polymer solutions - startseite · colligative properties in nonideal solutions 164 2.a.1 osmotic...

POLYMER SOLUTIONS

This page intentionally left blank

An Introduction to Physical Properties

IWAO TERAOKAPolytechnic UniversityBrooklyn, New York

WILEY-INTERSCIENCE

A JOHN WILEY & SONS, INC., PUBLICATION

POLYMER SOLUTIONS

This book is printed on acid-free paper. @

Copyright © 2002 by John Wiley & Sons, Inc., New York. All rights reserved.

Published simultaneously in Canada.

No part of this publication may be reproduced, stored in a retrieval system or transmitted in anyform or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise,except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, withouteither the prior written permission of the Publisher, or authorization through payment of theappropriate per-copy fee to the Copyright Clerance Center, 222 Rosewood Drive, Danvers, MA01923, (978) 750-8400, fax (978) 750-4744. Requests to the Publisher for permission should beaddressed to the Presmissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York,NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: [email protected].

For ordering and customer service, call 1-800-CALL-WILEY.

Library of Congress Cataloging-in-Publication Data:

Teraoka, Iwao.Polymer solutions: an introduction to physical properties / Iwao Teraoka.

p. cm.Includes bibliographical references and index.ISBN 0-471-38923-3 (cloth: alk. paper)

1. Polymer solutions. I. Title

QD381.9.S65T47 2001 2001045656547'.70454—dc21

Printed in the United States of America.1 0 9 8 7 6 5 4 3 2 1

To my wife, Sadae

CONTENTS

Preface xv

1 Models of Polymer Chains 1

1.1 Introduction 11.1.1 Chain Architecture 11.1.2 Models of a Linear Polymer Chain 2

1.1.2.1 Models in a Continuous Space 21.1.2.2 Models in a Discrete Space 4

1.1.3 Real Chains and Ideal Chains 5

1.2 Ideal Chains 71.2.1 Random Walk in One Dimension 7

1.2.1.1 Random Walk 71.2.1.2 Mean Square Displacement 91.2.1.3 Step Motion 101.2.1.4 Normal Distribution 10

1.2.2 Random Walks in Two and Three Dimensions 121.2.2.1 Square Lattice 121.2.2.2 Lattice in Three Dimensions 131.2.2.3 Continuous Space 14

1.2.3 Dimensions of Random-Walk Chains 151.2.3.1 End-to-End Distance and Radius of Gyration 151.2.3.2 Dimensions of Ideal Chains 181.2.3.2 Dimensions of Chains with Short-Range Interactions 19

1.2.4 Problems 20

1.3 Gaussian Chain 231.3.1 What is a Gaussian Chain? 23

1.3.1.1 Gaussian Distribution 231.3.1.2 Contour Length 25

1.3.2 Dimension of a Gaussian Chain 251.3.2.1 Isotropic Dimension 251.3.2.2 Anisotropy 26

vii

viii CONTENTS

1.3.3 Entropy Elasticity 281.3.3.1 Boltzmann Factor 281.3.3.2 Elasticity 30

1.3.4 Problems 31

1.4 Real Chains 331.4.1 Excluded Volume 33

1.4.1.1 Excluded Volume of a Sphere 3 31.4.1.2 Excluded Volume in a Chain Molecule 34

1.4.2 Dimension of a Real Chain 361.4.2.1 Flory Exponent 361.4.2.2 Experimental Results 37

1.4.3 Self-Avoiding Walk 391.4.4 Problems 40

1.5 Semirigid Chains 411.5.1 Examples of Semirigid Chains 411.5.2 Wormlike Chain 43

1.5.2.1 Model 431.5.2.2 End-to-End Distance 441.5.2.3 Radius of Gyration 451.5.2.4 Estimation of Persistence Length 46

1.5.3 Problems 47

1.6 Branched Chains 491.6.1 Architecture of Branched Chains 491.6.2 Dimension of Branched Chains 501.6.3 Problems 52

1.7 Molecular Weight Distribution 551.7.1 Average Molecular Weights 55

1.7.1.1 Definitions of the Average Molecular Weights 551.7.1.2 Estimation of the Averages and the Distribution 57

1.7.2 Typical Distributions 581.7.2.1 Poisson Distribution 581.7.2.2 Exponential Distribution 591.7.2.3 Log-Normal Distribution 60

1.7.3 Problems 62

1.8 Concentration Regimes 631.8.1 Concentration Regimes for Linear Flexible Polymers 631.8.2 Concentration Regimes for Rodlike Molecules 651.8.3 Problems 66

