introduction to physical polymer science

INTRODUCTION TO
PHYSICAL POLYMER
SCIENCE
FOURTH EDITION
L.H. Sperling
Lehigh University
Bethlehem, Pennsylvania
A JOHN WILEY & SONS, INC. PUBLICATION
Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved
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Library of Congress Cataloging-in-Publication Data:
Sperling, L. H. (Leslie Howard), 1932–
Introduction to physical polymer science / L.H. Sperling.—4th ed.
p. cm.
Includes index.
ISBN-13 978-0-471-70606-9 (cloth)
ISBN-10 0-471-70606-X (cloth)
1. Polymers. 2. Polymerization. I. Title.
QD381.S635 2006
668.9—dc22
2005021351
Printed in the United States of America
10987654321
This book is dedicated to the many wonderful graduate and
undergraduate students, post-doctoral research associates, and visiting
scientists who carried out research in my laboratory, and to the very many
more students across America and around the world who studied out of
earlier editions of this book. Without them, this edition surely would not
have been possible. I take this opportunity to wish all of them continued
good luck and good fortune in their careers.
Preface to the Fourth Edition xv
Preface to the First Edition xvii
Symbols and Definitions xix
1 Introduction to Polymer Science 1
1.1 From Little Molecules to Big Molecules / 2
1.2 Molecular Weight and Molecular Weight Distributions / 4

1.3 Major Polymer Transitions / 8
1.4 Polymer Synthesis and Structure / 10
1.5 Cross-Linking, Plasticizers, and Fillers / 18
1.6 The Macromolecular Hypothesis / 19
1.7 Historical Development of Industrial Polymers / 20
1.8 Molecular Engineering / 21
References / 22
General Reading / 22
Handbooks, Encyclopedias, and Dictionaries / 24
Web Sites / 24
Study Problems / 25
Appendix 1.1 Names for Polymers / 26
2 Chain Structure and Configuration 29
2.1 Examples of Configurations and Conformations / 30
2.2 Theory and Instruments / 31
2.3 Stereochemistry of Repeating Units / 36
2.4 Repeating Unit Isomerism / 42
CONTENTS
vii
2.5 Common Types of Copolymers / 45
2.6 NMR in Modern Research / 47
2.7 Multicomponent Polymers / 51
2.8 Conformational States in Polymers / 55
2.9 Analysis of Polymers during Mechanical Strain / 56
2.10 Photophysics of Polymers / 58
2.11 Configuration and Conformation / 63
References / 63
General Reading / 65
Study Problems / 65
Appendix 2.1 Assorted Isomeric and Copolymer

Macromolecules / 67
3 Dilute Solution Thermodynamics, Molecular Weights,
and Sizes 71
3.1 Introduction / 71
3.2 The Solubility Parameter / 73
3.3 Thermodynamics of Mixing / 79
3.4 Molecular Weight Averages / 85
3.5 Determination of the Number-Average Molecular Weight / 87
3.6 Weight-Average Molecular Weights and Radii of Gyration / 91
3.7 Molecular Weights of Polymers / 103
3.8 Intrinsic Viscosity / 110
3.9 Gel Permeation Chromatography / 117
3.10 Mass Spectrometry / 130
3.11 Instrumentation for Molecular Weight Determination / 134
3.12 Solution Thermodynamics and Molecular Weights / 135
References / 136
General Reading / 139
Study Problems / 140
Appendix 3.1 Calibration and Application of Light-Scattering
Instrumentation for the Case Where P(q) = 1 / 142
4 Concentrated Solutions, Phase Separation Behavior,
and Diffusion 145
4.1 Phase Separation and Fractionation / 145
4.2 Regions of the Polymer–Solvent Phase Diagram / 150
viii CONTENTS
4.3 Polymer–Polymer Phase Separation / 153
4.4 Diffusion and Permeability in Polymers / 172
4.5 Latexes and Suspensions / 184
4.6 Multicomponent and Multiphase Materials / 186
References / 186

