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  • Hiromi YamakawaModern Theoryof Polymer Solutions

  • Harpers Chemistry SeriesUnder the Editorship of Stuart Alan Rice

  • Modern Theoryof Polymer Solutions

    Professor Emeritus Hiromi Yamakawa

    Department of Polymer ChemistryKyoto UniversityKyoto 606-8501, Japan

    Electronic Edition

    Laboratory of Polymer Molecular ScienceDepartment of Polymer ChemistryKyoto UniversityKyoto 606-8501, Japan

  • Preface tothe Electronic Edition

    It has been just thirty years since this volume, Modern Theory ofPolymer Solutions, was published by Harper & Row, Publishers. It isnow out of print but is still in some demand. Furthermore, it is alsoan introduction to the authors new book, Helical Wormlike Chainsin Polymer Solutions, published by Springer-Verlag in 1997, althoughsome parts of it are now too old and classical and have only the sig-nificance of historical survey. Under these circumstances, the authorapproved of the preparation of this electronic edition without revisionat the Laboratory of Polymer Molecular Science, Department of Poly-mer Chemistry, Kyoto University. On that occasion, however, he at-tempted his efforts to correct errors as much as possible. Finally, it isa pleasure to thank Prof. T. Yoshizaki, Mrs. E. Hayashi, and his othercollaborators for preparing this electronic edition.

    Hiromi YamakawaKyotoJuly 2001

  • Modern Theory of Polymer Solutions

    Copyright c 1971 by Hiromi YamakawaPrinted in the United States of America. All rights reserved. No part of this bookmay be used or reproduced in any manner whatever without written permissionexcept in the case of brief quotaions embodied in critical articles and reviews.For information address Harper & Row, Publishers, Inc., 49 East 33rd Street,New York, N. Y. 10016.

    Standard Book Number: 06-047309-6

    Library of Congress Catalog Card Number: 71-141173

  • Contents

    Chapter I. Introduction 11. Survey of the Field 12. Scope and Introductory Remarks 3References 4

    Chapter II. Statistics of Ideal Polymer Chains: Random-FlightProblems 5

    3. Introduction 54. The Markoff Method for the General Problem of Random

    Flights 85. Distribution of the End-to-End Distance and Related

    Quantities 105a. Exact Expression for the Bond Probability 115b. Approximate Expression for the Bond Probability 16

    6. The Wang-Uhlenbeck Method for Multivariate GaussianDistributions 18

    7. Distribution of a Segment About the Center of Mass andRelated Quantities 217a. Distribution of a Segment About the Center of Mass 217b. Radius of Gyration 237c. Radii of Gyration with R Fixed 25

    8. Distribution of the Radius of Gyration 268a. Distribution of the Quasi-radius of Gyration 268b. Distribution of the Radius of Gyration 28

    9. Remarks 359a. Short-Range Interferences and Unperturbed Molecular

    Dimensions 35

  • vi CONTENTS

    9b. Branched and Ring Polymers 479c. Wormlike Chain Model for Stiff Chains 52

    Appendix II A. Method of Steepest Descents 57Appendix II B. Orthogonal Transformations 59Appendix II C. Distribution of the Quasi-radius of Gyration

    with S Fixed 62References 65

    Chapter III. Statistics of Real Polymer Chains:Excluded-Volume Effect 69

    10. Introduction 6911. The Flory Theory 7112. The Direction of Developments Following the Flory Theory 75

    12a. Ideal-Chain Type 7512b. Production-Chain Type 7812c. Real-Chain Type 79

    13. Perturbation Theory (A): Distribution Function Method 8114. Perturbation Theory (B): Cluster Expansion Method 8715. Approximate Closed Expressions 96

    15a. Approximate Expressions Derived from the Potential ofMean Force with R or S Fixed 97

    15b. The Differential-Equation Approach 10416. Asymptotic Solution at Large z 11317. Remarks 121

    17a. Branched and Ring Polymers 12117b. Numerical Calculations on Lattice Chains 12217c. General Comments 130

    Appendix III A.The Distribution Function, Markoff Process,and Diffusion Equation 131

    Appendix III B.The Probability Densities for Segment Contacts 133Appendix III C.Perturbation Theory for a Two-Dimensional

    Chain 133References 134

    Chapter IV. Thermodynamic Properties of Dilute Solutions 13718. Introduction 13719. The McMillanMayer General Theory of Solutions 13920. The Second Virial Coefficient (A):Random-Flight Chains 149

    20a. Perturbation Theory 14920b. Approximate Closed Expressions 157

    21. The Second Virial Coefficient (B):Real Polymer Chains withIntramolecular Interactions 16821a. Perturbation Theory 16921b. Approximate Treatments 171

    22. The Third Virial Coefficient 17422a. Perturbation Theory 17422b. Approximate Closed Expressions 176

    23. Remarks 179

  • Contents vii

    23a. Heterogeneous Polymers 17923b. Branched and Ring Polymers 18123c. General Comments 182

