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Some of the topics covered include:

. diffusion of free particles on the basis of the Langevin equation

.the separation of time, length and angular scales;

. the fundamental Fokker-Planck and Smoluchowski equations derived for interacting particles

. friction of spheres and rods, and hydrodynamic interaction of spheres (including three body interactions)

. diffusion, sedimentation, critical phenomena and phase separation kinetics

. experimental light scattering results.

For universities and research departments in industry this textbook makes vital reading.

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- 1. STUDIES IN INTERFACE SCIENCE An Introduction to Dynamics of Colloids
- 2. STUDIES IN INTERFACE SCIENCE SERIES EDITORS D. M6bius and R. Miller 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
- 3. 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 Amsterdam- Boston- London- New York-Oxford- Paris San Diego - San Francisco- Singapore - Sydney- Tokyo
- 4. ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands 9 1996 Elsevier Science B.V. All rights reserved. This work is protected under copyright by Elsevier Science, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier's Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail:permissions@elsevier.com. You may also complete your request on-line via the Elsevier Science homepage (http://www.elsevier.com), by selecting 'Customer Support' and then 'Obtaining Permissions'. In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) (978) 7508400, fax: (+1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London WlP 0LP, UK; phone: (+44) 207 631 5555; fax: (+44) 207 631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of Elsevier Science is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronicallyany material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier's Science & Technology Rights Department, at the phone, fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. First edition 1996 Second impression 2003 Library of Congress Cataloging in Publication Dam _ _ _ Ohont, Jan K. O. An introduction to dynamics of colloids / Jan K.G. Dhont. p. cm. -- (Studies tn interface science ; vol. 2) Includes bibliographical references (p. ISBN 0-444-82009-4 (acid-Free paper) 1. Colloids. 2. Rolecular dynamics. Studies in interface science ; v. 2. OD549.D494 1996 530.4'2--dc20 - ) and index. I. Title. II. Series: 96-12846 CIP ISBN: 0 444 82009 4 The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). Printed in The Netherlands.
- 5. To my mother In memory of my father
- 6. This Page Intentionally Left Blank
- 7. PREFACE This book is a self-contained treatment of the fundamentals of a number of aspects of colloid physics. It is intended to bridge the gap that exists between more or less common knowledge to researchers in this field and existing textbooks for graduate students and beginning researchers. For many aspects of the theoretical foundation of modern colloid physics one has to resort to original research papers, which are not always easy to comprehend. This book is aimed to provide the theoretical background necessary to understand (most of) the new literature in the field of colloid physics. Needless to say that the topics treated in this book are biased by my own interests (this is especially true for the last two chapters). There are roughly two kinds of theoretical considerations to be distin- guished 9those 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 self-explanatory section "Further Reading and References", in which some literature is collected. It is virtually impossible, nor is it my vii
- 8. intention, to provide each chapter with a complete list of references. I must apologize to those not referred to, who contributed significantly to subjects treated in this book. I am grateful to my colleaques at the van 't Hoff laboratory for giving me the opportunity to write this book. Special thanks go to Arnout Imhof, Luis Liz-Marz~, Henk Verduin and Anieke Wierenga, who made a number of suggestions for improvement of most of the chapters. I am especially grateful to Gerhard N~igele (University of Konstanz), not only for his constructive criticism, but also for providing me with some additional exercises. Many of the weekends I could have spent together with my wife were used to work on this book. I would not have managed to finish this book without her continuous encouragement. Utrecht, 4 January 1996 Jan K.G. Dhont viii
- 9. CONTENTS CHAPTERS : 1 : INTRODUCTION 2 : BROWNIAN MOTION OF NON-INTERACTING PARTICLES 3 : LIGHT SCATTERING 4 : FUNDAMENTAL EQUATIONS OF MOTION 5 : HYDRODYNAMICS 6 : DIFFUSION 7 : SEDIMENTATION 8 : CRITICAL PHENOMENA 9 : PHASE SEPARATION KINETICS 1-68 69-106 107-170 171-226 227-314 315-442 443-494 495-558 559-634 CHAPTER 1 : INTRODUCTION 1-68 1.1 An Introduction to Colloidal Systems 1.1.1 Definition of Colloidal Systems 1.1.2 Model Colloidal Systems and Interactions 1.1.3 Properties of Colloidal Systems 1.2 Mathematical Preliminaries 1.2.1 Notation and some Definitions 1.2.2 Integral Theorems 1.2.3 The Delta Distribution 1.2.4 Fourier Transformation 1.2.5 The Residue Theorem The Cauchy-Riemann relations Integration in the complex plane Cauchy's theorem The residue theorem An application of the residue theorem and Fourier transformation 1.3 Statistical Mechanics 2 2 5 11 13 13 16 17 19 22 22 24 25 26 28 31 ix
- 10. 1.3.1 Probability Density Functions (pdf's) Conditional pdf's Reduced pdf's The pair-correlation function 1.3.2 Time dependent Correlation Functions 1.3.3 The Density Auto-Correlation Function 1.3.4 Gaussian Probability Density Functions Appendix Exercises Further Reading and References 31 33 35 37 40 43 46 49 51 64 CHAPTER 2 : BROWNIAN MOTION OF NON-INTERACTING PARTICLES 69-106 2.1 Introduction 2.2 The Langevin Equation 2.3 Time Scales 2.4 Chandrasekhar's Theorem 2.5 The pdf on the Diffusive Time Scale 2.6 The Langevin Equation on the Diffusive Time Scale 2.7 Diffusion in Simple Shear Flow 2.8 Rotational Brownian Motion 2.8.1 Newton's Equations of Motion 2.8.2 The Langevin Equation for a Long and Thin Rod 2.8.3 Translational Brownian Motion of a Rod 2.8.4 Orientational Correlations Exercises Further Reading and References 70 70 74 79 80 81 83 88 88 91 96 97 102 105 CHAPTER 3 : LIGHT SCATTERING 107-170 3.1 Introduction 3.2 A Heuristic Derivation 3.3 The Maxwell Equation Derivation 3.4 Relation to Density Fluctuations 108 109 113 122
- 11. 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 Fokker-Planck Equation 4.4 The Smoluchowski Equation 4.5 Diffusion of non-Interacting Particles 4.5.1 Linear Fokker-Planck Equations 4.5.2 Diffusion on the Brownian Time Scale 4.5.3 Diffusion on the Fokker-Planck Time Scale 4.6 The Smoluchowski Equation with Simple Shear Flow 4.6.1 Hydrodynamic Interaction in Shear Flow 4.6.2 The Smoluchowski Equation with Shear Flow 4.6.3 Diffusion of non-Interacting Particles in Shear Flow 171-226 172 177 179 183 186 187 189 191 195 196 197 199 xi
- 12. 4.7 The Smoluchowski Equation with Sedimentation 204 4.7.1 Hydrodynamic Interaction with Sedimentation 204 4.7.2 The Smoluchowski Equation with Sedimentation 206 4.8 The Smoluchowski Equation for Rigid Rods 4.8.1 Hydrodynamic Interaction of Rods 4.8.2 The Smoluchowski Equation for Rods 4.8.3 Diffusion of non-Interacting Rods Exercises Further Reading and References 208 209 212 218 220 225 CHAPTER 5 : HYDRODYNAMICS 227-314 5.1 Introduction 5.2 The Continuity Equation 5.3 The Navier-Stokes Equation 5.4 The Hydrodynamic Time Scale Shear Waves Sound Waves 5.5 The Creeping Flow Equations 5.6 The Oseen Matrix 5.7 Flow past a Sphere 5.7.1 Flow past a Uniformly Translating Sphere 5.7.2 Flow past a Uniformly Rotating Sphere 5.8 Leading Order Hydrodynamic Interaction 5.9 Faxen's Theorems 5.10 One step further : the Rodne-Prager Matrix 5.11 Rotational Relaxation of Spheres 5.12 The Method of Reflections 5.12.1 Calculation of Reflected Flow Fields 5.12.2 Definition of Mobility Functions 5.12.3 The First Order Iteration 5.12.4 Higher Order Reflections 5.12.5 Three Body Hydrodynamic Interaction 5.13 Hydrodynamic Interaction in Shear Flow 5.13.1 Flow past a Sphere in Shear Flow 5.13.2 Hydrodynamic Interaction of two Spheres in Shear Flow 228 229 231 234 235 237 238 241 244 245 248 250 253 255 257 258 262 266 267 268 273 276 277 278 xii