General Reading / 190
Study Problems / 190
Appendix 4.1 Scaling Law Theories and Applications / 192
5 The Amorphous State 197
5.1 The Amorphous Polymer State / 198
5.2 Experimental Evidence Regarding Amorphous Polymers / 199
5.3 Conformation of the Polymer Chain / 211
5.4 Macromolecular Dynamics / 217
5.5 Concluding Remarks / 227
References / 227
General Reading / 230
Study Problems / 230
Appendix 5.1 History of the Random Coil Model for Polymer
Chains / 232
Appendix 5.2 Calculations Using the Diffusion Coefficient / 236
Appendix 5.3 Nobel Prize Winners in Polymer Science and
Engineering / 237
6 The Crystalline State 239
6.1 General Considerations / 239
6.2 Methods of Determining Crystal Structure / 245
6.3 The Unit Cell of Crystalline Polymers / 248
6.4 Structure of Crystalline Polymers / 256
6.5 Crystallization from the Melt / 260
6.6 Kinetics of Crystallization / 271
6.7 The Reentry Problem in Lamellae / 290
6.8 Thermodynamics of Fusion / 299
6.9 Effect of Chemical Structure on the Melting Temperature / 305
6.10 Fiber Formation and Structure / 307
6.11 The Hierarchical Structure of Polymeric Materials / 311
6.12 How Do You Know It’s a Polymer? / 312

References / 314
General Reading / 320
Study Problems / 320
CONTENTS ix
7 Polymers in the Liquid Crystalline State 325
7.1 Definition of a Liquid Crystal / 325
7.2 Rod-Shaped Chemical Structures / 326
7.3 Liquid Crystalline Mesophases / 326
7.4 Liquid Crystal Classification / 331
7.5 Thermodynamics and Phase Diagrams / 338
7.6 Mesophase Identification in Thermotropic Polymers / 341
7.7 Fiber Formation / 342
7.8 Comparison of Major Polymer Types / 344
7.9 Basic Requirements for Liquid Crystal Formation / 345
References / 346
General Reading / 347
Study Problems / 348
8 Glass–Rubber Transition Behavior 349
8.1 Simple Mechanical Relationships / 350
8.2 Five Regions of Viscoelastic Behavior / 355
8.3 Methods of Measuring Transitions in Polymers / 366
8.4 Other Transitions and Relaxations / 375
8.5 Time and Frequency Effects on Relaxation Processes / 377
8.6 Theories of the Glass Transition / 381
8.7 Effect of Molecular Weight on T
g
/ 397
8.8 Effect of Copolymerization on T
g
/ 399

8.9 Effect of Crystallinity on T
g
/ 404
8.10 Dependence of T
g
on Chemical Structure / 408
8.11 Effect of Pressure on T
g
/ 410
8.12 Damping and Dynamic Mechanical Behavior / 412
8.13 Definitions of Elastomers, Plastics, Adhesives, and Fibers / 415
References / 415
General Reading / 420
Study Problems / 420
Appendix 8.1 Molecular Motion near the Glass Transition / 423
9 Cross-linked Polymers and Rubber Elasticity 427
9.1 Cross-links and Networks / 427
9.2 Historical Development of Rubber / 430
9.3 Rubber Network Structure / 432
9.4 Rubber Elasticity Concepts / 434
x CONTENTS
9.5 Thermodynamic Equation of State / 437
9.6 Equation of State for Gases / 439
9.7 Statistical Thermodynamics of Rubber Elasticity / 442
9.8 The “Carnot Cycle” for Elastomers / 450
9.9 Continuum Theories of Rubber Elasticity / 453
9.10 Some Refinements to Rubber Elasticity / 459
9.11 Internal Energy Effects / 469
9.12 The Flory–Rehner Equation / 472
9.13 Gelation Phenomena in Polymers / 473

9.14 Gels and Gelation / 478
9.15 Effects of Strain on the Melting Temperature / 479
9.16 Elastomers in Current Use / 480
9.17 Summary of Rubber Elasticity Behavior / 488
References / 489
General Reading / 494
Study Problems / 495
Appendix 9.1 Gelatin as a Physically Cross-linked Elastomer / 497
Appendix 9.2 Elastic Behavior of a Rubber Band / 501
Appendix 9.3 Determination of the Cross-link Density of Rubber by
Swelling to Equilibrium / 503
10 Polymer Viscoelasticity and Rheology 507
10.1 Stress Relaxation and Creep / 507
10.2 Relaxation and Retardation Times / 515
10.3 The Time–Temperature Superposition Principle / 529
10.4 Polymer Melt Viscosity / 533
10.5 Polymer Rheology / 538
10.6 Overview of Viscoelasticity and Rheology / 547
References / 548
General Reading / 550
Study Problems / 550
Appendix 10.1 Energy of Activation from Chemical Stress Relaxation
Times / 552
Appendix 10.2 Viscoelasticity of Cheese / 553
11 Mechanical Behavior of Polymers 557
11.1 An Energy Balance for Deformation and Fracture / 557
11.2 Deformation and Fracture in Polymers / 560
11.3 Crack Growth / 585
11.4 Cyclic Deformations / 588
CONTENTS xi