CONTENTS ix

2 Thermodynamics of Dilute Polymer Solutions 69

2.1 Polymer Solutions and Thermodynamics 692.2 Flory-Huggins Mean-Field Theory 70

2.2.1 Model 702.2.1.1 Lattice Chain Model 702.2.1.2 Entropy of Mixing 722.2.1.3 x Parameter 722.2.1.4 Interaction Change Upon Mixing 74

2.2.2 Free Energy, Chemical Potentials, andOsmotic Pressure 752.2.2.1 General Formulas 752.2.2.2 Chemical Potential of a Polymer Chain in Solution 77

2.2.3 Dilute Solutions 772.2.3.1 Mean-Field Theory 772.2.3.2 Virial Expansion 78

2.2.4 Coexistence Curve and Stability 802.2.4.1 Replacement Chemical Potential 802.2.4.2 Critical Point and Spinodal Line 812.2.4.3 Phase Separation 822.2.4.4 Phase Diagram 84

2.2.5 Polydisperse Polymer 872.2.6 Problems 89

2.3 Phase Diagram and Theta Solutions 992.3.1 Phase Diagram 99

2.3.1.1 Upper and Lower Critical Solution Temperatures 992.3.1.2 Experimental Methods 100

2.3.2 Theta Solutions 1012.3.2.1 Theta Temperature 1012.3.2.2 Properties of Theta Solutions 103

2.3.3 Coil-Globule Transition 1052.3.4 Solubility Parameter 1072.3.5 Problems 108

2.4 Static Light Scattering 1082.4.1 Sample Geometry in Light-Scattering

Measurements 1082.4.2 Scattering by a Small Particle 1102.4.3 Scattering by a Polymer Chain 1122.4.4 Scattering by Many Polymer Chains 1152.4.5 Correlation Function and Structure Factor 117

2.4.5.1 Correlation Function 1172.4.5.2 Relationship Between the Correlation

Function and Structure Factor 117

X CONTENTS

2.4.5.3 Examples in One Dimension 1192.4.6 Structure Factor of a Polymer Chain 120

2.4.6.1 Low-Angle Scattering 1202.4.6.2 Scattering by a Gaussian Chain 1212.4.6.3 Scattering by a Real Chain 1242.4.6.4 Form Factors 125

2.4.7 Light Scattering of a Polymer Solution 1282.4.7.1 Scattering in a Solvent 1282.4.7.2 Scattering by a Polymer Solution 1292.4.7.3 Concentration Fluctuations 1312.4.7.4 Light-Scattering Experiments 1322.4.7.5 ZimmPlot 1332.4.7.6 Measurement of dn/dc 135

2.4.8 Other Scattering Techniques 1362.4.8.1 Small-Angle Neutron Scattering (SANS) 1362.4.8.2 Small-Angle X-Ray Scattering (SAXS) 139

2.4.9 Problems 139

2.5 Size Exclusion Chromatography and Confinement 1482.5.1 Separation System 1482.5.2 Plate Theory 1502.5.3 Partitioning of Polymer with a Pore 151

2.5.3.1 Partition Coefficient 1512.5.3.2 Confinement of a Gaussian Chain 1532.5.3.3 Confinement of a Real Chain 156

2.5.4 Calibration of SEC 1582.5.5 SEC With an On-Line Light-Scattering Detector 1602.5.6 Problems 162

APPENDIXES2. A: Review of Thermodynamics for

Colligative Properties in Nonideal Solutions 1642.A.1 Osmotic Pressure 1642.A.2 Vapor Pressure Osmometry 164

2.B: Another Approach to Thermodynamics ofPolymer Solutions 165

2.C: Correlation Function of a Gaussian Chain 166

3 Dynamics of Dilute Polymer Solutions 167

3.1 Dynamics of Polymer Solutions 1673.2 Dynamic Light Scattering and Diffusion of Polymers 168

3.2.1 Measurement System and Autocorrelation Function 1683.2.1.1 Measurement System 1683.2.1.2 Autocorrelation Function 1693.2.1.3 Photon Counting 170

CONTENTS xi

3.2.2 Autocorrelation Function 1703.2.2.1 Baseline Subtraction and Normalization 1703.2.2.2 Electric-Field Autocorrelation Function 172

