Physical Propertiesof MacromoleculesLaurence A. BelfioreDepartment of Chemical and Biological Engineering, Colorado State UniversityFort Collins, CO 80523

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Physical Propertiesof Macromolecules

Physical Propertiesof MacromoleculesLaurence A. BelfioreDepartment of Chemical and Biological Engineering, Colorado State UniversityFort Collins, CO 80523

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Belfiore, Laurence A.Physical properties of macromolecules / Laurence A Belfiore.p. cm.

Includes index.ISBN 978-0-470-22893-7 (cloth)1. Polymers. 2. Physical chemistry. I. Title.QC173.4.P65B45 2010547.7045—dc22

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Contents

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 31.2 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 81.4 Physical Aging, Densification, and Volume and Enthalpy Relaxation 81.5 Temperature–Pressure Differential Phase Equilibrium Relations for

First-Order Processes: The Clapeyron Equation 101.6 Temperature–Pressure Differential Phase Equilibrium Relations for

Second-Order Processes: The Ehrenfest Equations 111.7 Compositional Dependence of Tg via Entropy Continuity 151.8 Compositional Dependence of Tg via Volume Continuity 181.9 Linear Least Squares Analysis of the Gordon–Taylor Equation and

Other Tg–Composition Relations for Binary Mixtures 201.10 Free Volume Concepts 211.11 Temperature Dependence of Fractional Free Volume 221.12 Compositional Dependence of Fractional Free Volume and

Plasticizer Efficiency for Binary Mixtures 231.13 Fractional Free Volume Analysis of Multicomponent

Mixtures: Compositional Dependence of the Glass TransitionTemperature 25

1.14 Molecular Weight Dependence of Fractional Free Volume 261.15 Experimental Design to Test the Molecular Weight

Dependence of Fractional Free Volume and Tg 271.16 Pressure Dependence of Fractional Free Volume 291.17 Effect of Particle Size or Film Thickness on the Glass Transition

Temperature 31

vii

1.18 Effect of the Glass Transition on Surface Tension 34References 35Problems 36

2. Diffusion in Amorphous Polymers Near the Glass TransitionTemperature 49

2.1 Diffusion on a Lattice 492.2 Overview of the Relation Between Fractional Free Volume and

Diffusive Motion of Liquids and Gases Through PolymericMembranes 50

2.3 Free Volume Theory of Cohen and Turnbull for Diffusion in Liquidsand Glasses 51

2.4 Free Volume Theory of Vrentas and Duda for Solvent Diffusion inPolymers Above the Glass Transition Temperature 55

2.5 Influence of the Glass Transition on Diffusion in AmorphousPolymers 58

2.6 Analysis of Half-Times and Lag Times via the Unsteady StateDiffusion Equation 61

2.7 Example Problem: Effect of Molecular Weight DistributionFunctions on Average Diffusivities 66

References 69

3. Lattice Theories for Polymer–Small-Molecule Mixtures and theConformational Entropy Description of the Glass TransitionTemperature 71

3.1 Lattice Models in Thermodynamics 723.2 Membrane Osmometry and the Osmotic Pressure Expansion 723.3 Lattice Models for Athermal Mixtures with Excluded Volume 763.4 Flory–Huggins Lattice Theory for Flexible Polymer Solutions 793.5 Chemical Stability of Binary Mixtures 893.6 Guggenheim’s Lattice Theory of Athermal Mixtures 1053.7 Gibbs–DiMarzio Conformational Entropy Description of the Glass

Transition for Tetrahedral Lattices 1173.8 Lattice Cluster Theory Analysis of Conformational Entropy and the

Glass Transition in Amorphous Polymers 1233.9 Sanchez–Lacombe Statistical Thermodynamic Lattice Fluid Theory

of Polymer–Solvent Mixtures 126Appendix: The Connection Between Exothermic Energetics and Volume

Contraction of the Mixture 128References 131Problems 132

viii Contents

4. dc Electric Field Effects on First- and Second-Order PhaseTransitions in Pure Materials and Binary Mixtures 137

4.1 Electric-Field-Induced Alignment and Phase Separation 1374.2 Overview 1384.3 Electric Field Effects on Low-Molecular-Weight Molecules and

Their Mixtures 1384.4 Electric Field Effects on Polymers and Their Mixtures 1394.5 Motivation for Analysis of Electric Field Effects on Phase

Transitions 1414.6 Theoretical Considerations 1414.7 Summary 166Appendix: Nomenclature 167References 168

5. Order Parameters for Glasses: Pressure and CompositionalDependence of the Glass Transition Temperature 171

5.1 Thermodynamic Order Parameters 1715.2 Ehrenfest Inequalities: Two Independent Internal Order Parameters

Identify an Inequality Between the Two Predictions for the PressureDependence of the Glass Transition Temperature 172

5.3 Compositional Dependence of the Glass Transition Temperature 1775.4 Diluent Concentration Dependence of the Glass Transition Temperature

via Classical Thermodynamics 1815.5 Compositional Dependence of the Glass Transition Temperature via

