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Transcript of An Introduction to Dynamics of Colloids J.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
ELSEVIER
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�9 1996 Elsevier Science B.V. All rights reserved.
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First edition 1996 Second impression 2003
Library of Congress Cataloging in Publication Dam _ _ _
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 .
S t u d i e s in i n t e r f a c e s c i e n c e ; v. 2. OD549.D494 1996 5 3 0 . 4 ' 2   d c 2 0
 ) and i ndex .
I. Title. II. Series:
9 6  1 2 8 4 6 CIP
ISBN: 0 444 82009 4
The paper used in this publication meets the requirements of ANSI/NISO Z39.481992 (Permanence of Paper). Printed in The Netherlands.
To my mother
In memory of my father
This Page Intentionally Left Blank
PREFACE
This book is a selfcontained 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 distin guished �9 those aimed to predict equilibrium properties and equilibrium mi crostructure 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 selfexplanatory 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 LizMarz~, 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
NONINTERACTING PARTICLES 3 : LIGHT SCATTERING 4 : FUNDAMENTAL EQUATIONS OF MOTION 5 : HYDRODYNAMICS 6 : DIFFUSION 7 : SEDIMENTATION 8 : CRITICAL PHENOMENA 9 : PHASE SEPARATION KINETICS
168
69106 107170 171226 227314 315442 443494 495558 559634
CHAPTER 1 : INTRODUCTION 168
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 CauchyRiemann 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 paircorrelation function
1.3.2 Time dependent Correlation Functions 1.3.3 The Density AutoCorrelation 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 NONINTERACTING PARTICLES 69106
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 107170
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
Exercises Further Reading and References
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 FokkerPlanck Equation 4.4 The Smoluchowski Equation 4.5 Diffusion of nonInteracting Particles
4.5.1 Linear FokkerPlanck Equations 4.5.2 Diffusion on the Brownian Time Scale 4.5.3 Diffusion on the FokkerPlanck 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 nonInteracting Particles in
Shear Flow
171226
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 nonInteracting Rods
Exercises Further Reading and References
208 209 212 218 220 225
CHAPTER 5 : HYDRODYNAMICS 227314
5.1 Introduction 5.2 The Continuity Equation 5.3 The NavierStokes 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 RodnePrager 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 315442
6.1 Introduction 6.2 Collective Diffusion
The zero wavevector limit Shorttime and longtime collective diffusion Light scattering
6.3 Self Diffusion Shorttime and longtime self diffusion
6.4 Diffusion in Stationary Shear Flow 6.5 Shorttime Diffusion
6.5.1 Shorttime Self Diffusion 6.5.2 Shorttime Collective Diffusion 6.5.3 Concluding Remarks on Shorttime Diffusion
6.6 Gradient Diffusion 6.7 Longtime Self Diffusion
6.7.1 The Effective Friction Coefficient 6.7.2 The Distorted PDF 6.7.3 Evaluation of the Longtime 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 LongTime Tails
6.10 Diffusion of Rigid Rods
368
369 372 372 373 374 380
381 383 388 392
6.10.1 The Intensity AutoCorrelation 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 fortheTcoefficients 410 Evaluation of the scattered intensity 412
415 416 418 420 421 424 437
CHAPTER 7 : SEDIMENTATION 443494
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 HardCore
Interaction 7.2.4 Sedimentation of Spheres with very Long
Ranged Repulsive PairInteractions 7.3 Nonuniform Baektlow
The effective creeping flow equations Solution of the effective creeping flow equations
7.4 The SedimentationDiffusion 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
Exercises Further Reading and References
450
457
459 461 462
465 468
469 472 473
476 479 481 490
CHAPTER 8 : CRITICAL PHENOMENA 495558
8.1 Introduction 8.2 Long Ranged Interactions
8.2.1 The OrnsteinZernike Approach Asymptotic solution of the OmsteinZernike equation
8.2.2 Smoluchowski Equation Approach 8.2.3 A Static Light Scattering Experiment
8.3 The OrnsteinZernike 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 nonNewtonian 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 559634
9.1 Introduction 9.2 Initial Spinodai Decomposition Kinetics
9.2.1 The CahnHilliard 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
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Chapter I
INTRODUCTION
2 Chapter 1.
This introductory chapter consists of three sections. The first section in troduces colloidal systems. The various common kinds of pairinteraction 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 chap ter 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 math ematical 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 physicalchemistry education. In courses on mathematics for chemists, the residue theorem is often not in cluded. Special attention is therefore given to that theorem, which is derived in a more or less selfcontained 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 nonequilibrium and dy namical 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 rec ognized 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 mem brane, 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 lightmicroscopic 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 mo tion). 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 sur face 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 sur face 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 mo tion (=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 dis persions, 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, socalled 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 fdvirus). Polymer solutions may behave as dispersions of the first kind, when the polymer chain in a poor
1.1. Colloidal Systems
solvent is shrunk to a rigid spherically shaped object. An example of colloids of the third kind are microemulsions, 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 nonmetallic 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 crosslinked (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 hardcore 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 hardcore 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 (3methacryloxypropyltrimethoxysilane)). 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 socalled 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 pairinteraction 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 DVLOpotential, where DVLO stands for DerjaguinVerweyLandau 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 pairinteraction 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[ 12hydroxy 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 pairinteraction 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 hardcore 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 hardcore 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 pairinteraction potential, superimposed onto a hardcore 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 thetaconditions (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
1.1. Colloidal Systems
_• ::~"~:~:~: .. ::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 soca!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 pairinteraction potential on the distance between the colloidal particles is derived. On several occasions in this book we will use simple expressions for the pairinteraction 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 pairinteraction 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 poly dispersity, 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 TMVvirus (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 socalled fdvirus, 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 af fects 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 parti cles 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 pairinteraction 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 pairinteraction 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 mo tion. 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 nonInteracting 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 reflec tion can be observed indirectly for example by means of Xray 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. Fluidgas phase separation (into a concentrated and dilute colloidal fluid) can occur in case of attractive interactions. Also, thermody namically metastable states exist, like gel states, where colloidal particles are permanently but reversibly attached into strings which span the entire con tainer, 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 insta bilities, 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 trans lational 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, viscoelastic 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 hydrodyna mically. The question then is how these phenomena can be described and how predictions can be made on the basis of a given pairinteraction 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 selfcontained 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 physicalchemistry background. Special attention is given to the residue theo rem. 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 zcoordinate, r2 the ycoordinate and ra the zcoordinate" 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
nonboldfaced 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 socalled Kronecker delta. Thus, the elements of I on the diagonal, where i  j are equal to 1, while the offdiagonal 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 socalled 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 3dimensional 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
1.2. MathematicalPreliminaries 15
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 Ndimensional 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 3dimensional 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 ijthelement 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,m1
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 3dimensional position vector and F(X) is a 3dimensional 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 Ndimensional space NN. Gauss's integral theorem states that for continuous differentiable Ndimensional vector fields r(x),
fw dX V x . F(X)  ~ w dS F ( X ) , (1.4)
where the integral on the right handside ranges over the surface OW that encloses the volume W, and dS is the Ndimensional 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.