- 13. 5.14 Hydrodynamic Interaction in Sedimenting Suspensions 5.15 Friction of Long and Thin Rods 5.15.1 Translational Friction of a Rod 5.15.2 Rotational Friction of a Rod Appendix A Appendix B Appendix C Appendix D Appendix E Exercises Further Reading and References 281 282 285 286 288 294 295 296 300 302 311 CHAPTER 6 : DIFFUSION 315-442 6.1 Introduction 6.2 Collective Diffusion The zero wavevector limit Short-time and long-time collective diffusion Light scattering 6.3 Self Diffusion Short-time and long-time self diffusion 6.4 Diffusion in Stationary Shear Flow 6.5 Short-time Diffusion 6.5.1 Short-time Self Diffusion 6.5.2 Short-time Collective Diffusion 6.5.3 Concluding Remarks on Short-time Diffusion 6.6 Gradient Diffusion 6.7 Long-time Self Diffusion 6.7.1 The Effective Friction Coefficient 6.7.2 The Distorted PDF 6.7.3 Evaluation of the Long-time Self Diffusion Coefficient 6.8 Diffusion in Stationary Shear Flow 6.8.1 Asymptotic Solution of the Smoluchowski Equation The inner solution- K < v/Pe ~ 316 317 321 323 324 324 327 329 331 332 339 349 351 356 356 359 360 363 366 366 xiii
- 14. The outer solution" K > x/Pe ~ Match of the inner and outer solution and structure of the boundary layer An experiment 6.9 Memory Equations 6.9.1 Slow and Fast Variables 6.9.2 The Memory Equation 6.9.3 The Frequency Functions 6.9.4 An Alternative Expression for the Memory Functions 6.9.5 The Weak Coupling Approximation 6.9.6 Long-Time Tails 6.10 Diffusion of Rigid Rods 368 369 372 372 373 374 380 381 383 388 392 6.10.1 The Intensity Auto-Correlation Function (IACF) 392 The effect of translational and rotational coupling 6.10.2 Rotational Relaxation The equation of motion for P (fi1,t) Appendix A Appendix B Appendix C Appendix D Appendix E Exercises Further Reading and References 398 400 405 Evaluation of h(k, IAI1,62) and TI (1~!1,1~12) 407 Solution of the equation of motion for P(fi 1, t) 409 Mean field approximation fortheT-coefficients 410 Evaluation of the scattered intensity 412 415 416 418 420 421 424 437 CHAPTER 7 : SEDIMENTATION 443-494 7.1 Introduction Sedimentation at infinite dilution 7.2 Sedimentation Velocity of Interacting Spheres 7.2.1 Probability Density Functions (pdf's) for Sedimenting Suspensions 444 445 446 447 xiv
- 15. 7.2.2 The Sedimentation Velocity of Spheres 7.2.3 Sedimentation of Spheres with Hard-Core Interaction 7.2.4 Sedimentation of Spheres with very Long Ranged Repulsive Pair-Interactions 7.3 Non-uniform Baektlow The effective creeping flow equations Solution of the effective creeping flow equations 7.4 The Sedimentation-Diffusion Equilibrium 7.4.1 Barometric Height Distribution for Interacting Particles 7.4.2 Why does the Osmotic Pressure enter eq.(7.70)? 7.5 The Dynamics of Sediment Formation A simple numerical example of sediment formation The sedimentation velocity revisited Exercises Further Reading and References 450 457 459 461 462 465 468 469 472 473 476 479 481 490 CHAPTER 8 : CRITICAL PHENOMENA 495-558 8.1 Introduction 8.2 Long Ranged Interactions 8.2.1 The Ornstein-Zernike Approach Asymptotic solution of the Omstein-Zernike equation 8.2.2 Smoluchowski Equation Approach 8.2.3 A Static Light Scattering Experiment 8.3 The Ornstein-Zernike Static Structure Factor with Shear Flow Scaling Correlation lengths of the sheared system 8.4 The Temperature and Shear Rate Dependence of the Turbidity The definition and an expression for the turbidity 496 501 501 505 508 513 515 520 523 525 525 XV
- 16. A scaling relation for the turbidity 8.5 Collective Diffusion 8.6 Anomalous Behaviour of the Shear Viscosity 8.6.1 Microscopic expression for the Effective Shear Viscosity 8.6.2 Evaluation of the Effective Viscosity The contribution ~c~ The contribution q~ The contribution q~ The contribution ~ A scaling relation for the non-Newtonian shear viscosity Appendix A Appendix B Exercises Further Reading and References 527 530 535 536 538 539 541 541 543 545 548 549 550 555 CHAPTER 9 : PHASE SEPARATION KINETICS 559-634 9.1 Introduction 9.2 Initial Spinodai Decomposition Kinetics 9.2.1 The Cahn-Hilliard Theory 9.2.2 Smoluchowski Equation Approach 9.2.3 Some Final Remarks on Initial Decomposition Kinetics 561 567 567 572 577 The mechanism that renders a system unstable 579 9.3 Initial Spinodal Decomposition of Sheared Suspensions 580 9.4 Small Angle Light Scattering by Demixing Suspensions 586 9.5 Demixing Kinetics in the Intermediate Stage 590 9.5.1 Decomposition Kinetics without Hydrodynamic Interaction 591 Evaluation of the ensemble averages in terms of the static structure factor 594 Simplification of the equation of motion 596 Shift of kin(t) and k~(t) with time 597 xvi
- 17. 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
- 18. This Page Intentionally Left Blank
- 19. Chapter I INTRODUCTION
- 20. 2 Chapter 1. This introductory chapter consists of three sections. The first section in- troduces colloidal systems. The various common kinds of pair-interaction potentials of mean force are discussed. In further chapters the various pair- interaction potentials between the colloidal particles are modelled by simple expressions. The origin of these interactions is discussed in the present 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 physical-chemistry 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 self-contained manner. The third section is on basic notions from statistical mechanics and introduces the concept of probability density functions and time dependent correlation functions. Although this book is concerned with dynamical aspects, equilibrium probability density functions play an important role. For explicit evaluation of non-equilibrium and 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 9solutions 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".
- 21. 1.1. Colloidal Systems through a membrane, simply because the dissolved species is too large, that is, their linear dimension is larger than the pores of the membrane. These large particles are nowadays referred to as colloidal particles. Before Graham's publications, in 1827, the Botanist Robert Brown observed irregular motion of pollen grains in water, which grains happen to have a colloidal size. There has been a considerable disagreement about the origin of this irregular motion, which played an important role in the establishment of the molecular nature of matter. The irregular motion observed by Brown is referred to as Brownian motion, and is the result of random collisions of solvent molecules with the colloidal particles. The molecular nature of the solvent is thus observable through the irregular Brownian motion of colloidal particles. Although it was generally accepted around 1910 that molecules were more than the theorists invention, the experimental work of Jean Perrin (1910) definitely settled this issue. He confirmed the earlier theoretical predictions of Einstein (1906) and Langevin (1908), and verified that colloidal particles are nothing but "large molecules". Their irregular motion is then identified with thermal motion, common to all molecules, but only visible by light-microscopic techniques for colloidal particles. Graham's colloids are solutions of such large molecules exhibiting Brownian motion, so that colloidal particles are also referred to as Brownian particles. The interested reader is referred to the section Further Reading and References for detailed accounts on the history of colloid science. Colloidal systems are thus solutions of "large molecules". The large molecules are the colloidal or Brownian particles. These should be large compared to the solvent molecules, but still small enough to exhibit thermal motion (in the present context more commonly referred to as Brownian 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).
