PHENOMENOLOGY OF POLYMERSOLUTION DYNAMICS

Presenting a completely new approach to examining how polymers move innondilute solution, this book focuses on experimental facts, not theoretical specula-tions, and emphasizes nondilute polymer solutions, not dilute solutions or polymermelts.

From centrifugation and solvent dynamics to viscosity and diffusion, exper-imental measurements and their quantitative representations are the core of thediscussion. The book reveals several experiments never before recognized asrevealing polymer solution properties. A novel approach to relaxation phenomenaaccurately describes viscoelasticity and dielectric relaxation, and how they dependon polymer size and concentration.

Ideal for graduate students and researchers interested in the properties ofpolymer solutions, the book covers real measurements on practical systems, includ-ing the very latest results. Every significant experimental method is presentedin considerable detail, giving unprecedented coverage of polymers in solution.

george d. j . phillies is a Professor in the Worcester Polytechnic Institute,Massachusetts. He has attained international recognition for his scientific studiesof light scattering spectroscopy and polymer solutions.

PHENOMENOLOGY OF POLYMERSOLUTION DYNAMICS

GEORGE D. J. PHILLIESWorcester Polytechnic Institute, Massachusetts

cambridge university pressCambridge, New York, Melbourne, Madrid, Cape Town,

Singapore, So Paulo, Delhi, Tokyo, Mexico City

Cambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by Cambridge University Press, New York

www.cambridge.orgInformation on this title: www.cambridge.org/9780521875554

G. Phillies 2011

This publication is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the written

permission of Cambridge University Press.

First published 2011

Printed in the United Kingdom at the University Press, Cambridge

A catalog record for this publication is available from the British Library

Library of Congress Cataloging in Publication dataPhillies, George D. J.

Phenomenology of polymer solution dynamics / George D. J. Phillies.p. cm.

Includes bibliographical references and index.ISBN 978-0-521-87555-4 (hardback)

1. Polymer solutions. I. Title.QD381.9.S65P45 2011

547.7dc232011024274

ISBN 978-0-521-87555-4 Hardback

Cambridge University Press has no responsibility for the persistence oraccuracy of URLs for external or third-party internet websites referred to

in this publication, and does not guarantee that any content on suchwebsites is, or will remain, accurate or appropriate.

This volume is dedicated tothe late

Daniel Kivelson,Professor of Chemistry and Biochemistry,

University of California, Los Angeles.

Contents

Preface page xiii1 Introduction 1

1.1 Plan of the work 11.2 Classes of model for comparison with experiment 51.3 Interpretation of literature experimental results 8References 9

2 Sedimentation 102.1 Introduction 102.2 Homogeneous sedimentation 122.3 Probe sedimentation 182.4 General properties: sedimentation 26References 28

3 Electrophoresis 303.1 Introduction 303.2 Basis of electrophoretic studies 313.3 Electrophoresis using nucleic acid probes 333.4 Videomicroscopy of DNA electrophoresis 433.5 Electrophoresis of denatured polypeptides 493.6 Particulate probes 503.7 Triblock copolymer matrices 563.8 Other electrophoretic experiments 573.9 General properties: electrophoresis 59References 64

4 Quasielastic light scattering and diffusion 694.1 Introduction 69

vii

viii Contents

4.2 Scattering and particle positions 704.3 Nomenclature for diffusion coefficients 734.4 Diffusion coefficients 754.5 Calculation of diffusion coefficients 764.6 Rotational diffusion; segmental diffusion 864.7 Interpretation of spectra 87References 91

5 Solvent and small-molecule motion 945.1 Introduction 945.2 Motion in large-viscosity simple solvents 945.3 Small-molecule translational diffusion in polymer solutions 975.4 Small-molecule rotational diffusion in polymer solutions 1055.5 High-frequency viscoelasticity 1105.6 General properties: solvent dynamics 111References 112

6 Segmental diffusion 1166.1 Introduction 1166.2 Depolarized light scattering 1166.3 Time-resolved optical polarization 1176.4 Magnetic resonance experiments 1266.5 General properties: segmental diffusion 129References 131

7 Dielectric relaxation and chain dimensions 1347.1 Introduction 1347.2 End-to-end distances and relaxation times 1377.3 Chain dimensions and chain contraction 1447.4 Relaxation spectra single mode 1497.5 Relaxation spectra multiple modes and mode decompositions 1557.6 General properties: dielectric relaxation 162References 168

8 Self- and tracer diffusion 1718.1 Introduction 1718.2 Self-diffusion 1728.3 Tracer diffusion 1858.4 Other experimental studies 2048.5 General properties: single-chain dynamics 207References 213

Contents ix

9 Probe diffusion 2189.1 Introduction 2189.2 Light scattering spectroscopy 2199.3 Large probes 2219.4 Small probes 2309.5 Re-entrant phenomena 2339.6 Multiple relaxation modes 2369.7 Polyelectrolyte matrices 2409.8 Solvent quality 2439.9 Temperature dependence 2449.10 Hydroxypropylcellulose solutions 2479.11 Probe rotational diffusion 2579.12 Comparison of probe diffusion and polymer self-diffusion 2609.13 Particle tracking methods 2619.14 True microrheological measurements 2649.15 Probes in gels and biological systems 2679.16 Probe spectra interpreted with the Gaussian assumption 2699.17 General properties: probe diffusion 271References 280

10 Dynamics of colloids 28710.1 Introduction 28710.2 Single-particle diffusion 29010.3 Dynamic structure factor and mutual diffusion 29310.4 Rotational diffusion 29710.5 Viscosity 30110.6 Viscoelastic properties 30710.7 General properties: colloid dynamics 311References 315

11 The dynamic structure factor 32011.1 Introduction 32011.2 Near-dilute polymers and internal modes 32111.3 Neutral polymer slow modes 32911.4 The polyelectrolyte slow mode 33711.5 Thermal diffusion and Soret coefficients 33911.6 Nondilute ternary systems 34111.7 Inelastic neutron scattering 34311.8 General properties: dynamic structure factor 344References 350

x Contents

12 Viscosity 35512.1 Introduction 35512.2 Phenomenology 35712.3 General properties: viscosity 38512.4 Conclusions 392References 393

13 Viscoelasticity 39713.1 Remarks 39713.2 Temporal scaling ansatz for viscoelastic behavior 39813.3 Phenomenology of the dynamic moduli 40313.4 Phenomenology of shear thinning 41813.5 Concentration and molecular weight effects 42713.6 Optical flow birefringence 43613.7 General properties: viscoelasticity 437References 441

14 Nonlinear viscoelastic phenomena 44514.1 Normal stress differences 44514.2 Memory-effect phenomena 44814.3 Modern nonlinear behaviors 45214.4 Remarks 455References 456

15 Qualitative summary 45915.1 Introduction 45915.2 Sedimentation 45915.3 Electrophoresis 46115.4 Light scattering spectroscopy 46315.5 Solvent and small-molecule motion 46415.6 Segmental dynamics 46515.7 Dielectric relaxation and chain dimensions 46615.8 Single-chain diffusion 46715.9 Probe diffusion 46815.10 Colloid dynamics 46915.11 The dynamic structure factor 47115.12 Low-shear viscosity 47215.13 Viscoelasticity 473

16 Phenomenology 47516.1 Introduction 47516.2 Comparison with scaling and exponential models 475

Contents xi

16.3 Parametric trends 47716.4 Transitions 47816.5 Comparison of colloid and polymer dynamics 48116.6 How do polymers move in nondilute solution? 48416.7 Hydrodynamic interactions in solution 48616.8 Length scales in polymer solutions 48816.9 Effect of chain topology 48916.10 Other constraints 490References 491

17 Afterword: hydrodynamic scaling model for polymer dynamics 494References 497

Index 499

Preface

There are already vast numbers of reviews, monographs, edited collections,conference proceedings, and web pages on polymer diffusion, light scattering,electrophoresis, rheology, and almost every topic I cover, other than optical probediffusion. Why does the world need another book about polymers in solution?

On one hand, the chosen topic has reached a certain degree of maturity. Overthe past decade the spate of new research papers on polymer dynamics has greatlyslackened, so in the half-decade I needed to write this volume the first-writtenchapters did not date badly. On the other hand, there are some radically new methodsand results whose significance for polymer physics does not seem to be widelyrecognized.

What do I offer that has not been said many times before?First and foremost, my focus is phenomenology. There are bits of theoretical

discussion hither and thither throughout the volume, but most chapters discussexperiment. If you want to read about models for polymer motion or the formalbasis of particular experimental methods, you must for the most part look elsewhere.Except for light scattering spectroscopy, I give very little background on experi-mental methods and interpretation. The extremely extensive theoretical literatureon polymer dynamics in solution is not reviewed. For such reviews see, for exam-ple, Graessley(1), Tirrell(2), Pearson(3), Skolnick and Kolinski(4), Lodge, et al.(5),and (more recently but less directly) McLeish(6). Recent papers by Schweizer andcollaborators include extensive background references (79)

Second, my major interest lies with concentrated solutions. Readers may recallother, excellent reviews that skip from dilute solutions to the melt, leaving almostunmentioned the intervening nondilute gap. The Phenomenology does precisely theopposite; the focus here is on nondilute solutions. Dilute solutions do appear becausemany good studies of concentrated polymer solutions have wisely continued theirmeasurements down into the dilute range.

xiii

xiv PrefaceThird, I cover a wider range of experimental methods than is sometimes tra-

ditional. Note chapters on electrophoresis, sedimentation, the dynamic structurefactor, solvent and segmental dynamics, and optical probe diffusion, not to men-tion sections on magnetic resonance and neutron spin echo spectroscopy. Viscosityand viscoelasticity are examined at length, but they are among the last topics tobe reached. The chapter on nonlinear viscoelasticity, the very last to be finished,includes the revolutionary discoveries of the past half-decade, such as shear bandingand nonquiescent relaxation. I have tried to unite disparate methods of measuring thesame parameter. For example, the well-known tendency of random-coil polymersto contract with increasing polymer concentration is usually referenced to neutronscattering measurements, but the effect has also been quantified with static lightscattering and much more extensively with dielectric relaxation; all three methodsappear in Chapter 7.