1.2. MathematicalPreliminaries 17
The proof of these two theorems can be found in standard texts on mathematics, and should be part of the mathematics education of any physicalchemist.
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 nonzero. Indeed this is not a function in the usual sense but belongs to the class of socalled 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 testfunction f . The testfunction is essentially equal to f (xo) in the range of xvalues where ~n (x) for large n is nonzero.
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
mO
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 testfunctions, 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~.
1.2. MathematicalPreliminaries 19
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 1dimensional 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 Ndimensional 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 func tions. 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 socalled 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 socalled 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 intro duced 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 ge neralized to functions of Ndimensional vectors X  (x~, x2 , . . . , XN). First decompose the xidependence 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 handside now as a function of z2 , . . . , XN, and decompose the x2dependence, 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 Ndimensional kspace 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 com plex variable, socalled 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 CauchyRiemann 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 '
1.2. Mathematical Preliminaries 23
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 xaxis, 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 yaxis, 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 handside is equal to d f ( z ) / d x , while the right handside 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 CauchyRiemann 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 CauchyRiemann relations are continuous. That is, when the CauchyRiemann
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 fn, each of which
1.2. Mathematical Preliminaries 25
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 writ ten 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 anticlockwise 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 Ox Ov(x' Y) }  ~o {dx v(x y) + dy u(x y)} .
b y S ~
When u and v satisfy the CauchyRiemann relations (1.22), the left handsides
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 socalled firstorder 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).
1.2. Mathematical Preliminaries 27
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 dockwise 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 anticlockwise orientation is understood. For example, "7 is traversed anticlockwise, 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 anticlockwise direction. The integral on the right handside 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 )  Q6(r), (1.28)
1.2. Mathematical Preliminaries 29
with a a constant with dimension m 1, c the dielectric constant of the solvent, and 5(r) the 3dimensional 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 zdirection, 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{4ikr}. (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 anticlockwise, but that "7 is clockwise). The integrals in eq.(1.31) are thus equal to,
I•  ~+ dz Z
(z + ix)(z  i ~ ) exp{+izr}  47ri 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 6Ndimensional 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.
1.3. Statistical Mechanics 33
The brackets <  > are nothing but a shorthand notation for the integral on the right handside. 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 dis cussed 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)
1.3. Statistical Mechanics 35
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 exam ple, 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 twoparticle 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 paircorrelation 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 tempera ture) 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
1.3. Statistical Mechanics 37
equilibrium pdf's are often a necessary input for explicit evaluation of non equilibrium ensemble averages also. Since this book is on nonequilibrium 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 paircorrelation 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 socalled paircorrelation 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 threeparticle 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 paircorrelation function attains its value without interactions, which is 1 by definition. The threeparticle 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 paircorrelation function for systems which are so dilute that events where three or more particles interact simultaneously are unlikely. According to eqs.(1.4951), 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 pairinteraction 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 pairinteraction potential, it is found that,
go(r)  exp{  f lV(r )} . (1.55)
In this book we will use the phrase "on the pairlevel ", whenever interactions between three or more (colloidal) particles simultaneously are disregarded. Hence, eq.(1.55) is the paircorrelation function on the pairlevel, and can be used to calculate ensemble averages for dilute systems. In general, the paircorrelation 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 Mayerfunctions, 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 Mayerfunction 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 hardsphere interactions, with the result (the subscript "hs" stands for "hardspheres"),
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 hardcore and r ~aa/~ is the fraction of the total volume that is occupied by the cores of the particles, the socalled volume fraction. This paircorrelation function is plotted in fig. 1. lOa for ~  O. 1. At
1.3. Statistical Mechanics 39
1
0.5 ~ _ iI IJ 2 4 2 4 21+
Figure 1.10: The paircorrelation function to first order in concentration for hardspheres (see eq.(1.57)) with qo  O. 1, (a), a sketch for hardspheres at larger concentra tions (b), and for charges spheres with a long ranged repulsive pairinteraction potential (c).
larger concentrations, the paircorrelation 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 paircorrelation 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 deionized 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 paircorrelation 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 hardcore 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 pairinteraction potential on the pairlevel. 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 hardspheres near contact there is thus attraction, the depletion mechanism for which was already explained above. Around the peak in the paircorrelation function the effective force changes from strongly repulsive to attractive. Multi particle interactions may thus lead to attractions even if the pairinteraction 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
1.3. Statistical Mechanics 41
=  < >
The second pair of brackets < .. > on the left handside 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 pre pared in a certain state before the experiment is started. The most simple unconditional ensemble average that contains information concerning the dy namics 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 handside 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 inde pendent 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 exponen tial 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 ) n1 /~n
n = l ( n  1 ) !
_ s ( t  to)"
n"O
/~n _ ~ exp{/~(t  to)}.
1.3. Statistical Mechanics 43
Substitution into eq.(1.62) and integrating with respect to X0, using the defini tion (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 dif ference tto 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 shorttime 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 AutoCorrelation Function
A particularly important stochastic variable is the microscopic number den sity 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 socalled 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 handside 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 trans formation 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 microsco pic densities, the socalled density autocorrelation function, which is denoted here as S(k', k, t  to) for brevity,
1 S(k', k, t  to)  ~ < p (X(to) I k') p (x(t) ! k) >
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 autocorrelation 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
1.3. Statistical Mechanics 45
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(Rt to)] • d R _ e x p {  ~ ( . . . . ,  �9
An obvious shorthand 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 autocorrelation function is zero. a2 For translationally invari ant systems the meaningful density autocorrelation 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 ~,31 <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 socalled 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 autocorrelation 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 trans form eq.(1.29) to eq.(1.30). Rotational invariance has been assumed here to write the paircorrelation function as a function of R  I R [ . This equal time correlation function is the socalled static structure factor, and is essentially the Fourier transform of the paircorrelation 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)
1.3. Statistical Mechanics 47
where < X > and < X 2 > are generally time dependent averages. When the stochastic variable X is a Ndimensional 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 nonzero 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 handside 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 n2
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 nonzero. 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 handside 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 the orem 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 SX _ 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 { S1 } 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~dj1A~}  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 per form 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 antisymmetric 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 handside 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 righthand 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
54 Exercises Chapter I
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 nondifferentiable complex function Consider the function f (z)  x  iy. Verify that this function does not
satisfy the CauchyRiemann 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 nonanalytic function, its integral over a closed contour is not necessarily nonzero.
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 CauchyRiemann 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 3dimensional 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, hardcore interactions between the ions are not accounted for). In this "mean field
56 Exercises Chapter I
approximation", the number density of such "pointlike" 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 nonlinear PoissonBoltzmann equation. This nonlinear 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 PoissonBoltzmann 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 pairinteraction potential. The pairinteraction 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
Exercises Chapter 1 57
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)}. j1
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 , 1x 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 .
58 Exercises Chapter I
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 pairinteraction 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{ikR2}] 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 pairinteraction 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 nonlinear PoissonBoltzmann equation should generally be considered.
