Yamakawa Modern Theory of Polymer Solution 0060473096

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-flight model. The major purpose of this book is to give a systematicdescription of the advances made during the past two decades in thedistribution-function theory of random-flight models for dilute polymersolutions; this is indicated directly by the title of the book.

The modern theory of polymer solutions has depended on the ad-vances made since the 1940s in the statistical mechanical theory ofsystems of simple molecules. The random-flight model, which was firstinvestigated by Lord Rayleigh, can now be treated very convenientlyby the methods of Markoff and of Wang and Uhlenbeck. Specifically,the latter method has facilitated several advances in the theory of theexcluded-volume effect. Except in an ideal state, now called the thetastate, all equilibrium and nonequilibrium properties of dilute polymersolutions are influenced by the excluded-volume effect. A large part of

xii PREFACE

this book is concerned with the excluded-volume effect.The most elementary treatment of the interactions between chain

units which leads to the excluded-volume effect is an application of thestatistical mechanics of many-particle systems, e.g., the cluster theoryof Mayer and McMillan and also the equilibrium theory of liquids. Thenonequilibrium properties of the solution are treated by an applicationof the theory of Brownian motion, or in other words, on the basis ofgeneralized diffusion equations of the FokkerPlanck type. The moderntheory of polymer solutions may be classified from the point of view ofmethodology into these two main subdivisions. Another classification ofthe several problems is given in Chapter I. We shall consistently empha-size the dependence of dilute-solution properties on polymer molecularweight, as this is an important experimental observable.

The writing of this book was suggested by Professor S. A. Rice in1961 when the author visited the Institute for the Study of Metals, nowthe James Franck Institute, of the University of Chicago. Drafts of thefirst few chapters were written during 19611963, but the final draftwas not completed then. As in other active fields of science, a largebody of new results has accumulated so rapidly that the original designof the book had to be modified. It now seems appropriate to providea coherent and comprehensive description of the theory of polymersolutions. This does not mean that the theory has been completelyestablished. Rather the author hopes that the appearance of this bookat this time will stimulate new developments in the theory of polymersolutions.

This book is not for the beginner, but rather for graduate studentsand research workers. Much of the research presented has been devel-oped by specialists, and probably has not been read in original form bymany research workers and nonspecialists. Thus, the book is intended,on the one hand, to provide an understanding of the modern theory ofpolymer solutions for these nonspecialists and, on the other hand, tofacilitate the research work of specialists in the field and also of physi-cists and chemists who wish to enter the field. The derivations of mostmathematical equations are given in sufficient detail to elucidate thebasic physical ideas and the theoretical methods. Although as manyreferences as possible are cited, accidental omissions will occur and arethe authors responsibility.

The text is written in the technical terms used in polymer scienceand statistical mechanics, with only a few exceptions. For instance, theterm configuration is used instead of the term conformation, widelyemployed by polymer chemists at the present time. We choose thisnotation because this book does not deal with stereochemistry, and theterm configuration integral or configurational partition function isordinarily used in the statistical mechanics of many-particle systems.

There remains now only the pleasant task of thanking all those whohave rendered assistance to the author. The author is indebted to Pro-fessor S. A. Rice for his suggestion of this project, constant interest andencouragement for many years, and critical reading of the manuscript

Preface xiii

with corrections of the English in it. Thanks are also tendered to Pro-fessor W. H. Stockmayer for his reading of the manuscript and hislecture at Kyoto University in 1966, which influenced Chapter VI; andto Professor B. H. Zimm for his valuable criticisms of Chapter I and II.

It is a great pleasure to thank Professor Emeritus I. Sakurada for hisguidance in polymer chemistry and his general interest and encourage-ment for many years. The author has carried out much of the researchreported with Professor M. Kurata, and benefited from numerous dis-cussions with him. His constant interest and encouragement over theyears must also be acknowledged. Professor H. Fujita provided valu-able criticisms of the manuscript and ideas in Chapter VII, for whichthe author wishes to thank him. Needless to say, the writing of thisbook was made possible by the many papers published in many scien-tific journals. The author is grateful to these journals and the authorsconcerned.

Finally, it is a pleasure to acknowledge the assistance of Dr. G.Tanaka who read the manuscript and prepared the figures and theindices, and of Miss S. Sugiura who prepared the typescript and theindices.

Hiromi Yamakawa

Modern Theoryof Polymer Solutions

Chapter One

Introduction

1. Survey of the Field

At the very outset of the study of the properties of polymer solutions,in the 1920s, pioneering studies of solution viscosities were carried outby Staudinger. It was his intent to provide evidence supporting thehypothesis that polymers are composed of simple low-molecular-weightcompounds connected linearly by covalent bonds. Indeed, the principalobjective of early investigations of polymer solutions was to establishmethods of molecular weight determination or molecular characteri-zation, and theories of dilute polymer solutions which were developedsubsequently are related to this problem in many cases. Emphasis is fo-cused on solution properties because polymer molecules cannot exist ina gaseous state and the characterization of a single polymer molecule istherefore possible only in solutions so dilute that the polymer moleculesare well separated from one another.

The first theoretical investigations of the properties of polymersoccurred in the 1930s. The foremost of these advances is a statis-tical treatment of the configurational description of a polymer chaindeveloped independently by Kuhn and by Guth and Mark. These in-vestigators arrived at the significant conclusion that the mean-squareend-to-end distance of the chain is proportional to the number of ele-ments constituting it. This deduction provides the foundation for thepresently accepted random-flight model in the theory of dilute polymersolutions. Further application of statistical mechanics to the study ofthe properties of polymer solutions was delayed until the early 1940s.The new developments evolved in two directions. First, there were a

2 INTRODUCTION

number of direct applications of the theory of random-flight statisticsto dilute solution properties. The notable results are represented bythe theory of light scattering (Debye and Zimm), the theory of intrin-sic viscosities (Debye, Kirkwood, and others), and the theory of virialcoefficients (Zimm) based on the first general molecular theory of solu-tions presented by McMillan and Mayer in 1945. The other directionis represented by the lattice theory of polymer solutions, which was de-veloped first by Flory and independently by Huggins in 1942 in orderto explain the very large deviations from ideality exhibited by polymersolutions. The FloryHuggins theory became a standard starting pointin the statistical thermodynamics of concentrated polymer solutions.In 1949 and 1950, Flory introduced two new important concepts, nowcalled the excluded-volume effect and the theta state or theta point.Both effects arise from the fact that two elements, possibly remote fromeach other in sequence along the chain, interact with each other. Forexample, two elements cannot occupy the same point in space at thesame time, thereby generating an excluded volume. The theta stateis defined as a sort of ideal state, in the sense that in that state thevolume effect apparently vanishes and the chain behaves like an idealrandom-flight chain. Thus, by the early 1950s, there had been estab-lished the basic physical description of the polymer solution along withthe first step in the development of a theory of dilute polymer solutions.The advances made during the last two decades have delineated the re-lationship between the theory of polymer solutions and other branchesof the molecular sciences.

Now, the molecular weight of a given polymeric compound may varyalmost continuously from small to very large values, whereas a givenlow-molecular-weight compound possesses a definite molecular weightcharacteristic of that compound. In general, the properties of dilutepolymer solutions, or more generally polymeric systems, are dependenton their molecular weight. This is an important aspect characteristicof polymeric systems, which is never observed in systems consistingof low-molecular-weight compounds. Thus the molecular weight is animportant variable which may in fact be regarded as continuous. Thisis the reason why a single polymer molecule may be considered a sys-tem of the statistical-mechanical ensemble; it is the most fundamentalsystem in the development of the theory. It should be noted that inthe theory of dilute polymer solutions the solvent is usually treated asa continuous medium, and not on the molecular level. The principaltool is, of course, classical equilibrium statistical mechanics; nonequi-librium statistical mechanics has not yet been applied to the study ofthe frictional properties of dilute polymer solutions.

The main part of the theory of polymer solutions consists of a groupof theories, now called the two-parameter theory. Within the frame-work of the two-parameter theory, the properties of dilute polymer solu-tions, such as average molecular dimensions, second virial coefficients,and intrinsic viscosities, may be expressed in terms of two basic pa-rameters; one is the mean-square end-to-end distance R20 of a chain

SEC. 2. Scope and Introductory Remarks 3

in the theta state, and the other is the excluded-volume parameter,which is usually designated by z. The parameter z is proportional tothe effective excluded volume for a pair of chain elements at infinitedilution and also to the square root of the number of elements in thechain. The excluded volume, and hence the parameter z, vanish at thetheta point. This is, indeed, the definition of the theta point or thetastate. The central problem in the theory is to derive interrelations be-tween the dilute-solution properties and the parameters R20 and/orz, in particular for linear flexible chains. Thus this group of prob-lems is ultimately concerned with the exploration of the dependence ofdilute-solution properties on molecular weight.

