Physical Properties of Macromolecules

Physical Properties of Macromolecules Laurence A. Belfiore Department of Chemical and Biological Engineering, Colorado State University Fort Collins, CO 80523

Copyright # 2010 by John Wiley & Sons, Inc. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: 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 2010 547.7045—dc22 2009019348 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

This book is dedicated to all who have attempted and successfully climbed by bicycle the following mountain passes in the northern Italian Alps: Passo del Mortirolo, Passo di Gavia, Passo dello Stelvio, Tre Cime di Lavaredo, Passo Fedaia at the base of the Marmolada, and Monte Bondone above Pizza Pazza in Piedicastello, where Monika, Petra, and Alessandro wait for Lorenzo to return per cena. Buon appetito! Royalties from this book will be donated to support all activities at the San Patrignano rehabilitation center, located in the small village of San Vito Valsugana, Italy. This organization was founded in 1978 to rehabilitate unfortunate individuals who have experienced life on the fringe of society. The San Vito Center, under Andrea Pesenti’s supervision and four decades of frame-building experience, helps these individuals develop skills required to design and produce high-end carbon-fiber-reinforced epoxy bicycle frames.

Contents

Preface

xix

Part One Glass Transitions in Amorphous Polymers 1. Glass Transitions in Amorphous Polymers: Basic Concepts 1.1 1.2

1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13

1.14 1.15 1.16 1.17

3 Phase Transitions in Amorphous Materials Volume – Temperature and Enthalpy – Temperature Relations in the Vicinity of First-Order and Second-Order Phase Transitions: 4 Discontinuous Thermophysical Properties at Tm and Tg The Equilibrium Glassy State 8 Physical Aging, Densification, and Volume and Enthalpy Relaxation Temperature – Pressure Differential Phase Equilibrium Relations for 10 First-Order Processes: The Clapeyron Equation Temperature – Pressure Differential Phase Equilibrium Relations for 11 Second-Order Processes: The Ehrenfest Equations Compositional Dependence of Tg via Entropy Continuity 15 Compositional Dependence of Tg via Volume Continuity 18 Linear Least Squares Analysis of the Gordon –Taylor Equation and 20 Other Tg – Composition Relations for Binary Mixtures Free Volume Concepts 21 Temperature Dependence of Fractional Free Volume 22 Compositional Dependence of Fractional Free Volume and Plasticizer Efficiency for Binary Mixtures 23 Fractional Free Volume Analysis of Multicomponent Mixtures: Compositional Dependence of the Glass Transition 25 Temperature Molecular Weight Dependence of Fractional Free Volume 26 Experimental Design to Test the Molecular Weight Dependence of Fractional Free Volume and Tg 27 Pressure Dependence of Fractional Free Volume 29 Effect of Particle Size or Film Thickness on the Glass Transition Temperature 31

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

34

2. Diffusion in Amorphous Polymers Near the Glass Transition Temperature

49

2.1 2.2

Diffusion on a Lattice 49 Overview of the Relation Between Fractional Free Volume and Diffusive Motion of Liquids and Gases Through Polymeric 50 Membranes 2.3 Free Volume Theory of Cohen and Turnbull for Diffusion in Liquids 51 and Glasses 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 58 Polymers 2.6 Analysis of Half-Times and Lag Times via the Unsteady State 61 Diffusion Equation 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 3.1 3.2 3.3 3.4 3.5 3.6 3.7

Lattice Models in Thermodynamics 72 Membrane Osmometry and the Osmotic Pressure Expansion 72 Lattice Models for Athermal Mixtures with Excluded Volume 76 Flory – Huggins Lattice Theory for Flexible Polymer Solutions 79 Chemical Stability of Binary Mixtures 89 Guggenheim’s Lattice Theory of Athermal Mixtures 105 Gibbs – DiMarzio Conformational Entropy Description of the Glass Transition for Tetrahedral Lattices 117 3.8 Lattice Cluster Theory Analysis of Conformational Entropy and the 123 Glass Transition in Amorphous Polymers 3.9 Sanchez – Lacombe Statistical Thermodynamic Lattice Fluid Theory 126 of Polymer – Solvent Mixtures Appendix: The Connection Between Exothermic Energetics and Volume Contraction of the Mixture 128 References 131 Problems 132

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

137

4.1 4.2 4.3

Electric-Field-Induced Alignment and Phase Separation 137 Overview 138 Electric Field Effects on Low-Molecular-Weight Molecules and 138 Their Mixtures 4.4 Electric Field Effects on Polymers and Their Mixtures 139 4.5 Motivation for Analysis of Electric Field Effects on Phase 141 Transitions 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 5.2

171 Thermodynamic Order Parameters Ehrenfest Inequalities: Two Independent Internal Order Parameters Identify an Inequality Between the Two Predictions for the Pressure 172 Dependence of the Glass Transition Temperature 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 183 Lattice Theory Models 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 6.1 6.2 6.3 6.4 6.5

191

Ligand Field Stabilization 191 Overview 192 Methodology of Transition-Metal Coordination in Polymeric 193 Complexes Pseudo-Octahedral d8 Nickel Complexes with Poly(4-vinylpyridine) 209 d6 Molybdenum Carbonyl Complexes with Poly(vinylamine) that Exhibit 216 Reduced Symmetry Above the Glass Transition Temperature

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Cobalt, Nickel, and Ruthenium Complexes with Poly(4-vinylpyridine) and Poly(L-histidine) that Exhibit Reduced Symmetry in the Molten 224 State 6.7 Total Energetic Requirements to Induce the Glass Transition via Consideration of the First-Shell Coordination Sphere in Transition Metal 238 and Lanthanide Complexes 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 243 Mixtures References 243

Part Two Semicrystalline Polymers and Melting Transitions 7. Basic Concepts and Molecular Optical Anisotropy in Semicrystalline Polymers

249

7.1 7.2 7.3

Spherulitic Superstructure 249 Comments about Crystallization 250 Spherulitic Superstructures that Exhibit Molecular Optical 255 Anisotropy 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 263 Polymers 7.8 Birefringence of Rubbery Polymers Subjected to External Force 278 Fields 7.9 Chain Folding, Interspherulitic Connectivity, and Mechanical Properties 279 of Semicrystalline Polymers References 282 Problems 283

8. Crystallization Kinetics via Spherulitic Growth 8.1 8.2 8.3 8.4

287 Nucleation and Growth Heterogeneous Nucleation and Growth Prior to Impingement 288 Avrami Equation for Heterogeneous Nucleation that Accounts for 289 Impingement of Spherulites Crystallization Kinetics and the Avrami Equation for Homogeneous Nucleation of Spherulites 292

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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 307 Blends that Exhibit Multiple Melting Endotherms 8.12 Melting Point Depression in Polymer –Polymer and Polymer – Diluent Blends that Contain a High-Molecular-Weight Semicrystalline 317 Component References 322 Problems 322 9. Experimental Analysis of Semicrystalline Polymers

329

9.1 9.2

Semicrystallinity 329 Differential Scanning Calorimetry: Thermograms of Small Molecules that Exhibit Liquid Crystalline Phase Transitions Below the Melting 330 Point 9.3 Isothermal Analysis of Crystallization Exotherms via Differential 331 Scanning Calorimetry 9.4 Kinetic Analysis of the Mass Fraction of Crystallinity via the Generalized 335 Avrami Equation 9.5 Measurements of Crystallinity via Differential Scanning 337 Calorimetry 9.6 Analysis of Crystallinity via Density Measurements 339 9.7 Pychnometry: Density and Thermal Expansion Coefficient 340 Measurements of Liquids and Solids References 344 Problems 344

Part Three

Mechanical Properties of Linear and Crosslinked Polymers

10. Mechanical Properties of Viscoelastic Materials: Basic Concepts in Linear Viscoelasticity 10.1 10.2 10.3

Mathematical Models of Linear Viscoelasticity 355 Objectives 356 Simple Definitions of Stress, Strain, and Poisson’s Ratio

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10.4 10.5 10.6 10.7 10.8 10.9

10.10 10.11 10.12 10.13 10.14 10.15 10.16 10.17 10.18 10.19 10.20 10.21 10.22 10.23 10.24 10.25 10.26 10.27 10.28 10.29 10.30 10.31 10.32 10.33 10.34

Stress Tensor 357 Strain and Rate-of-Strain Tensors 358 Hooke’s Law of Elasticity 359 Newton’s Law of Viscosity 360 Simple Analogies Between Mechanical and Electrical Response 360 Phase Angle Difference Between Stress and Strain and Voltage and Current in Dynamic Mechanical and Dielectric 361 Experiments Maxwell’s Viscoelastic Constitutive Equation 362 Integral Forms of Maxwell’s Viscoelastic Constitutive Equation 364 Mechanical Model of Maxwell’s Viscoelastic Constitutive 366 Equation Four Well-Defined Mechanical Experiments 367 Linear Response of the Maxwell Model during Creep Experiments 368 Creep Recovery of the Maxwell Model 369 Linear Response of the Maxwell Model during Stress Relaxation 370 Temperature Dependence of the Stress Relaxation Modulus and 372 Definition of the Deborah Number Other Combinations of Springs and Dashpots 373 Equation of Motion for the Voigt Model 374 Linear Response of the Voigt Model in Creep Experiments 376 Creep Recovery of the Voigt Model 376 Creep and Stress Relaxation for a Series Combination of Maxwell and 377 Voigt Elements The Principle of Time – Temperature Superposition 385 Stress Relaxation via the Equivalence Between Time and 385 Temperature Semi Theoretical Justification for the Empirical Form of the WLF Shift Factor aT (T; Treference) 389 Temperature Dependence of the Zero-Shear-Rate Polymer Viscosity 390 via Fractional Free Volume and the Doolittle Equation Apparent Activation Energy for aT and the Zero-Shear-Rate Polymer 392 Viscosity Comparison of the WLF Shift Factor aT at Different Reference Temperatures 393 Vogel’s Equation for the Time – Temperature Shift Factor 394 Effect of Diluent Concentration on the WLF Shift Factor aC in Concentrated Polymer Solutions 394 Stress Relaxation Moduli via the Distribution of Viscoelastic Time 397 Constants Stress Relaxation Moduli and Terminal Relaxation Times 400 The Critical Molecular Weight Required for Entanglement 403 Formation Zero-Shear-Rate Viscosity via the Distribution of Viscoelastic 403 Relaxation Times

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10.35 The Boltzmann Superposition Integral for Linear Viscoelastic Response 405 10.36 Alternate Forms of the Boltzmann Superposition Integral 406 for s (t) 10.37 Linear Viscoelastic Application of the Boltzmann Superposition 407 Principle: Elastic Free Recovery 10.38 Dynamic Mechanical Testing of Viscoelastic Solids via Forced Vibration Analysis of Time-Dependent Stress and Dynamic 410 Modulus E (t ; v) 10.39 Phasor Analysis of Dynamic Viscoelastic Experiments via Complex Variables 413 10.40 Fourier Transformation of the Stress Relaxation Modulus Yields 415 Dynamic Moduli via Complex Variable Analysis 10.41 Energy Dissipation and Storage During Forced Vibration Dynamic 417 Mechanical Experiments 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 11.1 11.2 11.3

Nonlinear Viscoelasticity 469 Overview 470 Relevant Background Information about Palladium Complexes with Macromolecules that Contain Alkene Functional 471 Groups 11.4 Effect of Palladium Chloride on the Stress – Strain Behavior of 471 Triblock Copolymers Containing Styrene and Butadiene 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 480 Times 11.11 Summary 482 References 483

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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 490 Polymerization 12.4 The Degree of Polymerization for Polycondensation Reactions 491 12.5 Moments-Generating Functions for Discrete Distributions via 496 z-Transforms 12.6 Kinetics, Molecular Weight Distributions, and Moments-Generating 498 Functions for Free Radical Polymerizations 12.7 Anionic “Living” Polymerizations and the Poisson Distribution 508 12.8 Connection Between Laplace Transforms and the Moments-Generating 515 Function for any Distribution in the Continuous Limit 12.9 Expansion of Continuous Distribution Functions via Orthogonal 521 Laguerre Polynomials Appendix A: Unsteady State Batch Reactor Analysis of the Most Probable Distribution Function 524 Appendix B: Mechanism and Kinetics of Alkene Hydrogenation Reactions via 527 Transition-Metal Catalysts Appendix C: Alkene Dimerization and Transition-Metal Compatibilization of 1,2-Polybutadiene and cis-polybutadiene via Palladium(II) Catalysis: 534 Organometallic Mechanism and Kinetics 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 550 Chain Distance 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 555 Dimensions 13.6 Properties of Three-Dimensional Gaussian Distributions and Their 557 Moments-Generating Function 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

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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” 578 Solvents 13.12 Intrinsic Viscosity of Dilute Polymer Solutions and Universal Calibration 579 Curves for Gel Permeation Chromatography 13.13 Scaling Laws for Intrinsic Viscosity and the Mark – Houwink Equation 582 13.14 Intrinsic Viscosities of Polystyrene and Poly(ethylene 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 586 Concentrated Polymer Solutions and the Concept of “Blobs” 13.17 Entropically Elastic Retractive Forces via Statistical Thermodynamics 587 of Gaussian Chains Appendix: Capillary Viscometry 595 References 600 Problems 601 14. Classical and Statistical Thermodynamics of Rubber-Like Materials 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 14.10 14.11

14.12 14.13

609

Affine Deformation 609 Overview 610 Analogies 610 Classical Thermodynamic Analysis of the Ideal Equation of State for 610 Retractive Force from Chapter 13 Analogous Development for the Effect of Sample Length on Internal 614 Energy: The Concept of Ideal Rubber-Like Solids Thermoelastic Inversion 616 Temperature Dependence of Retractive Forces that Accounts for 617 Thermal Expansion Derivation of Flory’s Approximation for Isotropic Rubber-Like Materials that Exhibit No Volume Change upon Deformation 619 Statistical Thermodynamic Analysis of the Equation of State for Ideal 623 Rubber-Like Materials Effect of Biaxial Deformation at Constant Volume on Boltzmann’s 630 Entropy and Stress versus Strain Effect of Isotropic Chain Expansion in “Good” Solvents on the Conformational Entropy of Linear Macromolecules due to Excluded 631 Volume Effect of Polymer – Solvent Energetics on Chain Expansion via the Flory –Huggins Lattice Model 633 Gibbs Free Energy Minimization Yields the Equilibrium Chain 639 Expansion Factor

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Appendix A: Chemical or Diffusional Stability of Polymer – Solvent Mixtures 640 Appendix B: Generalized Linear Least Squares Analysis for Second-Order 641 Polynomials with One Independent Variable Appendix C: Linear versus Nonlinear Least Squares Dilemma 643 References 646 Problems 646

Part Four

Solid State Dynamics of Polymeric Materials

15. Molecular Dynamics via Magnetic Resonance, Viscoelastic, and Dielectric Relaxation Phenomena

651

15.1 15.2 15.3 15.4 15.5

Fluctuation– Dissipation 651 Overview 652 Brief Introduction to Quantum Statistical Mechanics 652 The Ergodic Problem of Statistical Thermodynamics 655 NMR Relaxation via Spin Temperature Equilibration with the Lattice 656 15.6 Analysis of Spin– Lattice Relaxation Rates via Time-Dependent 661 Perturbation Theory and the Density Matrix 15.7 Classical Description of Stress Relaxation via Autocorrelation of the End-to-End Chain Vector and the Fluctuation – Dissipation 673 Theorem 15.8 Comparisons Among NMR, Mechanical, and Dielectric Relaxation via Molecular Motion in Polymeric Materials: Activated Rate 684 Processes 15.9 Activation Energies for the Aging Process in Bisphenol-A 691 Polycarbonate 15.10 Complex Impedance Analysis of Dielectric Relaxation Measurements via Electrical Analogs of the Maxwell and Voigt Models of Linear 693 Viscoelastic Response 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 16.2 Overview 706

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16.3 16.4

The Spin-Diffusion Problem 706 Interdomain Communication via Magnetic Spin Diffusion: 707 Description of the Modified Goldman – Shen Experiment 16.5 Materials 709 16.6 Magnetic Spin-Diffusion Experiments on Random Copolymers that 709 Contain Disorganized Lamellae 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 715 with Spherical Polystyrene Domains in a Polybutadiene Matrix 16.9 Solid State NMR Analysis of Molecular Complexes 728 16.10 High-Resolution Solid State NMR Spectroscopy of PEO Molecular 730 Complexes: Correlations with Phase Behavior 16.11 Carbon-13 Solid State NMR Spectroscopy: Laboratory Experiments 738 and Data Analysis 16.12 Summary 762 References 763 Index

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Postface

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Preface

The task of writing this book has truly been a labor of love. The motivation required to deliver these lines of wisdom was not catalyzed by socioeconomic impact, success, promotion, or acceptance, because I realize that none of these will be achieved. So, why did I embark on this immense task? Long after I am capable of riding the roads on this planet, generation after generation of motivated students will be able to trace my path and hopefully comprehend the physical properties of polymers by following the words and equations that support all of the concepts discussed. I will experience the utmost gratification if this book influences and enhances the learning experience of only a select few, because I do not require acceptance on a large scale to justify my decision to pursue this project. I thoroughly enjoyed all of the time and effort that was invested to produce this product. Following one of the responsibilities outlined by Professor Olaf A. Hougen, eminent chair of Chemical Engineering at the University of Wisconsin, as transmitted by Professor Bob Bird, “textbook writing has a welcomed home in academia and faculty have a responsibility to produce these documents.” There are colleagues and students in the Department of Chemical Engineering at Colorado State University, as well as those nationwide and globally, who must be acknowledged for their assistance. A significant fraction of this book follows the notes and supplementary handouts that I acquired as a graduate student in the Spring of 1978 at the University of Wisconsin, during an unofficial audit of Macromolecular Chemistry taught by Professor Hyuk Yu in the Department of Chemistry. Professor Yu considered every step of elaborate statistical derivations, providing superior insight about all of the underlying assumptions that are not obvious from inspection of the final result. I also acknowledge recent communications with Professor Yu and wish him well in his retirement. This textbook project could not have been completed without more than a decade of generous support from the Polymers Program in the Division of Materials Research at NSF, which allowed me to investigate macromolecule – metal complexes and include some of these concepts in selected chapters. I was discussing polymer courses and textbook writing with Professor Dick Stein at the 1985 Elastomers Gordon Conference in New London, New Hampshire, when Dick graciously provided copies of his unpublished polymer notes. Extrapolations of Professor Stein’s notes appear in several sections of this book, including spherulite impingement, critical spherulite size required for spontaneous growth, excluded volume and expansion factors for realistic chain dimensions, biaxial orientation of rubber-like materials, terminal relaxation times, and the exponential integral for the effect of molecular weight on zero-shear rate polymer viscosities. Professor Erik Thompson in the Department of Civil Engineering at Colorado State University xix

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helped me formulate an analysis of the torsion pendulum for free vibration damping based on the Voigt model with mass, not the Maxwell model. Professor Grzegorz Szamel in the Physical Chemistry Division of Colorado State University’s Chemistry Department is acknowledged for his assistance in solving the Liouville equation for a simple two-state system. Professor David (Qiang) Wang provided assistance and encouragement for several years as a colleague in the Department of Chemical and Biological Engineering at Colorado State University. Professor Sonia Kreidenweis in the Department of Atmospheric Science (Colorado State University) provided information about pollution-based aerosols that become nucleation sites within storm clouds and their effect on snowpack in mountainous regions near polluted metropolitan areas, as an application of crystallization kinetics in the presence of nucleating agents. Dr. Pronab Das generated significant results for macromolecule – metal complexes, and some of the results from his PhD thesis at Colorado State University appear in Chapter 11. Students in the classroom provided many thought-provoking questions that begged for a response. For example, Kevin Fisher posed questions about interspherulitic connectivity and the mechanical properties of semicrystalline polymers that I redirected to members of the discussion list maintained by the American Chemical Society’s Division of Polymer Chemistry. Kevin’s questions and eight detailed answers appear near the end of Chapter 7. Ryan Senger suggested that linear least squares analysis should be applied to the logarithmic form of the Tg – composition relation that one obtains from entropy continuity in binary mixtures, prior to invoking any additional assumptions. Derek Johnson suggested that crystallization half-times should be compared with the time that corresponds to the maximum rate of crystallization for several Avrami exponents. Shane Bower requested additional information about volume and enthalpy relaxation below Tg and the sequence of nonequilibrium states traversed by densified glasses upon heating in the vicinity of Tg. Mike Floren saw polarized optical micrographs of PEO 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 on young impressionable students during the critical years when novice students wrestle with the formidable task of “learning how to learn,” Professor Costas Gogos at Stevens Institute of Technology in Hoboken NJ introduced me to this fascinating subject and told me that I had a “future in polymers” after my performance on his first exam in the spring of 1976—sounds somewhat similar to the advice that Dustin Hoffman received in The Graduate. It is my desire that two young sisters, Emily Marie Lighthart and Kimberly Renee Lighthart, will mature and find fulfillment and pleasure upon reading the Physical Properties of Macromolecules. And last but not least, for Pookie, who died in 2005, my super friend and companion for more than a decade, who accompanied me through snow and on dirt trails to the highest elevations possible in the Colorado Rockies . . . thanks for the memories. LAURENCE A. BELFIORE Fort Collins, Colorado [email protected]

Part One

Glass Transitions in Amorphous Polymers

Chapter

1

Glass Transitions in Amorphous Polymers: Basic Concepts A window shatters, into a cloud of uncertainty. —Michael Berardi

G

lass transitions in amorphous materials are described primarily from a thermodynamic viewpoint, but the kinetic nature of Tg is mentioned also. The pressure dependence of first- and second-order phase transitions is compared via the Clapeyron and Ehrenfest equations, respectively. Compositional dependence of Tg in single-phase mixtures is addressed from volume and entropy continuity. The connection between fractional free volume and Tg is introduced. Then, physical variables that affect Tg are discussed in terms of their influence on free volume. Effects of molecular weight, particle size, film thickness, and surface free energy on the glass transition are also considered.

1.1 PHASE TRANSITIONS IN AMORPHOUS MATERIALS Unlike crystalline solids with long-range order, glasses transform to highly viscous liquids upon heating. Amorphous materials exhibit some short-range order, but essentially no long-range order. Whereas melting is reserved for materials that exhibit some crystal structure, glass – liquid phase transitions are characterized by the continuous behavior of several thermodynamic state functions, including enthalpy, entropy, and volume. From a rigorous viewpoint, glasses do not melt, and their flow behavior is evident during the time scale of centuries in the vertical colored glass windows of medieval churches. Plasticizing additives shift the glass transition to lower temperature and increase the utility of relatively inexpensive brittle polymers. Physical Properties of Macromolecules. By Laurence A. Belfiore Copyright # 2010 John Wiley & Sons, Inc.

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Chapter 1 Glass Transitions in Amorphous Polymers: Basic Concepts

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 The glass transition temperature (i.e., Tg) is one of the most important thermophysical properties of a polymeric material. In the glassy state below Tg, materials are usually brittle with an elastic modulus on the order of 1010 dynes/cm2 and a fracture strain of 5% or 10%. Molecular vibrations and micro-Brownian motions that produce local conformational rearrangements of the chain backbone are characteristic of glasses. In the highly 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 entanglements are operative, then fracture strains easily exceed 100%. Viscous liquids exhibit molecular vibrations, conformational rearrangements of the chain backbone, and translational 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’s elastic modulus over a wide temperature range. Semicrystalline polymers exhibit a melting transition. However, all materials, regardless of their molecular weight, exhibit a glass transition. It might be necessary to quench a low-molecular-weight material very rapidly from the molten state so that Tg can be observed without complications due to crystallinity. The primary objectives of this chapter are to (i) observe and measure Tg in amorphous polymers and (ii) recognize several factors that affect Tg. It is instructive to compare the temperature dependence of intensive thermodynamic properties, like specific volume v (i.e., 1/r, where r represents density) or specific enthalpy h, in the vicinity of Tg and Tm. For a low-molecular-weight solid that is essentially 100% crystalline, the temperature dependence of its density or specific enthalpy exhibits an abrupt discontinuity at the melting temperature (i.e., Tm). Some of the discontinuous intensive thermodynamic properties at Tm are D(1/rmelt), Dvmelt, Dhmelt, and Dsmelt, where s is specific entropy and D signifies the difference between a thermodynamic property slightly above and slightly below the transition temperature. These discontinuous observables exhibit a step increase at Tm 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 derivatives of the chemical potential are discontinuous at Tm. This is illustrated as follows via the extensive Gibbs free energy of a pure material, G(T, p, N ), in terms of its natural variables: temperature T, pressure p, and total moles N, which represent complete thermodynamic information about the system. According to the phase rule, three degrees of freedom (i.e, T, p, and N ) must be specified for a unique description of extensive thermodynamic properties when a pure material exists as a single phase. The phase rule stipulates that there are two degrees of freedom for single-phase behavior of a pure material, but extensive properties require one additional degree of freedom associated with total system mass. The total differential of the Gibbs potential is       @G @G @G dT þ dp þ dN dG ¼ @T p,N @p T,N @N T,p

1.2 Volume–Temperature and Enthalpy–Temperature Relations

5

The temperature, pressure, and mole number coefficients of G are defined as follows:       @G @G @G ¼ S; ¼ V; ¼m @T p,N @p T,N @N T,p 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, as described above, Euler’s integral theorem for homogeneous functions of the first degree with respect to system mass yields the following result:   @G ¼ Nm G¼N @N T,p Hence, the total differential of G, based on Euler’s integration, is dG ¼ N dm þ m dN and this should be compared with the previous result for the total differential of G: dG ¼ S dT þ V dp þ m dN One arrives at the Gibbs– Duhem equation via this comparison, which reveals that an intensive quantity, like the chemical potential of a pure material, requires specification of two independent variables (i.e., T and p) for a unique description of single-phase behavior:     S V dT þ dp ¼ s dT þ v dp dm ¼  N N From the previous equation, the first derivatives of chemical potential with respect to either temperature or pressure are     @m @m ¼ s; ¼v @T p @p T If the system contains several components, then the temperature and pressure coefficients of the chemical potential of species i are written in terms of partial molar entropy and partial molar volume of component i, respectively. Since molar entropy and molar volume of pure materials and mixtures are discontinuous at Tm, and these intensive properties are obtained from the first derivatives of m, melting is classified as a firstorder thermodynamic phase transition. If all first derivatives of m are discontinuous at 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 discontinuous thermodynamic properties at the phase transition temperature. By definition, the zeroth-order derivatives of m are continuous at Tm, and the following statements represent the criterion of chemical equilibrium for pure materials, based on the integral

6

Chapter 1 Glass Transitions in Amorphous Polymers: Basic Concepts

and differential methods, respectively:

mSolid (Tm ) ¼ mLiquid (Tmþ ) dmSolid (Tm ) ¼ dmLiquid (Tmþ ) Now, consider the temperature dependence of specific volume and specific enthalpy in the vicinity of Tg. These thermodynamic properties are continuous at Tg, but their slopes change at the phase transition in the following manner:  

@v @T @h @T



 .

p;Liquid



 .

p;Liquid

@v @T @h @T

 p;Glass



p;Glass

The first inequality suggests that thermal expansion coefficients a increase abruptly upon heating at Tg, because     @ ln v 1 @v ¼ @T p v @T p

a¼

fvagLiquid . fvagGlass 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 properties slightly above Tg (i.e., highly viscous liquid) and slightly below Tg (i.e., rigid glass). Since the temperature dependence of specific enthalpy increases above Tg, and  Cp ¼

@h @T





@s ¼T @T p

 p

it follows directly that specific heats are larger for liquids than they are for the corresponding glasses. Hence DCp . 0. There are no known exceptions to the previous two inequalities, which indicate that coefficients of thermal expansion and specific heats experience step increments at Tg when materials are heated from the glassy state into the highly viscous liquid state. By definition, Tg is a second-order thermodynamic phase transition because volume, enthalpy, and entropy are continuous but the temperature derivatives of these 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

1.2 Volume–Temperature and Enthalpy–Temperature Relations

7

discontinuous observable properties at Tg. Two of these properties—specific heat and the coefficient of thermal expansion—have been identified above. The following thermodynamic relations provide a rigorous summary of all discontinuous thermophysical properties at a second order phase transition: "

#   @m @s Cp ¼ ¼ @T p T @T p p      @ @m @v ¼ ¼ vb @p @p T T @p T      @ @m @v ¼ ¼ va @T @p T p @T p @ @T



where a is the isobaric coefficient of thermal expansion and b is the isothermal compressibility. Since m is an exact differential, the order of mixed second partial differentiation with respect to T and p can be reversed without affecting the final result. Hence, one obtains the third equation above (i.e., va) upon taking the pressure derivative first, and the temperature derivative second. Da, Db, and DCp are greater than zero for all materials at Tg, where D represents the difference between thermophysical properties in the liquid and glassy states. Even though one typically assumes that liquids are incompressible, liquid state compressibilities are greater than the compressibility of glasses, or amorphous solids. All of the results discussed above are applicable to pure materials and mixtures. Since there are r þ 1 degrees of freedom for a single-phase mixture of r components, r þ 1 independent variables are required for a unique description of the chemical potential of component i. Hence, the rigorous definition of a second-order phase transition stipulates that all of the second partial derivatives 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 twice with respect to the same independent variable (i.e., @ 2mi =@x2i , 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 be reversed without affecting the final result, there are r(r þ 1)/2 mixed second partial derivatives of each mi that yield useful independent information. Most of these discontinuous quantities can be expressed in terms of the concentration dependence of (i) partial molar volume, (ii) partial molar entropy, and (iii) the chemical potential of each component. In summary, there are (r þ 1)(1 þ r/2) discontinuous thermophysical properties at Tg per component, and the total number of discontinuous quantities for a mixure of r components is r(r þ 1)(1 þ r=2) As expected, this analysis indicates that there are three discontinuous observables for a pure material (i.e., r ¼ 1).

8

Chapter 1 Glass Transitions in Amorphous Polymers: Basic Concepts

1.3

THE EQUILIBRIUM GLASSY STATE

Most glasses are not in a state of thermodynamic equilibrium. In fact, one can argue that the glass transition is not an equilibrium second-order phase transition because the measured value of Tg depends on the experimental rate of heating or cooling. If a viscous liquid achieves thermodynamic equilibrium above Tg and the temperature decreases at an infinitesimally slow rate, then conformational rearrangements of the chain backbone via rotation about carbon– carbon single bonds should allow the material to contract macroscopically on a time scale that is on the order of, or faster than, the experimental cooling rate. Under these conditions, the system traverses a sequence of equilibrium states and the coefficient of thermal expansion should decrease 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 infinitesimally slow rate. If one assumes typical values for specific volume of hydrocarbon polymers (i.e., r  1.2 g/cm3) and the coefficient of thermal expansion of a common liquid at ambient conditions (i.e., aLiquid  5 –6  1024 K21), and extrapolates the liquidus line to 0 K at a slope dictated by aLiquid, v(T) ¼ v(Treference ) exp{aLiquid (T  Treference )} Treference ¼ 300 K v(Treference ) ¼

1 1:2 g=cm3

aLiquid  (56)  104 K1 then predictions yield unacceptably low specific volume at absolute zero. This anomaly is prevented if a decreases when materials are cooled below Tg. Hence, the following theorem summarizes these observations: There must be an equilibrium glass transition temperature Tg,equil below which aGlass , aLiquid. Otherwise, equilibrium liquidus volume – temperature curves extrapolate to unrealistically low volume at absolute zero. The slope of the volume – temperature curve must exhibit a discontinuity at Tg,equil when experiments are conducted on an infinite time scale. In practice, finite rates of heating or cooling are required to measure Tg, and these kinetic measurements yield pseudo-phase-transition temperatures that are greater than Tg,equil.

1.4 PHYSICAL AGING, DENSIFICATION, AND VOLUME AND ENTHALPY RELAXATION Comments from the previous section provide support for the existence of an equilibrium 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 temperature of fluids moving through heat exchangers can approach but never achieve the

1.4 Physical Aging, Densification, and Volume and Enthalpy Relaxation

9

temperature of the surroundings unless the area of the exchanger is infinitely large or the flow rate is infinitesimally slow. Materials are aware of the specific volume and enthalpy that they might achieve as equilibrium glasses. The difference between their nonequilibrium and equilibrium properties provides the driving force for volume and enthalpy relaxation. The former is also known as densification or physical aging. Enthalpy relaxation affects the shape of a calorimeter trace during subsequent heating segments. For example, upon heating in the vicinity of the glass transition, one observes the superposition of a second-order phase transition (as expected for amorphous materials) and a first-order phase transition as materials recover from the decrease in specific enthalpy that occurred below Tg during relaxation toward the equilibrium glassy state. This type of response for Cp versus T occurs, to some extent, because glasses exhibit time-dependent (i.e., kinetic) behavior under experimental conditions. One does not follow the same sequence of nonequilibrium states upon heating and cooling amorphous materials in the vicinity of the glass transition temperature when kinetic processes occur below Tg that produce densified glasses. When an equilibrium liquid densifies via slow cooling, the material follows a sequence of states that exhibit reduced volume according to aLiquid, and this behavior continues to much lower temperature relative to the sequence of states that is traversed at faster cooling. Hence, the glass transition temperature is lower when cooling occurs at a slower rate, and the glass that forms is densified relative to glasses that form at faster cooling rates. Now, when less dense glasses, that are produced at faster cooling rates, relax to a densified glass, the heating trace for this densified glass follows a different set of states in the vicinity of Tg relative to the sequence of states that is traversed from the liquid phase when this densified glass is formed upon cooling. The densified glass that forms via slower cooling from the liquid state reveals a lower Tg. In contrast, when nonequilibrium densified glasses form via enthalpy and volume relaxation from a less dense glass, the heating traces for these densified glasses reveal a Tg (upon heating) that exceeds Tg (upon cooling) for the less dense glass. In general, densified glasses exhibit more restricted mobility and reduced fractional free volume relative to less dense glasses, and this morphological difference is reflected in the measurement of Tg upon heating because materials must achieve a certain level of chain mobility, including translation and reptation of the backbone, before one observes a secondorder transition to the viscous liquid state. Enthalpy relaxation effects on differential scanning calorimetry (DSC) heating traces are most prominent when materials are held isothermally in the glassy state, approximately 20 – 50 8C below Tg, for a significant duration of time. Above Tg, material behavior follows a sequence of equilibrium liquid states that do not depend on heating or cooling rates. Below Tg, experimental cooling rates, defined by r ¼2dT/dt, affect the magnitude of the driving force for volume and enthalpy relaxation, as illustrated below: Volume relaxation Driving force ¼ v(T; r)  vequilibrium (T) Enthalpy relaxation Driving force ¼ h(T; r)  hequilibrium (T)

10

Chapter 1 Glass Transitions in Amorphous Polymers: Basic Concepts

If conformational rearrangements of a single polymer chain have a strong influence on macroscopic volume, then higher rates of cooling produce a glass with larger specific volume because the rate of volume contraction becomes sluggish at lower temperature and occurs on a time scale that is much longer than the experimental cooling rate. In other words, materials that undergo thermal contraction according to aLiquid along the equilibrium liquidus line, experience a glass transition at higher temperature when the cooling rate increases. Now, thermal contraction during further cooling follows aGlass. Hence, the driving force for volume (and enthalpy) relaxation below Tg increases at higher cooling rates. On the other hand, the sluggishness of volume contraction at lower temperature is consistent with a decrease in mobility, which causes volume and enthalpy relaxation to occur at a slower rate. Temperature and cooling rate produce competing effects on relaxation processes because faster cooling increases the driving force, but materials are usually cooled to lower temperatures at higher rates of cooling, which decrease mobility. Qualitatively, it is acceptable to envision these relaxation processes as a product of (i) temperature-dependent mobility and (ii) temperatureand rate-dependent driving force.