Fourth, I consider solutions of hard-sphere colloids. Neutral polymers and neu-tral colloids interact through precisely the same forces. They have hydrodynamicinteractions, and they cannot interpenetrate. They differ only in their geometry. Aswill be seen, their dynamic behaviors are also quite similar, speaking to the possiblesignificance of topological interactions in polymer dynamics.

I could have written a much shorter book by selecting a few experiments usingeach experimental method, while substantially ignoring the bulk of the publishedliterature. That much shorter book would have been a failure. Its conclusions nomatter what they were would have been disbelieved, based on assertions that theselected experiments represent atypical systems. Phenomenology of Polymer Solu-tion Dynamics has therefore followed precisely the opposite approach. In almostevery chapter, I have sought to represent the bulk of the available literature on theproperty in question. Undoubtedly, I missed a few papers here or there.

Were my literature searches complete? For most topics I reached the point atwhich searches for citations to papers already in hand, back-tracing through foot-notes, searches for additional papers by key authors, and systematic browsing oftables of contents of key journals ceased to locate new articles. One area in whichmy coverage does not pretend to completeness is capillary electrophoresis. Theliterature on electrophoresis in polymer solutions is vast. There has been almost norecognition prior to this volume that electrophoresis supplies information on thesupport medium, as opposed to using the support medium to supply informationon the electrophoresing objects. The analysis in Chapter 3 thus began as a blankpage late in this books writing. Yes, there are papers that interpret electrophoreticmobilities in cross-linked true gels in terms of particular theoretical models forpolymer dynamics. However, those well-done papers assume the validity of theirtheoretical models, so they represent an interaction between a particular theory and

Preface xvelectrophoretic experiment, not an examination of what might be learned aboutpolymer motion by examining electrophoretic studies carefully.

To control the scope of the work, especially the number of years that I spentwriting, a few sacrifices had to be made. My concern is solutions of neutral poly-mers, primarily linear chains and star polymers. Melt properties are not considered.A prolonged discussion of rodlike polymers was dropped. Except in the discussionof the dynamic structure factor, polyelectrolyte properties are omitted. The chapteron colloids centers on neutral hard spheres. Proteins and other charged systemsare mentioned only to the extent that they illuminate neutral colloid behaviors.Solutions of charged biopolymers are largely absent from the discussion, thoughmotions of DNA, RNA, and protein probes in polymer solutions appear in vari-ous chapters. It has recently become apparent that modern biotechnology permitsthe synthesis of totally monodisperse, truly large, polymers of controlled topology,including rings and stars; use of these materials in polymer dynamics is just begin-ning. Cross-linking reactions can convert polymer solutions to gels: the dynamicsof true, covalently cross-linked gels are beyond the scope of the work. Mixed-amphiphile systems form long linear micelles resembling linear chains, except thatthey can interpenetrate; they are beyond my scope. The treatment of nonlinear vis-coelastic effects is a zoological collection with limited quantitative analysis: Herebe rod climbing, shear banding, multiple step stress relaxation, and a host of otherphenomena. A review of major theoretical models, a review I did not write, wouldreadily have doubled the length of this volume.

This volume represents the culmination of four decades study of the dynamicsof macromolecules in nondilute solution. I am profoundly grateful to the staff ofthe Library, Worcester Polytechnic Institute, for their assistance with my more eso-teric search inquiries. I am very grateful to my few graduate students and modestlymore numerous undergraduates for their research on exemplary polymer solutions,as described and cited below in the appropriate chapters. The work here bene-fited from interactions over four decades with many colleagues. The treatment ofsmall-molecule and ion diffusion in extremely viscous liquids grew largely fromconversations many years ago with Dr. Bret Berner. Preliminary studies on aspectsof viscoelasticity represent a collaboration with Dr. P. Peczak(10). However, theanalysis in this volume is almost entirely my own work.

At some point a writing project must come to an end, or it will continue forever.The following is the end that I reached.

George PhilliesWorcester, Massachusetts

xvi Preface

References[1] W. W. Graessley. The entanglement concept in polymer rheology. Adv. Polym. Sci.,

16 (1974), 1179.[2] M. Tirrell, Polymer self-diffusion in entangled systems. Rubber Chem. Tech., 57

(1984), 523556.[3] D. S. Pearson. Recent advances in the molecular aspects of polymer viscoelasticity.

Rubber Chem. Tech., 60 (1987), 439496.[4] J. Skolnick and A. Kolinski. Dynamics of dense polymer systems. Computer

simulations and analytic theories. Adv. Chem. Phys., 78 (1989), 223278.[5] T. P. Lodge, N. A. Rotstein, and S. Prager. Dynamics of entangled polymer liquids.

Do entangled chains reptate? Adv. Chem. Phys., 79 (1990), 1132.[6] T. C. B. McLeish. Tube theory of entangled polymer dynamics. Advances in Physics,

51 (2002), 13791527.[7] M. Fuchs and K. S. Schweizer. Mode-coupling theory of the slow dynamics of poly-

meric liquids: fractal macromolecular architectures. J. Chem. Phys., 106 (1997),347375.

[8] K. S. Schweizer. Microscopic theory of the dynamics of polymeric liquids. Gen-eral formulation of a modemode-coupling approach. J. Chem. Phys., 91 (1989),58025821.

[9] K. S. Schweizer and G. Szamel. Crossover to entangled dynamics in polymer solutionsand melts. J. Chem. Phys., 103 (1995), 19341945.

[10] G. D. J. Phillies and P. Peczak. The ubiquity of stretched-exponential forms in polymerdynamics. Macromolecules, 21 (1988), 214220.

1Introduction

This volume presents a systematic analysis of experimental studies on the dynamicsof polymers in solution. I cover not only classical methods, e.g., rheology, andmore modern techniques, e.g., self-diffusion, optical probe diffusion, but alsoradically innovative methods not generally recognized as giving information onpolymer dynamics, e.g., capillary zone electrophoresis. Actual knowledge comesfrom experiment. The intent is to allow the data to speak for themselves, not toforce them into a particular theoretical model in which they do not fit; freed of theProcrustean bed of model-driven analysis, the data do speak, loudly and clearly.

The Phenomenology examines what we actually know about polymer motion insolution. The objective has been to include every significant physical property andexperimental method, and what each method shows about polymer motion. Thelist of methods includes several that have not heretofore been widely recognizedas revealing the dynamics of polymer solutions. Undoubtedly there are omissionsand oversights, for which I apologize. The reader will note occasional discussionsthat speak to particular models, but experiment comes first, while comparison withvarious hypotheses is postponed.

The following dozen chapters demonstrate that the vast majority of measure-ments on polymer dynamics can be reduced to a very modest number of parameters.These parameters have simple relationships with underlying polymer propertiessuch as polymer molecular weight. The relationships in turn speak to the validity ofseveral possible models for polymer dynamics, models whose validity is also testedby a number of more qualitative observations on how polymers move in solution.

1.1 Plan of the workWhat methods are treated here, and in which order are they presented? I begin byconsidering experiments based on applying forces directly to individual moleculesand observing their resulting motions. I then turn to diffusive processes of single

1

2 Introduction

molecules or their parts, proceeding from the smallest to the largest mobile units:Solvent motion, segmental dynamics, dielectric relaxation, single-entire-chaindiffusion, and optical probe diffusion are examined. In each of these processes,the motions of the object being studied are altered by the presence of other polymermolecules in solution, but each experimental technique corresponds to a single-object correlation function. Finally, experiments measuring collective effects areconsidered, including results on mutual diffusion and the dynamic structure factor,the polymer slow mode, zero-shear viscosity, and linear and nonlinear viscoelas-ticity. A short chapter summarizes what has been said. A concluding chapter ofanalysis and interpretation unifies earlier presentations. Several topics that lie apartfrom this general arrangement, notably chapters on light scattering and diffusion,and on properties of colloidal systems, are inserted at seemingly convenient places.The discussion of diffusion and methods of measuring it is sensibly placed beforethe chapters on diffusive properties. Colloidal dynamics are treated prior to con-sidering the dynamic structure factor of polymer solutions. We turn now to shortsketches of the later chapters.

We begin with two experimental methods, sedimentation and electrophoresis,that measure the driven motion of polymer chains and colloidal particles. In eachmethod, an external force is applied directly to particular molecules in solution, andparticle motion is observed. The forces are buoyancy and the Coulomb force. Lightpressure (optical tweezers) has also been used to move particles; this methodappears in Chapter 9. Chapter 2 presents phenomenology associated with sedimen-tation by polymers and sedimentation of particulates through polymer solutions.The sedimentation rate of polymers in homogeneous solution, and the sedimenta-tion of particulate probes through polymer solutions, both depend on the polymerconcentration and molecular weight and the size of the particulates.

Chapter 3 takes us from one of the oldest techniques for the study of polymerdynamics sedimentation to one of the newest capillary electrophoresis. Aprimary theme of this chapter is the unity of behavior shown by the electrophoreticmobility over a wide range of concentrations and molecular weights of the poly-meric support media. As an experimental method in biochemistry, electrophoresisis almost as old as sedimentation. Discussions of electrophoresis center on how theseparation process can be improved. It was recently recognized that one can invokeparticular models for polymer dynamics to describe the progress of a separation.However, prior to the discussion in this chapter it does not appear to have been rec-ognized that electrophoretic separations, in addition to separating charged species,are at the same time measuring properties of the support medium.