Lattice Theory Models 1835.6 Comparison with Other Theories 1845.7 Model Calculations 1865.8 Limitations of the Theory 188References 188Problem 189

6. Macromolecule–Metal Complexes: Ligand Field Stabilizationand Glass Transition Temperature Enhancement 191

6.1 Ligand Field Stabilization 1916.2 Overview 1926.3 Methodology of Transition-Metal Coordination in Polymeric

Complexes 1936.4 Pseudo-Octahedral d8 Nickel Complexes with

Poly(4-vinylpyridine) 2096.5 d6 Molybdenum Carbonyl Complexes with Poly(vinylamine) that Exhibit

Reduced Symmetry Above the Glass Transition Temperature 216

Contents ix

6.6 Cobalt, Nickel, and Ruthenium Complexes with Poly(4-vinylpyridine)and Poly(L-histidine) that Exhibit Reduced Symmetry in the MoltenState 224

6.7 Total Energetic Requirements to Induce the Glass Transition viaConsideration of the First-Shell Coordination Sphere in Transition Metaland Lanthanide Complexes 238

6.8 Summary 2416.9 Epilogue 241Appendix: Physical Interpretation of the Parameters in the Kwei Equation for

Synergistic Enhancement of the Glass Transition Temperature in BinaryMixtures 243

References 243

Part Two Semicrystalline Polymers and Melting Transitions

7. Basic Concepts and Molecular Optical Anisotropyin Semicrystalline Polymers 249

7.1 Spherulitic Superstructure 2497.2 Comments about Crystallization 2507.3 Spherulitic Superstructures that Exhibit Molecular Optical

Anisotropy 2557.4 Interaction of a Birefringent Spherulite with Polarized Light 2587.5 Interaction of Disordered Lamellae with Polarized Light 2607.6 Interaction of Disordered Lamellae with Unpolarized Light 2617.7 Molecular Optical Anisotropy of Random Coils and Rigid Rod-Like

Polymers 2637.8 Birefringence of Rubbery Polymers Subjected to External Force

Fields 2787.9 Chain Folding, Interspherulitic Connectivity, and Mechanical Properties

of Semicrystalline Polymers 279References 282Problems 283

8. Crystallization Kinetics via Spherulitic Growth 287

8.1 Nucleation and Growth 2878.2 Heterogeneous Nucleation and Growth Prior to Impingement 2888.3 Avrami Equation for Heterogeneous Nucleation that Accounts for

Impingement of Spherulites 2898.4 Crystallization Kinetics and the Avrami Equation for Homogeneous

Nucleation of Spherulites 292

x Contents

8.5 Linear Least Squares Analysis of the Kinetics of Crystallization via theGeneralized Avrami Equation 293

8.6 Half-Time Analysis of Crystallization Isotherms 2968.7 Maximum Rate of Isothermal Crystallization 2978.8 Thermodynamics and Kinetics of Homogeneous Nucleation 2998.9 Temperature Dependence of the Crystallization Rate Constant 3028.10 Optimum Crystallization Temperatures: Comparison Between Theory

and Experiment 3048.11 The Energetics of Chain Folding in Semicrystalline Polymer–Polymer

Blends that Exhibit Multiple Melting Endotherms 3078.12 Melting Point Depression in Polymer–Polymer and Polymer–Diluent

Blends that Contain a High-Molecular-Weight SemicrystallineComponent 317

References 322Problems 322

9. Experimental Analysis of Semicrystalline Polymers 329

9.1 Semicrystallinity 3299.2 Differential Scanning Calorimetry: Thermograms of Small Molecules that

Exhibit Liquid Crystalline Phase Transitions Below the MeltingPoint 330

9.3 Isothermal Analysis of Crystallization Exotherms via DifferentialScanning Calorimetry 331

9.4 Kinetic Analysis of the Mass Fraction of Crystallinity via the GeneralizedAvrami Equation 335

9.5 Measurements of Crystallinity via Differential ScanningCalorimetry 337

9.6 Analysis of Crystallinity via Density Measurements 3399.7 Pychnometry: Density and Thermal Expansion Coefficient

Measurements of Liquids and Solids 340References 344Problems 344

Part Three Mechanical Properties of Linear and Crosslinked Polymers

10. Mechanical Properties of Viscoelastic Materials: Basic Concepts inLinear Viscoelasticity 355

10.1 Mathematical Models of Linear Viscoelasticity 35510.2 Objectives 35610.3 Simple Definitions of Stress, Strain, and Poisson’s Ratio 356

Contents xi

10.4 Stress Tensor 35710.5 Strain and Rate-of-Strain Tensors 35810.6 Hooke’s Law of Elasticity 35910.7 Newton’s Law of Viscosity 36010.8 Simple Analogies Between Mechanical and Electrical Response 36010.9 Phase Angle Difference Between Stress and Strain and