There is another important group of problems. In these attention isfocused primarily on relationships between the chain structure and thedilute-solution properties, especially the average molecular dimensionsin the theta state. The chain structure is considered on the atomic levelor on the subchain level. The description of the conformational statis-tics of polymer chains belongs to the first case, in which the local chem-ical structure of a chain is considered in detail; that is, restrictions onthe angles between successive bonds in the chain and steric hindrancesto internal rotation about the bonds are explicitly taken into accountin a calculation of the quantity R20 itself. Intramolecular interfer-ences of this sort are of short-range nature, while the excluded-volumeeffect is of long-range nature. In fact, there have been a number ofsignificant advances in the theory of conformational statistics as wellas in the two-parameter theory. For the second case, in which the chainstructure on the subchain level is considered, attention is focused onthe differences between the dilute-solution properties of linear flexiblechains and chains of other types, such as branched and ring polymersand stiff chains.

In the first group of problems, many-body problems are often en-countered, and also the self-consistency of theories is an importantfactor to be discussed. Thus, these have the nature of purely phys-ical problems. On the other hand, the second group of problems isof importance for molecular characterization. It should be noted thatboth groups of problems have been studied in order to emphasize theirinterrelations.

2. Scope and Introductory Remarks

Many of the chapters of this book will be devoted to the description ofthe two-parameter theory for linear flexible chain polymers. The reasonfor this is that the two-parameter theory for linear flexible chains isgood enough to provide an understanding of the basic physical processesin dilute polymer solutions and of the theoretical procedures. Studiesof the properties of branched and ring polymers and stiff chains willbe found in the sections of chapters entitled Remarks or Remarks andTopics. The analysis of the effects of short-range interferences in a chain

4 INTRODUCTION

will be omitted, although not completely, since it must be discussed inthe study of conformational statistics. Thus, the basic molecular modelused in this book is the random-flight model or its modifications.

Chapter II presents in detail the mathematical foundation of thestatistics of ideal random-flight chains, and is also an introduction tothe later chapters. In Chapter III there is a detailed description of thetheory of the excluded-volume effect, including both the perturbationtheory and the other approximate treatments. The present status ofthe theory is, of course, discussed in detail. No complete solution hasas yet been obtained, because the problem is similar to the many-bodyproblem in the theory of simple liquids. Chapter III is, indeed, the coreof this book. Chapter IV deals with the theory of virial coefficients onthe basis of the McMillanMayer general theory of solutions. In par-ticular, the theory of the second virial coefficient is described in detail.Chapter V covers the ordinary theory of light scattering (involving thefluctuation theory), the distribution function theory, and some othertopics. Chapter VI deals with the theory of transport properties, suchas intrinsic viscosities and friction coefficients. Although the dynami-cal properties are also discussed, the theory of viscosities is worked outmainly for the case of steady shear rate. The theory of viscosity is verydifficult, and no available representation of the viscosity seems to besatisfactory. In Chapter VII, a comparison of the two-parameter theorywith experiment is made. In doing this, a fundamental difficulty arisesfrom the fact that the parameter z is not directly observable. Thisproblem is discussed in detail.

The reader who wishes to learn some of the elementary conceptsof polymer chemistry, conformational statistics, or the fundamentals ofstatistical mechanics is referred to some of the books listed below.

References

1. P. J. Flory, Principles of Polymer Chemistry, Cornell UniversityPress, Ithaca, New York, 1953.

2. H. Tompa, Polymer Solutions, Butterworths Scientific Publica-tions, London, 1956.

3. C. Tanford, Physical Chemistry of Macromolecules, John Wiley& Sons, New York, 1961.

4. M. V. Volkenstein, Configurational Statistics of Polymeric Chains,Interscience Publishers, New York, 1963.

5. T. M. Birshtein and O. B. Ptitsyn, Conformations of Macro-molecules, Interscience Publishers, New York, 1966.

6. P. J. Flory, Statistical Mechanics of Chain Molecules, John Wiley& Sons, New York, 1969.

7. T. L. Hill, Introduction to Statistical Thermodynamics, Addison-Wesley Publishing Company, Reading, Massachusetts, 1960.

8. T. L. Hill, Statistical Mechanics, McGraw-Hill Book Company,New York, 1956.

Chapter Two

Statistics of IdealPolymer Chains:Random-Flight Problems

3. Introduction

The theory of dilute polymer solutions begins with a formulation ofthe distribution for a single linear flexible polymer chain in an infinitesolvent medium. This simplified system may be realized at infinite di-lution where all intermolecular interactions between solute polymersmay be neglected. Now, there exist many degrees of internal-rotationalfreedom about the single bonds in the chain in addition to the irregulartranslational and rotational motions of the entire molecule due to ther-mal Brownian motion. As a consequence of this and the great numberof elements constituting the chain, an almost limitless number of chainconfigurations may be realized by the polymer molecule. The instanta-neous configuration of the entire chain can be specified by all internaland external coordinates of the molecule. The Cartesian coordinates ofthe centers of elements in the chain may be chosen as such coordinates.We denote the coordinates of element j by Rj , assuming that the chainis composed of n+1 elements joined successively and the elements arenumbered 0, 1, 2, . . ., n from one end to the other. In this book, aportion of the chain belonging to the representative point Rj will beoften referred to as the segment instead of the element. In the case of apolymethylene chain, for example, Rj represents the coordinates of thejth carbon atom and the methylene group is considered a segment.

It is evident that the statistical properties of an isolated chain areindependent of its external coordinates. Accordingly, suppose that seg-

In later chapters, by the term segment we do not necessarily mean that onlyone carbon atom is contained in a segment of the backbone chain.

6 STATISTICS OF IDEAL POLYMER CHAINS: RANDOM-FLIGHT PROBLEMS

ment 0 is fixed at the origin of the Cartesian coordinate system, asdepicted in Fig. II. 1. The configurational partition function Z for thesystem under consideration may then be written in the form,

Z =

exp

[U

({Rn})kT

]d{Rn} , (3.1)

where {Rn} is a shorthand notation for a set of coordinatesR1, R2, . . .,Rn, and U is the potential energy of the system and may be expressedas

U({Rn}) = n

j=1

uj(Rj1,Rj) +W({Rn}) . (3.2)

The potential uj takes account formally of the fact that the j 1thand jth segments (or carbon atoms) are connected through a valencebond, and therefore the potential W includes interactions of all othertypes such as bond angle restrictions and steric hindrances to inter-nal rotation, and also long-range interferences between segments. InEq. (3.1), it should be understood that integration over coordinates ofthe solvent molecules has already been performed, and therefore thatW plays the role of the potential of mean force. The instantaneousdistribution P

({Rn}) for the entire chain is given byP({Rn}) = Z1 exp[U({Rn})

kT

]. (3.3)

It is often convenient to use, instead of the set of Rj , coordinatesrj called the bond vector and defined by

rj = Rj Rj1 . (3.4)Further, we define a distribution function j(rj) by

j(rj) = exp[uj(rj)

kT

](3.5)

with uj(rj) = uj(Rj1,Rj). The zero of the potential uj is chosen insuch a way that j is normalized to unity:

j(rj)drj = 1 . (3.6)

The function j is referred to as the bond probability since jdrj rep-resents the probability that the (vector) length of the jth bond liesbetween rj and rj + drj . Note that by definition uj and hence j arespherically symmetric, i.e., functions of rj = |rj | only. Equation (3.1)may then be rewritten in the form,

Z = n

j=1

j(rj)

exp(WkT

)d{rn} . (3.7)

SEC. 3. Introduction 7

Fig. II.1. Configurations of a polymer chain with the 0th segment fixed at the

origin of a coordinate system.

The distribution function P (R) of the end-to-end distance R ( Rn)can be obtained by integrating P

({Rn}) over {rn} under the restric-tion,

nj=1

rj = R . (3.8)

That is,

P (R) = Z1 n

j=1

j(rj)

exp(WkT

)d{rn}dR

, (3.9)

which is normalized as P (R)dR = 1 . (3.10)

In this chapter, we shall deal with the ideal case in which W = 0.With this assumption, Eq. (3.9) becomes

P (R) = n

j=1

j(rj)

d{rn}dR

(3.11)

with Z = 1. Thus the problem of evaluating P (R) of Eq. (3.11) isequivalent to that of random flights or the Brownian motion of a free


particle, which may be stated as follows: A particle undergoes a se-quence of displacements r1, r2, . . ., rn, the magnitude and directionof each displacement being independent of all the preceding ones. Theprobability that the displacement rj lies between rj and rj + drj isgoverned by a distribution function j(rj) assigned a priori. We ask:What is the probability P (R)dR that after n displacements the posi-tion R(= r1+r2+ +rn) of the particle lies between R and R+dR.For this reason, a chain whose distribution can be determined in termsof only j s is called a random-flight chain. The statistical propertiesof random-flight chains may be completely analyzed by the use of theMarkoff method (Section 4) and the WangUhlenbeck method (Sec-tion 6).

4. The Markoff Method forthe General Problem of Random Flights

Historically the problem of random flights was formulated first byPearson1 in terms of the wanderings of a drunkard, and the solution inthree dimensions was obtained for small and very large values of n, thenumber of steps, by Rayleigh.2 In its most general form the problem wasformulated by Markoff3 and subsequently by Chandrasekhar.4 In thissection, we consider a slight generalization of the problem presentedin the last section and the method for obtaining its general solution,which can readily be applied to other problems besides that of findingP (R) in later sections. However, this formulation is less general thanthat of Chandrasekhar.

