Preface ....................................................... xix
Part One Glass transitions in Amorphous Polymers
1 Glass Transitions in Amorphous Polymers: Basic Concepts ...... 3
1.1 Phase Transitions in Amorphous Materials ................ 3
1.2 Volume-Temperature and Enthalpy-Temperature Relations
in the Vicinity of First-Order and Second-Order Phase
Transitions: Discontinuous Thermophysical Properties
at Tm and Tg ............................................ 4
1.3 The Equilibrium Glassy State ............................ 8
1.4 Physical Aging, Densification, and Volume and Enthalpy
Relaxation .............................................. 8
1.5 Temperature-Pressure Differential Phase Equilibrium
Relations for First-Order Processes: The Clapeyron
Equation ............................................... 10
1.6 Temperature-Pressure Differential Phase EquiUbrium
Relations for Second-Order Processes: The Ehrenfest
Equations .............................................. 11
1.7 Compositional Dependence of Tg via Entropy
Continuity ............................................. 15
1.8 Compositional Dependence of Tg via Volume Continuity ... 18
1.9 Linear Least Squares Analysis of the Gordon-Taylor
Equation and Other Tg-Composition Relations for
Binary Mixtures ........................................ 20
1.10 Free Volume Concepts ................................... 21
1.11 Temperature Dependence of Fractional Free Volume ....... 22
1.12 Compositional Dependence of Fractional Free Volume
and Plasticizer Efficiency for Binary Mixtures ......... 23
1.13 Fractional Free Volume Analysis of Multicomponent
Mixtures: Compositional Dependence of the Glass
Transition Temperature ................................. 25
1.14 Molecular Weight Dependence of Fractional Free
Volume ................................................. 26
1.15 Experimental Design to Test the Molecular Weight
Dependence of Fractional Free Volume and Tg ............ 27
1.16 Pressure Dependence of Fractional Free Volume .......... 29
1.17 Effect of Particle Size or Film Thickness on the
Glass Transition Temperature ........................... 31
1.18 Effect of the Glass Transition on Surface Tension ...... 34
References .................................................. 35
Problems .................................................... 36
2 Diffusion in Amorphous Polymers Near the Glass Transition
Temperature ................................................. 49
2.1 Diffusion on a Lattice ................................. 49
2.2 Overview of the Relation Between Fractional Free
Volume and Diffusive Motion of Liquids and Gases
Through Polymeric Membranes ............................ 50
2.3 Free Volume Theory of Cohen and Turnbull for
Diffusion in Liquids and Glasses ....................... 51
2.4 Free Volume Theory of Vrentas and Duda for Solvent
Diffusion in Polymers Above the Glass Transition
Temperature ............................................ 55
2.5 Influence of the Glass Transition on Diffusion in
Amorphous Polymers ..................................... 58
2.6 Analysis of Half-Times and Lag Times via the Unsteady
State Diffusion Equation ............................... 61
2.7 Example Problem: Effect of Molecular Weight
Distribution Functions on Average Diffusivities ........ 66
References .................................................. 69
3 Lattice Theories for Polymer-Small-Molecule Mixtures and
the Conformational Entropy Description of the Glass
Transition Temperature ...................................... 71
3.1 Lattice Models in Thermodynamics ....................... 72
3.2 Membrane Osmometry and the Osmotic Pressure
Expansion .............................................. 72
3.3 Lattice Models for Athermal Mixtures with Excluded
Volume ................................................. 76
3.4 Flory-Huggins Lattice Theory for Flexible Polymer
Solutions .............................................. 79
3.5 Chemical Stability of Binary Mixtures .................. 89
3.6 Guggenheim's Lattice Theory of Athermal Mixtures ...... 105
3.7 Gibbs-DiMarzio Conformational Entropy Description of
the Glass Transition for Tetrahedral Lattices ......... 117
3.8 Lattice Cluster Theory Analysis of Conformational
Entropy and the Glass Transition in Amorphous
Polymers .............................................. 123
3.9 Sanchez-Lacombe Statistical Thermodynamic Lattice
Fluid Theory of Polymer- Solvent Mixtures ............. 126
Appendix: The Connection Between Exothermic Energetics
and Volume Contraction of the Mixture ............ 128