11.5 Molecular Aspects of Fracture and Healing in Polymers / 593
11.6 Friction and Wear in Polymers / 601
11.7 Mechanical Behavior of Biomedical Polymers / 603
11.8 Summary / 606
References / 607
General Reading / 610
Study Problems / 611
12 Polymer Surfaces and Interfaces 613
12.1 Polymer Surfaces / 614
12.2 Thermodynamics of Surfaces and Interfaces / 615
12.3 Instrumental Methods of Characterization / 619
12.4 Conformation of Polymer Chains in a Polymer Blend
Interphase / 644
12.5 The Dilute Solution–Solid Interface / 646
12.6 Instrumental Methods for Analyzing Polymer
Solution Interfaces / 652
12.7 Theoretical Aspects of the Organization of Chains at Walls / 659
12.8 Adhesion at Interfaces / 667
12.9 Interfaces of Polymeric Biomaterials with Living
Organisms / 675
12.10 Overview of Polymer Surface and Interface Science / 677
References / 679
General Reading / 683
Study Problems / 684
Appendix 12.1 Estimation of Fractal Dimensions / 686
13 Multicomponent Polymeric Materials 687
13.1 Classification Schemes for Multicomponent Polymeric
Materials / 688
13.2 Miscible and Immiscible Polymer Pairs / 692
13.3 The Glass Transition Behavior of Multicomponent Polymer

Materials / 693
13.4 The Modulus of Multicomponent Polymeric Materials / 698
13.5 The Morphology of Multiphase Polymeric Materials / 706
13.6 Phase Diagrams in Polymer Blends (Broad Definition) / 710
13.7 Morphology of Composite Materials / 721
13.8 Nanotechnology-Based Materials / 723
13.9 Montmorillonite Clays / 728
xii CONTENTS
13.10 Fracture Behavior of Multiphase Polymeric Materials / 736
13.11 Processing and Applications of Polymer Blends and Composites
/ 741
References / 748
General Reading / 753
Study Problems / 754
14 Modern Polymer Topics 757
14.1 Polyolefins / 757
14.2 Thermoset Polymer Materials / 762
14.3 Polymer and Polymer Blend Aspects of Bread Doughs / 765
14.4 Natural Product Polymers / 769
14.5 Dendritic Polymers and Other Novel Polymeric
Structures / 773
14.6 Polymers in Supercritical Fluids / 779
14.7 Electrical Behavior of Polymers / 782
14.8 Polymers for Nonlinear Optics / 786
14.9 Light-Emitting Polymers and Electroactive Materials / 789
14.10 Optical Tweezers in Biopolymer Research / 794
14.11 The 3-D Structure and Function of Biopolymers / 795
14.12 Fire Retardancy in Polymers / 807
14.13 Polymer Solution-Induced Drag Reduction / 811
14.14 Modern Engineering Plastics / 814

14.15 Major Advances in Polymer Science and Engineering / 815
References / 817
General Reading / 822
Study Problems / 823
Index 827
CONTENTS xiii
PREFACE TO THE
FOURTH EDITION
“So, what’s new in polymer science?”“Much more than most people realize!”
Yes, polymer science and engineering is marching on as it did in the 20th
century, but the emphasis is on new materials and applications.
Two of the most important advances are in the fields of nanocomposites
and biopolymers. The nanocomposites are of two basic types, carbon nan-
otubes and montmorillonite clay exfoliated platelets. The biopolymer aspects
can be traced to such Nobel Prize winning research as Watson and Crick’s
discovery of the double helix structure of DNA and an understanding of how
proteins work in muscles.
Computers are playing increasingly important roles in physical polymer
science. Polymer chain structures may be made to undergo Monte Carlo sim-
ulations to gain new insight as to how polymers crystallize, for example.
Polymer science was born of the need to understand how rubber and plas-
tics work. This speaks of the practicality of the subject from the beginning.
Today, polymers form the basis of clothing, automobile parts, etc. Yet, in fact,
today we are seeing a shift from theory to new applications, to such topics as
electronics and fire resistance.
All of these topics are covered in this fourth edition.There are,as the reader
might imagine, many other topics demanding consideration.Alas, my goal was
to create a readable introductory textbook, and not an encyclopedia!
I want to take this opportunity to thank the many students who helped in
proofreading the manuscript for this book. Many thanks must also be given