1.5 TEMPERATURE –PRESSURE DIFFERENTIAL PHASE EQUILIBRIUM RELATIONS FOR FIRST-ORDER PROCESSES: THE CLAPEYRON EQUATION The statement of chemical equilibrium for pure materials is useful to develop phase coexistence relations. The integral approach to phase equilibrium is based on

mSolid (Tm ) ¼ mLiquid (Tmþ ) whereas the differential approach, dmSolid (Tm ) ¼ d mLiquid (Tmþ ) is the method of choice to calculate the pressure dependence of Tm via the Clapeyron equation. The final result is also valid for mixtures because the pressure dependence of Tg is evaluated at constant composition. The Gibbs– Duhem equation (i.e., see Section 1.2) is employed to express the total differential of the chemical potential in terms of temperature and pressure. For example, sSolid dTSolid þ vSolid dpSolid ¼ sLiquid dTLiquid þ vLiquid dpLiquid implies that differential changes in mSolid and mLiquid must be balanced along the solid– liquid boundary on a temperature – pressure phase diagram. Differential statements of thermal and mechanical equilibrium in the absence of external fields are dTSolid ¼ dTLiquid dpSolid ¼ dpLiquid The latter statement of mechanical equilibrium must be modified if an external field is present. For example, when external fields exert a species-specific force on each component in a mixture (i.e., N components), the statements of thermal and chemical equilibrium require that temperature and field-dependent chemical potential gradients

1.6 Temperature–Pressure Differential Phase Equilibrium Relations

11

must vanish (i.e., rT ¼ 0, rmi ¼ 0 for 1  i  N ), but the Gibbs –Duhem equation yields the following result for the pressure gradient: rp ¼ 

N X

r i rw i

i¼1

where ri is the mass density of species i in the mixture and wi is the specific external potential with dimensions of energy per mass that exerts force rwi on the ith component. In a gravitational field where all species experience the same force, the specific external potential w is given by the product of the gravitational acceleration constant g and a position variable that increases vertically upward (i.e., opposite to the gravitational acceleration vector). Now, the pressure gradient is balanced by the gravitational force, rp ¼ rg, which is consistent with the momentum balance under hydrostatic conditions. The statement of complete thermodynamic solid – liquid equilibrium for simple systems in the absence of external force fields with no gradients in temperature, pressure, or chemical potential yields the following relation between temperature and pressure along a solid – liquid phase boundary:   vLiquid vSolid Dvmelt @T ¼ ¼ @p @Tmelt sLiquid sSolid Dsmelt This is the Clapeyron equation for first-order solid – liquid phase transitions, and it applies to any two phases in equilibrium that exhibit discontinuities in the first derivatives of the chemical potential at the transition temperature. Since Dvmelt and Dsmelt represent two discontinuous thermophysical properties at Tm that increase abruptly as materials are heated from the solid state to the liquid state, it is generally true that Tm is higher when the pressure increases. However, the melting temperature decreases at higher pressure for H2O because Dvmelt is negative (i.e., the liquid phase slightly above Tm is more dense than the solid phase slightly below Tm). The Clapeyron equation does not yield useful information for a second-order thermodynamic phase transition because volume and entropy are continuous at Tg. If one differentiates numerator and denominator of the Clapeyron equation (i) with respect to temperature at constant pressure and then (ii) with respect to pressure at constant temperature, it is possible to generate two Ehrenfest equations that predict the pressure dependence of Tg [Ehrenfest, 1933; Goldstein, 1963]. The first approach is equivalent to invoking entropy continuity at Tg, and the second approach is equivalent to invoking volume continuity at Tg.

1.6 TEMPERATURE –PRESSURE DIFFERENTIAL PHASE EQUILIBRIUM RELATIONS FOR SECONDORDER PROCESSES: THE EHRENFEST EQUATIONS 1.6.1

Volume Continuity

The pressure dependence of Tg is developed by expressing the specific volume of the liquid and the glass in terms of T and p for a pure material. This is sufficient for

12

Chapter 1 Glass Transitions in Amorphous Polymers: Basic Concepts

two-phase equilibrium of pure materials because there are two degrees of freedom, and T and p can be chosen independently in each separate phase. The final results are also valid for mixtures because one can focus on the relation between temperature and pressure changes along the glass – liquid phase boundary at constant composition. The integral approach to volume continuity at Tg for a pure material is based on vGlass (T, p) ¼ vLiquid (T, p) and the differential approach begins with the following statement: dvGlass (T, p) ¼ dvLiquid (T, p) Since temperature and pressure must be the same in each phase, unless strong external fields are present, the differential statement of volume continuity becomes         @vLiquid @vLiquid @vGlass @vGlass dT þ dp ¼ dT þ dp @T p @p T @T p @p T Now, the temperature and pressure coefficients of specific volume are expressed in terms of volumetric thermal expansion and isothermal compressibility, respectively:     @v @v ¼ va; ¼ vb @T p @p T Hence, vGlass aGlass dT  vGlass bGlass dp ¼ vLiquid aLiquid dT  vLiquid bLiquid dp There is only one degree of freedom for two-phase equilibrium of a pure material, and the previous restriction indicates that temperature and pressure changes are not independent on the glass transition phase boundary. Since vGlass ¼ vLiquid, the differential relation between T and p along a glass – liquid boundary is   bLiquid  bGlass Db @T ¼ ¼ @p @Tglass aLiquid  aGlass Da As mentioned earlier, this Ehrenfest equation is equivalent to applying l’Hoˆpital’s rule to the Clapeyron equation via differentiation with respect to pressure at constant temperature. Since liquids are more compressible and thermally more expandable than glasses, Tg increases invariably at higher pressure (i.e., by about 20– 30 8C per kilobar). There are no exceptions to this rule. Typical values for the discontinuity in thermal expansion and isothermal compressibility at the glass transition temperature are 5  1024 K21 for Da and 1  1025 atm21 for Db. Dense glasses are produced when molten polymers in the highly viscous liquid state are subjected to high pressure and cooled below Tg. This densified amorphous structure is essentially “frozen” upon cooling below the glass transition temperature. However, materials lose all memory of prior processing history when they are heated above Tg in a differential scanning calorimeter at ambient pressure.

1.6 Temperature–Pressure Differential Phase Equilibrium Relations

1.6.2

13

Entropy Continuity

If one invokes entropy continuity at Tg via the differential approach to two-phase equilibrium, then another Ehrenfest equation describes the pressure dependence of Tg. Once again, T and p represent two independent variables for a complete description of pure-component specific entropies in the liquid and glassy states. Hence, sGlass (T, p) ¼ sLiquid (T, p) dsGlass (T, p) ¼ dsLiquid (T, p) and the differential statement of entropy continuity at Tg becomes         @sLiquid @sLiquid @sGlass @sGlass dT þ dp ¼ dT þ dp @T p @p T @T p @p T The temperature and pressure coefficients of specific entropy are 

 @s Cp ¼ T @T p     @s @v ¼ ¼ va @p T @T p

The temperature coefficient of s at constant pressure is based on the total differential of specific enthalpy: dh ¼ T ds þ v dp þ composition-dependent terms for mixtures and the definition of specific heat, Cp ¼ (@h/@T )p,composition. The pressure coefficient of specific entropy at constant temperature is derived from a Maxwell relation using the Gibbs potential: dg ¼ s dT þ v dp þ composition-dependent terms for mixtures because the order of mixed second partial differentiation can be reversed without affecting the final result since g is a state function (i.e., exact differential). Hence, Cp,Glass d ln T  vGlass aGlass dp ¼ Cp,Liquid d ln T  vLiquid aLiquid dp The phase rule for glass –liquid equilibrium of a pure material indicates that fluctuations in T and p cannot occur independently. As one traverses the two-phase boundary between the liquid and glassy states, differential changes in T and p must follow: 

@ ln Tg @p

 ¼ constant composition

vLiquid aLiquid  vGlass aGlass Cp,Liquid  Cp,Glass

14

Chapter 1 Glass Transitions in Amorphous Polymers: Basic Concepts

Since volume continuity at Tg requires that vGlass ¼ vLiquid, the previous expression reduces to   @ ln Tg v Da ¼ @p constant composition DCp where the discontinuity in specific heat (i.e., DCp) at the glass transition temperature is approximately 0.2– 0.3 J/(g-K). Volume and entropy continuity at a second-order thermodynamic phase transition yield two Ehrenfest equations for the pressure dependence of Tg. These are summarized below.   @Tg Db ¼ (1:1) @p constant composition Da   @Tg vTg Da ¼ (1:2) @p constant composition DCp Division of Eq. (1.1) by (1.2) yields the Prigogine – Defay ratio, which should approach unity based on volume and entropy continuity at the glass transition [Prigogine and Defay, 1954]. However, when two order parameters are required to describe the morphological structure of glasses at an equilibrium second-order phase transition, the developments in Chapter 5 reveal that predictions from Eq. (1.1) yield a larger pressure dependence of Tg relative to Eq. (1.2). Now, the Prigogine – Defay ratio exceeds unity, and entropy continuity provides a better estimate of the pressure dependence of Tg, given by Eq. (1.2) above. If the discontinuous thermophysical properties at Tg are very weak functions of pressure, then the Ehrenfest equations can be integrated to predict Tg at different pressures: Db ( ppreference ) Da   ( ppreference )v preference Da Tg ( p)  Tg ( preference ) exp DCp Tg ( p)  Tg ( preference ) þ

(1:3) (1:4)

If one accounts for the pressure dependence of specific volume via the coefficient of isothermal compressibility as follows:   @ ln v ¼ b @p T v( p)  v( preference ) exp{b(ppreference )} then Ehrenfest equation (1.2) in differential form, based on entropy continuity, is written as     aLiquid vLiquid ( preference ) exp bLiquid ( ppreference ) @ ln Tg  DCp @p constant composition aGlass vGlass ( preference ) expfbGlass ( ppreference )g

1.7 Compositional Dependence of Tg via Entropy Continuity

15

1.7 COMPOSITIONAL DEPENDENCE OF Tg VIA ENTROPY CONTINUITY Glass transition temperature measurements can be used as a diagnostic probe of the phase behavior of mixtures. Completely miscible blends exhibit only one concentration-dependent Tg. If phase separation occurs, then a different Tg is characteristic of each phase. The formalism presented in this section applies to polymer – polymer and polymer – diluent blends, as well as random copolymers, which are considered to be miscible. Diluents can be plasticizers, antiplasticizers, ultraviolet stabilizers, antioxidants, dissolved supercritical CO2, and so on. Most practical applications involve binary or ternary blends. However, completely miscible multicomponent systems are addressed later. The total specific entropy of a mixture of r components is expressed as a weight-fraction-weighted sum of pure-component specific entropies for each component, and a contribution due to the mixing process. The total specific entropy of the liquid at the mixture Tg is stotal,Liquid (Tg,mixture ) ¼

r X

vi,Liquid si,Liquid (Tg,mixture ) þ Dsmixing,Liquid (Tg,mixture )

i¼1

where vi,Liquid is the mass fraction of component i in the equilibrium liquid phase, si,Liquid is the pure-component specific entropy of component i, and Dsmixing,Liquid is the conformational entropy change due to mixing, which is best described in terms of lattice theories (i.e., Flory – Huggins, Guggenheim, Stavermann, Sanchez – Lacombe, etc.) according to Chapter 3. The total specific entropy of the glass at the mixture Tg is stotal,Glass (Tg,mixture ) ¼

r X

vi,Glass si,Glass (Tg,mixture ) þ Dsmixing,Glass (Tg,mixture )

i¼1

where all variables described above in the liquid state have similar definitions in the glassy state. The mixture is ideal with partition coefficients of unity because it is assumed that the mass fraction of component i is the same in both phases. Hence,

vi,Glass ¼ vi,Liquid ¼ vi One typically equates the chemical potential of component i in both phases and develops relations between concentration variables in the corresponding phases at equilibrium. This tedium is circumvented by assuming ideality and the equality of component i’s mass fraction in both phases. Entropy continuity is invoked at Tg,mixture and the conformational entropy of mixing is assumed to be the same in the liquid and glassy states at the mixture Tg. Hence, stotal,Liquid (Tg,mixture ) ¼ stotal,Glass (Tg,mixture ) Dsmixing,Liquid (Tg,mixture ) ¼ Dsmixing,Glass (Tg,mixture ) r X   vi si,Liquid (Tg,mixture )  si,Glass (Tg,mixture ) ¼ 0 i¼1

16

Chapter 1 Glass Transitions in Amorphous Polymers: Basic Concepts

If second-order equilibrium phase transitions occur at constant pressure, which could be controversial, then the temperature dependence of specific entropy is evaluated using the total differential of specific enthalpy: dh ¼ T ds þ v dp þ composition-dependent terms for mixtures At constant pressure and composition,     @h @s ¼ Cp ¼ T @T p,composition @T p,composition The specific entropy of component i is evaluated with respect to reference temperature, Treference:

si,Liquid (Tg,mixture ) ¼ si,Liquid (Treference ) þ

Tg,mixture ð

Cpi,Liquid d ln T Treference

si,Glass (Tg,mixture ) ¼ si,Glass (Treference ) þ

Tg,mixture ð

Cpi,Glass d ln T Treference

For temperature-independent specific heats, or temperature-averaged values of Cp, the two previous equations are simplified as follows:   Tg,mixture si,Liquid (Tg,mixture ) ¼ si,Liquid (Treference ) þ Cpi,Liquid ln Treference   Tg,mixture si,Glass (Tg,mixture ) ¼ si,Glass (Treference ) þ Cpi,Glass ln Treference Treference is chosen as the glass transition temperature of pure component i, Tg,i. Furthermore, entropy continuity is invoked for each pure component at Tg,i: si,Liquid (Tg,i ) ¼ si,Glass (Tg,i ) Hence,   Tg,mixture si,Liquid (Tg,mixture )si,Glass (Tg,mixture ) ¼ DCpi ln Tg,i where DCpi is the discontinuous increment in specific heat (i.e., Cpi,Liquid – Cpi,Glass) of component i at its pure-component glass transition temperature, Tg,i. Entropy continuity for a multicomponent mixture at Tg,mixture yields   r X Tg,mixture ¼0 vi DCpi ln Tg,i i¼1

1.7 Compositional Dependence of Tg via Entropy Continuity

17

Rearrangement of the previous equation allows one to predict the mixture Tg in terms of composition variables and pure-component thermophysical properties: Xr   v DC ln Tg,i   i pi i¼1 Xr ln Tg,mixture ¼ v DCpi i¼1 i This is the Couchman – Karasz equation for the compositional dependence of the glass transition temperature in miscible multicomponent mixtures via entropy continuity [Couchman and Karasz, 1978]. Most Tg – composition relations for miscible mixtures can be obtained from the Couchman – Karasz equation by invoking additional approximations or assumptions. This is illustrated below for the classic Gordon– Taylor and Fox equations. For example, if component k is chosen arbitrarily, then Xr      v DC ln T  ln Tg,k     i pi g,i i¼1 Xr ln Tg,mixture  ln Tg,k ¼ v DCpi i¼1 i   Xr Tg,i   v DC ln pi i¼1 i Tg,mixture Tg,k Xr ln ¼ Tg,k v DCpi i¼1 i Even though Tg,mixture and Tg,k are different, if an absolute temperature scale is employed, as required, then the ratio of Tg,mixture to Tg,k is not very different from unity. Hence, it is acceptable to expand the log of the Tg ratios on each side of the previous equation and truncate the series after the linear term. In other words, Tg,mixture ¼ Tg,k þ DTg,mix=k DTg,mix=k Tg,mixture ¼ 1 Tg,k Tg,k

1¼ ln

  Tg,mixture Tg,mixture ¼ ln(1 þ 1)  1 ¼ 1 Tg,k Tg,k 

Similarly, ln

Tg,i Tg,k

 

Tg,i 1 Tg,k

The previous two approximations for the logarithm of a temperature ratio are used to arrive at the Gordon – Taylor equation for the compositional dependence of Tg,mixture:   Xr Tg,i v DC  1 pi i¼1 i Tg,mixture Tg,k Xr 1¼ Tg,k v DCpi i¼1 i The final result predicts curvature in Tg,mixture versus composition for binary mixtures: Xr v DCpi Tg,i i¼1 i Tg,mixture ¼ X r v DCpi i¼1 i

18

Chapter 1 Glass Transitions in Amorphous Polymers: Basic Concepts

The following assumptions were invoked to derive the Gordon – Taylor equation: (a) Entropy continuity: si,Liquid (Tg,i ) ¼ si,Glass (Tg,i ) stotal,Liquid (Tg,mixture ) ¼ stotal,Glass (Tg,mixture ) Dsmixing,Liquid (Tg,mixture ) ¼ Dsmixing,Glass (Tg,mixture ) (b) Ideal mixtures: vi,Glass ¼ vi,Liquid (c) Upon heating, Tg occurs at constant pressure. (d) Temperature-independent pure-component specific heats: Cpi,Liquid = f (T)

(e) ln



Tg,i Tg,k

Cpi,Glass = f (T)

 

Tg,i  1 þ  Tg,k

If the following assumption is included in the previous list, (f1) Tg,i DCpi (@ Tg,i) ¼ a constant that is the same for all components then the Gordon– Taylor equation reduces to the Fox equation: r X 1 vi ¼ T Tg,mixture i¼1 g,i which only requires knowledge of pure-component glass transition temperatures and the mixture composition. The Fox equation predicts curvature in Tg,mixture versus composition for binary mixtures (see Problem 1.10d and Figure 1.1). If assumption (f1) is replaced by (f2) DCpi (@ Tg,i) ¼ a constant that is the same for all components then the Gordon– Taylor equation reduces to the following additive rule of mixtures (i.e., linear weight-fraction-weighted sum of pure-component glass transition temperatures), which does not predict curvature in the compositional dependence of Tg,mixture: r X Tg,mixture ¼ vi Tg,i i¼1

1.8 COMPOSITIONAL DEPENDENCE OF Tg VIA VOLUME CONTINUITY A modified version of the Gordon – Taylor equation is developed by invoking volume continuity of each component and the mixture at their respective glass transition temperatures. The specific volume of the liquid mixture at Tg,mixture is calculated as a volume-fraction-weighted sum of pure-component specific volumes, and the volume change due to mixing; r  X  vtotal,Liquid (Tg,mixture ) ¼ wi vi,Liquid (Tg,mixture ) þ Dvmixing,Liquid (Tg,mixture ) i¼1

1.8 Compositional Dependence of Tg via Volume Continuity

19

where wi is the volume fraction of component i in both phases, vi,Liquid is the specific volume of pure component i, and Dvmixing,Liquid is the volume change upon mixing due to conformational changes and energetic interactions in the equilibrium liquid. Likewise, the specific volume of the glass at Tg,mixture is calculated as follows: r  X  wi vi,Glass (Tg,mixture ) þ Dvmixing,Glass (Tg,mixture ) vtotal,Glass (Tg,mixture ) ¼ i¼1

If second-order phase transitions occur at constant pressure, then the following equation is integrated with the assumption that thermal expansion coefficients are essentially independent of temperature or specific volume:   @ ln v a¼ @T p The reference temperature for component i is its pure-component glass transition, Tg,i. Hence,   vi,Liquid (T)  vi,Liquid (Tg,i ) exp ai,Liquid (T  Tg,i )   vi,Glass (T)  vi,Glass (Tg,i ) exp ai,Glass (T  Tg,i ) Now, one invokes volume continuity of the mixture at Tg,mixture and assumes that volume changes upon mixing are the same for the equilibrium liquid and glass. Hence, vtotal,Liquid (Tg,mixture ) ¼ vtotal,Glass (Tg,mixture ) Dvmixing,Liquid (Tg,mixture ) ¼ Dvmixing,Glass (Tg,mixture ) The result is r X   wi vi,Liquid (Tg,i ) exp ai,Liquid (Tg,mixture  Tg,i ) i¼1

¼

r X

  wi vi,Glass (Tg,i ) exp ai,Glass (Tg,mixture  Tg,i )

i¼1

The exponentials are expanded in a Taylor series and truncation is performed after the linear terms. This is reasonable because thermal expansion coefficients are on the order of 51024 K21. Since temperatures are in the 300– 500 K range, the argument of each exponential is between 0.15 and 0.25. Under these conditions,   exp ai (Tg,mixture  Tg,i )  1 þ ai (Tg,mixture  Tg,i ) þ    Volume continuity of the mixture assumes the following form: r X   wi vi,Liquid (Tg,i ) 1 þ ai,Liquid (Tg,mixture  Tg,i ) i¼1



r X   wi vi,Glass (Tg,i ) 1 þ ai,Glass (Tg,mixture  Tg,i ) i¼1

Since volume continuity of pure component i at Tg,i is also operative, vi,Liquid (Tg,i ) ¼ vi,Glass (Tg,i ) ¼ vi (Tg,i )

20

Chapter 1 Glass Transitions in Amorphous Polymers: Basic Concepts

one obtains r X wi vi (Tg,i ){ai,Liquid  ai,Glass}{Tg,mixture  Tg,i }  0 i¼1

Hence, one predicts the compositional dependence of Tg,mixture via the following volume continuity modification of the Gordon – Taylor equation for miscible r-component mixtures: Xr wT v Dai,@Tg,i i¼1 i g,i i,@Tg,i Tg,mixture ¼ X r wv Dai,@Tg,i i¼1 i i,@Tg,i where volume fractions wi replace weight fractions vi and the discontinuous increment in thermal expansion coefficient, Dai ¼ ai,Liquid 2 ai,Glass, for pure component i at Tg,i replaces DCpi.

1.9 LINEAR LEAST SQUARES ANALYSIS OF THE GORDON –TAYLOR EQUATION AND OTHER Tg – COMPOSITION RELATIONS FOR BINARY MIXTURES The original Gordon– Taylor equation for multicomponent mixtures, with lnTg replaced by Tg, Xr v {DCp,i@Tg,i}Tg,i i¼1 i Tg,mixture ¼ X r v DCp,i@Tg,i i¼1 i is written for binary mixtures as Tg,mixture ¼

v1 {DCp,1@Tg,1}Tg,1 þ v2 {DCp,2@Tg,2}Tg,2 v1 DCp,1@Tg,1 þ v2 DCp,2@Tg,2

which predicts that Tg,mixture lies between Tg,1 and Tg,2. If one defines the ratio of specific heat discontinuities for both pure components at their respective glass transition temperatures as DCp,2@Tg,2 h¼ DCp,1@Tg,1 then algebraic manipulation of the Tg – composition relation yields a linear form that is useful for actual data analysis: Tg,mixture ¼ Tg,1  h

v2 {Tg,mixture  Tg,2 } v1

Hence, one experimentally measures Tg,mixture versus v2, where component 2 can be viewed as the plasticizer that depresses the glass transition temperature of the polymer. The first data pair at v2 ¼ 0 is used to “force” the linear analysis described below to yield an intercept of Tg,1, which corresponds to the glass transition temperature of the

1.10 Free Volume Concepts

21

undiluted polymer. A first-order polynomial is required (i.e., y ¼ bx þ c) with dependent variable y ¼ Tg,mixture, independent variable x¼

v2 {Tg,mixture  Tg,2} v1

and slope b ¼2h. The pure-component data pair at v2 ¼ 1 is excluded from the analysis. Linear least squares analysis is also possible if one employs the Tg – composition relation for binary mixtures that includes logarithmic temperatures instead of incorporating any of the approximations that yield the Gordon– Taylor equation. For example, X2 ln(Tg,mixture ) ¼

¼

v DCp,i@Tg,i ln(Tg,i ) i¼1 i X 2 v DCp,i@Tg,i i¼1 i

v1 DCp,1@Tg,1 ln(Tg,1 ) þ v2 DCp,2@Tg,2 ln(Tg,2 ) v1 DCp,1@Tg,1 þ v2 DCp,2@Tg,2

where component 2 is viewed as the plasticizer. Rearrangement of the previous equation allows one to identify dependent and independent variables of a first-order polynomial for linear least squares analysis: 

 Tg,mixture v2 ln ln(Tg,mixture ) ¼ ln(Tg,1 )  h v1 Tg,2 The appropriate polynomial model is y(x) ¼ bx þ c, and (i) the dependent variable is y ¼ ln(Tg,mixture );   (ii) the independent variable is x ¼ v 2 =v1 ln Tg,mixture =Tg,2 ; (iii) the zeroth-order coefficient is forced to be c ¼ ln(Tg,1 ); (iv) the first-order coefficient is b ¼2h. The “fitting parameter” h should be interpreted as a ratio of discontinuous increments in specific heat for pure plasticizer relative to the undiluted polymer at their respective glass transition temperatures.

1.10 FREE VOLUME CONCEPTS Qualitative and quantitative aspects of free volume are useful to analyze effects on Tg due to plasticizers, molecular weight, and pressure. Diffusion of solvents and gases in polymers occurs via the empty space between molecules that redistributes itself with little or no energy change. This empty space in a material results from the formation of holes or vacancies. The specific free volume associated with this empty space is vfree (T) ¼ vactual (T)  voccupied (T)

22

Chapter 1 Glass Transitions in Amorphous Polymers: Basic Concepts

where vactual represents the experimentally measured specific volume and voccupied, defined by voccupied (T) ¼ voccupied (T ¼ 0 K) þ v free,interstitial (T) represents the volume at absolute zero that is occupied by all of the atoms, as well as the space between the atoms known as interstitial free volume. Large amounts of energy are required to redistribute interstitial free volume and, hence, it is not very useful for molecular transport (i.e., diffusion of solvents and gases through polymers). In reference to the free space in a material that can be manipulated with relative ease, one defines fractional free volume as vfree f ¼ vactual Manipulation of f can be achieved when one understands how fractional free volume depends on a variety of physicochemical parameters. Time-dependent effects on fractional free volume, such as densification and physical aging below the glass transition temperature, are not considered in the analyses below.

1.11 TEMPERATURE DEPENDENCE OF FRACTIONAL FREE VOLUME The strategy is to expand the temperature dependence of actual and occupied specific volumes in a Taylor series about a reference temperature and truncate the nonlinear terms. The reference temperature is chosen as Tg. Above the glass transition temperature, actual volume expands according to aLiquid, whereas occupied volume expands according to aGlass. In other words,   @ ln vactual ¼ aLiquid @T p   @ ln voccupied ¼ aGlass @T p At constant pressure, the polynomials that describe the temperature dependence of vactual and voccupied are   @vactual (T  Tg ) þ    vactual (T) ¼ vactual (Tg ) þ @T p,T¼Tg   @voccupied (T  Tg ) þ    voccupied (T) ¼ voccupied (Tg ) þ @T p,T¼Tg These linear polynomials are written in terms of the appropriate coefficients of thermal expansion as follows: vactual (T) ¼ vactual (Tg ) þ (T  Tg )aLiquid vactual (T ¼ Tg ) þ    voccupied (T) ¼ voccupied (Tg ) þ (T  Tg )aGlass voccupied (T ¼ Tg ) þ   

1.12 Compositional Dependence of Fractional Free Volume

23

By definition, the useful free volume is constructed by subtracting the previous two equations:   vfree (T)  vactual (Tg )  voccupied (Tg )   þ (T  Tg ) aLiquid vactual,T¼Tg  aGlass voccupied,T¼Tg This linear function for vfree(T) is divided by vactual(Tg) to generate an approximate expression for fractional free volume. Hence, vfree (T) f (T)  vactual (Tg ) Since vfree(T ) is a linear function of temperature, and vactual(Tg) is a zeroth-order function of temperature (i.e., a constant), the previous equation for fractional free volume should be linear in T. The result is   vactual (Tg )  voccupied (Tg ) f (T)  vactual (Tg )   aLiquid vactual (Tg )  aGlass voccupied (Tg ) (T  Tg ) þ    þ vactual (Tg ) The first term for f(T ) on the right side of the previous equation is, by definition, the fractional free volume at the glass transition temperature, f (Tg). The coefficient of aGlass in large brackets { } in the second term on the right side of f(T ) is the ratio of occupied to actual specific volumes at Tg, which is equivalent to voccupied (Tg ) vactual (Tg )  vfree (Tg ) ¼ ¼ 1  f (Tg ) vactual (Tg ) vactual (Tg ) Hence,

   f (T)  f (Tg ) þ aLiquid  1  f (Tg ) aGlass (T  Tg ) þ     f (Tg ) þ Da(T  Tg ) þ   

where Da ¼ aLiquid 2 {12f(Tg)}aGlass, which reduces to aLiquid 2 aGlass if the actual and occupied specific volumes are approximately equal at the glass transition temperature.

1.12 COMPOSITIONAL DEPENDENCE OF FRACTIONAL FREE VOLUME AND PLASTICIZER EFFICIENCY FOR BINARY MIXTURES This analysis of the compositional dependence of fractional free volume is applicable to binary mixtures. When one of the components is a low-molecular-weight plasticizer, the dependence of fractional free volume on diluent volume fraction for trace amounts of diluent is identified as the plasticizer efficiency. In other words, "  # @fmixture ¼ G(T ) lim w diluent )0 @ wdiluent T, p

24

Chapter 1 Glass Transitions in Amorphous Polymers: Basic Concepts

where wdiluent is the volume fraction of the small molecule, and G(T) is the plasticizer efficiency parameter, or the difference between the pure-component fractional free volumes of diluent and polymer. This linear theory assumes that the actual volume and the useful free volume of a mixture of polymer and diluent can be obtained as a contribution from each pure component, with no change in actual volume or free volume upon mixing. This assumption might not be justified if small molecules occupy the empty space between chain molecules that originates from packing imperfections. In terms of extensive properties (i.e., volume) of individual components upon mixing, Vactual,mixture ¼ Vactual,polymer þ Vactual,diluent Vfree,mixture ¼ Vfree,polymer þ Vfree,diluent then, by definition, the fractional free volume of the mixture is Vfree,polymer þ Vfree,diluent Vactual,polymer þ Vactual,diluent



 Vfree,polymer Vfree,diluent ¼ w þ w Vactual,polymer polymer Vactual,diluent diluent

fmixture ¼

where the volume fraction of component i is defined, in general, for a mixture of N components as Vactual,component i Vactual,component i wi ¼ XN ¼ Vactual,mixture V j¼1 actual,component j The previous expression for fmixture is simply a linear volume-fraction-weighted sum of the fractional free volume for each pure component in the mixture. Since all volume fractions must sum to unity, and the pure-component fractional free volume is defined by Vfree,component i fcomponent i ¼ Vactual,component i the temperature and compositional dependence of a binary mixture’s fractional free volume is fmixture (T, wdiluent ) ¼ fpolymer (T) þ wdiluent G(T) The plasticizer efficiency parameter is defined by G(T) ¼ fdiluent (T)  fpolymer (T) From the previous section, the fractional free volume for each pure component is essentially a linear function of temperature. Hence,   G(T) ¼ fdiluent (Tg,diluent )  fpolymer (Tg,polymer ) þ Dadiluent (T  Tg,diluent )  Dapolymer (T  Tg,polymer ) If (i) the glass transition is an “iso-free-volume” state such that any material exhibits about 2.5% useful empty space at its Tg when it is cooled from the molten

1.13 Fractional Free Volume Analysis of Multicomponent Mixtures

25

state at 5 – 10 8C/min, and (ii) the universal value for the discontinuous increment in thermal expansion coefficient at Tg is approximately 4.81024 K21, then the plasticizer efficiency parameter reduces to G  4:8104 K1 (Tg,polymer  Tg,diluent ) Diluents with lower glass transition temperatures are better plasticizers for a given polymer. They induce larger reductions in Tg of the mixture because, at any temperature and diluent volume fraction, the fractional free volume of the mixture is larger when the plasticizer efficiency parameter increases. If the fractional free volume of a polymer – diluent mixture is larger, and presumably greater than 2.5%, then the material must experience a greater reduction in temperature to achieve the glassy state where f  0.025. The concept of an “iso-free-volume” state at the glass transition temperature does not consider the rate dependence of Tg during heating or cooling traces that are required in actual experiments. A standardized cooling rate from the equilibrium viscous liquid state (i.e., 5 –10 8C/min) is necessary to compare the effects of external factors on Tg via fractional free volume. The concept of plasticizer efficiency is useful if the primary objective is to decrease the glass transition temperature of an amorphous polymer. The only restriction is that polymer and diluent must be miscible so their mixture will yield a single Tg. Negative plasticizer efficiency implies that rigid aromatic-containing diluents with relatively high glass transition temperatures will increase a flexible polymer’s Tg when the diluent’s Tg is higher than that of the polymer. All theories of the compositional dependence of the glass transition temperature for mixtures predict that Tg,mixture is somewhere between the purecomponent second-order phase transition temperatures. Thermal synergy is operative when Tg,mixture is higher than the glass transition temperature of each pure component. This is discussed in Chapter 6 when low-molecular-weight metal complexes from the d-block in the Periodic Table induce synergistic increases in the glass transition temperature of functional polymers that contain a lone pair of electrons on nitrogen in the side group. Exothermic energetic interactions are operative when transition-metal complexes coordinate to these functional sidegroups and increase Tg.

1.13 FRACTIONAL FREE VOLUME ANALYSIS OF MULTICOMPONENT MIXTURES: COMPOSITIONAL DEPENDENCE OF THE GLASS TRANSITION TEMPERATURE Results from the previous section for binary polymer – diluent blends can be extended to mixtures of N components. The fractional free volume of the mixture is written as a linear volume-fraction-weighted sum of the fractional free volume for each pure component, neglecting any changes in actual and useful free volumes due to the mixing process. For example, fmixture ¼

N X wi fcomponent i i¼1

26

Chapter 1 Glass Transitions in Amorphous Polymers: Basic Concepts

The temperature dependence of the fractional free volume of each pure component is fcomponent i (T)  fcomponent i (Tg,i ) þ Dai (T  Tg,i ) þ    where the iso-free-volume assumption suggests that the leading term in the truncated series is approximately 0.025 for any material (i.e., fcomponent i (Tg,i)  0.025). If fmixture is evaluated at Tg,mixture, then the previous two equations yield fmixture (T ¼ Tg,mixture )  0:025 ¼

N X   wi fcomponent i (Tg,i ) þ Dai (Tg,mixture  Tg,i ) þ    i¼1

Rearrangement yields the compositional dependence of the glass transition temperature of an N-component mixture via linear additivity of fractional free volume: XN w T Dai,@Tg,i i¼1 i g,i Tg,mixture ¼ X N w Dai,@Tg,i i¼1 i The compositional dependence of Tg via volume continuity in Section 1.8 contains additional factors of pure-component specific volumes at Tg,i in each term of the numerator and denominator. The previous equation compares well with the Gordon – Taylor equation via entropy continuity: XN v T DCpi,@Tg,i i¼1 i g,i Tg,mixture ¼ X N v DCpi,@Tg,i i¼1 i if volume fractions wi are replaced by mass fractions vi and Dai is replaced by DCpi.

1.14 MOLECULAR WEIGHT DEPENDENCE OF FRACTIONAL FREE VOLUME There is a considerable amount of useful free volume in the vicinity of the chain ends due to packing imperfections. The concentration of chain ends, as determined by endgroup titration, increases at lower molecular weights. The number-average molecular weight is most appropriate to account for the chain ends when polymers exhibit a broad distribution of molecular weights. In other words, it is better to use a molefraction-weighted average (i.e., the number-average molecular weight, Mn) instead of a weight-fraction-weighted average (i.e., the weight-average molecular weight, Mw) when all chains do not contain the same number of repeat units. One postulates the molecular weight dependence of fractional free volume to agree with these claims: f (Mn )  f (Mn ) 1) þ

A Mn

where the leading term on the right side of the previous equation represents the molecular-weight-insensitive fractional free volume for very high molecular weight polymers, and A is a positive constant on the order of 10– 20 daltons for polymers that are produced via condensation mechanisms, and 200– 500 daltons for polymers

1.15 Experimental Design to Test the Molecular Weight Dependence

27

that are produced via free radical mechanisms. If linear temperature dependence is included in the previous equation, then   f (T, Mn )  f (Tg , Mn ) þ Da@Tg (Mn ) T  Tg (Mn )   A  f (Tg , Mn ) 1) þ Da@Tg (Mn )1) T  Tg (Mn ) 1) þ Mn where the discontinuity in thermal expansion coefficient Da at the corresponding Tg is assumed to be independent of molecular weight. Since the glass transition is an iso-free-volume state, the leading terms, f (Tg, Mn) and f(Tg, Mn ) 1), are approximately 0.025, and rearrangement yields the following expression for the molecular weight dependence of Tg: A Tg (Mn )  Tg (Mn ) 1)  Mn D a Tg(Mn ) 1) represents the molecular-weight-insensitive glass transition temperature that is tabulated in handbooks for many polymers. The previous equation suggests that Tg exhibits strong dependence on Mn at lower molecular weights, and approaches an asymptote when the concentration of chain ends diminishes significantly at very high molecular weights. The coefficient of the molecular-weightsensitive term on the right side of the previous equation (i.e., A/Da) is on the order of (i) 2 –4104 Da-K for condensation-type polymers and (ii) 4 – 10  105 Da-K for free-radical-type polymers. In general, number-average molecular weights on the order of A/Da, with dimensions of daltons, are required to reach the molecular-weight-insensitive plateau on a graph of Tg versus Mn. When Mn approaches the magnitude of A/Da, there is at most a 1 – 2 degree difference between Tg(Mn ) 1) and the actual glass transition temperature, and this difference is below the detection limits of conventional calorimeters used to measure Tg.