Consider n, three-dimensional vectors,

j = (jx, jy, jz) (j = 1, . . . , n) , (4.1)

where the components are assumed functions of three coordinates in aCartesian coordinate system:

js = js(xj , yj , zj) (s = x, y, z) . (4.2)

Further, the probability that the xj , yj , zj occur in the range,

xj xj + dxj ; yj yj + dyj ; zj zj + dzj ,is assumed to be given a priori by

j(xj , yj , zj)dxjdyjdzj j(rj)drj , (4.3)j satisfying the normalization condition of (3.6). That is, j is inde-pendent of all the preceding vectors rk with k < j, and the processunder consideration is just a simple Markoff process. Now the problemis to find the distribution function P () of the resultant of n originalvectors,

=nj=1

j , (4.4)

SEC. 4. The Markoff Method for the General Problem of Random Flights 9

that is, the probability P ()d that lies between and + d.Obviously the probability that j lies between j and j+dj is equalto the probability that rj lies between rj and rj+drj . In our notation,P () may therefore be written as

P () = n

j=1

j(rj)

d{rn}d

, (4.5)

where d{rn} = dr1 . . . drn, and the integration goes over all magnitudesand directions of all rj at constant with the condition of (4.4).

The restriction on the integration of (4.5) can be removed by in-troducing Dirichlets cutoff integral or a Fourier representation of athree-dimensional Dirac delta function as follows,

P () =(r)

nj=1

j(rj)drj

, (4.6)where

r = nj=1

j , (4.7)

(r) = (2pi)3

exp(ir )d . (4.8)

Equation (4.6) with (4.7) and (4.8) can be rewritten in the form,

P () = (2pi)3K() exp(i )d (4.9)

with

K() =nj=1

j(rj) exp(i j)drj . (4.10)

From Eq. (4.9) and the Fourier inversion formula, K() is seen to be thethree-dimensional Fourier transform of the distribution function P ():

K() =P () exp(i )d . (4.11)

Thus, the advantage of the method introduced is that P () can beobtained by an inverse transformation of K() which can be evaluatedfrom Eq. (4.10) without any restriction on the range of integration. Ingeneral, the Fourier transform, K, of a distribution function is calledthe characteristic function of that distribution function.

We now examine the properties of the characteristic function. Ex-panding the exponential in Eq. (4.11), we have

K() =k=0

1k!

(i )kP ()d . (4.12)


If is the component of in the direction of , Eq. (4.12) becomes

K() =k=0

1k! k (i)k (4.13)

with k =

k P ()d . (4.14)

In particular, if P () is spherically symmetric, the integral of (4.14)can be evaluated by using polar coordinates with the z axis in thedirection of as follows,

k =1

k + 1k for k = 2p ,

= 0 for k = 2p+ 1 , (4.15)

where

2p =

2pP ()d

= 0

2pP ()4pi2d . (4.16)

Equation (4.13) may then be rewritten in the form,

K() =p=0

(1)p(2p+ 1)!

2p2p . (4.17)

Thus the expansion coefficients of the characteristic function yield themoments, 2p, of the distribution function (assuming spherical sym-metry). In turn, when all the moments are given the distribution func-tion can be determined completely.

5. Distribution of the End-to-EndDistance and Related Quantities

It is evident that when we put = R and j = rj Eqs. (4.9) and(4.10) become the basic equations from which the distribution functionof the end-to-end distance of a polymer chain can be evaluated. Forsimplicity, we consider a polymer chain, an example of which is a vinylpolymer, whose bond probabilities j(rj) are identical for all j, andomit the subscript j. The basic equations may then be written in theforms,

P (R) = (2pi)3K() exp(iR )d , (5.1)

K() =[

(r) exp(i r)dr]n

. (5.2)

To carry out the calculation the form of must be specified.Most of the problems in this book will be discussed on the basis of this model.

SEC. 5. Distribution of the End-to-End Distance and Related Quantities 11

5a. Exact Expression for the Bond Probability

We may assume that each bond has a constant length a, neglecting thedeviation of the length of a valence bond from its mean value a dueto atomic vibrations in the spine of the chain. The constancy of bondlength may be taken into account by expressing the bond probabilityin terms of a delta function;

(r) =1

4pia2(|r| a) . (5.3)

By using polar coordinates with the z axis in the direction of , theintegral in Eq. (5.2) can then be easily evaluated to give

K() =[sin(a)a

]n. (5.4)

Substitution of Eq. (5.4) into Eq. (5.1) leads to

P (R) =1

2pi2R

0

sin(R)[sin(a)a

]nd , (5.5)

where use has been made again of polar coordinates. Note that P (R) isspherically symmetric. In general, if the bond probability is sphericallysymmetric, so are both the characteristic and distribution functions,provided j and rj are in the same direction for all j.

We first evaluate the moments R2p of the distribution of (5.5)from Eq. (5.4) by the method of cumulants. The logarithm of thecharacteristic function of (4.13) may be expanded in the form,5, 6

lnK() = ln

[ k=0

kk!(i)k

]

=k=1

kk!(i)k (5.6)

with k R k the kth moment. The coefficient k is called the kthcumulant (semiinvariant). The kth cumulant can be explicitly repre-sented in terms of only the moments j with j k, and vice versa.Without proof, we use the result,7

k = k!m

kj=1

1mj !

(jj!

)mj, (5.7)

where

m means the summation over all sets ofm1,m2, . . . compatiblewith

kj=1

jmj = k .

Sometimes the moment generating functionM() K(/i), the Laplace trans-form of the distribution function, is used instead of the characteristic function.


Now the logarithm of Eq. (5.4) can be expanded as

lnK() = nl=1

Bl(2a)2l

(2l)!2l2l (5.8)

with Bl the Bernoulli numbers (Bl = 1/6, B2 = 1/30, . . .). A compar-ison of Eq. (5.8) with (5.6) leads to

k = (1)l1(2l)1Bln(2a)2l for k = 2l ,= 0 for k = 2l + 1 . (5.9)

Substituting Eq. (5.9) into Eq. (5.7) and recalling that R2p =(2p+ 1)2p, we obtain

R2p = (1)p(2p+ 1)!m

pl=1

1ml!

[Bln(2a)

2l

(2l)!2l

]ml(5.10)

withpl=1

lml = p .

In particular, when p = 1, Eq. (5.10) yields the mean-square end-to-end distance R2 or root-mean-square end-to-end distance R21/2as a measure of the average size of the random-flightchain,

R2 = na2 , (5.11)

orR21/2 = n1/2a . (5.12)

It is important to observe that the mean-square end-to-end distance isproportional to the number of bonds or segments in the chain. Thischaracteristic will be referred to as the Markoff nature of a chain.

We now proceed to evaluate the integral in Eq. (5.5). This will bedone for three cases of interest.

5a(i). Exact Solution

The exact solution was obtained by Treloar,8 by Wang and Guth,9 byNagai,10 and by Hsiung et al.,11 the mathematical techniques beingdifferent from one another. The evaluation is made conveniently usingthe theory of functions.

Equation (5.5) may be rewritten in the form,

P (R) = i4pi2a2R

+

exp(iR

a

)(sin

)nd (5.13)

with = a . (5.14)


Using the development,

sinn =1

(2i)n

np=0

(1)p(np

)exp(in 2ip) , (5.15)

Eq. (5.13) may be further rewritten in the form,

P (R) = 12n+2in1pi2a2R

np=0

(1)p(np

) +

ei(n2p+R/a)

n1d .

(5.16)We now consider the integral,

I =

eib

( + i)n1d (5.17)

with > 0. When b > 0, the contour of integration consists of a largehalf circle of radius r in the upper complex plane and the real axisbetween r and r; when b 0, a large half circle of radius r in thelower complex plane and the real axis between r and r. When b > 0,application of Cauchys integral formula leads to +

eib

n1d = lim

r0

I = 0 (b > 0) . (5.18)

On the other hand, when b 0, application of Goursats theorem leadsto +

eib

n1d = lim

r0

I

= lim0

2pii(n 2)!

[dn2

dn2eib]=i

= 2piin1

(n 2)!bn2 (b 0) . (5.19)

Substituting Eqs. (5.18) and (5.19) with b = n2p+R/a into Eq. (5.16)and putting n p = k, we obtain the result,

P (R) =1

2n+1(n 2)!pia2Rk(nR/a)/2

k=0

(1)k(n

k

)(n 2k R

a

)n2.

(5.20)Although Wang and Guth9 have also used the theory of functions toarrive at Eq. (5.20), their procedure is incorrect.

The series displayed in Eq. (5.20) contains, as special cases, theexpressions derived by Rayleigh2 and Chandrasekhar4 for the first fewvalues of n. Since this series is not useful for practical computationsin our problems (with n large), we introduce some approximations toobtain a closed expression.