References ................................................. 131
Problems ................................................... 132
4 de Electric Field Effects on First- and Second-Order
Phase Transitions in Pure Materials and Binary Mixtures .... 137
4.1 Electric-Field-Induced Alignment and Phase
Separation ............................................ 137
4.2 Overview .............................................. 138
4.3 Electric Field Effects on Low-Molecular-Weight
Molecules and Their Mixtures .......................... 138
4.4 Electric Field Effects on Polymers and Their
Mixtures .............................................. 139
4.5 Motivation for Analysis of Electric Field Effects on
Phase Transitions ..................................... 141
4.6 Theoretical Considerations ............................ 141
4.7 Summary ............................................... 166
Appendix: Nomenclature ..................................... 167
References ................................................. 168
5 Order Parameters for Glasses: Pressure and Compositional
Dependence of the Glass Transition Temperature ............. 171
5.1 Thermodynamic Order Parameters ........................ 171
5.2 Ehrenfest Inequalities: Two Independent Internal
Order Parameters Identify an Inequality Between the
Two Predictions for the Pressure Dependence of the
Glass Transition Temperature .......................... 172
5.3 Compositional Dependence of the Glass Transition
Temperature ........................................... 177
5.4 Diluent Concentration Dependence of the Glass
Transition Temperature via Classical Thermodynamics ... 181
5.5 Compositional Dependence of the Glass Transition
Temperature via Lattice Theory Models ................. 183
5.6 Comparison with Other Theories ........................ 184
5.7 Model Calculations .................................... 186
5.8 Limitations of the Theory ............................. 188
References ................................................. 188
Problem .................................................... 189
6 Macromolecule-Metal Complexes: Ligand Field Stabilization
and Glass Transition Temperature Enhancement ............... 191
6.1 Ligand Field Stabilization ............................ 191
6.2 Overview .............................................. 192
6.3 Methodology of Transition-Metal Coordination in
Polymeric Complexes ................................... 193
6.4 Pseudo-Octahedral d8 Nickel Complexes with Poly(4-
vinylpyridine) ........................................ 209
6.5 d6 Molybdenum Carbonyl Complexes with
Poly(vinylamine) that Exhibit Reduced Symmetry Above
the Glass Transition Temperature ...................... 216
6.6 Cobalt, Nickel, and Ruthenium Complexes with Poly(4-
vinylpyridine) and Poly(L-histidine) that Exhibit
Reduced Symmetry in the Molten State .................. 224
6.7 Total Energetic Requirements to Induce the Glass
Transition via Consideration of the First-Shell
Coordination Sphere in Transition Metal and
Lanthanide Complexes .................................. 238
6.8 Summary ............................................... 241
6.9 Epilogue .............................................. 241
Appendix: Physical Interpretation of the Parameters in
the Kwei Equation for Synergistic Enhancement
of the Glass Transition Temperature in Binary
Mixtures ......................................... 243
References ................................................. 243
Part Two Semicrystalline Polymers and Melting Transitions
7 Basic Concepts and Molecular Optical Anisotropy in
Semicrystalline Polymers ................................... 249
7.1 Spherulitic Superstructure ............................ 249
7.2 Comments about Crystallization ........................ 250
7.3 Spherulitic Superstructures that Exhibit Molecular
Optical Anisotropy .................................... 255
7.4 Interaction of a Birefringent Spherulite with
Polarized Light ....................................... 258
7.5 Interaction of Disordered Lamellae with Polarized
Light ................................................. 260
7.6 Interaction of Disordered Lamellae with Unpolarized
Light ................................................. 261
7.7 Molecular Optical Anisotropy of Random Coils and
Rigid Rod-Like Polymers ............................... 263
7.8 Birefringence of Rubbery Polymers Subjected to
External Force Fields ................................. 278
7.9 Chain Folding, Interspherulitic Connectivity, and
Mechanical Properties of Semicrystalline Polymers ..... 279