1.15 EXPERIMENTAL DESIGN TO TEST THE MOLECULAR WEIGHT DEPENDENCE OF FRACTIONAL FREE VOLUME AND Tg The following design strategy represents a logical sequence of experiments and analyses to quantify the information presented in the previous section. 1. Collaborate with a polymer chemist and synthesize several molecular weight fractions of an amorphous polymer. Molecular weight control and polydispersity are important considerations in the synthetic procedure. 2. Generate at least 10 different samples of the same polymer (i.e., polystyrene), where each sample contains chains with a different average number of repeat units. Hence, at least 10 different molecular weight fractions of the same polymer will be tested, as described below. 3. Each sample exhibits a distribution of chain lengths, which is analogous to a distribution of molecular weights. The first characterization technique

28

Chapter 1 Glass Transitions in Amorphous Polymers: Basic Concepts

measures the molecular weight distribution of each sample, from which the number-average molecular weight is calculated. Hence, at least 10 different number-average molecular weights are calculated from the molecular weight distributions. 4. Gel permeation chromatography (GPC) is the separation technique that measures the molecular weight distribution of each sample. Larger chain molecules have smaller diffusion coefficients via the Stokes – Einstein equation, and they experience difficulty diffusing into the pores of the particles that are used to pack the chromatographic column. Hence, convective transport “sweeps” the larger chain molecules through the column before they have time to explore the internal structure of the packing via intrapellet diffusion. Consequently, larger chain molecules exit the column and contribute to the detector output curve before the smaller chain molecules. GPC is a separation technique based on the fact that molecules of different size have different residence times in the column. There is an inverse relation between molecular size and residence time, similar to the inverse relation between molecular size and intrapellet diffusion coefficients in porous catalytic pellets. 5. The GPC output curve for each sample reveals the distribution of chain lengths within each sample. This output curve is essentially the molecular weight distribution for the sample, from which the number-average molecular weight can be calculated via statistical analysis of the distribution. 6. Now that the number-average molecular weight of each sample is known, it is necessary to collaborate with a thermal analysis expert and obtain a differential scanning calorimetry (DSC) trace of each sample. These data correspond to specific heat versus temperature. If each sample is heated from the glassy state to the molten state, then one can calculate the glass transition temperature where the specific heat exhibits a discontinuity. More specifically, Tg is measured during the second or third heating trace and reported as the temperature where the discontinuity in Cp is approximately one-half of the total step increment in Cp. For example, one extends the baseline heat capacity of the glass into the molten state. A similar extrapolation of Cp,liquid below Tg is required so that both baseline heat capacities encompass a broad temperature range above and below Tg. Graphical evaluation of the glass transition temperature is obtained by (i) identifying two midpoints between the baseline heat capacities—one midpoint is above Tg and the other midpoint is below the phase transition; (ii) connecting these midpoints with a straight line; and (iii) locating the temperature where the straight line that connects the two midpoints intersects the actual calorimetric data. Now, at least 10 Mn – Tg data pairs are available. 7. Perform linear least squares analysis of the Mn – Tg data pairs, realizing that the molecular weight dependence of the glass transition is Tg (Mn )  Tg (Mn ) 1) 

A Mn D a

(1:5)

1.16 Pressure Dependence of Fractional Free Volume

29

This model of Tg was obtained by postulating the molecular weight dependence of fractional free volume as follows: f (Mn )  f (Mn ) 1) þ

A Mn

(1:6)

where the molecular-weight-independent constant A is positive. Hence, linear least squares analysis of Tg versus 1/Mn via a first-order polynomial should exhibit a negative slope because (i) A . 0 and (ii) the discontinuity in thermal expansion coefficient at Tg, Da ¼ aLiquid 2 aGlass, is positive. The zeroth-order coefficient obtained from linear least squares analysis yields a good estimate of the molecular-weight-insensitive Tg, given by Tg(Mn ) 1) in Eq. (1.5). 8. Next, it is necessary to locate a dilatometer and measure the specific volume of each sample as a function of temperature from the glassy state to the highly viscous liquid state. The slope of ln vspecific versus temperature corresponds to thermal expansion a ¼ (@ ln vspecific/@T )p, which must be measured above and below Tg to calculate the discontinuity Da at the glass transition. 9. These two tests of the data should be self-consistent. If Tg versus 1/Mn is linear, then the slope is constant and independent of molecular weight. Since the slope is 2A/Da, the Mn 2Tg data suggest that Da is not a function of molecular weight. This claim should be verified by calculating Da from dilatometry for each sample with a different number-average molecular weight. 10. If the discontinuity in thermal expansion is truly independent of molecular weight, then   dTg A ¼ Da@Tg  dMn1 Reasonable values for Da(@Tg) are on the order of 51024 K21, and typical ranges for A have been discussed previously for both free-radical and condensation polymers. Now, quantitative expressions are available for the molecular weight dependence of (i) Tg via Eq. (1.5) and (ii) fractional free volume via Eq. (1.6).

1.16 PRESSURE DEPENDENCE OF FRACTIONAL FREE VOLUME Since the actual volume of a material decreases isothermally upon compression, one expects that the same is true for free volume. This is consistent with predictions from the Ehrenfest equations, which indicate that Tg increases at higher pressure. The strategy is to expand the pressure dependence of actual and occupied volumes in a Taylor series about a reference pressure, denoted by pref, and truncate the nonlinear terms. Above the glass transition temperature, actual volume compresses according to

30

Chapter 1 Glass Transitions in Amorphous Polymers: Basic Concepts

bLiquid, whereas occupied volume compresses according to bGlass. In other words     @ ln voccupied @ ln vactual ¼ bLiquid ; ¼ bGlass @p @p T T At constant temperature, the polynomials that describe the pressure dependence of vactual and voccupied are   @vactual ( p  pref ) þ    vactual ( p)  vactual ( pref ) þ @p T¼Tg   @voccupied ( p  pref ) þ    voccupied ( p)  voccupied ( pref ) þ @p T¼Tg These linear polynomials are written in terms of the appropriate coefficients of isothermal compressibility as follows: vactual ( p)  vactual ( pref )  ( p  pref )bLiquid vactual@pref þ    voccupied ( p)  voccupied ( pref )  ( p  pref )bGlass voccupied@pref þ    By definition, the useful free volume is constructed by subtracting the previous two equations:   vfree ( p)  vactual ( pref )  voccupied ( pref )    ( p  pref ) bLiquid vactual@pref  bGlass voccupied@pref þ    This linear function for vfree(p) is divided by vactual( pref ) to generate an approximation for fractional free volume. Hence f ( p) 

vfree ( p) vactual@pref

Since vfree( p) is a linear function of temperature, and vactual( pref ) is a zeroth-order function of temperature (i.e., a constant), the previous equation for fractional free volume should be linear in p. The result is   vactual ( pref )  voccupied ( pref ) f ( p)  vactual ( pref )   bLiquid vactual ( pref )  bGlass voccupied ( pref ) ( p  pref ) þ     vactual ( pref ) The first term for f( p) on the right side of the previous equation is, by definition, the fractional free volume at the reference pressure, f ( pref ). The coefficient of bGlass in large brackets { } in the second term on the right side of f( p) is the ratio of occupied to actual specific volumes at pref, which is equivalent to voccupied ( pref ) vactual ( pref )  vfree ( pref ) ¼ ¼ 1  f ( pref ) vactual ( pref ) vactual ( pref )

1.17 Effect of Particle Size or Film Thickness on the Glass Transition Temperature

31

Hence,   f ( p)  f ( pref )  bLiquid  [1  f ( pref )]bGlass ( p  pref ) þ     f ( pref )  Db( p  pref ) þ    where Db ¼ bLiquid 2 {1 2 f ( pref )}bGlass, which reduces to bLiquid 2 bGlass if the actual and occupied specific volumes are approximately equal at pref.

1.17 EFFECT OF PARTICLE SIZE OR FILM THICKNESS ON THE GLASS TRANSITION TEMPERATURE Question: Is Tg for a powder sample of a polymer the same as that for a thin film of the same polymer? How does particle size (i.e., for powders) or film thickness affect the glass transition temperature? The 21 responses that follow were obtained from selected members of a discussion list that is maintained by the Division of Polymer Chemistry of the American Chemical Society. The responses have been edited for clarity and to ensure anonymity. RESPONSE #1: The glass transition should not vary as a function of the physical state of the polymer. What might change is the ability to transfer heat adequately. For example, if one tests a thick sample via DSC, then there will be inefficient heat transfer between the bottom of the sample pan and the upper surface of the film. This is particularly important for most polymers that have low thermal conductivity. Consequently, the glass transition is broadened and shifted to higher temperatures, depending on the heating rate. Technically, all thermal transitions should not depend on particle size or film thickness. RESPONSE #2: This is a debatable issue and a very interesting question. If glass transition temperatures are different for powders and films, then one should formulate a fundamental explanation of this phenomenon. Remember that Tg is typically a broad transition (maybe 10 8C) and the reported value depends on the experimental protocol, especially test frequency (i.e., Tg is approximately 6 8C higher per decade increase in frequency) and midpoint-versusonset measurement. RESPONSE #3: The glass transition temperature depends on film thickness and molecular weight. As film thickness increases, it affects cure and powder properties. RESPONSE #4: The difference between Tg values of powders and films depends on whether the powder is a thermosetting formulation (i.e., fusion-bonded epoxy) or a thermoplastic such as polyethylene or nylon. One measures a glass transition during the first heating trace that depends very much on thermal history, aging, stored mechanical energy from the powdering process, moisture content, plasticizing solvents, and so on. Annealing at temperatures just below Tg (i.e., enthalpy relaxation) will identify the “real” glass transition much more quickly than aging the sample for a few months at ambient temperature. Some of this information is available in the research literature on powder coatings. The work of Turi [1997] is described in definitive texts on this subject. An entire chapter is dedicated to the discussion of thin films. RESPONSE #5: Glass transition temperature differences between powders, films, and fibers might be due to the fact that the amount of residual solvent is different in each sample.

32

Chapter 1 Glass Transitions in Amorphous Polymers: Basic Concepts

Residual solvent content probably follows the following trend: film . powder . fiber, which will exert a plasticizing effect on the polymer and lower Tg. This trend is observed in the conversion of polyamic acids to polyimides, which is facilitated by residual solvent. RESPONSE #6: This is a very active and controversial area of research, and universal trends have not been established. The lack of universality is related to different effects of the substrate and air interface on free volume and local packing of the chains. The effect of film thickness on the glass transition temperature should be important when the thickness is on the order of a few radii of gyration, at most. RESPONSE #7: Tg decreases precipitously when film thicknesses are less than the dimensions of a single chain. This can be achieved by spin-coating ultrathin films onto substrates at sufficiently high rotational velocity such that centrifugal forces influence anisotropic chain conformations with larger projections of the segment vectors in the plane of the substrate. RESPONSE #8: This research problem has not been resolved yet. In 1994, the glass transition temperature of polymer thin films was found to be different from that in the bulk for polystyrene spin-coated on silicon and PMMA on gold-coated silicon. Tg decreased by a maximum of 20% from the bulk value when film thickness was decreased. More recently, the same group of researchers investigated PMMA thin films on silicon and measured an increase in Tg as film thickness decreased. In the earlier work, interactions between the polymer and the substrate were weakly favorable, at most, whereas in the more recent study, PMMA –silicon interactions are strongly attractive. In other studies of the same phenomenon, random copolymers of styrene and methyl methacrylate form an interface between polystyrene homopolymer and the substrate for quantitative control of interfacial energy. Results suggest that the effect of film thickness on the glass transition temperature is not driven by interfacial energy considerations. Perturbations in local density of the polymer at the interface with the substrate are the most probable cause of different Tg values for bulk polymers and thin films, when film thickness is below approximately ˚ . Similar effects for powdered samples of a given polymer are possible if particle size is 500 A ˚ with no aggregation among particles upon heating. The following references below 500 A represents an introduction to the effects of particle size and film thickness on the glass transition temperature: Keddie et al. [1994a, b], Mayes [1994], Wallace et al. [1995], Forrest et al. [1996], Kajiyama et al. [1997], and Tsui et al. [2001]. RESPONSE #9: The glass transition temperature depends on molecular weight, crosslink density, comonomer composition, phase separation, degree of crystallinity, chemical structure, pressure, and the time scale of the measurements. Tg should be similar if the same polymer is studied as a film or a powder, and all of the above-mentioned parameters are constant. RESPONSE #10: The glass transition temperature of a polymer depends on the heating rate employed. Film thickness affects the rate of heat transfer rate into the film. When films are produced from powders, the morphology of the polymer might change, which will influence Tg. For semicrystalline polymers, sample preparation procedures will change the size and perfection of the microcrystalline regions. RESPONSE #11: The following points must be considered: (i) thin films yield more “definitive glass transition temperatures” than powdered samples of the same polymer; (ii) annealing below Tg will affect any subsequent measurements of the transition; (iii) thermal contact between the aluminum sample pan and the bottom of the sample is important; and (iv) thin films cast from solution might contain residual solvent that will plasticize the polymer. RESPONSE #12: The glass transition temperature of a polymer is an inherent property of the material, but the actual value that one measures could depend on the experimental method used,

1.17 Effect of Particle Size or Film Thickness on the Glass Transition Temperature

33

thermal history of the sample, and molecular weight. If powders and films of the same polymer have identical average molecular weight and polydispersity, then the second DSC heating trace after rapid thermal quenching should yield similar values that are independent of the sample’s original physical state. RESPONSE #13: If DSC is used to measure the glass transition temperature, then Tg might depend on the physical form of a polymer. Contact between the polymer and the bottom of the aluminum sample pan is required for heat transfer to increase the sample temperature. It is best to obtain several different measurements via DSC scans at a heating rate of 2 8C/min with approximately 7–10 milligrams of sample in a sealed environment. It might be difficult to observe a glass transition for powdered samples during the first heating trace. When thermoplastics are heated into the highly viscous liquid state above Tg and then cooled to lower temperatures, the physical state of the solid sample will change from a powder to a film, and this introduces unwanted complexity into the problem. RESPONSE #14: The glass transition temperature depends on film thickness. There are indications from gas permeation measurements that very thin films on the order of 100 nm, or less, exhibit accelerated physical aging via volume and enthalpy relaxation, which yield higher Tg values than thicker films. RESPONSE #15: The glass transition temperature should be the same for a fully amorphous sample of the same polymer. Since Tg depends on thermal and preparation history, different values of Tg will be measured for samples cast from solution versus cooling from the molten state. Thermal history can be erased by heating samples to temperatures above the highest thermal transition (i.e., Tg or Tm), but below the decomposition temperature. The degree of crystallization has an important role, and the rate of cooling must be considered. Rapid cooling or quenching will produce a glassy material, whereas slow cooling might allow some crystallization to occur. Hence, one expects to measure different glass transition temperatures during the second heating trace for powders, pellets, and films. RESPONSE #16: Even though Tg is an intrinsic property of a material, it probably depends on film thickness or particle size because a discontinuous powder influences the rate of heat transfer through the aggregate differently than a solid film, thus affecting the rate at which the polymer experiences temperature changes. RESPONSE #17: The answer depends on how the glass transition temperature is measured. If DSC experiments are performed, then when a powder reaches its Tg, it will fuse together and cause some thermal disturbance in the aluminum sample pan. Data are not reported from the first heating scan. On the second and subsequent heating scans, the sample is no longer in powder form. The effect of particle size or film thickness on Tg is observed when these dimensions are less than approximately 0.1 mm (i.e., 100 nm). RESPONSE #18: The glass transition temperature is a material constant and should not depend on sample thickness when it is measured using the correct tools, like precision calorimetry. Films might have different Tg values due to the effects of thermal history that can lead to different degrees of crystallinity. RESPONSE #19: The glass transition temperature of a polymer depends on chemical structure and chain microstructure, but it is independent of the macroscopic form of the material (i.e., powder or thin film). If measurements nevertheless indicate differences, then the presence of contaminants, such as residual solvents in the case of cast films, might be the cause. Any species that is soluble in the polymer (plasticizer, surfactant, or other additives) should depress Tg.

34

Chapter 1 Glass Transitions in Amorphous Polymers: Basic Concepts

RESPONSE #20: The physical state (i.e., powder or thin film) of any material should not affect the glass transition temperature. However, the measurement of Tg might reflect some dependence on size. Since Tg depends on heating or cooling rate, and the rate of heat transfer should be different for the different physical states, the apparent Tg might be different. RESPONSE #21: The previous responses are fairly diverse and address very different questions. The powder question and questions related to history or method of measurement are only marginally related to the question of Tg in nanoscale structures. The latter is an area of active research. The problem in polymers should not be fundamentally different from the problem in small molecules if the glass transition event has “universal” features. Work on liquids confined to pores, or microemulsions, is very relevant to the question. The advantage of polymers is that stable films of nanometer thickness can be made. When this is done, supported films of polystyrene reveal modest Tg depression, similar to ortho-terphenyl in nanopores. Initial results on free-standing films describe rather large changes in Tg, and this transformed the paradigm from an interesting but not fully explained question to one of great interest because polymers are used routinely at temperatures that are more than 70%, and often greater than 90%, of their glass transition temperature. The problem of interest is: why do we see such a range of results? The glass transition temperature decreases, increases, remains the same, or even disappears, depending upon details of the experimental or molecular simulation conditions. Different behaviors have been observed for the same material subjected to different testing methods. In many cases, experiments have been performed carefully, and the results are reproducible. Existing theories of the glass transition cannot explain the range of behaviors seen at the nanometer scale, because the glass transition phenomenon is not fully understood. A mobile layer on the surface might explain some observations, but if there is a mobile layer that provides a complete explanation, then why does polystyrene show such large effects, whereas poly(methyl methacrylate) and poly(vinyl acetate) reveal a very small change, or no change, in Tg for the same free-standing film geometry? Furthermore, why do some systems, such as ortho-terphenyl and a few thermosetting polymers, reveal two glass transitions upon confinement in pores, where, in some cases, one Tg is higher than, and the other is lower than, the bulk Tg? Or, why do polystyrene films on a glycerol surface exhibit no significant changes in Tg, as this geometry seems to be very similar to free-standing films? The question remains of considerable interest, currently there is no full explanation, and it may suggest that the “glass-is-a-glass” paradigm of the glass transition is not correct, as suggested by nanoscale measurements and results. Courtesy of Greg McKenna, Department of Chemical Engineering, Texas Tech University, Lubbock, Texas, USA; November 2009.

1.18 EFFECT OF THE GLASS TRANSITION ON SURFACE TENSION Question: How does the surface tension (or surface free energy) of polymers at an air interface vary with temperature as one passes through the glass transition? The responses that follow were obtained from selected members of a discussion list that is maintained by the Division of Polymer Chemistry of the American Chemical Society. The responses have been edited for clarity and to insure anonymity. RESPONSE #1: Surface and interfacial energies have been measured by Wu [1970], who found that surface tension decreases at higher temperature, but there does not appear to be a significant discontinuity at the glass transition. The surface energy of glasses should be higher than that of rubber-like materials.

References

35

RESPONSE #2: Surface tension is described more appropriately as “surface equilibrium free energy.” Since glasses exist in a nonequilibrium state, surface tension is not well defined for these materials from the viewpoint of rigorous equilibrium thermodynamics. However, the temperature dependence of contact angle measurements should provide some useful insight (Neumann and Tanner [1970]). Kwok and Neumann [1999] review contact angle measurements and discuss various pitfalls. Della Volpe et al. (2006) discuss a method to obtain the equilibrium contact angle for water on poly(methylmethacrylate). Depending on the type of interface (i.e., polymer–air, polymer– water, polymer– solvent), surface tension exhibits time dependence as the polymer is exposed to the “other phase” until equilibrium is achieved. One expects that the surface tension dynamics of glassy polymers (i.e., the approach to equilibrium) is much weaker than that of rubbery polymers, due to reduced mobility below the glass transition temperature. Experimental problems will be encountered above Tg because surface tension measurements suffer from significant instabilities that are related directly to the high degree of mobility in the rubbery state. Since temperature changes have no effect on surface chemistry, thermal energy differences (i.e., kT ) do not contribute much to the equilibrium surface free energy.

REFERENCES COUCHMAN PR, KARASZ FE. Classical thermodynamic discussion of the effect of composition on glass transition temperatures. Macromolecules 11(1):117– 119 (1978). DELLA VOLPE C, BRUGNARA M, MANIGLIO D, SIBONI S, WANGDU T. About the possibility of experimentally measuring equilibrium contact angles and their theoretical and practical consequences. In Contact Angle, Wettability, and Adhesion, Mittal KL, editor, Vol. 4, 2006, 1– 20. EHRENFEST P. Phase changes in the ordinary and extended sense, classified according to the corresponding singularities of the thermodynamic potentials. Proceedings of the Academy of Sciences (Amsterdam) 36:153 (1933). FORREST JA, KALNOKI-VERESS K, STEVENS JR, DUTCHER JR. Effect of free surfaces on the glass transition temperature of thin polymer films. Physical Review Letters 77:2002 (1996). GOLDSTEIN M. Some thermodynamic aspects of the glass transition: free volume, entropy, and enthalpy theories. Journal of Chemical Physics 39:3369 (1963). KAJIYAMA T, TANAKA K, TAKAHARA A. Surface molecular motion of monodisperse polystyrene films. Macromolecules 30:280 (1997). KEDDIE JL, JONES RAL, CORY RA. Size dependent depression of the glass transition temperature in polymer films. Europhysics Letters 27:59 (1994a). KEDDIE JL, JONES RAL, CORY RA. Interface and surface effects on the glass-transition temperature in thin polymer films. Faraday Discusssions 98:219 (1994b). KWOK DY, NEUMANN AW. Contact angle measurements and their interpretation. Advances in Colloid and Interface Science 81:167 –249 (1999). MAYES AM. Glass transition of amorphous polymer surfaces. Macromolecules 27:3114 (1994). NEUMANN AW, TANNER W. Temperature dependence of contact angles, polytetrafluoroethylene with normal decane. Journal of Colloid and Interface Science 34(1):1 (1970). PRIGOGINE I, DEFAY R. Chemical Thermodynamics. Longmans Green, New York, 1954, Chap. 19. TSUI OKC, RUSSELL TP, HAWKER CJ. Effect of interfacial interactions on the glass transition temperature of polymer thin films. Macromolecules 34(16):5535– 5539 (2001). TURI E. editor. Thermal Analysis and Thermal Characterization of Polymeric Materials. 2nd edition; Academic Press, New York, 1997. WALLACE WE, VAN ZANTEN JH, WU WL. Influence of an impenetrable interface on polymer glass transition temperatures. Physical Review E 52:3329 (1995). WU S. Surface and interfacial tension of polymer melts. Journal of Physical Chemistry 74:632 (1970).

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Chapter 1 Glass Transitions in Amorphous Polymers: Basic Concepts

PROBLEMS 1.1. Integrate both of the Ehrenfest equations and obtain explicit expressions for the pressure dependence of the glass transition temperature. Your answers should be cast in the following form: Tg ¼ f (p). In both cases, it is appropriate to assume that discontinuous observables at an equilibrium second-order phase transition are independent of temperature and pressure. Answer See the developments in Section 1.6.2, particularly Eqs. (1.3) and (1.4). The final results are Volume continuity Tg ( p)  Tg ( pref ) þ

Db@Tg Da@Tg

( ppref )

Entropy continuity ( ) ( p  pref )v@pref Da@Tg Tg ( p)  Tg ( pref ) exp DCp,@Tg 1.2. The following data are available for completely amorphous atactic polystyrene from various textbooks, the Polymer Handbook, and refereed journal literature: Glass transition temperature at ambient pressure: Tg ¼ 105 8C Density of the amorphous polymer: ramorphous ¼ 1.052 g/cm3 Coefficient of thermal expansion below the glass transition: aGlass ¼ 1.91024 K21 Coefficient of thermal expansion above the glass transition: aLiquid ¼ 5.61024 K21 Discontinuity in specific heat at the glass transition: DCp ¼ 0.27 J/(g-K) Universal gas constant: Rgas ¼ 8.31 J/g-mol-K ¼ 0.0823 L-atm/(g-mol-K) Begin with the statement of entropy continuity at Tg and calculate the equilibrium glass transition temperature of polystyrene subjected to an external pressure of 2500 atmospheres. Answer Use the second answer to Problem 1.1 and substitute numerical values for all of the thermophysical properties, as provided in the problem statement: ( ) ( p  pref )v@pref Da@Tg Tg (p ¼ 2:5  10 atm) ¼ Tg (pref ) exp DCp,@Tg 8 9 1 4 1 > > (2499atm) (5:6  1:9)10 K > > < = 1:052g=cm3 ¼ [(105 þ 273) K] exp  273 K 3 > > J 82:3 (cm -atm)=(mol-K) > > : ; 0:27 g-K 8:31J=(mol-K) 3

¼ 252 8C

Problems

37

1.3. Estimate the magnitude of the discontinuity in the coefficient of isothermal compressibility,   @ ln v Db ¼ bLiquid  bGlass ; b ¼  @p T for an amorphous polymer at its glass transition temperature. Express your answer in units of inverse atmospheres. Answer Use the differential form of the Ehrenfest equation, based on volume continuity in Section 1.6.1: @Tg Db@Tg  0:020 8C=atm ¼ @p Da@Tg The universally accepted value for the pressure dependence of the glass transition temperature is that Tg increases by 20 K per 103 atmospheres, and the discontinuous increment in thermal expansion coefficients at Tg is Da  4.81024 K21. Hence, Db at Tg is approximately 1025 inverse atmospheres, which corresponds to the coefficient of isothermal compressibility of viscous liquids, because glasses are extremely difficult to compress. For comparison, the coefficient of isothermal compressibility for ideal gases is 1/p, approximately five or six orders-of-magnitude larger than that for liquids, which are typically assumed to be incompressible. 1.4. Sketch v{r, Tg 250 8C}2vequilibrium{Tg 250 8C} versus the experimental cooling rate r, defined by r ¼ 2dT/dt, for an amorphous polymer that has been cooled from the equilibrium liquid state into the nonequilibrium glassy state at several different cooling rates. In other words, compare the difference between specific volume of the nonequilibrium and equilibrium glass, 50 8C below the glass transition temperature, as a function of the experimental cooling rate. Is this a linear relation, or does the slope increase or decrease at high cooling rates? 1.5. (a) Is it possible to invoke DG ¼ DH – TgDS ¼ 0 at the glass transition temperature and rearrange this thermodynamic equation to calculate Tg? Answer No, because enthalpy H and entropy S are continuous at second-order phase transitions, so rearrangement of DG ¼ 0 yields an indeterminate ratio for Tg. The chemical potentials of the glass and viscous liquid are equivalent at the glass transition temperature, which yields DG ¼ 0 at Tg. However, rearrangement of this equation does not provide useful information about the glass transition temperature, as it does at the melting temperature. In other words, Tm ¼ DHfusion/DSfusion can be analyzed qualitatively to identify trends that affect the melting temperature (see Section 7.2.4). (b) Is it possible to invoke DG ¼ DH 2 TgDS ¼ 0 at equilibrium second-order phase transitions, rearrange this thermodynamic equation, apply l’Hoˆpital’s rule, and obtain an expression that allows one to analyze qualitative trends that affect the glass transition temperature? The extensive Gibbs free energy is denoted by G.

38

Chapter 1 Glass Transitions in Amorphous Polymers: Basic Concepts

Answer The answer is negative, regardless if one applies l’Hoˆpital’s rule via differentiation of numerator and denominator, separately, with respect to temperature T or pressure p. In both cases, one obtains the identity Tg ¼ Tg, which is correct, but not useful. Obviously, rearrangement of DG ¼ 0 yields an indeterminate ratio for the glass transition temperature that can be analyzed further by invoking l’Hoˆpital’s rule and differentiating numerator and denominator, separately, with respect to temperature at constant pressure:

Tg ¼

DHGlass)Liquid DSGlass)Liquid

  @H D @T p,Glass)Liquid DCp,Glass)Liquid  )   ¼ @S DCp,Glass)Liquid limT!Tg D @T p,Glass)Liquid Tg

If one differentiates numerator and denominator, separately, with respect to pressure at constant temperature, then another trivial expression is obtained:

Tg ¼

DHGlass)Liquid DSGlass)Liquid

"   #   @S @H D T þV D @p T @p T,Glass)Liquid Glass)Liquid     ) ) @S @V limT!Tg limT!Tg D D @p T,Glass)Liquid @T p,Glass)Liquid   @V Tg D DT¼0 @T DV¼0   p,Glass)Liquid ) @V limT!Tg D @T p,Glass)Liquid

where dH ¼ T dSþV dp and the Maxwell relation, {@S/@p}T ¼ 2{@V/@T}p, via the Gibbs free energy have been employed to implement l’Hoˆpital’s rule via differentiation with respect to pressure. (c) Are the Gibbs free energies of the glass and the highly viscous liquid equivalent at the glass transition temperature? Answer Yes, if second-order phase transitions can be described by equilibrium thermodynamics. The Ehrenfest equations can be developed at the differential level by equating differential changes in the chemical potentials of the glass and viscous liquid on the transition line between these two phases. An indeterminate ratio is obtained for the pressure dependence of the glass transition temperature that can be analyzed further by invoking l’Hoˆpital’s rule and differentiating numerator and denominator, separately, with respect to either temperature or pressure. It should be emphasized that the chemical potential of a pure material is synonymous with the molar Gibbs free energy. 1.6. (a) A miscible binary polymer– polymer blend exhibits a third-order thermodynamic phase transition at constant composition. Draw idealistic data from the heating trace in a

Problems

39

differential scanning calorimeter (i.e., heat capacity Cp vs. temperature) that allows one to identify this transition temperature. Answer Enthalpy versus temperature is discontinuous at first-order melting transitions, and idealistic heat capacities can be represented by delta functions. At second-order glass transitions, enthalpy versus temperature is continuous, but its temperature derivative (i.e., Cp ¼ {@H/@T}p), or heat capacity, versus temperature is discontinuous. At third-order thermodynamic phase transitions, heat capacity versus temperature is continuous, but its temperature derivative (i.e., {@Cp/@T}p) versus temperature is discontinuous. Since DSC data are presented as Cp versus temperature, idealistic third-order phase transitions exhibit identical heat capacities for both phases, but there is a discontinuous increment in the slope of Cp versus temperature as materials are heated through the phase transition. It is impossible to identify third-order phase transitions via DSC because, under realistic conditions, the abrupt discontinuity in slope of Cp versus temperature is broadened such that positive curvature in the baseline cannot be distinguished from a phase transition of this nature. (b) True or false: The temperature dependence of specific enthalpy for a first-order phase transition is analogous to the temperature dependence of specific heat for a secondorder phase transition. Answer True (c) True or false: The temperature dependence of specific enthalpy for a second-order phase transition is analogous to the temperature dependence of specific heat for a third-order phase transition. Answer True 1.7. Instead of invoking entropy continuity, the compositional dependence of the glass transition temperature of a miscible ternary polymer blend is developed by using the concept of volume continuity at Tg,mix. Without performing the tedius derivation, what is your best estimate of the final expression to calculate Tg,mix? Answer The appropriate expression for the glass transition temperature of the ternary mixture is developed completely in Section 1.8. The final result is X3 i¼1 Tg,mixture ¼ X 3

wi Tg,i vi,@Tg,i Dai,@Tg,i

i¼1

wi vi,@Tg,i Dai,@Tg,i

1.8. Identify seven assumptions that must be satisfied before one can invoke the Fox equation to describe the compositional dependence of ternary mixtures that contain two polymers and one UV stabilizer.

40

Chapter 1 Glass Transitions in Amorphous Polymers: Basic Concepts

Answer Use of the Fox equation for the compositional dependence of the glass transition temperature of binary or multicomponent mixtures requires that the following assumptions must be reasonable: (i) Entropy continuity: si,Liquid (Tg,i ) ¼ si,Glass (Tg,i ): stotal,Liquid (Tg,mixture ) ¼ stotal,Glass (Tg,mixture ) Dsmixing,Liquid (Tg,mixture ) ¼ Dsmixing,Glass (Tg,mixture ) (ii) Ideal mixtures: vi,Glass ¼ vi,Liquid. (iii) Upon heating, Tg occurs at constant pressure. (iv) Temperature-independent specific heats: Cpi,Liquid = f (T ), Cpi,Glass = f (T ).   (v) Expansion and truncation of ln Tg,i =Tg,k  (Tg,i =Tg,k )  1 þ    . (vi) The product Tg,iDCpi (@Tg,i) must be the same for all pure components. (vii) Homogeneous single-phase behavior is necessary. 1.9. (a) Consider homogeneous binary mixtures of a glassy polymer and a low-molecularweight flexible plasticizer. The diluent concentration dependence of Tg,mixture is described by the Gordon–Taylor equation based on linear additivity of fractional free volume, as indicated below: Tg,mixture ¼

wDiluent Tg,Diluent DaDiluent,@Tg,Diluent þ (1  wDiluent )Tg,Polymer DaPolymer,@Tg,Polymer wDiluent DaDiluent,@Tg,Diluent þ (1  wDiluent )DaPolymer,@Tg,Polymer

Obtain an expression for the initial slope of Tg,mixture versus plasticizer volume fraction wDiluent. In other words, calculate   @Tg,mixture lim wDiluent )0 @ wDiluent p and simplify your answer as much as possible, such that it contains a total of four thermophysical properties in two terms (i.e., two thermophysical properties per component). Answer

  DaDiluent@Tg,Diluent  @Tg,mixture ¼ Tg,Polymer  Tg,Diluent w Diluent )0 @ wDiluent p DaPolymer@Tg,Polymer lim

(b) Is this initial slope from part (a) positive, negative, zero or too complex to determine? Identify any conditions that must be satisfied to support your answer. Answer The initial slope is negative if the glass transition temperature of the additive is less than that of the polymer and the binary mixtures do not exhibit phase separation, as they shouldn’t in the limit of pure polymer. As pure materials are heated through their second-order phase transitions, they exhibit discontinuous increments in the coefficient of thermal expansion, so Da is greater than zero for polymer and diluent.

Problems

41

1.10. Dibutyl phthalate is a low-molecular-weight additive that functions as a plasticizer for polystyrene. The glass transition temperature of the pure polymer is 105 8C, and Tg,Diluent of the plasticizer is 291 8C. (a) Estimate the glass transition temperature of plasticized polystyrene when the volume fraction of dibutyl phthalate is 20%. Volume fractions wi and mass fractions vi are not significantly different when the densities of the two components are similar. A numerical answer is required here. Answer In the absence of thermophysical property data, like discontinuous increments in thermal expansion and specific heat at pure-component glass transition temperatures, use absolute temperature and apply the Fox equation: 1 v Diluent 1  v Diluent 0:20 0:80 þ ; ¼ þ ¼ Tg,mixture Tg,Diluent (91 þ 273) K (105 þ 273) K Tg,Polymer

Tg,mixture  38 8C

(b) Estimate the fractional free volume of the polymer –diluent blend at 105 8C when the volume fraction of plasticizer, dibutyl phthalate, is 20%. Answer The equations that are required to analyze this problem can be found in Sections 1.11 and 1.12. Temperature and compositional dependence of the binary mixture’s fractional free volume is fmixture (T ¼ 105 8C, w Diluent ¼ 0:20)  fPolymer (T) þ w Diluent G  0:044 fPolymer (T)  fPolymer (Tg,Polymer ) þ DaPolymer@Tg,Polymer (T Tg,Polymer ) ¼ fPolymer (Tg,Polymer )  0:025 G  4:8  104 K1 (Tg,Polymer Tg,Diluent )  0:094 Note that the fractional free volume of the mixture at 105 8C (i.e., 4.4%) is greater than the universally accepted 2.5% empty space between molecules that all materials possess at their glass transition temperatures, because T ¼ 105 8C is greater than Tg,mixture  38 8C. (c) Does “volume relaxation” of the polymer –diluent blend via physical aging in the glassy state at ambient temperature (i.e., 20 8C) result primarily in a decrease in (1) the space between molecules (i.e., free volume available for molecular transport), or (2) the space between the atoms of each molecule (i.e., interstitial free volume)? Answer Densification, or physical aging in the glassy state, decreases the empty space between molecules that is available for molecular transport of gases and solvents through the glassy matrix. (d) Sketch Tg,mixture versus mass fraction of dibutyl phthalate vDiluent in binary mixtures for the following five values of the “fitting parameter” h:

h ¼ DCp,Diluent =DCp,Polymer ¼ 0:25, 0:5, 1, 2, 4, 8

42

Chapter 1 Glass Transitions in Amorphous Polymers: Basic Concepts in the Gordon–Taylor equation. Put all five graphs on one set of axes and compare your predictions of Tg,mixture with those from the Fox equation.