to the Department of Chemical Engineering and the Department of Ma-
terials Science and Engineering, as well as the newly renamed Center for
Advanced Materials and Nanotechnology, and the Center for Polymer Science
and Engineering at Lehigh University. Special thanks are due to Prof.
Raymond Pearson, who made valuable suggestions for this edition. Special
thanks are also due to Ms. Gail Kriebel, Ms. Bess King, and the staff at the
E. W. Fairchild-Martindale Library, who helped with literature searching, and
provided me with a carrel right in the middle of the stacks.
xv
To all of the students, faculty, and industrial scientists and engineers who
read this textbook, good luck in your careers and lives!
Bethlehem, Pennsylvania L. H. Sperling
February 2005
xvi PREFACE TO THE FOURTH EDITION
PREFACE TO THE
FIRST EDITION
Research in polymer science continues to mushroom, producing a plethora of
new elastomers, plastics, adhesives, coatings, and fibers. All of this new infor-
mation is gradually being codified and unified with important new theories
abut the interrelationships among polymer structure, physical properties, and
useful behavior.Thus the ideas of thermodynamics,kinetics, and polymer chain
structure work together to strengthen the field of polymer science.
Following suit, the teaching of polymer science in colleges and universities
around the world has continued to evolve. Where once a single introductory
course was taught, now several different courses may be offered.The polymer
science and engineering courses at Lehigh University include physical polymer
science, organic polymer science, and polymer laboratory for interested
seniors and first-year graduate students, and graduate courses in emulsion
polymerization, polymer blends and composites, and engineering behavior of
polymers. There is also a broad-based introductory course at the senior level

for students of chemical engineering and chemistry. The students may earn
degrees in chemistry, chemical engineering, metallurgy and materials engi-
neering, or polymer science and engineering, the courses being both inter-
disciplinary and cross-listed.
The physical polymer science course is usually the first course a polymer-
interested student would take at Lehigh, and as such there are no special pre-
requisites except upper-class or graduate standing in the areas mentioned
above. This book was written for such a course.
The present book emphasizes the role of molecular conformation and con-
figuration in determining the physical behavior of polymers. Two relatively
new ideas are integrated into the text. Small-angle neutron scattering is doing
for polymers in the 1980s what NMR did in the 1970s, by providing an entirely
new perspective of molecular structure. Polymer blend science now offers
thermodynamics as well as unique morphologies.
Chapter 1 covers most of the important aspects of the rest of the text in a
qualitative way. Thus the student can see where the text will lead him or her,
having a glimpse of the whole. Chapter 2 describes the configuration of
xvii
polymer chains, and Chapter 3 describes their molecular weight. Chapter 4
shows the interactions between solvent molecules and polymer molecules.
Chapters 5–7 cover important aspects of the bulk state, both amorphous and
crystalline, the glass transition phenomenon, and rubber elasticity.These three
chapters offer the greatest depth. Chapter 8 describes creep and stress relax-
ation, and Chapter 9 covers the mechanical behavior of polymers, emphasiz-
ing failure, fracture, and fatigure.
Several of the chapters offer classroom demonstrations, particularly
Chapters 6 and 7. Each of these demonstrations can be carried out inside a
50-minute class and are easily managed by the students themselves. In fact, all
of these demonstrations have been tested by generations of Lehigh students,
and they are often presented to the class with a bit of showmanship. Each