Chapter 4 presents an extended treatment of scattering techniques and diffu-sion coefficients. There is a variety of diffusion coefficients, a variety of namesthat have been assigned to those coefficients, and a need for consistency. Several

1.1 Plan of the work 3approaches, including the Onsager regression hypothesis, the Langevin equation,and statistico-mechanical averaging over intermacromolecular forces, have beenused to compute diffusion coefficients. The emphasis is on colloids, generatingresults needed in Chapter 10. For a solution of dilute diffusing particles, the rela-tionship between the measured dynamic structure factor g(1)(q, t) and the statisticalmoments of the probability distribution P(r, t) for particle displacements r issometimes misunderstood; the relationship is therefore examined at length. Finally,methods for extracting parameters from measurements of g(1)(q, t) are examined.Computational information-theoretic methods confirm what has long been knownpractically, namely that the number of independent parameters that can be extractedfrom a light scattering spectrum is quite modest.

The book next turns to dynamic properties determined by motions of singlemacromolecules. Chapters 5 and 6 consider the smallest molecular motions, namelymotions of single solvent molecules and motions of molecular bonds and polymersegments. Until recently, it was assumed that the solvent had the same physicalproperties in a polymer solution and in the neat liquid. It has now become clearthat just as solvents modify polymer properties such as chain radius, so also dopolymer molecules modify properties of nearby solvent molecules. The relation-ship between the small-molecule diffusion coefficient and the solvent viscosity hassometimes been assumed to follow Waldens rule D T/, T being the absolutetemperature. The experimental literature as developed in Chapter 5 leads to alter-native relationships, different for small and large diffusing objects in low and highviscosity simple liquids and in dilute and highly concentrated polymer solutions.The subsequent Chapter 6 on segmental diffusion considers VH light scattering,time-resolved polarization measurements, and NMR as paths to determining howfast chain segments move, each technique being sensitive to motions on its ownlength scale. A generalized Kramers relation for segment orientation times is found,the relationship plausibly being the one that would have been obtained by Kramersif the phenomenology demonstrated in Chapter 5 had been recognized.

Dielectric relaxation is the primary topic in Chapter 7. Dielectric relaxationaffords information on a plethora of different polymer properties, including (forappropriately chosen materials) the average mean-square length of the end-to-endvector r, the relaxation time for end-to-end vector reorientation, the dynamic dielec-tric and dielectric loss functions () and (), and cross-correlations betweenmotions of different parts of the same chain. Parametric dependences of these quan-tities on polymer properties, and several cross-correlations, are noted. Comparisonis made with other techniques for measuring polymer chain extent, including staticlight scattering and elastic neutron scattering.

Single-chain diffusion, the motion of an identified chain through a uniformpolymer solution, is treated in Chapter 8. The diffusion coefficients for polymer

4 Introduction

self-diffusion (the motion of a single probe chain through a solution of substantiallyidentical matrix chains) and tracer diffusion (the diffusion of a single probe chainthrough a solution of matrix chains that are not the same as the chain of interest)in general depend on the molecular weight P of the probe chain, the concentrationc and molecular weight M of the matrix chains, solvent quality, temperature, andother physical variables.

Probe diffusion is the subject of Chapter 9. Probe motion in polymer solutionshas long been studied with light scattering spectroscopy. Interest in the method wasenhanced by the early observation that the probe diffusion coefficient is often notdetermined by the solutions shear viscosity. In some systems slow probe modesare seen; these are not the same as the polymer slow modes seen in light scat-tering from binary polymer : solvent mixtures. More recently, computer and videotechniques permit tracking the motion of individual particles, permitting determina-tion of hitherto-inaccessible statistical properties of particle motion. Chapter 9 alsoconsiders the few true microrheological studies in which the motion of mesoscopicparticles subject to outside forces is examined.

Chapter 10 is nearly unique in a volume on polymer dynamics, namely it assignsto the dynamics of rigid colloidal particles an importance equal to the dynamicsof nonrigid polymer coils. There are few precedents for such an assignment. How-ever, polymer and colloid dynamics are governed by the same forces and the samegeneral dynamic equations, so it should not be surprising that polymer and colloiddynamics have many fruitful points of comparison. In particular, any nondilutesolution property that qualitatively is exhibited both by colloid and by polymersolutions cannot arise from topological interactions unavailable to colloids.

No location for Chapter 10 was entirely satisfactory. It seemed critical to intro-duce the concentration dependence of the colloidal mutual diffusion coefficientDm, in particular the fundamental issue that Dm of diffusing macromolecules can-not meaningfully be represented in terms of a scaling length , before reachingChapter 11 on the dynamic structure factor. On the other hand, the functions usedto represent the zero-shear viscosity and the dynamic moduli of colloidal sus-pensions are taken from the chapters on zero-shear viscosity and viscoelasticityof polymer solutions, and those chapters were best placed toward the end of thebook. Discussions of colloids might have been dispersed throughout chapters onrandom-coil polymers, but that alternative would have lost the impact of a unifieddisplay of properties of colloidal preparations. These contrary needs were resolvedby allowing Chapter 10 to invoke results from later chapters.

Finally the book reaches properties that are determined by the collective proper-ties of the dissolved polymers, including the dynamic structure factor, the polymerslow mode, the zero-shear viscosity, and linear and nonlinear viscoelasticity.Chapter 11 treats the dynamic structure factor S(q, t) of polymer solutions as

1.2 Classes of model for comparison with experiment 5obtained by the scattering of light, neutrons, or other coherent waves. In dilute solu-tion, S(q, t) measures a translational diffusion coefficient; equivalently, it measuresa hydrodynamic radius rH . At large q, S(q, t) reflects polymer internal motions.In not-quite-dilute solutions, the initial relaxation rate K1 of S(q, t) depends onq and c in simple ways; the observed q-dependence of K1 has implications for ref-erences to hydrodynamic screening hypotheses. At elevated concentration, S(q, t)sometimes shows a very slow relaxational mode. A discussion of polyelectrolytes,which sometimes have spectral slow modes, is included; recent experimentsappear to clarify the physical nature of the polyelectrolyte slow modes.

The low-shear viscosity of polymer solutions is considered in Chapter 12. Themajor effort in the chapter is demonstrating the functional form of the dependenceof on c and M . A large-concentration transition in the functional form of (c) isfound for some but certainly not all systems. We finally consider the behavior ofthe parameters obtained from an accurate functional description of (c,M).

Chapter 13 examines the dependence of viscoelastic behavior, including the stor-age and loss moduli and shear thinning, on solution properties. Historically, it hasbeen difficult to obtain a simple description of the dependence of G and G onc, M , or other parameters. Traditional reduced-variable methods have been dis-appointing; experimental results remained confusing. Chapter 13 presents a novelansatz and set of functional forms that describe G(), G(), and () accuratelyat all frequencies and shear rates, while reducing measurements to a very smallnumber of parameters. These parameters are found to have simple dependences onc and M , reinforcing the belief that the ansatz description has a fundamental basis.

Chapter 14 sketches nonlinear properties of polymer solutions, some classicaland some quite modern. Strange behaviors can arise in polymer solutions becausethe normal stress differences are nonzero, i.e., the diagonal components of thepressure tensor can be unequal. Memory effect properties, such as stress and strainrelaxations, and responses to imposing multiple strains, are noted. Finally we con-sider very recent developments in the study of nonlinear effects, such as shearbanding and nonquiescent relaxation following imposition of a sudden strain.

A summary chapter presents briefly what was done in each of the prior chapters.Results from different experimental properties are then united, showing how theyare interrelated and drawing additional conclusions that would not have beenobvious from a single experimental method.

1.2 Classes of model for comparison with experimentThe approach here is to compare experimental measurements of transport coeffi-cients with functional forms and parametric dependences predicted by models ofpolymer dynamics. There is a very large number of proposed models. Most models

6 Introduction

fall into two major phenomenological classes, distinguished by the functional formsthey give for the transport coefficients. These phenomenological classes are not thesame as theoretical classes categorized by assumptions as to the dominant forcesin solution. This section sketches the predictions of these classes in preparation forthe comparison.

(1) In scaling models (1), the relationship between, e.g., the self-diffusioncoefficient and polymer properties is described by power laws such as

Ds =D1M cx, (1.1)where here and x are scaling exponents, and D1 is a scaling prefactor, namelythe nominal diffusion coefficient at unit molecular weight and concentration. Insome cases, scaling laws are proposed to be true only over some range of theirvariables, or only to be true asymptotically in some limit. On moving away fromthe limit, corrections to scaling then arise. Some models of melts derive a scalinglaw for Ds(M) from model dynamics, and then predict numerical values for .For polymer solutions, more typically a scaling-law form is only postulated; thetheoretical objective is limited to calculating the exponents.

Many scaling-type models propose a transition in solution behavior betweena lower-concentration dilute regime and a higher-concentration nondilute regime.Scaling arguments do not usually supply numerical coefficients, so there is no guar-antee that an interesting transition actually occurs at unit value of a hypothesizedtransition concentration ct rather than at, say, 2ct . Correspondingly, the observationthat a transition is found at 2ct rather than ct is generally in no sense a disproofof a scaling model, because in most cases scaling models do not supply numericalprefactors required for a disproof. (Some level of rationality must be preserved. Ifa physical model leads to ct as the transition concentration, and the transition isfound at 30150 ct , and then only in some systems, one must ask why one shouldbelieve that the observed transition is related to the transition in the model.)

Two transition concentrations are often identified in the literature. The firsttransition concentration is the overlap concentration c, formally defined as theconcentration c = N/V at which 4R3gN/(3V ) = 1. Here N is the number ofmacromolecules in a solution having volume V and Rg is the macromoleculeradius of gyration. In many cases, c is obtained from the intrinsic viscosity viac = n/[] for some n in the range 14. The second transition concentration is theentanglement concentration ce. In some papers, the entanglement concentration isobtained from a loglog plot of viscosity against concentration by extrapolating anassumed low-concentration linear behavior and an assumed higher-concentrationpower-law behavior (e.g., cx for, e.g., x = 4) to an intermediate concentration atwhich the two forms predict the same viscosity, this intermediate concentrationbeing taken to be ce. In other papers, the entanglement concentration is inferred

1.2 Classes of model for comparison with experiment 7from the behavior of the viscoelastic moduli, for example, the onset of viscousrecovery.