References ................................................. 282
Problems ................................................... 283
8 Crystallization Kinetics via Spherulitic Growth ............ 287
8.1 Nucleation and Growth ................................. 287
8.2 Heterogeneous Nucleation and Growth Prior to
Impingement ........................................... 288
8.3 Avrami Equation for Heterogeneous Nucleation that
Accounts for Impingement of Spherulites ............... 289
8.4 Crystallization Kinetics and the Avrami Equation
for Homogeneous Nucleation of Spherulites ............. 292
8.5 Linear Least Squares Analysis of the Kinetics of
Crystallization via the Generalized Avrami
Equation .............................................. 293
8.6 Half-Time Analysis of Crystallization Isotherms ....... 296
8.7 Maximum Rate of Isothermal Crystallization ............ 297
8.8 Thermodynamics and Kinetics of Homogeneous
Nucleation ............................................ 299
8.9 Temperature Dependence of the Crystallization Rate
Constant .............................................. 302
8.10 Optimum Crystallization Temperatures: Comparison
Between Theory and Experiment ......................... 304
8.11 The Energetics of Chain Folding in Semicrystalline
Polymer-Polymer Blends that Exhibit Multiple Melting
Endotherms ............................................ 307
8.12 Melting Point Depression in Polymer-Polymer and
Polymer-Diluent Blends that Contain a High-
Molecular-Weight Semicrystalline Component ............ 317
References ................................................. 322
Problems ................................................... 322
9 Experimental Analysis of Semicrystalline Polymers .......... 329
9.1 Semicrystal Unity ..................................... 329
9.2 Differential Scanning Calorimetry: Thermograms of
Small Molecules that Exhibit Liquid Crystalline
Phase Transitions Below the Melting Point ............. 330
9.3 Isothermal Analysis of Crystallization Exotherms
via Differential Scanning Calorimetry ................. 331
9.4 Kinetic Analysis of the Mass Fraction of
Crystallinity via the Generalized Avrami Equation ..... 335
9.5 Measurements of Crystal Unity via Differential
Scanning Calorimetry .................................. 337
9.6 Analysis of Crystallinity via Density Measurements .... 339
9.7 Pychnometry: Density and Thermal Expansion
Coefficient Measurements of Liquids and Solids ........ 340
References ................................................. 344
Problems ................................................... 344
Part Three Mechanical Properties of Linear Crosslinked
Polymers
10 Mechanical Properties of Viscoelastic Materials: Basic
Concepts in Linear Viscoelasticity ......................... 355
10.1 Mathematical Models of Linear Viscoelasticity ......... 355
10.2 Objectives ............................................ 356
10.3 Simple Definitions of Stress, Strain, and Poisson's
Ratio ................................................. 356
10.4 Stress Tensor ......................................... 357
10.5 Strain and Rate-of-Strain Tensors ..................... 358
10.6 Hooke's Law of Elasticity ............................. 359
10.7 Newton's Law of Viscosity ............................. 360
10.8 Simple Analogies Between Mechanical and Electrical
Response .............................................. 360
10.9 Phase Angle Difference Between Stress and Strain
and Voltage and Current in Dynamic Mechanical and
Dielectric Experiments ................................ 361
10.10 Maxwell's Viscoelastic Constitutive Equation ......... 362
10.11 Integral Forms of Maxwell's Viscoelastic
Constitutive Equation ................................ 364
10.12 Mechanical Model of Maxwell's Viscoelastic
Constitutive Equation ................................ 366
10.13 Four Well-Defined Mechanical Experiments ............. 367
10.14 Linear Response of the Maxwell Model during Creep
Experiments .......................................... 368
10.15 Creep Recovery of the Maxwell Model .................. 369
10.16 Linear Response of the Maxwell Model during Stress
Relaxation ........................................... 370
10.17 Temperature Dependence of the Stress Relaxation
Modulus and Definition of the Deborah Number ......... 372
10.18 Other Combinations of Springs and Dashpots ........... 373
10.19 Equation of Motion for the Voigt Model ............... 374
10.20 Linear Response of the Voigt Model in Creep
Experiments .......................................... 376