chapter is also accompanied by a problem set.
The author thanks the armies of students who studied from this book in
manuscript form during its preparation and repeatedly offer suggestions rel-
ative to clarity, organization, and grammar. Many researchers from around the
world contributed important figures. Dr. J. A. Manson gave much helpful
advice and served as a Who’s Who in highlighting people, ideas, and history.
The Department of Chemical Engineering, the Materials Research Center,
and the Vice-President for Research’s Office at Lehigh each contributed sig-
nificant assistance in the development of this book. The Lehigh University
Library provided one of their carrels during much of the actual writing. In par-
ticular, the author thanks Sharon Siegler and Victoria Dow and the staff at
Mart Library for patient literature searching and photocopying. The author
also thanks Andrea Weiss, who carefully photographed many of the figures in
this book.
Secretaries Jone Susski, Catherine Hildenberger, and Jeanne Loosbrock
each contributed their skills. Lastly, the person who learned the most from the
writing of this book was
Bethlehem, Pennsylvania L. H. Sperling
November 1985
xviii PREFACE TO THE THIRD EDITION
SYMBOL DEFINITION SECTION
English Alphabet
AA
2
= second virial coefficient 3.3.2
A
1
= first virial coefficient 3.5.3.3
A
3

= third virial coefficient 3.5.3.3
A
4
= fourth virial coefficient 3.5.3.3
A (with various subscripts) = area under a Bragg 6.5.4
diffraction line
Angular amplitude 8.3.3
A
T
= reduced variables shift factor 8.6.1.2
Surface area (with various subscripts) 12.2.3
B Bulk modulus 8.1.1.2, 8.1.4
CC* = chiral center, optically active carbon 2.3.2, 2.4.1
C
m
= constant 3.3.2
C
N
= neutron scatting equivalent of H 5.2.2.1
C
•
= characteristic ratio 5.3.1.1
DC
p
= change in heat capacity 6.1
CI = crystallinity index 6.5.4
C
p
= heat capacity 8.2.9
C

1
, C
2
= Mooney–Rivlin constants 9.9.1
C
100
, C
010
, C
200
, C
400
, C, C¢, C≤, = generalized strain 9.9.2
energy constants
C
1
¢, C
2
¢=WLF constants 10.4.1
C
A
= concentration of A A10.1
C
p
, C
v
= capacitance of polymer and vacuum 14.7.1
SYMBOLS AND
DEFINITIONS
xix

SYMBOL DEFINITION SECTION
D Diffusion coefficient 3.6.6,
4.4.2,
5.4.2.1
Disk diameter 13.9.3
D
e
= Deborah number 10.2.4
D¢=fractal dimension 12.7.3
D
2
= IPN phase domain size 13.5.4
D = Tensile compliance 8.1.6
E Young’s modulus 1.3, 8.1.1.1
DE = change in energy 2.2.4
E
act
= energy of activation 2.8, 8.6.1.2
E* = complex Young’s (tensile) modulus 8.1.8
E¢=storage modulus 8.1.8
E≤=loss modulus 8.1.8
E
1
, E
2
, etc. = spring moduli 10.1.2.1
Elongational compliance 8.1.6
F Helmholtz free energy 9.5
G Gibbs’s free energy 3.2
Group molar attraction constant 3.2.3

DG
M
= change in free energy on mixing 3.2
G
N
0
= steady-state rubbery shear modulus 5.4.2.1
Radial growth rate of crystal 6.6.2.2
Shear modulus 8.1.1.1
G* = complex shear modulus 8.1.8
G¢=shear storage modulus 8.2.9
G≤=shear loss modulus 8.2.9
G = fracture energy 11.1.2
G
c
= critical energy of crack growth 11.1.2
G
lc
= critical energy of crack growth on extension 11.5.2.4
G
s
= surface free energy 12.2.1
HH
0
= magnetic field 2.2.4
Optical constant 3.6
DH
f
= enthalpy of fusion 6.1
dH = NMR absorption line width 8.3.4

Heat energy per unit volume per cycle 8.12
Wool’s general function 11.5.3
H
s
= surface enthalpy 12.2.1
II
D
= dimer emission intensity 2.10.3.1
I
M
= single mer emission intensity 2.10.3.1
xx SYMBOLS AND DEFINITIONS
SYMBOL DEFINITION SECTION
II
1
, I
2
, I
3
= strain invarients 9.9.2
Current 14.7.1
J Flux 4.4.2
J
n
= de Gennes defect current 5.4.2.1
Compliance (with various subscripts) 5.4.2.1,
8.1.1.2
Shear compliance 8.1.1, 8.1.6
J* = complex compliance 8.1.6
J¢, J≤=storage and loss compliance 10.2.4