Exercises Chapter I 59
We also assumed a constant total charge on the colloidal particles, indepen dent 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 deionization, 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
60 Exercises Chapter I
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 expres sion (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 paircorrelation 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 hardspheres
The hardsphere pairinteraction 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 hardcore. Verify that the Mayerfunction for this pairpotential 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 nonzero 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
62 Exercises Chapter I
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 shorthand 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 paircorrelation 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 socalled totalcorrelation 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 totalcorrelation function). Conclude that the relative standard deviation <(NcJV>)~> 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 noninteracting 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
Exercises Chapter I 63
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 noninteracting particles are by definition statistically indepen dent, 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 handside 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 noninteracting 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 noninteracting particles.
Show that Z~ exp{ik �9 r}  Do k 2 exp{ik �9 r}, and hence,
/~n exp{ik, r}  ( Do 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 noninteracting particles, the paircorrelation function is identi cally 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, McGrawHill, New York, 1934. �9 J. Alexander (ed.), Colloid Chemistry, volumes 16, Reingold Publishing
Corporation, New York, 1946. �9 H.B. Weiser, Colloid Chemistry, volumes 13, 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, AddisonWesley 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, KolloidZ. 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, KolloidZ. 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 la tex 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 1521, 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, McGrawHill, 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.
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Chapter 2
BROWNIAN MOTION OF NONINTERACTING PARTICLES
69
70 Chapter 2.
2.1 Introduction
As discussed in the previous chapter, a colloidal particle exerts socalled 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 dis tinction 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 mole cules 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 1014,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 righthand 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 • 3dimensional 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 1014s.
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 1014s. 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,
N1 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 3dimensional, so that D is 6 x 6dimensional. Using eqs.(2.4,5), the ensemble averages < xixj >
74 Chapter 2.
are easily calculated,
MG [Xexp(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 < p ( t ) p ( t ) >  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 obser vables 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 righthand 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 FokkerPlanck time scale, which we shall denote as TFp.
At the end of the previous section we have seen that the ensemble ave rage < p ( t ) p ( t ) > 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 st. 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,
1014s   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 parti cle, one finds that M/7 ~ 109s (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 FokkerPlanck time scale, the solvent coordinates are long relaxed to thermal equilibrium, and only the momentum and position coordi nate 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 ( t ) >  p(O) e x p {  ~ t } ,
so that,
[~176 1< p ( t )> [ 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/<I p 12>  ~/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),
lDD ~ 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 pro cesses 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 FokkerPlanck 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 noninteracting Brownian particles the diffusive time scale is the largest time scale of interest. As soon as interactions amongst Brownian par ticles come into play there are two further time scales. These two time scales are related to direct and hydrodynamic interactions between the Brownian par ticles 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 hydrody namic 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 mdimensional 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 mdimensional 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 mdimensional 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 • mdimensional 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 statementChandrasekhar'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  ]: 277.
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 com monly referred to as an Einstein relation, and when an explicit expression for the friction coefficient is substituted it is referred as a StokesEinstein relation. Einstein and StokesEinstein relations apply also to rotational and transla tional diffusion of rigid rods, as will be seen in section 2.8. Chandrasekhar's theorem (2.32) thus yields,
1 [ Irr(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 im mediately 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 handside 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,
2.7. Diffusion in Shear Flow 85
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 handside 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 FokkerPlanck 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+ = ~ il ~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)
2.7. Diffusion in Shear Flow 87
(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 handside is a socalled 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 nonspherical objects.
2.8.1 Newton's Equations of Motion
Let us first recall Newton's equations of motion for nonspherical rigid parti cles. 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 dt 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 handside can be written as a matrix multiplication of f~,
J  Vfl, (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
2.8. Rotational Motion 91
 ~~ ~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 righthand 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 angu lar 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 tiffv 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).
2.8. Rotational Motion 93
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 trans lational 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.(2.101103) 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 (2.8689). 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~ 24 "
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, 112 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 handside is written as a matrix multipli cation,
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)
2.8. Rotational Motion 99
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 fourfold 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 fourfold integrals turn out to be zero, because the arguments of the delta functions are nonzero in the entire integration range. Only products
2.8. Rotational Motion 101
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 ndimensional 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
Exercises Chapter 2 103
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 den sity 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 equipar tition theorem (2.15) must now be taken with respect to the coordinate frame attached to the particle, that is, in the left handside 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 posi tions 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 zaxis 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 zdirection, the different components of the displacement in eq.(2.123) can be analysed separately.)
Further Reading 105
Further Reading and References
Early papers on Brownian motion of noninteracting 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 mo tion, 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 equa tions discussed here are simple examples, is a separate discipline of mathe matics. 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, SpringerVerlag,
Berlin, 1983.
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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 ex perimental 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 di rection 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 noninteracting 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,zx . 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 oointlike 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 monochro matic 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 nonspherical 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 nonspherical 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 as sumed is usually referred to as the Rayleigh Gans Debye scattering theory. Although these conditions seem quite restrictive, scattering data of many col loidal 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)
0D , 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 elec tric 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 in stantaneously 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 handside 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 adependence of the numerator is of no importance, and we may set a equal to zero there. It is the adependence of the denominator which is essential to be able to invert the matrix for all real valued wavevectors. Therefore, only the adependence 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 handside 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~(kko)
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 nonspherical 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 ' ) j1
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 kintegral in eq.(3.36).
The kintegral on the right handside 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 kplane 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 handside of eq.(3.37) may be approxi mated 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, kok~)  47r k~ exp{ikor}r T(k,).~~ Bj(kok~)exp{i(kok , ) . 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 handside 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 configura tion 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 polariza tion 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 par ticles (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 propor tional to the squared amplitude of the corresponding Fourier component of the instantaneous density. For nonspherical 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 prop erties 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 par ticles, is established in the next two sections. Nonspherical 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 wave length, 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 con stants, independent of orientation. The factor V~ I o / r 2 on the right handside of the expression (3.55) is an apparatus constant. The socalled Rayleigh ratio, defined as,
R  I r 2/Io V~, (3.56)
is independent of apparatus constants, and is determined solely by the pro perties 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 sys tem (both translational and rotational invariant), this pdf is a function of the
3.5. Static Light Scattering 127
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 paircorrelation 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 ampli tude at zero wavevector can be written as,
I B(k  0 ) 1 2  Vp 2 ! ep  ef 12 . (3.63) e]
Substitution of eqs.(3.5963 ) 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 handside 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 paircorrelation 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 interfe rence 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 paircorrelation 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),
3.5. Static Light Scattering 129
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 kavalues is re/erred to in subsec tion 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
3.5. Static Light Scattering 131
Similarly, the form factor (3.61) can be written in terms of refractive index differences as,
~ ~r ~ ~ ~~jZ] , (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 handside 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 handside 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 paircorrelation 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 expo nential 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 nonzero wavevector, the last integral on the right handside 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 cor responding 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 mea sured in a static light scattering experiment. The simplest function that charac terizes the fluctuations of the intensity is the intensity autocorrelation 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 fourpoint 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 statisti cally 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 paircorrelation function tends to unity. These clusters of parti cles 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 fourpoint ensemble average in eq.(3.75) can thus be written as a sum of products of twopoint 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 autocorrelation 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 nonzero 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 nor malized 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 nonzero 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 autocorrelation function.