Voltage and Current in Dynamic Mechanical and DielectricExperiments 361

10.10 Maxwell’s Viscoelastic Constitutive Equation 36210.11 Integral Forms of Maxwell’s Viscoelastic Constitutive Equation 36410.12 Mechanical Model of Maxwell’s Viscoelastic Constitutive

Equation 36610.13 Four Well-Defined Mechanical Experiments 36710.14 Linear Response of theMaxwell Model during Creep Experiments 36810.15 Creep Recovery of the Maxwell Model 36910.16 Linear Response of the Maxwell Model during Stress Relaxation 37010.17 Temperature Dependence of the Stress Relaxation Modulus and

Definition of the Deborah Number 37210.18 Other Combinations of Springs and Dashpots 37310.19 Equation of Motion for the Voigt Model 37410.20 Linear Response of the Voigt Model in Creep Experiments 37610.21 Creep Recovery of the Voigt Model 37610.22 Creep and Stress Relaxation for a Series Combination of Maxwell and

Voigt Elements 37710.23 The Principle of Time–Temperature Superposition 38510.24 Stress Relaxation via the Equivalence Between Time and

Temperature 38510.25 Semi Theoretical Justification for the Empirical Form of the WLF

Shift Factor aT(T; Treference) 38910.26 Temperature Dependence of the Zero-Shear-Rate Polymer Viscosity

via Fractional Free Volume and the Doolittle Equation 39010.27 Apparent Activation Energy for aT and the Zero-Shear-Rate Polymer

Viscosity 39210.28 Comparison of the WLF Shift Factor aT at Different Reference

Temperatures 39310.29 Vogel’s Equation for the Time–Temperature Shift Factor 39410.30 Effect of Diluent Concentration on the WLF Shift Factor aC in

Concentrated Polymer Solutions 39410.31 Stress Relaxation Moduli via the Distribution of Viscoelastic Time

Constants 39710.32 Stress Relaxation Moduli and Terminal Relaxation Times 40010.33 The Critical Molecular Weight Required for Entanglement

Formation 40310.34 Zero-Shear-Rate Viscosity via the Distribution of Viscoelastic

Relaxation Times 403

xii Contents

10.35 The Boltzmann Superposition Integral for Linear ViscoelasticResponse 405

10.36 Alternate Forms of the Boltzmann Superposition Integralfor s (t) 406

10.37 Linear Viscoelastic Application of the Boltzmann SuperpositionPrinciple: Elastic Free Recovery 407

10.38 Dynamic Mechanical Testing of Viscoelastic Solids via ForcedVibration Analysis of Time-Dependent Stress and DynamicModulus E�(t ; v) 410

10.39 Phasor Analysis of Dynamic Viscoelastic Experiments via ComplexVariables 413

10.40 Fourier Transformation of the Stress Relaxation Modulus YieldsDynamic Moduli via Complex Variable Analysis 415

10.41 Energy Dissipation and Storage During Forced Vibration DynamicMechanical Experiments 417

10.42 Free Vibration Dynamic Measurements via the TorsionPendulum 419

Appendix A: Linear Viscoelasticity 425Appendix B: Finite Strain Concepts for Elastic Materials 435Appendix C: Distribution of Linear Viscoelastic Relaxation Times 443Further Reading 453References 453Problems 454

11. Nonlinear Stress Relaxation in Macromolecule–Metal Complexes 469

11.1 Nonlinear Viscoelasticity 46911.2 Overview 47011.3 Relevant Background Information about Palladium Complexes

with Macromolecules that Contain Alkene FunctionalGroups 471

11.4 Effect of Palladium Chloride on the Stress–Strain Behavior ofTriblock Copolymers Containing Styrene and Butadiene 471

11.5 Crosslinked Polymers and Limited Chain Extensibility 47211.6 Nonlinear Stress Relaxation 47211.7 Results from Stress Relaxation Experiments on Triblock

Copolymers 47611.8 Effect of Strain on Stress Relaxation 47811.9 Time–Strain Separability of the Relaxation Function 47911.10 Characteristic Length Scales for Cooperative Reorganization

and the Effect of Strain on Viscoelastic RelaxationTimes 480

11.11 Summary 482References 483

Contents xiii

12. Kinetic Analysis of Molecular Weight Distribution Functionsin Linear Polymers 485

12.1 All Chains Do Not Contain the Same Number of Repeat Units 48512.2 The “Most Probable Distribution” for Polycondensation Reactions:

Statistical Considerations 48612.3 Discrete versus Continuous Distributions for Condensation

Polymerization 49012.4 The Degree of Polymerization for Polycondensation Reactions 49112.5 Moments-Generating Functions for Discrete Distributions via

z-Transforms 49612.6 Kinetics, Molecular Weight Distributions, and Moments-Generating

Functions for Free Radical Polymerizations 49812.7 Anionic “Living” Polymerizations and the Poisson Distribution 50812.8 Connection Between Laplace Transforms and the Moments-Generating

Function for any Distribution in the Continuous Limit 51512.9 Expansion of Continuous Distribution Functions via Orthogonal

