632 Further Reading

�9 J. Lai, G.G. Fuller, J. Pol. Sci.: part B: Pol. Physics 32 (1994) 2461. In most of these papers the tendency for concentration fluctuations to acquire two dimensional character as time proceeds is explicitly mentioned, in accordance with the results of section 9.3. The approach developed in section 9.3 is taken from the paper by Dhont and Duyndam.

Experiments on spinodal decomposition in binary fluids are reported in, �9 P. Guenoun, R. Gastaud, E Perrot, D. Beysens, Phys. Rev. A 36 (1987)

4876. �9 A. Cumming, P. Wiltzius, F.S. Bates, J.H. Rosendale, Phys. Rev. A 45

(1992) 885. �9 N. Kuwahara, K. Kubota, M. Sakazume, H. Eda, K. Takiwaki, Phys.

Rev. A 45 (1992) 8324. �9 K. Kubota, N. Kuwahara, H. Eda, M. Sakazume, K. Takiwaki, J. Chem.

Phys. 97 (1992) 9291. �9 A.E. Bailey, D.S. Cannell, Phys. Rev. lett. 70 (1993) 2110.

Experiments on polymer systems can be found in, �9 C.A. Smolders, J.J. van Aartsen, A. Steenbergen, Kolloid-Z.u.Z. Poly-

mere 243 ( 1971) 14. �9 I.G. Voigt-Martin, K.-H. Leister, R. Rosenau, R. Koningsveld, J. Pol.

Sci.: Part B: Pol. Phys. 24 (1986) 723. �9 P. Wiltzius, ES. Bates, W.R. Heffner, Phys. Rev. lett. 60 (1988) 1538. �9 ES. Bates, P. Wiltzius, J. Chem. Phys. 91 (1989) 3258. �9 H. Lee, T. Kyu, A. Gadkari, J.P. Kennedy, Macromolecules 24 (1991)

4852. �9 M. Takenaka, T. Hashimoto, J. Chem. Phys. 96 (1992) 6177. �9 N. Kuwahara, H. Sato, K. Kubota, J. Chem. Phys. 97 (1992) 5905,

Phys. Rev. E 47 (1993) 1132. �9 M. Takenaka, T. Hashimoto, Macromolecules 27 (1994) 6117. �9 C.C. Lin, H.S. Jeon, N.P. Balsara, J. Chem. Phys. 103 (1995) 1957.

The data in fig.9.14 are taken from Wiltzius and Bates (1988). Spinodal decomposition in other systems, like alloys (Komura) and surfactant systems (Mallamace et al.) is discussed in,

�9 S. Komura, K. Osamura, H. Fujii, T. Takeda, Phys. Rev. B 31 (1985) 1278.

�9 E Mallamace, N. Micali, S. Trusso, S.H. Chen, Phys. Rev. E 51 (1995) 5818. The data in fig.9.13 are taken from Malamace et al. (1995).

Further Reading 633

Experiments on the effect of steady and oscillatory shear flow on the spinodal decomposition kinetics of binary fluids can be found in,

�9 D. Beysens, M. Gbadamassi, L. Boyer, Pys. Rev. Lett. 43 (1979) 1253. �9 D. Beysens, M. Gbadamassi, B. Moncef-Bouanz, Phys. Rev. A 28

(1983) 2491. �9 D. Beysens, E Perrot, J. Physique-Lettres 45 (1984) 31. �9 C.K. Chan,E Perrot, D. Beysens, Phys. Rev. Lett. 61 (1988)412. �9 E Perrot, C.K. Chan, D. Beysens, Europhysics lett. 9 (1989) 65. �9 T. Baumberger, E Perrot, D. Beysens, Physica A 174 (1991) 31. �9 C.K. Chart, E Perrot, D. Beysens, Phys. Rev. A 43 (1991) 1826. �9 T. Baumberger, F. Perrot, D. Beysens, Phys. Rev. A 46 (1992) 7636.

Similar experiments on polymer systems are reported in, �9 T. Hashimoto, T. Takebe, K. Fujioka (eds. A. Onuki, K. Kawasaki), in

Dynamics and Patterns in Complex Fluids, Springer Proceedings in Physics vol.52, Springer Verlag, Berlin, Heidelberg, 1990.