10.21 Creep Recovery of the Voigt Model .................... 376
10.22 Creep and Stress Relaxation for a Series
Combination of Maxwell and Voigt Elements ............ 377
10.23 The Principle of Time-Temperature Superposition ...... 385
10.24 Stress Relaxation via the Equivalence Between Time
and Temperature ...................................... 385
10.25 Semi Theoretical Justification for the Empirical
Form of the WLF Shift Factor αT(T; Treference) ........ 389
10.26 Temperature Dependence of the Zero-Shear-Rate
Polymer Viscosity via Fractional Free Volume and
the Doolittle Equation ............................... 390
10.27 Apparent Activation Energy for αT and the Zero-
Shear-Rate Polymer Viscosity ......................... 392
10.28 Comparison of the WLF Shift Factor αT at Different
Reference Temperatures ............................... 393
10.29 Vogel's Equation for the Time-Temperature Shift
Factor ............................................... 394
10.30 Effect of Diluent Concentration on the WLF Shift
Factor ac in Concentrated Polymer Solutions .......... 394
10.31 Stress Relaxation Moduli via the Distribution of
Viscoelastic Time Constants ......................... 397
10.32 Stress Relaxation Moduli and Terminal Relaxation
Times ................................................ 400
10.33 The Critical Molecular Weight Required for
Entanglement Formation ............................... 403
10.34 Zero-Shear-Rate Viscosity via the Distribution of
Viscoelastic Relaxation Times ........................ 403
10.35 The Boltzmann Superposition Integral for Linear
Viscoelastic Response ................................ 405
10.36 Alternate Forms of the Boltzmann Superposition
Integral for σ(t) .................................... 406
10.37 Linear Viscoelastic Application of the Boltzmann
Superposition Principle: Elastic Free Recovery ....... 407
10.38 Dynamic Mechanical Testing of Viscoelastic Solids
via Forced Vibration Analysis of Time-Dependent
Stress and Dynamic Modulus E*(t; ω) .................. 410
10.39 Phasor Analysis of Dynamic Viscoelastic
Experiments via Complex Variables .................... 413
10.40 Fourier Transformation of the Stress Relaxation
Modulus Yields Dynamic Moduli via Complex Variable
Analysis ............................................. 415
10.41 Energy Dissipation and Storage During Forced
Vibration Dynamic Mechanical Experiments ............. 417
10.42 Free Vibration Dynamic Measurements via the
Torsion Pendulum ..................................... 419
Appendix A: Linear Viscoelasticity ......................... 425
Appendix B: Finite Strain Concepts for Elastic
Materials ...................................... 435
Appendix C: Distribution of Linear Viscoelastic
Relaxation Times ............................... 443
Further Reading ............................................ 453
References ................................................. 453
Problems ................................................... 454
11 Nonlinear Stress Relaxation in Macromolecule-Metal
Complexes .................................................. 469
11.1 Nonlinear Viscoelasticity ............................. 469
11.2 Overview .............................................. 470
11.3 Relevant Background Information about Palladium
Complexes with Macromolecules that Contain Alkene
Functional Groups ..................................... 471
11.4 Effect of Palladium Chloride on the Stress-Strain
Behavior of Triblock Copolymers Containing Styrene
and Butadiene ......................................... 471
11.5 Crosslinked Polymers and Limited Chain
Extensibility ......................................... 472
11.6 Nonlinear Stress Relaxation ........................... 472
11.7 Results from Stress Relaxation Experiments on
Triblock Copolymers ................................... 476
11.8 Effect of Strain on Stress Relaxation ................. 478
11.9 Time-Strain Separability of the Relaxation
Function .............................................. 479
11.10 Characteristic Length Scales for Cooperative
Reorganization and the Effect of Strain on
Viscoelastic Relaxation Times ......................... 480
11.11 Summary .............................................. 482
References ................................................. 483
12 Kinetic Analysis of Molecular Weight Distribution
Functions in Linear Polymers ............................... 485
12.1 All Chains Do Not Contain the Same Number of Repeat
Units ................................................. 485
12.2 The "Most Probable Distribution" for