An expression for the EACF for spherical particles, in terms of an ensem ble 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 par ticles 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 instanta neous 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 experi mental 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 rt'
 d t ' + dt ' g I ( k t ' ) a t 4 T 2 2 r 4T2 ' " (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 Jt2r 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 expe rimental 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 exper imental 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 unimpor tant) 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 socalled 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 nonspherical 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 noninteracting 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 handside 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 nonzero wavevectors these averages are therefore zero. Only the "dia gonal terms" i  j in eq.(3.83) for the EACF survive for noninteracting Brownian particles. Furthermore, the static structure factor in eq.(3.67) is equal to 1 for noninteracting 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 subsec tion 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 StokesEinstein 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 eP  ~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 lognormal 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.
3.9. Polydispersity Effects 147
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 devia tion 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 scat tered 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 StokesEinstein 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 expli citly.
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 lognormal 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
3.9. Polydispersity Effects 149
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 par ticles are important, there are two fundamental correlation functions which are of interest. The socalled 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 suspen sion 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 pairinteraction potentials. All particles are then statistically equivalent. The optical properties, however, are assumed different, that is, the scattering am plitudes Bj (k) are generally different for different j 's. Since the particles are
3.9. Polydispersity Effects 151
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
j1
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 approxi mately 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 cyclohex ane. The solid curves are fits to the data points with a sum of two exponentials. This figure is taken from KopsWerkhoven, Fijnaut (1982).
There is thus an extra contribution (proportional to the self dynamic struc ture factor) to the EACF in comparison to the monodisperse case. Intuitively this contribution may be understood as follows. When two optically dis tinct particles interchange their positions, the microscopic density remains unchanged, but nevertheless, the scattered intensity changes. In the monodis perse 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 rodshaped 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 de pending 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)
3.10. Scattering by Rigid Rods 155
,~ ~ 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 handside 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 jofunction 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 jofunctions 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 orienta tional 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
3.10. Scattering by Rigid Rods 157
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 jofunctions 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,j1
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 jofunctions 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 autocorrelation 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))} > ,
3.10. Scattering by Rigid Rods 159
( [ 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 socalled 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
160 Exercises Chapter 3
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 WoW~ k o  k ~ _ 7(mo4m~)
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.
Exercises Chapter 3 161
3.2) * Calculate the inverse of the matrix,
which appears on the left handside 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 handside 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 foo ( ' " ) . 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 the orem, after closing the integration range in the upper complex half plane (for the first integral on the right handside 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 handside of eq.(3.37).
162 Exercises Chapter 3
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 II 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 paircorrelation function g tends to unity), the above integral is to be replaced by,
f Io(r) ] dr exp{ik, r} << 1
I0
Exercises Chapter 3 163
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.
164 Exercises Chapter 3
These are the wavevectors at which the curves in fig.3.11, for a small polydis persity, 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 scatte ring data as if the system were monodisperse (and assuming optically homo geneous 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)
Exercises Chapter 3 165
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)
166 Exercises Chapter 3
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 displace ment. 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 tempe rature 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
Exercises Chapter 3 167
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 zaxis, and the wavevector oriented in the x, zplane. 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
168 Exercises Chapter 3
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 socalled 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.)
Further Reading and References
The data shown in fig.3.13 are taken from, �9 M.M. KopsWerkhoven, 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 ana lysed, on the basis of the OrnsteinZernike 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 socalled 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 macro molecules 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 sys tems was developed in chapter 2 for noninteracting Brownian particles. The starting point there is a stochastic equation of motion for the phase space coordinates of a Brownian particle, the socalled Langevin equation. Such an approach is also feasible for systems of interacting particles. However, the Langevin equations for interacting particles are nonlinear 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 parti cles (such as positions, momenta, orientations and angular velocities). As we have seen in section 1.3 in the introductory chapter, once the equation of mo tion for the probability density function (pdf) of the phase space coordinates is known, time dependent correlation functions (such as the important den sity autocorrelation 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 proba bility 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 subse quently 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 socalled FokkerPlanck equations, which are the equa tions of motion for the corresponding pdf's which we are after here. For the equivalence of stochastic differential equations and FokkerPlanck 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 NavierStokes 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 equa tion. 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 mdimensional vector. One can think of X as the set of momentum and position coordinates of the Brownian particles (on the FokkerPlanck 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 mdimensional 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 mdimensional 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 mdimensional 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 mdimensional 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 nonequilibrium 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, outwardnormal to 0W (see fig.4.2). The minus sign on the right handside 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 handside of eq.(4.1) can be taken inside the integral and the integral on the right handside can be written as a volume integral over ]41 with the use of Gauss's integral theorem (in m dimensions).
4.1. Introduction 175
dS
_j
~W
%
Figure 4.2: W is an arbitrary volume in the mdimensional 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 mdimensional 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 mdimensional 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 mdimensional 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 coor dinates of spherical Brownian particles on the FokkerPlanck 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 noninteracting particles. The effects of shear flow and sedi mentation 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 molecu lar 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 socalled 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 socalled hydrodynamic interactions between Brownian particles immersed in a fluid. The origin of the interactions is not difficult to under stand. 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 NavierStokes 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 ma croscopic 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 FokkerPlanck time scale. On the FokkerPlanck time scale the above approximation in ques tionable. On the Brownian time scale, however, the validity of instantaneous hydrodynamic interaction is beyond doubt.
Secondly, it turns out that the NavierStokes 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 nonlinear inertial terms in the NavierStokes 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 3dimensional 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 interac tion is the explicit calculation of these matrices. A minus sign is added to the right handside 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. FokkerPlanck Equation 179
with 7 the friction coefficient of an isolated sphere, which is shown in subsec tion 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 "offdiagonal" friction matrices, Tij with i ~ j , describe the hy drodynamic 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 FokkerPlanck Equation
The FokkerPlanck 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 FokkerPlanck time scale, where the phase space coordinates of the solvent molecules are long relaxed (see chapter 2).
In colloid science the phrase "FokkerPlanck 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 "FokkerPlanck 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 6Ndimensional 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 FokkerPlanck 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 j1
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) i1
4.3. FokkerPlanck 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 handside of eq.(4.15) is zero in equili brium, the right handside 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 FokkerPlanck 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 socalled super vector notation. The "super vectors" are the two 3Ndimensional 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 3Ndimensional 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 FokkerPlanck operator now reads,
,~FP(''') = P V,.(..) (4.25) M
p / { oj +, + +
The explicit notation of indices is lost in the super vector notation. The FokkerPlanck equation can be used to describe Brownian motion
(on the FokkerPlanck 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 FokkerPlanck 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 FokkerPlanck 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 FokkerPlanck 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 ex pressed 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 handside of eq.(4.14) is equal to zero on the Brownian time scale,
N Pj
0   ~ T i j ( r l , . . . , r N ) . M j1
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 (4.2024) 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 • 3dimensional 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 FokkerPlanck 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 FokkerPlanck 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 StokesEinstein diffusion coefficient that we have already met in chapter 2 on diffusion of noninteracting 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 diago nal matrix,
ri o . . . o o  A . . . o
T  . . . . , (4.45)
0 0 . . . 7I
where I is the 3 x 3dimensional identity matrix and 0 the 3 x 3dimensional 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 FokkerPlanck and Brownian time scale of noninteracting 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 noninteracting particles both are identical. There is a diffe rence 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 noninteracting particles belong to a single class of differential equations �9 linear FokkerPlanck equations. The Smoluchowski equation for noninteracting 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 FokkerPlanck
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 FokkerPlanck 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, expres sions 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 FokkerPlanck 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 FokkerPlanck equation. In case these matrices are Xdependent, the resulting equation is referred to as nonlinear. For interacting particles, the equations of motion which were derived in the previous sections are nonlinear FokkerPlanck equations through their hy drodynamic 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 FokkerPlanck 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 d7 < 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 FokkerPlanck equation (4.49) can be verified by substitution. This is most easily done in terms of Fourier transforms. The Fourier transform of the FokkerPlanck 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 FokkerPlanck 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 FokkerPlanck 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 noninteracting 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{ikr} • 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 dt < r2(t) >  6Do. (4.72)
Integration leads to the result in eq.(4.70).