�9 T. Hashimoto, T. Takebe, K. Asakawa, Physica A 194 (1993) 338. �9 T. Hashimoto, K. Matsuzaka, E. Moses, A. Onuki, Phys. Rev. lett. 74

(1995) 126.

Experiments on the influence of sedimentation due to gravitational forces on spinodal decompisition kinetics are described in,

�9 D. Beysens, P. Guenoun, E Perrot, Phys. Rev. A 38 (1988) 4173. �9 G. Schmitz, H. Klein, D. Woermann, J. Chem. Phys. 99 (1993) 758.

Overview articles, where in some cases nucleation is also discussed, and which contain additional references, are,

�9 K. Binder, Rep. Prog. Phys. 50 (1987) 783. �9 W.I. Goldburg (eds. S.H. Chen et al.), Scattering Techniques Applied

to Supramolecular and Nonequilibrium Systems, Plenum Press, New York, 1981, page 383.

�9 J.D. Gunton, M. San Miquel, P.S. Sahni (eds. C. Domb, J.L. Lebowitz), Phase Transitions and Critical Phenomena, vol. 8, Academic Press, New York, 1983, page 267.

�9 S.W. Koch (eds. H. Araki et al.), Dynamics of First-order Phase Transi- tions in Equilibrium and Nonequilibrium systems, Lecture Notes in Physics, Springer Verlag, Berlin, 1984.

�9 K. Binder, D.W. Heermann (eds. R. Pynn, A. Skjeltorp), Scaling Phe- nomena in Disordered Systems, Plenum Press, New York, 1985, page 207.

634 Further Reading

�9 H. Fumkawa, Adv. Phys. 34 (1985) 703. �9 P. Guyot, J.P. Simon, Joumal de Chim. Phys. 83 (1986) 703.

The derivation of the free energy functional of a van der Waals fluid, used in exercise 9.1, and a description of two-phase equilibrium can be found in,

�9 N.G. van Kampen, Phys. Rev. 135 (1964) A362.

INDEX

635

A Associated Legendre functions 422

B Backflow 204,461-468 Barometric height distribution 469 Binoda1497,561 Boltzmann exponential 36 Boundary layer 366,430 Brownian force 183 Brownian oscillator 220 Brownian torque 216

C Cage of particles 40,390,391 Cahn-Hilliard plot 607 Cahn-Hilliard square gradient coefficient 569 Cahn-Hilliard theory 567 Cauchy-Riemann relations 22,55 Cauchy's formula 28 Cauchy's theorem 25 Central limit theorem 48 Chandrasekhar's theorem 79 Collective diffusion

introductory 317 near critical point 530 short-time 339

Collective dynamic structure factor definition 45,149,324 rods, non-interacting 398 spheres, non-interacting 63,186

Colloids, definition 2 ' Condensation 497,562,563 Configurational partition function 36

van der Waals fluid 622 Connectors 264

Continuity equation 229 Contraction 14 Contrast

dynamical 135 optical 129 variation 152,166

Convolution theorem 52 Correlation function

definition 40 density auto- 44

Correlation length 510,507,514 in sheared systems 523

Coupling function 399 Covariance matrix

definition 47 equation of motion for 188

Creeping flow equations 238 effective 462

Critical point 498 scattering close to 500,514

Critical slowing down 531 Cumulant expansion 426 Curves in the complex plane 24

Delta distribution 17,302,418,424 Delta sequence 17,51,418 Density wave 317,318,564 Dielectric constant of a rod 153 Diffusion coefficient

collective 321 light scattering 324 long-time 323

zero wavevector 322 short-time 323,340,341/347

infinite wavevector 349 zero wavevector 347

gradient 321,347,355,474,475,488

636

polydisperse 148 self 325

light scattering 325 long-time 327,361,363,430

weak coupling approximation for 387

short-time 327,333,339 near critical point 554

Stokes-Einstein, rods 97,101,211 Stokes-Einstein, spheres 81,185

Diffusive angular scale 104 Dimension of a vector/matrix 13 Direct correlation function 504,550 Disturbance matrix 197,277,280,537 Double layer 7,28 DVLO theory 7,28,55 Dyadic product 14 Dynamic light scattering 132

and optical polydispersity 149 and size polydispersity 147,164 heterodyne 168 rods 158 spheres 143,324,325