5a(ii). Case for n 1The asymptotic solution for large n can be obtained by the methodof steepest descents9 (see Appendix II A). Equation (5.13) may berewritten in the form,

P (R) =1

4ipi2a2R

+

exp[nf()

]d , (5.21)

where

f() = i(R

na

) + ln

(sin

). (5.22)

The integrand of Eq. (5.21) is analytic in the entire finite complex plane.The saddle point, given by the condition

f () = 0 , (5.23)

can be shown to be the point 0 = iy0 on the positive imaginary axiswith

coth y0 1y0

= L(y0) = Rna

, (5.24)

where L is the Langevin function. Further, in the vicinity of the saddlepoint, f() may be expanded as

f() = f(0) + 12f(0)( 0)2 + (5.25)

withf (0) = cosech2y0 1

y 20< 0 , (5.26)

and the contour of integration should therefore be the line through 0and parallel to the real axis. Thus Eq. (5.21) can be approximated by

P (R) =exp[nf(iy0)

]4ipi2a2R

+

(x+ iy0) exp[12nf

(iy0)x2]dx

=(

y04pi2a2R

)[ 2pinf (iy0)

]1/2exp[nf(iy0)

]. (5.27)

That is,

P (R) =

[L1(t)]2(2pina2)3/2t

{1 [L1(t)cosech L1(t)]2}1/2

{

sinhL1(t)L1(t) exp[tL1(t)]

}n(5.28)

witht R

na,

where L1 is the inverse Langevin function.


It should be noted that Eq. (5.28) is valid over the whole range ofR, 0 R na (full extension). The problem was also solved by Kuhnand Grun12 and by James and Guth13 in a different manner. Theirprocedure is based on the fact that the problem is essentially equivalentto that of finding the polarization of a gas due to the orientation ofpermanent magnetic or electric dipoles in an external field of sufficientstrength to produce effects approaching complete orientation.

Now, expanding Eq. (5.28) for R/na 1 and renormalizing it, weobtain

P (R) = C(

32pina2

)3/2exp

( 3R

2

2na2

)(1 +

3R2

2n2a2 9R

4

20n3a4+

),

(5.29)where C is the normalizing constant. This expansion is equivalent tothat obtained by Kuhn and Grun and by James and Guth. We notethat originally Eqs. (5.20), (5.28), and (5.29) were derived with theaim of explaining the behavior of polymeric network systems (rubberelasticity) at high extensions.14

5a(iii). Case for n 1 and R/na 1Equation (5.29) is, of course, valid for this case. However, the sameresult can be obtained in a simpler manner, which we first describe.Equation (5.8) may be rewritten in the form,

K() = exp( 16na22) exp[n

l=2

Bl(2a)2l

(2l)!2l2l]

= exp( 16na22) [1 1180n(a)4 +

]. (5.30)

Integration after substitution of Eqs. (5.4) and (5.30) into Eq. (5.5)leads to

P (R) =(

32pina2

)3/2exp( 3R

2

2na2

)[1 3

20n

(5 10R

2

na2+

3R4

n2a4

)+O(n2)

]. (5.31)

This equation satisfies the normalization condition of (3.10), and Eq.(5.29) becomes identical with Eq. (5.31) when the normalizing constantC is determined.

Evidently the expansion of (5.31) is valid under the present condi-tions, and asymptotically approaches

P (R) =(

32pina2

)3/2exp

( 3R

2

2na2

). (5.32)

This Gaussian asymptotic solution is originally due to Rayleigh.2 Nowwe consider the moments of (5.10) in the limit of n . Recalling


that the term with the highest power of n in the sum in Eq. (5.10)corresponds to the set, m1 = p, m2 = = mp = 0, we retain onlythis term (for n) to obtain

R2p = (2p+ 1)!6pp!

(na2)p . (5.33)

Substitution of Eq. (5.33) into Eq. (4.17) with = R leads to

K() = exp( 16na22) . (5.34)The inverse Fourier transform of Eq. (5.34) just gives Eq. (5.32). Theseresults state the central limit theorem for the random-flight chain. ThusEqs. (5.33) and (5.34) yield the moments and the characteristic functionof the Gaussian distribution. Note that in the particular case of p = 1,Eq. (5.33) reduces exactly to Eq. (5.11). When we approximate thedistribution function P (R) for the random-flight chain by Eq. (5.32),it is sometimes called the Gaussian chain.

5b. Approximate Expression for the Bond Probability

Let us approximate the bond probability by a Gaussian function suchthat it gives the correct mean-square length a2 of the bond; that is,

(r) =(

32pia2

)3/2exp

(3r

2

2a2

). (5.35)

Then the Fourier transform of this function is exp(a22/6) and thecharacteristic function becomes equal to Eq. (5.34). Accordingly thedistribution function of the end-to-end distance becomes the Gaussianfunction, Eq. (5.32). This implies that in the statistics of the Gaussianchain the exact bond probability may be replaced by the Gaussian bondprobability. In fact, most of the problems in this book will be treatedon the basis of the Gaussian chain model with the use of Eq. (5.35).Thus it is worthwhile to describe here further details pertinent to theGaussian chain.

Using Eq. (5.11), Eq. (5.32) may also be written in the form,

P (R) =(

32piR2

)3/2exp

( 3R

2

2R2). (5.36)

For illustrative purposes, the functions P (R) and 4piR2P (R) (the prob-ability density that R lies between R and R + dR irrespective of thedirection of R) for a Gaussian chain with R21/2 = 300 (A) are plottedagainst R in Fig. II.2. The maximum of 4piR2P (R) occurs at

R =(23

)1/2 R21/2 , (5.37)which represents the most probable value of R and is somewhat smallerthan the root-mean-square end-to-end distance.


Fig. II.2. Gaussian distribution functions of the end-to-end vector R and the

end-to-end distance R of a polymer chain with the root-mean-square end-to-end

distance of 300 A.

In Fig. II.3 a comparison is made of the Gaussian distribution func-tion 4piR2P (R) (broken curve) and the corresponding exact distribu-tion function from Eq. (5.20) (full curve) for a = 1 and n = 10. It isseen that even for n = 10 the asymptotic Gaussian distribution givessurprising accuracy over the whole range. It must however be recog-nized that the Gaussian distribution has finite, although small, valueseven for R > na, where the exact value is zero; Eq. (5.32) or (5.36)becomes invalid at values of R approaching full extension of the chain.

In general, a linear chain may be considered to be composed ofsubchains joined successively at their ends. It is then evident that thesesubchains are mutually independent for the random-flight chain. Thusthe distribution function P (Rij) of the distance Rij between segmentsi and j (j > i) in the Gaussian chain can readily be written as

P (Rij) =

(3

2piR 2ij

)3/2exp

( 3R

2ij

2R 2ij

)(5.38)

withR 2ij = (j i)a2 , (5.39)

where segment i, instead of 0, may be supposed to be fixed at theorigin.

From the previous discussion, it seems quite adequate to approx-imate the random-flight chain, which is a basic model for polymer


Fig. II.3. Comparison of the distribution functions of the end-to-end distances

of an exact random-flight chain and of a Gaussian chain, each with a = 1 A and

n = 10. Full curve: exact random-flight chain. Broken curve: Gaussian chain.

chains, by the Gaussian chain in the theoretical treatment of the prop-erties of dilute polymer solutions. The reasons for this may be summa-rized as follows: (1) the deviation of the Gaussian distribution from theexact one is small over the range ordinarily of interest, (2) the Gaus-sian distribution gives the exact value for the mean-square end-to-enddistance (the deviation at high extensions has no influence on averagechain dimensions), and (3) various calculations can be greatly simpli-fied by the use of the Gaussian distribution, in particular, the Gaussianbond probability.

6. The Wang-Uhlenbeck Method forMultivariate Gaussian Distributions

This method is an extension of Markoffs method in the particularcase for which the distribution function j(rj) is given by the Gaussianfunction,

j(rj) =

(3

2pia 2j

)3/2exp

(3r

2j

2a 2j

), (6.1)

SEC. 6. The Wang-Uhlenbeck Method for Multivariate Gaussian Distributions 19

where all aj are assumed not to be equal, for convenience. Let usconsider ns, three-dimensional vectors kj defined by

kj = kjrj

(j = 1, 2, . . . , nk = 1, 2, . . . , s; s n

), (6.2)

where the kj are constants, and the components of each of these vec-tors are just functions of xj , yj , and zj as before. The problem is tofind the multivariate distribution function P

({s}) = P (1, . . . ,s)or the simultaneous probability density of s resultant vectors,

k =nj=1

kj =nj=1

kjrj (k = 1, 2, . . . , s) , (6.3)

each being a linear combination of n vectors rj . The problem in the caseof a one-dimensional rj was formulated first by Wang and Uhlenbeck,15

and the extension to the three-dimensional case was made by Fixman.16

Before solving this problem, we note that P({s}) is normalized as

P({s})d{s} = 1 , (6.4)

and that the distribution function P({r}) of a subset {r} of the set

{s} is given by

P({r}) = P ({s})d{s}

d{r} , (6.5)

Now we many write P({s}) in the form,

P({s}) =

nj=1

j(rj)

d{rn}d{s} , (6.6)

which, by the use of Eq. (6.1) and s, three-dimensional Dirac deltafunctions, reduces to