    Appendix IV A.The Second Virial Coefficient for RigidMacromolecules 184

    Appendix IV B.The Third Virial Coefficient for RigidSphere Molecules 187

    References 188

    Chapter V. Light Scattering from Dilute Solutions 19124. Introduction 19125. Scattering by Independent Small Isotropic Particles 19326. Fluctuation Theory 198

    26a. General Theory 19826b. Heterogeneous Polymers 20426c. Mixed-Solvent Systems 206

    27. Distribution Function Theory 21127a. General Theory 21227b. Intramolecular Interferences and Angular Dissymmetries 21627c. Intermolecular Interferences 22027d. Heterogeneous Polymers 22227e. Mixed-Solvent Systems 226

    28. Remarks and Topics 23128a. Effects of the Optical Anisotropies 23128b. Copolymers 23628c. Critical Opalescence 24128d. Some Other Topics 246

    Appendix V A. The Electromagnetic Field Due to anOscillating Electric Dipole 248

    Appendix V B. Angular Distributions for Rigid Sphere andRod Molecules 250

    Appendix V C. The Space-Time Correlation Function 253References 254

    Chapter VI. Frictional and Dynamical Properties ofDilute Solutions 257

    29. Introduction 25730. Some Fundamentals 258

    30a. The Viscosity Coefficient 25930b. The Friction Coefficient 26530c. Brownian Motion 267

    31. The Hydrodynamic Interaction:The KirkwoodRiseman Theory 26931a. Intrinsic Viscosities 27031b. Translational and Rotatory Friction Coefficients 275

    32. The Diffusion-Equation Approach (A): The Kirkwood GeneralTheory 278

    33. The Diffusion-Equation Approach (B): The Spring and BeadModel 285

  • viii CONTENTS

    34. The Nonaveraged Oseen Tensor and the Viscosity Constant 0 29635. Excluded-Volume Effects 304

    35a. Intrinsic Viscosities 30535b. Friction Coefficients 315

    36. Remarks and Some Other Topics 31636a. Concentration Dependence 31636b. Non-Newtonian Viscosities 32036c. Branched and Ring Polymers 32236d. Rigid Rods and Stiff Chains 33036e. Some Other Problems 343

    Appendix VI A.The Equation of Motion for Viscous Fluids 353Appendix VI B.The Oseen Hydrodynamic Interaction Tensor 355Appendix VI C.The Intrinsic Viscosity and Friction Coefficient

    of Rigid Sphere Macromolecules 357References 359

    Chapter VII. Comparison with Experiment 36537. Introduction 36538. Determination of Molecular Weights,Molecular Dimensions,

    and Second Virial Coefficients 36639. Determination of Unperturbed Molecular Dimensions 37140. Correlations Between the Expansion Factor and the Second

    Virial Coefficient 37941. Correlations Between the Expansion Factor and the Intrinsic

    Viscosity 38642. The Two Molecular Parameters 393

    42a. The Conformation Factor 39442b. The Binary Cluster Integral 397

    References 399

    Chapter VIII. Concluding Remarks 403

    Author Index 409

    Subject Index 418

  • Foreword

    One of the rewards of academic life is the opportunity to meet and workwith talented individuals from all over the world. In 1961 Hiromi Yamakawacame to the University of Chicago to work in my laboratory. We had afruitful collaboration and I learned much from him. I have followed hissubsequent work with interest; there are, for me, few pleasures that cancompare with the witnessing of the intellectual evolution and continuingcontributions of former colleagues.

    It is in this spirit that I welcome the writing of this book by ProfessorHiromi Yamakawa. He has made many contributions to the theory ofpolymer solutions, and writes from the balanced point of view of researchworker and teacher. I believe this book complements those dealing withpolymer solutions already published. No other text available so consistentlyincludes the effect of excluded volume on the properties of dilute polymersolutions, and no other so fully develops the distribution function theoryapproach. For these reasons I recommend the text to research workers andstudents. Although the presentation is concise, and continuous effort isrequired to extract all the information implicit in the theory, the reward forsuch concentration is large.

    Stuart A. Rice

  • Preface

    It is well known that statistical mechanics provides a tool for thedescription of the relationship between the macroscopic behavior of sub-stances and their atomic and/or molecular properties. Clearly, the sameprinciples apply to polymer science as to the study of small molecules.However, polymeric systems are too complicated to treat rigorously onthe basis of molecular mechanics, because polymer molecules have anexceedingly great number of internal degrees of freedom, and therebyalso very complicated intramolecular and intermolecular interactions.Thus, it is only for dilute solutions that a molecular theory of poly-mers can be developed in the spirit of, for instance, the equilibriumand nonequilibrium statistical mechanical theory of simple fluids. Infact, the physical processes which occur in dilute polymer solutionscan be described in terms of only a few parameters using the random-fligh