Answer The additive rule of mixtures (i.e., linear relation from Tg,Polymer at vDiluent ¼ 0 to Tg,Diluent at vDiluent ¼ 1) is obtained for Tg,mixture via the Gordon–Taylor equation when h ¼ 1. Predictions for Tg,mixture lie (1) below the additive rule of mixtures when h . 1, and (2) above the additive rule of mixtures when 0 , h , 1, as illustrated in Figure 1.1. The Fox equation exhibits weak nonlinear compositional dependence of the mixture’s glass transition temperature, such that Tg,mixture lies below the additive rule of mixtures and matches predictions for this binary mixture via the Gordon–Taylor equation when h ¼ 2. (e) Is it possible for a graph of Tg,mixture versus

 vDiluent  Tg,mixture  Tg,Diluent vPolymer

to exhibit a positive slope if the Gordon–Taylor equation adequately describes the compositional dependence of the glass transition temperature in miscible binary mixtures? The mass fraction of component i in the mixture is denoted by vi. Answer No. These two quantities represent the dependent (i.e., Tg,mixture) and independent variables for linear least squares analysis of the Gordon–Taylor equation. Consult the discussion in

h = 0.25 h = 0.50 h = 1.0 Fox Equation h = 2.0 h = 4.0 h = 8.0

Glass Transition Temperature (°C)

100 80 60 40 20 0 –20 –40 –60 –80 –100 0.0

0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 Diluent Mass Fraction (dibutyl phthalate)

0.9

1.0

Figure 1.1 Predictions from the Gordon–Taylor and Fox equations for the compositional dependence of the glass transition temperature of polystyrene in the presence of a miscible plasticizer, like dibutyl phthalate. The empirical fitting parameter h in the Gordon– Taylor equation is defined above in terms of the ratio of discontinuous increments in specific heats (i.e., diluent relative to polymer). The Fox equation only contains pure-component glass transition temperatures and mass fraction. Both equations are essentially identical when h ¼ 2.

Problems

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Section 1.9 for binary mixtures, and algebraically rearrange     vPolymer DCp,Polymer@Tg,Polymer Tg,Polymer þ vDiluent DCp,Diluent@Tg,Diluent Tg,Diluent Tg,mixture ¼ vPolymer DCp,Polymer@Tg,Polymer þ vDiluent DCp,Diluent@Tg,Diluent to obtain the following linear form that is useful for actual data analysis: Tg,mixture ¼ Tg,Polymer 

 DCp,Diluent@Tg,Diluent vDiluent  Tg,mixture  Tg,Diluent DCp,Polymer@Tg,Polymer vPolymer

Hence, the slope of the graph under consideration is negative because it corresponds to the ratio of discontinuous increments in specific heat of the diluent relative to that of the polymer at their respective pure-component glass transition temperatures. Similar to thermal expansion and isothermal compressibility coefficients, there are no exceptions to the fact that specific heat exhibits a discontinuous increase as materials are heated through their glass transition temperatures. Hence, positive slopes are not allowed for the graph under consideration. However, if the Gordon–Taylor equation does not provide realistic predictions of Tg,mixture because diluents increase the glass transition temperature of the polymer, then positive slopes are possible but, now, the functional form of the compositional dependence of Tg should be reformulated. 1.11. Five possible plasticizers for polystyrene are under evaluation. The glass transition temperature of polystyrene is 105 8C, and {DCp}Polystyrene is 0.27 J/(g-K). The glass transition temperature of each plasticizer is approximately 295 8C. The discontinuous increment in specific heat for each additive at its pure-component glass transition temperature is summarized in Table 1.1. (a) Which diluent is most efficient in plasticizing polystyrene when the diluent mass fraction is 5%? In other words, which plasticizer induces the largest decrease in polystyrene’s Tg when all five polymer –diluent systems are compared at the same diluent mass fraction (i.e., vDiluent ¼ 0.05)? Answer Plasticizer #4 is most efficient at decreasing the glass transition temperature of polystyrene, because h ¼ DCp,Diluent/DCp,Polymer is largest for the five polymer –diluent combinations. (b) Provide support for your choice in part (a). Hint: Consider the answer to Problem 1.9a and Figure 1.1.

Table 1.1 Representative Glass Transition Temperatures and Discontinuous Increments in Specific Heat for Five Small-Molecule Additives that Function as Plasticizers for Polystyrene Diluent Plasticizer Plasticizer Plasticizer Plasticizer Plasticizer

#1 #2 #3 #4 #5

Tg (8C)

DCp,Diluent @ Tg (J/(g-K))

295 295 295 295 295

0.35 0.20 0.30 0.40 0.30

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Chapter 1 Glass Transitions in Amorphous Polymers: Basic Concepts

Answer Plasticizers with larger discontinuous increments in specific heat at their pure-component glass transition temperature, DCp,Diluent@Tg , exhibit larger fitting parameters h in the Gordon –Taylor equation for the compositional dependence of Tg,mixture in miscible blends with a given polymer. If the Gordon –Taylor equation yields adequate predictions of the glass transition temperature in these polymer–diluent blends, and all diluents have approximately the same pure-component glass transition temperature Tg,Diluent, then it should be obvious from Figure 1.1 that plasticizers with larger DCp;Diluent@Tg and larger h will be more efficient at decreasing the glass transition temperature of polystyrene when comparisons are made at the same diluent mass fraction. This conclusion is consistent with the fact that plasticizers with larger DCp;Diluent@Tg , but the same Tg,Diluent, induce steeper initial slopes of Tg,mixture versus vDiluent in binary mixtures with the same polymer. The initial slope of the Gordon– Taylor equation, when analysis is based on entropy continuity instead of linear additivity of fractional free volume, can be obtained by analogy with the solution to problem 1.9a if one replaces (i) Dai@Tg,i by DCpi@Tg,i and (ii) volume fraction wi by mass fraction vi. The desired initial slope is

  DCp,Diluent@Tg,Diluent  @Tg,mixture lim ¼ Tg,Polymer Tg,Diluent vDiluent )0 @ vDiluent p DCp,Polymer@Tg,Polymer Hence, more efficient plasticizers with steeper initial slopes of Tg,mixture versus diluent concentration in miscible binary blends with the same polymer induce greater depression of the polymer’s glass transition temperature. (c) Use only one set of axes and sketch the diluent concentration dependence of Tg for (i) polystyrene with plasticizer #1, and (ii) polystyrene with plasticizer #2. Put two curves on one set of axes. Answer These graphs of Tg,mixture versus plasticizer mass fraction are contained in Figure 1.1. The curve for plasticizer #1 lies below the additive rule of mixtures, because h . 1, and that for plasticizer #2 lies above the additive rule of mixtures, with h , 1. Plasticizer #1 is more efficient than plasticizer #2 at decreasing the glass transition temperature of polystyrene. 1.12. Experimental data are available for the compositional dependence of the glass transition temperature for single-phase homogeneous binary mixtures of polystyrene and dibutyl phthalate. This low-molecular-weight additive acts as a plasticizer and lowers the glass transition temperature of polystyrene at higher concentrations of dibutyl phthalate. The glass transition temperature of the plasticized polymer is described accurately by the Gordon –Taylor equation with fitting parameter h. This parameter represents the ratio of discontinuous increments in specific heat for the pure components, dibutyl phthalate relative to the polymer, at their respective pure-component glass transition temperatures, which are 105 8C for polystyrene and 291 8C for dibutyl phthalate. (a) Describe a data manipulation procedure based on linear least-squares analysis (LLSA) to calculate the fitting parameter h. (b) Estimate the fractional free volume of the polymer– diluent blend at 105 8C when the volume fraction of plasticizer, dibutyl phthalate, is 5%. (c) Estimate the glass transition temperature of plasticized polystyrene when the volume fraction of dibutyl phthalate is 5%. Volume fractions and weight fractions are not very different when the densities of the two components are similar.

Problems

45

(d) Polystyrene and poly(phenylene oxide) represent a classic example of two highmolecular-weight polymers that form miscible binary mixtures in the solid state with a single composition-dependent glass transition temperature. Write an expression based on the Gordon– Taylor equation to estimate Tg,mixture for miscible ternary mixtures of polystyrene (PS) and poly(phenylene oxide) (PPO) that are plasticized by dibutyl phthalate (DBP). Answer The generalized Gordon –Taylor equation, based on entropy continuity at the glass transition, can be written explicitly for miscible ternary mixtures as follows: Tg,mixture ¼       vPS DCp,PS@Tg,PS Tg,PS þ vPPO DCp,PPO@Tg,PPO Tg,PPO þ vDBP DCp,DBP@Tg,DBP Tg,DBP vPS DCp,PS@Tg,PS þ vPPO DCp,PPO@Tg,PPO þ vDBP DCp,DBP@Tg,DBP 1.13. The glass transition temperatures in Table 1.2 have been measured for four different molecular weight (i.e., Mn) fractions of a polymer that could be used as an oxygen barrier in food packaging applications. When the sample with number-average molecular weight Mn ¼ 2  104 Daltons was tested for oxygen permeability at ambient temperature, it was suggested that 50% reduction in its fractional free volume could achieve the desired oxygen barrier in the packaging material. Qualitatively describe the methodology to determine the lowest number-average molecular weight of this polymer that meets the desired specification at 25 8C. You should summarize a logical sequence of at least four steps to solve this problem. Answer Mn ¼ 1.45  105 Daltons, Tg ¼ 19 8C Step 1: Linear least squares analysis of Tg(K) versus 1/Mn via a first-order polynomial yields a slope of 2A/Da ¼ 1:45  106 K-Daltons. Step 2: The free volume parameter A is calculated from Step (1) via multiplication of 2dTg/ d(1/Mn) by the universal value for the discontinuity in thermal expansion coefficient (i.e., Da  4.8  1024 K21). Hence, A ¼ 696 Daltons.

Table 1.2 Representative Glass Transition Temperatures and Fractional Free Volume at 25 8C for Four Different Molecular Weight Fractions of an Amorphous Polymer Mn (Daltons) 2  104 5  104 2  105 5  105

Tg (8C) 244 0 þ22 þ26

Fractional free volume at 25 8C 0.060 (i.e., 6%)

46

Chapter 1 Glass Transitions in Amorphous Polymers: Basic Concepts

Step 3: When Mn ¼ 2  104 Daltons, f ¼ 0.060 ¼ f (Mn ) 1) þ A/Mn, allows one to calculate the ambient-temperature fractional free volume at very high molecular weight (i.e., f (Mn ) 1)). Step 4: A 50% reduction in fractional free volume at 25 8C, based on the sample with Mn ¼ 2  104 Daltons yields f ¼ 0.030 ¼ f(Mn ) 1) þ A/Mn, which allows one to estimate the minimum molecular weight of this polymer that could be useful for packaging applications. 1.14. The following molecular weight dependence of fractional free volume for an amorphous polymer is postulated to agree with the fact that there is more empty space between polymer chains at lower molecular weight: f (MW)  f (MW ) 1) þ

B MW a

where the exponent a is positive, and B is a constant on the order of (500 Daltons)a. (a) Use this model for the molecular weight dependence of fractional free volume and predict the molecular weight dependence of Tg. Answer Express fractional free volume as a linear function of temperature, expanded about the molecular-weight-dependent glass transition temperature:       f Tg (MW) þ Da T  Tg (MW)  f Tg (MW ) 1)   B þ Da T  Tg (MW ) 1) þ MW a Neglect any dependence of the discontinuity in thermal expansion coefficients on molecular weight and rearrange the previous equation, subjected to the approximation that the glass transition is an iso-free-volume state. Hence, the first terms on each side of the previous equation cancel. It might be necessary to measure specific volume versus temperature above and below Tg in a dilatometer for samples of the same polymer that have different number-average molecular weights. Then, the discontinuity in (@ ln vspecific/@T )p at Tg can be calculated for each sample to verify or disprove the approximation that Da is molecular-weight independent. If all of the approximations mentioned above are valid, then Tg (MW)  Tg (MW ) 1) 

B (Da)MW a

which should be compared with similar predictions in Sections 1.14 and 1.15. (b) Since Tg(MW ) 1) is tabulated in handbooks and textbooks, rearrange your Tg versus MW expression from part (a) and explain how linear least squares analysis can be implemented to calculate the exponent a from experimental data similar to those in Table 1.2. Answer Rearrange the previous equation for the molecular weight dependence of the glass transition temperature and take the logarithm of both sides. Be sure that the argument of each logarithm

Problems

47

is greater than zero.

log[Tg (MW ) 1)  Tg (MW)]  log (i) (ii) (iii) (iv)

 B  a log(MW) Da

Polynomial model is y(x) ¼ a0 þ a1x. Independent variable x is log(MW ). Dependent variable y is log[Tg(MW)1) 2 Tg(MW)]. The first-order coefficient, or slope, of the polynomial is a1 ¼2a.

(c) Sketch Tg versus MW when a ¼ 1, 2, and 3. Put all three curves on one set of axes. Answer When the exponent a is larger, molecular weight has a smaller effect on decreasing the glass transition temperature of the polymer. Furthermore, one achieves the molecular-weightinsensitive asymptotic limiting value of Tg at lower molecular weights when the exponent a is larger. 1.15. Consider the following experimental description for completely amorphous polystyrene and then sketch (i) the dilatometer trace and (ii) the DSC trace for this material during the heating and subsequent cooling cycle. Be as quantitative as possible on the temperature axis. “High molecular weight atactic polystyrene is subjected to 2500 atm pressure in the molten state at 300 8C. The high-pressure material is cooled rapidly to ambient temperature at 2500 atmospheres. Some of the material is placed in a dilatometer, and another piece of the same sample is tested in a differential scanning calorimeter. In both experiments, the polystyrene sample is heated at a rate of 10 8C/min to 300 8C and immediately cooled from 300 8C to ambient at the same rate.” (a) Sketch the temperature dependence of the data that are generated from the dilatometer. (b) Sketch the temperature dependence of the data that are generated from the differential scanning calorimeter.

Chapter

2

Diffusion in Amorphous Polymers Near the Glass Transition Temperature Shapeless whispers, shimmering membranes, mumble about love. —Michael Berardi

A

dvanced concepts in fractional free volume are discussed in connection with a primitive lattice model to evaluate hard-sphere diffusion coefficients via entropy maximization with constraints. This Lagrange multiplier diffusion model is extended to include solvent size in polymer solutions, as well as the effect of temperature on diffusivities above and below Tg via the discontinuity in thermal expansion coefficients. The unsteady state diffusion equation is analyzed to (i) measure half-times, (ii) calculate lag-times, and (iii) predict membrane diffusion coefficients.

2.1

DIFFUSION ON A LATTICE

Small molecules successfully jump to adjacent lattice sites when polymer chains undergo (i) thermally induced molecular motion and (ii) conformational rearrangements such that empty space between large molecules becomes available. This molecular picture of diffusion is appropriate for solvents and solubilized gases in rigid and mobile nanoporous matrices. Temperature-dependent diffusional rate processes described by Fick’s second law exhibit activation energies that increase abruptly at the glass transition.

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

49

50

Chapter 2 Diffusion in Amorphous Polymers

2.2 OVERVIEW OF THE RELATION BETWEEN FRACTIONAL FREE VOLUME AND DIFFUSIVE MOTION OF LIQUIDS AND GASES THROUGH POLYMERIC MEMBRANES Free volume concepts from the previous chapter are employed to analyze the effects of structural characteristics of polymer – penetrant combinations on the diffusion process. Experimental methods to measure diffusion include techniques known as the half-time and the lag time, as described in this chapter. Predictions are based on solutions of the unsteady state diffusion equation, which is better known as Fick’s second law. A qualitative lattice model description of this process focuses on penetrant molecules that are surrounded by chain segments which occupy adjacent sites in three dimensions. If the penetrant molecules of interest acquire sufficient thermal energy and adjacent lattice sites are vacant, then these small molecules can jump to an empty site via stochastic Brownian motion. Diffusive transport occurs when neighboring chain segments undergo conformational rearrangements to occupy sites vacated by the molecules of interest before these penetrants return to their original positions. If diffusing molecules make w jumps per time, on average, then the molecular diffusion coefficient D in Fick’s law is related to the jump frequency w and the jump distance d by the following equation: D  16wd2 Question: Is the factor of 6 in the previous equation due to the fact that diffusing molecules on a three-dimensional cubic lattice have conformational freedom of jumping to n adjacent cells, and n ¼ 6 for cubic lattices? For example, one must replace the factor of 61 by (1) 14 to describe diffusion on two-dimensional surfaces, or (2) 12 for one-dimensional diffusion on a line. Hint: Consult Section 13.5 for mean-square displacements based on random-walk statistics in one, two, and three dimensions, and the analogy with Fick’s second law for transient diffusion in response to a Dirac delta-function tracer input. Since jump distances d are quite similar for diffusing species, the primary factor affecting diffusion coefficients for various polymer – penetrant combinations is the jump frequency, which depends directly on penetrant size and the local fractional free volume of the lattice. Creation of new void space in polymers requires the localization of considerable thermal energy that is of the same order of magnitude as, or greater than, the polymer’s cohesive energy density, where the latter is overwhelmingly larger than the energy barriers that penetrants must overcome to move from one equilibrium position to another. It should be emphasized that molecules vibrate at a frequency of 1012 hertz about their equilibrium positions. Hence, it is reasonable to assume that there is a sufficient number of vibrations in the proper direction for penetrants to move to a new position on the lattice when the redistribution of existing void space produces vacancies that are large enough to accommodate small-molecule jumps between adjacent cells. Consequently, the probability that diffusional jumps occur is proportional to the probability that void space of sufficient volume v on the lattice exists adjacent to the penetrant molecules of interest. It is instructive to introduce the concept of a distribution of void space C(v) and represent the void volume

2.3 Free Volume Theory of Cohen & Turnbull for Diffusion in Liquids & Glasses

51

fraction associated with holes of sufficient volume v required for penetrant molecules to jump to adjacent lattice sites by ð1 C(v) dv  ðv1 C(v) dv 0

If penetrant diffusion through polymeric membranes were described as an activated rate process, then it seems reasonable to (i) identify the energy E(v) required to create holes in the matrix with void volume v, and (ii) postulate that C(v) should follow a Boltzmann distribution:   E(v) C(v)  exp RT where R is the universal gas constant and T represents absolute temperature. If void volume must be greater than v for penetrants to jump to an adjacent lattice site, then the probability that diffusional jumps occur and the jump frequency w are proportional to ð1 exp{E(v)=RT} dv  ðv1 exp{E(v)=RT} dv 0

It should be emphasized that structural modifications in the lattice, which increase the difficulty of creating voids with sufficient volume v , will have a negative impact on the jump frequency w, causing diffusivities to decrease. In other words, equal amounts of thermal energy will create smaller holes in more cohesive lattices that restrict diffusive motion of the penetrants. This picture of diffusion through polymeric membranes neglects the concept of free volume cooperativity. For example, the void space in an adjacent lattice cell is not large enough to accommodate a penetrant, so diffusive motion should be hindered based on the model described above because the energy required to create voids with threshold volume v is prohibitive. However, cooperative reorganization of several adjacent segments of the polymer chain occurs in harmony to produce holes with volume greater than v , allowing penetrant molecules to jump to adjacent lattice sites. This redistribution of empty space between chain segments that occurs with little or no energy requirement to assist the diffusion process is discussed quantitatively in the next section.

2.3 FREE VOLUME THEORY OF COHEN AND TURNBULL FOR DIFFUSION IN LIQUIDS AND GLASSES Molecular diffusion in liquids and glasses depends strongly on the distribution of useful free volume that can be manipulated and rearranged with little or no energy input. If D(v) represents the diffusion coefficient of solubilized liquids or gases in a

52

Chapter 2 Diffusion in Amorphous Polymers

material with specific free volume v, and C(v) dv accounts for the normalized probability of finding specific free volume between v and v þ dv in this material, then the average diffusivity kDl is hD i ¼

ð1 D(v)C(v) dv v

where v is the critical specific free volume required for molecular “jumps” to occur. There is no diffusion of solubilized species in liquids or glasses if all of the available free volume exists as holes or vacancies smaller than v and cooperative reorganization of the matrix is severely prohibited. When micro-Brownian motion and thermal fluctuations create a hole greater than v adjacent to a molecule of the diffusing species, that molecule jumps to a new pseudo-equilibrium position in the lattice. The theory of Cohen and Turnbull [1959, 1961] does not describe molecular diffusion as an activated rate process. Instead, diffusion is considered to occur as a result of the redistribution of free volume, which occurs with essentially no energy input to generate holes that are greater than v . Let Ni be the number of lattice cells in a fluid with free volume given by vi per cell, where vi is much smaller than the size of the cell itself. The total number of cells that represent a model for the liquid or glass is X Ni NTotal ¼ i

and the total amount of free volume, dictated by cooling rate and packing of the molecules, is X Vfree,Total ¼ g Ni vi i

where 0.5 , g , 1 is an overlap factor that accounts for free volume that might be shared by adjacent cells in the lattice. One seeks to maximize the number V of distinguishable ways that free volume vi per cell can be redistributed among Ni cells, subject to the previous two constraints. Alternatively, one seeks to maximize the number V of distinguishable ways that groups of Ni cells, each containing free volume vi per cell, can be redistributed among the total number of cells that constitute the lattice. If the energy of each permutation were an important consideration, then the calculation described below would be much more complex. However, this useful free volume under consideration can be manipulated and redistributed rather easily with little or no energy input such that each permutation is essentially “equally likely.” In other words, the system seeks a state of maximum entropy via the following “counting problem” according to Boltzmann’s equation, S ¼ k ln V, where k is Boltzmann’s constant. The multiplicity V of permutations of NTotal cells that characterize the entire lattice such that there are always Ni cells with free volume vi per cell is V¼

(NTotal )! Pi N i !

2.3 Free Volume Theory of Cohen & Turnbull for Diffusion in Liquids & Glasses

53

Division by each Ni! is required so that identical permutations are not counted multiple times. The previous expression for V invariably yields an integer for this “counting problem,” but the results below for Ni as a function of vi are not restricted to whole numbers, particularly when the discrete solution Ni (vi ) is extended to the continuous limit, yielding N(v). The strategy focuses on maximizing ln V after using Stirling’s approximation for the factorial of a large argument (i.e., n 1):   pffiffiffiffiffiffiffiffiffi 1 1 þ   n! ¼ n 2p n exp(n) 1 þ 12n 288n2 n

ln n!  12 ln(2p) þ (n þ 12) ln n  n The objective function that must be maximized is X ln(Ni !) ln V ¼ ln(NTotal !)  i

¼ 12 ln(2p) þ (NTotal þ 12) ln NTotal  NTotal  ¼

1 2 ln(2p)

þ (NTotal þ

1 2) ln NTotal



X

X

1 2 ln(2p)

þ (Ni þ 12) ln Ni  Ni



i 1 2 ln(2p)

þ (Ni þ 12) ln Ni



i

The method of Lagrange multipliers accounts for the two constraints mentioned above and yields the following modification of the objective function:    X  X Ni þ r Vfree,Total  g Ni vi G ¼ ln V þ l NTotal  i

i

where l . 0 and r . 0 represent generic Lagrange multipliers. The extremum conditions are 

   @G 1 ¼ 1þ þ ln Nj  l  rgvj ¼ 0 @Nj NTotal ,l,r,Nk[k=j] 2Nj   X @G ¼ NTotal  Ni ¼ 0 @ l NTotal ,r,Ni i   X @G ¼ Vfree,Total  g Ni vi ¼ 0 @ r NTotal ,l,Ni i

In the first extremum condition, it is not rigorously possible to vary a certain group of lattice cells Nj with free volume v j per cell, while all other groups of cells Nk and the total number of cells NTotal remain constant, because the first constraint is not satisfied. However, for very large numbers of lattice cells, NTotal is approximately constant when Nj varies slightly. Furthermore, since the number of cells in the lattice with free volume vi per cell is, in general, quite large also, the first extremum condition given above

54

Chapter 2 Diffusion in Amorphous Polymers

simplifies to 

@G @Nj



  1 ¼ 1þ þ ln Nj  l  rg vj 2Nj

NTotal ,l,r,Nk[k=j]

  ln Nj  l  rg vj ¼ 0 which provides an estimate for each discrete Nj in terms of vj . In the continuous limit, the number of lattice cells with individual cell free volume v is given by the following function: N(v) ¼ expfl  rg vg Lagrange multipliers l and r are determined from continuous representations of the two constraints mentioned above: X

1 ð

Ni )

i

g

X i

1 ð

N(v) dv ¼ exp(l) exp(rg v) dv ¼

0

1 exp(l) ¼ NTotal rg

0

1 ð

1 ð

0

0

Ni vi ) g vN(v) dv ¼ g exp(l) v exp(rg v) dv

¼

1 exp(l) ¼ Vfree,Total r2 g

where the second constraint is evaluated via integration by parts. The solution is

r¼ exp(l) ¼ N(v) ¼

NTotal 1 ¼ Vfree,Total vfree,average

gNTotal vfree,average gNTotal vfree,average

 exp 

gv



vfree,average

The average free volume per lattice cell (i.e., Vfree,Total/NTotal ) is denoted by vfree;average . The normalized probability of finding specific free volume between v and v þ dv per lattice cell in the continuous limit is   N(v) g gv dv C(v) dv ¼ dv ¼ exp  vfree,average NTotal vfree,average As one should expect, this result for C(v) is consistent with the following requirement: ð1 0

C(v) dv ¼ 1

2.4 Free Volume Theory of Vrentas and Duda for Solvent Diffusion in Polymers

55

Due to the exponential dependence of C on v, the weak dependence of diffusivity D on v (i.e., D increases at larger v), and the fact that the critical specific free volume v required for molecular “jumps” to occur is on the order of 10vfree,average, one calculates the average diffusivity of solubilized species in liquids and glasses by evaluating D(v) at v ¼ v: hD i ¼

1 ð v

 D(v)C(v) dv  D(v ) C(v) dv ¼ D(v ) exp  

1 ð

v



g v



vfree,average

Diffusion coefficients are larger at higher temperature primarily because the average specific free volume per lattice cell vfree,average undergoes thermal expansion faster than the occupied volume expands as the temperature increases. Under isothermal conditions, smaller molecules exhibit larger diffusivities in the same liquid or glass because a smaller critical free volume per lattice cell v is required for the diffusing species to jump to a new lattice site. Hence, v is characteristic of the diffusing molecules, and vfree,average is a property of the matrix.

2.4 FREE VOLUME THEORY OF VRENTAS AND DUDA FOR SOLVENT DIFFUSION IN POLYMERS ABOVE THE GLASS TRANSITION TEMPERATURE For binary mixtures of polymer and solvent, energetic interactions between the two components do not affect the random distribution of free volume in the system. When trace amounts of solvent are present, the mutual polymer –solvent diffusion coefficient, which is dominated by free volume characteristics of the polymer and the molecular size of the solvent, can be expressed in terms of results from the previous section:   g vPolymer hDi  D(vPolymer ) exp  vfree,Polymer where vPolymer is the critical specific free volume of the polymer required for a solvent molecule to jump to an adjacent pseudo-equilibrium position in the lattice, and vfree,Polymer is the average specific free volume of the polymeric matrix which exhibits the following temperature dependence:  vfree,Polymer (T)  vactual,Polymer (T g,Polymer ) f Polymer (T g,Polymer )  þ DaPolymer (T Tg,Polymer ) þ    The coefficient of the first-order term in the Taylor series expansion of the fractional free volume of the polymer was evaluated in Section 1.11: DaPolymer ¼ aPolymer,Liquid  f1  fPolymer (Tg,Polymer )gaPolymer,Glass

56

Chapter 2 Diffusion in Amorphous Polymers

where aPolymer represents the coefficient of thermal expansion above or below Tg,Polymer, and fPolymer(Tg,Polymer) is the fractional free volume of the undiluted polymer at its glass transition temperature (i.e., 2.5%). If the weak temperature dependence of the average specific free volume of the polymer is considered below the glass transition temperature, then DaPolymer requires aPolymer,Glass instead of aPolymer,Liquid for the first term on the right side of the previous equation. This modification is discussed in the next section, where DaPolymer ¼ fPolymer(Tg,Polymer)aPolymer,Glass is employed to analyze activation energies for viscoelastic diffusion slightly below the glass transition temperature. After algebraic manipulation of the defining equation for kDl, it is possible to employ linear least squares analysis and determine some of the parameters in the free volume description of molecular diffusion for trace amounts of solvent in a polymer matrix above its glass transition temperature. For example, when Treference . Tg,Polymer,   hD(T)i lnhD(T)ilnhD(Treference )i ¼ ln hD(Treference )i

1 1  ¼ g vPolymer vfree,Polymer (T) vfree,Polymer (Treference ) Substitution for vfree,Polymer at temperatures T and Treference yields, upon algebraic manipulation and rearrangement, the following linear model for the temperature dependence of solvent diffusion in polymers: T  Treference G T  Tg,Polymer  ¼ þ hD(T)i H H ln hD(Treference )i with G¼

fPolymer (Tg,Polymer ) DaPolymer

H¼

g vPolymer vactual,Polymer (Tg,Polymer )fPolymer (Treference )

Hence, temperature-dependent diffusion data for one organic solvent in one polymer above its glass transition can be analyzed via linear least squares analysis. If temperatures T and Treference are greater than Tg,Polymer such that the average specific free volume of the polymeric matrix expands according to DaPolymer,Liquid rather than DaPolymer,Glass, then the preferred procedure is summarized below: (i) Polynomial model: y ¼ bx þ c. (ii) Independent variable: x ¼ T 2 Tg,Polymer. (iii) Dependent variable: y ¼

T  Treference  : hD(T)i ln hD(Treference )i

2.4 Free Volume Theory of Vrentas and Duda for Solvent Diffusion in Polymers

57

(iv) The first-order coefficient in the model (i.e., slope) is b ¼ 1/H, which is dimensionless and depends on the particular polymer – solvent combination. (v) The zeroth-order coefficient in the model (i.e., intercept) is c ¼ G/H, which has dimensions of absolute temperature and also depends on the polymer – solvent combination. However, the parameter G depends only on the free volume characteristics of the polymer. (vi) If one analyzes temperature-dependent diffusion data for a different organic solvent with larger molar volume in the same polymer, then the slope of y versus x in the linear model described above decreases. This is reasonable because the critical specific free volume of the polymer vPolymer required for a solvent molecule to jump to a new pseudo-equilibrium position in the lattice is larger. In other words, H increases and b decreases. It should be obvious from the defining equation for kDl in this section, together with the temperature dependence of fractional free volume, that lnkDl versus reciprocal absolute temperature for one polymer –solvent combination will not yield a straight line from which the activation energy for diffusive transport can be obtained. However, linear behavior with slope 2J is obtained when lnkDl is correlated with 1={G þ T  Tg,Polymer}, where G is the ratio of intercept to slope from the previous linear least squares analysis with dimensions of absolute temperature. This claim is justified below:   lnhDi ¼ ln D(vPolymer ) 

g vPolymer vfree,Polymer (T)

   ln D(vPolymer ) 

g vPolymer vactual,Polymer (Tg,Polymer )fPolymer (T)

   ln D(vPolymer ) 

J G þ T  Tg,Polymer

where the slope J¼

g vPolymer vactual,Polymer (Tg,Polymer )DaPolymer

depends on the particular combination of polymer and solvent because J contains the critical specific free volume of the polymer required for a solvent molecule to jump to an adjacent pseudo-equilibrium position in the lattice. In fact, the slope of lnkDl versus 1={G þ T  Tg,Polymer} is steeper for solubilized mobile components with larger molar volume in a given polymer because the critical specific volume of the polymer required for the diffusing species to jump to a new lattice site must be larger. This trend is supported by the fact that slope J scales linearly with vPolymer . Actual data from the research contributions of Vrentas and Duda [1976, 1977, 1978] for diffusion of 17 different species, from hydrogen to ethylbenzene, in atactic polystyrene reveal that J is a linear function of the molar volume at 0 K for the

58

Chapter 2 Diffusion in Amorphous Polymers

diffusing species. Apparent Arrhenius-like activation energies for diffusion are calculated as follows:   d lnhDi JRT 2 ¼ Eactivation (T . Tg,Polymer ) ¼ RT 2 dT (G þ T  Tg,Polymer )2 These activation energies in the highly viscous liquid state invariably decrease at higher temperature. This claim is justified by the following calculations: dEactivation 2JRT 2JRT 2 ¼  dT (G þ T  Tg,Polymer )2 (G þ T  Tg,Polymer )3   2JRT T ,0 ¼ 1 T  (Tg,Polymer  G) (G þ T  Tg,Polymer )2 G¼

fPolymer (Tg,Polymer ) 0:025   50 K DaPolymer 5  104 K1

In other words, the parameter G in the theory of Vrentas and Duda [1976, 1977, 1978] is approximately 50 K via typical estimates of fractional free volume at the glass transition temperature (i.e., 0.025) and the discontinuity in thermal expansion coefficients at Tg,Polymer (i.e., 5  1024 K21), as calculated above. For example, diffusion of methane through polystyrene is described by G ¼ 45.3 K. Since all nongaseous materials have glass transition temperatures above 50 K, the temperature derivative of activation energies in the viscous liquid state is negative. Hence, activation energies for viscous diffusion decrease at higher temperatures, which implies that the magnitude of the slope of lnkDl versus reciprocal absolute temperature decreases as 1/T decreases. More importantly, the diffusion process for larger solvents in a given polymer is characterized by larger activation energies that decrease faster at higher temperature, due to the fact that Eactivation is directly proportional to J, and J scales linearly with the critical specific free volume of the polymer vPolymer required for solvent molecules to jump to an adjacent pseudo-equilibrium position in the lattice.

2.5 INFLUENCE OF THE GLASS TRANSITION ON DIFFUSION IN AMORPHOUS POLYMERS Above the glass transition, material response times are much shorter than characteristic time constants for viscous diffusion, and mass transfer occurs through an equilibrium liquid structure. Below the glass transition, material response times increase significantly, and elastic diffusion occurs through a nonequilibrium rigid liquid. There is negligible structural variation in glassy materials on the time scale of the diffusion process. Viscoelastic diffusion occurs in the vicinity of Tg,Polymer, when the time constants for diffusion and molecular rearrangements of the chains are similar, which produces

2.5 Influence of the Glass Transition on Diffusion in Amorphous Polymers

59

anomalous effects near the glass transition temperature. As mentioned in the previous section, activation energies for diffusion decrease at higher temperature, either above or below Tg,Polymer. However, Eactivation experiences a discontinuous increase at the glass transition temperature that is proportional to the discontinuity in thermal expansion coefficients for the polymer DaPolymer at Tg,Polymer. Justification for this statement is provided by the analysis below. Begin by evaluating the activation energy for diffusion in the highly viscous liquid state, slightly above the glass transition temperature where, for all practical purposes, T  Tg,Polymer. Results from the previous section yield the following expressions for viscoelastic diffusion:  J Liquid þ RT 2 ¼ 2 Eactivation T ¼ Tg,Polymer GLiquid g,Polymer JLiquid ¼

g vPolymer vactual,Polymer (Tg,Polymer )DaPolymer,Liquid

fPolymer (Tg,Polymer ) DaPolymer,Liquid   ¼ aPolymer,Liquid  1  fPolymer (Tg,Polymer ) aPolymer,Glass

GLiquid ¼ DaPolymer,Liquid

Now, consider the temperature dependence of solvent diffusion through the same polymer matrix at temperatures that are slightly below Tg,Polymer. The process is also classified as viscoelastic diffusion, but the polymer’s fractional free volume expands and contracts according to DaPolymer,Glass instead of DaPolymer,Liquid. This difference between the temperature coefficients of fractional free volume above and below the glass transition temperature significantly affects activation energies for diffusion. For example,

 JGlass  2 ¼ 2 RTg,Polymer Eactivation T ¼ Tg,Polymer GGlass JGlass ¼

g vPolymer vactual,Polymer (Tg,Polymer )DaPolymer,Glass

fPolymer (Tg,Polymer ) DaPolymer,Glass   ¼ aPolymer,Glass  1fPolymer (Tg,Polymer ) aPolymer,Glass

GGlass ¼ DaPolymer,Glass

¼ fPolymer (Tg,Polymer )aPolymer,Glass Either the ratio of or difference between these diffusional activation energies above and below the glass transition temperature reveals that Eactivation experiences a discontinuous increase at Tg. The ratio of activation energies does not depend on any

60

Chapter 2 Diffusion in Amorphous Polymers

molecular characteristics of the solvent, as illustrated below:  þ    Eactivation T ¼ Tg,Polymer JLiquid GGlass 2 DaPolymer,Liquid ¼

 ¼  JGlass GLiquid DaPolymer,Glass Eactivation T ¼ Tg,Polymer   aPolymer,Liquid  1  fPolymer (Tg,Polymer ) aPolymer,Glass ¼ fPolymer (Tg,Polymer )aPolymer,Glass   aPolymer,Liquid 1 ¼1þ 1 .1 fPolymer (Tg,Polymer ) aPolymer,Glass The difference between these activation energies above and below Tg depends on the (i) molecular size of the solvent, (ii) glass transition temperature of the polymer, (iii) free volume characteristics of the polymer, and (iv) difference between the polymer’s thermal expansion coefficient in the liquid and glassy states. Detailed calculations are provided below:   þ   Eactivation T ¼ Tg,Polymer Eactivation T ¼ Tg,Polymer ( ) JLiquid JGlass 2  ¼ RTg,Polymer G2Liquid G2Glass ¼ ¼

2 g vPolymer RTg,Polymer

vactual,Polymer (Tg,Polymer )[ fPolymer (Tg,Polymer )] 2 g vPolymer RTg,Polymer

vactual,Polymer (Tg,Polymer )[ fPolymer (Tg,Polymer )]



 DaPolymer,Liquid DaPolymer,Glass



 aPolymer,Liquid  aPolymer,Glass . 0

2

2

Hence, the theory of Vrentas and Duda [1976, 1977, 1978] for diffusion of trace amounts of solvent in polymers predicts that the difference between diffusional activation energies slightly above and slightly below the glass transition temperature of the polymer scales linearly with the (i) discontinuity in the polymer’s thermal expansion coefficient at Tg,Polymer and (ii) critical specific free volume of the polymer required for a solvent molecule to jump to an adjacent pseudo-equilibrium position in the lattice. This latter quantity vPolymer is directly related to the molecular dimensions of the solvent. For a particular polymer whose glass transition temperature is approximately 350 K, the discontinuity in activation energies for diffusive transport at Tg is on the order of 2 105 (vPolymer ), where the critical specific free volume of the polymer required for solvent molecules to jump to a new pseudo-equilibrium position in the lattice must be specified in cm3/g and the activation energy difference is given in cal/mol.