(2) In exponential models, the concentration dependence is an exponentialor stretched exponential in concentration(2, 3). For self-diffusion, the stretchedexponential form is

Ds =Do exp(c). (1.2)Here Do is the diffusion coefficient in the limit of infinite dilution of the polymer, is a scaling prefactor, and is a scaling exponent; = 1 for a simple exponential.If the probe and matrix polymers have unequal molecular weights P and M , anelaborated form of the stretched exponential is

Dp =DoPa exp(cP M), (1.3)where a, , and are additional scaling exponents, Do now represents the diffusioncoefficient in the limit of zero matrix concentration of a hypothetical probe poly-mer having unit molecular weight, and Pa describes the dependence on probemolecular weight of the diffusion coefficient of a dilute probe molecule.

In derivations leading to stretched-exponential models, functional forms andnumerical values for exponents and prefactors are obtained, subject to variousapproximations (25).Some derivations assume that chain motion is adequatelyapproximated by whole-body translation and rotation, which may be appropriateif P M , but which is not obviously appropriate if P and M are substantiallyunequal.

Some exponential models also include a transition concentration, namely a tran-sition between a lower-concentration regime in which some transport coefficientsshow stretched-exponential concentration dependences and a higher-concentrationregime in which the same transport coefficients show power-law concentrationdependences(6, 7). This transition concentration is here denoted c+. The lower-concentration regime is the solutionlike regime, the higher-concentration regimeis the meltlike regime. Power-law and exponential forms can both follow from arenormalization-group approach, depending on the location of the supporting fixedpoint(8). The stretched-exponential form is an invariant of the AltenbergerDahler(8) positive-function renormalization group(5).

Our analysis will examine whether either of these classes of model describesexperiment. While a power law and a stretched exponential both can represent anarrow range of measurements to within experimental error, on a loglog plot apower law is always a straight line, while a stretched exponential is always a smoothcurve of nonzero curvature. Neither form can fit well data that are described well bythe other form, except in the sense that in real measurements with experimental scat-ter a data set that is described well by either function is tangentially approximatedover a narrow region by the other function.

8 Introduction

1.3 Interpretation of literature experimental resultsPhenomenology of Polymer Solution Dynamics presents and systematizes theresearches of hundreds of researchers who employed a large number of experi-mental techniques, including centrifugation, electrophoresis, light scattering spec-troscopy, neutron scattering, electrical conductivity, depolarized light scattering,time-resolved polarization, nuclear magnetic resonance, dielectric relaxation, elas-tic neutron scattering, fluorescence recovery, optical probe diffusion, particletracking, true microrheology, viscometry, and multiple methods for examiningviscoelastic response and shear thinning, among others.

What was done here was to extract the original measurements and provide auniform phenomenological description. Numerical values for each property wereobtained from the literature, rarely from tabulated data but usually by scanning andpoint-by-point digitization of individual figures in the original papers. A modestobstacle was that some authors report only measurements that have been heavilyprocessed with respect to particular theoretical models, so for a few papers it wasimpossible to determine the fundamental underlying measurements. A nonlinearleast squares fitting program employing the simplex algorithm was then used tofit possible functional forms to measurements, thereby extracting fitting parame-ters that were studied further(9). Possible functional forms for each property wereinferred from the measurements or drawn from the theoretical literature. The quan-tity minimized by the fitting algorithm was the mean-square difference betweenthe data and the fitting function, expressed as a fraction of the value of the fittingfunction. This quantity is the appropriate choice for minimization if the error inthe measurement is some constant fraction of the value of the quantity being mea-sured. In some cases, one or more potentially free parameters were held constant(frozen) during the fitting process.

The approach here differs from much valid analysis in the earlier literature.Historically, there has been great interest in reducing variables and superpositionplots. A starting point for applying reducing variables is a set of measurements of,for example, the viscosity (c,M) at a series of concentrations c and molecularweights M . With an appropriate choice of molecular-weight-dependent reducingfactors ac and perhaps a, a plot of a against acc reduces (c,M) at differentM to a single master curve for a(acc). When it works, reduction transforms aseries of very different curves into a single line. A master curve predicts dynamicproperties at concentrations and molecular weights that were not studied.

In this work, we advance from reducing variables to numerical curve fitting.Numerical fitting methods afford strong advantages over reducing variables andsuperposition plots. Numerical fits reveal weak dependences not readily apparentto the naked eye. Furthermore, reducing variables can only lead to superposition

References 9plots if the underlying experimental variable has appropriate scaling properties.Numerical fitting can handle parametric dependences far more complex than sim-ple scaling. For example, if the functional form of the concentration dependencedepends on the polymer molecular weight, in general no reducing variable can leadto a master plot covering multiple molecular weights. This challenge to reducingvariables was long known to be an issue for the viscoelastic functions. As Ferrywrote: It is evident that the concentration reduction scheme for the transitionzone described above cannot be applied in the plateau zone, and indeed thatno simple method for combining data at different concentrations can exist; theshapes of the viscoelastic functions change significantly with dilution (10). Here analternative method for reducing measurements to a few parameters will be revealed.

References[1] P.-G. de Gennes. Scaling Concepts in Polymer Physics. Third Printing, (Ithaca, NY:

Cornell UP, 1988).[2] R. S. Adler and K. F. Freed. On dynamic scaling theories of polymer solutions at

nonzero concentrations. J. Chem. Phys., 72 (1980), 41864193.[3] G. D. J. Phillies. Dynamics of polymers in concentrated solution, the universal scaling

equation derived. Macromolecules, 20 (1987), 558564.[4] G. D. J. Phillies. Quantitative prediction of in the scaling law for self-diffusion.

Macromolecules, 21 (1988), 31013106.[5] G. D. J. Phillies. Derivation of the universal scaling equation of the hydrody-

namic scaling model via renormalization group analysis. Macromolecules, 31 (1998),23172327.

[6] G. D. J. Phillies. Range of validity of the hydrodynamic scaling model. J. Phys. Chem.,96 (1992), 1006110066.

[7] G. D. J. Phillies and C. A. Quinlan. Analytic structure of the solutionlike-meltliketransition in polymer solution dynamics. Macromolecules, 28 (1995), 160164.

[8] A. R. Altenberger and J. S. Dahler. Application of a new renormalization group to theequation of state of a hard-sphere fluid. Phys. Rev. E, 54 (1996), 62426252.

[9] J. H. Noggle. Physical Chemistry on a Microcomputer, (New York, NY: Little,Brown & Company, 1985).

[10] J. D. Ferry. Viscoelastic Properties of Polymers, (New York, NY: Wiley, 1980),506507.

2Sedimentation

2.1 IntroductionThe importance of sedimentation to the study of macromolecules has been apparentsince the early 1920s, when Theodor Svedberg invented the ultracentrifuge andused it to demonstrate that proteins are monodisperse macromolecules and not,as he had originally believed, colloidal aggregates formed from amino acids. Theapplication of sedimentation studies in the analytic ultracentrifuge to determine themolecular weight of polymers is well known. This chapter considers sedimentationin nondilute polymer solutions, including both the sedimentation of polymers ina homogeneous monodisperse preparation and the sedimentation of probe chainsand particles through a background (matrix) polymer.

In a dilute solution, the sedimentation rate of a polymer is characterized by itssedimentation constant s, which is related to other solution properties by

s = M(1 v2)NAf s

, (2.1)

where M is the polymer molecular weight, v2 is the polymers specific volume, is the solvent density, NA is Avogadros number, and f s is the drag coefficient forsedimentation.

At elevated concentrations, hydrodynamic and other interactions between sed-imenting molecules become important. Two sorts of sedimentation measurement,involving respectively a binary and a ternary system, then suggest themselves. First,s in binary polymer systems may depend on polymer concentration and molecu-lar weight. Second, the sedimentation rate of colloidal particles or probe polymermolecules through a solution of a second polymer, as might occur in ternary systems,may depend on the second polymers properties.

Polymers at sufficiently large concentrations overlap uniformly. It has beenasserted that polymers in a binary solution may be envisioned as an amorphous

10

2.1 Introduction 11

porous plug through which the solvent passes, at a rate that determines the sedi-mentation coefficient of the polymer. Brochard and deGennes(1) and Pouyet andDayantis(2) offer a scaling relationship for s of a polymer, namely

s cxMy. (2.2)Here c is the polymer concentration, while x and y are scaling exponents. Brochardand deGennes, and Pouyet and Dayantis proposed y = 0, with x = 0.5 in a goodsolvent and x = 1 under Theta conditions.

Modern discussions of probe sedimentation are traced to the seminal study ofLangevin and Rondelez(3). They determined s of bovine serum albumin, bushystunt virus, eggplant mosaic virus, ludox, and polystyrene latex spheres in solutionsof polyethylene oxide. Langevin and Rondelez proposed a stretched-exponentialdependence of s on c, M , and R, namely

s = s0(exp((R/))+0/), (2.3)where s0 is the dilute-solution limit of s, here R is the probe radius, is a hypothe-sized dynamic scaling length, is a scaling exponent, and 0 and are the solventand solution viscosities, respectively. The Langevin and Rondelez form is writtento comply with several of their expectations, notably (i) if R , the exponentialterm is negligible and s is determined by the solution viscosity, and (ii) if R and0/ 1 (which appears to require that the matrix is a high-molecular-weight poly-mer dissolved at nondilute but not extremely large concentrations), s is determinedby R/ .