- 22. 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, so-called macromolecules, and (iii) they are assemblies of small molecules which are in thermodynamic equilibrium with their environment. Examples of the second kind of colloids are polymer solutions, solutions of large protein molecules, very long virusses (like fd-virus). Polymer solutions may behave as dispersions of the first kind, when the polymer chain in a poor
- 23. 1.1. Colloidal Systems solvent is shrunk to a rigid spherically shaped object. An example of colloids of the third kind are micro-emulsions, which mostly exist of droplets of water (or some apolar fluid) in an apolar fluid (or water) together with stabilizing surfactant molecules which are nested in the interface between the droplets and the solvent. The droplets consitute the colloidal particles which can exchange matter with each other. In this book the first kind of suspensions will be discussed. Furthermore, the discussion is limited to spherical, and to some extent, to rigid rod like Brownian particles. This may seem a severe restriction, and indeed it is, but these seemingly simple systems have a rich dynamic (and static) behaviour, about which many features are still poorly understood. The things that can be learned from these seemingly simple systems are a prerequisite to the study of more complicated colloidal systems of the second and third kind mentioned above. There are many industrial colloidal systems of the first kind which are extremely complicated due to the variety of colloidal particles that is present in the suspension, and due to the complicated interactions between the colloidal particles (for example as the result of an inhomogeneous charge density on the surfaces of the colloidal particles or their complicated anisometric geometry). In this book, relatively simple colloidal systems are treated, where the colloidal particles are mostly assumed identical and the interaction is modelled by simple functions. Again this is a severe restriction, but a quantitative treatment of most of the complicated industrial systems is as yet hardly feasible. The theories discussed in this book can be, and in some cases have been tested, using model dispersions which are chemically prepared specially for that purpose. The behaviour of industrial systems can often be understood on the basis of these model experiments and calculations, although on a qualitative level. 1.1.2 Model Colloidal Systems and Interactions There are many colloidal model systems consisting of metallic particles, such as gold, silver, copper, lead, mercury, iron and platinum particles. Examples of non-metallic colloidal systems are carbon, sulfur, selenium, tellurium and iodine particles. ~ There are many different methods to prepare these kinds of particles, including chemical, electrochemical and mechanical methods. 2Mostof these particleshavea radiuslargerthan 10 #m, whichis actuallybeyondthe maximumsizeof whatwe wouldclassifynowadaysas colloidal.
- 24. 6 Chapter 1. The two most widely used spherical model particles, in order to understand the microscopic basis of macroscopic phenomena, are latex and amorphous silica particles. Latex particles consist ofPMMA (poly-[methylmethacrylate]) chains. In water, which is a poor solvent for PMMA, these particles are compact rigid spheres, while in for example an apolar solvent like benzene, which is a good solvent for PMMA, the particle swells to a soft and deformable sphere. In the latter case the individual polymer PMMA chains must be chemically cross-linked (with for example ethylene glycol dimethacrylate) while otherwise the particles fall apart and one will end up with a solution of free polymers. The silica model particles consist of a rigid amorphous Si02 core. The solubility in particular solvents depends on the surface properties of these particles, which can be modified chemically in various ways. Different chemical modifications of the surface give rise to different kinds of interaction potentials between the colloidal particles. Two forces that are always present are the attractive van der Waa/s force and a repulsive hard-core interaction. The destabilizing attractive van der Waals force is of a relatively short range and can be masked by longer ranged repulsive forces due to charges on the surface of the particles, polymer chains grafted on the surface or a solvation layer (for example, silica particles in water are surrounded by a 3 nm thick structured water layer, which makes these particles relatively insensative to van der Waals attractions). The strength of these van der Waals forces is related to the refractive index difference between the particle cores and the solvent. The refractive index difference at the frequency of light is usually chosen small in order to be able to perform meaningful light scattering experiments. In most cases this minimizes the van der Waals forces.3 For large particles or for particles with a large refractive index difference with the solvent, van der Waals forces can lead to irreversible aggregation of the colloidal particles. The repulsive hard-core interaction is simply due to the enormous increase in energy when the cores of two colloidal particles overlap. This is an interaction potential that is zero for separations between the centers of the two spherical colloidal particles larger than twice their radius, and is virtually infinite for smaller separations. For spheres "with a soft core", such as swollen latex particles in a good solvent, the repulsive interaction increases more gradually with decreasing distance between the colloidal particles (compare figs. lb and c). aThe van der Waals force is actually related to a sum of the refractiveindex over all frequencies, so that minimizingthe refractive index at one particularfrequency does not necessarilyimplysmallvander Waalsforces.
- 25. 1.1. ColloidalSystems The surface of a colloidal particle may carry ionized chemical groups. The core material of the colloidal particles itself may carry such charged groups, or one can chemically attach charged polymers to the surface of the particles when it is favorable to use more apolar solvents (for example silica particles coated with TPM (3-methacryloxypropyltrimethoxysilane)). The charged surfaces of such colloidal particles repel each other. The pair- interaction potential of such charged colloidal particles is not a Coulomb repulsion (,,~ 1/r, with r the distance between the centers of the two spherical colloidal particles), but is screened to some extent by the free ions in the solvent. When the surface of a colloidal particle is negatively charged, free ions with a negative charge are expelled from the region around the particle while positive ions are attracted towards the particle. In this way a charge distribution is formed around the colloidal particle, the so-called double layer, which partly screenes the surface charges. The asymptotic form of the pair- interaction potential for large distances, where the potential energy is not too large, is a screened Coulomb potential, or equivalently, a Yukawa potential, ,~ exp{-~r}/r, where ~ measures the effectiveness of screening, that is, the extent of the double layer. Screening is more efficient (n is larger) for larger concentrations of free ions, and addition of salt can diminish the double layer repulsion such that van der Waals forces become active, which can lead to aggregation of the colloid. When the potential energy is large, the Yukawa form for the pair-interaction potential no longer holds, and is a more complicated function of the distance. The total potential, being the sum of the van der Waals energy and the interaction energy due to the charges on the surfaces, including the role of the free ions in solution, is commonly referred to as the DVLO-potential, where DVLO stands for Derjaguin-Verwey-Landau- Overbeek, the scientists who established the theory concerning these kind of interactions. For low concentrations of free ions in the solvent, and negligible van der Waals attractions, the DVLO pair-interaction potential is a long ranged repulsive interaction as sketched in fig. 1.1a. The surfaces of the colloidal particles may be coated with polymer chains, where the polymer chains are either chemically attached to the surface ("grafted polymers") or physically adsorbed. Examples are silica particles coated with stearylalcohol and latex particles coated with PHS (poly-[ 12-hydroxy stearic acid]). The length of these polymer chains is usually very small in comparison to the size of the core of the colloidal particles. When the solvent is a good solvent for the polymer, the polymer brushes on two colloidal particles are repulsive, since the polymer rather dissolves in solvent than in its own melt.