Dynamic scaling 606,612 Dynamic similarity scaling 606,609

E

F Far field approximation for

electric field 120 hydrodynamic interaction 253,307

Fax6n's theorems 253 rods 284,311 rotational motion 255 translational motion 255

Fick's law 323,355 Fluctuation strength

rods, rotational 95 rods, translational 94 spheres 71,74 spheres in shear flow 84-86

Fluid flow past a rotating sphere 244,248 sphere in shear flow 277 translating sphere 244,245

Fokker-Planck equation derivation of 179 linear 187

Fokker-Planck operator 181 Form factor

rods 155,167,393 spheres 127

polydisperse 146

Effective interaction potential40,60,502 Fourier inversion 52 Effective medium approach 429 Fourier transformation 19

Electric field auto-correlation function (EACF)

definition 133 polydispersity 151 rods, general 158 rods, non-interacting 396,622

Ensemble, definition 32 Equipartition theorem 102 Extensional flow 87

Friction of rod in shear flow 309 Friction coefficient

effective 356 rods, rotational 92,210,286,310 rods, translational 92,210,285 spheres, rotational 71,250 spheres, translational 71,247

Frequency functions collective 380

637

self 381 Incompressibility 230 Functional differentiation 570,612-615 Indexrank 13

G Gauss's theorem 16,53 Gaussian variables 46,64 Gradient diffusion

attractive spheres 428 hard-spheres 351

Gradient operator 15 Green's theorems 17 Guinier approximation 142

H Hard-core repulsion 8 Hydrodynamic interaction

in shear flow 276,278 leading order 250 Rodne-Prager level 255 spheres 271 three body 273 unequal spheres 308 with sedimentation 281

Hydrodynamic interaction, introductory

in shear flow 196

Inner product 14,375,425 Integral theorems 16,17 Intensity auto-correlation function (IACF)

definition 132 rods, non-interacting 392

Intensity cross-correlation function (ICCF) in shear flow 201 Interaction

direct 5 effective 40,60,502 hydrodynamic 177 long ranged 501

Intermediate scattering function see : collective dynamic structure factor

J Jordan's lemma 59

K Kawasaki function 533,534 Kramer's equation 182 Kronecker delta 14

rods 209 spheres 177,222 with sedimentation 204

Hydrodynamic mobility function 340,347,348

near critical point 531-535 relation with sedimentation 487

I Identity matrix 14 Incident wavevector 110

L Langevin equation

on diffusive time scale 81 rods 91 spheres 70

Laplace operator 15 Legendre polynomials 421 Length of a vector 13 Length scale, diffusive 77 Light scattering 107 Local equilibrium

638

statistical analoque 574 thermodynamic 568

Long-time tail 388 function 391 mean squared displacement 392 self memory function 390 velocity auto-correlation

Lubrication theory 272

M Mathematical notations 13 Mean squared displacement

rotational 101,219 translational, of rods 97,218 translational, of spheres 77,191,325

long time tail of 392 Memory equations 372

collective 378 self 379

Memory functions alternative expression for 383 collective 377,386 self 379,387

long time tail of 390 weak coupling approximation for 386,387

Method of reflections 258 Microscopic diffusion matrices 184,228 Microscopic friction matrices 178,228 Mobility functions

definition 266 higher order 271,272 Rodne-Prager level 267

Multiple scattering 112 near critical point 618-622

Multivariate Gaussian pdf 47

N Navier-Stokes equation 231 Non-Gaussian displacements 424 Nucleation 496,562,563

0 Onsager's equation 406 Operator exponential 42,85 Orientational correlations 97 Orientational relaxation

rods 223,400,435 spheres 257

Ornstein-Zernike correlation function 506,513 equation 504,505 static structure factor 508,519

with shear flow 515 theory 501

Oseen matrix 241 Oseen approximation 253,307 Outer product 14

P Pair-correlation function 37,61 Pair-interaction potential 5 Parseval's theorem 52 Peclet number

sedimentation 477 shear flow 366

bare 517 dressed 520

Phase function, definition 32 Phase space, definition 32,173 Poisson-Boltzmann equation 56 Polyadic product 14 Polydispersity 9

and light scattering 144,163

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Porod's law 629 Pressure vector 242 Probability density function 31