P({s}) = (2pi)3s n

j=1

(3

2pia 2j

)3/2exp(3r

2j

2a 2j

)drj

s

k=1

exp

ik ( nj=1

kjrj k) dk . (6.7)

That is,

P({s}) = (2pi)3s K({s}) exp

(i

sk=1

k k)d{s} (6.8)

with

K({s}) = n

j=1

(3

2pia 2j

)3/2 exp(3r

2j

2a 2j+i

sk=1

kjk rj)drj . (6.9)


In this case, the characteristic function K({s}) is the 3s-dimensional

Fourier transform of the distribution function P({s});

K({s}) = P ({s}) exp

(i

sk=1

k k)d{s} . (6.10)

The integral in Eq. (6.9) can be easily evaluated to give

K({s}) = exp

[ 16 a2

sk=1

sl=1

Cklk l], (6.11)

where

Ckl =nj=1

kjlja 2ja2 , (6.12)

a2 = 1n

nj=1

a 2j . (6.13)

Substitution of Eq. (6.11) into Eq. (6.8) leads to

P({s}) = (2pi)3s exp[16 a2

sk=1

sl=1

Cklk l

is

k=1

k k]d{s} . (6.14)

After an orthogonal transformation of the coordinates the integral inEq. (6.14) can readily be evaluated, and we obtain the final result (seeAppendix II B),

P({s}) = ( 32pia2

)3s/2|C|3/2

exp[(

32a2|C|

) sk=1

sl=1

Cklk l], (6.15)

where Ckl is the cofactor of the element Ckl of the s s symmetricmatrix C, and |C| is the determinant of C. That is, P ({s}) is amultivariate Gaussian distribution.

In the particular case of s = 1 and aj = a for all j, Eq. (6.15)reduces to

P () =(

32piCa2

)3/2exp( 3

2

2Ca2

)(6.16)

with

=nj=1

jrj , (6.17)

SEC. 7. Distribution of a Segment About the Center of Mass and Related Quantities 21

C =nj=1

2j . (6.18)

This result can also be obtained by direct application of Markoffsmethod. Further, if j = 1 for all j, then C = n and Eq. (6.16)reduces to Eq. (5.32).

It will be seen that Eq. (6.15) is of considerable value in developingthe theories of the excluded-volume effect (Chapter III) and of thesecond virial coefficient (Chapter IV), based on the cluster-expansionmethod.

7. Distribution of a Segment Aboutthe Center of Mass and Related Quantities

We now proceed to examine the distribution of segments about thecenter of mass of a polymer chain. The results of this study provideanother important measure of the average molecular dimensions, di-rectly related to the properties of dilute solutions. Evaluation of distri-bution functions will be carried out on the assumption of the Gaussianbond probability of Eq. (5.35). Needless to say, in the present case, themolecular center of mass, instead of segment 0, may be considered tobe fixed at the origin.

7a. Distribution of a Segment About the Center of Mass

We first consider the distribution function Pj(Sj) of the distance Sjfrom the center of mass to segment j. Obviously there is a relationbetween Sj and the bond vectors rk,

Sj Si =j

k=i+1

rk for j > i ,

= i

k=j+1

rk for j < i . (7.1)

If all the segments have the same mass, by the definition of the centerof mass the sum of all the Sj must be zero,

ni=0

Si = 0 . (7.2)

By summation of both sides of Eq. (7.1) over i, we obtain

Sj =ni=1

jiri (7.3)

withji = H(j i) + i

n+ 1 1 (7.4)


where H(x) is a unit step function defined as

H(x) = 1 for x 0 ,= 0 for x < 0 . (7.5)

The use of Eq. (6.16) with = Sj and j = ji leads to

Pj(Sj) =(

32piS 2j

)3/2exp( 3S

2j

2S 2j )

(7.6)

with

S 2j = a2ni=1

2ji . (7.7)

Since n is large, the summation may be replaced by integration, andS 2j can then be evaluated to be

S 2j = 13na2[1 3j(n j)

n2

]. (7.8)

Equation (7.6) with (7.8) is the formula obtained by Isihara17 and byDebye and Bueche.18 It is seen from Eq. (7.8) that S 2j takes themaximum value 13na

2 at j = 0 or n, and the minimum value 112na2 at

j = 12n. In other words, the end segments are located, on the average,at the positions most remote from the center of mass, while the middlesegment is nearest to the center of mass.

The function Pj(Sj) has the meaning of a specific distribution func-tion, since Pj(Sj)dSj is the probability of finding a particular segment(the jth segment) in the volume element dSj at the distance Sj fromthe center of mass. On the other hand, a generic distribution function(s) is defined as

(s) =nj=0

Pj(s) (7.9)

with the normalization condition,(s)ds = n . (7.10)

That is, (s)ds is the probability of finding any one of n segments inthe volume element ds at the distance s from the center of mass; inother words, (s) is the average segment density at s. This function isreferred to as the segment-density distribution function. It is sometimesconvenient to use the distribution function P (s) defined by

P (s) = n1(s) , (7.11)The summations of a function of the indices i, j, . . . will be replaced by integra-

tions throughout the remainder of this book except where specified otherwise. Inaddition, small numbers occurring in such a function will be suppressed comparedto n.


so that P (s) satisfies the ordinary normalization condition, as definedby Eq. (3.10).

Now P (s) may be written in the form,

P (s) =1n

j

(3

2piS 2j )3/2

exp( 3s

2

2S 2j ). (7.12)

Using Eq. (5.33), the moments of P (s) can readily be expressed as

s2p = (2p+ 1)!6pp!n

j

S 2j p . (7.13)

Substituting Eq. (7.8) into Eq. (7.13) and carrying out the summation(integration), we have

s2p = (2p+ 1)!2p 62p p!

[pl=0

(pl

)3l

2l + 1

](na2)p . (7.14)

The first three moments are

s2 = 16na2 = 16 R2 ,s4 = 1

18(na2)2 =

130R4 ,

s6 = 29972

(na2)3 =293780

R6 . (7.15)

The form of the distribution function P (s) will be considered in Sec-tion 8a.

7b. Radius of Gyration

We define the radius of gyration S of a polymer chain by the equation,

S2 =1n

j

S 2j . (7.16)

From Eq. (7.13), the mean-square radius of gyration can then be easilyobtained as19

S2 = 16 R2 = s2 . (7.17)This is an important relationship between the mean-square radius ofgyration and the mean-square end-to-end distance, indicating their in-terchangeability. However, it is the former that has a direct relationto the solution properties, as will be seen in later chapters. It must benoted that the distribution function, P (S), of the radius of gyration is

The term radius of gyration is used in physics with a different meaning, namely,the mean of the square of the radius of a body about an axis (not a center). However,to avoid confusion, in this book we shall often use this incorrect nomenclature whichis widely used in the extant literatures. Note that S is a scalar.


essentially different from the distribution function 4pis2P (s) of s; theform of P (S) will be discussed in Section 8b. For convenience, s willbe called the quasi-radius of gyration.

Now we derive two other useful formulas for the radius of gyration.From Eqs. (7.7) and (7.16), we have

S2 = a2

n

i

j

2ji . (7.18)

On performing the summation only over j after substitution of Eq. (7.4),Eq. (7.18) becomes

S2 = a2

n2

i

i(n i) . (7.19)

This is the formula derived by Kramers.20 The meaning of this formulais the following: divide the chain at segment i into two parts, and S2can then be evaluated by summing up the product of the numbers ofsegments contained in the two parts over all possible divisions. Ofcourse, summation over i in Eq. (7.19) recovers Eq. (7.17).

By definition, there holds the relation,i

j

Si Sj = 0 . (7.20)

On the other hand, from the rule of cosines,

Si Sj = 12 (S 2i + S 2j R 2ij ) ,we have

i

j

Si Sj = n2i

S 2i +n

2

j

S 2j 12

i

j

R 2ij

= n2S2 12

i

j

R 2ij . (7.21)

From Eqs. (7.20) and (7.21), we obtain

S2 =12n2

i

j

R 2ij =1n2

i


7c. Radii of Gyration with R Fixed

By the definition of a conditional probability, the distribution functionPj(Sj |R) of Sj with the end-to-end vector R fixed is given by

Pj(Sj |R) = Pj(Sj ,R)P (R)

(7.23)

with the normalization condition,Pj(Sj |R)dSj = 1 . (7.24)

The bivariate distribution function Pj(Sj ,R) may be evaluated bymeans of the WangUhlenbeck theorem (Section 6). After a simplealgebraic calculation with the use of Eqs. (3.8) and (7.3) with (7.4), wecan readily obtain, from Eqs. (6.15) and (7.23),

Pj(Sj |R) =[

32pi(na2/12)

]3/2exp

[ 3L

2

2(na2/12)

](7.25)

with

L = Sj +12n

(n 2j)R , (7.26)

where we have used Eq. (5.32).Now suppose a Cartesian coordinate system (ex, ey, ez) to be fixed

in space, ex, ey, ez being the unit vectors in the directions of the x, y,z axes, respectively. Then the mean-square component (Sj ex)2R ofSj in the direction of the x axis with R fixed can be evaluated, fromEq. (7.25) with (7.26), as