- 26. Chapter 1. |174 9 | 9 *-| o e_~o & _|174 | | .~. o.'1: "l ~ .'.;; 9O | 9 I r () ' | | Figure 1.1" The most common kinds of pair-interaction potentials for spherical colloidal particles: (a) the screened Coulomb potential, that is, the DVLO potential with negligible van der Waals attraction, (b) an almost ideal hard-core interaction, (c) steric repulsion of long polymers in a good solvent, grafted on the surface of the colloidal particles, ((t) short ranged attraction of polymers in a marginal solvent. These kind of interactions are referred to as steric repulsion. The interaction is then an almost ideal hard-core repulsion, as sketched in fig. lb. In practice such steric repulsions are often essential to screen the destabilizing van der Waals attractions. For very long polymers (such as poly-[isobutylene]), the range of the repulsive interaction is of course larger, and resembles that of swollen latex particles in a good solvent. This longer ranged repulsive potential is sketched in fig.l.lc. If on the other hand the solvent is a marginal solvent for the polymer, the energetically more favorable situation is overlap of two polymer brushes. This then results in a very short ranged attractive pair-interaction potential, superimposed onto a hard-core repulsion, as sketched in fig.l.ld. An example of such a system is a dispersion of silica particles coated with stearyl alcohol with benzene as solvent. The strength of the attraction may be increased by lowering the quality of the solvent for the polymer at hand, for example by changing the temperature, and may lead to phase separation. Attractive interactions of short range can also be induced by the addition of free polymer under theta-conditions (such as polystyrene in cyclohexane at 34.5~ The origin of this attraction is that free polymer is expelled from regions between nearby colloidal particles, for geometric and entropic
- 27. 1.1. Colloidal Systems _• ::~"~:~:~:.. ::i~i!~. :~i::!:;:;i!~i:.~:~ii~':i~: :~:~i~i~:,~:~i!!i!:~:;~i!:.~ ..iiii ~ ~!! ii~ ~ iii ~ i~~%i!:!~ii~~ii ~ i:~'..~~~:.:::~::~:'~..........:,::::~:~~.....................~::~ 5 ]- I-~~-i" ~ I ' '~,. , ..:::;;:_...............i;ii........~~i::~;;;.... 1O0 120 140 160 180 Diameter [nm] Figure 1.2: An electron micrograph of silica particles (a) and the histogram of the size distribution (b). The horizontal bar corresponds to 100 rim. reasons, leading to an uncompensated osmotic pressure that drives the colloidal particles towards each other. This so-ca!led depletion attraction is of a range that is comparable to the size of the polymers, and a strength that depends on the concentration of the polymers. These attractions can be strong enough to give rise to phase separation. The potentials described above may be treated on a quantitative level, where the sometimes complicated dependence of the pair-interaction potential on the distance between the colloidal particles is derived. On several occasions in this book we will use simple expressions for the pair-interaction potential. For example, for charged particles we will use a Yukawa potential and for particles coated with polymers in a marginal solvent we use a simple square well potential, the depth of which is considered as a variable parameter. We will not go into the derivation of precise formulas for pair-interaction potentials. The section Further Reading and References contains a list of some of the books that deal with these subjects in detail. The above mentioned model systems do have a certain degree of 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
- 28. 10 Chapter 1. of colloidal particles is mostly such that the relative spread in size decreases as the reaction proceeds, that is, as the average size of the particles increases. Typically the relative spread in size is about 5 - 10%. Model rigid rod like particles are much more difficult to prepare than the above mentioned spherical particles. Rigid rod like colloidal particles that are most frequently being used for experimentation up to now is TMV-virus (where TMV stands for Tobacco Mosaic Virus, which is a plant virus). These are charged hollow cylindrical particles with a length of 300 nm and a diameter of 18 n m. Another virus that is used is the so-called fd-virus, which is a very long and thin particle. This is not really a rigid rod, but has a considerable amount of flexibility. The advantage of these virus systems is that they are quite monodisperse. A considerable effort is needed to isolate larger amounts of these virusses and fresh samples must be prepared about every two weeks. Rod like particles of latex can be synthesized by stretching elastic sheets which contain deformable spherical latex inclusions. In this way almost identical charged rods with a well defined shape are obtained. The amount of colloidal material is however very small. Classical examples of inorganic colloidal rods are vanadiumpentoxide and iron(hydr)oxide colloids. Recently, rigid rod like particles with a core consisting of boehmite (A1OOH) have been synthesized. These particles can be coated with polymers, like the spherical silica particles mentioned above. The disadvantage here is the relatively large spread in size, and the, up to now, poorly understood interactions between the rods that play a role. Besides the potential interactions, which also exist in molecular systems, there are interactions which are special to colloidal systems. As a colloidal particle translates or rotates, it induces a fluid flow in the solvent which 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
- 29. 1.1. CoHoidalSystems 11 is concerned. 4 For colloidal systems one cannot simply speak of"interactions" without specifying the kind of interaction, direct or indirect, that is, potential interaction or hydrodynamic interaction. As a result of the large size difference between the Brownian particles and the solvent molecules (and free ions and possibly small polymers that may be present), the time scale on which the colloidal particles move is much larger than those for the solvent molecules. That is, during a time interval in which Brownian particles have hardly changed their positions, the solvent molecules are thermally displaced over distances many times their own size) This means that the fluid (free ions and polymers) are in instantaneous equilibrium in the field generated by the Brownian particles on a time scale that is relevant for the subsystem of Brownian particles. The pair-interaction potential for Brownian particles is, by definition, proportional to the reversible work needed to realize an infinitesimal displacement of one colloidal particle relative to a second colloidal particle. Due to the above mentioned separation in time scales, the solvent molecules (free ions, polymers) may be assumed in equilibrium with the field generated by the colloidal particles during their displacement. This reversible work is then equal to the change of the Helmholtz free energy of the total system of two Brownian particles and the solvent (free ions, polymers), and therefore consists of two parts : a part due to the change of the total internal energy of the system of two Brownian particles and the solvent, plus a change related to the change in entropy of the solvent (free ions, polymers). This free energy change, which is the relevant energy on the forementioned time scale, is usually referred to as the potential of mean force. The above discussed pair-interaction potentials for colloidal particles are such potentials of mean force. 1.1.3 Properties of Colloidal Systems Since colloidal particles are nothing but large molecules, exhibiting thermal motion, colloidal systems undergo phase transitions just as molecular systems do. For example, colloidal systems can crystallize spontaneously, where the Brownian particles reside on lattice sites around which they exert thermal 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 non-Interacting Particles.
- 30. 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 X-ray experiments. Investigations on the structure of colloidal fluids can be done by means of light scattering for the same reason 9structures extend over distances of the order of the wavelength of visible light. Besides crystallization, many other types of phase transitions in colloidal systems are observed that also occur for molecular systems. Fluid-gas phase separation (into a concentrated and dilute colloidal fluid) can occur in case of attractive interactions. Also, thermody- namically meta-stable 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, visco-elastic behaviour, and phase separation kinetics. All these phenomena are affected by interactions between the colloidal particles, both direct and indirect, that is, both energetically and hydrodyna- mically. The question then is how these phenomena can be described and how predictions can be made on the basis of a given pair-interaction potential and hydrodynamic interaction functions. This is roughly the question with which statistical mechanics is concerned. It is the aim of the present book to establish, in a self-contained manner, the statistical mechanical theory for dynamical phenomena of interacting colloids. Needless to say that a detailed treatment all the above mentioned topics is not feasible in a single book. I had to make a choice, which is to a large extent dictated by the aim to write an introductory text, and is of course
- 31. 1.2. MathematicalPreliminaries 13 also biased by my own interests. This book treats translational and rotational Brownian motion, sedimentation, light scattering, effects of shear flow, critical phenomena, and to some extent the kinetics of phase separation. 1.2 Mathematical Preliminaries The purpose of this mathematical section is to provide a concise treatment of subjects that may not have been part of mathematics courses of readers with a physical-chemistry background. Special attention is given to the residue 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 z-coordinate, r2 the y-coordinate and ra the z-coordinate" r - (ra, r2, ra) - (x, y, z). A vector may have more than just three entries. The number of entries is the dimension of the vector. The length of a vector a - (a a,..., aN) of dimension N is given by the Pythagorian formula ~/~jU 1 [aj [2, and is simply denoted by a non-boldfaced a or by [a 1. The length of the forementioned position vector,, is thus r - x/'x2 + y2 .q_z2. 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.