3.2.3 Dynamic Structure Factor of Suspended Particles 1723.2.3.1 Autocorrelation of Scattered Field 1723.2.3.2 Dynamic Structure Factor 1743.2.3.3 Transition Probability 174

3.2.4 Diffusion of Particles 1763.2.4.1 Brownian Motion 1763.2.4.2 Diffusion Coefficient 1773.2.4.3 Gaussian Transition Probability 1783.2.4.4 Diffusion Equation 1793.2.4.5 Concentration 1793.2.4.6 Long-Time Diffusion Coefficient 180

3.2.5 Diffusion and DLS 1803.2.5.1 Dynamic Structure Factor and Mean

Square Displacement 1803.2.5.2 Dynamic Structure Factor of a Diffusing

Particle 1813.2.6 Dynamic Structure Factor of a Polymer Solution 182

3.2.6.1 Dynamic Structure Factor 1823.2.6.2 Long-Time Behavior 183

3.2.7 Hydrodynamic Radius 1843.2.7.1 Stokes-Einstein Equation 1843.2.7.2 Hydrodynamic Radius of a Polymer Chain 185

3.2.8 Particle Sizing 1883.2.8.1 Distribution of Particle Size 1883.2.8.2 Inverse-Laplace Transform 1883.2.8.3 Cumulant Expansion 1893.2.8.4 Example 190

3.2.9 Diffusion From Equation of Motion 1913.2.10 Diffusion as Kinetics 193

3.2.10.1 Pick's Law 1933.2.10.2 Diffusion Equation 1953.2.10.3 Chemical Potential Gradient 196

3.2.11 Concentration Effect on Diffusion 1963.2.11.1 Self-Diffusion and Mutual Diffusion 1963.2.11.2 Measurement of Self-Diffusion Coefficient3.2.11.3 Concentration Dependence of the

Diffusion Coefficients 1983.2.12 Diffusion in a Nonuniform System 2003.2.13 Problems 201

3.3 Viscosity 2093.3.1 Viscosity of Solutions 209

xii CONTENTS

3.3.1.1 Viscosity of a Fluid 2093.3.1.2 Viscosity of a Solution 211

3.3.2 Measurement of Viscosity 2133.3.3 Intrinsic Viscosity 2153.3.4 Flow Field 2173.3.5 Problems 219

3.4 Normal Modes 2213.4.1 Rouse Model 221

3.4.1.1 Model for Chain Dynamics 2213.4.1.2 Equation of Motion 222

3.4.2 Normal Coordinates 2233.4.2.1 Definition 2233.4.2.2 Inverse Transformation 226

3.4.3 Equation of Motion for the NormalCoordinates in the Rouse Model 2263.4.3.1 Equation of Motion 2263.4.3.2 Correlation of Random Force 2283.4.3.3 Formal Solution 229

3.4.4 Results of the Normal-Coordinates 2293.4.4.1 Correlation of q,(0 2293.4.4.2 End-to-End Vector 2303.4.4.3 Center-of-Mass Motion 2313.4.4.4 Evolution of q,(0 231

3.4.5 Results for the Rouse Model 2323.4.5.1 Correlation of the Normal Modes 2323.4.5.2 Correlation of the End-to-End Vector 2343.4.5.3 Diffusion Coefficient 2343.4.5.4 Molecular Weight Dependence 234

3.4.6 Zimm Model 2343.4.6.1 Hydrodynamic Interactions 2343.4.6.2 Zimm Model in the Theta Solvent 2363.4.6.3 Hydrodynamic Radius 2383.4.6.4 Zimm Model in the Good Solvent 238

3.4.7 Intrinsic Viscosity 2393.4.7.1 Extra Stress by Polymers 2393.4.7.2 Intrinsic Viscosity of Polymers 2413.4.7.3 Universal Calibration Curve in SEC 243

3.4.8 Dynamic Structure Factor 2433.4.8.1 General Formula 2433.4.8.2 Initial Slope in the Rouse Model 2473.4.8.3 Initial Slope in the Zimm Model, Theta Solvent 2473.4.8.4 Initial Slope in the Zimm Model, Good Solvent 2483.4.8.5 Initial Slope: Experiments 249