Laguerre Polynomials 521Appendix A: Unsteady State Batch Reactor Analysis of the Most Probable

Distribution Function 524Appendix B: Mechanism and Kinetics of Alkene Hydrogenation Reactions via

Transition-Metal Catalysts 527Appendix C: Alkene Dimerization and Transition-Metal Compatibilization of

1,2-Polybutadiene and cis-polybutadiene via Palladium(II) Catalysis:Organometallic Mechanism and Kinetics 534

References 543Problems 544

13. Gaussian Statistics of Linear Chain Molecules and CrosslinkedElastomers 547

13.1 Gaussian Chains and Entropy Elasticity 54713.2 Summary of Three-Dimensional Gaussian Chain Statistics 54813.3 Vector Analysis of the Mean-Square End-to-End

Chain Distance 55013.4 One-Dimensional Random Walk Statistics via Bernoulli Trials and

the Binomial Distribution 55213.5 Extrapolation of One-Dimensional Gaussian Statistics to Three

Dimensions 55513.6 Properties of Three-Dimensional Gaussian Distributions and Their

Moments-Generating Function 55713.7 Mean-Square Radius of Gyration of Freely Jointed Chains 56113.8 Mean-Square End-to-End Distance of Freely Rotating Chains 56513.9 Characteristic Ratios and Statistical Segment Length 568

xiv Contents

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 CalibrationCurves for Gel Permeation Chromatography 579

13.13 Scaling Laws for Intrinsic Viscosity and the Mark–HouwinkEquation 582

13.14 Intrinsic Viscosities of Polystyrene and Poly(ethylene oxide) 58313.15 Effect of pH During Dilute-Aqueous-Solution Preparation of Solid

Films on the Glass Transition 58413.16 deGennes Scaling Analysis of the Threshold Overlap Molar Density c� in

Concentrated Polymer Solutions and the Concept of “Blobs” 58613.17 Entropically Elastic Retractive Forces via Statistical Thermodynamics

of Gaussian Chains 587Appendix: Capillary Viscometry 595References 600Problems 601

14. Classical and Statistical Thermodynamics of Rubber-Like Materials 609

14.1 Affine Deformation 60914.2 Overview 61014.3 Analogies 61014.4 Classical Thermodynamic Analysis of the Ideal Equation of State for

Retractive Force from Chapter 13 61014.5 Analogous Development for the Effect of Sample Length on Internal

Energy: The Concept of Ideal Rubber-Like Solids 61414.6 Thermoelastic Inversion 61614.7 Temperature Dependence of Retractive Forces that Accounts for

Thermal Expansion 61714.8 Derivation of Flory’s Approximation for Isotropic Rubber-Like

Materials that Exhibit No Volume Change upon Deformation 61914.9 Statistical Thermodynamic Analysis of the Equation of State for Ideal

Rubber-Like Materials 62314.10 Effect of Biaxial Deformation at Constant Volume on Boltzmann’s

Entropy and Stress versus Strain 63014.11 Effect of Isotropic Chain Expansion in “Good” Solvents on the

Conformational Entropy of Linear Macromolecules due to ExcludedVolume 631

14.12 Effect of Polymer–Solvent Energetics on Chain Expansion via theFlory–Huggins Lattice Model 633

14.13 Gibbs Free Energy Minimization Yields the Equilibrium ChainExpansion Factor 639

Contents xv

Appendix A: Chemical or Diffusional Stability of Polymer–SolventMixtures 640

Appendix B: Generalized Linear Least Squares Analysis for Second-OrderPolynomials with One Independent Variable 641

Appendix C: Linear versus Nonlinear Least Squares Dilemma 643References 646Problems 646

Part Four Solid State Dynamics of Polymeric Materials

15. Molecular Dynamics via Magnetic Resonance, Viscoelastic, andDielectric Relaxation Phenomena 651

15.1 Fluctuation–Dissipation 65115.2 Overview 65215.3 Brief Introduction to Quantum Statistical Mechanics 65215.4 The Ergodic Problem of Statistical Thermodynamics 65515.5 NMR Relaxation via Spin Temperature Equilibration with the

Lattice 65615.6 Analysis of Spin–Lattice Relaxation Rates via Time-Dependent

Perturbation Theory and the Density Matrix 66115.7 Classical Description of Stress Relaxation via Autocorrelation of

the End-to-End Chain Vector and the Fluctuation–DissipationTheorem 673

15.8 Comparisons Among NMR, Mechanical, and Dielectric Relaxationvia Molecular Motion in Polymeric Materials: Activated RateProcesses 684

15.9 Activation Energies for the Aging Process in Bisphenol-APolycarbonate 691

15.10 Complex Impedance Analysis of Dielectric Relaxation Measurements viaElectrical Analogs of the Maxwell and Voigt Models of LinearViscoelastic Response 693

15.11 Thermally Stimulated Discharge Currents in Polarized DielectricMaterials 696

15.12 Summary 702References 703

16. Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers andMolecular Complexes 705