4.5.3 Diffusion on the FokkerPlanck Time Scale
The FokkerPlank 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 FokkerPlanck 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{ikro} V (27rMkBT) 3/2
x epo e p P ( p , k , t I po, ro, t  0),
where the kdependence refers to the Fourier transformation with respect to r. Now let us define the following 6m 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 3dimensional 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 FokkerPlanck equation is known.
For the particular FokkerPlanck equation (4.73), where X  (p, r), the matrices A and B appearing in the more general FokkerPlanck 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 3dimensional zero matrix. The equation of motion (4.50) for the mean is thus,
d  '  ~ < p , r >   ~ < p > , ~ < P > �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 1exp{  rT t} . "7
Equation (4.51) for the average of the bilinear product yields,
d dt < p( t )p( t )> = d dt < r(t)p(t) > = d d~ < r(t)r(t) > =
27 M
< p(t)p(t) > t2fl1"),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.
(Ss1)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 FokkerPlanck 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 FokkerPlanck 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 FokkerPlanck time scale. The initial slope of S, versus time is zero on the FokkerPlanck 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 nonzero 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 zdirection with its gradient in the ydirection. 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 hydrodyna mics.
The Smoluchowski equation changes, not only due to the different hy drodynamic 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 noninteracting particles, Brownian motion is severely affected by shear flow. Diffusion in shear flow, for noninteracting Brownian particles, is analysed in subsection 4.6.3. The Smoluchowski equation for this case belongs to the class of linear FokkerPlanck 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 nonzero.
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 handside 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.(4.2024)). 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 3Ndimensional 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)
4.6. Smoluchowski Equation with Shear Flow 199
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 nonInteracting 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 FokkerPlanck 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)
4.6. Smoluchowski Equation with Shear Flow 201
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 autocorrelation 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 handside of eq.(4.113). It is possible, however, to devise a twodetector 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 crosscorrelation 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 twodetector 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 noninteracting 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.(4.110112) 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 crosscorrelation 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 Ffunctions 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 crosscorrelation 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 nonperfect 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 sec tional 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 socalled 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 socalled 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 nonzero velocity of the fluid surrounding the particles (the socalled 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 socalled "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 hydro dynamic 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 sedimen ting 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 handside, 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.(4.2024). 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 0t 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 <I>]  V~ In{P}] + D,j .flF ~t . (4.124)
j1 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 par ticles 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 nonspherical 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 noninteracting 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 6Ndimensional 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 (4.25), 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 noninteracting 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 3Ndimensional 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 (per pendicular) 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 fric tion matrix, which is referred to as the microscopic diffusion matrix,
TRl a I oTT I T RT T RR DRT D n n
4.8. Smoluchowski Equation for Rigid Rods 211
(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 3dimensional microscopic diffusion matrices D ij is a mix of an the 3 x 3dimensional microscopic friction matrices.
Without hydrodynamic interaction, the "offdiagonal 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 noninteracting 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 interac tion, couples the translational dynamics of a rod to its rotational motion. For spherical particles this is not the case, which circumstance allows for an anal ysis 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 6Ndimensional vector which cannot attain arbitrary values in 6Ndimensional 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 speak ing 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 dSn
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 co ordinates 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,
4.8. Smoluchowski Equation for Rigid Rods 215
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 0t 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 nonzero, 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)
4.8. Smoluchowski Equation for Rigid Rods 217
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 trans lational 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 0t 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 autocorrelation function (EACF), as defined in the previous chapter on light scattering, for a system of noninteracting 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 nonInteracting 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 handside of the Smoluchowski operator (4.155),
d dt < 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 trans lational 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.
4.8. Smoluchowski Equation for Rigid Rods 219
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 handside 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 FokkerPlanck 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 handside 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 mdimensions,
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 noninteracting 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 "
Exercises Chapter 4 221
The center of mass thus diffuses as a single sphere with a diffusion coefficient equal to half the StokesEinstein diffusion coefficient of the separate spheres of the Brownian oscillator. The Smoluchowski equation for the pdf of the separation R is a linear FokkerPlanck 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
222 Exercises Chapter 4
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 3Ndimensional 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
Exercises Chapter 4 223
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 zdirection by means of an external field. At time t  0 the external field is turned off. The rods attain an isotropic
224 Exercises Chapter 4
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 zdirection, which is the alignment direction of the rods at time zero. The mean scattered intensity, with a polarization direction perpendicular to the zdirection, say in the xdirection, 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 ~ < u z u~ > .
1 kL < 0 5, and the "cross The jofunctions 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 xaxis (zaxis).
In this exercise, the time dependence of this depolarized small angle scat tered intensity is calculated from the Smoluchowski equation (4.154,155), along similar lines followed in subsection 4.8.3 to calculate the time depen dence of < 6(t) > in eq.(4.158).
In the following, the indices 1, 2 and 3 refer to the x, y and zdirection, 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 FokkerPlanck equa tions 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, SpringerVerlag, Am
sterdam, 1983. �9 H. Risken, The FokkerPlanck Equation, SpringerVerlag, Berlin, 1984.
The original papers on the derivation of the FokkerPlanck 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 FokkerPlanck 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. Twodetector 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 StokesEinstein 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 StokesEinstein 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 macro scopic 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 con sider 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 con sidered 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 handside is added, because the mass in ],V decreases when u is along the outward normal. The time derivative on the left handside can be taken inside the integral, while the integral on the right handside 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 0t 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 ne cessary 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. NavierStokes Equation 231
5.3 The NavierStokes Equation
The NavierStokes 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 com ponent 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 xdirection 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. NavierStokes Equation 233
with I the 3 x 3dimensional 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 handside 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 handside. 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 NavierStokes 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 NavierStokes equa tion 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 distur bance due to the motion of Brownian particles is instantaneous on the time scale under consideration (the FokkerPlanck 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 coor dinates of the Brownian particles. Here we discuss the propagation velocity of disturbances, and compare the outcome with the Brownian and FokkerPlanck 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 pres sure 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 xyplane. At time zero the plate's velocity is zero, and from that time on the plate is pulled along the xaxis with a certain nonzero 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 zdirection. We solve the NavierStokes 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 NavierStokes 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 NavierStokes 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 onedimensional 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 xyplane, induces shear waves on displacement parallel to the xyplane (a), and sound waves on displacement along the zdirection (b).