3.4.9 Motion of Monomers 250

CONTENTS xiii

3.4.9.1 General Formula 2503.4.9.2 Mean Square Displacement:

Short-Time Behavior Between aPair of Monomers 251

3.4.9.3 Mean Square Displacement of Monomers 2523.4.10 Problems 257

3.5 Dynamics of Rodlike Molecules 2623.5.1 Diffusion Coefficients 2623.5.2 Rotational Diffusion 263

3.5.2.1 Pure Rotational Diffusion 2633.5.2.2 Translation-Rotational Diffusion 266

3.5.3 Dynamic Structure Factor 2663.5.4 Intrinsic Viscosity 2693.5.5 Dynamics of Wormlike Chains 2693.5.6 Problems 270

APPENDICES

3.A: Evaluation of <q,2)eq 2713.B: Evaluation of <exp[ik • (Aq - fip)]> 2733.C: Initial Slope of Si(k,f) 274

4 Thermodynamics and Dynamics of Semidilute Solutions 277

4.1 Semidilute Polymer Solutions 2774.2 Thermodynamics of Semidilute Polymer Solutions 278

4.2.1 Blob Model 2784.2.1.1 Blobs in Semidilute Solutions 2784.2.1.2 Size of the Blob 2794.2.1.3 Osmotic Pressure 2824.2.1.4 Chemical Potential 285

4.2.2 Scaling Theory and Semidilute Solutions 2864.2.2.1 Scaling Theory 2864.2.2.2 Osmotic Compressibility 2894.2.2.3 Correlation Length and Monomer

Density Correlation Function 2894.2.2.4 Chemical Potential 2944.2.2.5 Chain Contraction 2954.2.2.6 Theta Condition 296

4.2.3 Partitioning with a Pore 2984.2.3.1 General Formula 2984.2.3.2 Partitioning at Low Concentrations 2994.2.3.3 Partitioning at High Concentrations 300

4.2.4 Problems 301

xiv CONTENTS

4.3 Dynamics of Semidilute Solutions 3074.3.1 Cooperative Diffusion 3074.3.2 Tube Model and Reptation Theory 310

4.3.2.1 Tube and Primitive Chain 3104.3.2.2 Tube Renewal 3124.3.2.3 Disengagement 3134.3.2.4 Center-of-Mass Motion of the Primitive Chain 3154.3.2.5 Estimation of the Tube Diameter 3184.3.2.6 Measurement of the Center-of-Mass Diffusion

Coefficient 3194.3.2.7 Constraint Release 3204.3.2.8 Diffusion of Polymer Chains in a Fixed Network 3214.3.2.9 Motion of the Monomers 322

4.3.3 Problems 324

References 325Further Readings 326Appendices 328

Al Delta Function 328A2 Fourier Transform 329A3 Integrals 331A4 Series 332

Index 333

PREFACE

The purpose of this textbook is twofold. One is to familiarize senior undergraduateand entry-level graduate students in polymer science and chemistry programs withvarious concepts, theories, models, and experimental techniques for polymer solu-tions. The other is to serve as a reference material for academic and industrialresearchers working in the area of polymer solutions as well as those in charge ofchromatographic characterization of polymers. Recent progress in instrumentation ofsize exclusion chromatography has paved the way for comprehensive one-stop char-acterization of polymer without the need for time-consuming fractionation. Size-exclusion columns and on-line light scattering detectors are the key components inthe instrumentation. The principles of size exclusion by small pores will be explained,as will be principles of light-scattering measurement, both static and dynamic.

This textbook emphasizes fundamental concepts and was not rewritten as a re-search monograph. The author has avoided still-controversial topics such as poly-electrolytes. Each section contains many problems with solutions, some offered toadd topics not discussed in the main text but useful in real polymer solution systems.

The author is deeply indebted to pioneering works described in the famed text-books of de Gennes and Doi/Edwards as well as the graduate courses the authortook at the University of Tokyo. The author also would like to thank his advisorsand colleagues he has met since coming to the U.S. for their guidance.

This book uses three symbols to denote equality between two quantities A and B.

1) 'A = B' means A and B are exactly equal.2) 'A = B' means A is nearly equal to B. It is either that the numerical coefficient

is approximated or that A and B are equal except for the numerical coefficient.3) 'A ~ B' and 'A °c B' mean A is proportional to B. The dimension (unit) may

be different between A and B.