To relate to solution properties Langevin and Rondelez used a scaling relation

cM , (2.4)leading to

s = s0(exp(aRcM )+0/). (2.5)Here , , = , and = are scaling exponents, notation being chosenfor consistency with the remainder of the book, and a is a scaling prefactor. InLangevin and Rondelezs particular theoretical model, = 3/4 and = 0. Theyattempted to confirm = 0 experimentally.

Langevin and Rondelez used scaling assumptions to reach their final answer, buttheir result for s(c,M) is not a scaling law. Langevin and Rondelez predict for sof probe particles not power-law but stretched-exponential dependences on c, M ,andR. Furthermore, those dependences appear as part of a sum with0/, a quantityitself dependent on c and M , as the other term.

The functional form proposed by Langevin and Rondelez was in part anticipatedby Laurent, Ogston, and their collaborators (47). Laurent and collaborators found

12 Sedimentation

an empirical form similar to Eq. 2.5, with = 1 and = 1/2. Most work that may belinked back to Langevin and Rondelez has, in applying Eq. 2.5 to various transportcoefficients, omitted the term 0/. For an exception retaining this term note Buand Russo on polymer tracer diffusion(8).

The following two sections treat sedimentation of polymers in homogeneoussolution, and sedimentation of probe chains and spheres through polymer solutions.A final section offers a systematic discussion of these results. Throughout, theanalysis matches the experimental concentration dependence against the simplestretched-exponential form

s = s0 exp(c) (2.6)of Laurent, Ogston, and collaborators.

2.2 Homogeneous sedimentationWhat experimental results are extant on the sedimentation of monodisperse random-coil polymers?

Brown, et al. report the sedimentation of 64.2 kDa Mw dextran, Mw/Mn =1.5, in water(9), where s accurately follows Eq. 2.6, see Figure 2.1. In nondilutesolutions the drag coefficients f s and fs for sedimentation and self-diffusion (fromPFGNMR) were found to be unequal, f s consistently being substantially less thanfs . This result, coming at a period when it was often assumed that these two dragcoefficients were equal to each other and to the drag coefficient for mutual diffusion,served to clarify that the equality of these drag coefficients in dilute solution didnot imply that they were all equal in nondilute solutions(10, 11).

0 100 200 300

c (g/l)

0

1

2

3

4

s (S

)

Figure 2.1 Sedimentation of 64.2 kDa dextran in water, as measured by Brown, et al.(9).

2.2 Homogeneous sedimentation 13

0.1 1 10 100c (g/l)

0.1

1

10

s (10

13

s1 )

Figure 2.2 Sedimentation of () 110, () 390, and () 1800 kDa polystyrenesin toluene and () 1800 kDa polystyrene in trans-decalin at close to 25 C frommeasurements by Nystrom, et al.(12).

Nystrom, et al. consider sedimentation of polystyrene in the good solvent toluene,and in trans-decalin under Theta conditions, as seen in Figure 2.2(12). PolymerMw/Mn extended from 1.06 to 1.2, increasing with increasing Mw; s(c) for eachpolymer follows approximately a stretched exponential in c. With increasing poly-mer M , increases markedly and decreases slightly; transferring the 1800 kDachains from good to Theta conditions reduces three fold and increases . At con-centrations near 10 g/l, s(c) is approximately the same in all systems. Nystrom, et al.focus on the fluctuating curvature in plots of s0/s against c for polystyrene : toluene.Because the authors interpreted their results as showing unexpected behavior,they used multiple methods to analyze their measurements, all methods findingsubstantially the same results.

Nystrom and Roots consider a 390 kDa polystyrene sedimenting through trans-decalin, which over the observed temperature range 20 40C offers both Thetaand good solvent behavior(13). Figure 2.3 plots s against c at each of five temper-atures, together with stretched-exponential fits to measurements at temperaturesabove 20 C. At the near-Theta temperature 20 C, s(c) does not follow a stretchedexponential: At 20 C and c < 4 g/l, s(c) decreases weakly with increasing c.Between two data points s(c) falls twofold; at larger concentrations, s(c) falls rathermore quickly with increasing c. The discontinuity of s(c) is the sort of change that

14 Sedimentation

0.1 1 10c (g/l)

0.1

1

10

s (10

13

s1 )

Figure 2.3 Sedimentation of 390 kDa polystyrene in trans-decalin at temperatures20 (), 25 (), 30 (), 35 (), and 40 () C, from measurements by Nystrom andRoots showing sedimentation under near-Theta to good solvent conditions(13).

might occur if there had been a sudden change in the dominant forces controllingpolymer motion.Atransition to entanglement-dominated dynamics might show thisbehavior. However, one might have expected that the expanded chains in good sol-vent conditions would entangle more readily than unexpanded chains in near-Thetaconditions; the observed transition occurs under near-Theta conditions.

Nystrom, et al. examine the effect of polymer branching on sedimentation bycomparing sedimentation rates of linear polymethylmethacrylate and branchedpoly-2-triphenylmethoxyethylmethacrylate (PTEMA) samples(14). For each sam-ple, s(c) is described reasonably accurately by Eq. 2.6. In dilute solutions, s0varies by no more than 50% between these polymers. At larger concentrations,s(c) for PTEMA is nearly independent of M . Over the concentrations seen inFigure 2.4, s(c) falls tenfold. At elevated concentration, nondilute linear chainssediment one-third as fast as the branched chains of close to the same molecu-lar weight, while for the branched chains is a quarter or a third of for thelinear chain.

Nystrom and Roots reviewed sedimentation and mutual diffusion in semidilutepolymer solutions, making comparison with then-current ideas about scaling lawsfor these solutions(15). They collect an extensive series of studies on good andTheta polymer : solvent systems, including their own unpublished data, some ofwhich are seen in Figures 2.5 and 2.6. Works that did not cover both dilute andnondilute regimes are not considered farther. Stretched exponentials universallydescribe the measurements, including (Figure 2.6) sedimentation of polymers underTheta conditions. The peculiar concentration dependences reported by Nystrom

2.2 Homogeneous sedimentation 15

1 10 100c (g/l)

1

10s

(1013

s1 )

Figure 2.4 Sedimentation of 1.4 MDa linear polystyrene () and 1.0 () and 1.35() MDa branched PTEMA, all in toluene, from results of Nystrom, et al.(14),showing that branched chains sediment much more rapidly than linear chainshaving about the same concentration and molecular weight.

1 10 100c (g/l)

100

1000

s

Figure 2.5 Sedimentation in good polymer : solvent systems: 312 kDa ()branched polystyrene in benzene; 1.8 MDa () and 390 kDa () polystyrenein benzene. Arbitrary vertical shifts for clarity were present in the originalreference(15).

16 Sedimentation

1 10 100c (g/l)

1

10

100

1000

10000

s

Figure 2.6 Sedimentation in Theta polymer : solvent systems: from top to bottom,950 (), 390 (), and 110 () kDa polystyrene : cyclopentane(16), 2.2 MDa ()and 213 kDa () branched polystyrene : cyclohexane(15), 1.8 MDa () and 390kDa () polystyrene : cyclohexane(15), 6.6 () and 1.0 () MDa and 234 ()kDa poly(-methyl styrene) in cyclohexane(17), with stretched-exponential fits.Arbitrary vertical shifts for clarity (hence, arbitrary units) were already suppliedby Ref. (15).

and Roots(13) are not seen, suggesting that those dependences reflect a specificchemical property of polystyrene : trans-decalin, while results in Figure 2.6 reflectthe general behavior of polymers in Theta solvents.

Roots and Nystrom observed sedimentation of polystyrenes, Mw/Mn 1.15, inthe marginal solvent butan-2-one(18). As visible in Figure 2.7 and noted by Rootsand Nystrom, the plot of log(s) against log(c) shows continuous curvature at allconcentrations and molecular weights studied. Roots and Nystrom interpret theirresult in terms of scaling models by asserting, citing Weill and des Cloizeaux(19),that dynamic scaling exponents only reach their asymptotic limits slowly: under thisinterpretation, concentration scaling for s has an apparent concentration-dependentscaling exponent because simple scaling is obtained only at concentrations andmolecular weights much higher than those they studied. Measurements on the twolargest molecular weight polymers were not extended to low concentrations, so fitsto those measurements are less reliable than the others.

Sedimentation of polystyrenes in cyclohexane under Theta conditions wasobserved by Vidakovic, et al.(20); see Figure 2.8. Extremely elaborate precautions

2.2 Homogeneous sedimentation 17

1 10 100c (g/l)

1

10

100

s (10

13

s1 )

Figure 2.7 Sedimentation of polystyrene in the marginal polymer butan-2-one forpolymer molecular weights 0.11(), 0.67 (), 0.90 (), 4.48 (), and 20.6 ()MDa, based on measurements from Roots and Nystrom(18).

1

10

100

s (10

13

s

1 )

101 100 101 102c (g/l)

Figure 2.8 Sedimentation of polystyrene in cyclohexane under Theta conditionsfor polymer molecular weights (+) 0.422, () 1.26, () 3.84, () 8.42, and ()20.6 MDa, based on measurements from Vidakovic, et al.(20).

18 Sedimentation

were taken to avoid potential sources of systematic error, such as radial dilutionduring centrifugation and quadratic deviations of sedimentation from simple linearprogress. Vidakovic, et al.(20) propose that at large c the sedimentation coefficientscollapse onto a master curve, irrespective of chain molecular weight. The actualdata points are not inconsistent with a large-c master curve. However, extrapola-tion of the stretched-exponential fitting functions to larger c (bottom right of thefigure) implies that at large c the sedimentation coefficients do not collapse ontoan M-independent master curve; instead, the large chains sediment more slowlythan the small chains, as actually observed in other systems. The apparent mastercurve arises because measurements of s for each chain were terminated at differentlargest concentrations. It should be recognized that at large c these are very difficultexperiments, reasonably terminated if an expected master curve appears to havebeen demonstrated.