The solution of the problem (5.911) 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 FokkerPlanck 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 zdirection (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 zdependent. Furthermore, the fluid flow is along the zdirection. 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 handside 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 NavierStokes equation (5.7), with r/o  0 and ~0  0, gives,
0 0 u ( z , t )  o 07 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 substi tution 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  ffPo" (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 FokkerPlanck 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 FokkerPlanck time scale.
5.5 The Creeping Flow Equations
The different terms in the NavierStokes 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
NavierStokes 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 NavierStokes 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 NavierStokes 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 socalled 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 handside 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 NavierStokes 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 handside of the NavierStokes 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 NavierStokes equation can be ne glected 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 FokkerPlanck equation. On the FokkerPlanck time scale only the term ,~ u. Vu on the right handside of the NavierStokes equation can be omitted (for small Reynolds numbers), but the time derivative a u / a t must be kept. Hydrodynamic friction functions on the FokkerPlanck 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 FokkerPlanck 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, sub stitute 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 3dimensional 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~r1~ 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 ex pressions for the fluid flow as a result of translation or rotation of a spherical Brownian particle. The following two subsections are devoted to the calcula tion 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 nonrotating 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 handside 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 ')(rrp)]} +4 ]r rp] rrpi2 .V.
This expression can also be obtained directly from the creeping flow equa tions 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 con stant 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 sepa rations between the Brownian particles, these particles can be considered as pointlike. For such pointlike 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 3dimensional 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 handside 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(rr')  i 2a v, ~o' for r' e 017/. (5.52)
The first term on the right handside 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 handside 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 handside 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 StokesEinstein 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 dif fusion 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 handside,
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)
tV2pV2p [(...)uo(rp)I...I(..)V2p .. . V2uo(rp) I...] .
The last term on the right handside 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 RodnePrager 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')] .
147ra2 JOV JOV
Only the second term in the Taylor expansion (5.59) survives in the first term on the right handside. 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 handside 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 handside 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 RodnePrager 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 handside of eq.(5.65) is usually referred to as the RodnePrager 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 RodnePrager 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 temp ting 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 nonzero. 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 RodnePrager result (5.64,65) for the microscopic dif fusion 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 boun dary 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 RodnePrager 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,.
5.12. Method of Reflections 261
The add/tiona/velocity of sphere j , in order to sustain the prescribed hydro dynamic 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)+11! 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 (2n1)(ri)] , (5.73) 6.0/3DoF~'+(16,,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 nfold 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 nfold 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 handside of the contraction symbol is of higher indexrank than the matrix B, then the number of leftover 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 boun dary 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)
5.12. Method of Reflections 265
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 pro perties 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
5.12. Method of Reflections 267
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 selfmobility functions, and those with i ~ j , distinct or crossmobility 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 RodnePrager 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 RodnePrager expression (5.64,65) for the micro scopic 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
5.12. Method of Reflections 271
center at the origin. Thus, r is replaced by r + ri. The term in the boundary condition (5.70) containing the angular velocity 11!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 ap proximation (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 be tween 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 be tween 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
5.12. Method of Reflections 273
(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 pairinteraction potential and also for mild attractive pairinteraction potentials. For stronger attractive pairpotentials, 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 pairpotential, 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 pairinteractions. 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.
5.12. Method of Reflections 275
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 socalled velocity gradient matrix is,
O 1 0 / r  ~ o o o ,
0 0 0 (5.102)
representing a fluid flow along the xdirection, linearly increasing with position in the ydirection and independent of the zcoordinate. 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 in dexrank 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 con jectured 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 im mersed 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 handside 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 handside 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 Rodneprager level,
v } l )  r . rji + ()5
8 a E . r j i . E .  5
(5.111) Up to the RodnePrager 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 (rrj) u rj .
/=2
(5.112)
The first term r . r on the right handside 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 threebody 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 sedi ment 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 socalled 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 suspen sion, 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 handside 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 handside 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 + 12 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) ~ ggiF 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 StokesEinstein 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 li miting 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) 12 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 trans formed 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 kspace. Let us now introduce the socalled 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 kintegral 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 dzexp{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 handside 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 zplane. 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 zplane (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 zplane, 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 handside of eq.(5.142) is evaluated similarly, with the result,
79 dk1 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 kspace. 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 zaxis. Let the matrix A denote the rotation that maps r onto the zaxis (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 xyplane,
~ , o}    < k 3 <  , ()5.148
r r
where k~ is the zcomponent 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 handside 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 r3 +27r r + , f o r n   r n  3 .
Hence,
2,~ [a/,. fo d~' aa~,, 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 zaxis,
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 da/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 kintegral on the right handside 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 handside 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 handside 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 handside 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 Voperators", 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~
m1
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 (nl) .
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 differen tiation yields,
H(O) = 1_ r
H ! 1 )  r i /,3
H!~) 6ij rirj  r S + 3 rT, (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~ + $imrjr~ § ~jmrir~ + ~inrjrm + ~j~rirm I ~m~rirj
H!. 4) ~3mn
r7
Let us now consider the construction of the connectors, which are the solutions of the boundary value problem (5.82). We shall need explicit ex pressions for H ('~)  (V)~uo. These quantities may be obtained simply by first calculating II (n) and then contracting with (V)nuo. The explicit expres sions for II ('~), however, become quite formidable for n > 5. It requires an enormous effort to calculate the desired contractions for n > 5 in this way. On performing the contraction, many terms yield identical contributions due to the symmetry of (V)'~Uo in its first n indices, and many terms vanish due to the creeping flow equation V2V2uo  0. The easy way to obtain explicit expressions for H ('~)  (V)'~Uo, without having to calculate H (n) first, is as follows. Since (V)'~Uo is symmetric in its first n indices, we may interchange any of the last n indices in any term in the expression for II ('~). Many terms become equal by performing such interchanges of indices, which considerably simplifies the explicit expression for H (n). Secondly, since V2V2uo  0, all
terms in H !n). which are proportional to a product of two or more Kronecker ~1 ""$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 II ('~) 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 II ('~) 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~ + 3r2176 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 r2II (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(n2)(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)n2uo(r) . The constants
' and ' cn, cn2, 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 j1 I I j j+l j42
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/Dratios, 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).
302 Exercises Chapter 5
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  a77   ; r .
angular coordinates, to obtain,
f f/r/V [ r  ro [  4~" f(ro) �9
Exercises Chapter 5 303
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 (5.911) 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(zzo) 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 1dimensional 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
304 Exercises Chapter 5
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 NavierStokes 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 parti cles, 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 mi croscopic stress matrix depends on the positions of the spherical Brownian particles within the volume V. For N noninteracting 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
Exercises Chapter 5 305
/
\
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
306 Exercises Chapter 5
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 + ~
Exercises Chapter 5 307
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 pointlike 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 op posite 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 in version 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 ~,
308 Exercises Chapter 5
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 handside 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 inter action 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 RodnePrager matrix is now given by,
D i j  6ryoaj rij 4 r~j
Exercises Chapter 5 309
(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 pointlike 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,
310 Exercises Chapter 5
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, nonBrownian 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 • [Itiff] . ( 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 handside 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.