(Sj ex)2

R=(Sj ex)2Pj(Sj |R)dSj

= [

(L ex)2 + 14n2 (n 2j)2(R ex)2

]Pj(Sj |R)dL

=136na2 +

14n2

(n 2j)2(R ex)2 . (7.27)

Defining the mean-square radius of gyration S 2x R in the direction ofthe x axis with R fixed by the equation,

S 2x R =1n

j

(Sj ex)2

R, (7.28)

We obtain, from Eqs. (7.27) and (7.28),

S 2x R =136na2

[1 +

3(R ex)2na2

]. (7.29)


Since similar results apply to the y and z components, respectively, wehave for the mean-square radius of gyration S2R with R fixed

S2R = S 2x R + S 2y R + S 2z R=

112na2

(1 +

R2

na2

), (7.30)

which reduces to Eq. (7.17), averaged over all values of R. Equations(7.29) and (7.30) are the formulas derived by Hermans and Overbeek22

using a different method.In particular, if the x axis is chosen in the direction of R, the three

components of S2R become

S 2x R =136na2

(1 +

3R2

na2

),

S 2y R = S 2z R =136na2 (7.31)

withR = Rex. Thus, when the end segments are fixed, the distributionof segments about the center of mass is no longer spherically symmetric,but may be regarded as approximately ellipsoidal, the major axis ofthis ellipsoid being in the direction of the end-to-end vector. In thisconnection, Eq. (7.25) is also to be compared with Eq. (7.6) or (7.12).At full extension of the chain (R = na), the component of S2R in thedirection of R takes the value (na)2/12, which is the exact value forS2 = S 2x R of the rod of length na. The components perpendicularto R are predicted to be independent of R, whereas the exact valuesare zero for the rod. Clearly this discrepancy is due to the use of aGaussian bond probability.

8. Distribution of the Radius of Gyration

In this section, we shall discuss the forms of the distribution functionsP (s) and P (S). In anticipation of the results, however, we note thatclosed expressions for neither P (s) nor P (S) can be derived, which arevalid over the whole range, even for the Gaussian chain.

8a. Distribution of the Quasi-radius of Gyration

The original discussion of the form of the distribution function P (s) ofthe quasi-radius of gyration s is due to Debye and Bueche.18 Replacingthe summation by integration in Eq. (7.12), we obtain for P (s) theseries forms,

P (s) = 21/2(

32pis2

)3/2exp

(3t4

)[1 1

1 3(9t2

)+

11 3 5

(9t2

)2

]for small t , (8.1)

SEC. 8. Distribution of the Radius of Gyration 27

Fig. II.4. Comparison of the exact distribution and the approximate Gaussian

distribution of the quasi-radius of gyration of a polymer chain. Full curve: exact.

Broken curve: approximate.

P (s) = 21/2(

32pis2

)3/2exp

(3t4

)(29t

)[1 +

(29t

)+ 1 3

(29t

)2+ 1 3 5

(29t

)3+

]for large t , (8.2)

where

t s2

s2 =6s2

na2. (8.3)

The integration has also been carried out graphically, and the result isdisplayed by the full curve in Fig. II.4. For comparison, the Gaussiandistribution of s, (

32pis2

)3/2exp

( 3s

2

2s2), (8.4)

the second moment of which coincides with that of P (s), is also plottedin the figure.

As seen from Eqs. (8.1) and (8.2) and Fig. II.4, the distribution ofthe quasi-radius of gyration is not strictly Gaussian in the same ap-proximation which is employed in obtaining the Gaussian distributionof the end-to-end distance and of the distance from the center of massto a particular segment. However, the agreement between the exactdistribution and the approximate Gaussian distribution is fairly satis-factory except at large values of s. Thus the Gaussian approximation


for P (s) will still be useful insofar as the dilute-solution properties aredescribed in a crude approximation.

8b. Distribution of the Radius of Gyration

The distribution function P (S) of the radius of gyration was first inves-tigated by Fixman23 and by Forsman and Hughes.24 For convenience,we describe the procedure of Fixman. However, it should be anticipatedthat Fixmans calculation involves some error, and the complete numer-ical results have recently been obtained by Koyama,25 by Hoffman andForsman,26 and by Fujita and Norisuye.27 The conditional segment-density distribution function with S fixed will also be discussed in Ap-pendix II C. In this section, it is therefore convenient to begin witha formulation of the bivariate distribution function Pj(Sj , S2) ratherthan the desired function P (S). The distribution function P (S2) of S2

can be obtained from

P (S2) =Pj(Sj , S2)dSj . (8.5)

Evidently the probability P (S)dS that S lies between S and S + dS isequal to P (S2)dS2, and we therefore have the relation,

P (S) = 2SP (S2) . (8.6)

Now, Sj is given by the linear combination (7.3) of n bond vectorsrj , while from Eqs. (7.3), (7.4), and (7.16) S2 is given by the quadraticform,

S2 =n

k=1

nl=1

gklrk rl (8.7)

with

gkl =1n

j

jkjl = glk

= n2{n[kh(l k) + lh(k l)] kl} , (8.8)

where h(x) is a unit step function defined as

h(x) = 1 for x > 0 ,= 12 for x = 0 ,= 0 for x < 0 . (8.9)

The distribution function Pj(Sj , S2) may be written in the form,

Pj(Sj , S2) = n

j=1

(rj)

d{rn}dSjdS2

, (8.10)

where we assume the Gaussian bond probability. The range of S2 istaken as < S2 < + in the absence of an explicit demonstration


that Eq. (8.10) yields the inevitable result, Pj(Sj , S2) 0 for S2 < 0.By a slight modification of the WangUhlenbeck method, the integralof (8.10) can be treated easily.

Introducing Fourier representations of the one-and three-dimensionalDirac delta functions and using Eqs. (7.3) and (8.7), we may rewriteEq. (8.10) in the form,

Pj(Sj , S2) = (2pi)4Kj(, ) exp(iS2 i Sj)d d (8.11)

with

Kj(, ) =(

32pia2

)3n/2 exp[ 32a2

k

r 2k

+ik

l

gklrk rl + ik

jk rk]d{rn} . (8.12)

A resolution of the rk and into x, y, and z components simplifies theevaluation of the integral in Eq. (8.12). That is,

Kj(, ) =(

32pia2

)3n/2 x,y,z

j(x, ) (8.13)

with

j(x, ) =

exp

[k

l

Aklxkxl + ixk

jkxk

]dx1 dxn ,

(8.14)

Akl =(

32a2

)kl igkl , (8.15)

where kl is the Kronecker delta. The integral of (8.14) is of a formsimilar to that of Eq. (6.14), and therefore it can be easily evaluatedby an orthogonal transformation of the coordinates (Appendix II B).If k are the eigenvalues of the matrix A with elements Akl and j isthe column vector of the jk with transpose Tj = (j1 jn), anorthogonal transformation Q which diagonalizes A gives

j(x, ) =n

k=1

+

exp[k 2k + ix( Tj Q)kk]dk . (8.16)

As shown below, the real parts of the k are all positive, and hence theintegral in Eq. (8.16) is convergent. Thus we find

j(x, ) = pin/2(

nk=1

1/2k

)exp

[

2x

4

k

1k (Tj Q)

2k

]. (8.17)

From Eqs. (8.13) and (8.17), the characteristic function Kj(, ) isobtained as

Kj(, ) = K() exp(Wj2) (8.18)


with

K() = Kj(0, ) =(

32a2

)3n/2 nk=1

3/2k , (8.19)

Wj =14

nk=1

1k (Tj Q)

2k

=14

k

1k

(l

jlQlk

)2, (8.20)

where the Qkl are the normalized components of Q. On substitutionof Eq. (8.18) into Eq. (8.11) and integration over , we have

Pj(Sj , S2) = (16pi5/2)1K()W3/2j exp

(iS2 S

2j

4Wj

)d ,

(8.21)and integration over Sj gives

P (S2) =12pi

+

K() exp(iS2)d . (8.22)

Note that K() given by Eq. (8.19) is the characteristic function ofP (S2).