- 32. 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 all a12 a13 9 9 9 alN all a21 a31 9 9 9 aN1 a21 a22 a23 9 9 9 a2N a12 a22 a32 "" 9 aN2 a31 a32 a33 9 9 9 a3N -- a13 a23 a33 9 9 9 aN3 aN1 aN2 aN3 "'' aNN alN a2N a3N " 9 aNN .(1.1) A special matrix is the identity matrix or unit matrix I, which has elements 6ij - 1 for i - j, and 6ij - 0 for i ~ j. The 6ij is the so-called Kronecker delta. Thus, the elements of I on the diagonal, where i - j are equal to 1, while the off-diagonal elements, where i ~ j are all equal to 0. This matrix leaves vectors unchanged, that is, I. a = a for any vector a. Vectors can be multiplied with other vectors in several ways. Two vectors a and b can be multiplied to form a matrix of indexrank 2, which matrix is denoted as ab, and has per definition components (ab)ij - aibj. Such a product is referred to as a dyadic product. Similar products of more than two vectors are referred to as polyadic products. The so-called inner product N . a. b is defined as ~j=l ajbj, where * denotes complex conjugation, and is itself a scalar quantity (a real or complex number). The inner product of a vector with itself is nothing but its squared length. Two vectors are said to be perpendicular when their inner product vanishes. In case a and b are 3-dimensional vectors, the outer product a x b is defined as the vector perpendicular to both a and b, with a direction given by the cork screw rule, and a length equal to ab Isin{ 0 } [, with 0 the angle between a and b. The three components of this vector are a2b3 - aab2, aabl - alb3 and alb2 - a2bl. The usual multiplication of a vector a by a matrix M is denoted as M 9a, where the dot indicates summation with respect to adjacent indices. M 9a 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)ij N9 - ~n=l A~,~Bnj. Such summations over adjacent indices can be generalized to more than simply one index. For example, A 9B denotes the summation over two indices, indicated by the two dots, A 9B - N A,~mBm~. Such summations are generally referred to asErr,m--1 contractions. The number of indices with respect to which the contraction
- 33. 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 ~, 999 xN) denote a N-dimensional vector. Functions of the variables x~, 999 XN can be interpreted as being functions of the vector X. The most common examples are functions of the position vector X - r - (x, y, z) in 3-dimensional space. Functions of vectors which are real or complex valued are called scalar fields or simply scalar functions. Functions of vectors which are vectors or matrices are called vector tields. For example, f(X) - X is a scalar field, while F(X) - XX is a vector field. Vector fields are usually (but not always) denoted by a capital boldfaced letter. The gradient operatorV x is a differential vector operator defined as Vx - (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 Vxf(X) of a scalar field f is thus a vector field with entries Of(X)/Oxj. Similarly, the dyadic product VxF(X) is a matrix with the ijth-element equal to OFj(X)/Oxi. The divergenceof a vector field of indexrank 1 is a scalar field equal to the inner product of the gradient operator and the vector field" Vx F(X) - U9 ~j=l OFj(X)/Oxj. Analogous to a dyadic product of two vectors, the dyadic product Vx Vx 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)+ ~. Vxf(X) + -~AA'VxVxf(X) -1 +6~A&A'VxVxVxf(X) +.... (1.3) Contractions are defined as before for vectors and matrices, except that here differentiation is understood. For example, N a z 'v v.v f(x) - E l,n,m---1 0 3 Am AnAt OXtOXnOXmf(X).
- 34. 16 Chapter 1. A specially important operator is the Laplace operatorV~c, which is a short- hand notation for Vx 9Vx - ~Y=I 02/Ox~. In case X is the 3-dimensional position vector and F(X) is a 3-dimensional vector field with indexrank 1, the outer product V x F(X) is defined in analogy with the outer product of two vectors, where again differentiation is understood. We always use square brackets to indicate to which part in an expression the action of a differentiation is limited. For example, the action of the first gradient operator in the combination V xf(X) 9V 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 Vx. [f(X) Vxg(X)]. 1.2.2 Integral Theorems Two very important theorems are the integral theorems of Gauss and Stokes. Let W be some volume in the N-dimensional space NN. Gauss's integral theorem states that for continuous differentiable N-dimensional vector fields r(x), fw dX Vx. F(X) - ~w dS- F(X), (1.4) where the integral on the right hand-side ranges over the surface OW that encloses the volume W, and dS is the N-dimensional vector with a length equal to an infinitesimally small surface area on 014;, and with a direction perpendicular to that surface, pointing away from the volume. In eq.(1.4), dX is an abbreviation for dxl dx2"" dxN, an infinitesimally small volume element in NN. Stokes's theorem states that, again for continuous differentiable fields, fs dS. (V x F(r)) - ~s dl. F(r), (1.5) where S is a surface in ~3, OS its boundary, and dl is a vector with a length equal to an infinitesimal length segment on the curve OS and a direction that is related to the direction of dS by the cork screw rule. Volume and surface integrals are thus expressed in terms of integrals ranging over their boundaries.
- 35. 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 physical-chemist. Two further integral theorems, referred to as Green's integral theorems, are an almost immediate consequence of Gauss's integral theorem. The vector field F in Gauss's integral theorem (1.4) is now chosen as F(X) - f(X)Vxg(X), with f and g scalar functions. Using that, Vx. [f(X)Vxg(X)] - f(X)V~cg(X ) + [Vxf(X]. [Vxg(X)], immediately yields Green's first integral theorem, :wdX {f(X)V~cg(X)+ [Vxf(X)]. [Vxg(X)]} - ~owdS. f(X)Vxg(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)Vxg(X) - g(X)Vxf(X)} . These integral theorems play an important role in the various mathematical aspects of dynamics of colloids. 1.2.3 The Delta Distribution On several occasions we will make use of an "infinitely sharply peaked" scalar function with a normalized surface area. This function is zero everywhere except in one particular point x - x0 in ~ where it is infinite in such a way that its integral equals 1. Being zero everywhere except in one point seems in contradiction with the condition that its integral is non-zero. Indeed this is not a function in the usual sense but belongs to the class of so-called generalized functions, or equivalently, distributions. In this subsection we will not give the general definition of a distribution but rather specialize to the delta distribution, since this is the only distribution that is used in this book. Consider a sequence of scalar functions Cn(x), n - 1, 2,.--, with the properties, L: lim,~_..~ f-~oodx t~,(x) f(x) - 1 , for all n, } - f(zo), (1.8)
- 36. 18 Chapter 1. Xo X Figure 1.3: A sketch of a delta sequence together with a test-function f. The test-function is essentially equal to f (xo) in the range of x-values where ~n (x) for large n is non-zero. for any well behaved function f.6 Such a sequence of functions is referred to as a delta sequence, centered at xo. The probably simplest example of a delta sequence is, 1 1 - <x<zo+--Cn(x) n , for xo 2n 2n' = 0 , elsewhere. (1.9) The first condition in (1.8) is trivially satisfied. around z - xo yields, Taylor expansion of f(z) Flim 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 {~71)i [1 - (-1) ] U moon m--O where f(m) (Zo) is the ruth derivative of f(x) in x - zo. Only the term with m - 0 survives the limit where n ~ oc, so that also the second condition (1.8) is satisfied. Hence, the sequence (1.9) is a delta sequence. General- ly a delta sequence can be recognized by observing that the functions are increasingly sharply peaked around some x0. As sketched in fig. 1.3, for large 6The functions f for which this property is assumed to hold are referred to as test-functions, and are most commonly assumed to be infinitely continuous differentiable, with a compact support, meaning that they are zero everywhere except in a closed and bounded subset of !l~.