16.1 Magnetic Resonance 70516.2 Overview 706

xvi Contents

16.3 The Spin-Diffusion Problem 70616.4 Interdomain Communication via Magnetic Spin Diffusion:

Description of the Modified Goldman–Shen Experiment 70716.5 Materials 70916.6 Magnetic Spin-Diffusion Experiments on Random Copolymers that

Contain Disorganized Lamellae 70916.7 Magnetic Spin-Diffusion Experiments on Triblock Copolymers that

Contain Spherically Dispersed Hard Segments 71116.8 Phenomenological Transient Diffusion Models for Two-Phase Systems

with Spherical Polystyrene Domains in a Polybutadiene Matrix 71516.9 Solid State NMR Analysis of Molecular Complexes 72816.10 High-Resolution Solid State NMR Spectroscopy of PEO Molecular

Complexes: Correlations with Phase Behavior 73016.11 Carbon-13 Solid State NMR Spectroscopy: Laboratory Experiments

and Data Analysis 73816.12 Summary 762References 763

Index 765

Postface 799

Contents xvii

Preface

The task of writing this book has truly been a labor of love. The motivation requiredto deliver these lines of wisdom was not catalyzed by socioeconomic impact, success,promotion, or acceptance, because I realize that none of thesewill be achieved. So, whydid I embark on this immense task? Long after I am capable of riding the roads on thisplanet, generation after generation of motivated students will be able to trace my pathand hopefully comprehend the physical properties of polymers by following thewordsand equations that support all of the concepts discussed. I will experience the utmostgratification if this book influences and enhances the learning experience of only aselect few, because I do not require acceptance on a large scale to justify my decisionto pursue this project. I thoroughly enjoyed all of the time and effort that was investedto produce this product. Following one of the responsibilities outlined by ProfessorOlaf A. Hougen, eminent chair of Chemical Engineering at the University ofWisconsin, as transmitted by Professor Bob Bird, “textbook writing has a welcomedhome in academia and faculty have a responsibility to produce these documents.”

There are colleagues and students in the Department of Chemical Engineeringat Colorado State University, as well as those nationwide and globally, who mustbe acknowledged for their assistance. A significant fraction of this book follows thenotes and supplementary handouts that I acquired as a graduate student in the Springof 1978 at the University of Wisconsin, during an unofficial audit of MacromolecularChemistry taught by Professor Hyuk Yu in the Department of Chemistry. ProfessorYu considered every step of elaborate statistical derivations, providing superior insightabout all of the underlying assumptions that are not obvious from inspection of thefinal result. I also acknowledge recent communications with Professor Yu andwish him well in his retirement. This textbook project could not have been completedwithout more than a decade of generous support from the Polymers Program in theDivision of Materials Research at NSF, which allowed me to investigate macromol-ecule–metal complexes and include some of these concepts in selected chapters.

I was discussing polymer courses and textbook writing with Professor Dick Steinat the 1985 Elastomers Gordon Conference in New London, New Hampshire, whenDick graciously provided copies of his unpublished polymer notes. Extrapolationsof Professor Stein’s notes appear in several sections of this book, including spheruliteimpingement, critical spherulite size required for spontaneous growth, excludedvolume and expansion factors for realistic chain dimensions, biaxial orientation ofrubber-like materials, terminal relaxation times, and the exponential integral for theeffect of molecular weight on zero-shear rate polymer viscosities. Professor ErikThompson in the Department of Civil Engineering at Colorado State University

xix

helped me formulate an analysis of the torsion pendulum for free vibration dampingbased on the Voigt model with mass, not the Maxwell model. Professor GrzegorzSzamel in the Physical Chemistry Division of Colorado State University’sChemistry Department is acknowledged for his assistance in solving the Liouvilleequation for a simple two-state system. Professor David (Qiang) Wang provided assis-tance and encouragement for several years as a colleague in the Department ofChemical and Biological Engineering at Colorado State University. Professor SoniaKreidenweis in the Department of Atmospheric Science (Colorado State University)provided information about pollution-based aerosols that become nucleation siteswithin storm clouds and their effect on snowpack in mountainous regions near pol-luted metropolitan areas, as an application of crystallization kinetics in the presenceof nucleating agents. Dr. Pronab Das generated significant results for macromol-ecule–metal complexes, and some of the results from his PhD thesis at ColoradoState University appear in Chapter 11. Students in the classroom provided manythought-provoking questions that begged for a response. For example, Kevin Fisherposed questions about interspherulitic connectivity and the mechanical properties ofsemicrystalline polymers that I redirected to members of the discussion list maintainedby the American Chemical Society’s Division of Polymer Chemistry. Kevin’s ques-tions and eight detailed answers appear near the end of Chapter 7. Ryan Sengersuggested that linear least squares analysis should be applied to the logarithmicform of the Tg–composition relation that one obtains from entropy continuityin binary mixtures, prior to invoking any additional assumptions. Derek Johnsonsuggested that crystallization half-times should be compared with the time thatcorresponds to the maximum rate of crystallization for several Avrami exponents.Shane Bower requested additional information about volume and enthalpy relaxationbelow Tg and the sequence of nonequilibrium states traversed by densified glassesupon heating in the vicinity of Tg. Mike Floren saw polarized optical micrographs ofPEO spherulites in my laboratory and found a home for them in Chapters 7 and 8,as well as on the cover of this book. As an example of a professor’s influence onyoung impressionable students during the critical years when novice students wrestlewith the formidable task of “learning how to learn,” Professor Costas Gogos at StevensInstitute of Technology in Hoboken NJ introduced me to this fascinating subject andtold me that I had a “future in polymers” after my performance on his first exam in thespring of 1976—sounds somewhat similar to the advice that Dustin Hoffman receivedin The Graduate. It is my desire that two young sisters, Emily Marie Lighthart andKimberly Renee Lighthart, will mature and find fulfillment and pleasure upon readingthe Physical Properties of Macromolecules.