Polycondensation Reactions: Statistical
Considerations ........................................ 486
12.3 Discrete versus Continuous Distributions for
Condensation Polymerization ........................... 490
12.4 The Degree of Polymerization for Polycondensation
Reactions ............................................. 491
12.5 Moments-Generating Functions for Discrete
Distributions via z-Transforms ........................ 496
12.6 Kinetics, Molecular Weight Distributions, and
Moments-Generating Functions for Free Radical
Polymerizations ....................................... 498
12.7 Anionic "Living" Polymerizations and the Poisson
Distribution .......................................... 508
12.8 Connection Between Laplace Transforms and the
Moments-Generating Function for any Distribution in
the Continuous Limit .................................. 515
12.9 Expansion of Continuous Distribution Functions via
Orthogonal Laguerre Polynomials ....................... 521
Appendix A: Unsteady State Batch Reactor Analysis of the
Most Probable Distribution Function ............ 524
Appendix B: Mechanism and Kinetics of Alkene
Hydrogenation Reactions Transition-Metal
Catalysts ...................................... 527
Appendix C: Alkene Dimerization and Transition-Metal
Compatibilization of 1,2-Polybutadiene and
cis-polybutadiene via Palladium(II) Catalysis:
Organometallic Mechanism and Kinetics .......... 534
References ................................................. 543
Problems ................................................... 544
13 Gaussian Statistics of Linear Chain Molecules and
Crosslinked Elastomers ..................................... 547
13.1 Gaussian Chains and Entropy Elasticity ................ 547
13.2 Summary of Three-Dimensional Gaussian Chain
Statistics ............................................ 548
13.3 Vector Analysis of the Mean-Square End-to-End Chain
Distance .............................................. 550
13.4 One-Dimensional Random Walk Statistics via
Bernoulli Trials and the Binomial Distribution ........ 552
13.5 Extrapolation of One-Dimensional Gaussian
Statistics to Three Dimensions ........................ 555
13.6 Properties of Three-Dimensional Gaussian
Distributions and Their Moments-Generating
Function .............................................. 557
13.7 Mean-Square Radius of Gyration of Freely Jointed
Chains ................................................ 561
13.8 Mean-Square End-to-End Distance of Freely Rotating
Chains ................................................ 565
13.9 Characteristic Ratios and Statistical Segment
Length ............................................... 568
13.10 Excluded Volume and the Expansion Factor a for
Real Chains in "Good" Solvents: Athermal Solutions ... 570
13.11 deGennes Scaling Analysis of Flory's Law for Real
Chains in "Good" Solvents ............................ 578
13.12 Intrinsic Viscosity of Dilute Polymer Solutions
and Universal Calibration Curves for Gel Permeation
Chromatography ....................................... 579
13.13 Scaling Laws for Intrinsic Viscosity and the Mark-
Houwink Equation ..................................... 582
13.14 Intrinsic Viscosities of Polystyrene and
Polyethylene oxide) .................................. 583
13.15 Effect of pH During Dilute-Aqueous-Solution
Preparation of Solid Films on the Glass Transition ... 584
13.16 deGennes Scaling Analysis of the Threshold Overlap
Molar Density c* in Concentrated Polymer Solutions
and the Concept of "Blobs" ........................... 586
13.17 Entropically Elastic Retractive Forces via
Statistical Thermodynamics of Gaussian Chains ........ 587
Appendix: Capillary Viscometry ............................. 595
References ................................................. 600
Problems ................................................... 601
14 Classical and Statistical Thermodynamics of Rubber-Like
Materials .................................................. 609
14.1 Affine Deformation .................................... 609
14.2 Overview .............................................. 610
14.3 Analogies ............................................. 610
14.4 Classical Thermodynamic Analysis of the Ideal
Equation of State for Retractive Force from Chapter
13 .................................................... 610
14.5 Analogous Development for the Effect of Sample
Length on Internal Energy: The Concept of Ideal
Rubber-Like Solids .................................... 614
14.6 Thermoelastic Inversion ............................... 616