2.3 Probe sedimentationThis section examines sedimentation in selected ternary solutions, namely sedimen-tation of colloids and chain tracers through polymer solutions. Probe sedimentationthrough polymer solutions is in many ways analogous to probe electrophoresis usinga polymeric solution support, or to studies of probe diffusion and polymer tracerdiffusion in polymer solutions, as studied with quasielastic light scattering, fluo-rescence recovery after photobleaching, and other techniques. It is therefore notsurprising that the phenomenology of probe sedimentation as seen here is verysimilar to the phenomenology of other probe studies, to be described below. Whatis perhaps surprising is that the phenomenology for probe sedimentation was estab-lished by pioneering studies of Laurent and collaborators (46) nearly a half-centuryago, well before modern theoretical and experimental studies on tracer and probediffusion and electrophoresis in nondilute polymer solutions, but these pioneeringstudies had only a limited influence on the trajectory of modern research.

Laurent and Pietruszkiewicz examine the sedimentation of bovine serum albu-min, yellow turnip virus, and four diameters of polystyrene latex spheres throughaqueous 0.14 and 1.7 MDa hyaluronic acid(4). They report that

s

s0=Aexp(Bc). (2.7)

Here A and B are substance-dependent constants, with A in the range 11.6 andB increasing more than 20-fold with a c. 60-fold increase in probe radius. In theoriginal papers was assumed to be constant at = 0.5. The matrix concentrationwas no larger than 7 g/l, which was enough to reduce s of the larger particles (latexspheres, radii 88365 nm) by 23 orders of magnitude. Limited measurements

2.3 Probe sedimentation 19

indicated that B depends at least weakly on the matrix polymer molecular weight,but does not depend on solution pH or ionic strength.

Laurent, et al. examine ten biomacromolecules and colloidal silica in 1.7 MDahyaluronic acid(5). For all probe particles, s had a stretched-exponential depen-dence on matrix c, with = 0.5. s was independent of the centrifugal acceleration,i.e., experiments were in a linear domain in which solution viscoelastic propertieswere not evident. Contrast is to be made with the electrophoretic mobility, as dis-cussed in the next chapter, in which one can enter a nonlinear domain in whichparticle mobility depends on the applied field E. The constant B was found, togood accuracy, to be linear in the particle radius, except for the very largest spheres.Laurent, et al. also measured the diffusion coefficients D of four smaller probes inthe same polymer solutions. Within experimental error, s/D was independent ofmatrix concentration.

Laurent, et al. specifically promise a subsequent paper in which it will beshown that this sieving effect can be used for the separation of various compoundsthat would otherwise sediment together in the ultracentrifuge (5). The polymersolution sieving effect is the differential effect of polymer solutions on the sedi-mentation rate of particles of different size, as reflected in the dependence of Bon particle radius. This proposal to use polymer solutions as a path to biochemicalseparations has come to fruition a half-century later in the use of polymer solutionsas support media in electrophoresis.

Laurent and Persson examine the effect of a dozen biopolymers, molecu-lar weights 10 kDa 25 MDa, on the sedimentation of serum albumin and-crystallin(6). The effect of the polymer on probe mobility increases withincreasing polymer molecular weight, but is reduced by the introduction of chainbranching, i.e., s increases when the matrix hydrodynamic radius is reduced with-out change in matrix polymer molecular weight. On comparing sedimentationthrough solutions of dextran or dextran sulfate, and through solutions of methyl-cellulose or carboxymethylcellulose, Laurent and Persson concluded that chargedpolymers are substantially more effective at retarding sedimentation than are theirneutral analogs. Laurent, et al. had previously shown that pH and solution ionicstrength do not affect s significantly, leading them toward the conclusion thatthe effect of polymer charge must proceed through its effect on polymer sizeand rigidity, not through electrostatic interactions between the polymer moleculesand the probes. Laurent and Persson also measured solution viscosities, findingnon-StokesEinsteinian behavior, i.e., s was not a constant. From their shortsummary of their findings, large-molecular-weight matrix polymers are more effec-tive at increasing than at reducing s. The non-StokesEinsteinian behavior waslargest for linear polyelectrolytes, for which s/s00 reached a largest value of3.9. Laurent and Persson also demonstrate that several of their polymers retard

20 Sedimentation

0 2 4 6 8c (g/l)

1

10

100

1000

10000

s (10

13

s)

Figure 2.9 Sedimentation coefficients of (left edge, top to bottom) 365 and 188 nmdiameter polystyrene latex, turnip yellow mosaic virus, -crystallin, -globulin,serum albumin, and -crystallin in solutions of 1.7 MDa hyaluronic acid, demon-strating that the ability of the polymer to retard sedimentation (as reflected inthe concentration dependence of the normalized sedimentation coefficient s/s0)increases with increasing size of the sedimenting particle. Measurements are fromLaurent and Pietruszkiewicz(4) and Laurent, et al.(5).

the sedimentation of serum albumin even when the polymers are sedimentingmore rapidly than the albumin is. The ability of a rapidly sedimenting polymer atmodest absolute polymer concentrations to retard sedimentation of a more slowlysedimenting probe is not readily consistent with images of probe sedimentationas involving the passage of probe particles through a substantially motionlessmatrix.

Figure 2.9 shows sedimentation coefficient against matrix concentration for aseries of small and large particles in solutions of 1.7 MDa hyaluronic acid. Atfixed c, with increasing probe size s0 increases but s/s0 declines. As confirmed byFigure 2.10, Laurent, et al. found that the constant B of Eq. 2.7 is linear in probehydrodynamic radius rh. Figure 2.11 shows Laurent and Perssons determinationof the effect of matrix molecular weight on s of -crystallin. The matrix was ahomologous series of dextrans having molecular weights ranging from 10 kDa to25 MDa. Even with very large dextrans, increasing the dextran molecular weightat fixed dextran concentration increases the extent to which sedimentation of the-crystallin is retarded by the matrix.

Tong, et al.(21) and Ye, et al.(22,23) report sedimentation of 4.0 nm surfactant-coated calcium carbonate particles through monodisperse hydrogenated polyiso-prenes in decane. Unmodified hydrogenated polyisoprene is nonabsorbing to thecoated carbonate spheres. Addition of an end-terminal amino group yields an

2.3 Probe sedimentation 21

1 10 100rh (nm)

0.1

1

10

B

Figure 2.10 Constant B from fits of Eq. 2.7 to Laurent, et al.s determinations ofsedimentation coefficients of various proteins and polystyrene latex spheres, someseen in the previous figure(4,5). () B from fits to Eq. 2.7 with fixed at 0.5. ()B from fits to Eq. 2.7 with a free parameter. Solid lines are power-law fits toB rj for = 0.92 ( = 0.5 points) and 1.007 ( a free parameter point).

0 1 2 3 4c (g/l)

0

4

8

12

16

s (10

13

s

1 )

Figure 2.11 Sedimentation of -crystallin, an 830 kDa globular protein, throughsolutions of dextrans, molecular weights (top to bottom) 0.01, 0.08, 0.5, 2, and 25MDa, showing the progressive reduction in s on increasing the polymer matrixsmolecular weight, even for matrices much larger than the sedimenting protein,incorporating results of Laurent and Persson(6). The scaling exponent falls from0.83 to 0.75 with increasing matrix molecular weight.

end-absorbing polymer species. Figure 2.12 shows s/s0 of the spheres in solutionsof nonadsorbing polymer. Stretched exponentials in c provide approximate but notprecise descriptions for s/s0 at each M . At all c, s/s0 clearly depends on the matrixmolecular weight.

22 Sedimentation

0 50 100 150c (g/l)

0

0.2

0.4

0.6

0.8

1

s/s 0

Figure 2.12 Sedimentation coefficients of 4.0 nm radius coated calcium carbon-ate microspheres through solutions of nonadsorbing hydrogenated polyisoprene,molecular weights () 17.5, () 26, () 33, and () 88 kDa, based on experimentsof Tong, et al.(21).

Tong, et al. (21) and Ye, et al.(22) compare s with the solution viscosity . Forprobes with the nonabsorbing polymer, s sometimes depends on c.At intermediatec, s becomes larger as c is increased, but returns at still larger c to its small-c values00. One sees here an example of the re-entrant transport behavior discussed atgreater length in the chapter on probe diffusion. Ye, et al. find re-entrance for probesin 17.5 and in 26 kDa polymer solutions(22). The deviation from simple StokesEinstein behavior is the largest at the same c for both matrix molecular weights.Here s(c) was measured only to concentrations barely larger than the concentrationat which s returned to s00, so it might be over-ambitious to claim that s doesnot continue below s00 at larger c.

Ye, et al. note an odd viscosity dependence associated with re-entrant behav-ior(22). At the lower concentrations at which s/s00 has not yet reached itsmaximum, s tracks the single-chain viscosity 1 = 0(1 + []c), [] being theintrinsic viscosity. The single-chain viscosity is a mathematical construct, not a sim-ple physical measurable, namely it is the lead two terms in a Taylor series expansionfor . Mathematical constructs are not usually real physical quantities. It is thenpuzzling that, other than in the initial slope ds(c)/dc|c0, in these systems s(c) isdetermined accurately by 1.

Probes in solutions of the amine-terminal adsorbing polymer have the same con-centration dependence for s and for , as seen in Figure 2.13. Ye, et al. propose that

2.3 Probe sedimentation 23

0 50 100 150c (g/l)

0

0.4

0.8

1.2

s/s 0

Figure 2.13 Sedimentation coefficient of 4.0 nm radius coated CaCO3 micro-spheres in (i) solutions of adsorbing 25 kDa amine-terminal hydrogenatedpolyisoprene (), and (ii) solutions of nonadsorbing unmodified polyisoprene(, dashed line), based on experiments of Ye, et al.(22). The two solid lines,which are very nearly indistinguishable on the scale of the figure, show stretched-exponential fits to s(c) and to the reported viscosity of these solutions(22).

the difference between the amine-terminal and nonmodified polyisoprene behav-iors is (i) R/ is a significant variable, (ii) a sphere attached to an amine-terminalchain is, hydrodynamically, always much larger than the hypothesized polymertransient lattice spacing , but (iii) a probe with no adsorbed chains might be eithersmaller or larger than (22). In a separate paper Ye, et al. determined the depen-dence of s (expressed as the sedimentation velocity) on the concentration of probespheres(23). The dependence is substantial, but the probes in their earlier paperswere sufficiently dilute that probeprobe interactions were not significant(21, 22).