2.6 Analysis of Half-Times and Lag Times

61

2.6 ANALYSIS OF HALF-TIMES AND LAG TIMES VIA THE UNSTEADY STATE DIFFUSION EQUATION Equations and methodology are discussed in this section to predict binary molecular diffusion coefficients of penetrant gases in polymeric membranes. The pressure of gas A is maintained constant at pA above the membrane such that interfacial equilibrium is achieved on the upper surface of the polymer (i.e., x ¼ 0). Application of Henry’s law at x ¼ 0 provides an expression for the molar density of solubilized gas as a product of the gas – polymer solubility constant SAB and pA. The downstream side of the membrane is exposed to ultrahigh vacuum (i.e., 1028 torr). Molecules of gas A that traverse the membrane are analyzed via mass spectrometry, such that selected fragments of the penetrant achieve dynamically stable trajectories within this analytical device. It is possible to detect the increase in current due to the appropriate mass-to-charge ratio for dominant fragments of the penetrant. The transient current response is proportional to the flux of gas A across the lower surface of the membrane (i.e., at x ¼ L), driven by the established concentration gradient between x ¼ 0 and x ¼ L. One calculates the diffusion half-time from measurements of transient and steady state flux at x ¼ L, as illustrated in Figure 2.1. Typical binary molecular diffusion coefficients D for diatomic gases permeating through glassy or rubbery polymers can be found in the following range: 1026 – 1025 cm2/s, where D increases as both penetrant size and the glass transition temperature decrease. When the film thickness is 0.03 cm, typical diffusion half-times t1/2 are

JAx (t, x = L)

Typical half-time plot

JS 1 2

O

JS

t½ Time

Figure 2.1 Schematic illustration of the transient flux of gas A across the lower surface of thin polymeric membranes. The diffusion half-time t1/2 is identified when the magnitude of the flux J(t1/2) achieves 50% of its steady state value (i.e., JS). Half-times typically range from tens of seconds to a few minutes, and they can be controlled by membrane thickness.

62

Chapter 2 Diffusion in Amorphous Polymers

15 s for permeation of O2 through rubbery membranes (i.e., D  1025 cm2/s) and 125 s for H2 permeation through glassy membranes (i.e., D  1026 cm2/s). Since half-times scale as the square of film thickness, thicker films will increase t1/2 and reduce experimental error when diffusion is relatively fast.

2.6.1 Solution of the Diffusion Equation: Analysis of Half-Times Fick’s second law of diffusion (i.e., the diffusion equation) describes the transient and spatial dependence of the molar density of gas A throughout the membrane. For one-directional flux in the x-direction across the thinnest dimension (e.g., L  0.02– 0.04 cm) of the polymeric film and constant diffusivity D, one must solve the following equation for CA(t, x): @CA @ 2 CA ¼D @t @x2 Initial degassing of the membrane and exposure of its downstream side at x ¼ L to ultrahigh vacuum, together with maintaining constant pressure of gas A above the upper surface of the membrane, produce three required boundary conditions: CA ¼ SAB pA ¼ constant CA ¼ 0 CA ¼ 0

x ¼ 0; t . 0 x ¼ L; all t t ¼ 0; x . 0

Introduction of dimensionless variables for molar density, spatial position in the thinnest dimension of the sample, and time yields Molar density of gas A: FA ¼

CA SAB pA

Spatial coordinate in thinnest dimension: h ¼ Dimensionless diffusion time: t ¼

x L

tD L2

allows one to re-express the diffusion equation and its boundary conditions in dimensionless form: @FA @ 2 FA ¼ @t @ h2 FA ¼ 1; h ¼ 0; t . 0 FA ¼ 0; h ¼ 1; all t FA ¼ 0; t ¼ 0; h . 0 It is possible to construct a well-posed Sturm – Liouville problem for QA(t, h), based on the diffusion equation and its boundary conditions, by forcing both boundary

2.6 Analysis of Half-Times and Lag Times

63

conditions at h ¼ 0 and h ¼ 1 to be homogeneous such that QA ¼ 0. This is accomplished as follows: Q A ( t, h ) ¼ F A ( t, h ) þ h  1 This transient membrane diffusion problem is reformulated in terms of QA(t, h) with homogeneous spatial boundary conditions as follows: @QA @ 2 QA ¼ @t @ h2 QA ¼ 0; h ¼ 0; t . 0 QA ¼ 0; h ¼ 1; all t QA ¼ h  1; t ¼ 0; h . 0 Postulate a separation of variables solution to the previous partial differential equation (PDE), such that QA(t, h) ¼ T(t)N(h), and substitute this expression into the PDE. After division by T(t)N(h), one obtains two ordinary differential equations for T(t) and N(h) in terms of a negative separation constant (i.e., 2l2) that is consistent with transient decay of T(t): 1 dT 1 d2 N ¼ ¼ l2 T dt N d h2 T(t)  exp(l2 t) N(h)  A cos(lh) þ B sin(lh) Boundary conditions at h ¼ 0 and h ¼ 1 are satisfied when A ¼ 0 and l ¼ kp for all integers k. The eigenvalues, eigenfunctions, and general solution for this problem are

lk ¼ kp ; k ¼ 0, 1, 2, 3, . . . QA,k (t, h) ¼ Tk (t)Nk (h) ¼ Bk sin(kph) exp(k 2 p 2 t) Q A ( t, h ) ¼

1 X k¼0

QA,k (t, h) ¼

1 X

Bk sin(kph) exp(k 2 p 2 t)

k¼1

The general solution of the diffusion equation is satisfied initially, at t ¼ 0, when Bk is given by the Fourier sine coefficients of h 2 1. Since the sin(kph) is orthogonal, but not normalized, for different values of k, one (i) evaluates the general solution at t ¼ 0, (ii) multiplies the expression by sin(mph), and (iii) integrates with respect to h from h ¼ 0 to h ¼ 1. These steps are illustrated below:

QA (t ¼ 0, h) ¼ h  1 ¼

1 X k¼1

Bk sin(k ph)

64 hð ¼1

Chapter 2 Diffusion in Amorphous Polymers

(h  1) sin(mph)d h ¼

1 X

hð ¼1

Bk

k¼1

h¼0

Bm ¼ 2

hð ¼1

sin(kph) sin(mph)dh ¼

h¼0

(h  1) sin(mph)d h ¼

1 1X 1 Bk dkm ¼ Bm 2 k¼1 2

2 mp

h¼0

The complete solution for the dimensionless molar density profile of gas A throughout the membrane is 1 CA (t, x) 2X 1 sin(k ph) exp(k 2 p 2 t) ¼ 1  h þ Q A ( t, h ) ¼ 1  h  FA (t, h) ¼ p k¼1 k SAB pA Analysis of the diffusion half-time proceeds by evaluating the flux of species A across the lower surface of the membrane at x ¼ L (i.e., h ¼ 1) during the transient response and after steady state conditions are achieved. Fick’s first law of diffusion yields an expression for this flux, where the product of D and SAB is known as the permeability of gas A in polymer B:     @CA DSAB pA @FA ¼ JAx (t, h ¼ 1) ¼ D @x x¼L L @ h h ¼1 " ( )# 1 DSAB pA @ 2X 1 2 2 sin(k ph) exp(k p t) ¼ 1h L p k¼1 k @h h ¼1 ( ) 1 X DSAB pA 1þ2 ¼ (1)k exp(k 2 p 2 t) L k¼1 ( ) 1 X k 2 2 ¼ JA x (t ) 1, h ¼ 1) 1 þ 2 (1) exp(k p t) k¼1

Now, one compares measurements and predictions of the dimensionless half-time t1/2 when JA x(t1/2, h ¼ 1) achieves 50% of the steady state flux of gas A across the lower surface of the membrane. Predictions from the previous equation yield 1 1 X þ (1)k exp(k 2 p 2 t1=2 ) ¼ 0 4 k¼1 If z ¼ expf2p 2t1/2g, then successive approximations to the root of this nonlinear equation are given by z þ 14 ¼ 0;

z ¼ 0:25000

z4  z þ 14 ¼ 0;

z ¼ 0:25417

k ¼ 1: k ¼ 1, 2: k ¼ 1, 2, 3: k ¼ 1, 2, 3, 4:

9

4

1 4

z þ z  z þ ¼ 0;

z ¼ 0:25417

z16  z9 þ z4  z þ 14 ¼ 0;

z ¼ 0:25417

2.6 Analysis of Half-Times and Lag Times

65

The asymptotic solution for the dimensionless half-time is expf2p 2t1/2g ¼ 0.25417, which yields the following relation between the binary molecular diffusion coefficient D, membrane thickness L, and the dimensional half-time t1/2: D¼

L2 7:2t1=2

2.6.2 Membrane Diffusion Coefficients via the Analysis of Lag Times Let’s begin with the solution to the previous problem for the transient flux of gas A across the lower surface of the membrane relative to the steady state flux: 1 X JA x (t, h ¼ 1) ¼1þ2 (1) k exp(k 2 p 2 t) JA x (t ) 1, h ¼ 1) k¼1

The previous equation exhibits zero initial slope with respect to t and asymptotically approaches unity at long dimensionless diffusion times t. The behavior of JA x(t, h ¼ 1) is illustrated qualitatively in Figure 2.1 and numerically in Figure 2.2. The dimensionless diffusion lag time tLag is defined as the area between the graph of the previous equation with respect to t and a horizontal line that is one unit above

Dimensionless Lag-Time = 0.167 via Integration 1.0

Transient Flux of Penetrant

0.9

Lag-Time Area

0.8 0.7 0.6 0.5

Dimensionless Half-Time = 0.14 at 50% Flux

0.4 0.3 0.2 0.1 0.0 0.0

0.1

0.2 0.3 0.4 Dimensionless Diffusion Time (t )

0.5

0.6

Figure 2.2 Quantitative evaluation of the dimensionless transient flux of penetrant across the lower surface of thin polymeric membranes versus dimensionless diffusion time t, as defined in the text. The dimensionless half-time t1/2 is 1/7.2 when the transient flux achieves 50% of its steady state value (i.e., JA x(t ) 1, h ¼ 1)). The dimensionless lag time tLag is 1/6 via integration area on the upper left side of this graph. Numerical calculations were performed by including 50 terms in the infinite series expression for JA x(t, h ¼ 1) to reveal zero initial slope versus t at t ¼ 0, whereas analytical evaluation of f@JA x/@ tgt ¼0, h ¼1 is not immediately obvious.

66

Chapter 2 Diffusion in Amorphous Polymers

zero on the vertical axis. For example,

tLag

tLag D ¼ 2 ¼ L

t )1 ð  t ¼0

¼ 2

1 X

(1)

k¼1

k

 JA x (t, h ¼ 1) 1 dt JA x (t ) 1, h ¼ 1)

t )1 ð

exp(k 2 p 2 t) dt ¼

t ¼0

1 2 X (1)kþ1 2 p k¼1 k2

Since the alternating infinite series converges to p 2/12 [Gradshteyn and Ryzhik, 1980], one obtains the following relation between the binary molecular diffusion coefficient D, membrane thickness L, and the dimensional lag time tLag: 1 X (1)kþ1 k¼1

k2

¼

D¼

p2 12 L2 6tLag

If one compares the two previous developments that allow one to predict membrane diffusion coefficients D via measurement/calculation of half-times t1/2 and lag times tLag, then it should be obvious that the complete transient response for the flux of gas A across the lower surface of the membrane is analyzed (i.e., via integration) to calculate tLag. In contrast, only the steady state flux JA x(t ) 1,h ¼ 1) and 50% of this steady state flux are used to identify t1/2. Hence, lag-time calculations incorporate more information about the transient response, relative to measurements of diffusion half-times, to estimate D. Analytical evaluation of the initial slope of the transient flux of penetrants across the lower surface of the membrane with respect to dimensionless diffusion time t yields the following infinite series expression: 

@JA x (t, h ¼ 1) @t



¼ 2p 2 JA x (t ) 1, h ¼ 1) t¼0

1 X

(1)kþ1 k 2 ) 0

k¼1

This slope vanishes due to the alternating nature of the infinite series, but numerical analysis of JA x(t, h ¼ 1) versus t in Figure 2.2 provides better illustration the time delay required for penetrants to traverse the thickness of the membrane.

2.7 EXAMPLE PROBLEM: EFFECT OF MOLECULAR WEIGHT DISTRIBUTION FUNCTIONS ON AVERAGE DIFFUSIVITIES The mass fraction distribution function for a polydisperse polystyrene sample is given by the following normalized triangular profile that is continuous in each segment

2.7 Example Problem: Effect of Molecular Weight Distribution Functions

67

(i.e., a x b and b x c): W(x) ¼ dx  e; a x b W(x) ¼ 2e  dx; b x c a ¼ 200 b ¼ 300 c ¼ 400 d ¼ 104 e ¼ 2  102 where W(x) is the mass fraction of chains that contain x monomer units, and the repeat unit molecular weight MWrepeat is 104 daltons for styrene (i.e., ZCH2CH(C6H5)Z), which can be approximated by 102 daltons. The molecular-weight dependence of diffusion coefficients for this polymer in cyclohexane at 37 8C conforms to the following scaling law: 1:2d D(x) [cm2 =s]  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xMWrepeat 0:12d  pffiffiffi x (a) Obtain an expression for the continuous mole fraction distribution P(x) of chains that contain x monomer units.

Answer Begin with the relation between mole fraction distribution functions and mass fraction distribution functions in the continuous limit (see Section 12.3): W(x) ¼ ð 1

xP(x) xP(x) dx

¼

xP(x) Normalization constant

0

where the normalization constant in the denominator of the previous equation is the number-average degree of polymerization (i.e., first moment of P(x)). Rearrangement of the previous expression yields the mole fraction distribution function: 1 P(x) ¼ fNormalization constantg W(x) x

68

Chapter 2 Diffusion in Amorphous Polymers

Normalization of P(x) allows one to evaluate the constant in the previous equation: 1 ð

2b 3 ðc ð P(x) dx ¼ fNormalization constantg4 W(x)d ln x þ W(x)d ln x5 a

0

b

2b 3  ðc  ðn o e 2e  d dx5 ¼ 1 ¼ fNormalization constantg4 d  dx þ x x a

Normalization constant ¼ 2:94  10 h 8 ei < 2:94  102 d  ; a x b x i h P(x) ¼ : 2:94  102 2e  d ; b x c x

b 2

(b) Calculate the number-average diffusivity Dn of polystyrene in cyclohexane at 37 8C. Answer Dn is defined as the average value of D(x) with respect to the normalized mole fraction distribution function P(x), where the latter was evaluated explicitly in part (a). Consideration of each continuous section of the distribution separately yields the following result: 9 8b 1

= ðc ð <ð d e 2e d pffiffiffi  3=2 dx þ  pffiffiffi dx Dn ¼ D(x)P(x)dx ¼ 2:94  102 (0:12d) : x x x ; x3=2 a

0

7

¼ 7:34  10

b

2

cm =s

(c) Calculate the weight-average diffusivity Dw of polystyrene in cyclohexane at 37 8C. Answer Dw is defined as the average value of D(x) with respect to the normalized mass fraction distribution function W(x). Once again, consideration of each continuous section of the distribution separately yields 8b 9 1



ðc ð <ð pffiffiffi pffiffiffi = e 2e Dw ¼ D(x)W(x)dx ¼ (0:12d) d x  pffiffiffi dx þ pffiffiffi  d x dx : ; x x 0

a

¼ 7:04  107 cm2 =s

b

References

69

(d) Demonstrate that the weight-average diffusion coefficients cannot be obtained simply by inserting the weight-average degree of polymerization (i.e., hxw i ¼ 300) into the molecular-weight scaling law provided above for D(x). Answer 1:2d 0:12d D(x ¼ hxw i ¼ 300) [cm2 =s]  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffi ¼ 6:93  107 cm2 =s hxw iMWrepeat hxw i Dw ¼

1 ð

D(x)W(x) dx = D(x ¼ hxw i)

0

Thought-Provoking Exercise How should one modify the analyses in Sections 2.4 and 2.5 for solvent diffusion in amorphous polymers near the glass transition when solubilized solvent molecules decrease the glass transition temperature of the matrix?

REFERENCES COHEN MH, TURNBULL D. Molecular transport in liquids and glasses. Journal of Chemical Physics 31(5): 1164– 1169 (1959). COHEN MH, TURNBULL D. Free volume model of the amorphous phase—glass transition. Journal of Chemical Physics 34(1):120–125 (1961). GRADSHTEYN IS, RYZHIK IS. Tables of Integrals, Series, and Products, corrected and enlarged edition. Academic Press, San Diego, CA, 1980, p. 7. VRENTAS JS, DUDA JL. Diffusion of small molecules in amorphous polymers. Macromolecules 9(5): 785–790 (1976). VRENTAS JS, DUDA JL. Solvent and temperature effects on diffusion in polymer– solvent systems. Journal of Applied Polymer Science 21:1715–1728 (1977). VRENTAS JS, DUDA JL. A free volume interpretation of the influence of the glass transition on diffusion in amorphous polymers. Journal of Applied Polymer Science 22:2325–2339 (1978).

Chapter

3

Lattice Theories for Polymer – Small-Molecule Mixtures and the Conformational Entropy Description of the Glass Transition Temperature Our joy and our pain dissipate like pale breath into a clear glass sky. —Michael Berardi

Several lattice theories are presented to analyze thermodynamic properties of polymer solutions, plasticized polymer – diluent blends, and phase-separated binary mixtures. Comparisons between binodal and spinodal points on temperature – composition phase diagrams are discussed. The entropy of mixing on the Flory – Huggins and Guggenheim lattices is compared. These statistical analyses provide the background required to appreciate the Gibbs –DiMarzio conformational entropy description of the glass transition in binary systems, where the mixture’s entropy vanishes at Tg upon cooling. Structural characteristics of additives that increase and decrease a polymer’s Tg are simulated and discussed. Qualitative comparison between the Gibbs – DiMarzio theory and lattice cluster theory spans more than 40 years from the early lattice models to current research topics.

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

71

72

3.1

Chapter 3 Lattice Theories for Polymer– Small-Molecule Mixtures

LATTICE MODELS IN THERMODYNAMICS

Envision the pattern of alternately colored squares on a chess board and the placement of point particles on all sites to simulate mixtures of two or more low-molecularweight components. Statistical methods have been employed to understand the thermodynamic properties of small-molecule mixtures using point particles and the chess board, where the latter is better known as the lattice. Now, introduce the restriction that one of the components contains hundreds or thousands of interconnected segments to simulate linear chain molecules. Polymer solutions (i.e., polymer – solvent mixtures) have been modeled in this fashion, and their thermodynamic properties have been calculated by combining classical and statistical methods. If each species in the mixture exhibits a distinct color, then the “state of mixing” can be addressed qualitatively and quantitatively in terms of miscibility and phase separation. In many cases, the lattice provides assistance to generate macroscopic thermodynamic properties of mixtures that do not depend on geometric details of the lattice.

3.2 MEMBRANE OSMOMETRY AND THE OSMOTIC PRESSURE EXPANSION A porous membrane is inserted in the bottom of a capillary tube. The capillary leg on the left side of the membrane contains distilled water. A dilute aqueous solution of cellulase enzyme is placed on the right side of the membrane and both legs of the capillary tube are filled with liquid to the same height. The air – liquid interface in both legs of the capillary is exposed to ambient pressure pambient, and the temperature remains constant at 37 8C. Scanning electron micrographs of the membrane reveal that ˚ (i.e., 0.1 mm), which translates into the pores have an average diameter of 1000 A a “molecular-weight cutoff” of 50,000 Da. Since cellulase is a biological macromolecule with an average molecular weight of 200,000 Da, the membrane prohibits the enzyme from diffusing through the pores. However, solvent molecules are allowed to pass through the pores of the membrane. Question: Does distilled water diffuse from left-to-right or from right-to-left through the membrane? ANSWER: Water molecules diffuse from left-to-right. EXPLANATION: Diffusion of water molecules from left-to-right occurs through the porous membrane in response to a gradient in the chemical potential of water at the water –membrane interface on both sides of the membrane. The chemical potential of pure water on the left side is greater than that in the aqueous solution of the enzyme on the right side. The passage of water from left-to-right, across the membrane, causes the liquid level to decrease on the left side and increase on the right side of the capillary. This “capillary rise” in the right leg generates additional hydrostatic pressure that opposes the diffusion of water through the membrane. When equilibrium is established, this increase in hydrostatic pressure on the right side prevents further diffusion of the solvent (water) and the flow process ceases. Alternatively, one exposes the right leg of the capillary to an increased pressure, given by pambient þ p, where p is the osmotic pressure. Now, the height of liquid in both legs of the

3.2 Membrane Osmometry and the Osmotic Pressure Expansion

73

capillary remains the same because the hydrostatic pressure gradient opposes the chemical potential gradient that was attributed initially to differences in solvent concentration on both sides of the membrane.

3.2.1

The Consequence of Chemical Equilibrium

When equilibrium is established, the pressure at the water – membrane interface on the left side is pLeft, which is greater than ambient pressure due to the hydrostatic contribution. On the right side, the pressure at the interface between the membrane and the cellulase enzyme solution is pRight, which is also greater than ambient pressure, but pRight is greater than pLeft due to the osmotic pressure effect described above. The statement of chemical equilibrium for the solvent on both sides of the membrane is

mPure Water (T, pLeft ) ¼ mSolvent (T, pRight , xSolute ) where the chemical potential of pure water on the left side of the membrane is evaluated at 37 8C and pressure pLeft. On the right side of the membrane, the chemical potential of water in the aqueous solution is evaluated at 37 8C, pressure pRight (i.e., pRight – pLeft ¼ p), and cellulase enzyme mole fraction xSolute. The pressure dependence of the solvent’s chemical potential at constant temperature and composition is described by the partial molar volume of the solvent via a Maxwell relation based on the extensive Gibbs free energy G of a binary mixture. There are three degrees of freedom for single-phase behavior of binary mixtures. Consequently, four independent variables are required for a complete description of an extensive thermodynamic potential, such as G. The appropriate independent variables are temperature T, pressure p, and mole numbers of the solute and solvent (i.e., NSolute and NSolvent). Hence, dG ¼ S dT þ V dp þ mSolvent dNSolvent þ mSolute dNSolute where S and V are the extensive entropy and volume, respectively, of the binary mixture. The appropriate Maxwell relation at composition xSolute that allows one to evaluate the pressure dependence of the chemical potential is     @ mSolvent @V ¼ ¼ V Solvent @p @NSolvent T,p,NSolute T,NSolvent ,NSolute The previous equation is based on the fact that second mixed partial derivatives of an exact differential, like the extensive Gibbs free energy of binary mixtures, are not affected by reversing the order in which differentiation is performed. One integrates the previous equation from pressure pLeft to pRight at constant temperature and composition to obtain the effect of pressure on the chemical potential of the solvent in the aqueous solution on the right side of the membrane: mSolvent (T, pðRight , xSolute ) mSolvent (T, pLeft , xSolute )

d mSolvent ¼

pRight ð

V Solvent dp pLeft

mSolvent (T, pRight , xSolute )  mSolvent (T, pLeft , xSolute )  pV Solvent

74

Chapter 3 Lattice Theories for Polymer– Small-Molecule Mixtures

Pressure dependence of the partial molar volume of the solvent was neglected in the previous integration. Now, the statement of chemical equilibrium for the solvent on both sides of the membrane reduces to

mPure Water (T, pLeft ) ¼ mSolvent (T, pRight , xSolute ) ¼ mSolvent (T, pLeft , xSolute ) þ pV Solvent The dilute aqueous cellulase enzyme solution in the right leg of the capillary can be approximated as an ideal liquid mixture because Raoult’s law is applicable at small solute concentrations. Hence, the chemical potential of the solvent at temperature T, pressure pLeft, and mole fraction xSolute is evaluated as follows:

mSolvent (T, pLeft , xSolute ) ¼ mPure Water (T, pLeft ) þ RT lnf1  xSolute g In the dilute solution regime, where 12xSolute is very close to unity, one expands the logarithmic term about xSolute ¼ 0 in the previous equation: 2 3 lnf1  xSolute g  xSolute  12 xSolute  13 xSolute  

The statement of chemical equilibrium yields the following osmotic pressure expansion:

mPure Water (T, p)  mSolvent (T, p, xSolute ) V Solvent  RT  2 3  xSolute þ 12 xSolute þ 13 xSolute þ  V Solvent

p¼

Problem: Sketch the difference between the liquid height in both legs of the capillary as a function of cellulase enzyme concentration in dilute aqueous solutions at 25 8C and 37 8C. Both liquids are incompressible to a good approximation. Their densities are roughly the same and equal to that of pure water.

The polymer concentration in aqueous solution, CPolymer, with dimensions of grams solute per solution volume, is related to the mole fraction of solute, xSolute, in the dilute solution regime as follows:

xSolute ¼

CPolymer MWPolymer CPolymer 1 þ MWPolymer V Solvent

Dilute

) Solution

CPolymer V Solvent MWPolymer

The osmotic pressure expansion is rewritten in canonical form for molecular weight and second virial coefficient determination via linear least squares analysis of

3.2 Membrane Osmometry and the Osmotic Pressure Expansion

75

p/RTCPolymer versus CPolymer:

! p 1 V Solvent CPolymer  þ 2 RTCPolymer MWPolymer 2MWPolymer ! 2 V Solvent 2 CPolymer þ þ  3 3MWPolymer

The intercept yields the inverse of the polymer’s molecular weight (i.e., numberaverage molecular weight for a distribution of chain lengths) and the initial slope corresponds to the second virial coefficient B, where B¼

V Solvent 2 2MWPolymer

The second virial coefficient and all higher virial coefficients in the osmotic pressure expansion vanish for dilute polymer solutions at the Q-temperature, where the chains exhibit unperturbed dimensions and p is a linear function of polymer concentration.

3.2.2

Analogy with the Virial Expansion of Real Gases

Consider the equation of state for a van der Waals gas with attractive interaction parameter a and excluded volume parameter b that are calculated from the critical constants. In terms of molar volume v, which is equivalent to the ratio of molecular weight MW and mass density r, the following equations are applicable when b is significantly less than the molar volume v:  a p þ 2 (v  b) ¼ RT v ( )  1  2 RT b a RT bh a i b 1 1þ 1 þ  2¼ þ p¼ v v v v v bRT v  2  p 1 b h a i b ¼ þ 1 rþ r2 þ    rRT MW MW 2 MW 3 bRT One invokes the following correspondences between the virial expansion of a van der Waals gas and the osmotic pressure expansion for a dilute polymer solution: (i) Gas pressure p is analogous to osmotic pressure p. (ii) Gas density r is analogous to polymer concentration in solution, CPolymer. (iii) Molecular weight of the gas, MW, is analogous to polymer molecular weight MWPolymer. (iv) The second virial coefficient vanishes (and also changes sign) at the Boyle temperature, defined by TBoyle ¼ a/bR ¼ (27/8)TCritical, where ideal behavior is achieved for a van der Waals gas. The second virial coefficient vanishes (and also changes sign) at the Q-temperature for dilute polymer solutions,

76

Chapter 3 Lattice Theories for Polymer– Small-Molecule Mixtures

where chains exhibit unperturbed dimensions because polymer – solvent interactions are indistinguishable from segment – segment interactions in the undiluted polymer. Gas pressure scales linearly with density at TBoyle and osmotic pressure scales linearly with CPolymer at the Q-temperature.

3.3 LATTICE MODELS FOR ATHERMAL MIXTURES WITH EXCLUDED VOLUME The major objectives of this section are to introduce the concepts of lattices, multiplicity of states, and Boltzmann’s entropy expression to calculate thermodynamic properties of mixtures. One of the most important results is that the second virial coefficient in the osmotic pressure expansion of a dilute polymer solution scales linearly with the excluded volume per chain molecule, identified by g. Excluded volume identifies lattice sites that are inaccessible to long chain molecules because these sites are occupied by chains that were inserted previously on the lattice. Each solvent molecule is modeled as a structureless point particle that occupies one lattice site. There is no entropic contribution associated with the placement of solvent molecules in empty lattice sites. The entropy of mixing is due solely to the multiplicity of placing polymer molecules on interconnected lattice sites. Let’s begin by counting the number of ways that each polymer molecule can occupy the lattice. The total volume of the lattice corresponds to solution volume V and g represents the volume per chain that is not available to other molecules. If A, with dimensions of inverse volume, is the proportionality constant that relates the available lattice volume to the number of conformations, or multiplicity, for placing each chain on the lattice, then Number of conformations for the first polymer chain ¼ AV Number of conformations for the second polymer chain ¼ A(V  g) Number of conformations for the third polymer chain ¼ A(V  2g) Number of conformations for the ith polymer chain ¼ A{V  [i  1]g} The multiplicity, or total number of distinguishable ways that NPolymer molecules can be inserted into the lattice, is obtained via multiplication of the conformational freedom for all chains, as summarized above, and subsequent division by (NPolymer)! because all chains are identical in chemical structure. Hence, the number of distinguishable states for this binary polymer – solvent mixture is V¼

1



NPolymer !

(AV)NPolymer  ¼ NPolymer !

NPolymer Y i¼1

NY h (AV)NPolymer Polymer gi  A[V  (i  1)g ] ¼  1  (i  1) V NPolymer ! i¼1

NPolymer Y1 i¼0

ig 1 V



The corresponding entropy of the polymer solution is given by Boltzmann’s expression, where S ¼ k ln V. The multiplicity of states V can be evaluated from the previous expression for the solution, pure polymer, and pure solvent (i.e., NPolymer ¼ 0). Explicit evaluation of the entropy of pure solvent should not be

3.3 Lattice Models for Athermal Mixtures with Excluded Volume

77

performed until one invokes the approximation that NPolymer 2 1  NPolymer:

 NPolymer X1   ig ln 1  S ¼ k ln V ¼ kNPolymer ln(AV)  k ln NPolymer ! þ k V i¼0 For dilute polymer solutions, ig/V is much less than unity for all values of i in the previous summation, so it is acceptable to expand the logarithmic terms and retain only the linear contribution. Since ln (1 2 x)  2x 2 O(x 2), one obtains the following result: X1   kg NPolymer i S  kNPolymer ln(AV)  k ln NPolymer !  V i¼1   kg 1 NPolymer (NPolymer  1)  kNPolymer ln(AV)  k ln NPolymer !  V 2 2   kgNPolymer  kNPolymer ln(AV)  k ln NPolymer !  2V

Next, one evaluates the extensive entropy of mixing for a binary system that contains the following number of molecules of polymer and solvent, respectively, NPolymer and NSolvent. The calculation proceeds as follows: DSmixing ¼ S(NPolymer , NSolvent )  S(NPolymer , NSolvent ¼ 0)  S(NPolymer ¼ 0, NSolvent ) where the last two terms in the previous equation represent pure polymer and pure solvent, respectively. As mentioned above, structureless solvent molecules do not contribute to the entropy of mixing because there is only one way to insert them in the lattice, both in the pure state and in solution. When one evaluates the entropy of the binary mixture, S(NPolymer, NSolvent ) requires partial molar volumes of polymer and solvent in the expression for solution volume V (i.e., VSolution): VSolution ¼ nPolymer V Polymer þ nSolvent V Solvent where nPolymer and nSolvent represent mole numbers for polymer and solvent, respectively. When one evaluates the entropy of pure polymer in the absence of solvent, extensive volume V (i.e., VPolymer) can be calculated from the previous equation if the solvent term is omitted and the partial molar volume of the polymer is replaced by its molar volume vPolymer. Boltzmann’s entropy expression yields the following entropy change upon mixing:

 VSolution DSmixing ¼ kNAvogadro nPolymer ln VPolymer

 1 1 1 2 þ kgNAvogadro n2Polymer  2 VPolymer VSolution Statistical concepts have been employed to calculate a macroscopic thermodynamic property, as illustrated above. The specific details of the lattice are no longer required, as evidenced by the fact that the lattice parameter A does not appear in the final

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Chapter 3 Lattice Theories for Polymer– Small-Molecule Mixtures

expression for DSmixing. The following sequence of steps is required to relate the second virial coefficient in the osmotic pressure expansion to the excluded volume parameter g: Step 1: The Gibbs free energy of mixing DGmixing for athermal solutions is calculated from the entropy of mixing via 2T DSmixing because DHmixing ¼ 0. Step 2: The chemical potential difference between the solvent in its pure state and in the polymer solution is calculated from the solvent mole number derivative of DGmixing at constant temperature T, pressure p, and mole numbers of polymer nPolymer (i.e., via partial molar properties of the solvent). Step 3: There is a direct relation between osmotic pressure and the chemical potential difference between the solvent in its pure state and in the polymer solution. Replacing kNAvogadro by the gas constant and multiplying DSmixing by 2T yields an expression for the Gibbs free energy of mixing for athermal binary polymer solutions:

 VSolution DGmixing ¼ RTnPolymer ln VPolymer

 1 1 1 2   RT gNAvogadro nPolymer 2 VPolymer VSolution Now, one calculates the required chemical potential difference between the solvent in its pure state and in solution via introduction of the partial molar volume of the solvent:   @DGmixing ¼ mSolvent (T, p, xPolymer )  m0Pure Solvent (T, p) @nSolvent T,p,nPolymer     V Solvent 1 V Solvent 2 ¼ RTnPolymer  RT gNAvogadro nPolymer 2 2 VSolution VSolution Division of the previous expression by the product of RT and the partial molar volume of the solvent yields a relation between excluded volume and osmotic pressure:

m0 (T, p)  mSolvent (T, p, xPolymer ) p ¼ Pure Solvent RT RTV Solvent   nPolymer 1 nPolymer 2 ¼ þ gNAvogadro VSolution 2 VSolution The molar concentration of polymer in solution, given by the ratio of nPolymer and VSolution is equivalent to CPolymer, with dimensions of grams polymer per solution volume, divided by the polymer’s molecular weight MWPolymer. The lattice model with excluded volume yields the following osmotic pressure expansion in which the coefficient of the term that is linear with respect to CPolymer relates the second virial

3.4 Flory–Huggins Lattice Theory for Flexible Polymer Solutions

79

coefficient and g :

! gNAvogadro p 1 CPolymer ¼ þ 2 RTCPolymer MWPolymer 2MWPolymer

The product of g and NAvogadro can be interpreted as the excluded volume per mole of chain molecules, where a mole is based on the molecular weight of the entire chain, not simply the repeat unit. As mentioned from the analysis of membrane osmometry, linear least squares analysis of p/RTCPolymer versus CPolymer yields the polymer’s molecular weight from the intercept and the second virial coefficient from the slope. Generalization of these results suggests that the excluded volume vanishes when the second virial coefficient is identically zero. For example, when polymer chains exhibit unperturbed dimensions in a Q-solvent, p/RTCPolymer is not a function of the polymer concentration in solution and the excluded volume vanishes.