conditional 33 deformation of, due to

external force 358,359 sedimentation 447 shear flow 364

for position 80 for position, in shear flow 87 reduced 35

Projection operator 475

Quench 564 Q

R Radius

hydrodynamic radius 144 optical radius of gyration 142

Rayleigh ratio 126 Reflected flow fields 262 Residue theorem 22,26 Resolvent operator 434 Rodne-Prager matrix 256 Rotational Brownian motion

non-interacting rods 88 Rotational flow 87 Rotational relaxation

rods 223,400,435 spheres 257

Rotation operator 216

S Scaling of

non-Newtonian viscosity near critical point 545

static structure factor for demixing suspension 605-607 under shear near critical point 520

turbidity near critical point 530 Scattered field strength 112,121

depolarization of 435 heuristic derivation 109 Maxwell equation derivation 113 relation to density fluctuations 122

Scattered intensity 122 by demixing systems 586-590 close to critical point 500,514

Scattered wavevector 110,162 Scattering amplitude 117,121 Scattering angle 110 Scattering by rods 153,167,392,412

depolarized small angle 223,401 Scattering strength 110 Scattering volume 111 Second cumulant 164,426 Sedimentation 4

hydrodynamic interaction 204,281 relation with hydrodynamic mobility function 487 Smoluchowski equation with 207,447

Sedimentation of rods 104,487 spheres 307,445-457,479

charged 459 hard 457 sticky 481 superparamagnetic 482,484

Sedimentation-diffusion equilibrium 468 Sediment formation 473,488 Self diffusion

introductory 324

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long-time 356,430 short-time 332

near critical point 554 Self dynamic structure factor

definition 46,149 on Fokker-Planck time scale 191 rods, non-interacting 398 spheres, non-interacting 60,186 with shear flow 201

Self intermediate scattering function see : self dynamic structure factor Shear flow

diffusion in 83,103,199,329,363 disturbance matrix 197,277,280,537 effect near critical point 515-530 friction of rod in 309 hydrodynamic interaction in 276 Smoluchowski equation with 195 sphere in 277

Shear thinning 546 Shear waves 235 Short-time diffusion 331

collective 339 self 332

Siegert relation 134 Smoluchowski equation

rods 208,212 spheres 183 with sedimentation 204,207 with shear flow 195,197

Smoluchowski operator Hermitian conjugate of 332,425 rods 216,217 spheres 184 with sedimentation 208 with shear flow 198

Sound waves 237 Spherical harmonics 402,422

Spinoda1497,498,561 Spinodal decomposition 497,552

Cahn-Hilliard theory 567 confined suspensions 628 final stage 566 initial stage 566,567-580,607

experiments 607 under shear 580-586

intermediate stage 566,590-605 experiments 609-612 interaction 599-602 role of hydrodynamic

introductory 561-567 transition stage 566 van der Waals fluid 622-627

Static light scattering 125 and of size polydispersity 145,163 Porod's law 629 rods 154 spheres 141

near critical point 514 Static structure factor

definition 46 demixing suspension 588

scaling 605-607 in shear flow 368,369 Ornstein-Zernike 508,519

with shear flow 519,520 rods 154 scattering 128

Steric repulsion 8 Stochastic variables, definition 32 Stokes's theorem 16,54 Stress matrix 232

deviatoric part of 234 Structure factor see : static structure factor Superposition approximation 509

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improved 511 solvent 233

T Taylor expansion 15 Three body hydrodynamic interaction 273 Three-particle correlation function 37 Time evolution operator 42 Time scale

and dynamic light scattering 140 Brownian 76 diffusive 76 Fokker-Planck 75 hydrodynamic 78,234 interaction 78 solvent 70,72,75,76 Smoluchowski 76

Torque averaged 406,408 Brownian 216 direct 222 hydrodynamic 92,209

Translational diffusion of rods 96 Transpose of a matrix 13 Turbidity 525,553

scaling near critical point 527,554

U Uphill diffusion 571

V van der Waals fluid 622-627 Viscosity

anomalous behaviour of 535 effective 304 Newtonian/non-Newtonian 546 scaling near critical point 545 shear thinning 546

W Weak coupling approximation 383 Wick's theorem 49

Y Yukawa potential 7,28,55 Yvon's identity 426

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An Introduction to Dynamics of Colloids - Jan K. G. Dhont

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