Further Reading and References
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, ButterworthHeinemann, 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 expan sion 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 socalled 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/Dratios can be done by the socalled OseenBurgers 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.
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Chapter 6
DIFFUSION
315
316 Chapter 6.
6.1 Introduction
In most cases, experimental data are macroscopic, ensemble averaged quanti ties. 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 en semble 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 previ ous 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 microstruc ture 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 nonequilibrium steady state mi crostructure.
After the heuristic and introductory sections 6.24, quantitative results are derived from the Smoluchowski equation. We start with the evaluation of shorttime 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 longtime 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 socalled 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 autocorrelation function is considered in section 6.10 for non interacting rods, as well as rotational relaxation to first order in concentration for rods with hardcore 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 phe nomenon, since many Brownian particles are displaced simultaneously. No tice that p(k, t  0) is the amplitude of the sinusoidal variation with wavevec tor 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 0Tp(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 co efficient", 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 den sity 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 neigh bouring points nor with preceding states of the system, the diffusion coeffi cient 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 dif ferent 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 transforma tion 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)
6.2. Collective Diffusion 321
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 si nusoidal 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 co efficient, 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 pairinteraction potentials for arbitrary concentra tions (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)
6.2. Collective Diffusion 323
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 0t 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 noninteracting Brow nian particles is of the form of Fick's law, except that the gradient diffusion coefficient is replaced by the StokesEinstein diffusion coefficient Do. For very dilute suspensions, where interactions are of no importance, the gradient coefficient thus becomes equal to the StokesEinstein diffusion coefficient.
Shorttime and longtime 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) shorttime collective diffusion coefficient D~(k),
D ~ ( k )  lim D ~ ( k , t )  D ( k , t  0 ) . t,O
(6.15)
In practice, the shorttime 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 de scribed by the longtime 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 longtime limit is reached, if it reached at some finite time at all.
Notice that the conjecture (6.12) implies that the long and shorttime 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 autocorrelation function is equal to the normalized electric field autocorrelation 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 monodis perse 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 tracerparti cle 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 noninteracting 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 StokesEinstein diffusion coefficient. For very small times, W(t ) ,~ t 2, while for larger times, W(t) ,.~ t. The crossover between these two limiting forms occurs for times larger than the FokkerPlanck 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 connec tion 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 ex ercise 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 obtai ning 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).
Shorttime and longtime self diffusion
On average, the tracer particle resides at positions where the "flee energy landscape", created through interactions with other Brownian particles, ex hibits minima (see fig.6.4). Shorttime 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 shorttime 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 dis placements, 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 longtime 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 longtime 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 longtime limit is approached like ,~ t 1/2 within the socalled 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 handside is approximately the time that it takes a density wave of wavelength Am to fully relax. The time scale 7i is called the interaction time scale. The longtime 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
6.4. Diffusion in Shear Flow 329
t > ri. The actual displacement of the tracer particle need not be large to reach the longtime 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 noninteracting 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 longtime self diffusion coefficient is smaller than the shorttime 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 noninteracting particles there is no such energy landscape, and there is no difference between long and shorttime self diffusion. Both the long and shorttime self diffusion coefficient are then equal to the StokesEinstein diffusion coefficient Do.
Once fig.6.5 is constructed experimentally, the longtime 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 equi librium. 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 xdirection with its gradient in the ydirection 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 paircorrelation 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, vpI 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. Shorttime 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 ydirection, 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 struc tures which extend over large distances in the gradient direction leads to socalled singularly perturbed equations of motion for the paircorrelation 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 conse quence 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 Shorttime Diffusion
Shorttime 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 3Ndimensional 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 instan taneous 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 3Ndimensional gradient operator with respect to r, D(r) is the 3N x 3Ndimensional 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 con jugate 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 handside 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 shorttime self diffusion coefficient first.
6.5.1 Shorttime Self Diffusion
The shorttime 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 3Ndimensional super vector r. The tracer
6.5. Shorttime Diffusion 333
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 },
(g1)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 3dimensional microscopic diffusion matrix Dll (r) is a function of all the coordinates r~ .. rN. Notice that the shorttime 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 twoparticle 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 selfmobility functions are given by (see eq.(5.95)),
A~(rxj) = 154 a +211 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+(N1)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 twoparticle pdf,
/ / 1 dr3 . . , drNP(r)  P2(rl,r2)  V 2 g ( r l , r2 ) , (6.48)
where the last equation defines the paircorrelation function g (see also sub section 1.3.1 in the introductory chapter). For the homogeneous and isotropic system under consideration, the paircorrelation 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 paircorrelation 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 shorttime self diffusion coefficient is depending on the form of the pairinteraction potential through the pairdistribution function. One of the most simple pairpotentials is that of socalled hardsphere systems. The pairpotential 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 hardsphere 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 contribu tion comes from eq.(6.49) with the paircorrelation 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 hardsphere interactions, the first order in volume fraction expansion of the paircorrelation 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 threeparticle 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)
6.5. Shorttime Diffusion 337
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 threepanicle pdf,
/ / 1 d r y . . , drN P(r )  P~(r:, r~, r~)  Y~ g~(r~, r~, r~). (6.6O)
The last equation defines the threeparticle correlation function, which, for the special case of hardsphere 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 shorttime selfdiffusion coefficient as a function of the volume fraction for hardsphere colloids. The solid curve is eq.(6.63), the dashed curve is the linear approximation, where the ~2term 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 zaxis for the r13integration 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/
6.5. Shorttime Diffusion 339
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 shorttime 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 Shorttime Collective Diffusion
The shorttime collective diffusion coefficient is defined in terms of the col lective 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 threefold 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 nonzero. Secondly, the choice of the upper limits for the zig and z~sintegration 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 O2sintegration 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 selfdiffusion 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{• = m1
+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)
6.5. Shorttime Diffusion 341
As for self diffusion, the shorttime 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 depen dent. Let us calculate the coefficients for hardsphere interactions. The static structure factor is calculated with the use of eq.(6.56) for the paircorrelation 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 mobi lity 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 shorttime 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 twoparticle microscopic diffusion matrices to the remaining "nondiagonal" terms. The three body hydrodynamic matrices also contribute to the ~2coefficient �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 paircorrelation 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 crossmobility 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, cor responding 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
6.5. Shorttime Diffusion 343
The first integral on the right handside 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 shorttime collective diffusion coefficient. Hence,
H(k)  1  1.734r + 0.910qp 2
+ ~ 1 ~ . dr (gh~(r) 1) exp{ik, r}ar [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 handside 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 substitu tion (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)3terms in the combination A~ + 2B~ cancel. Substitution of the expression (5.56) for the paircorrelation 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 [75zSh1(zx)(zX~ 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 frac tion 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 m3

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 nonzero for k ~ 0. Whenever H(k  0) appears, what is meant is its limiting value for k ~ 0.
6.5. Shorttime Diffusion 345
This expression can be substituted into eq.(6.70) for H(k). This leads, how ever, to a nonconvergent integral. Precisely as in the case of the twopartic!e contribution in eq.(6.78), there are nonconvergent 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 sub stituting 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 paircorrelation function g(r:, r3) is not subtracted from the threeparticle 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 hardsphere interactions this function of 2ka can be evaluated by numerical integration using Simpson's quadrature. The numerical integration is a bit tricky" the x12integral converges slowly in an oscillatory fashion.