And last but not least, for Pookie, who died in 2005, my super friend and com-panion for more than a decade, who accompanied me through snow and on dirt trailsto the highest elevations possible in the Colorado Rockies . . . thanks for the memories.

LAURENCE A. BELFIOREFort Collins, Coloradobelfiore@engr.colostate.edu

xx Preface

Part One

Glass Transitions inAmorphous Polymers

Chapter 1

Glass Transitions inAmorphous Polymers:Basic Concepts

A window shatters, into a cloud of uncertainty.—Michael Berardi

Glass transitions in amorphous materials are described primarily from athermodynamic viewpoint, but the kinetic nature of Tg is mentioned also. Thepressure dependence of first- and second-order phase transitions is compared viathe Clapeyron and Ehrenfest equations, respectively. Compositional dependence ofTg in single-phase mixtures is addressed from volume and entropy continuity. Theconnection between fractional free volume and Tg is introduced. Then, physicalvariables that affect Tg are discussed in terms of their influence on free volume.Effects of molecular weight, particle size, film thickness, and surface free energyon the glass transition are also considered.

1.1 PHASE TRANSITIONS IN AMORPHOUSMATERIALS

Unlike crystalline solids with long-range order, glasses transform to highly viscousliquids upon heating. Amorphous materials exhibit some short-range order, but essen-tially no long-range order. Whereas melting is reserved for materials that exhibit somecrystal structure, glass–liquid phase transitions are characterized by the continuousbehavior of several thermodynamic state functions, including enthalpy, entropy, andvolume. From a rigorous viewpoint, glasses do not melt, and their flow behavior isevident during the time scale of centuries in the vertical colored glass windows ofmedieval churches. Plasticizing additives shift the glass transition to lower temperatureand increase the utility of relatively inexpensive brittle polymers.

Physical Properties of Macromolecules. By Laurence A. BelfioreCopyright # 2010 John Wiley & Sons, Inc.

3

1.2 VOLUME–TEMPERATURE AND ENTHALPY–TEMPERATURE RELATIONS IN THE VICINITY OFFIRST-ORDER AND SECOND-ORDER PHASETRANSITIONS: DISCONTINUOUS THERMOPHYSICALPROPERTIES AT Tm AND Tg

The glass transition temperature (i.e., Tg) is one of the most important thermophysicalproperties of a polymeric material. In the glassy state below Tg, materials are usuallybrittle with an elastic modulus on the order of 1010 dynes/cm2 and a fracture strain of5% or 10%.Molecular vibrations andmicro-Brownianmotions that produce local con-formational rearrangements of the chain backbone are characteristic of glasses. In thehighly viscous liquid state above Tg, materials are rubbery with an elastic modulus of�107 dynes/cm2 that exhibits strong dependence on molecular weight. If chain entan-glements are operative, then fracture strains easily exceed 100%. Viscous liquids exhi-bit molecular vibrations, conformational rearrangements of the chain backbone, andtranslational motion of the chain along its contour, which is called reptation.Knowledge of Tg allows one to develop a reasonably accurate picture of a material’selastic modulus over a wide temperature range. Semicrystalline polymers exhibit amelting transition. However, all materials, regardless of their molecular weight, exhibita glass transition. It might be necessary to quench a low-molecular-weight materialvery rapidly from the molten state so that Tg can be observed without complicationsdue to crystallinity. The primary objectives of this chapter are to (i) observe andmeasure Tg in amorphous polymers and (ii) recognize several factors that affect Tg.