KK
˜
= constant 3.6.1
Wave vector 3.6.1,
5.2.2.1,
12.3.8.1
Equilibrium constant of polymerization 3.7.2.1
Constant in the Mark–Houwink–Sakurada 3.8.3
equation
K
d
= distribution coefficient 3.9.2
K
¯
= constant relating end-to-end distance to 4.3.9
molecular weight
K
1
, K
2
= measures of free volume 8.6.1.1
K
L
, K
H
= constants in melt viscosity 10.4.2.1
K = stress intensity factor 11.2.4.1,
11.3.2
K
1c

, K
2c
, K
3c
= critical stress intensity factor in the 11.2.4.1,
extension, shear, and tearing modes 11.3.2
DK = stress intensity factor range 11.4.2
L Sample length 9.4
L(x), L(b) = inverse Langevin function 9.10.1
L
1
, L
2
= transverse lengths 12.3.8.1
2L
0
= separation length 12.6.1
M Molecular weight
M
n
= number-average molecular weight 1.2.1, 3.4
M
w
= weight-average molecular weight 3.4
M
z
= z-average molecular weight 3.4
M
v
= viscosity-average molecular weight 3.4

M
1
, M
2
= mass fractions 8.8.1
M
c
= number-average molecular weight between 9.4
cross-links
SYMBOLS AND DEFINITIONS xxi
SYMBOL DEFINITION SECTION
MM
e
= molecular weight between entanglements 9.4
M
c
¢=entanglement molecular weight 9.4, 10.4.2.3
Mass 11.1.2
NN
i
= number of molecules of molecular weight M
i
1.2.2
Number of cells 3.3.1.2
N
A
= Avogadro’s number 3.3.2
N
c
= number of molecules in 1cm

3
4.3.2
N
e
= number of mers between entanglements 11.5.2.2
O
PP(q) = single-chain form factor 3.6.1
P
c
= critical extent of reaction at the gel point 3.7.4
P
˜
= reduced pressure 4.3
P* = characteristic value 4.3
Permeability coefficient 4.4.2
P
1
= probability of a chain arm folding back 5.4.3
on itself
Probability of barriers being surmounted 8.6.1.2
P
i
= induced polarization 14.8.1
Q Partition function 8.6.3.1
Q
I
, Q
II
= amounts of heat released 9.8.3
R Gas constant 1.3

R• = free radical 1.4.1.2
R = generalized organic group 1.4.1.2
R(q) = Rayleigh’s ratio 3.6
R
e
= Reynolds number 14.13.1
R
g
= radius of gyration 3.6.1
R
i
= rate of initiation 3.7.2.2
R
p
= rate of propagation 3.7.2.2
R
t
= rate of termination 3.7.2.2
R
e
= hydrodynamic sphere equivalent radius 3.8.2
R = ratio of radii of gyration 9.10.6
R
¯
= fracture resistance 11.1.2
Resistance in ohms 14.7.1
S Entropy 3.2
DS
M
= change in entropy on mixing 3.2

Solubility coefficient 4.4.2
Scaling variable A4.1
S
k
= mean separation distance 6.6.2.5
xxii SYMBOLS AND DEFINITIONS
SYMBOL DEFINITION SECTION
S Disclination strength 7.6
S
s
= surface entropy 12.2.1
S
th
= interphase surface thickness 12.3.7.2
T Absolute temperature 1.3
T
f
= fusion or melting temperature 1.1, 6.1
T
g
= glass transition temperature 1.3
DT
b
= boiling point elevation 3.5.2
DT
f
= freezing point depression 3.5.2
T
˜
= reduced temperature 4.3

T* = characteristic temperature 4.3
T = Fraction of light transmitted 5.2.1
T
f
* = equilibrium melting temperature of crystals 6.8.5
T
ll
= liquid–liquid transition 8.4
T
0
= generalized transition temperature 8.6.1.2
T
s
= arbitrary WLF temperature 8.6.1.2
T
2
= unifying treatment of the second-order glass 8.6.3.4
transition temperature
T
e
= fraction of trapped entanglements 9.10.5.1
T
e
, T
R
, T
d
= relaxation times 10.2.5
T
r