Appendices for some mathematics formulas have been included at the end of thebook. The middle two chapters have their own appendices. Equations in the book-end appendices are cited as Eq. Ax.y; equations in the chapter-end appendices arecited as Eq. x.A.y; all the other equations are cited as Eq. x.y. Important equationshave been boxed.

xv

1Models of Polymer Chains

1.1 INTRODUCTION

1.1.1 Chain Architecture

A polymer molecule consists of the same repeating units, called monomers, or ofdifferent but resembling units. Figure 1.1 shows an example of a vinyl polymer, anindustrially important class of polymer. In the repeating unit, X is one of the mono-functional units such as H, CH3, Cl, and C6H5 (phenyl). The respective polymerswould be called polyethylene, polypropylene, poly(vinyl chloride), and poly-styrene. A double bond in a vinyl monomer CH2=CHX opens to form a covalentbond to the adjacent monomer. Repeating this polymerization step, a polymer mol-ecule is formed that consists of n repeating units. We call n the degree of polymer-ization (DP). Usually, n is very large. It is not uncommon to find polymers with nin the range of 104-105.

In the solid state, polymer molecules pack the space with little voids either in aregular array (crystalline) or at random (amorphous). The molecules are in closecontact with other polymer molecules. In solutions, in contrast, each polymer mole-cule is surrounded by solvent molecules. We will learn in this book about propertiesof the polymer molecules in this dispersed state. The large n makes many of theproperties common to all polymer molecules but not shared by small molecules. Adifference in the chemical structure of the repeating unit plays a secondary role.The difference is usually represented by parameters in the expression of each physi-cal property, as we will see throughout this book.

1

2 MODELS OF POLYMER CHAINS

Figure 1.1. Vinyl polymer.

Figure 1.2 shows three architectures of a polymer molecule: a linear chain (a), abranched chain (b), and a cross-linked polymer (c). A bead represents amonomer here. A vinyl polymer is a typical linear polymer. A branched chain hasbranches, long and short. A cross-linked polymer forms a network encompassingthe entire system. In fact, there can be just one supermolecule in a container. In thebranched chain, in contrast, the branching does not lead to a supermolecule. Across-linked polymer can only be swollen in a solvent. It cannot be dissolved. Wewill learn linear chain polymers in detail and about branched polymers to a lesserextent.

Some polymer molecules consist of more than one kind of monomers. An A-Bcopolymer has two constituent monomers, A and B. When the monomer sequenceis random, i.e., the probability of a given monomer to be A does not depend on itsneighbor, then the copolymer is called a random copolymer. There is a differentclass of linear copolymers (Fig. 1.3). In an A-B diblock copolymer, a whole chainconsists of an A block, a B block, and a joint between them. In a triblock copoly-mer, the chain has three blocks, A, B, and C. The C block can be another A block. Apolymer consisting of a single type of monomers is distinguished from the copoly-mers and is called a homopolymer.

1.1.2 Models of a Linear Polymer Chain

1.1.2.1 Models in a Continuous Space A polymer chain in the solution ischanging its shape incessantly. An instantaneous shape of a polymer chain in

a linear chain b branched chain c cross-linked polymer

Figure 1.2. Architecture of polymer chain: a linear chain (a), a branched chain (b), and across-linked polymer (c).

INTRODUCTION 3

homopolymer A—A A~A

diblock copolymer A A—B B

triblock copolymer A A ~ B B ~ C C

Figure 1.3. Homopolymer and block copolymers.

solution (Fig. 1.4a) is called a conformation. To represent the overall chain confor-mation, we strip all of the atoms except for those on the backbone (Fig. 1.4b).Then, we remove the atoms and represent the chain by connected bonds (Fig. 1.4c).In linear polyethylene, for instance, the chain is now represented by a link ofcarbon-carbon bonds only. We can further convert the conformation to a smoothedline of thread (Fig. 1.4d). In the last model, a polymer chain is a geometrical objectof a thin flexible thread.

We now pull the two ends of the skeletal linear chain to its full extension(Fig. 1.5). In a vinyl polymer, the chain is in all-trans conformation. The distancebetween the ends is called the contour length. The contour length (Lc) is propor-tional to DP or the molecular weight of the polymer. In solution, this fully stretchedconformation is highly unlikely. The chain is rather crumpled and takes a confor-mation of a random coil.

Several coarse-grained geometrical models other than the skeletal chain modelare being used to predict how various physical quantities depend on the chainlength, the polymer concentration, and so forth, and to perform computer simula-tions. Figure 1.6 illustrates a bead-stick model (a), a bead-spring model (b), and apearl-necklace model (c).