Nemoto, et al. report sedimentation and tracer diffusion of dilute poly-methylmethacrylate probes in dilute and nondilute polystyrene : thiophenol(24,25).Polystyrene and thiophenol are isopycnic, permitting s of PMMA probes inpolystyrene : thiophenol to be determined unambiguously with ultracentrifugation.

In Ref. (24), a single 343 kDa probe polymer is examined in matrix polymerswith molecular weights ranging from 43.9 kDa to 8.42 MDa. Figure 2.14 shows s ofthe probe. At each matrix molecular weight, as originally noted by Nemoto, et al.,Ds and s decrease monotonically with increasing polystyrene concentration(24).Over the observed range of matrix M , increases more than twofold, while declines from near 0.7 to 0.57.

Nemoto, et al. compared the concentration dependences of s and Ds(24). Thesedependences were very nearly the same with the 43.9 and 186 kDa matrix polymers,

24 Sedimentation

0.01

0.1

1

s (10

13

s

1 )

0.01 0.1 1 10 100c (g/l)

Figure 2.14 Sedimentation of 343 kDa polymethylmethacrylate through () 44,() 186, () 775 , and () 8420 kDa polystyrene in thiophenol, from measurementsby Nemoto, et al.(24).

but with increasing polymer concentration s/s0 became larger than Ds/Ds0. The775 and 8420 kDa matrix polymers were up to 30% less effective at retardingprobe sedimentation than at retarding probe diffusion. Nemoto, et al. interpret thisdifference as arising from the great difference in the temporal and spatial scales towhich light scattering spectroscopy and ultracentrifugation are sensitive. In supportof this conclusion, these workers measured Ds in the ultracentrifuge via a syntheticboundary method, finding Ds as measured in the ultracentrifuge is considerablylarger than Ds measured with dynamic light scattering. When probe diffusion andsedimentation were measured over the same time and distance scales,Ds and s werefound to have the same concentration dependence. That is, in this system Ds asmeasured over long time and distance scales is faster than Ds as measured at smalltime and distance scales, a trend opposite to the trend expected for hypotheticalparticles that diffuse rapidly within a transient cage in a pseudolattice, but are slowto migrate from cage to cage. A mechanistic interpretation for this surprising resultappears to be lacking.

At fixed matrix concentration, Ds and s both decline with increasing matrixM . Even when M is 20-fold larger than the probe weight P , Ds and s decreasewith increasing M . This decrease in s and Ds at large M/P constrains allowabletheoretical models.

In a second study, Nemoto, et al. report s of eight probe polymers throughsolutions of three matrix polystyrenes(25). Their results extend fromP M toP

2.3 Probe sedimentation 25

0

0 2 4 6

10 20c (wt %)

0.01

0.1

1

s (10

13

s)

(a)

c (wt %)

0.01

0.1

1

s (10

13

s)

(b)

Figure 2.15 Sedimentation of () 107, () 260, () 265, () 401, () 844, ()1100, and () 2140 kDa polymethylmethacrylate through solutions of (a) 43.9(dashed lines) and 775 (solid lines), and (b) 8420 kDa polystyrene : thiophenol,using data from Nemoto, et al.(25).

M . Figure 2.15 shows their measurements for the 16 probe-matrix combinationson which measurements were made at four or more concentrations. The data wererefit in order to assure a uniformity of statistical weights in the fitting process.As previously reported by Nemoto, et al., Eq. 2.6 uniformly describes well theconcentration dependence of s at all P/M ratios. As also seen in other figures,observe that multiple s(c) curves cross at nearly a single point, so that there is amatrix concentration at which s(c) is nearly independent of probe molecular weight.

26 Sedimentation

There is no obvious explanation for this pseudoisosbestic behavior, whose existenceis not generally recognized. Nemoto, et al. also report limited measurements, inthe 40 kDa matrix, on s of six probe species other than those shown in the figure.Note that s(c) of probes in this small-M matrix polymer uniformly tracks theconcentration dependence of the solution fluidity 1.

2.4 General properties: sedimentationSedimentation in homogeneous polymer systems exhibits a series of prominentfeatures. First, the sedimentation coefficient depends strongly on c, decreasingfivefold or more between dilute solution and 100 g/l polymer. Second, in almostall systems, the concentration dependence of s is a stretched exponential in c,regardless of M or solvent quality. There is no indication of a transition to scaling(cx) behavior, except in the sense that a power-law curve could be nearly tan-gential to measurements over a modest range of c. Nemoto, et al. searched forsuch a transition(24), finding that there is . . .no sharp break near [the criticaloverlap concentration] cPS . Nemoto, et al. report that s and Ds follow stretched-exponential forms, similar to those found in the model of Ogston, et al.(7), butno region in which a power-law concentration dependence replaces the stretchedexponential. In a few systems, s(c) has a more complex concentration dependence.Third, at elevated concentrations, in some systems s(c) for homologous polymersconverges towards M-independent master curve behavior (cf., e.g., Figure 2.4 orthe topmost curves in Figure 2.6). In other systems, e.g., bottom curves of Figure 2.6and Figure 2.7), extrapolation of s(c) to higher concentrations suggests that large-Mpolymers may sediment more slowly than small-M polymers. These two behaviorsappear to be correlated with good and Theta conditions, respectively. For simplepolymer : solvent systems, there do not appear to be comparisons of s(c) with thesolution viscosity.

From limited results, s(c) of sedimenting probes in a polymer matrix generallyfollows a stretched exponential in c. With a small matrix polymer, s(c) of a probechain simply tracks the solution viscosity. In solutions of large matrix polymers, sand do not show the same concentration dependence. With probe spheres, s(c)may track the solution viscosity or may show re-entrant behavior. The agreementof s(c) with a stretched-exponential form is less outstanding when re-entrance isobserved. The literature describes too few probe : polymer pairs to be able to say ifre-entrant behavior is common or rare.

How do the fitting parameters, the scaling prefactor and the scaling exponent ,depend on solution properties? For ternary probe polymer : matrix polymer : solventsystems, the dependence of and on probe molecular weight is vividly revealedin Figure 2.16, which shows and against P for polymethylmethacrylate

2.4 General properties: sedimentation 27

100 1000 10000P (kDa)

0.1

1

10

100 1000 10000P (kDa)

1

0.9

0.8

0.7

0.6

0.5

Figure 2.16 Dependence of and on probe molecular weight for polymethyl-methacrylate probes in polystyrene : thiophenol with matrix molecular weights() 44, () 775, and () 8420 kDa. Straight lines represent fits to Eqs. 2.8 and2.9.

probes in 44, 775, and 8420 kDa polystyrenes in thiophenol, as studied by Nemoto,et al.(24, 25). For , one finds

P , (2.8)with being a scaling exponent for the probe molar mass dependence. For the threematrices, was 0, 0.3, and 0.4, respectively. Values of , and at fixed P , bothincrease with increasing M .

With increasing matrix molecular weight, falls monotonically. For , thestraight lines in Figure 2.16 indicate

= 0 + a log(P ), (2.9)with the reasonable expectation that Eq. 2.9 may not apply over much wider rangesof P .

Figure 2.17 shows and for the homogeneous sedimentation studies. The valueof tends to increase with increasing M , but the variation in from homologousseries to homologous series tends to mask trends in this parameter. For almost allhomologous series, declines with increasing M , but almost never was a homol-ogous series studied at enough concentrations over enough of a range of M toform definite conclusions as to the quantitative behavior of on matrix molecularweight.

In conclusion, s(c) almost always follows the stretched-exponential form ofEq. 2.6. When there are measurements on enough members of a homologous seriesof polymers, and can show clear dependences on probe and matrix molecularweights. It appears that the dominant variable determining is the polymers extent,not the polymers molecular weight, so that transferring a polymer from a good toa Theta solvent substantially reduces .

28 Sedimentation

100 1000 10000M (kDa)

0.01

0.1

1

10

100 1000 10000M (kDa)

0.1

1

0.2

0.5

2

Figure 2.17 Dependence of and on matrix molecular weight for binarysystems dextran : water(9) (), polystyrene : toluene(12) (), polystyrene : trans-decane at Theta point(13) (), polystyrene : trans-decane as good sol-vent(13) (), PTEMA: toluene(14) (), polystyrene : benzene(15) (),polystyrene : cyclopentane(16) (), branched polystyrene : cyclohexane(15) (+),polystyrene : cyclohexane(15, 20) (), poly--methylstyrene : cyclohexane(17)(), and polystyrene : butan-2-one(18) ().

References[1] F. Brochard and P. G. deGennes. Dynamical scaling for polymers in Theta solvents.

Macromolecules, 10 (1977), 11571161.[2] G. Pouyet and J. Dayantis. Velocity sedimentation in the semidilute concentration

range of polymers dissolved in good solvents. Macromolecules, 12 (1979), 293296.[3] D. Langevin and F. Rondelez. Sedimentation of large colloidal particles through

semidilute polymer solutions. Polymer, 19 (1978), 875882.[4] T. C. Laurent and A. Pietruszkiewicz. The effect of hyaluronic acid on the sed-

imentation rate of other substances. Biochimica et Biophysica Acta, 49 (1961),258264.

[5] T. C. Laurent, I. Bjork, A. Pietruszkiewicz, and H. Persson. On the interactionbetween polysaccharides and other macromolecules: II. The transport of globular par-ticles through hyaluronic acid solutions. Biochimica et Biophysica Acta, 78 (1963),351359.