3.4 FLORY – HUGGINS LATTICE THEORY FOR FLEXIBLE POLYMER SOLUTIONS More detailed considerations of the lattice are provided in this section for polymer – solvent mixtures that exhibit entropic and enthalpic changes upon mixing. Interconnectivity of lattice sites that are occupied by the same polymer chain imposes severe restrictions on the conformational entropy of mixing, and these constraints become more severe when two polymers are mixed, to the extent that DSmixing is essentially zero for polymer – polymer blends. This interconnectivity was mentioned but not addressed in much detail in the previous section.

3.4.1

Nomenclature

The following notation is employed to construct the Flory – Huggins lattice model. NTotal ¼ total number of individual lattice sites NSolvent ¼ number of solvent molecules, modeled as structureless point particles NPolymer ¼ number of monodisperse polymer molecules, each one contains xNPolymer

x segments ¼ number of sites in the lattice that are occupied by polymer chains

NTotal ¼ NSolvent þ xNPolymer

3.4.2

Assumptions

Each site in the three-dimensional lattice is large enough for either one solvent molecule or one segment of the polymer chain to reside. Hence, solvent molecules are approximately the same size as the polymer repeat unit. Segments of the polymer

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Chapter 3 Lattice Theories for Polymer– Small-Molecule Mixtures

chain interact nonspecifically with solvent molecules. In other words, there are no polar functional groups or charged species in the main chain or side group of the polymer that interact preferentially with the solvent. Each segment of the polymer is equally accessible to the solvent. Each polymer chain contains x segments or repeat units. As illustrated in the next section, the occupational probability of each lattice site depends on the number of polymer molecules that have already been placed on the lattice. In other words, the fact that prior segments of the same chain necessarily occupy lattice sites does not affect the calculations below when subsequent segments of the chain are placed on the lattice. All of these assumptions are considered to calculate the combinatorial entropy of mixing via statistical analysis of the “counting problem” because each permutation is considered to be equally likely, analogous to the microcanonical ensemble in statistical thermodynamics. The interaction free energy of mixing accounts for energetic effects between similar and dissimilar molecules but “Boltzmann weighting factors” are not employed to distinguish different permutations of polymer chain conformations and structureless solvent molecules on the lattice.

3.4.3

Conformational Entropy of Mixing, DSmixing

This development begins by placing i polymer molecules on the lattice and calculating the occupational probability for all x segments of the (i þ 1)st chain. For example, if there are NTotal lattice sites and x interconnected sites are required for each chain molecule that has already been placed on the lattice, then there are NTotal 2 ix sites available for the first segment of the next polymer molecule. Now, interconnectivity must be considered for segments 2 through x via the coordination number z, or the number of nearest neighbor sites. Typical coordination numbers are 4 for twodimensional lattices and 6 for three-dimensional lattices. Hence, there are a maximum of z sites available for the second segment of the chain and z 2 1 sites available for segments 3 through x, but some of these sites might be occupied by polymer molecules that have already been placed on the lattice. The fraction of vacant sites on the lattice ki after i polymer chains occupy sites is {NTotal 2 ix}/NTotal, and this probability factor ki together with the coordination number (i.e., z or z 2 1) is required to calculate the number of sites that are available for segments 2 through x in the (i þ 1)st chain. Changes in the probability factor ki are considered for different chains, but not for different segments of the same chain. In summary, the number of ways that each segment of the (i þ 1)st chain can be arranged on the lattice is First segment: NTotal  ix Second segment: zki Third segment: (z  1)ki , same for all remaining segments Since all NPolymer chains are structurally identical, the multiplicity of states, or number of distinguishable ways of placing these chains on the lattice, is obtained via multiplication of the occupational freedom of all x segments in all chains, and subsequent

3.4 Flory–Huggins Lattice Theory for Flexible Polymer Solutions

81

division by NPolymer!, as follows: V¼

NPolymer ! 



NPolymer Y1

1

z1 NTotal

(NTotal  ix)zki f(z  1)ki gx2

i¼0

NPolymer (x1)

1

NPolymer Y1

NPolymer !

i¼0

(NTotal  ix)x

This expression for V is not restricted to the dilute solution regime, but it is almost impossible to evaluate the combinatorial entropy of mixing, as written, because the total number of lattice sites NTotal and the number of polymer molecules NPolymer are exceedingly large. Boltzmann’s equation, S ¼ k ln V, is employed to calculate the conformational entropy of mixing after substantial manipulation of the multiplicity of states. In addition to replacing z by z21, the following approximations are invoked at low polymer concentrations: Step 1: The term within the product on the far right side of the previous expression for V (i.e., (NTotal 2 ix)x) is written as a ratio of two factorials, which simplifies to a product of x terms that are almost identical in the dilute solution regime (i.e., NSolvent  xNPolymer): (NTotal  ix)x 

(NTotal  ix)! fNTotal  x(i þ 1)g!

¼ (NTotal  ix)(NTotal  ix  1)(NTotal  ix  2)    (NTotal  ix  x þ 1) Step 2: Explicit evaluation of the product in the expression for V, with assistance from the dilute solution approximation in Step 1, reduces to NPolymer Y1 i¼0

¼

  NTotal !(NTotal  x)!(NTotal  2x)!    NTotal  (NPolymer  1)x ! (NTotal  x)!(NTotal  2x)!       NTotal  (NPolymer  1)x ! NTotal  xNPolymer !

¼ Step 3:

(NTotal  ix)! fNTotal  x(i þ 1)g!

NTotal ! N !  ¼ Total NTotal  xNPolymer ! NSolvent !

Stirling’s approximation for the factorial when n is large,

 pffiffiffiffiffiffiffiffiffi 1 1 n þ   n! ¼ n 2pn exp(n) 1 þ 12n 288n2

 ln n!  12 ln(2p) þ n þ 12 ln n  n  n ln n  n

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Chapter 3 Lattice Theories for Polymer– Small-Molecule Mixtures

is employed to evaluate ln V as follows:   NTotal ! z  1 NPolymer (x1) V NPolymer !NSolvent ! NTotal ln V  (NSolvent þ xNPolymer ) ln(NSolvent þ xNPolymer )  (NSolvent þ xNPolymer )  NPolymer ln NPolymer þ NPolymer  NSolvent ln NSolvent þ NSolvent þ (x  1)NPolymer ln(z  1)  (x  1)NPolymer ln(NSolvent þ xNPolymer ) Now, Boltzmann’s equation yields the following expression for the combinatorial entropy of the Flory – Huggins lattice model: S ¼ (NSolvent þ NPolymer ) ln(NSolvent þ xNPolymer )  NSolvent ln NSolvent k  (x  1)NPolymer f1  ln(z  1)g  NPolymer ln NPolymer Once again, the extensive entropy of mixing is calculated by evaluating the previous equation for (i) dilute polymer solutions, (ii) pure polymer, and (iii) pure solvent. In the spirit of maintaining an incompressible system, the size of the lattice for the sum of the precursors (i.e., NPolymer molecules in the absence of solvent and NSolvent molecules without polymer) is the same as the size of the lattice required to accommodate NSolvent þ NPolymer molecules in dilute solution. Hence, NSolvent and NPolymer do not change from their respective undiluted states to the solution in the expressions below. DSmixing ¼ S(NPolymer , NSolvent )  S(NPolymer , NSolvent ¼ 0)  S(NPolymer ¼ 0, NSolvent ) ¼ k(NSolvent þ NPolymer ) ln(NSolvent þ xNPolymer )  kNSolvent ln NSolvent  kNPolymer ln(xNPolymer )



 xNPolymer NSolvent  kNPolymer ln ¼ kNSolvent ln NSolvent þ xNPolymer NSolvent þ xNPolymer There is no combinatorial entropy associated with placing identical structureless point particles on all of the lattice sites, to simulate pure solvent. The indistinguishability of solvent molecules reduces the multiplicity of this process to unity, but justification of this statement requires l’Hoˆpital’s rule for the last term in S when NPolymer ¼ 0. Coordination number z appears in the expression for the combinatorial entropy of the Flory – Huggins lattice, but the final expression for the entropy of mixing does not depend on any lattice structural parameters. This fact is reassuring because the lattice is only a crutch that allows one to simulate the mixing process. From a different viewpoint, however, the coordination number of the lattice appears in the final expressions for the entropy of mixing in Guggenheim’s theory that has been applied by DiMarzio and Gibbs to predict either increases or decreases in the glass transition temperature of amorphous polymers at higher diluent volume fractions. Statistical details about the lattice, or the coordination numbers of transition metal complexes, are included in the conformational entropy description of the glass transition, based on Guggenheim’s theory of mixtures as described below. The quantities in curly

3.4 Flory–Huggins Lattice Theory for Flexible Polymer Solutions

83

brackets { } in the previous expression for the Flory – Huggins entropy of mixing represent volume fractions w of solvent and polymer, respectively, as defined by

wSolvent ¼

xNPolymer NSolvent ; wPolymer ¼ NSolvent þ xNPolymer NSolvent þ xNPolymer

If the numbers of molecules of polymer and solvent, NPolymer and NSolvent, are reexpressed in terms of mole numbers and Avogadro’s number, with kNAvogadro ¼ R, and y represents mole fraction, then the previous mixture composition variables can be solved for mole fractions in terms of volume fraction as follows: ySolvent ¼

x(1  wPolymer ) wPolymer ; yPolymer ¼ wPolymer þ x(1  wPolymer ) wPolymer þ x(1  wPolymer )

The molar entropy of mixing Dsmixing is given by DSmixing Dsmixing ¼ ¼ ySolvent ln wSolvent þ yPolymer ln wPolymer k(NSolvent þ NPolymer ) R There is one major difference between this expression for the combinatorial entropy of mixing of a dilute binary polymer solution and the ideal Dsmixing for regular solutions of low-molecular-weight molecules. The chain-like nature of one of the components restricts chaotic mixing and dictates the need for volume fractions in the final expression for Dsmixing. Volume fractions are replaced by mole fractions for the ideal entropy of mixing in regular solutions of small molecules. The extent of randomness and chaos due to the mixing process is reduced by the chain-like nature of one or both components. Comparison of equimolar binary mixtures reveals that Dsmixing is largest for regular solutions of small molecules, intermediate for polymer solutions, and smallest for polymer – polymer blends.

3.4.4 Interaction Free Energy of Mixing, DGmixing, and the Flory– Huggins Thermodynamic Interaction Parameter x This calculation focuses on nonspecific pairwise energetic interactions between species that occupy nearest neighbor sites on the lattice. These interactions are exclusively intermolecular for the solvent, but intramolecular interactions between different segments of the same polymer chain are allowed. The following nomenclature is used to describe the various types of pairwise interactions: 1SS 1PP 1SP z zwSolvent zwPolymer

Interaction energy between two solvent molecules Interaction energy between two polymer segments on the same chain (i.e., intramolecular) or on different chains (i.e., intermolecular) Interaction energy between one solvent molecule and one polymer segment Number of nearest neighbor sites on the two- or three-dimensional lattice Number of nearest neighbor sites occupied by solvent Number of nearest neighbor sites occupied by polymer segments

84

Chapter 3 Lattice Theories for Polymer– Small-Molecule Mixtures

Energetic Interactions Within the Pure Solvent (w Solvent 5 1) The lattice is occupied completely by NSolvent molecules. For each site on the lattice, excluding the boundaries, there are z nearest neighbors that contain solvent molecules that interact with a solvent molecule in the site of interest, and the energy of interaction is 1SS. Hence, the interaction energy is z1SS per site, which must be multiplied by the total number of sites that contain solvent (i.e., NSolvent). However, each pairwise interaction is counted twice by this procedure, so the cumulative interaction energy when solvent molecules occupy the entire lattice is Pairwise interaction energy for pure solvent ¼ 12 NSolvent z1SS Energetic Interactions Within the Undiluted Polymer (w Polymer 5 1) Now, the lattice is occupied completely by NPolymer molecules with x segments per chain. Hence, xNPolymer lattice cells are required to describe the undiluted polymer. Once again, excluding the boundaries of the lattice, there are z nearest neighbor sites that contain polymer segments that interact with the segment of interest according to 1PP. The interaction energy is z1PP per lattice site, which must be multiplied by xNPolymer sites and divided by 2 because each pairwise segment– segment interaction has been counted twice. The total interaction energy for the undiluted polymer is Pairwise interaction energy for pure polymer ¼ 12 xNPolymer z1PP Energetic Interactions Within the Polymer – Solvent Mixture The lattice contains both solvent and polymer with volume fractions given by wSolvent and wPolymer, respectively. Each calculation of a pairwise interaction energy per solvent site must be multiplied by NSolvent and each calculation per site that is occupied by a polymer segment must be multiplied by xNPolymer. Energetic interactions occur between similar molecules and dissimilar molecules. As mentioned above, interactions between similar molecules are necessarily intermolecular for the solvent, but they can be either intrachain or interchain for the polymer. Consider all lattice sites that are occupied by solvent molecules: Pairwise solvent–solvent interaction energy ¼ 12 NSolvent zwSolvent 1SS Pairwise polymer–solvent interaction energy ¼ 12 NSolvent zwPolymer 1SP Now, consider all lattice sites that are occupied by segments of a polymer chain: Pairwise polymer–solvent interaction energy ¼ 12 xNPolymer zwSolvent 1SP Pairwise polymer–polymer interaction energy ¼ 12 xNPolymer zwPolymer 1PP The extensive interaction free energy of mixing is constructed by subtracting pairwise interaction energies for pure solvent and undiluted polymer from all of the

3.4 Flory–Huggins Lattice Theory for Flexible Polymer Solutions

85

possible pairwise interaction energies that exist within the solution. One obtains the following result:  z  NSolvent wPolymer þ xNPolymer wSolvent 1SP DGmixing,interaction ¼ 2  NSolvent wPolymer 1SS  xNPolymer wSolvent 1PP Substitution for polymer and solvent volume fractions yields a rather simple expression after rearrangement that illustrates how some, but not all, solvent – solvent and both intrachain and interchain polymer segment – segment interactions are disrupted in favor of nonspecific solvent/polymer – segment interactions: DGmixing,interaction ¼ z

xNSolvent NPolymer f1SP  12(1SS þ 1PP )g NSolvent þ xNPolymer

The Flory – Huggins dimensionless polymer – solvent thermodynamic interaction parameter x is defined in terms of the difference between pairwise interaction energies on the right side of the previous equation: z x ¼ f1SP  12(1SS þ 1PP )g kT Final expressions for the extensive (DGmixing,interaction) and molar (Dgmixing,interaction, per total moles of both components) interaction free energies of mixing in binary polymer – solvent solutions, provided below, are not limited to the dilute solution regime: DGmixing,interaction ¼ kT xNSolvent wPolymer DGmixing,interaction Dgmixing,interaction ¼ ¼ xySolvent wPolymer kTfNSolvent þ NPolymer g RT This is analogous to the two-parameter van Laar model for the excess nonideal Gibbs free energy of mixing, where the van Laar quantities of interest represent a dimensionless energetic interaction parameter and the ratio of molar volumes of both components such that the effect of composition on Dgmixing,interaction (i.e., per mole of mixture) requires the mole fraction of the smaller component and the volume fraction of the larger component. For regular (i.e., nonideal) binary mixtures of essentially equisized small molecules with nonzero interaction energies, both van Laar parameters are identical and it is necessary to replace volume fraction by mole fraction in the previous equation, which reduces to the Margules one-parameter symmetric model for nonideal mixing where the Flory – Huggins thermodynamic interaction parameter is analogous to the temperature-dependent Margules constant. If the extensive interaction free energy of mixing DGmixing,interaction were divided by the total number of lattice sites (i.e., NSolvent þ xNPolymer) instead of the total number of molecules (i.e., NSolvent þ NPolymer) in the mixture, then the effect of composition on the interaction free energy of mixing, per mole of lattice sites, requires a product of the volume fractions of both components in the binary mixture. The temperature dependence of x governs the entropic and enthalpic contributions to the interaction free energy of

86

Chapter 3 Lattice Theories for Polymer– Small-Molecule Mixtures

mixing in the previous equation. For example, standard formalism from classical thermodynamics yields the following results for the molar entropy and enthalpy of mixing due to energetic interactions:   @Dgmixing,interaction Dsmixing,interaction ¼  @T p,composition

 @(T x) ¼ RySolvent wPolymer @T p,composition

  @ Dgmixing,interaction 2 Dhmixing,interaction ¼ T T @T p,composition

 @x ¼ RT 2 ySolvent wPolymer @T p,composition If the pairwise interaction energies are constant and x varies inversely with temperature, then the interaction free energy of mixing is completely enthalpic in origin, whereas a temperature-independent value for x (i.e., due to interaction energies that vary linearly with temperature) implies that Dgmixing,interaction is solely due to entropic effects. The distinction between entropic and enthalpic contributions to the Flory – Huggins thermodynamic interaction parameter for polymer – solvent mixtures is discussed further in the Appendix at the end of this chapter, Problem 3.2, and Section 5.3.2.

3.4.5 Complete Expression for the Gibbs Free Energy of Mixing, Partial Molar Properties, and the Osmotic Pressure Expansion Developments from the previous two sections for the conformational entropy of mixing and the interaction free energy of mixing are combined to calculate the extensive Gibbs free energy of mixing: DGmixing ¼ DGmixing,interaction  T DSmixing   ¼ kT NSolvent ln wSolvent þ NPolymer ln wPolymer þ xNSolvent wPolymer The following logical sequence of calculations is based on the previous expression for DGmixing: (i) Chemical potential difference between the solvent in solution and in the pure state. (ii) Osmotic pressure expansion and identification of the second virial coefficient. (iii) Stability criteria based on the chemical potential or activity of the solvent. (iv) Critical value of the Flory – Huggins polymer –solvent thermodynamic interaction parameter at the upper critical solution temperature (UCST), above which homogeneous single-phase behavior exists. (v) Relation between the UCST and the Q-temperature.

3.4 Flory–Huggins Lattice Theory for Flexible Polymer Solutions

87

Initially, one replaces the number of molecules of polymer and solvent (i.e., NPolymer and NSolvent) by the product of Avogadro’s number NAvogadro and mole numbers (i.e., nPolymer and nSolvent). Now, the partial molar Gibbs free energy of mixing of the solvent can be evaluated in a straightforward manner via partial differentiation of DGmixing with respect to nSolvent at constant temperature T, pressure p, and moles of polymer nPolymer. The chemical potential difference between the solvent in solution and in its pure state is

 @DGmixing 0 mSolvent (T, p, wPolymer )  mPure Solvent (T, p) ¼ @nSolvent T,p,nPolymer (     nPolymer @ wPolymer nSolvent @ wSolvent ¼ RT ln wSolvent þ þ wSolvent @nSolvent T,p,nPolymer wPolymer @nSolvent T,p,nPolymer " #)   @ wPolymer þx wPolymer þ nSolvent @nSolvent T,p,nPolymer  

 1 2 ¼ RT ln(1  wPolymer ) þ 1  wPolymer þ xwPolymer x Whereas the combinatorial entropic contribution to the previous expression is restricted to the dilute solution regime, the contribution from energetic interactions is applicable to both dilute and concentrated solutions. This limitation on the conformational entropy of mixing is modified below, so that the Gibbs free energy of mixing and the chemical potentials of both components can be employed in an order parameter model for polymer – diluent blends at vanishingly small diluent concentration (see Sections 3.4.6, 5.3.2, and 5.5). The osmotic pressure expansion and identification of the second virial coefficient for dilute polymer solutions is obtained via (i) division of the previous equation by the partial molar volume of the solvent, (ii) expansion of the logarithmic term when wPolymer  1 (i.e., ln(1  w)  w  12 w2  13 w3     ), (iii) expression of the polymer volume fraction in terms of its mass concentration in solution CPolymer and molecular weight MWPolymer, and (iv) rearrangement to arrive at a virial expansion or power series for osmotic pressure with respect to CPolymer. For example,

m0Pure Solvent (T, p)  mSolvent (T, p, wPolymer ) V Solvent  

 RT 1 2 ¼ ln(1  wPolymer ) þ 1  wPolymer þ xwPolymer x V Solvent  

 RT 1 1   x w2Polymer þ    wPolymer þ 2 V Solvent x

p¼

When very high-molecular-weight polymer chains exhibit unperturbed dimensions in a Q-solvent, due to the fact that polymer –solvent interactions are indistinguishable from segment – segment interactions in the undiluted polymer, osmotic pressure scales linearly with polymer concentration and the Flory – Huggins dimensionless interaction parameter x approaches a value of 0.5 in dilute solution.

88

Chapter 3 Lattice Theories for Polymer– Small-Molecule Mixtures

3.4.6 Flory –Huggins Entropy of Mixing for Concentrated Polymer Solutions Let’s return to the initial formulation of the multiplicity of states for placing NPolymer molecules on the lattice, where each chain contains x segments. Without invoking any dilute solution approximations, the initial analysis in Section 3.4.3 yields,  V

z1 NTotal

NPolymer (x1)

1

NPolymer Y1

NPolymer !

i¼0

(NTotal  ix)x

The corresponding extensive entropy via Boltzmann’s equation is 9 8   1 =
The combinatorial entropy of mixing is constructed from the previous equation by adjusting the size of the lattice so that it accommodates (i) {NSolvent þ NPolymer} molecules for the mixture, (ii) NPolymer molecules in the absence of solvent, and (iii) NSolvent molecules in the absence of polymer. Hence, the same number of polymer or solvent molecules is present in solution and in the respective “pure states.” Since the multiplicity of states V was based on the placement of polymer chains on the lattice, without consideration of the solvent, the previous two equations are not applicable to calculate the combinatorial entropy when the polymer is absent. As illustrated by two previous lattice examples in this chapter, stuctureless solvent molecules do not contribute to the entropy of mixing because there is only one way to insert them in the lattice, both in the pure state and in solution. This convention is adopted here. Realizing that the total number of lattice sites is either (i) NSolvent þ xNPolymer for the mixture or (ii) xNPolymer in the absence of solvent, one obtains the following expression for the extensive entropy of mixing: DSmixing ¼ S(NPolymer , NSolvent )  S(NPolymer , NSolvent ¼ 0)  S(NPolymer ¼ 0, NSolvent )    z1  ln(NPolymer !) ¼ k (x  1)NPolymer ln NSolvent þ xNPolymer (N )# Polymer Y1  x NSolvent þ x NPolymer  i þln i¼0

 z1  ln(NPolymer !)  k (x  1)NPolymer ln xNPolymer (N )# Polymer Y1 x xðNPolymer  iÞ þln 

i¼0



8 YN 93 Polymer 1 x= < (N þ x[N  i]) Solvent Polymer i¼0 5 ¼ k 4(x  1)NPolymer ln wPolymer þ ln YNPolymer 1 x : ; x(N  i) Polymer i¼0 2

3.5 Chemical Stability of Binary Mixtures

89

The denominator of the logarithmic term in the previous equation can be simplified with assistance from Stirling’s approximation for the factorial of large numbers: NPolymer Y1



x(NPolymer  i)

x

¼ [xNPolymer (NPolymer )!]x

i¼0

(N

Polymer 1

ln

Y 

x(NPolymer  i)

x

) ¼ xNPolymer ln x þ x ln(NPolymer !)

i¼0

¼ xNPolymer fln(xNPolymer )  1g Once again, the combinatorial entropy of mixing does not depend on coordination number z of the hypothetical lattice used to obtain thermodynamic results. The following expression for DSmixing is useful in Sections 5.3.2 and 5.5 when the compositional dependence of the glass transition temperature is analyzed for trace amounts of plasticizer, where the slope of Tg versus diluent concentration is largest: " DSmixing ¼ k (x  1)NPolymer ln wPolymer 93 8 YN Polymer 1 x= < (N þ x[N  i]) Solvent Polymer i¼0 5 þ ln YNPolymer1  x ; : x(N  i) Polymer i¼0 "   ¼ k (x  1)NPolymer ln wPolymer  xNPolymer ln(xNPolymer )  1

þx

NPolymer X1



ln NSolvent þ x(NPolymer  i)



#

i¼0

For concentrated polymer solutions, Flory – Huggins analysis of the extensive combinatorial entropy of mixing can be written concisely as DSmixing ¼ f (NPolymer , NSolvent ; x) where the generalized function f is temperature independent.

3.5

CHEMICAL STABILITY OF BINARY MIXTURES

Homogeneous binary mixtures that do not exhibit phase separation impose a few restrictions on the Gibbs free energy and its concentration dependence. One of the necessary conditions for miscibility is that the Gibbs free energy of a mixture should be less than a weighted sum of pure-component Gibbs free energies. In other words, DGmixing must be negative. However, this condition is not sufficient to achieve a single homogeneous phase. This section focuses on the chemical stability of binary mixtures via thermodynamic analysis of the Gibbs free energy. Consider

90

Chapter 3 Lattice Theories for Polymer– Small-Molecule Mixtures

Ni moles of pure component i. If each component exists independently as a pure single phase, then the phase rule (i.e., f ¼ c 2 p þ 2 ¼ 2) suggests that extensive properties like the Gibbs free energy of pure component i, Gi,pure, enjoy three degrees of freedom, where an additional degree of freedom is required because extensive functions depend on total mass of the system. If temperature T, pressure p, and mole numbers Ni are chosen as three independent variables for a unique description of Gi,pure, then Euler’s integral theorem for thermodynamic state functions that are homogeneous to the first degree with respect to their extensive independent variables yields the following result [Belfiore, 2003, Ch. 29]:

 @Gi,pure ¼ Ni mi,pure (T, p) Gi,pure (T, p, Ni ) ¼ Ni @Ni T,p where mi,pure(T, p) is the chemical potential of pure component i in its reference state. Intensive thermodynamic properties like mi,pure, which is equivalent to the molar Gibbs free energy of pure component i, are homogeneous functions of the zeroth order with respect to molar mass. Hence, Euler’s theorem reveals that mi,pure is not a function of Ni. This is consistent with the phase rule, which predicts that only two degrees of freedom are required for a unique description of mi,pure. If an r-component homogeneous mixture contains Ni moles of component i, then the phase rule (i.e., f ¼ r 2 1 þ 2 ¼ r þ 1) indicates that r þ 2 degrees of freedom are required for a unique description of extensive properties. Euler’s integral theorem focuses on the extensive independent variables, yielding the following expansion in terms of partial molar properties for the Gibbs free energy of the mixture (i.e., intensive variables, like T and p, are not included in the expansion):

 r r X X @Gmixture Ni ¼ Ni mi (T, p, composition) Gmixture (T, p, all Ni ) ¼ @Ni T,p,all N j[ j=i] i¼1 i¼1 where mi is the chemical potential of component i in the mixture. The extensive Gibbs free energy of mixing is constructed as follows: DGmixing ¼ Gmixture 

r X

Gi,pure

i¼1

¼

r X

n o Ni mi (T, p, composition)  mi,pure (T, p)

i¼1

P Division by the total number of moles of all components, NTotal ¼ 1 jr fNjg, yields the molar Gibbs free energy of mixing: r n o DGmixing X ¼ yi mi (T, p, composition)  mi,pure (T, p) Dgmixing ¼ NTotal i¼1 where yi is the mole fraction of component i in the mixture. With the aid of activity coefficient correlations, the previous equation is useful to generate graphs of Dgmixing versus mole fraction of either component in binary mixtures at constant temperature and pressure. Chemical stability analysis of these graphs is discussed below.

3.5 Chemical Stability of Binary Mixtures

91

3.5.1 Shape of Dgmixing versus Composition in Binary and Multicomponent Mixtures Based on the previous equation, the Gibbs free energy of mixing, per total moles of both components in binary mixtures, is Dgmixing ¼ (1  y2 )fm1  m1,pureg þ y2 fm2  m2,pureg and the instantaneous slope of Dgmixing versus the mole fraction of component 2, y2, at constant temperature and pressure is calculated as follows:





 @ @ m1 @ m2 Dgmixing ¼ fm2  m2,pureg  fm1  m1,pureg þ y1 þ y2 @y2 T, p @y2 T, p @y2 T, p The last two terms on the right side of the previous equation cancel because y1 dm1 þ y2 d m2 ¼ 0 via the Gibbs – Duhem equation at constant temperature and pressure [Belfiore, 2003, Ch. 29]. Hence,

 n o n o @ Dgmixing ¼ m2  m2,pure  m1  m1,pure @y2 T, p If one introduces activities ai and activity coefficients gi such that, at constant T and p, ai (T, p, composition) ¼ yi gi (T, p, composition)

mi  mi,pure (T) ¼ RT ln ai (T, p, composition) where mi,pure(T) is a pure-component chemical potential in the reference state at one atmosphere total pressure, then it is possible to evaluate the slope of Dgmixing versus composition (i.e., mole fraction of species 2, y2) in the concentration limits because



 @ a2 Dgmixing ¼ RT ln a1 @y2 T,p lim fai g ¼ 1; lim fai g ¼ 0

lim

y2 )1



yi )1

@ Dgmixing @y2

@ Dgmixing lim y2 )0 @y2



yi )0

¼ þ1 T,p



¼ 1 T,p

Hence, if Dgmixing is plotted versus y2, then the graph begins at pure component 1 with Dgmixing ¼ 0 and an infinitely negative slope, and culminates at pure component 2 with Dgmixing ¼ 0 and an infinitely positive slope. Some consequences of this result are that (i) Dgmixing must be negative near the compositional limits for all mixtures that achieve thermodynamic equilibrium, and (ii) when mixtures separate into distinct phases, these phases can be highly concentrated in one of the components, but

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pure-component phases are thermodynamically disallowed. These results can be extended to multicomponent mixtures. Begin with the molar Gibbs free energy of mixing, recast in terms of activities: r r n o X X yi mi (T, p, composition)  mi,pure (T, p) ¼ RT yi ln ai Dgmixing ¼ i¼1

i¼1

If one envisions a multidimensional plot and focuses on the slope of Dgmixing with respect to the mole fraction of component k, then it is necessary to vary the mole fraction of one other component (e.g., yr). Changes in yr are not independent, but they are equal and opposite to the changes in yk to ensure that all mole fractions sum to unity. The following partial derivative is of interest: " #





 r X @ @yi @ ln ai þyi Dgmixing ¼ RT ln ai @yk @yk T,p,all y j[ j=k,r] @yk T,p,all y j[ j=k,r] T,p,all y j[ j=k,r] i¼1 and the second term in the summation of the previous equation vanishes via the Gibbs –Duhem equation at constant temperature and pressure [Belfiore, 2003, Ch. 29]: r X

yi d mi ¼ RT

i¼1

r X

yi d ln ai ¼ 0

i¼1

Hence,

@ Dgmixing @yk

 ¼ RT

r X

T,p,all y j[ j=k,r]

ln ai

i¼1

@yi @yk

 T,p,all y j[ j=k,r]

and yk þ yr þ

r1 X

yj ¼ 1

j¼1[ j=k]

Since all mole fractions in the summation of the previous equation remain constant during differentiation with respect to yk, it follows that dyr ¼ 2dyk, and one obtains the following mole fraction derivatives: 8

 < 1; i ¼ k @yi ¼ 1; i ¼ r @yk T,p,all y j[ j=k,r] : 0; i = k, r The compositional dependence of Dgmixing in multicomponent mixtures is



 @ ak Dgmixing ¼ RT ln ar @yk T,p,all y j[ j=k,r] This slope is (i) infinitely positive in the limit of pure component k (i.e., yk ) 1, ak ) 1, yr ) 0, ar ) 0) and (ii) infinitely negative in the limit of extremely dilute mixtures of component k (i.e., yk ) 0, ak ) 0, yr . 0, ar . 0). Hence, if phase separation is inevitable and thermodynamic equilibrium is achieved, then multicomponent mixtures will not separate into pure-component phases because a lower Dgmixing can be achieved if the phases are slightly impure.

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3.5.2 Intercepts and Common Tangents to Dgmixing versus Composition in Binary Mixtures As illustrated in the previous subsection, Euler’s integral theorem and the Gibbs – Duhem equation provide the tools to obtain expressions for Dgmixing, per total moles of both components, and (@Dgmixing/@y2)T,p in binary mixtures. This information allows one to evaluate the tangent line at any mixture composition via the point-slope formula. For example, if m1 ¼ m1 and m2 ¼ m2 when the mole fraction of component 2 is y2, then Dgmixing ( y2 ¼ y2 ) ¼ (1  y2 )fm1  m1,pureg þ y2 fm2  m2,pureg

     @ Dgmixing ¼ m2  m2,pure  m1  m1,pure @y2 T,p,y2 ¼y 2

The Taylor series expansion for the tangent line at y2 is truncated after the first-order term without introducing any error:

 @ Dgmixing tangent( y2 ; y2 ) ¼ Dgmixing (y2 ¼ y2 ) þ [y2  y2 ] @y2 T,p,y2 ¼y2     ¼ (1  y2 ) m1  m1,pure þ y2 m2  m2,pure     þ ( y2  y2 ) m2  m2,pure  m1  m1,pure where tangent( y2; y2) represents a linear function of y2 that is tangent to Dgmixing versus composition at mole fraction y2. Simplification of the previous equation yields     tangent(y2 ; y2 ) ¼ (1  y2 ) m1  m1,pure þ y2 m2  m2,pure Evaluation of the tangent line at the pure-component intercepts provides useful information about the chemical potentials of both components in the mixture at composition y2. For example, tangent( y2 ¼ 0; y2 ) ¼ m1  m1,pure tangent( y2 ¼ 1; y2 ) ¼ m2  m2,pure Chemical stability of binary mixtures is addressed via the shape of Dgmixing versus composition. The tangent line is critical in this analysis because the pure-component intercepts of a common tangent that contacts Dgmixing versus y2 at two different points on the curve provide the conditions for chemical equilibrium of a two-phase mixture. Homogeneous single-phase behavior occurs at all mixture compositions when both of the following conditions are satisfied: Condition 1: Dgmixing , 0

2  @ Dgmixing .0 Condition 2: @y22 T, p

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Chapter 3 Lattice Theories for Polymer– Small-Molecule Mixtures

If Condition 1 is violated, then the mixture splits into at least two phases. However, mixtures that achieve thermodynamic equilibrium will not violate Condition 1 near pure-component boundaries, because the slope of Dgmixing versus y2 is infinitely negative near pure component 1, and infinitely positive near pure component 2. This can only occur if Dgmixing is negative near the pure-component boundaries since, by definition, Dgmixing ¼ 0 when y2 ¼ 0 and y2 ¼ 1. The most interesting situations occur when Condition 1 is satisfied and a region of negative curvature exists between two regions of positive curvature, such that (@ 2 Dgmixing =@y22 )T, p changes sign smoothly. Under these conditions, mixtures exhibit composition-dependent miscibility and one must consider stable, metastable, and unstable states that are separated by binodal and spinodal points, respectively. These terms are defined below within the context of chemical stability for binary mixtures. Spinodal points represent the boundary between positive and negative curvature of Dgmixing versus y2 at constant T and p. An equilibrium state on the spinodal curve is defined by (@ 2 Dgmixing =@y22 )T, p ¼ 0. Regions between the spinodal points, where (@ 2 Dgmixing =@y22 )T, p , 0, are intrinsically unstable and violate the second criterion of chemical stability. If one graphs Dgmixing versus y2 isothermally at several different temperatures, then the locus of spinodal points is known as the spinodal curve on temperature– composition axes. The phrase spinodal decomposition describes the kinetics of phase separation within the spinodal region where binary molecular diffusion coefficients are negative, such that mass transfer of species 2 occurs from the phase that is more diluted with respect to species 2 toward the phase that is more concentrated with respect to species 2. Hence, spinodal decomposition describes a mechanism for the kinetics of phase separation where diffusion occurs in the direction of increasing concentration gradient, but decreasing chemical potential or decreasing Dgmixing. Binodal points represent the points of contact of a common tangent to Dgmixing versus y2 at constant T and p when a region of negative curvature exists between two regions of positive curvature. If one graphs Dgmixing versus y2 isothermally at several different temperatures, then the locus of binodal points is known as the binodal curve on temperature– composition axes, or the two-phase envelope, which represents the experimentally observed phase boundary under normal conditions. The binodal region exists between the binodal and spinodal curves, where (@ 2 Dgmixing =@y22 )T, p . 0. Mixtures between the binodal and spinodal curves are intrinsically stable with respect to spontaneous phase separation, because the curvature stability criterion is satisfied, but these mixtures can achieve the lowest Dgmixing if they separate into more than one phase, as identified by points of contact of the common tangent to Dgmixing versus composition. The phase separation mechanism in this regime between the binodal and spinodal curves is described by nucleation and growth. Stable states exist when Dgmixing versus y2 exhibits positive curvature at constant T and p. Hence, (@ 2 Dgmixing =@y22 )T, p . 0 and it is not possible for mixtures to achieve a lower Dgmixing by splitting into more than one phase. Stable equilibrium states exist

3.5 Chemical Stability of Binary Mixtures

95

outside the binodal region where both requirements of chemical stability are satisfied and single-phase behavior prevails. Unstable states exist when Dgmixing versus y2 exhibits negative curvature at constant T and p, between the spinodal points. In this region, (@ 2 Dgmixing =@y22 )T,p , 0. Single-phase equilibrium states of this nature are completely disallowed, even if the first stability criterion is satisfied. Metastable states exist in the binodal region, between the binodal and spinodal curves, where (@ 2 Dgmixing =@y22 )T,p . 0, but mixtures can achieve a lower Dgmixing by splitting into more than one phase. When a region of negative curvature exists between two regions of positive curvature on a graph of Dgmixing, per total moles of both components, versus y2, and composition-dependent miscibility prevails, the points of contact of the common tangent (i.e., binodal points) identify two different phases (i.e., a and b) that are in thermodynamic equilibrium. Since chemical stability is analyzed at constant T and p, both coexisting phases exhibit the same temperature (i.e., Ta ¼ Tb) and the same pressure in the absence of external fields (i.e., pa ¼ pb). These are requirements for thermal and mechanical equilibrium, respectively. The requirement of chemical equilibrium for component k in a two-phase mixture is

mk (phase a) ¼ mk (phase b) Since (i) states a and b correspond to binodal points and (ii) the common tangent, by definition, not only implies that these states have the same slope but also share common intercepts at y2 ¼ 0 and y2 ¼ 1, the requirement for chemical equilibrium is satisfied via the properties of the tangent line to Dgmixing versus composition. For all metastable and unstable states between two binodal points on a common tangent line, single-phase mixtures achieve a lower Dgmixing by splitting into phases a and b. All properties of two-phase mixtures, which lie on the common tangent to Dgmixing versus composition between points a and b, are obtained from a linear combination of the properties of phase a and phase b. Eubank and Barrufet [1988] discuss simple algorithms for the calculation of phase separation. This publication addresses Dgmixing versus composition at constant T and p for binary mixtures that exhibit the following properties: (i) Four spinodals, four thermodynamically allowed binodal points, doublephase separation (i.e., a/b and g/d), and one global minimum in Dgmixing. (ii) Four spinodals, three local minima in Dgmixing, two thermodynamically allowed binodal points, single-phase separation (i.e., a/b), and one thermodynamically disallowed binodal point. Tanford [1961] illustrates the coalescence of spinodal and binodal points in partially miscible mixtures at the critical temperature, where the spinodal and binodal curves are tangent to each other. If this phenomenon occurs upon raising the temperature, then the critical point is identified as an upper critical solution temperature (UCST), and homogeneous single-phase behavior exists above the UCST.