It is instructive to compare the temperature dependence of intensive thermodyn-amic properties, like specific volume v (i.e., 1/r, where r represents density) orspecific enthalpy h, in the vicinity of Tg and Tm. For a low-molecular-weight solidthat is essentially 100% crystalline, the temperature dependence of its density orspecific enthalpy exhibits an abrupt discontinuity at the melting temperature (i.e.,Tm). Some of the discontinuous intensive thermodynamic properties at Tm areD(1/rmelt), Dvmelt, Dhmelt, and Dsmelt, where s is specific entropy and D signifies thedifference between a thermodynamic property slightly above and slightly below thetransition temperature. These discontinuous observables exhibit a step increase atTm for all materials, except H2O in which D(1/rmelt) and Dvmelt are negative.Melting is classified as a first-order phase transition because all first and higher deriva-tives of the chemical potential are discontinuous at Tm. This is illustrated as follows viathe extensive Gibbs free energy of a pure material, G(T, p, N ), in terms of its naturalvariables: temperature T, pressure p, and total moles N, which represent complete ther-modynamic information about the system. According to the phase rule, three degreesof freedom (i.e, T, p, and N ) must be specified for a unique description of extensivethermodynamic properties when a pure material exists as a single phase. The phaserule stipulates that there are two degrees of freedom for single-phase behavior of apure material, but extensive properties require one additional degree of freedomassociated with total system mass. The total differential of the Gibbs potential is

dG ¼ @G@T

� �p,N

dT þ @G@p

� �T ,N

dpþ @G@N

� �T ,p

dN

4 Chapter 1 Glass Transitions in Amorphous Polymers: Basic Concepts

The temperature, pressure, and mole number coefficients of G are defined as follows:

@G

@T

� �p,N

¼ �S; @G@p

� �T ,N

¼ V ; @G@N

� �T ,p

¼ m

where S is extensive entropy, V is extensive volume, and m is the chemical potential.Since total moles N represents the only extensive independent variable for G, asdescribed above, Euler’s integral theorem for homogeneous functions of the firstdegree with respect to system mass yields the following result:

G ¼ N @G@N

� �T ,p

¼ Nm

Hence, the total differential of G, based on Euler’s integration, is

dG ¼ N dmþ m dN

and this should be compared with the previous result for the total differential of G:

dG ¼ �S dT þ V dpþ m dN

One arrives at the Gibbs–Duhem equation via this comparison, which reveals that anintensive quantity, like the chemical potential of a pure material, requires specificationof two independent variables (i.e., T and p) for a unique description of single-phasebehavior:

dm ¼ � SN

� �dT þ V

N

� �dp ¼ �s dT þ v dp

From the previous equation, the first derivatives of chemical potential with respect toeither temperature or pressure are

@m

@T

� �p

¼ �s; @m@p

� �T

¼ v

If the system contains several components, then the temperature and pressure coeffi-cients of the chemical potential of species i arewritten in terms of partial molar entropyand partial molar volume of component i, respectively. Since molar entropy and molarvolume of pure materials and mixtures are discontinuous at Tm, and these intensiveproperties are obtained from the first derivatives of m, melting is classified as a first-order thermodynamic phase transition. If all first derivatives of m are discontinuousat Tm, then all higher-order derivatives of m are also discontinuous upon melting.An nth-order phase transition is defined as one in which the nth derivatives of m(including mixed nth partial derivatives) are the first ones that yield discontinuousthermodynamic properties at the phase transition temperature. By definition, thezeroth-order derivatives of m are continuous at Tm, and the following statements rep-resent the criterion of chemical equilibrium for pure materials, based on the integral

1.2 Volume–Temperature and Enthalpy–Temperature Relations 5

and differential methods, respectively:

mSolid(T�m ) ¼ mLiquid(Tþm )

dmSolid(T�m ) ¼ dmLiquid(Tþm )

Now, consider the temperature dependence of specific volume and specificenthalpy in the vicinity of Tg. These thermodynamic properties are continuous atTg, but their slopes change at the phase transition in the following manner:

@v

@T

� �p;Liquid

.@v

@T

� �p;Glass

@h

@T

� �p;Liquid

.@h

@T

� �p;Glass

The first inequality suggests that thermal expansion coefficients a increase abruptlyupon heating at Tg, because

a ¼ @ ln v@T

� �p

¼ 1v

@v

@T

� �p

vaf gLiquid . vaf gGlass

Since specific volume is continuous at Tg, the previous inequality reveals that

aLiquid . aGlass

Da ¼ aLiquid � aGlass . 0

where, in this case, D represents the difference between thermodynamic propertiesslightly above Tg (i.e., highly viscous liquid) and slightly below Tg (i.e., rigidglass). Since the temperature dependence of specific enthalpy increases above Tg, and

Cp ¼ @h@T

� �p

¼ T @s@T

� �p

it follows directly that specific heats are larger for liquids than they are for the corre-sponding glasses. Hence DCp . 0.