- 37. 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, " lim" 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~__.oor (z) does not exist in the usual sense. That is why in eq.(1.10) we used the notation" lim" : it means that integrals should be evaluated first for finite n's, after which the limit where n ~ co is taken. Such a limit is called a distributional limit. Two somewhat more complicated delta sequences are discussed in exercise 1.3. The particular sequence in exercise 1.3a plays an important role in the theory of Fourier transformation, while the sequence in 1.3b is important in relation to Brownian motion. The delta distribution 6(X - X0) in higher dimensions is simply defined as a product of the above defined 1-dimensional delta distributions, ~(X--Xo) -- ~(X 1 --XlO ) X "'" X t~(XN --XNO), (1.12) with Xo - (Xl 0, " " " , XN 0). Equation (1.11) immediately carries over to the N-dimensional case, / dX 6(X - Xo)f(X) - f(Xo), (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)
- 38. 20 Chapter 1. The so-called wavenumber or wavevectorr k is equal to 27r/A, with A the wavelength of the particular sinusoidal contribution. The functions f~(k) and f~(k) are the sine and cosine Fourier transforms of f(x), respectively. These functions measure the contribution of the particular sine and cosine contributions to f(x). If for example f~(ko) is relatively large for a particular wavevector k - ko, the function f(x) has much the character of cos{ kox}. The above decomposition in sine and cosine functions can be written more compactly as, 1 f_" dk f(k) exp{ikx} (1 15) f(x)- 2---~ o~ " " Contrary to the sine and cosine transforms f, and f~, the so-called Fourier transform f(k) of f(z) is a complex valued function,s Using that the complex exponential exp{ikx} is equal to cos{kx} + i sin{kx}, it is easily seen that the two above formulas (1.14,15) are equivalent, with f~(k) - [f(k)+ f(-k)l/2r 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) exp{-ikx}. (1.16) O0 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 _Fourierinversion. The above decomposition into sinusoidally varying functions can be ge- neralized to functions of N-dimensional vectors X - (x~, x2,..., XN). First decompose the xi-dependence of f(X) in sinusoidal functions as discussed above, with x = xi and k = kl, 1 dkx f(k~,x2,x3,., xN)exp{ikxxl}.f (zi, z2, z3, . . . , zN) - 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.
- 39. 1.2. Mathematical Preliminaries 21 Regard the right hand-side now as a function of z2,..., XN, and decompose the x2-dependence, with z = x2 and k = k2, to obtain, f(xl,x2, x3,''.,XN) J_ i?_-- 1 dkl dk2 f(kl k2, x3,"" XN)exp{i [klXl -[- k2x2]} . (27r)2 ~ ~ ' , This procedure is repeated N times, leading to, 1 / /(X) - (27r)N dkf(k) exp{ik. X}, (1.17) with the wavevector k equal to (kl, k~,..., kN). The integral in understood to range over the entire N-dimensional k-space NN. Successively applying eq.(1.16) N times yields the Fourier transform in terms of the function itself, f(k) - f dX/(X)exp{-ik. 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 9F(X), fdX [Vx. F(X)] exp{-ik. X} = ]" dX Vx" IF(X)exp{-/k. X}] + ik. f dX F(X)exp{-ik. X} - fdS.F(X) exp{-ik.X} + ik.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 ), If dS. F(X)exp{-ik. X} i_< lids. F(x)exp{-ik.X}! <_ maxlxl=R IF(X) I dS - 2 ir(X) l,maxlxl=RJ
- 40. 22 Chapter 1. the surface integral is zero when the product of the maximum value of IF(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, so-called complex functions. The complex variable is denoted by z = x + iy, 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) + iv(z). (1.19) Both real valued functions u and v may be regarded as functions of x and y, and we can also write, f(z) - u(x, y) + iv(x, y). (1.20) For example, in case f (z) - z 2, we have u(x, y) - x 2- y2 and v(x, y) - 2xy. The complex number z may be visualized as the point (x, y) in ~2, which 2- dimensional space is in the present context referred to as the complex plane. The Cauchy-Riemann relations The derivative of a complex function is defined as for real functions of a real variable as, f'(z) - lira f(z + h)- f(z) (1.21) h..-.O h '
- 41. 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 x-axis, then the complex number z + h can be written as z + A = x + A + iy, with A a real number tending to zero as h goes to zero. Alternatively, z can be approached along a line parallel to the y-axis, in which case z + h - z + iA - 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, limf(X+A+iY)-f(x+iy) = limf(X+i(Y+A))-f(x+iy) A-.~o A a~o iA The left hand-side is equal to df(z)/dx, while the right hand-side is equal to df(z)/d(iy). 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 au(x,y) av(x,y) Ox Oy " (1.22) These are the Cauchy-Riemann relations, which are conditions under which the limit in eq.(1.21) taken in the two forementioned directions are equal. These relations thus provide necessary conditions for differentiability, that is, when the above relations are found invalid for a given function f at some point z, then that function is not differentiable at that point. An example of a function that is not differentiable can be found in exercise 1.6. The converse can also be proved, provided that the derivatives in the Cauchy-Riemann relations are continuous. That is, when the Cauchy-Riemann Figure 1.4: A point z in the complex plane can be approached from different directions. r~ , ' X "r
- 42. 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 7i'u of the closed curve. relations hold and all partial derivatives are continuous, then f is continuous differentiable. For the derivation of the residue theorem we do not need this converse statement and we therefore do not go into its proof. Integration in the complex plane Functions f(z) can be integrated over curves 7 in the complex plane. These integrals can be defined in terms of integrals that one is used to in 1 dimension. In case of curves 7 as depicted in fig.l.5a, which defines y as a function of x, the integral f.~ ranging over the curve 7 is simply defined as (with y'(x) - dy(x)/dx), f~ dz f (z) - f (dx + idy) [u(x, y)+ iv(x, y)] rl - (1 + + iv( , (1.23) The points a and b mark the smallest and largest value of x on 7, as indicated in fig. 1.5a. In this example, y is regarded as a function of x. This is not possible for the curve sketched in fig. 1.5b. The flick is now to write 7 as a sum of, in this example, two curves ~1 and 72, which separately allow to regard y as a function of x. The integral f.y is now the sum of f.y~ and f-n, each of which
- 43. 1.2. MathematicalPreliminaries 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 anti-clockwise direction. Consider fields F(r) of the form (-u(x, y), v(x, y), 0), with u and v continuous differentiable functions. Since in this case, V x F(r) - (0, 0, Ov/Ox + Ou/Oy), Stokes's integral theorem reduces to, fs dx dy { Ov(x, y) Ox + ~o,~{-dx u(x, y) + dy v(x, y)} . Replacing v by u and u by -v gives, {Ou(z,y)fs dx dy -O-x Ov(x'Y)} - ~o {dx v(x y) + dy u(x y)} . by S ~ When u and v satisfy the Cauchy-Riemann relations (1.22), the left hand-sides dS • Figure 1.6: The special choice of the surfaceS in Stokes's integral theorem.