14.7 Temperature Dependence of Retractive Forces that
Accounts for Thermal Expansion ........................ 617
14.8 Derivation of Flory's Approximation for Isotropic
Rubber-Like Materials that Exhibit No Volume Change
upon Deformation ...................................... 619
14.9 Statistical Thermodynamic Analysis of the Equation
of State for Ideal Rubber-Like Materials .............. 623
14.10 Effect of Biaxial Deformation at Constant Volume
on Boltzmann's Entropy and Stress versus Strain ...... 630
14.11 Effect of Isotropic Chain Expansion in "Good"
Solvents on the Conformational Entropy of Linear
Macromolecules due to Excluded Volume ................ 631
14.12 Effect of Polymer-Solvent Energetics on Chain
Expansion via the Flory-Huggins Lattice Model ........ 633
14.13 Gibbs Free Energy Minimization Yields the
Equilibrium Chain Expansion Factor ................... 639
Appendix A: Chemical or Diffusional Stability of
Polymer-Solvent Mixtures ....................... 640
Appendix B: Generalized Linear Least Squares Analysis
for Second-Order Polynomials with One
Independent Variable ........................... 641
Appendix C: Linear versus Nonlinear Least Squares
Dilemma ........................................ 643
References ................................................. 646
Problems ................................................... 646
15 Molecular Dynamics via Magnetic Resonance, Viscoelastic,
and Dielectric Relaxation Phenomena ........................ 651
15.1 Fluctuation-Dissipation ............................... 651
15.2 Overview .............................................. 652
15.3 Brief Introduction to Quantum Statistical
Mechanics ............................................. 652
15.4 The Ergodic Problem of Statistical Thermodynamics ..... 655
15.5 NMR Relaxation via Spin Temperature Equilibration
with the Lattice ...................................... 656
15.6 Analysis of Spin-Lattice Relaxation Rates via Time-
Dependent Perturbation Theory and the Density
Matrix ................................................ 661
15.7 Classical Description of Stress Relaxation via
Autocorrelation of the End-to-End Chain Vector and
the Fluctuation-Dissipation Theorem ................... 673
15.8 Comparisons Among NMR, Mechanical, and Dielectric
Relaxation via Molecular Motion in Polymeric
Materials: Activated Rate Processes ................... 684
15.9 Activation Energies for the Aging Process in
Bisphenol - A Polycarbonate ........................... 691
15.10 Complex Impedance Analysis of Dielectric Relaxation
Measurements via Electrical Analogs of the Maxwell
and Voigt Models of Linear Viscoelastic Response ..... 693
15.11 Thermally Stimulated Discharge Currents in
Polarized Dielectric Materials ....................... 696
15.12 Summary .............................................. 702
References ................................................. 703
16 Magnetic Spin Diffusion at the Nanoscale in Multiphase
Polymers and Molecular Complexes ........................... 705
16.1 Magnetic Resonance .................................... 705
16.2 Overview .............................................. 706
16.3 The Spin-Diffusion Problem ............................ 706
16.4 Interdomain Communication via Magnetic Spin
Diffusion: Description of the Modified Goldman-Shen
Experiment ............................................ 707
16.5 Materials ............................................. 709
16.6 Magnetic Spin-Diffusion Experiments on Random
Copolymers that Contain Disorganized Lamellae ......... 709
16.7 Magnetic Spin-Diffusion Experiments on Triblock
Copolymers that Contain Spherically Dispersed Hard
Segments .............................................. 711
16.8 Phenomenological Transient Diffusion Models for
Two-Phase Systems with Spherical Polystyrene
Domains in a Polybutadiene Matrix ..................... 715
16.9 Solid State NMR Analysis of Molecular Complexes ....... 728
16.10 High-Resolution Solid State NMR Spectroscopy of
PEO Molecular Complexes: Correlations with Phase
Behavior ............................................. 730
16.11 Carbon-13 Solid State NMR Spectroscopy: Laboratory
Experiments and Data Analysis ........................ 738
16.12 Summary .............................................. 762
References ................................................. 763
Index ......................................................... 765
Postface ...................................................... 799
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