We now determine the eigenvalues of A and the normalized com-ponents of Q. If x is an eigenvector of A, we have

Ax x = 0 , (8.23)or

nj=1

Aijxj = xi , (8.24)

from which there must result n different s, k, and n different xis,x(k)i . The orthogonal matrix Q may be composed of column vectorsx(k)i , which form a complete and orthogonal set for functions deter-mined at n points, and

k

x(k)i x

(k)j =

k

x(i)k x

(j)k = ij . (8.25)

Now it proves convenient to convert Eq. (8.24) to an integral equation.This treatment of i and j as continuous variables naturally gives aninfinite set of eigenvalues and eigenfunctions from which the proper setmay easily be extracted. With xi = x(z), Aij = A(z, y), and so on,where z = i/n and y = j/n, Eqs. (8.8), (8.15), and (8.24) give

32X2[ z

0

yx(y)dy + z 1z

x(y)dy z 10

yx(y)dy]= (a2 32 )x(z)

(8.26)


withX2 23 ina2 . (8.27)

Equation (8.26) can be solved by two successive differentiations withrespect to z;

32X2[ 1

z

x(y)dy 10

yx(y)dy]= (a2 32 )x(z) , (8.28)

32X

2x(z) = (a2 32 )x(z) . (8.29)Equation (8.29) has the solution,

x(z) = K1 exp[zX(23a

2 1)1/2]+K2 exp[zX( 23a2 1)1/2] .(8.30)

Substitution of Eq. (8.30) into Eqs. (8.26) and (8.28) gives

x(1) = x(0) = 0 . (8.31)

The condition x(0) = 0 is satisfied by the choice K2 = K1, and thereresults

x(z) = K sin[izX(23a

2 1)1/2] . (8.32)The condition x(1) = 0 then establishes the eigenvalues from the rela-tion,

iX(23a2 1)1/2 = kpi (k = 1, 2, . . .) , (8.33)

that is

k =32a2

(1 X

2

k2pi2

). (8.34)

On transformation back from the continuous space of the eigenvectorsx(z) to the discrete space of xi, it is evident that the values k > n inEq. (8.33) are redundant for the expansion of functions on n points. Ac-cordingly the original nnmatrixA has eigenvalues given by Eq. (8.34)with k = 1, . . ., n. From Eqs. (8.32) and (8.34), we find for the eigen-vectors

x(k)j = K sin

(pijk

n

), (8.35)

which are just the components Qjk of the matrix Q. The constant Kcan be determined from Eq. (8.25) as

1 =nj=1

K2 sin2(pijk

n

)= 12nK

2 , (8.36)

that is,

K =(2n

)1/2. (8.37)

The normalized components of Q are therefore given by

Qjk =(2n

)1/2sin(pijk

n

). (8.38)


From Eqs. (8.19) and (8.34), we can now obtain for the characteristicfunction K()

K() =n

k=1

(1 X

2

k2pi2

)3/2. (8.39)

It can be shown that the product in Eq. (8.39) may be extended tok = without introduction of any significant error if n 1. ThenEq. (8.39) becomes

K() =(sinXX

)3/2. (8.40)

Before proceeding to evaluate the distribution function P (S2), we com-pute its moments. The logarithm of Eq. (8.40) can be expanded as

lnK() =j=1

23j2Bj3j1(2j)!j

(na2)j(i)j , (8.41)

where Bj are the Bernoulli numbers (B1 = 1/6, B2 = 1/30, ) asbefore. We therefore have for the cumulants j

j =23j2(j 1)!Bj

3j1(2j)!(na2)j . (8.42)

Substituting Eq. (8.42) into Eq. (5.7), we obtain for the moments k =S2k of P (S2)

S2k = k!(na2)km

23k2m

3km

kj=1

1mj !

[Bj

j(2j)!

]mj(8.43)

with

jmj = m and

j jmj = k. The first three moments are

S2 = 16na2 ,S4 = 19

540(na2)2 =

1915S22 ,

S6 = 63168040

(na2)3 =631315

S23 . (8.44)

It is important to observe that the second moments of the distribu-tions of the radius of gyration and of the quasi-radius of gyration areidentical, while the fourth and higher moments are different from eachother.

We now turn to the evaluation of P (S2). Substitution of Eq. (8.40)into Eq. (8.22) leads to

P (S2) =12pi

+

(sinXX

)3/2exp( 14 tX2)d , (8.45)


where

t =S2

S2 =6S2

na2. (8.46)

Fujita and Norisuye27 transformed the integral of (8.45) to a contourintegral on the X complex plane, and evaluated it exactly, althoughonly in a series form. The derivation is so lengthy that we do notreproduce it here. If P (S2) is transformed back to P (S), the result is

P (S) =1

21/2piS21/2t3k=0

(2k + 1)!(2kk!)2

(4k + 3)7/2 exp(tk)

[(

1 58tk

)K1/4(tk) +

(1 3

8tk

)K3/4(tk)

], (8.47)

where Ks are the modified Bessel functions of the second kind and tkis defined by

tk =(4k + 3)2

8t. (8.48)

If we use the asymptotic expansions of K1/4 and K3/4, we have anexpansion of P (S) valid for small t,

P (S) = 18(

6piS2

)1/2t5/2 exp

( 94t

)(1 19

36t+

1051296

t2 )

(for small t) . (8.49)

The leading term of this expansion agrees with the result derived byFixman23 from Eq. (8.45) by the method of steepest descents.

On the other hand, it is impossible to derive an asymptotic formof P (S) valid for large t from Eq. (8.47). Thus, Fujita and Norisuyeturned back to Eq. (8.45) and reevaluated the integral by the choice ofa proper contour. The result is

P (S) =21/2pi5/2

S21/2 t exp(pi

2t

4

)[1 +

94pi2

(1t

)+8pi2 + 1532pi4

(1t

)2+

](for large t) . (8.50)

The leading term of this expansion agrees with Forsmans result24 de-rived by a different method, but does not agree with Fixmans result,23

P (S) =pi5/2e3/2

3S21/2 t exp(pi

2t

4

)(Fixman) , (8.51)

which has been derived by the method of steepest descents. However,the difference between the two results is rather small, since 21/2 = 1.414and e3/2/3 = 1.494. The error in Fixmans calculation arises from the


Fig. II.5. Comparison of the exact and approximate distributions of the radius of

gyration of a polymer chain. Curve 1: exact. Curve 2: the Gaussian distribution.

Curve 3: the FloryFisk function.

fact that the condition under which the method of steepest descentscan be applied is not satisfied at the point he specified as the saddlepoint for t 1. Note that P (S) becomes a Gaussian function at largeS, although the numerical coefficients are different from those of theusual Gaussian distribution function.

The values of S21/2P (S) as a function of t calculated from Eq. (8.47)are in excellent agreement with those calculated by Koyama25 fromEq. (8.45) by the use of a computer, and are shown in curve 1 inFig. II.5. For comparison, the Gaussian distribution of S is shown incurve 2. It is seen that the exact distribution of S converges extremelyrapidly to zero at small S, and goes more rapidly to zero at large Sthan does the Gaussian distribution, the former being sharper than thelatter. Flory and Fisk28 assumed as the closed form for P (S)

P (S) = const. S21/2tm exp[(m+ 12 )t] . (8.52)The value 3 was assigned to m, since this choice reproduces fairly wellthe exact moments of (8.44). Then the numerical constant in Eq. (8.52)is found to be (343/15)(14/pi)1/2. The normalized FloryFisk functionis shown in curve 3 in Fig. II.5. This function is seen to representfairly well the salient behavior of the exact distribution of S. Furtherdiscussion of P (S) will be given in Chapter III.

SEC. 9. Remarks 35

9. Remarks

In this chapter, so far, we have described the mathematical details of thestatistics of linear random-flight chains. Although the linear random-flight chain is the basic model in the theory of dilute polymer solutions,it is unrealistic and incomplete from the point of view of the structuralrestrictions or the conformational statistics of polymer chains. Thus, inthis section, we shall describe the statistical properties of those chainswhich are not placed in a category termed the linear random-flightchain; that is, chains with short-range interferences, branches, or highstiffness.

9a. Short-Range Interferences and Unperturbed MolecularDimensions

In order to make a random-flight chain model more realistic, accountshould be taken of the fact that the valence angle between successivebonds in the chain is actually a fixed quantity and the angle of rotationabout each bond is not uniformly distributed owing to steric hindrancescaused by interactions between atomic groups attached to the spine ofthe chain. In the case of vinyl polymers, for instance, the direction of agiven CC bond, for example, the jth bond, is most strongly affectedby the direction of its predecessor, the (j 1)th bond, due to bond an-gle restrictions, and is also influenced to some extent by the directionsof other neighbors, the (j 2), . . ., (j s)th bond, due to hinderedrotations. It is clear that one bond has no appreciable influence uponthe rotation of another bond when they are far apart, and therefore thevalue of s is relatively small. This is the reason why such interactionsbetween bonds are referred to as the short-range interference. In thestudy of this effect, it would be necessary to take into account a partof the potential W in Eq. (3.2). Then the probability j of each stepdepends on the past s steps, and therefore the present problem corre-sponds to that of random flights with correlations of order s, namelythe s-fold Markoff process. However, it is very difficult to derive gen-erally the distribution function P (R) of the end-to-end distance of thechain with short-range interferences. For convenience, we first evaluatethe mean-square end-to-end distance.

9a(i). Freely Rotating Chains

Squaring both sides of Eq. (3.8), we have the general expression for themean-square end-to-end distance of a linear polymer chain,

R2 =ni=1

r 2i + 21i


Fig. II.6. Schematic representation of a repeating unit of a freely rotating chain.

recover Eq. (5.11). In this subsection, we consider a chain in whichthe bond angles are restricted but rotations about single bonds areunrestricted. Such a chain is called the freely rotating chain.