In the bead-stick model, the chain consists of beads and sticks that connectadjacent beads. Many variations are possible: (1) the bead diameter and the stickthickness can be any nonnegative value, (2) we can restrict the angle between twoadjacent sticks or let it free, or (3) we can restrict the tortional angle (dihedralangle) of a stick relative to the second next stick. Table 1.1 compares two typicalvariations of the model: a freely jointed chain and a freely rotating chain. Whenthe bond angle is fixed to the tetrahedral angle in the sp3 orbitals of a carbon atomand the dihedral angle is fixed to the one of the three angles corresponding to trans,gauche +, and gauche — , the model mimics the backbone of an actual linear vinylpolymer. The latter is given a special name, rotational isometric state model(RIMS). A more sophisticated model would allow the stick length and the bond

TABLE 1.1 Bead-Stick Models

Model Bond Length Bond Angle Dihedral Angle

Freely jointed chain fixed free freeFreely rotating chain fixed fixed free


Figure 1.4. Simplification of chain conformation from an atomistic model (a) to main-chainatoms only (b), and then to bonds on the main chain only (c), and finally to a flexible threadmodel (d).

angle to vary according to harmonic potentials and the dihedral angle following itsown potential function with local minima at the three angles. In the bead-stickmodel, we can also regard each bead as representing the center of a monomer unit(consisting of several or more atoms) and the sticks as representing just theconnectivity between the beads. Then, the model is a coarse-grained version of amore atomistic model. A bead-stick pair is called a segment. The segment is thesmallest unit of the chain. When the bead diameter is zero, the segment is just astick.

In the bead-spring model, the whole chain is represented by a series of beadsconnected by springs. The equilibrium length of each spring is zero. The bead-spring model conveniently describes the motion of different parts of the chain. Thesegment of this model is a spring and a bead on its end.

In the pearl-necklace model, the beads (pearls) are always in contact withthe two adjacent beads. This model is essentially a bead-stick model with thestick length equal to the bead diameter. The bead always has a positive dia-meter. As in the bead-stick model, we can restrict the bond angle and the dihedralangle.

There are other models as well. This textbook will use one of the models thatallows us to calculate most easily the quantity we need.

1.1.2.2 Models in a Discrete Space The models described in the preceding sec-tion are in a continuous space. In the bead-stick model, for instance, the bead cen-ters can be anywhere in the three-dimensional space, as long as the arrangementsatisfies the requirement of the model. We can construct a linear chain on a discrete

Figure 1.5. A random-coil conformation is pulled to its full length Lc.

INTRODUCTION 5

Figure 1.6. Various models for a linear chain polymer in a continuous space: a bead-stickmodel (a), a bead-spring model (b), and a pearl-necklace model (c).

space as well. The models on a discrete space are widely used in computer simula-tions and theories.

The discrete space is called a lattice. In the lattice model, a polymer chain con-sists of monomers sitting on the grids and bonds connecting them. The grid point iscalled a site; Figure 1.7 illustrates a linear polymer chain on a square lattice (a) anda triangular lattice (b), both in two dimensions. The segment consists of a bond anda point on a site. In three dimensions, a cubic lattice is frequently used and also adiamond lattice to a lesser extent. Figure 1.8 shows a chain on the cubic lattice. Thediamond (tetrahedral) lattice is constructed from the cubic lattice and the bodycenters of the cubes, as shown in Figure 1.9. The chain on the diamond lattice isidentical to the bead-stick model, with a bond angle fixed to the tetrahedral angleand a dihedral angle at one of the three angles separated by 120°. There are otherlattice spaces as well.

The lattice coordinate Z refers to the number of nearest neighbors for a latticepoint. Table 1.2 lists Z for the four discrete models.

1.1.3 Real Chains and Ideal Chains

In any real polymer chain, two monomers cannot occupy the same space. Even apart of a monomer cannot overlap with a part of the other monomer. This effect iscalled an excluded volume and plays a far more important role in polymer solu-tions than it does in solutions of small molecules. We will examine its ramificationsin Section 1.4.

TABLE 1.2 Coordination Number

Dimensions Geometry Z

2 square 42 ' triangular 63 cubic 63 diamond 4


Figure 1.7. Linear chains on a square lattice (a) and a triangular lattice (b).