[6] T. C. Laurent and H. Persson. The interaction between polysaccharides and othermacromolecules VII. The effects of various polymers on the sedimentation ratesof serum albumin and a-crystallin. Biochimica et Biophysica Acta, 83 (1964),141147.

[7] A. G. Ogston, B. N. Preston, and J. D. Wells. On the transport of compact parti-cles through solutions of chain-polymers. Proc. Roy. Soc. London (A), 333 (1973),297316.

[8] Z. Bu and P. S. Russo. Diffusion of dextran in aqueous hydroxypropylcellulose.Macromolecules, 27 (1994), 11871194.

[9] W. Brown, P. Stilbs, and R. M. Johnsen. Diffusion and sedimentation of dextran inconcentrated solutions. J. Polym. Sci. Polym. Physics Ed., 20 (1982), 17711780.

[10] G. D. J. Phillies, G. B. Benedek, and N. A. Mazer. Diffusion in protein solutions athigh concentrations: a study by quasielastic light scattering spectroscopy. J. Chem.Phys., 65 (1976), 18831892.

References 29[11] R. G. Kitchen, B. N. Preston, and J. D. Wells. Diffusion and sedimentation of serum

albumin in concentrated solutions. J. Polym. Sci. Polym. Symp., 55 (1976), 3949.[12] B. Nystrom, B. Porsch, and L.-O. Sundelof. Sedimentation in concentrated polystyrene

solutions and the observation of an anomaly in a good solvent. Eur. Polym. J., 13(1977), 683687.

[13] B. Nystrom and J. Roots. Dilute and concentrated solutions of polystyrene close toand far from the -temperatureI. Velocity sedimentation measurements. Eur. Polym.J., 14 (1978), 551556.

[14] B. Nystrom, L.-O. Sundelof, M. Bohdanecky, and V. Petrus. Influence of branchingon sedimentation behavior in concentrated polymer solutions. J. Polym. Sci.: Polym.Lett. Ed., 17 (1979), 543551.

[15] B. Nystrom and J. Roots. Molecular transport in semidilute macromolecular solutions.J. Macromol. Sci.-Rev. Macromol. Chem. C, 19 (1980), 3582.

[16] Cited by (15) as B. Nystrom, J. Roots, and R. Bergman. Sedimentation velocity mea-surements close to the upper critical solution temperature and at Theta-conditions:polystyrene in cyclopentane over a large concentration interval. Polymer, 20 (1979),157161.

[17] Cited by (15) as P. F. Mijnlieff and W. J. M. Jaspers. Solvent permeability of dissolvedpolymer material. Its direct determination from sedimentation measurements. Trans.Far. Soc., 67 (1971), 18371854.

[18] J. Roots and B. Nystrom. Sedimentation in the semidilute concentration range ofpolystyrene in a marginal solvent. Chem. Soc. Faraday Trans. I, 77 (1981), 947952.

[19] G. Weill and J. des Cloizeaux. Dynamics of polymers in dilute solutions: an explanationof anomalous indices by cross-over effects. J. Phys.(Paris), 40 (1970), 99106.

[20] P. Vidakovic, C. Allain, and F. Rondelez. Sedimentation of dilute and semidilutepolymer solutions at the temperature. Macromolecules, 15 (1982), 15711580.

[21] P. Tong, X. Ye, B. J. Ackerson, and L. J. Fetters. Sedimentation of colloidal particlesthrough a polymer solution. Phys. Rev. Lett., 79 (1997), 23632366.

[22] X. Ye, P. Tong, and L. J. Fetters. Transport of probe particles in semidilute polymersolutions. Phys. Rev. Lett., 31 (1998), 57855793.

[23] X. Ye, P. Tong, and L. J. Fetters. Colloidal sedimentation in polymer solutions. Phys.Rev. Lett., 31 (1998), 65346540.

[24] N. Nemoto, T. Inoue, Y. Tsunashima, and M. Kurata. Dynamics of polymer-polymer-solvent ternary systems. 2. Diffusion and sedimentation of poly(methyl methacrylate)in semidilute solutions of polystyrene in thiophenol. Macromolecules, 18 (1985),25162522.

[25] N. Nemoto, S. Okada, T. Inoue, and M. Okada. Hydrodynamic and topologicalinteractions in sedimentation of poly(methylmethacrylate) in semidilute solutions ofpolystyrene in thiophenol. Macromolecules, 21 (1988), 15021508.

3Electrophoresis

3.1 IntroductionThe early electrophoresis experiments of Tiselius, first published in 1930, examinedthe motions of proteins in bulk solution as driven by an applied electrical field(1).In the original method, a mixture of proteins began at a fixed location. Under theinfluence of the field, different protein species migrated through solution at differ-ent speeds. In time, the separable species moved to distinct locations (bands).Electrophoresis is now a primary technique for biological separations(2, 3). Twoimprovements were critical to establishing the central importance of electrophore-sis in biochemistry: First, thin cells and capillary tubes replaced bulk solutions.Second, gels and polymer solutions replaced the simple liquids used by Tiseliusas support media. These two improvements greatly increased the resolution of anelectrophoretic apparatus. Electrophoresis in true gels is a long-established exper-imental method. The use of polymer solutions as support media is more recent. Anearly motivation for their use was the suppression of convection, but electrophoreticmedia that enhance selectivity via physical or chemical interaction with migratingspecies are now an important biochemical tool.

Electrophoresis and sedimentation have a fundamental similarity: in eachmethod, one observes how particular molecules move when an external forceis applied to them. In sedimentation, the enhancement of buoyant forces by theultracentrifuge causes macromolecules to settle or rise. In electrophoresis, theapplied electrical field causes charged macromolecules to migrate. The experi-mental observable is the drift velocity of the probe as one changes the molecularweight and concentration of the matrix, the size or shape of the probe, or thestrength of the external force. Historically, sedimentation and electrophoresis haveboth been viewed as methods for studying the properties of the migrating species.Chapter 2 shows that sedimentation studies also give information on the supportmedium, notably on how its ability to resist particle motion depends on the matrix

30

3.2 Basis of electrophoretic studies 31polymers concentration and molecular weight, and on the size of the sedimentingparticle.

We now come to electrophoresis as a probe of the dynamics of solutions of neutralpolymers. There is a huge literature on electrophoresis, including electrophoresis inpolymer solutions. That literature focuses on improving electrophoretic separationsfor practical applications. A few efforts have been made to use a particular model ofpolymer dynamics to illuminate particular experimental results(4), particularly forelectrophoresis in true gels, but the use of electrophoresis to understand polymerdynamics is little discussed. We thus have the happy circumstance that studies ofelectrophoretic motion through polymer solutions give an almost entirely freshphenomenological perspective on polymer dynamics.

3.2 Basis of electrophoretic studiesHow are electrophoretic motions described? Formally, one writes for the elec-trophoretic velocity v of a migrating species

v =E, (3.1)in which E is the applied electric field, and the electrophoretic mobility is definedby Eq. 3.1.

As initially sharp concentration boundaries migrate, they also become broader. Insedimentation, broadening is due entirely to the diffusion of the migrating species. Inelectrophoresis, band broadening arises from multiple sources, diffusion creatinga lower limit for broadening. This chapter concentrates on measurements of .Fundamental information might also lurk in well-characterized band broadeningrates if these were obtained.

From a theoretical standpoint, electrophoresis is more complicated than sedi-mentation because an electrophoretic probe is subject to a long list of significantforces. What are those forces? A probe particle having charge q and drag coefficientf is subject to a hydrodynamic drag force f v and electrical force qE. The hydro-dynamic drag is modified by hydrodynamic interactions with other macromoleculesin solution. Migrating molecules are also perturbed by mechanical collisions withmatrix chains(5, 6). Finally, E acts equally on every ion in solution. Most of thesolution is very nearly electroneutral and experiences almost no net force from theapplied field. Because there is Debye screening, the volume near the probe particleis selectively filled with the probes counterions. The counterions are subject to a netforce, which is transmitted via hydrodynamic and electrostatic interactions to theprobe. The ratio of the hydrodynamic and counterion drag forces is determined bythe molecular size R, small probes being slowed more by direct hydrodynamic dragand large probes being slowed more by their interactions with their ion cloud(7).

32 Electrophoresis

The boundary between small and large is determined by the dimensionless quantityR, being the Debye screening inverse length. Probes are also subject to electro-osmotic solvent flow, which arises because the walls of the system are chargedand have associated Debye clouds. The net electrical force on these clouds createssolvent flows elsewhere in the system. Penetration of matrix chains into Debyeclouds will perturb the clouds and thus the hydrodynamic flows and the forces onthe migrating probe.

The experimental literature on electrophoresis in polymer solutions invokestheoretical models, including the LangevinRondelez treatment(8), the Ogstonmodel(9), and reptation-type models(4). The Langevin and Rondelez form treatedin the last chapter is modified for electrophoresis by replacing s with and omittingthe 0/ term, leading to

=0 exp(acMR), (3.2)with scaling exponents , , and , and scaling prefactor a: R can be replacedwith the probe molecular weight P , though not to the same power. Equation 3.2reduces to

=0 exp(c). (3.3)In contrast to these stretched-exponential forms, a simple scaling relationship for would be

= cx. (3.4)Corresponding expressions for the dependence of on P are

=0 exp(P ), (3.5)and as a scaling relation

= pP y. (3.6)Probe size might actually be reported as probe radius R, probe molecular weightP , or number of base pairs (bp) or bases (b). The probe charge q enters directly theprefactors 0 and p. The observed dependence of /0 on E reflects nonlineardynamics.

The Ogston model treats the moving body as a sphere and the matrix as a setof interlaced rods(9). The retardation is identified as the probability that a probe ofgiven radius will fail to encounter a pore a gap between the rods large enoughto permit its passage. The rods are immobile. No analysis is made of whe