= reptation time 10.2.5
UU
max
= maximum in scattering intensity in the 6.5.1
radial direction
Internal energy 9.5
dU
1
, dU
2
, dU
3
, dU
4
, = energies related to fracture 11.1.1
V Molar volume 3.2
V
s
= scattering volume 3.6
V
e
= hydrodynamic sphere volume 3.8.2
V = reduced volume 4.3
V* = characteristic volume 4.3
V
r
= volume of one cell 4.3.2
V
0
= occupied volume 8.6.2.1

V
t
= specific volume 8.6.2.1
V
1
= molar volume of solvent 9.12
Voltage 14.7.1
W Work on elongation 9.7.2
W
a
= work of adhesion 12.3.7.2
W
g
= Weight fraction gel 9.10.5.1
SYMBOLS AND DEFINITIONS xxiii
SYMBOL DEFINITION SECTION
X Brownian motion average distance traversed 5.2.2.1
X
t
= degree of crystallinity 6.6.2.1
X
B
= mole fraction of impurity 6.8.1
X
1
, X
2
= mole fractions 8.8.1
X(t) = average mer interpenetration depth 10.2.6
function of time

X(•) = interpenetration depth constant 10.2.6
Y
Z Avrami constant 6.6.2.1
Constant in cross-link density calculations 8.6.3.2
Number of carbon atoms in a chain’s backbone 10.4.2.1
Zw Backbone atoms per chain 10.4.2
a Exponent in the Mark–Houwink–Sakurada 3.8.3
equation
a
H
, a
D
= scattering lengths 5.2.2.1
End-to-end distance of a Rouse–Bueche segment 5.4.1
Cell axis distance 6.1.1
van der Wall’s constant 9.6
Half the crack length 11.3.1
Correlation distance 12.3.8.1
bb
1
, b
2
= polarizabilities 5.2.1
Kuhn segment length 5.3.1.2
Defect stored length 5.4.2.1
Cell axis distance 6.1.1
van der Wall’s constant 9.6
Statistical segment step length 12.3.7.2
c Solute concentration 3.5.2
Cell axis distance 6.1.1

d Thickness 5.2.1
Bragg distance 6.2.2
Domain period 13.6.2.1
e
f Functionality of branch units 3.7.4
f
0
= frictional coefficient 3.8.2
Function A4.1
f
I
= orientation function 5.2.1
Restoring force 5.4.1
Fractional free volume 8.6.1.2
xxiv SYMBOLS AND DEFINITIONS
SYMBOL DEFINITION SECTION
ff
0
= fractional free volume at T
g
8.6.1.2
Retractive force 9.5
f
e
= energetic portion of the retractive force 9.5
f
s
= entropic portion of the retractive force 9.5
Force on a chain 9.7.1
f* = network functionality 9.10.2

f
xx
, f
yy
, f
zz
= stress components 10.5.2
g Gauche 2.1.2
h Planck’s constant 2.10.2
ii
e
= Thomson scattering factor 3.6.1
Square root of minus one 8.1.8
j
k Boltzmann’s constant 2.8
k
i
= rate constant of initiation 3.7.2.2
k
p
= rate constant of propagation 3.7.2.2
k
t
= rate constant of termination 3.7.2.2
k¢=Huggins’s constant 3.8.4
k≤=Kraemer’s constant 3.8.4
l Length of a link or mer 5.3.1.1
ഞ = crystal thickness 6.4.2.1
m Meso, same side 2.3.3
Mass of a polymer chain 3.10

n Number of mers in the chain 1.1
Number of network chains per unit volume 1.3
Mole fraction 3.3.1.1
Refractive index 3.6, 5.2.1
Any whole number 6.2.2
Avrami constant 6.6.2.1
n
c
, n
p
= chemical and physical cross-links 9.10.5.1
n
tot
= total number of effective cross-links 9.10.5.1
Number of stress cycles 11.4.2
n(t), n
•
= number of chains intersecting a unit area 11.5.3
of interface at t and at infinite time
o
p Partial vapor pressure 3.3.1.1
Fractional conversion 3.7.2.3
Persistence length 4.2
SYMBOLS AND DEFINITIONS xxv
SYMBOL DEFINITION SECTION
p Number of pitches 6.3.2
Probability of Avrami crystal fronts crossing 6.6.2.1
1/p
2
= measure of stiffness 8.3.3