We often idealize the chain to allow overlap of monomers. In the latticemodel, two or more monomers of this ideal chain can occupy the same site.To distinguish a regular chain with an excluded volume from the ideal chain, wecall the regular chain with an excluded volume a real chain or an excluded-volume chain. Figure 1.10 illustrates the difference between the real chain(right) and the ideal chain (left) for a thread model in two dimensions. Thechain conformation is nearly the same, except for a small part where two partsof the chain come close, as indicated by dashed-line circles. Crossing is allowedin the ideal chain but not in the real chain. The ideal chain does not exist inreality, but we use the ideal-chain model extensively because it allows us tosolve various problems in polymer solutions in a mathematically rigorous way. Wecan treat the effect of the excluded effect as a small difference from the idealchains. More importantly, though, the real chain behaves like an ideal chain insome situations. One situation is concentrated solutions, melts, and glasses. Theother situation is a dilute solution in a special solvent called a theta solvent. We

Figure 1.8. Linear chain on a cubic lattice.

IDEAL CHAINS 7

Figure 1.9. Diamond lattice.

will learn about the theta solvent in Section 2.3 and the concentrated solution inChapter 4.

1.2 IDEAL CHAINS

1.2.1 Random Walk in One Dimension

1.2.1.1 Random Walk A linear flexible polymer chain can be modeled as a ran-dom walk. The concept of the random walk gives a fundamental frame for the con-formation of a polymer chain. If visiting the same site is allowed, the trajectory ofthe random walker is a model for an ideal chain. If not allowed, the trajectory re-sembles a real chain. In this section, we learn about the ideal chains in three dimen-sions. To familiarize ourselves with the concept, we first look at an ideal randomwalker in one dimension.

Figure 1.10. Conformations of an ideal chain (a) and a real chain (b) in two dimensions.


Figure 1.11. Step motion in one-dimensional random walk.

The random walker moves in each step by b either to the right or to the left, eachwith a probability of one-half (Fig. 1.11). Each time it decides where to move nextindependently of its preceding moves. The walker does not have a memoryregarding where it has come from. The latter property is called Markoffian instochastic process theory. The walker can come back to the sites previously visited(ideal). The JV-step trajectory of the random walker is a chain of length Nb foldedone-dimensionally, as illustrated in Figure 1.12. The movement of the randomwalker is specified by a sequence of "+" and "—," with + being the motion tothe right and — being that to the left. In this example the sequence is+ H 1 1 h + + —. Thus one arrangement of the chain foldingcorresponds to an event of having a specific sequence of + and —. Another way tolook at this sequence is to relate + to the head and — to the tail in a series of cointosses.

Suppose there are n "+" out of a total N trials (n — 0, 1,... ,N). Then the ran-dom walker that started at x = 0 on the jc-axis has reached a final position of x =nb + (N — «)(— b) = b(2n — N). How these n+ are arranged is irrelevant to the fi-nal position. What matters is how many + there are. If all are +, x = Nb; if all are—,x= —Nb. The probability Pn to have n+ is given by

The probability distribution is called a binomial distribution, because Pn is equal tothe nth term in the expansion of

Figure 1.12. One-dimensional random walk of 16 steps. The trajectory is a folded chain

IDEAL CHAINS 9

Figure 1.13. Probability distribution for the number n of positive moves. The correspondingfinal position x is also indicated.

with p = q = 1 /2. Thus we find that Pn given by Eq. 1.1 is normalized. An exam-ple of Pn is shown in Figure 1.13 for N = 16. The range of n is between 0 and N,which translates into the range of x between — N and N. Only every other integralvalues of x can be the final position of the random walker for any N.

1.2.1.2 Mean Square Displacement If we set p = q — 1 in Eq. 1.2, we have theidentity

Using the identity, the mean (expectation) of n is calculated as follows:

On the average, the random walker moves half of the steps to the right. Likewise,the average of n2 is calculated as


Then the variance, the mean square of An = n— (n), is

Its square root, (An2)172, called the standard deviation, is a measure for the broad-ness of the distribution. Note that both (n) and (An2) increase linearly with N.Therefore, the relative broadness, (An2}1/2/{n), decreases with increasing N.

Let us translate these statistical averages of n into those of x. Becausex = b(2n - N), the mean and the variance of x are

where Ax = x - (x) is the displacement of the random walker in TV steps. Becausex = 0 before the random walk, AJC = jc. The average of its square, <A;c2>, is calledthe mean square displacement.

1.2.1.3 Step Motion Now we look at the Af-step process from another perspec-tive. Let Ajcn be the displacement in the nth step. Then, A*,, is either b or —b withan equal probability. Therefore, (Ajtn) = 0 and <A;cn

2) = b2. Different steps are notcorrelated. Mathematically, it is described by (AjcnA;cm> = 0 if n ¥^ m. Combiningn = m and n ¥* m, we write

where 8nm is the Kronecker's delta (8nm— 1 if n = m; 8nm = 0 otherwise). In Nsteps, the random walker arrives at x, starting at x = 0. The total displacementAJC = jc - 0 of the N steps is given as

The mean and the variance of AJC are calculated as

As required, the results are identical to those in Eq. 1.7.