For convenience, suppose a chain composed of n/p repeating units,each consisting of p bonds of lengths a1, . . ., ap joined successively withangles 1, . . ., p, where the free rotation about every bond is permitted(see Fig. II.6). Then it is evident that the first sum in Eq. (9.1) is givenby

i

r 2i =n

p

pi=1

a 2i . (9.2)

By rearrangement of terms with account of the repeating nature, thesecond sum in Eq. (9.1) may be rewritten as

i

SEC. 9. Remarks 37

TABLE II.1. PARAMETERS IN EQ. (9.5) FOR VARIOUS TYPESOF POLYMERS

TYPEa p a1 a2 a3 a4 1 2 3 4

1 1 a 2 2 a a 1 2 3 2 a b 1 2 4 3 a a b 1 2 2 5 4 a a b b 1 2 1 2

aType 1: vinyl polymer. Type 2: polyoxymethylene and polydimethylsiloxane.Type 3: cis-1, 4-polysaccharide (amylose) and cis-polypeptide. Type 4: cis-polybutadiene. Type 5: polycarbonate.

C = 2(pi=1 i)

[1 (pi=1 i)n/p]

(1pi=1 i)2

pi=1

[a 2i + ai

i1j=1

aji1k=j k

+ aip

j=i+1

aj

j1k=i

k

]. (9.6)

When n is large, the term C becomes independent of n since |i| < 1,and it may therefore be suppressed in Eq. (9.5); thus R2 becomesproportional to n.

From Eq. (9.5), we can readily obtain expressions for R2 of varioustypes of freely rotating chains. Some of polymers whose backbonescontain structural features such as double bonds or rings can also bereduced to the above model, namely a linear sequence of freely rotatingbonds.For instance, vinyl polymer chains may be represented by thefollowing choice of the parameters: p = 1, a1 = a, and 1 = . For thiscase, we have

R2 = na2[1 cos 1 + cos

+2 cos n

1 ( cos )n(1 + cos )2

], (9.7)

which, for large n, becomes

R2 = na2 1 cos 1 + cos

. (9.8)

Equations (9.7) and (9.8) are the formulas derived by Eyring,29 Wall,30

and Benoit.31 In Table II.1 are summarized the parameter assignmentsfor various types of polymers; Type 1: vinyl polymers; Type 2: poly-oxymethylene and polydimethylsiloxane32; Type 3: cis-1,4-polysac-charide (amylose)31, 33 and cis-polypeptide34; Type 4: cis-polybuta-diene30, 31, 33; Type 5: polycarbonate. We note that the expressionused by Schulz and Horbach35 for polycarbonates is incorrect.


There are other types of polymers whose backbone structures can-not be reduced to the above model. However, a similar principle can beapplied after use of a maneuver. In fact, calculations have already beenmade for trans-1,4-polysaccharide (cellulose and pectic acid),31, 33, 34

trans-polybutadiene,30, 31, 33 and trans-polypeptide.34 In any case, themean-square end-to-end distance of a freely rotating chain can be easilyevaluated when the geometrical structure of the backbone of the chainis given.

9a(ii). Chains with Hindered Internal Rotations

For simplicity, we consider vinyl polymers with ai = a and i = forall i. It is convenient to choose rotation angles 1, 2, . . ., n = {n}as the internal coordinates, assuming a and to be fixed quantities.The configurational partition function of (3.1) may then be rewrittenin the form,

Z =

exp

[U

({n})kT

]d{n} , (9.9)

where U is the energy of internal rotation (strictly the potential of meanforce). The problem is to express the scalar product ri rj in Eq. (9.1)in terms of {n}, and then to evaluate its average. This can be doneby applying a method, originally suggested by Eyring.29, 30

Let us choose a set of Cartesian coordinate systems as follows: thepositive direction of the xj axis of the jth coordinate system coincideswith the vector rj , the positive direction of the yj axis makes an acuteangle with the vector rj1 in the plane containing the two vectorsrj1 and rj , and the zj axis constitutes a right-handed rectangularcoordinate system with the xj and yj axes. The angle between the twoplanes containing rj2 and rj1, and rj1 and rj , respectively, definesthe rotation angle j about the (j1)th bond, which is zero when rj2and rj are situated in the trans position with respect to each other andtakes a positive value when rj lies in the positive range of zj1 in the(j 1)th coordinate system (see Fig. II.7). Then the jth coordinatesystem is transformed into the (j 1)th one by the orthogonal matrix,

Aj =

cos sin 0sin cosj cos cosj sinjsin sinj cos sinj cosj

. (9.10)The vector rj which in its own coordinate system was given by rj =(a 0 0) a (column vector) is expressed in the (j1)th system byAja.By repeating this procedure, rj can be expressed in the ith coordinatesystem. Thus the average of the scalar product ri rj may be writtenas

ri rj = aT

jk=i+1

Ak

a , (9.11)

SEC. 9. Remarks 39

Fig. II.7. Transformation of a coordinate system associated with the jth bond

into a coordinate system associated with the (j1)th bond.

where the superscript T indicates the transpose. The average on theright-hand side is given by

j

k=i+1

Ak

= Z1

( jk=i+1

Ak

)exp

( UkT

)d{n} . (9.12)

The energy U may be decomposed as

U({n}) = n

i=1

u1i(i) +ni=1

u2i(i, i+1) + . (9.13)

According to the assumed form of U , the theories may be classified intothree types;1. independent rotation:

U =ni=1

u1i(i) . (9.14)

2. pairwise independent rotation:

U =ni=1

u1i(i) +n/2k=1

u2,2k(2k1, 2k) . (9.15)

3. interdependent rotation:

U =ni=1

u1i(i) +ni=1

u2i(i, i+1) . (9.16)


It is evident that the third type is most realistic and important since ittakes into account completely both first and second neighbor interac-tions. In the discussion that follows, we assume u1i = u1 and u2i = u2for all i.

We first consider the case of independent rotation. By assumption,Eq. (9.12) becomes

jk=i+1

Ak

=k

Ak Aji , (9.17)

where A is the transformation matrix of (9.10) with 1 = cos and2 = sin in place of cosj and sinj , respectively, where

cos = pipi cose

u1/kT d pipi e

u1/kT d. (9.18)

The matrix A can be transformed into a diagonal matrix by a simi-larity transformation with an appropriate matrix Q,

Q1AQ = (9.19)

the diagonal elements i of (eigenvalues of A) being the roots of thecharacteristic equation of A,

|EA| = 3 c12 + c2 c3 = 0 (9.20)with E the unit matrix. From Eqs. (9.1), (9.11), (9.17), and (9.19), wethen have

R2 = na2 + naTQQ1a , (9.21)where is the diagonal matrix given by

=2n

1i

SEC. 9. Remarks 41

If the potential u1() is an even function of , 2 vanishes and Eq. (9.25)reduces to4043

R2 = na2 1 cos 1 + cos

1 + 11 1 . (9.26)

This is a well-known classical formula.The case of pairwise independent rotation can in principle be treated

by the same procedure as above, and we do not reproduce the math-ematical details. It is to be noted that calculations belonging to thiscategory have been made by many workers.38, 39, 4446

We now consider the case of interdependent rotation. For this case,it is convenient to introduce the assumption of discrete rotational en-ergy levels or the so-called rotational-isomeric approximation. In thisapproximation, the rotation angle i is considered to take only a fi-nite set of fixed values (k)i (k = 1, 2, . . ., s) corresponding to theminima of the potential. In the following discussion we consider onlythree available states, T (trans, = 0), G (gauche, = 120), andG (another gauche, = 120), which are indicated by k = 1, 2, 3,respectively. The partition function of (9.9) may then be rewritten inthe form,

Z ={n}

ni=1

p(i, i+1) (9.27)

with

p(i, i+1) = exp{u1(i) + u2(i, i+1)

kT

}, (9.28)

where we assume n+1 = 1. The sum in Eq. (9.27) extends over allpossible values of 1, . . ., n. Thus the problem becomes equivalent tothat of a one-dimensional cooperative system, the Ising model.47 Letus introduce the 3 3 matrix p whose elements are

pkl = p((k)i ,

(l)i+1 ) , (9.29)

pkl being independent of i. Then p can be transformed into a diagonalmatrix Q1pQ = (i) by a similarity transformation, where i arethe eigenvalues of p, and Eq. (9.27) reduces to

Z = trace pn =3i=1

ni

= n (for large n) , (9.30)

where is the largest eigenvalue (assuming its nondegeneracy). Simi-larly the average of product of functions fi(i) may be expressed as

jk=i+1

fk(k)

= Z1

{n}

[j

k=i+1

fk(k)

][ni=1

p(i, i+1)

]

= Z1trace fi+1

[j

k=i+2

(pfk)

]pn(ji1)


= yf i+1

[j

k=i+2

(1pfk)

]x (for large n) ,(9.31)

where fi is the diagonal matrix with diagonal elements fi((k)) (k =1, 2, 3), and x and y are the normalized right-hand and left-handeigenvectors associated with , respectively; i.e.,

px = x , yp = y , yx = 1 . (9.32)

Recalling that the element Ars of t

Yamakawa Modern Theory of Polymer Solution 0060473096

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