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Chapter 3 Lattice Theories for Polymer– Small-Molecule Mixtures

If coalescence of spinodal and binodal points occurs upon lowering the temperature, then the critical point is identified as a lower critical solution temperature (LCST), and homogeneous single-phase behavior prevails below the LCST. Olabisi et al. [1979] illustrate some interesting temperature – composition phase diagrams that exhibit UCSTs, LCSTs, combinations of these two critical points, and “hourglass” shaped phase behavior. Van der Put [1998] illustrates correspondences between (i) the compositional dependence of the Gibbs free energy of mixing and (ii) the temperature– composition phase diagram for a binary mixture that exhibits single eutectic response, at temperatures that are above, below, and coincident with the eutectic temperature. Macromolecular conformational changes that occur in response to temperature variations above and below the LCST or UCST might be useful for biomedical applications when the critical point is in the vicinity of physiological temperatures (i.e., 37 8C). This phenomenon, which is thermally reversible, has been exploited in chemically crosslinked hydrogels based on poly(N-isopropylacrylamide) smart materials for drug delivery and controlled release. At low mass fractions of polymer in lightly crosslinked hydrogels, the conformational characteristics of network strands between crosslink junction points in the absence of stress (i.e., unstretched state) could be similar to the Gaussian-like behavior of linear chains in aqueous solution such that there is a direct correlation between the LCST of dilute solutions and temperature-dependent swelling of smart hydrogels [Zeni, 2010].

3.5.3 The Curvature Criterion for Chemical Stability of Binary Mixtures via the Compositional Dependence of the Chemical Potential Begin with the slope of Dgmixing, per total moles of both components, versus mole fraction of component 2 from Section 3.5.1 and construct the curvature requirement for chemical stability as follows:

     @ Dgmixing ¼ m2  m2,pure (T, p)  m1  m1,pure (T, p) @y2 T,p



2  @ @ Dgmixing ¼ (m  m 1 ) . 0 @y2 2 @y22 T,p T,p Now, differentiate the Gibbs – Duhem equation for binary mixtures [Belfiore, 2003, Ch. 29] with respect to y2 at constant temperature and pressure to evaluate (@ m1/ @y2)T,p and simplify the curvature criterion for chemical stability. For example, y1 dm1 þ y2 d m2 ¼ 0



 @ m1 y2 @ m2 ¼ @y2 T, p y1 @y2 T, p

3.5 Chemical Stability of Binary Mixtures

97

Binary mixtures exist as a single homogeneous phase when the following condition is satisfied:

2 







 @ @ y2 @ m2 1 @ m2 Dgmixing ¼ (m  m 1 ) ¼ 1 þ ¼ .0 y1 @y2 T,p y1 @y2 T,p @y2 2 @y22 T,p T,p One arrives at the same result by analyzing the curvature (i.e., second derivative) of the Gibbs free energy of binary mixtures, per total moles of both components, with respect to the mole fraction of either component, instead of Dgmixing. Since y1 . 0, chemical stability in binary mixtures requires that the chemical potential (and activity) of each component must increase as the system becomes more concentrated with respect to the same component at constant temperature and pressure. In other words, if the following conditions are violated,



 @ m1 @ m2 . 0; .0 @y1 T,p @y2 T,p then binary mixtures split into two phases of different composition (i.e., either liquid– liquid or solid – solid equilibrium). At constant T and p, chemical potentials and activities are related by

m1 (T, p, y1 ) ¼ m1,pure (T, p) þ RT ln a1 (T, p, y1 ) Now, the criterion for chemical stability at constant T and p can be expressed in terms of the compositional dependence of activities. For example,



 @ m1 @ ln a1 ¼ RT .0 @y1 T,p @y1 T,p

3.5.4 Chemical Stability of Polymer –Solvent Mixtures on the Flory –Huggins Lattice Begin with the solvent’s chemical potential in polymer solutions from Section 3.4.5 and apply the previous inequality that guarantees the existence of single-phase behavior. One identifies mixture compositions on the spinodal curve when the inequality becomes an equality. Phase separation occurs via spinodal decomposition when the inequality is reversed (i.e., the greater than sign in the previous equation is replaced by a less than sign) and the chemical potential of either species in binary mixtures decreases at higher concentrations of the same species. Chemical stability criteria have been developed in terms of the dependence of chemical potentials and the molar Gibbs free energy of mixing on mole fractions, whereas the solvent’s chemical potential on the Flory – Huggins lattice is expressed most conveniently in terms of the volume fraction of polymer, wPolymer. Since the polymer’s volume fraction decreases when the solvent’s mole fraction increases, the previous curvature criterion of chemical stability is reformulated as follows: ( ) @ mSolvent ,0 @ wPolymer T,p

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Chapter 3 Lattice Theories for Polymer– Small-Molecule Mixtures

mSolvent (T, p, wPolymer )  m0Pure Solvent (T, p)   o n 1 ¼ RT ln(1  wPolymer ) þ 1  wPolymer þ xw2Polymer x 1 RT

(

@ m Solvent @ w Polymer

) ¼ T, p

1 1 þ 1  þ 2xwPolymer , 0 1  wPolymer x

Hence, there is an upper limit of the dimensionless polymer – solvent interaction free energy of mixing x, imposed by chemical stability analysis, to guarantee homogeneous single-phase behavior of polymer solutions. The previous inequality reveals that the value of x at the critical point depends on the number of monomers x per chain (see Table 3.1). For the Flory – Huggins lattice model, this is described as an upper critical solution temperature, because single-phase behavior occurs at higher temperatures where the interaction parameter is less than the smallest value allowed by the previous composition-dependent inequality. For very high molecular weight chains (i.e., x ) 1), the critical value of the interaction parameter x is 0.5 for infinitely dilute solutions (i.e., wPolymer ) 0):

x,

1  f1  1=xg(1  wPolymer ) x1 1 1 ) ) x1 Dilute 2(1  wPolymer ) 2 2wPolymer (1  wPolymer ) Solution wPolymer ) 0

The upper limit of the dimensionless interaction free energy of mixing, xCritical, and the corresponding polymer volume fraction wPolymer,Critical at the critical point, are summarized below in Table 3.1 as a function of chain length x for binary mixtures on the Margules (i.e., x ¼ 1) and Flory – Huggins (i.e., x . 1) lattices. Binary mixtures will not exhibit phase separation at any overall mixture composition when the interaction parameter is less than xCritical. Values of xCritical and wPolymer,Critical in Table 3.1 can be obtained by identifying polymer concentrations that yield minimum values of the interaction parameter as a function of chain length x in the previous inequality. Rigorously, one invokes the thermodynamic criteria at the critical point, in terms of the compositional dependence of the chemical potential, and solves two equations for xCritical and wPolymer,Critical. For example, the first and second derivatives of mSolvent with respect to wPolymer must vanish at the critical point, the highest temperature and minimum value of x at which one predicts two-phase behavior (i.e., for an upper critical solution temperature), where both phases exhibit the same composition given by wPolymer,Critical. This is analogous to the requirement that the first and second derivatives of pressure with respect to volume must vanish along the critical p 2V isotherm, from the viewpoint of mechanical stability. Chemical stability requires that the solvent’s chemical potential must decrease at higher concentrations of polymer in binary solutions. The critical point occurs when binodal and spinodal points converge on a graph of the compositional dependence of the Gibbs free energy of mixing in binary systems.

3.5 Chemical Stability of Binary Mixtures

99

Table 3.1 Effect of Chain Length x, or the Number of Monomers per Chain, on the Flory–Huggins Interaction Parameter and the Polymer Volume Fraction at the Critical Point of Binary Polymer– Solvent Mixtures Chain length x

xCritical

wPolymer,Critical

2.00 1.46 1.24 1.13 1.05 0.87 0.79 0.72 0.65 0.62 0.61 0.57 0.55 0.53 0.51

0.50 0.41 0.37 0.33 0.31 0.24 0.21 0.17 0.12 0.10 0.09 0.07 0.04 0.03 0.01

1 2 3 4 5 10 15 25 50 75 100 200 500 1000 10000

The appropriate equations that define the critical point are ( ) @ mSolvent 1 1 ¼0 )  þ 1  þ 2xwPolymer ¼ 0 @ wPolymer 1  wPolymer x T,p ( ) @ 2 mSolvent 1 ¼0 )  þ 2x ¼ 0 @ w2Polymer (1  wPolymer )2 T,p

Simultaneous solution of these two equations yields xCritical and wPolymer,Critical in terms of the number of monomers x per chain, as follows:

xCritical

 1 1 2 1 þ pffiffiffi ¼ 2 x

wPolymer,Critical ¼

1 pffiffiffi 1þ x

As illustrated in Table 3.1 and supported by the previous two equations, xCritical ¼ 2 and wPolymer,Critical ¼ 50% for regular solutions of equisized molecules (i.e., x ¼ 1) on the Margules lattice. For very high molecular weight chains (i.e., x ) 1), xCritical ) 0.5 and wPolymer,Critical ) 0 on the Flory – Huggins lattice.

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Chapter 3 Lattice Theories for Polymer –Small-Molecule Mixtures

Question: Qualitatively describe the stable morphologies that are possible in A– B diblock copolymers when phase separation occurs upon lowering the temperature from the molten state. ANSWER: There are four possible morphologies of block copolymers (lamellae, cylinders, spheres, and gyroids) for the simplest case of A –B diblock copolymers. An O70 orthorhombic network phase was predicted recently [Tyler and Morse, 2005] in a narrow window for A –B diblocks when the volume fraction of one of the blocks ranges from 42% to 48%. This bicontinuous network phase is observed more readily in experiments on A –B –C triblocks, relative to A –B diblocks. In symmetric diblock copolymers with equivalent segment lengths and equal volume fractions for both blocks, phase separation with lamellar domains occurs when xABN ¼ 10.5, where N is the total number of copolymer segments and xAB is the Flory –Huggins binary interaction parameter. For more complex chain structures such as A –B –C triblock copolymers, all of the stable phases are not known yet, because only a few have been observed experimentally (some of these morphologies could represent metastable phases) and theoretical predictions based on free-energy minimization are computationally intensive (i.e., there are five degrees of freedom; volume fractions for two of the three segments, and three binary interaction parameters, xAB, xBC, xAC). Bates and Fredrickson [1999] provide an excellent introduction to phase diagrams and morphology in block copolymers.

3.5.5 Strategy to Determine if Phase Separation Occurs For monodisperse polymer chains, the Flory – Huggins lattice was used to develop thermodynamic properties of polymer solutions when the molar volume of the solvent matches the molar volume of one segment of the polymer. Since each segment of the polymer chain represents a repeat unit, the ratio of the molar volume of the polymer to that of the solvent corresponds to the degree of polymerization x, or the number of segments per chain. For a given number of segments x per chain and Flory – Huggins x parameter, polymer solutions will exhibit homogeneous single-phase behavior at all compositions if the dimensionless polymer – solvent interaction parameter is less than xCritical, as summarized in Table 3.1. Phase separation will occur somewhere on the temperature – composition phase diagram when x . xCritical such that binary mixtures split into two phases whose compositions lie on the binodal curve. As discussed in Section 3.5.2, pairs of equilibrium states on the binodal curve are identified as points of contact of a common tangent to Dgmixing, per total moles of both components, versus mole fraction at constant temperature and pressure, such that a weighted average of thermodynamic properties of the two-phase mixture yields a local minimum of Dgmixing for any overall mixture composition that lies between two binodal points. Graphs of Dgmixing, per total moles of both components in the binary mixture, versus mole fraction provide the most straightforward approach to identify points on the binodal curve and the composition of the coexisting phases (see Figure 4.1 in Section 4.6.7). Numerically, one identifies these binodal points by searching the complete phase space of polymer mole fractions yPolymer until two different compositions at polymer mole fractions yPolymer #1 and yPolymer #2 share a common tangent to Dgmixing versus composition with the same slope (i.e., (@Dgmixing/@yPolymer)T,p). Hence, one solves two nonlinear algebraic

3.5 Chemical Stability of Binary Mixtures

101

equations for yPolymer #1 and yPolymer #2 such that, at these two compositions of the binary mixture, the slope of Dgmixing versus yPolymer is the same as the slope of the tangent line that connects these two points. The relevant equations are summarized below. In some cases that correspond to extremely high-molecular-weight polymers, it could be necessary to prevent nonlinear equation solvers from gravitating toward yPolymer #1 ¼ yPolymer #2, instead of identifying an essentially pure-solvent phase, because the shape of Dgmixing might not be captured with sufficient accuracy near the pure-component limits:  Dgmixing ¼ RT (1  yPolymer )ln wSolvent þ yPolymer ln wPolymer  þx(1  yPolymer )wPolymer



 (1  yPolymer ) d wSolvent 1 @ Dgmixing ¼  ln wSolvent þ wSolvent dyPolymer RT @yPolymer T,p

 yPolymer dwPolymer þ ln wPolymer þ wPolymer dyPolymer

 d wPolymer  xwPolymer þ x(1  yPolymer ) dyPolymer xyPolymer wPolymer ¼ 1  yPolymer þ xyPolymer d wPolymer wPolymer wSolvent x ¼ ¼ 2 dyPolymer (1  yPolymer þ xyPolymer ) yPolymer ySolvent

wSolvent ¼

1  yPolymer ¼ 1  wPolymer 1  yPolymer þ xyPolymer

dwPolymer dwSolvent ¼ dyPolymer dyPolymer



 @ @ Dgmixing ¼ Dgmixing @yPolymer @yPolymer T,p,@yPolymer #1 T,p,@yPolymer #2 ¼

Dgmixing (@yPolymer #2 )  Dgmixing (@yPolymer #1 ) yPolymer #2  yPolymer #1

Application of the previous set of equations to binary mixtures of (i) equisized molecules (i.e., regular solutions with x ¼ 1) on the Margules lattice and (ii) monomers and dimers (i.e., x ¼ 2) on the Flory – Huggins lattice yields the phase compositions (i.e., mole fractions when x ¼ 1 and volume fractions when x ¼ 2) shown in Figures 3.1 and 3.2 as the dimensionless interaction free energy of mixing x increases systematically. There are two spinodal points for each mixture investigated because the smallest value of the x parameter on the horizontal axes of both graphs exceeds its critical value for x ¼ 1 and x ¼ 2, as summarized in Table 3.1.

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Chapter 3 Lattice Theories for Polymer –Small-Molecule Mixtures

Mole Fraction of Component 1

1.0 0.8

Component 1-rich phase 0.6 0.4

Component 2-rich phase 0.2 0.0 2.0

2.2 2.4 2.6 2.8 Flory–Huggins (i.e., Margules) Parameter, c

3.0

Figure 3.1 Effect of the Flory–Huggins or the Margules energetic interaction parameter x on the

1.0

1.0

0.8

0.8 Dimer-rich phase

0.6

0.6

0.4

0.4

0.2 0.0 1.5

Monomer-rich phase

1.7 1.9 2.1 2.3 Flory–Huggins Interaction Parameter, c

0.2

Dimer Volume Fraction

Dimer Volume Fraction

composition of each phase in two-phase mixtures of equisized small molecules (i.e., regular solutions). Both types of molecules occupy the same number (i.e., 1) of lattice sites. Homogeneous single-phase behavior occurs when x , 2, for any overall composition of the binary mixture. When phase separation occurs, the phases become purer in their respective components as the magnitude of x increases.

0.0 2.5

Figure 3.2 Effect of the Flory–Huggins energetic interaction parameter x on the composition of each phase in two-phase mixtures of monomers and dimers, where each dimer occupies twice as many lattice sites relative to a monomer. Homogeneous single-phase behavior occurs when x , 1.46, for any overall composition of the binary mixture. When phase separation occurs, the monomer-rich phases become extremely pure (i.e., much purer than the dimer-rich phase) as the magnitude of x increases.

3.5 Chemical Stability of Binary Mixtures

103

A summary of the phase behavior for binary mixtures of monomers and dimers with x ¼ 2 reveals that there are two spinodal points at dimer mole fractions corresponding to yDimer ¼ 0.10 and 0.53, where f@ mMonomer/@yDimergT,p ¼ 0. Points of contact of the common tangent to Dgmixing/RT versus yDimer identify two binodal points at yDimer ¼ 0.03 and 0.74, where the slope of the tangent line is approximately 20.2 (dimensionless). Hence, this binary mixture splits into a monomer-rich phase at yDimer ¼ 0.03 and a dimer-rich phase at yDimer ¼ 0.74 for all overall mixture compositions between these two values of yDimer. It is extremely important to emphasize that the curvature requirement for chemical stability of binary mixtures, as well as the identification of binodal and spinodal points when phase separation is favored, must be determined from an analysis of either (i) the Gibbs free energy of mixing, per total moles of both components in binary mixtures, versus mole fraction of one of the components, or (ii) the Gibbs free energy of mixing, per mole or volume of lattice sites, versus volume fraction of one of the components. Incorrect predictions about chemical stability occur when one analyzes the Gibbs free energy of mixing, per total moles of both components in binary mixtures, versus volume fraction of one of the components. The lever rule, based on simple material balances for (i) total moles of the mixture (i.e., MolesTotal ) and (ii) moles of dimers, allows one to determine the relative fractions of both phases that coexist at equilibrium. For example, MolesTotal ¼ MolesDimerrich þ MolesMonomerrich phase

phase

y Dimer MolesTotal ¼ yDimerrich MolesDimerrich þ yDimerpoor MolesMonomerrich (0:74)

Overall

phase

(

MolesDimerrich y Dimer ¼ yDimerrich (0:74)

Overall

MolesTotal

¼

Overall (0:74)

MolesTotal

þ yDimerpoor 1 

phase

MolesTotal

y Dimer  0:03

(0:03)

¼

Overall

0:74  0:03

(0:03)

yDimerrich  y Dimer

MolesDimerrich ¼1

MolesDimerrich )

(0:03)

yDimerrich  yDimerpoor

MolesMonomerrich phase

MolesTotal

y Dimer  yDimerpoor

MolesDimerrich phase

phase

phase

(0:03)

phase

MolesTotal

¼

(0:74)

Overall

yDimerrich  yDimerpoor (0:74)

0:74  y Dimer ¼

Overall

0:74  0:03

(0:03)

3.5.6 Phase Equilibrium Relations at the Upper Critical Solution Temperature (UCST) The locus of binodal points maps out the binodal curve, whereas the spinodal curve is obtained by connecting all of the spinodal points. The binodal and spinodal

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Chapter 3 Lattice Theories for Polymer –Small-Molecule Mixtures

curves are tangent to each other when the binodal and spinodal points coalesce at the critical point.

3.5.7 Kinetics of Phase Separation Within the Spinodal Region: Spinodal Decomposition via the Unsteady State Diffusion Equation with Negative Binary Molecular Diffusion Coefficients The thermodynamics of irreversible processes, for systems that are not too far removed from equilibrium, yield linear laws between molecular fluxes and driving forces (i.e., gradients) whose tensorial ranks are the same or differ by an even integer. This classic theorem is known as the Curie restriction for isotropic systems, as proposed by Pierre Curie in 1903. Hence, molecular fluxes for heat and mass transfer are coupled via gradients in temperature and chemical potential of one of the components in binary mixtures. An important consequence of these linear laws is that binary molecular diffusion coefficients are proportional to the mass fraction derivative of the difference between chemical potentials for both components. For example, if r is the overall mass density of the mixture, aT is the Onsager coefficient that couples diffusional mass flux to gradients in the chemical potential of species A at temperature T, vA represents the mass fraction of species A, yA is the corresponding mole fraction of species A, aA is the activity of species A, and mA and mB are the corresponding chemical potentials of A and B, whose molecular weights are MWA and MWB, respectively, with MWmixture as the molecular weight of the mixture, then binary molecular diffusion coefficients DAB are given by [Belfiore, 2003]

 

 @ mA m a MWmixture RT @ ln aA rDAB ¼ a  B ¼ @ vA MWA MWB T,p MWA MWB vA vB @ ln yA T,p Hence, binary molecular diffusion coefficients must be positive for homogeneous single-phase binary mixtures under irreversible conditions. Phase separation within the spinodal region at constant temperature and pressure occurs spontaneously. Mixtures are unstable with respect to small time-dependent fluctuations in composition and there is no thermodynamic barrier to the growth of two separate phases. The kinetics of this process based on the theory of diffuse interfaces, known as spinodal decomposition, obeys a modified version of the classic unsteady state diffusion equation (i.e., Fick’s second law of diffusion) with negative binary molecular diffusion coefficients, DAB , 0, because the curvature requirement for chemical stability is not satisfied [Cahn, 1965]. Regions of negative curvature for Dgmixing versus composition between two spinodal points imply that the system achieves lower Dgmixing by splitting into two separate phases of different composition, whose properties lie on a straight line connecting these two states. In other words, Dgmixing is lower for two-phase mixtures on the straight line between two states relative to the homogeneous curve connecting these two points when the curvature is negative. Fick’s second law is modified for inhomogeneous binary solutions because (i) the free energy density is expanded as a polynomial in mole fraction (i.e., density gradient

3.6 Guggenheim’s Lattice Theory of Athermal Mixtures

105

expansion) with truncation after the second-order term, (ii) the free energy functional is constructed in terms of an integral of this density gradient expansion over the entire solution volume, (iii) the chemical potential difference between both components is obtained by differentiating the free energy functional with respect to mole fraction, (iv) linear laws from the irreversible thermodynamics of binary mixtures provide the connection between diffusional mass flux and chemical potential gradient, and (v) the unsteady state diffusion equation (i.e., mass balance) relates the accumulation rate process to the divergence of diffusional mass flux. The second-order term of the density gradient expansion for the free energy density of inhomogeneous binary mixtures is responsible for the fact that Fick’s second law contains an additional term. Of particular importance, diffusional mass flux of one of the components occurs in the direction of increasing concentration gradient of the same component as the system splits into two phases. Even though this statement seems to contradict physical intuition, diffusional mass flux proceeds in the direction of decreasing chemical potential gradient during the kinetics of phase separation as species A diffuses into a region of higher vA to lower the Gibbs free energy of mixing.

3.5.8 Phase Separation Between the Binodal and Spinodal Curves via Nucleation and Growth Kinetics Binary molecular diffusion coefficients are positive in this regime of phase separation because the curvature requirement for chemical stability is satisfied between the binodal and spinodal curves. Now, small fluctuations within this metastable region decay because homogeneous mixtures are intrinsically stable. Regions of positive curvature for Dgmixing versus composition between the binodal and spinodal curves imply that the system achieves lower Dgmixing by remaining on the curve that is characteristic of homogeneous mixtures, instead of adopting the properties of a two-phase mixture, which can be found on a straight line connecting these two states. In other words, Dgmixing is higher for two-phase mixtures on the straight line between two states relative to the homogeneous curve connecting these two points when the curvature is positive. Larger fluctuations in mixture composition induce the nucleation of a second phase that grows by normal diffusion, and the kinetics of phase separation, which includes nucleation and growth, is described by a mechanism that is similar to the kinetics of crystallization, as discussed in Chapter 7.

3.6 GUGGENHEIM’S LATTICE THEORY OF ATHERMAL MIXTURES 3.6.1 Classical Thermodynamic Analysis of the Gibbs Free Energy and Entropy of Mixing The primary objective of this section is to develop the methodology and final expressions for the combinatorial entropy of mixing DSmixing based on classical

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Chapter 3 Lattice Theories for Polymer –Small-Molecule Mixtures

thermodynamics of athermal mixtures and statistical considerations. Since there are no energetic effects associated with the mixing process, the extensive Gibbs free energy change is given by DGmixing ¼ T DSmixing Initially, the Gibbs free energy of mixing is developed by invoking classical concepts for binary mixtures. Then, a probabilistic interpretation is employed to evaluate the ratio of absolute activities for polymer and solvent, such that the classical result for the athermal entropy of mixing is re-evaluated from a combinatorial viewpoint. Consider a binary mixture that contains NSolvent molecules of monomer, or solvent, and NPolymer molecules of polymer, where each chain has r segments. Each segment of the polymer chain occupies the same volume as one of the solvent molecules. In this single phase mixture of r-mers and solvent, the phase rule indicates that there are three degrees of freedom for intensive properties. Hence, the extensive Gibbs free energy of the mixture is described completely by specifying four independent variables (T, p, NSolvent, NPolymer). One writes dGmixture ¼ Smixture dT þ Vmixture dp þ mSolvent dNSolvent þ mPolymer dNPolymer where S and V represent extensive entropy and volume, respectively, of the mixture. When the mixture is analyzed at constant temperature T and pressure p, and the chemical potential of species i per molecule, mi, in the mixture is written relative to its standard state chemical potential m0i (T) using absolute activity ai, the following expression defines the starting point to calculate the mixture’s extensive Gibbs free energy: dGmixture ¼ (m0Solvent þ kT ln aSolvent ) dNSolvent þ (m0Polymer þ kT ln aPolymer ) dNPolymer The numbers of solvent and polymer molecules are related to the polymer volume fraction w and the total number of lattice sites NTotal ¼ NSolvent þ rNPolymer, which remains constant in Guggenheim’s model of athermal mixtures that contain molecules of different sizes. The following expressions allow one to relate NSolvent, NPolymer, and their differential changes to w: rNPolymer ¼ wNTotal ; r dNPolymer ¼ w dNTotal þ NTotal d w NSolvent ¼ (1  w)NTotal ; dNSolvent ¼ (1  w) dNTotal  NTotal d w Differential changes in the mixture’s extensive Gibbs free energy are written in terms of dw when the total number of lattice sites NTotal remains constant (i.e., dNTotal ¼ 0);

 1 dGmixture ¼ NTotal (m0Polymer þ kT ln aPolymer )  (m0Solvent þ kT ln aSolvent ) dw r Upon integration from pure solvent, where w ¼ 0, to variable polymer concentration at polymer volume fraction w, one obtains the following extensive free energy

3.6 Guggenheim’s Lattice Theory of Athermal Mixtures

107

function for Gmixture(w, NTotal ): Gmixture (w, NTotal ) ¼ GSolvent (w ¼ 0, NTotal ) þ NTotal

 m0Solvent þ kT ln aSolvent

 

ðw

 1 0 mPolymer þ kT ln aPolymer r

0

dw

Now, it is possible to construct an expression for the Gibbs free energy of mixing by subtracting a weighted sum of terms for pure solvent and pure polymer from the previous equation for the extensive Gibbs free energy of binary mixtures. Volume fractions of solvent, 12 w, and polymer, w, represent the appropriate weighting factors for GSolvent and GPolymer, respectively, because the total number, or volume, of lattice sites does not change for either of the pure components or the binary mixture. For example, DGmixing ¼ Gmixture (w, NTotal )  wGPolymer (w ¼ 1, NTotal )  (1  w)GSolvent (w ¼ 0, NTotal ) ¼ GSolvent (w ¼ 0, NTotal )  ðw

 0  1 0 m þ kT ln aPolymer  mSolvent þ kT ln aSolvent dw þ NTotal r Polymer 0

"  w GSolvent (w ¼ 0, NTotal ) þ NTotal

ð1

 1 0 mPolymer þ kT ln aPolymer r

0

 #

0   mSolvent þ kT ln aSolvent d w  (1  w)GSolvent (w ¼ 0, NTotal ) All pure-component terms cancel in the previous calculation of DGmixing, including m0Polymer , m0Solvent , and GSolvent (w ¼ 0, NTotal ). Upon rearrangement, the extensive Gibbs free energy of mixing is simplified as follows: 2w 3   ð

ð1

aPolymer aPolymer NTotal 4 kT ln r dw  w ln r d w5 DGmixing ¼ r aSolvent aSolvent 0

0

The argument of the natural logarithm in the previous equation for DGmixing can be described as the ratio of two occupational probabilities for sites on the lattice. The numerator corresponds to the occupational probability of a group of r-interconnected sites by a single linear polymer chain. The denominator describes the occupational probability of the same group of r-interconnected sites by r solvent molecules. Division of the previous equation for DGmixing by the total number of molecules yields the Gibbs free energy of mixing per molecule. Then, one arrives at the molar

108

Chapter 3 Lattice Theories for Polymer –Small-Molecule Mixtures

Gibbs free energy of mixing Dgmixing via multiplication by Avogadro’s number. Since each lattice site contains sufficient volume for one solvent molecule or one segment of a polymer chain, and each linear chain consists of r segments, the total number of molecules in the binary mixture is Total number of molecules ¼ (1  w)NTotal þ

w NTotal r

The molar Gibbs free energy Dgmixing and entropy Dsmixing of mixing are calculated in dimensionless form as follows: Dsmixing Dgmixing ¼ R RT

2w 3   ð

ð1

1 4 ln aPolymer d w  w ln aPolymer dw5 ¼ w þ r(1  w) arSolvent arSolvent 0

0

3.6.2 Occupational Probabilities of Two Adjacent Lattice Sites for Binary Mixtures of Monomers and Dimers The primary objective of this section is to illustrate the combinatorial aspects of placing two types of molecules on a lattice, which allows one to evaluate the argument of the natural logarithm in the previous equation for Dgmixing and Dsmixing. There are N1 monomer molecules and N2 dimer molecules that occupy a lattice which contains a fixed total number of sites given by NTotal ¼ N1 þ 2N2. Consider one pair of adjacent sites on the lattice that can be occupied or vacant. This pair of sites has a few restrictions that do not apply to any two sites on the lattice that are chosen randomly. The frequency of occupation of any given site by monomer or dimer is proportional to their volume fractions, 12 w or w, respectively. The occupational probability of the adjacent site, which is restricted, depends on the fraction of sites on the lattice that are nearest neighbors to monomers and dimers. The following possibilities exist. (i) Both sites are vacant and, eventually, they will be occupied by monomers. (ii) Both sites are occupied by one dimer. (iii) One site is occupied by monomer, and the other site is occupied by one segment of a dimer. The second segment of the dimer occupies a site that is not under consideration. When both sites under investigation are not occupied by the same dimer, the probabilities that these two sites are occupied by monomer and one segment of a dimer are independent of each other. (iv) Each site is occupied by one segment of a dimer, but the same dimer does not occupy both sites. The larger molecule (i.e., dimer or polymer in the examples below) enjoys conformational or orientational freedom such that each conformation/orientation is equally probable because they all exhibit the same energy. This is analogous to the microcanonical ensemble in statistical thermodynamics where entropy (i.e., S ¼ k ln V) is related directly to the “multiplicity of states” V via combinatorial aspects of the lattice.

3.6 Guggenheim’s Lattice Theory of Athermal Mixtures

109

Lattice Coordination Numbers and Nearest Neighbor Sites If the coordination number of the lattice is z, then each site, or each monomer, has z nearest neighbors. There are zN1 sites on the lattice that are adjacent to monomers. If one dimer occupies two adjacent sites under investigation, then each segment of the dimer has z 21 nearest neighbor sites that contain different molecules. There are 2(z 21) nearest neighbor sites per dimer that are occupied by different molecules. Hence, 2(z 21)N2 lattice sites are adjacent to dimers and contain different molecules. Probability that Two Adjacent Lattice Sites Are Occupied by Monomers The monomer volume fraction 12 w governs the probability that the first site contains monomer. If this first site that contains monomer must be adjacent to another site that contains monomer, then it is necessary to consider the fraction of lattice sites that are adjacent to monomers, based on the fact that zN1 sites are nearest neighbors to monomers and 2(z 21)N2 sites contain different molecules that are adjacent to dimers. For two adjacent lattice sites under investigation, the probability that both are occupied by monomers is (1  w)

z N1 z(1  w) z(1  w) ¼ (1  w) ¼ (1  w) zN1 þ 2(z  1)N2 z(1  w) þ w(z  1) zw

Probability that One Site Is Occupied by Monomer and the Adjacent Lattice Site Is Occupied by One Segment of a Dimer Begin with the monomer volume fraction 12 w to account for the occupational probability of the first site by monomer. Then, consider the fraction of lattice sites that are nearest neighbors to dimers, which requires that the adjacent site contains one segment of a dimer regardless where the second segment of the dimer resides. One obtains the following occupational probability for the sequence—monomer first and one segment of a dimer second: (1  w)

2(z  1)N2 w(z  1) w(z  1) ¼ (1  w) ¼ (1  w) zN1 þ 2(z  1)N2 z(1  w) þ w(z  1) zw

This probability is the same if the site occupation is reversed; the first site is occupied by one segment of a dimer and the second site is occupied by monomer. However, an additional factor must be considered. Begin with the dimer volume fraction w to account for the occupational probability of the first site by one segment of a dimer. Now, there are only z 21 possibilities from a total of z nearest neighbors for the adjacent site because the second segment of the same dimer must reside elsewhere. After one accounts for the situation where the adjacent lattice site contains another molecule, then it is necessary to consider the fraction of sites that are nearest neighbors to monomers. The second attempt to determine this occupational probability yields the

110

Chapter 3 Lattice Theories for Polymer –Small-Molecule Mixtures

following result for the sequence—one segment of a dimer first and monomer second:

w

z1 zN1 z1 z(1  w) 1w ¼ w(z  1) ¼w zw z zN1 þ 2(z  1)N2 z z(1  w) þ w(z  1)

Hence, one obtains the same result based on two different approaches. Probability that Two Adjacent Lattice Sites Are Occupied by Different Dimers Begin with the volume fraction of dimers, w, for the probability that the first site contains one segment of a dimer. Once again, there are only z 21 possibilities from a total of z nearest neighbors for the adjacent site because the second segment of the same dimer must reside elsewhere. Now, one accounts for the situation where the adjacent lattice site contains another molecule, specifically another dimer. Hence, it is necessary to consider the fraction of lattice sites that are nearest neighbors to dimers. This occupational probability is

w

z1 2(z  1)N2 z1 w(z  1) z  1 w(z  1) ¼w ¼w z z N1 þ 2(z  1)N2 z z(1  w) þ w(z  1) z zw

Probability that Two Adjacent Lattice Sites Are Occupied by the Same Dimer Now, the dimer volume fraction w dictates the probability that the first site contains one segment of a dimer. Then, there is only one possibility from a total of z nearest neighbors for the adjacent lattice site to contain the second segment of the same dimer. It is not necessary to consider neighboring site occupation by another molecule. Hence, the occupational probability is   1 w z Connection Between Thermodynamic Activities and Occupational Probabilities Yields Macroscopic Thermodynamic Mixing Properties For a chosen pair of adjacent lattice sites, one constructs the ratio of two different occupational probabilities; both sites are occupied by the same dimer versus the frequency of occupation of both sites by monomer. Then, one equates this result to the ratio of thermodynamic activities of dimer and monomer as follows:   1 w aDimer w(z  w) z ¼ ¼ z(1  w) fz(1  w)g2 a2Monomer (1  w) zw

3.6 Guggenheim’s Lattice Theory of Athermal Mixtures

111

This yields an expression for the entropy of mixing, per mole of molecules, for binary mixtures of monomers and dimers via the final equation in Section 3.6.1: 8 1 " # # 9 ð ðw " = Dsmixing 1 < w(z  w) w(z  w) ¼ w ln d d w  ln w ; R 2  w: fz(1  w)g2 fz(1  w)g2 0

0

1 ¼ f(z  w) ln(z  w)  w[ ln w þ (z  1) ln(z  1)] 2w (1  w)[z ln z þ 2 ln(1  w)]g Except at the pure-component limits (i.e., Dsmixing ¼ 0 at w ¼ 0 and 1), the entropy of mixing (i) is always positive, (ii) exhibits asymmetry about w ¼ 0.5 because the mixture contains molecules of different size, and (iii) unlike the Flory – Huggins lattice theory, Guggenheim’s lattice model yields macroscopic thermodynamic mixing properties that depend on the coordination number of the lattice, even though this dependence of Dsmixing on z is very weak. The maximum entropy of mixing occurs when the dimer volume fraction is w ¼ 0.66, independent of z, and fDsmixing/RgMaximum  0.70– 0.75, with slightly larger maximum Dsmixing when the coordination number of the lattice is higher (i.e., 3  z  12). The maximum entropy of mixing, per mole of lattice sites, occurs at lower dimer volume fraction (i.e., w ¼ 0.55). All of these results are summarized graphically and in tabular form for mixtures of monomers with dimers, trimers, tetramers, and pentamers at the conclusion of the next section that considers mixtures of monomers with straight-chain r-mers.