1.2.1.4 Normal Distribution Let us see how Pn or P(x) changes when Nincreases to a large number. Figure 1.14 compares Pn for N = 4, 16, and 64. As N

IDEAL CHAINS 11

Figure 1.14. Distribution of the final position x for 4-, 16-, and 64-step random walks.

increases, the plot approaches a continuously curved line. To predict the large Nasymptote of Pn, we use Stirling's formula \nN\ =N(lnN- 1). Equation 1.1 isrewritten to

With n = (N + x/b)/2, this equation is converted to a function of x:

where the Taylor expansion was taken up to the second order of x/(Nb) in the lastpart, because P(x) is almost zero except at small \x/(Nb)\. This equation does notsatisfy the normalization condition because we used a crude version of Stirling'sformula. Normalization leads Eq. 1.13 to

This probability distribution, shown in Figure 1.15, is a normal distribution with azero mean and a variance of Nb2. Note that the mean and the variance are the same


Figure 1.15. Distribution of the final position x for a random walk of infinite number ofsteps.

as those we calculated for its discrete version Pn. Now x is continuously distributed.The probability to find the walker between x and x + dx is given by P(x)dx.

For a large N, the binomial distribution approaches a normal distribution. Thisrule applies to other discrete distributions as well and, in general, is called the lawof large numbers or the central limit theorem. When N» I , the final position x ofthe random walker is virtually continuously distributed along x.

1.2.2 Random Walks in Two and Three Dimensions

1.2.2.1 Square Lattice We consider a random walk on a square lattice extendingin x and y directions with a lattice spacing b, as shown in Figure 1.7a. The randomwalker at a grid point chooses one of the four directions with an equal probabilityof 1/4 (Fig. 1.16). Each step is independent. Again, the random walker can visit thesame site more than once (ideal). The move in one step can be expressed by a dis-placement Arj = [A*,, Ay,]. Similarly to the random walker in one dimension,(Ax,) = <Av!> = 0 and hence (Ar,) = 0. The variances are {A^2} = (Ay,2) = b2/2;therefore, the mean square displacement is (Ar,2) = b2. In a total N steps starting atr = 0, the statistics for the final position r and the displacement Ar are: (x) =<Ax> - 0, <v> = (Ay) = 0 and hence <r> = <Ar> = 0; <jc2> = <Ax2) = M>2/2, (y2) =(Ay2) = M>2/2 and hence <r2> = <Ar2) = Nb2.

The x component of the position after the N-step random walk on the two-dimensional (2D) square lattice has a zero mean and a variance of Nb2/2. WhenN » I , the probability density Px(r) for the x component approaches a normal distri-bution with the same mean and variance. Thus,

IDEAL CHAINS 13

Figure 1.16. Step motion in a two-dimensional random walk on a square lattice.

The y-component Py(r) has a similar expression. When the two components arecombined, we have the joint probability density P(r) = Px(r)Py(r) as

Again, the mean and the variance are held unchanged in the limiting procedure.

1.2.2.2 Lattice in Three Dimensions We place the random walker on a cubiclattice with a lattice spacing b in three dimensions, as shown in Figure 1.8. In eachstep, the random walker chooses one of the six directions with an equal probabilityof 1/6 (Fig. 1.17). The displacement in one step is expressed by Arj = [Ajcl5 Aji,Az,]. Statistical properties of Art and their components are (Ar,) = 0, {A^2) =(Ay,2) = (Az,2> = b2/3; therefore, (Arj2) = b2. In a total N steps starting at r = 0,the statistics for the final position r and the displacement Ar are <r) = <Ar> = 0;(Ax2) = (A/) = <Az2) = Nb2/3 and <r2) = (Ar2) = Nb2.

The x component of the position after the TV-step random walk on the three-dimensional (3D) cubic lattice has a zero mean and a variance of Nb2/3. WhenN » 1, the probability density Px(r) for the x component approaches that of a normaldistribution with the same mean and variance. Thus,

Figure 1.17. Step motion in a three-dimensional random walk on a cubic lattice.

polymer solutions - startseite · colligative properties in nonideal solutions 164 2.a.1 osmotic...

Documents