- 44. 26 Chapter 1. of the two above equations vanish. Hence, o - g~ {d~ ~(~, y) - dy v(~, ~)}, 0 - ~s {dxv(x,y) + dyu(x,y)} . Identifying the (x, y)-plane with the complex plane, OS with a closed curve 7, and writing f - u + iv, it follows that, ~ dz f(z) - ~(d~ + i~y) (~(~, y) + iv(~, y)) {d~ ~(~, y) - d~ v(~, y)} + i ~ {d~ v(~, y) + dy u(~, y)} - O. We thus found that when f(z) is a continuous differentiable function, that, ~ dz f (z) - O. (1.24) This is Cauchy's theorem. Continuous differentiability of f is not required throughout the complex plane, but only within a set that contains 7 together with its interior .),int. 9 Cauchy's theorem can be used to show that a continuous differentiable complex function is infinitely continuous differentiable, meaning that all higher order derivatives are continuous. Such functions are also called analytic functions. We do not go into the proof of this statement. The residue theorem We are now in the position to derive the residue theorem. Only the simplest version of this theorem is used in this book, where only so-called first-order poles are encountered. The discussion of the residue theorem is therefore limited here to that simplest form. Consider the following integral over a closed contour 7, ~ dz a(z) Z -- ZO with g continuous differentiable (or equivalently, analytic) within a set that contains 7 and its interior 7 int. When z0 is not in ,,lint nor on 7, the integrand 9The interior of 7 is the entire region that is enclosed by 7 (the dashed area fig. 1.5c).
- 45. 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 dock-wise orientation. f(z) - g(z)/(z - zo) satisfies the conditions that go with Cauchy's theorem. Hence, ~ dz g(z) = 0 , for zo r T U Tint . (1.25) Z -- Zo When zo is in 7 int, however, the integrand is generally infinite at z - zo and therefore certainly not analytic. We can, however, exclude the point z0 from the interior of an extended contour, by deformation of the integration countour as depicted in fig.l.7. At some arbitrary point on 7, the contour is extended towards zo, by a curve 71 say, a circle C~(zo) with an arbitrary small radius e encloses z0, and the curve 0'2 closes the contour again. The superscript c on CC(zo) is used to indicate that the circle is traversed in clockwise direction. When no such superscript is added, an anti-clock-wise orientation is understood. For example, "7is traversed anti-clock-wise, while 7 ~ is the same curve traversed in clockwise direction. The extended contour ,.[ext __,.[ _~_71 "~- 72 "~- CCe(zo) does not contain the point z0 in its interior nor is Zo on ,,/ext, SOthat 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
- 46. 28 Chapter 1. direction. It thus follows that, d z g (z )_ _ d z .g(z )_ . Z- Zo ,(zo) Z- Zo (1.26) Notice that C, (zo) here is the circle traversed in anti-clock-wise direction. The integral on the right hand-side is evaluated by noting that points on the circle can be written as z - zo + e exp{iqo}, with 0 < ~ < 27r. Since e is arbitrarily small, we have, fodz 9(z) = lim ieexp{i~o}d~ 9(zo +e exp{i~}) = 27rig(zo) ,(~o) z- Zo ~lo ~ ~ 9 e exp{iq;} = dz 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(zo)- 27ri g(z) Z -- Z 0 , for all zo E 7i" (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, V2~(r)- x2~(r)- Q---6(r), (1.28)
- 47. 1.2. Mathematical Preliminaries 29 with a a constant with dimension m -1, c the dielectric constant of the solvent, and 5(r) the 3-dimensional delta distribution centered at r - O. The delta distribution describes the presence of the small charged particle, carrying a total charge Q, which is assumed to be located at the origin. As we have seen in the subsection on Fourier transformation, V is replaced by ik upon Fourier transformation (see also exercise 1.5). This implies that V 2 - V 9V 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 - exp{-ik 9r} ). Fourier inversion thus leads to, Q exp{ik, r} (I)(r) - (27r)3c fdk ~2 ~ xi . (1.29) We now transform the integration to spherical coordinates (k, O, q;). Jacobian of this standard transformation is k2sin{ 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} fo2~ fo~9 - dT dO sin{O} exp{ik, r}. A little thought shows that this integral is independent of the direction of the position vector r. We can therefore choose that vector along the z-direction, so that k. r - kr cos{O }. Hence, with x - cos{ O }, /~ld 2rr [exp{ikr} - exp{-ikr}] .dl~ exp{ikl~, r} - 27r x exp{ikrx}- ~r We thus obtain, (I)(r) - 87r2eirQ f_~~ dk ---7-k2 k tr2 [exp{ ikr } - exp{-ikr }] , (1.30)
- 48. 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 fo dk by 1 oo ~_~ dk. Consider the integrals, f_,o k1+ - oodk k2 + tc2 exp{4-ikr}. (1.31) The potential is then equal to 9 - 8~, i~[I+ - I_]. We are going to evaluate both integrals I+ with the help of the residue theorem. The first step is to transform the integrals into integrals ranging over a closed contour in the complex plane. This can be done by interpreting the integration over (-oo, +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 ~ dzf(z) exp{+izr} - 0 , R----~oo R4- when lim max~eca, If(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 {-Rr 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{-izr} 1- exp {-Rr sin{~o} }, and again the integrand tends to zero exponentially fast as R --o c~. We can thus
- 49. 1.3. Statistical Mechanics 31 add integrals over the semi circles at infinity since they are zero. The result is an integral over a closed contour in the complex plane, allowing for an application of the residue theorem to evaluate that integral. Such a procedure is called closing of the integration contour in the upper (or lower) complex half plane. Let "7+ denote the closed contour in the upper (+) or lower (-) complex half plane, as depicted in fig. 1.8a and 1.8b respectively (note that 7+ is anti-clockwise, but that "7- is clockwise). The integrals in eq.(1.31) are thus equal to, I• - ~+ dz Z (z + ix)(z -i~) exp{+izr} - 4-7riexp{-xr}, 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{-xr} (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.
- 50. 32 Chapter 1. Imagine a collection of macroscopically identical systems (for example, colloidal suspensions). Thermodynamic variables for each system are the same, but of course microscopically each of the systems is generally in a different state, that is, the position coordinates and momenta of the particles in each system at a certain instant in time are generally different. Such a collection of macroscopically identical systems is referred to as an ensemble. The phase space for spherical particles is defined as the 6N-dimensional space spanned by the position coordinates rl, 999rN and momenta Pl, 999 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," 99 rN, Pl," 9",PN) is now defined as, P(X,t)dX 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) 10For the determinationofthe momentaone shouldactuallymaketwophotographs.