There are no known exceptions to the previous two inequalities, which indicatethat coefficients of thermal expansion and specific heats experience step incrementsat Tg when materials are heated from the glassy state into the highly viscous liquidstate. By definition, Tg is a second-order thermodynamic phase transition becausevolume, enthalpy, and entropy are continuous but the temperature derivatives ofthese thermophysical properties are discontinuous. If m ¼ m(T, p) for a pure material,then there are three second partial derivatives of the chemical potential that yield

6 Chapter 1 Glass Transitions in Amorphous Polymers: Basic Concepts

discontinuous observable properties at Tg. Two of these properties—specific heat andthe coefficient of thermal expansion—have been identified above. The followingthermodynamic relations provide a rigorous summary of all discontinuous thermo-physical properties at a second order phase transition:

@

@T

@m

@T

� �p

" #p

¼ � @s@T

� �p

¼ �CpT

@

@p

@m

@p

� �T

� �T

¼ @v@p

� �T

¼ �vb

@

@T

@m

@p

� �T

� �p

¼ @v@T

� �p

¼ va

where a is the isobaric coefficient of thermal expansion and b is the isothermalcompressibility. Since m is an exact differential, the order of mixed second partialdifferentiation with respect to T and p can be reversed without affecting the finalresult. Hence, one obtains the third equation above (i.e., va) upon taking the pressurederivative first, and the temperature derivative second. Da, Db, and DCp are greaterthan zero for all materials at Tg, whereD represents the difference between thermophy-sical properties in the liquid and glassy states. Even though one typically assumes thatliquids are incompressible, liquid state compressibilities are greater than the compres-sibility of glasses, or amorphous solids. All of the results discussed above are appli-cable to pure materials and mixtures. Since there are r þ 1 degrees of freedom for asingle-phase mixture of r components, r þ 1 independent variables are required fora unique description of the chemical potential of component i. Hence, the rigorousdefinition of a second-order phase transition stipulates that all of the second partialderivatives of mi are discontinuous, where 1 � i � r. For each component, there are(r þ 1) second partial derivatives of mi, where differentiation is performed twicewith respect to the same independent variable (i.e., @2mi=@x

2i , with xi representing

an independent variable), and r(r þ 1) mixed second partial derivatives (i.e.,@2mi=@xj @xk, j = k). Since the order of mixed second partial differentiation can bereversed without affecting the final result, there are r(r þ 1)/2 mixed second partialderivatives of each mi that yield useful independent information. Most of these discon-tinuous quantities can be expressed in terms of the concentration dependence of (i)partial molar volume, (ii) partial molar entropy, and (iii) the chemical potential ofeach component. In summary, there are (r þ 1)(1 þ r/2) discontinuous thermophysi-cal properties at Tg per component, and the total number of discontinuous quantitiesfor a mixure of r components is

r(r þ 1)(1þ r=2)

As expected, this analysis indicates that there are three discontinuous observables fora pure material (i.e., r ¼ 1).

1.2 Volume–Temperature and Enthalpy–Temperature Relations 7

1.3 THE EQUILIBRIUM GLASSY STATE

Most glasses are not in a state of thermodynamic equilibrium. In fact, one can arguethat the glass transition is not an equilibrium second-order phase transition because themeasured value of Tg depends on the experimental rate of heating or cooling. If aviscous liquid achieves thermodynamic equilibrium above Tg and the temperaturedecreases at an infinitesimally slow rate, then conformational rearrangements of thechain backbone via rotation about carbon–carbon single bonds should allow thematerial to contract macroscopically on a time scale that is on the order of, or fasterthan, the experimental cooling rate. Under these conditions, the system traverses asequence of equilibrium states and the coefficient of thermal expansion shoulddecrease abruptly upon cooling at the equilibrium glass transition. Simple volume–temperature calculations reveal that this hypothetical “equilibrium glassy state”exists, and that a decreases abruptly at Tg,equil when materials are cooled at an infini-tesimally slow rate. If one assumes typical values for specific volume of hydrocarbonpolymers (i.e., r � 1.2 g/cm3) and the coefficient of thermal expansion of a commonliquid at ambient conditions (i.e., aLiquid � 5–6 � 1024 K21), and extrapolates theliquidus line to 0 K at a slope dictated by aLiquid,

v(T) ¼ v(Treference) exp{aLiquid(T � Treference)}Treference ¼ 300K

v(Treference) ¼ 11:2 g=cm3aLiquid � (5�6)� 10�4 K�1

then predictions yield unacceptably low specific volume at absolute zero. This ano-maly is prevented if a decreases when materials are cooled below Tg. Hence, thefollowing theorem summarizes these observations:

There must be an equilibrium glass transition temperature Tg,equil below whichaGlass , aLiquid. Otherwise, equilibrium liquidus volume–temperature curvesextrapolate to unrealistically low volume at absolute zero. The slope of thevolume–temperature curve must exhibit a discontinuity at Tg,equil when exper-iments are conducted on an infinite time scale. In practice, finite rates of heatingor cooling are required to measure Tg, and these kinetic measurements yieldpseudo-phase-transition temperatures that are greater than Tg,equil.

1.4 PHYSICAL AGING, DENSIFICATION, ANDVOLUME AND ENTHALPY RELAXATION

Comments from the previous section provide support for the existence of an equili-brium glassy state that can be discussed in principle, but never achieved in practice.This is equivalent to the well-known phenomenon in heat transfer where the tempera-ture of fluids moving through heat exchangers can approach but never achieve the

8 Chapter 1 Glass Transitions in Amorphous Polymers: Basic Concepts