p
1
= probability of finding a molecule 9.6
qq
1
, q
2
= heat absorbed and released 9.8.3
r Racmic-opposite side 2.3.3
End-to-end distance 3.6.2,
9.7.1,
10.2.7
r˜
0
2
= root-mean square end-to-end distance of 3.9.7
a chain
Reptation rate 6.6.2.5
r
i
¯
2
, r
0
¯
2
= mean square end-to-end distances of 9.10.4
swollen and relaxed chains
r
y

= crack-tip plastic zone radius 11.3.2
Exponent in interface theory 11.5.3
s
t Trans 2.1.2
Time 3.7.2.4
Exponent in interface theory 11.5.3
u Intermolecular excluded volume 3.3.2
u¯ = Stokes terminal velocity 10.5.4
v Volume fraction 3.2
v
2
= volume fraction of polymer 4.1.2,
9.10.4
Excluded volume parameter 4.2
v
2
* = critical volume concentration 7.5.1
v
f
= specific free volume 8.6.1.1
v
0
= occupied volume 8.6.1.2
Velocity of chain pullout 11.5.2.2
w Distance from source 3.6
x Mole fraction 4.3.6
General parameter A4.1
Number of mers in chain 3.3.1.2
Axial ratio of liquid crystalline molecule 7.5.1
y

z Charge on the polymer 3.10
xxvi SYMBOLS AND DEFINITIONS
SYMBOL DEFINITION SECTION
Greek Alphabet
A
B
G
D Logarithmic decrement 8.12
E
Z
H
Q
I
K
L
M
N
X
O
P
R
S dS/dW=scattering cross section 5.2.2.1
T
U
F Universal constant in intrinsic viscosity 3.8.3
C
Y Entropic factor 3.3.2
Y
1
= constant 4.1.2

W Number of possible arrangements in space 3.3.1.2
Solid angle 5.2.2.1
Probability of finding all the molecules 9.6
W
1
= angular velocity 10.5.4
aa
x
= mechanically induced peak frequency shift 2.9
Expansion of a polymer coil in a good solvent 3.8.2
a
A/B
= gas selectivity ratio 4.4.6.2
Volumetric coefficient of expansion 8.3
a
R
= cubic expansion coefficient in the rubbery 8.6.1.1
state
a
G
= cubic expansion coefficient in the glassy state 8.6.1.1
SYMBOLS AND DEFINITIONS xxvii
SYMBOL DEFINITION SECTION
aa
f
= expansion coefficient of the free volume 8.6.1.2
Extension ratio 9.4
bb
1
= lattice constant 3.3.2

Compressibility 8.1.4
b
f
= compressibility free volume 8.11
Gaussian distribution term 9.7.1
g Number of flexible bonds per mer 8.6.3.2
Shear strain 8.1.1.2
g
.
= shear rate 10.5.1
g
s
= surface tension (intrinsic surface energy) 11.3.1
g
p
= plastic deformation energy 11.3.2
Surface tension 12.2.1
g (r) = Debye correlation function 12.3.8.1
g
xx
Perpendicular stress 10.5.2
d Solubility parameter 3.2
Measure of internal structure 6.6.2.2
tan d = loss tangent 8.2.9
e Normal strain 8.1.1
e* = van der Waals energy of interaction 4.3.4
Tensile strain 8.1.1.1
e¢, e≤=dielectric storage and loss constants 8.3.4,
14.7.1
z

h Viscosity of a solution 3.8.1
h
0
= viscosity of the solvent 3.8.1
h
rel
= relative viscosity 3.8.1
h
sp
= specific viscosity 3.8.1
[h] = intrinsic viscosity 3.8.1
Shear viscosity (of a polymer melt) 8.1.1.2
h
g
= melt viscosity at T
g
8.6.1.2
h
2
, h
3
, etc. = dashpot viscosities 10.1.2.2
h¢, h≤=storage and loss viscosities 10.5.3
h* = complex viscosity 10.5.3
q Flory q-temperature 3.3.2
Angle of scatter 3.6.1
ı
k
xxviii SYMBOLS AND DEFINITIONS