Collecting results we find the following expression for the first and sec ond order volume fraction coefficients of the shorttime 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 hardsphere colloids. Also plotted in (c) are the first and second order in ~ coefficients in eqs.(6.90,91).
6.5. Shorttime Diffusion 347
coefficient for hardsphere 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 plot ted in fig.6.9 versus 2ka, together with the two coefficients a~ and a~. Note the functional similarity of the H and Sfunctions.
For the zero wavevector limit we find that, up to order ~2,
D:(k  0)D Do 16441 +67 2 = {1+,.559 14.S 1_8~+34~2
(6.92) Using more accurate twobody 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 40.2. 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 hardsphere 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 hardsphere 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 shorttime collective zero wavector diffusion coefficient (solid curve) with experiments on a hardsphere like dispersion. Data are from van KopsWerkhoven 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).
6.5. Shorttime Diffusion 349
Since Sx, $2, HI,//2(1) and 11(22) are zero for infinite values of their argu ment, the long wavelength limit of the shorttime collective diffusion coeffi cient is found to be equal to the shorttime 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 crossterms (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 hardcore diameter of the particles. In comparing with our theoretical result for hardspheres, an "effective volume fraction" and an "effective hardcore 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 hardcore 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 Shorttime Diffusion
A striking difference between the general expressions (6.44) for the short time self diffusion coefficient and (6.69) for the shorttime 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 shorttime 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 handside of eq.(6.94) is usually referred to as a "mobility" for shorttime 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 hardspheres, self diffusion is seen to be slowed down by interactions, while collective diffusion for long wavelengths is en hanced. 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 po sition 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 Nparticle 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 can not 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 twoparticle 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 pairwise additive, that is,
N (~(rl, r2~"~rN)  Z
i , j = a i < j
V(rij) . (6.98)
For spherically symmetric Brownian particles, the pairinteraction potential V is a function of the absolute distance rij  [ ri  rj [ between two particles. For many systems, the approximation of pairwise additivity of potential interac tions is a very good approximation (for monodispers hardsphere systems this is even exact). Hydrodynamic interactions, on the other hand, are certainly not pairwise 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)
+(N1)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 handside 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
6.6. Gradient Diffusion 353
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)(N2)flV.fdr2fdr3AD,(rr2)P3(r, r2, r3, t ) . VV(] r   r3 1).
The pair and threeparticle correlation function g and ga for an inhomoge neous 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 noninteracting particles. On the pair level considered here, the paircorrelation function is equal to the Boltzmann exponential of the pairinteraction 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 threeparticle 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 = iv1 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 NififlVVAp(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'
6.6. Gradient Diffusion 355
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 expres sions 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 hardsphere interactions, with the use of the following relations,
dg(~  ~ ( r  2a) dr
dV(r) g(O) (r) __ __fli dr dr
 f l  l ~ ( r  2 a ) , (6.110)
where 6 is the 1dimensional delta distribution. The first integral on the right handside 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 selfmobility functions. Its numerical value is 111/64  1 .734.  . The function f~(ax) is found from the expression (6.77) for the crossmobility
3r5 _s The third integral now turns out to be equal functions to be equal to Tx . 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 shorttime 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 hardsphere interactions.
6.7 Longtime Self Diffusion
In this section, the longtime 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 longtime self diffusion coefficient can be found from an Einstein relation, where the friction coefficient is that of the tracer particle. This is the propor tionality 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 longtime diffusion coefficient is calculated in subsection 6.7.3.
6.7.1 The Effective Friction Coefficient
In chapter 2 on diffusion of noninteracting 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 meansquared 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. Longtime Self Diffusion 357
assume an Einstein relation between the longtime 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 coeffi cient 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 longtime 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 hardsphere 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 longtime 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 paircorrelation function around the tracer particle due to its translational motion.
6. 7. Longtime 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 pairinteraction potential. The stationary Smoluchowski equation (4.40,41) thus reads,
0  1 " [Dll{ Trlg fJg rlgflg ezt} 2I D12"{ 7r2 g P Tr2g}]
The pdf P here is the twoparticle pdf P  P(r~2). The above differen tial 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 differ ential 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 hardsphere 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 longtime self dif fusion 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 Longtime 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 Longtime 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 handside 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 offdiagonal 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 shorttime 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 expres sions (5.95) for the diffusion matrices, and assuming a pairwise additive potential energy (6.98), gives, for identical Brownian particles,
q {As(r12)Ac(r,2)} i'1~i'124{Bs(r12)Bc(r,,)} [if12f,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 paircorrelation 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 inter actions, the Nparticle correlation function gN, defined as,
1 P(~  V N g N ( r l , ' ' ' r N ) , (6.125)
is a product of paircorrelation 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 per forming 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 longtime 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 paircorrelation 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 socalled nonanalytic 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 disre garding hydrodynamic interaction, the stationary Smoluchowski equation for the shear rate dependent paircorrelation 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 paircorrelation function (without shear). The paircorrelation function is therefore said to be a "singular", or equivalently, a "nonanalytic" 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 nonanalytic 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 paircorrelation
6.8. Diffusion in Stationary Shear Flow 365
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 paircorrelation 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 pairinteraction 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 paircorrelation 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 paircorrelation 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 pairinteraction 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 socalled 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 ycoordinate yp between two Brownian particles replaced by the range of the pairpotential 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 selfcontained, you may prefer to go through exercise 6.7 first.
6.8.1 Asymptotic Solution of the Smoluchowski Equation
The term on the left handside in eq.(6.136) is approximately of the same order of magnitude as the first term on the right handside, when K 2 ~ P e ~ For K >> ~/Pe ~ the perturbing term (the left handside 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 Kspace 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 per turbed equations like eq.(6.136), is to separately construct the solution in the inner and the outer region (the socalled 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
6.8. Diffusion in Stationary Shear Flow 367
us therefore introduce the rescaled wavevector,
q  K/(peO) ~' " (6.138)
Rewriting the Smoluchowski equation (6.136) in terms of this rescaled wavevec tor 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 nonequilibrium 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 handside 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 pairpotential 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 socalled singular pertur bation 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~(KK ' ) .
(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 handside of eq.(6.145), with Sx taken equal
6.8. Diffusion in Stationary Shear Flow 369
to the first iterated solution. This contribution is relatively small, due to the almost antisymmetric 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~ ~2S~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 Kspace 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 nonzero 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 nonzero). 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 nonzero 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 40o
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 Kspace. 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 Kspace. The boundary layer is quite asymmetric.
6.8. Diffusion in Stationary Shear Flow 371
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 PercusYevick equilibrium structure factor for an effective hardsphere 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 Kspace.
An experiment
Although the result (6.143) for the structure factor distortion is based on a simplified Smoluchowski equation, where only leading order direct inter actions 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 distor tion AS/S ~q. For the equilibrium structure factor, that is needed as an input to calculate the structure factor distortion, the PercusYevick structure factor for hardspheres is used, with an effective hardcore diameter and volume fraction which are determined by scaling the actual crystallization concentration to that of a monodisperse hardsphere 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 shorttime 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 socalled 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 socalled 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 u