- 51. 1.3. Statistical Mechanics 33 The brackets < --- > are nothing but a short-hand notation for the integral on the right hand-side. This average is the ensemble average of f. Alternatively one may introduce the pdf P (f, t) for a stochastic variable f instead of X, by rewriting the above equation as, < f >- /dfP(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 fdf 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 time t (1.39) and in (Xo, Xo + dXo) at time 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
- 52. 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 aparticular 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 lXo,to)dX the probability that positions and momenta are in (X, X + dX) at time t, given (1.41) that their values were Xo at time 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 lfo, to)df the probability that the phase function is in (f , f + df ) at time t, (1.42) given that its value was fo at time 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)
- 53. 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 >f0- /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(rl,... ,rN) (1.45) P(r3,..-, rN [rl, r2) -- P2(rl, 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> = fdrl...fdrNP(rx,...,rN, t)f(rx,r2) = f dr1 f dr2 P2(rx, r2, t)f(rx, r2), (1.46)
- 54. 36 Chapter 1. where, P2(rl, r2, t) - f dr3 . . . f drNP(rl , . . . , rN, t) . (1.47) P2 is referred to as the reduced pdf of order 2, the two-particle pdf or simply as the second order pdf. This equation can be regarded as a special case of P(f, t) - fd#P(f, t, #, to), with to - t, f - (rx, r2) and g - (r3,..., rN). Similarly, ensemble averages of phase functions ofjust one position coordinate are averages with respect the first order reduced pdf, P~(rl,t)-f dr2...f drNP(rl, ,rN, t). (1.48) Higher order reduced pdf's (such as Pa(r~, r2, ra, t) ) are similarly defined. The probability of finding a particle at some position r at time t is pro- portional to the macroscopic number density p(r, t), which is the average number of particles per unit volume at r and at time t. Normalization sets the proportionality constant, 1 Pl(r, t) - ~ p(r, t). (1.49) A similar relation for P2 will be discussed later, when the pair-correlation function is introduced. When the system is in thermal equilibrium, the time independent pdf for the position coordinates is proportional to the Boltzmann exponential of the total potential energy r 999 rN) of the assembly of N particles, P(rl,.-., rN) -- exp {--fl(X)(rl,--., rN)} Q(N,T,V) , (1.50) 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 .../drN 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
- 55. 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 non-equilibrium and dynamical phenomena we will not go into the various approximate methods to calculate these equilibrium pdf's, but merely mention some of their properties together with definitions of related functions. The pair-correlation function When particles do not interact with each other, all reduced pdf's are products of Pl'S. In particular, P2(ra, r~, t) - P1(rl, t) P1(r2, t). Interactions can formally be accounted for by an additional factor g(ra, r2, t), the so-called pair-correlation function, 1 P2(rl, r2, t) - P, (r,, t) e, (r2, t) g(ra, r2, t) - ~-sp(r,, t) p(r2, t) g(rl, r2, t). (1.52) Similarly, the three-particle correlation function g3 "corrects" for the effect of interactions for the third order pdf P3, P3(rl, r2, r3, t) - /91(rl, t) Pl(r2, t) Pl(r3, t) g3(rl, r2, r3, t) (1.53) 1 - N3 p(rl, t) p(r2, t) p(r3, t) g3(rl, r2, r3, t). For large distances [r~ -r2 I between two particles, the pair-correlation function attains its value without interactions, which is 1 by definition. The three-particle correlation function becomes equal to i when all three particles are well separated. In case of homogeneous and isotropic fluids in equilibrium, the pair- correlation function is a function of r -I r~ - r21 only, and can be expanded in a power series of the number density fi - N/V as, g(r) -- go(r)+ /~gl(r) + ~2g2(r) +'". (1.54) The leading term go describes interactions between two particles without the intervening effects of other particles. This then is nothing but the pair- correlation function for a system containing just two particles. It is the relevant pair-correlation function for systems which are so dilute that events where three or more particles interact simultaneously are unlikely. According to eqs.(1.49-51), with/91 - 1/V, we thus obtain, go(r -[ ra - r2 I) - V2 exp{-/3V(r)} f drx f dr2 exp(-~V(r)} '
- 56. 38 Chapter 1. where V(r) is the potential energy of an assembly of just two particles, the pair-interaction potential. Now noting that, f dr1 f dr2 exp{-/3V(r)} - f dr2 f d(ra- r2)exp{-flV(r)} = V (fdr[exp{-flV(r)}-1] + V} ~ V2 , since the integral in the last equation is of the order R~,, with Rv the range of the pair-interaction potential, it is found that, go(r) -- exp{-flV(r)}. (1.55) In this book we will use the phrase "on the pair-level ", whenever interactions between three or more (colloidal) particles simultaneously are disregarded. Hence, eq.(1.55) is the pair-correlation function on the pair-level, and can be used to calculate ensemble averages for dilute systems. In general, the pair-correlation function does include "higher order interactions", that is, it includes the intervening effects of the remaining particles on the interaction between two given particles. A systematic approach where the expression (1.50), after substitution into the definition (1.47) for P2, is expanded in terms of Mayer-functions, leads to, f gl (r) --I rl -- r2 [) - exp{-/3V(I ra - r21)}/dr3 f(I rx - r3 [) f([ r2 - r3 I), i I J (1.56) where f(r) is the Mayer-function f(r) - exp{-/3V(r)} - 1. The derivation of this result can be found in most standard texts on statistical mechanics, a few of which are collected in the section Further Reading and References at the end of this chapter. In exercise 1.12, 9x is calculated explicitly for hard-sphere interactions, with the result (the subscript "hs" stands for "hard-spheres"), ghs(r) -- go(r)-~-/~gl(r) - 1, for r>4a, [ ]= l+qo 8- 3- + for re[2a 4a),a ~ ~ = O, for r<2a, (1.57) where a is the radius of the hard-core and r ~aa/~ is the fraction of the total volume that is occupied by the cores of the particles, the so-called volume fraction. This pair-correlation function is plotted in fig.1.lOa for ~ - O.1. At
- 57. 1.3. Statistical Mechanics 39 1 0.5 ~ _ iI- I-J- 2 4 2 4 21+ Figure 1.10: The pair-correlation function to first order in concentration for hard-spheres (see eq.(1.57)) with qo- O.1, (a), a sketch for hard-spheres at larger concentra- tions (b), and for charges spheres with a long ranged repulsive pair-interaction potential (c). larger concentrations, the pair-correlation function develops a large contact value (defined as the value of g at r = 2a + e with e arbitrary small), and peaks appear at larger distances, as depicted in fig. 1.lOb. The pair-correlation function behaves quite differently in case of long ranged and strongly repulsive interacting particles, as depicted in fig. 1.10c. This may be the case for charged colloidal particles in de-ionized solvents. First of all, the contact value of g is zero : the probability that two particles touch is zero due to their strong repulsive interaction. Secondly, the peak position shifts to smaller distances for higher concentrations. This is due to the tendency of the particles to remain far apart from each other so as to minimize their (free) energy. The peak position varies approximately as 1/fi 1/3 for such systems. Consider a colloidal particle at the origin. One may ask about the average density around that particle, which density is a function of the distance from the particle due to interactions. This density is N PI, as in eq.(1.49), with the additional condition that there is a particle in the origin. According to eq.(1.43) (with t = to, X0 = 0 = the position of the particle at the origin and X - r) this conditional probability is equal to P2(r, r' - O, t)/P~ (r' - O, t). Hence, from the definition (1.52) of the pair-correlation function, Number density at r with a particle at the origin - N P2(r'r'-O't) r' /91(r'-O,t) = p(r,t) g(r, -O,t). (1.58)
- 58. 40 Chapter 1. Well away from the origin, where interaction with the particle at the origin is lost so that g(r, r' - 0, t) - 1, this is simply the macroscopic density p(r, t), as it should. The peaks in the figures 1.10b,c thus imply enhanced concentrations around a given particle at those distances. For hard-core interactions there is also an enhanced concentration close to contact. This enhancement is due to depletion : particles are expelled from the gap between two nearby particles leaving an uncompensated repulsive force from particles outside the gap that drives the two particles together. Each colloidal particle, charged or uncharged, is thus surrounded by a "cage" of other particles. The "effective interaction potential" veff(r) can be defined for isotropic and homogeneous systems in equilibrium as, g(r) - exp{-flV ~ff(r)} . (1.59) According to eq.(1.55) this effective potential is equal to the pair-interaction potential on the pair-level. The average force F~ff(r) between two particles for arbitrary concentrations can be shown to be equal to -VV ~ff(r) (see exercise 1.11), and includes the effects of intervening particles. Hence, by definition, Feff(r) _ /~-1~7 ln{g(r)} -- ~-1~. dln{g}(r)/dr, so that there is an attraction for those distances where dg(r)~dr < 0. For hard-spheres near contact there is thus attraction, the depletion mechanism for which was already explained above. Around the peak in the pair-correlation function the effective force changes from strongly repulsive to attractive. Multi particle interactions may thus lead to attractions even if the pair-interaction potential is purely repulsive. 1.3.2 Time dependent Correlation Functions Consider the conditional ensemble average, < g >So - fdgP(g, tlfo, to)g. (1.60) This ensemble average is a time dependent function, also for systems in equilibrium. It describes the average evolution of the phase function #, given that at time to < t the value of the phase function f was fo. When this conditional average is subsequently averaged with respect to fo, the result is simply the unconditional ensemble average < g > 9since P(g, t If o, to) - P(g, t, fo, to)/P(fo, to) we have, << g >/o > -- f dfo P(fo, to) f dg P(g, t Ifo, to) g

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