3.6.3 Combinatorial Aspects of Binary Mixtures of Solvent and Straight-Chain r-mers Overview The generalized methodology developed in the previous section for binary mixtures of monomers and dimers is repeated here when dimers are replaced by linear amorphous polymers that contain r segments per chain. The primary objective is to obtain a statistical expression for the molar entropy change upon mixing by considering the occupational probabilities of r-interconnected lattice sites. DiMarzio and Gibbs [1963] employed the final result of Guggenheim’s derivation for Dsmixing to explain the effect of plasticizers and other flexible and rigid small molecules on the glass transition temperature of linear polymers. The conformational entropy description of the glass transition temperature, proposed by DiMarzio and Gibbs, monitors the decrease in conformational entropy of a concentrated polymer – solvent mixture as temperature is reduced. Tg is identified as the temperature at which the conformational entropy of the mixture initially vanishes upon cooling. Statistical parameters of the lattice (i.e., z) and the low-molecular-weight additive can be chosen to predict either increases or decreases in the polymer’s Tg at higher diluent volume fractions. Most weakly interacting miscible additives that yield single-phase polymer –diluent mixtures, including plasticizers, antiplasticizers, antioxidants, and UV stabilizers,

112

Chapter 3 Lattice Theories for Polymer –Small-Molecule Mixtures

reduce the polymer’s Tg at higher diluent concentrations. The most notable exceptions to this trend are discussed in Chapter 6 from an energetic viewpoint by focusing on the stabilization of metal d-electrons when transition metal complexes coordinate to basic functional groups in the side chain of polar polymers and increase the glass transition temperature. The DiMarzio – Gibbs conformational entropy description of Tg provides a combinatorial lattice theory prediction of the enhancement in the glass transition temperature at higher concentrations of transition metal complexes from the d-block of the Periodic Table. Volume Fractions and Nearest Neighbor Considerations Once again, the analysis below focuses on two adjacent lattice sites, where each site has z nearest neighbors. However, complete results for the entropy change upon mixing polymer and solvent require a consideration of r-interconnected lattice sites. The mixture contains N1 solvent molecules and Nr linear polymers. The total number of lattice sites is given by NTotal ¼ N1 þ rNr. Volume fractions of polymer w and solvent 1 2 w are defined below, based on the restriction that each segment of the polymer occupies the same volume as a single solvent molecule:

wPolymer ¼ w ¼

rNr N1 ; wSolvent ¼ 1  w ¼ N1 þ rNr N1 þ rNr

These volume fractions, 1 2 w or w, describe the occupational probability of the first of two adjacent lattice sites by solvent or one segment of the polymer, respectively. The frequency of occupation of the second lattice site, which is adjacent to the first one, depends on nearest neighbor considerations. For example, z N1 lattice sites are adjacent to solvent molecules. The definition of q provides the number of nearest neighbor sites (i.e., qz) for all r segments of a single polymer chain that are occupied by another molecule, either solvent or a segment of a different polymer chain. Likewise, rz 2qz represents the number of nearest neighbor sites to an r-segment linear chain that are occupied by segments of the same polymer chain. Consideration of each segment of a single chain yields the following calculation of q. Since the chain is linear, there are two end segments that have z 2 1 nearest neighbor sites occupied by other molecules. For all r 2 2 interior segments, there are z 2 2 nearest neighbor sites per segment occupied by other molecules. Hence, q is related to r and z by the following equation: 2(z  1) þ (r  2)(z  2) ¼ qz 2 r  (r  1) ¼ q z As expected, qz ¼ 2(z 2 1) when r ¼ 2, which was employed in the previous section for binary mixtures of monomers and dimers. Since the mixture contains Nr polymer molecules, there are qzNr lattice sites that are adjacent to r-mers which contain different molecules. As mentioned above, each possible conformation of a polymer chain that contains r segments is assumed to have the same energy, which implies that all of these conformations (i.e., on the order of z(z 2 1)r22  z r21) are equally probable.

3.6 Guggenheim’s Lattice Theory of Athermal Mixtures

113

Probability that Two Adjacent Lattice Sites Are Occupied by Solvent The frequency of occupation of the first lattice site by a solvent molecule is given by its volume fraction 1 2 w. Occupation of the adjacent site by another solvent molecule is based on the fraction of sites that are nearest neighbors to solvent. Since there are zN1 sites adjacent to solvent molecules and qzNr sites adjacent to polymer chains that contain different molecules, the probability that two adjacent lattice sites are occupied by solvent is given by z N1 z(1  f) (1  f) ¼ (1  f) q zN1 þ qz Nr z(1  f) þ zf r Probability that One Lattice Site Is Occupied by Solvent and the Adjacent Site Is Occupied by One Segment of a Linear Polymer Begin with the frequency of occupation of the first site by solvent, which is governed by the solvent volume fraction 1 2 w. Now, consider the fraction of lattice sites that contain different molecules, which are nearest neighbors to all segment of linear polymer chains. The result of this sequence of occupational probabilities in which two adjacent lattice sites are populated by solvent first and a segment of the polymer chain second is q zw qz Nr r ¼ (1  w) (1  w) q z N1 þ qz Nr z(1  w) þ zw r This the problem could have been addressed by reversing the sequence of events, such that one considers the probability of occupying two adjacent lattice sites in the following order—one segment of a polymer chain first and solvent second. After the first lattice site is occupied by one of the r segments of a linear polymer, with frequency of occupation given by the polymer volume fraction w, it is necessary to ensure that the adjacent lattice site does not contain another segment of the same chain. Hence, (q/r) represents the fraction of lattice sites adjacent to any segment of an r-segment chain that is occupied by different molecules. Then, one considers the fraction of sites that are adjacent to solvent molecules. One obtains the identical final result based on this sequence of events:

w

q z N1 q z(1  w) ¼w r z N1 þ qz Nr r z(1  w) þ q zw r

Probability that Two Adjacent Lattice Sites Are Occupied by Segments of Two Different Polymer Chains Begin with the polymer volume fraction w to account for the frequency of occupation of the first lattice site by one segment of the polymer. Next, it must be guaranteed that the adjacent lattice site contains a different molecule, not another segment of the same

114

Chapter 3 Lattice Theories for Polymer –Small-Molecule Mixtures

chain. This consideration introduces a factor of (q/r). Finally, the fraction of lattice sites that are nearest neighbors to any segment of the polymer and occupied by different molecules yields the following result: q zw q qz Nr q r w ¼w r zN1 þ qz Nr r z(1  w) þ q zw r Probability that Two Adjacent Lattice Sites Are Occupied by Two Segments of the Same Polymer Chain Once again, the polymer volume fraction w governs the frequency of occupation of the first lattice site by one segment of an r-mer. The only remaining consideration is that the adjacent lattice site contains a segment of the same chain, not another molecule. The desired result is given by  rz  qz q ¼w 1 w rz r Extrapolation of Occupational Probabilities to r-Interconnected Lattice Sites, and the Relevant Thermodynamic Properties of Binary Polymer – Solvent Mixtures The analysis of two adjacent lattice sites, discussed directly above for four specific situations, is extended to a set of r-interconnected sites. Only two occupational probabilities are required to evaluate the classical thermodynamic expression for the Gibbs free energy and entropy of mixing for athermal binary mixtures. Begin with the probability that two adjacent lattice sites are occupied by solvent and extend this result to r lattice sites. Upon invoking the assumption that the additional r 2 2 sequential events for site occupation by solvent do not depend on the fraction of previous sites already occupied by solvent yields the following result: 2 3r1  r1 z N1 6 z(1  w) 7 ¼ (1  w)4 (1  w) q 5 zN1 þ qz Nr z(1  w) þ zw r Next, begin with the probability that two adjacent lattice sites are occupied by two segments of the same polymer chain. Invoke the assumption that sequential occupational probabilities of another (r 2 2)-interconnected lattice sites by segments of the same polymer chain are not a function of the number of previous sites already occupied by segments of the same chain. This sequence of r independent events yields    rz  qz r1 qr1 w ¼w 1 rz r The ratio of classical thermodynamic activities for polymer and solvent is constructed from the ratio of the previous two occupational probabilities; r-interconnected lattice sites occupied by r segments of the same polymer chain versus occupation of the same

3.6 Guggenheim’s Lattice Theory of Athermal Mixtures

115

set of sites by r solvent molecules. The following equations summarize thermodynamic mixing properties of athermal binary polymer – solvent systems:  qr1 w 1  aPolymer r ¼ 2 3r1 arSolvent 6 z(1  w) 7 (1  w)4 q 5 z(1  w) þ zw r 2 q 3r1 w 4 q (1  w) þ r w5 1 ¼ (1  w) (1  w) r 2 1 3   ð

ðw

Dsmixing Dgmixing a a 1 4w ln Polymer d w  ln Polymer d w5 ¼ ¼ R RT w þ r(1  w) arSolvent arSolvent 0

0

The final result of Guggenheim’s combinatorial lattice description of binary polymer – solvent mixtures for the entropy of mixing, per mole of molecules, is Dsmixing fw þ r(1  w)g ¼ w ln w  r(1  w) ln(1  w) R r(r  1) n q  q q qo 1  w þ w ln 1  w þ w  w ln þ rq r r r r 2 q ¼ r  (r  1) z This expression for the entropy of mixing, per mole of molecules, in binary polymer – solvent systems reduces to the final equation in Section 3.6.2 for binary mixtures of monomers and dimers when r ¼ 2. The previous equation is presented graphically in Figure 3.3 for mixtures of monomers with dimers, trimers, tetramers, and pentamers. The results are somewhat misleading because there are fewer molecules, but the same number of lattice sites, in the mixture as r increases, which causes the maximum entropy of mixing, per mole of molecules, to increase when the larger component in these binary mixtures contains more segments and occupies more lattice sites. A better comparison of the four binary mixtures in Figure 3.3 is illustrated in Figure 3.4, by calculating the entropy of mixing, per mole of lattice sites. To accomplish this task, it is necessary to return to Guggenheim’s original development of the extensive Gibbs free energy of mixing in Section 3.6.1, based on classical thermodynamics: 2w 3   ð1

ð

aPolymer aPolymer NTotal 4 kT ln r d w  w ln r d w5 DGmixing ¼ r aSolvent aSolvent 0

0

and divide by the total number of lattice sites, NTotal, not the total number of molecules. After multiplication by Avogadro’s number, one obtains the desired results for the

116

Chapter 3 Lattice Theories for Polymer –Small-Molecule Mixtures 1.0 Monomer/Pentamer Monomer/Tetramer Monomer/Trimer Monomer/Dimer

Entropy of Mixing (divided by R)

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 Volume Fraction of the Larger Species

0.9

1.0

Figure 3.3 Guggenheim’s dimensionless entropy of mixing, per mole of molecules, versus volume fraction of the larger component in binary mixtures of monomers and r-mers (i.e., 2  r  5) on a three dimensional cubic lattice (z ¼ 6).

Gibbs free energy of mixing Dgmixing and the entropy of mixing Dsmixing, per mole of lattice sites: 2w 3   ð1

ð

Dsmixing Dgmixing aPolymer aPolymer 1 ¼ ¼  4 ln r d w  w ln r d w5 R RT aSolvent aSolvent r 0

0

The factor of fw þ r(1 2 w)g on the left side of the final combinatorial expression for Dsmixing, per mole of molecules, must be replaced by r to obtain Dsmixing, per mole of lattice sites. Table 3.2 Comparison of the Maximum Dimensionless Entropy of Mixing a for Binary Mixtures of Monomers and r-mers on a Three-Dimensional Cubic Lattice (z ¼ 6) Via Guggenheim’s Theory of Athermal Mixtures

r-mer Dimer Trimer Tetramer Pentamer a

fDsmixinggMaximum per mole of molecules

wLarger Species @ fDsmixinggMaximum per mole of molecules

fDsmixinggMaximum per mole of lattice sites

wLarger Species @ fDsmixinggMaximum per mole of lattice sites

0.74 0.81 0.87 0.94

0.66 0.74 0.78 0.81

0.51 0.45 0.42 0.40

0.55 0.58 0.60 0.61

Divided by the gas constant, R.

3.7 Gibbs– DiMarzio Conformational Entropy Description of the Glass Transition

117

Entropy of Mixing (divided by R)

0.6 0.5 0.4 0.3

Monomer/Dimer Monomer/Trimer Monomer/Tetramer Monomer/Pentamer

0.2 0.1 0.0 0.0

0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 Volume Fraction of the Larger Species

0.9

1.0

Figure 3.4 Guggenheim’s dimensionless entropy of mixing, per mole of lattice sites, versus volume fraction of the larger component in binary mixtures of monomers and r-mers (i.e., 2  r  5) on a threedimensional cubic lattice (z ¼ 6). The larger species increases in size from the uppermost curve (i.e., dimer) to the lowermost curve (i.e., pentamer).

Table 3.2 compares the maximum entropy of mixing, and the volume fraction of the larger component in these binary mixtures where the maximum entropy of mixing occurs, when calculations are based on moles of molecules versus moles of lattice sites. It should be obvious from Table 3.2 and Figure 3.4 that the entropy of mixing, per mole of lattice sites, in binary systems of monomers (i.e., solvent or diluent) and r-mers decreases when the larger species contains more interconnected segments. This conclusion is independent of the lattice coordination number. In summary, chain connectivity restricts the conformational freedom associated with placing larger molecules on a lattice, and these restrictions become more severe when the molar mass of chain-like molecules increases.

3.7 GIBBS –DIMARZIO CONFORMATIONAL ENTROPY DESCRIPTION OF THE GLASS TRANSITION FOR TETRAHEDRAL LATTICES 3.7.1

Overview and Governing Equations

Guggenheim’s lattice theory, described in the previous section for polymer solutions and mixtures of monomers and dimers, was modified by DiMarzio and Gibbs [1963] to obtain an expression for the conformational entropy S of athermal polymer – solvent mixtures. One predicts the glass transition temperature of mixtures by (i) monitoring the decrease in S as systems are cooled and (ii) identifying the temperature at which S vanishes initially. Kauzmann’s [1948] paradox addresses the anomaly of “negative”

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Chapter 3 Lattice Theories for Polymer –Small-Molecule Mixtures

entropy, relative to perfectly ordered crystals, below the glass transition temperature. Within the framework of the Flory – Huggins approximations, the two-phase boundary (i.e., transition line) on a temperature – pressure diagram for pure materials, that separates liquids from glasses, is described by zero conformational entropy. Following Debenedetti and Shell [2004], it is also possible to equate the dimensionless conformational entropy of binary polymer – solvent mixtures at the glass transition temperature to a small positive nonzero constant (i.e., approximately 10% of the maximum conformational entropy of ideal gases at infinitely high temperature), due primarily to vibrational and crankshaft motion. However, it must be emphasized that the actual conformational entropy at Tg is not known. When small molecules occupy more than three sites on a tetrahedral lattice, the propensity for interconversion between various rotational isomers is summarized in terms of “diluent rigidity,” D1Diluent, via the energy difference between local minima on the potential energy surface for trans, gaucheþ, and gauche2 rotational isomeric states. Stiffer additives exhibit larger values of D1Diluent. The most important parameters that affect Tg are molecular size via rDiluent, rigidity via D1Diluent, and concentration (i.e., volume fraction, 12 w) of the additive, as well as the coordination number z of the lattice, which summarizes packing efficiency or the ability of functional groups in the side chain of polar polymers to occupy sites in the first shell of a transition metal complex. It is possible to describe accurately the effects of (i) molecular weight, (ii) additive concentration, (iii) random copolymer composition, (iv) pressure, (v) crosslink density, and (vi) mechanical strain on the glass transition temperature of amorphous polymers via the methodology summarized below. For example, one predicts an increase in Tg at higher strain via conformational entropy formalism because oriented chains have fewer accessible conformations and lower entropy relative to unstretched random coils. These predictions contrast those from the free volume approach, which suggests that Tg decreases upon stretching if Poisson’s ratio is less than 0.5 because system volume, as well as fractional free volume, increases at higher strain. The modified version of Guggenheim’s lattice model, employed by DiMarzio and Gibbs, is discussed below for the dimensionless conformational entropy S of polymer – solvent mixtures per segment of polymer. Statistical thermodynamics is required to obtain expressions for the conformational freedom of placing pure polymer and pure diluent on the lattice. Unlike the Flory –Huggins lattice, diluents are not necessarily modeled as structureless “point” molecules that occupy only one lattice site. Pure-component conformational entropies are calculated via partition function formalism and added to Guggenheim’s temperature-independent entropy of mixing Dsmixing that was discussed in the previous section via classical thermodynamics and “lattice-counting” considerations. The Gibbs– DiMarzio expression is restricted to tetrahedral lattices in which z ¼ 4, and diluent rigidity becomes important when these small molecules occupy at least four interconnected lattice sites. If tetrahedral bond angles are appropriate, then the conformational freedom of a three-segment two-bond “united atom” diluent molecule, with rDiluent ¼ 3, is not affected by diluent rigidity because free rotation about either bond does not affect the shape of the molecule. Hence, the third term on the right side of the following equation for the conformational entropy of miscible athermal binary polymer – diluent mixtures contains a factor of frDiluent 2 3g that

3.7 Gibbs– DiMarzio Conformational Entropy Description of the Glass Transition

119

is only applicable for tetrahedral lattices. Furthermore, the effect of diluent rigidity via the third term in the following equation for S should be excluded from all calculations when diluent molecules occupy one, two, or three interconnected lattice sites. All pure-component temperature-dependent terms in the conformational entropy expression that contain effects of molecular rigidity must be re-evaluated via statistical thermodynamic formalism, with a unique bond rotational energy assigned to each allowed state, when the lattice packing efficiency is described by a coordination number z that differs from 4. The diluent volume fraction (i.e., 12 w) and temperature dependence of the dimensionless conformational entropy of binary mixtures, per segment of polymer, for linear chains with infinitely large molecular weight (i.e., rPolymer ) 1) on a tetrahedral lattice (i.e., z ¼ 4) is

 SConformation (w, T; z, rDiluent ) z  2 (z  2)rDiluent þ 2(1  w) ¼ ln kBoltzmann rPolymer NPolymer 2w zrDiluent

 1w (z  2)rDiluent þ 2(1  w) z1 z2 (1  w)(rDiluent  3) þ ln þ 2 wrDiluent wrDiluent z(1  w)

    D1Diluent fDiluent D1Diluent þ

ln 1 þ (z  2) exp  kBoltzmann T kBoltzmann T    D1Polymer fPolymer D1Polymer þ þ ln 1 þ (z  2) exp  kBoltzmann T kBoltzmann T where

fDiluent=Polymer

 D1Diluent=Polymer (z  2) exp  k T

Boltzmann  ¼ D1Diluent=Polymer 1 þ (z  2) exp  kBoltzmann T

and z1 ¼ z, unless rDiluent ¼ 1 (z1 ¼ 1), and z2 ¼ z – 1, unless rDiluent ¼ 1 or 2 (z2 ¼ 1). DiMarzio’s expression for the conformational entropy of binary polymer – solvent mixtures considers additive molecules of different size, relative to the volume of one segment of the polymer chain. For example, the diluent volume fraction is 1w¼

rDiluent NDiluent rDiluent NDiluent þ rPolymer NPolymer

when NDiluent molecules of the additive each occupy rDiluent lattice sites and NPolymer molecules of the polymer each contain rPolymer segments. The Flory – Huggins lattice model is based on the fact that each additive, or solvent, molecule occupies one lattice site, and it is equivalent in size to one segment of the polymer chain. It should be obvious that larger additive molecules occupy more than one site and reduce the conformational entropy, which is consistent with the fact that rDiluent is greater than one. More rigid additives are described by larger values of D1Diluent, whereas D1Diluent ¼ 0 corresponds to infinitely flexible molecules with free rotation about the valence cone. The intramolecular interaction energy parameter for the

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Chapter 3 Lattice Theories for Polymer –Small-Molecule Mixtures

polymer is obtained by invoking the concept of vanishingly small conformational entropy of the undiluted polymer (i.e., w ¼ 1) at its pure-component glass transition temperature (i.e., Tg,Polymer). In other words,

 SConformation (w ¼ 1, Tg,Polymer ; z) 1 z2 ¼ (z  2) ln 2 z kBoltzmann rPolymer NPolymer    D1Polymer þ ln 1 þ (z  2) exp  kBoltzmann Tg,Polymer þ

fPolymer D1Polymer ¼0 kBoltzmann Tg,Polymer

Since D1Polymer and Tg,Polymer always appear together in the previous equation as the following dimensionless ratio, D1Polymer/fkBoltzmannTg,Polymerg, one solves for this dimensionless intramolecular interaction energy of the polymer on a tetrahedral lattice, with z ¼ 4. Hence, D1Polymer/fkBoltzmannTg,Polymerg ¼ 1.92 when z ¼ 4 and SConformation ) 0.

3.7.2 Flexible versus Rigid Additives that Decrease a Polymer’s Glass Transition Temperature When small-molecule plasticizers occupy four lattice sites (i.e., rDiluent ¼ 4) and the coordination number of the lattice is z ¼ 4, the polymer’s glass transition temperature 1.0 Glass Transition Temperature

4 Lattice sites per diluent molecule 0.9

0.8

0.7

0.6 0.00

Diluent energy = 5 Diluent energy = 3 Diluent energy = 2 Diluent energy = 1 Diluent energy = 0 0.05

0.10 0.15 Diluent Volume Fraction

0.20

0.25

Figure 3.5 Gibbs –DiMarzio predictions for the effects of diluent volume fraction and rigidity on the glass transition temperature of miscible polymer–diluent blends, relative to Tg of the undiluted polymer, on a tetrahedral lattice (i.e., z ¼ 4). Each diluent molecule occupies four lattice sites. Less glass transition temperature depression is predicted when the rigidity of the additive increases, as measured by the dimensionless diluent energy parameter in the legend: D1Diluent/kTg,Polymer.

3.7 Gibbs– DiMarzio Conformational Entropy Description of the Glass Transition

121

decreases more if the additive’s dimensionless energy barrier for intramolecular conversion between rotational isomers (i.e., D1Diluent/kTg,Polymer) is smaller. For these specifications (i.e., z ¼ 4, rDiluent ¼ 4), completely flexible additives correspond to the lowermost curve in Figure 3.5, where D1Diluent ¼ 0. Predictions for D1Diluent/kTg,Polymer ¼ 5 are indistinguishable from those when D1Diluent/kTg,Polymer ¼ 10, so the asymptotic limit of Tg depression for infinitely rigid small diluents (i.e., rDiluent ¼ 4) is revealed by the uppermost curve in Figure 3.5. Diluent concentration effects encompass diluent volume fractions (i.e., 1 2 w) from 0% to 25%. In Figures 3.5, 3.6, and 3.7, the vertical axis reveals the ratio of the glass transition temperature of miscible binary mixtures relative to that of the undiluted polymer (i.e., Tg,Polymer).

3.7.3 Effect of Molecular Size of Semiflexible Additives on Tg When the lattice coordination number is z ¼ 4 and the diluent is not infinitely flexible (i.e., D1Diluent/kTg,Polymer ¼ 1), glass transition temperature depression is larger for smaller plasticizers, as illustrated in Figure 3.6. Tg depression versus diluent volume fraction exhibits mildly nonlinear behavior over the compositional range investigated (i.e., 0  f12 fg  0.25), but this nonlinearity becomes less apparent as the

Glass Transition Temperature

1.0

0.9

0.8

0.7

0.6 0.00

1000 Lattice sites 100 Lattice sites per diluent 25 Lattice sites per diluent 10 Lattice sites per diluent 5 Lattice sites per diluent 3 Lattice sites per diluent 0.05

0.10 0.15 Diluent Volume Fraction

0.20

0.25

Figure 3.6 Gibbs –DiMarzio predictions for the effects of diluent size and volume fraction on the glass transition temperature of miscible polymer– diluent blends, relative to Tg of the undiluted polymer, on a tetrahedral lattice (i.e., z ¼ 4). Less glass transition temperature depression is predicted when the molecular size of the additive increases, as measured by the number of lattice sites per diluent (i.e., rDiluent) in the legend. In each case, the semiflexible diluent energy parameter for interconversion between rotational isomers is D1Diluent/kTg,Polymer ¼ 1.

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Chapter 3 Lattice Theories for Polymer –Small-Molecule Mixtures

molecular size of the additive increases. Predictions for semiflexible diluents with D1Diluent/kTg,Polymer ¼ 1 and rDiluent ¼ 1000 are indistinguishable from those when rDiluent ¼ 5000, so the asymptotic limit of Tg depression for extremely large semiflexible plasticizers is illustrated by the uppermost curve in Figure 3.6. Hence, if semiflexible additives are completely miscible with the polymer, then Tg should be depressed.

3.7.4 Infinitely Rigid Diluents that Increase a Polymer’s Glass Transition Temperature When liquid crystalline-like rigid additives occupy more than 10 lattice sites per molecule, glass transition temperatures increase monotonically at higher diluent volume fractions. Predictions in Figure 3.7 reveal that the increase in Tg is greater when rod-like diluents exhibit a larger Kratky – Porod persistence length, which is captured in the Gibbs – DiMarzio conformational entropy model by (i) the number of lattice sites per small molecule (i.e., rDiluent), and (ii) D1Diluent ) 1, with no possibility for interconversion from one rotational isomer to another. There exists a critical diluent volume fraction, approximately defined by rDiluentf12 wgCritical  8, which decreases for rigid molecules with larger persistence lengths, such that liquid crystalline 1.40 Glass Transition Temperature

1.35 1.30 1.25 1.20

50 Lattice sites per diluent 40 Lattice sites per diluent 25 Lattice sites per diluent 15 Lattice sites per diluent 10 Lattice sites per diluent 7 Lattice sites per diluent

1.15 1.10 1.05 1.00 0.95 0.00

0.05

0.10 0.15 Diluent Volume Fraction

0.20

0.25

Figure 3.7 Gibbs –DiMarzio predictions for the effects of diluent size and volume fraction on the glass transition temperature of miscible polymer– diluent blends, relative to Tg of the undiluted polymer, on a tetrahedral lattice (i.e., z ¼ 4). Larger glass transition temperature enhancement occurs when the molecular size of the additive increases, as measured by the number of lattice sites per diluent (i.e., rDiluent) in the legend. No Tg enhancement is predicted over the diluent volume fraction range investigated when rDiluent ¼ 7. In each case, diluent molecules are infinitely rigid such that no interconversion between rotational isomers is allowed.

3.8 Lattice Cluster Theory Analysis of Conformational Entropy

123

alignment of the additive occurs when its concentration exceeds f12 wgCritical. Conformational entropy simulations in Figure 3.7 reveal that the glass transition temperature of the mixture increases in response to this alignment.

3.8 LATTICE CLUSTER THEORY ANALYSIS OF CONFORMATIONAL ENTROPY AND THE GLASS TRANSITION IN AMORPHOUS POLYMERS Conformational entropy S is a basic quantity in theoretical descriptions of the glass transition. Karl Freed and co-workers at the University of Chicago discuss this concept from the viewpoint of high-dimensional lattice cluster theory, which represents a simple analytical extension of the Gibbs– DiMarzio model to mixtures of semiflexible, interacting polymers composed of structured monomers. This analysis provides a theoretical tool for investigating how the glass transition of amorphous polymers depends on chemical structure. Lattice cluster theory is based on two major modifications of the Flory – Huggins approximation to the free energy of binary polymer – solvent mixtures. First, united atom models are used to represent individual monomers that occupy several neighboring lattice sites. The second improvement involves a superior solution to the resulting lattice model. Systematic corrections to the Flory – Huggins free energy for nonrandom mixing are derived in the form of a high-temperature one-dimensional cluster expansion. This theoretical development yields analytical solutions by confining attention to the high-pressure, high-molecular-weight, and fully flexible-chain limit, which is identified as simplified lattice cluster theory. Some of the inadequacies of the Flory – Huggins lattice model for random mixing may be traced to inherent limitations of the simple mean-field approximation that was employed. In order to understand the origins of these deficiencies, Freed and co-workers developed perturbation methods for calculating corrections to the Flory – Huggins mean-field approximation. The theory emerges in the form of a cluster expansion that bears strong resemblance to Mayer’s cluster expansion for nonideal gases. Corrections to the Flory – Huggins lattice model arise from packingand interaction-induced local correlations between polymer and solvent molecules. The noncombinatorial portion of the Helmholtz free energy is generated analytically by lattice cluster theory as a single function of composition, molecular weight, nearest neighbor attractive van der Waals interactions, and temperature. Lattice cluster theory also includes the effects of individual small-molecule architecture that extend over several lattice sites and, therefore, more closely resemble the actual molecular structure of solvents, additives, or diluents. Hence, by combining a significantly more advanced solution to the lattice model of polymers that considers monomer molecular structure, lattice cluster theory exhibits the following characteristics: (i) It explains a temperature-independent “entropic” Flory – Huggins x parameter that originates from the packing of monomers with different sizes and shapes.

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Chapter 3 Lattice Theories for Polymer –Small-Molecule Mixtures

(ii) It reveals both composition and molecular-weight dependence of x as a consequence of monomer molecular structure, compressibility, and nonrandom mixing effects. (iii) It predicts the pressure dependence of x, which was subsequently verified experimentally. (iv) It describes the dependence of x on chain architecture and monomer sequences in copolymers. (v) It predicts novel phenomena, such as the ordering of certain block copolymers upon heating. Lattice cluster theory also provides a molecular explanation for the fact that poly(ethylene-co-propylene) exhibits different miscibility with atactic and head-tohead polypropylene. An understanding of the relationship between monomer molecular structure and glass formation is fundamental to numerous practical applications in polymer science and engineering. It should be emphasized that glass formation, or vitrification, is not a true underlying equilibrium transition because the phase transition exhibits dependence on kinetic phenomena, such as heating or cooling rate and excitation frequency. The equilibrium Gibbs – DiMarzio (GD) lattice model of glass formation has achieved significant success in explaining the influence of molecular parameters on glass formation in polymeric materials. In particular, the GD theory describes how the glass transition temperature Tg varies with polymer molecular weight, pressure, blend composition, the introduction of crosslinks and mechanical deformation in rubbers, and the addition of plasticizers. There is ample evidence from specific heat measurements that fluid entropy decreases precipitously near the glass transition, supporting the basic physical picture of the GD model that glass formation is fundamentally related to this entropy change. The success of the lattice model proposed by DiMarzio and Gibbs [1963] can be traced to its capacity for describing, at least qualitatively, this basic aspect of glass formation. However, the GD theory is based on a crude mean-field approximation for the thermodynamic entropy of semiflexible chain polymers that had been developed earlier by Flory for the description of polymer melting. This theory predicts that the conformational entropy extrapolates to zero at finite temperatures. Following the ideas of Kauzmann [1948], DiMarzio and Gibbs [1963] identify this total collapse of conformational entropy with the glass transition and describe Tg as a second-order phase transition whose underlying origin is thermodynamic. Several approximations embedded in the GD theory are controversial at low temperatures. For example, the GD theory violates a rigorous bound on the entropy of molten polymers. Simulations of polymeric liquids indicate that entropy S does not vanish at low temperatures, but S approaches a low-temperature plateau. These results do not invalidate the GD approach to glass formation if the theory is viewed qualitatively to imply that glass formation occurs when the conformational entropy becomes sufficiently small to cause structural arrest and solidification at low temperatures. It is reasonable to use the GD theory for extrapolating thermodynamic properties from the high-temperature regime to lower temperatures

3.8 Lattice Cluster Theory Analysis of Conformational Entropy

125

where the theory breaks down, thus providing an estimate of the point at which the conformational entropy becomes critically small at the glass transition temperature. In view of these inadequacies of the GD mean-field theory, the use of lattice models remains valid for estimating trends in the glass transition in terms of molecular details such as monomer structure. Other models, like the Adam – Gibbs theory, proposes that the barrier height associated with thermally activated relaxation processes is inversely proportional to conformational entropy due to the increased cooperative motion that must occur in cooled liquids. The Adam – Gibbs model of transport in cooled liquids agrees with measurements and simulations of cooled liquids, providing convincing evidence of a link between transport processes and conformational entropy. Proposed relations between structural relaxation rates and conformational entropy S imply that rapid changes in the temperature dependence of S should lead to significant changes in the apparent activation energies for viscous and diffusive transport in “sluggish” liquids that exhibit rather long relaxation times. Fragile glass-forming liquids exhibit a large slope for activation energy versus temperature near Tg, whereas strong glass-forming liquids exhibit weak temperature dependence of the activation energy. The fragility of glasses can be predicted by monitoring the temperature dependence of conformational entropy, excluding the vibrational contribution near Tg. The temperature derivative of S just below Tg can be used to estimate kinetic fragility that is deduced from activated transport. Hence, it seems reasonable that correlations should exist between changes in liquid entropy, or closely related quantities, and changes in liquid dynamics at higher temperatures where liquids achieve thermodynamic equilibrium. Mean-field studies indicate that there are two distinct transitions associated with glass formation: a dynamic transition with diverging relaxation times, that is prevalent in the mode-coupling theory of structural glass formation, followed by a thermodynamic glass transition at lower temperatures where conformational entropy vanishes, as described by DiMarzio and Gibbs [1963]. Simulations suggest that the first dynamic transition lies close to the temperature range where significant changes begin to occur in the fluid’s entropy. Since conformational entropy S is a central quantity in theories of the glass transition, lattice cluster theory analysis begins by constructing an expression for S from statistical thermodynamics. The temperature dependence of S is developed from the logarithm of the “density of states” via Boltzmann’s equation. It must be emphasized that the glass transition is discussed from lattice models, without any vibrational contribution to conformational entropy. Since the vibrational component might be substantial in some cases, corrections for the vibrational contribution to S can be used to modify or finetune any prediction of monomer molecular structure on Tg. It should also be emphasized that there are other definitions of conformational entropy, particularly the one based on excess entropy with respect to the crystal state. It is relevant to determine the connection between lattice cluster theory (LCT) and the classic GD theory, which is also based on a lattice model. The GD theory represents additives as structureless molecules, whereas LCT describes monomers as molecules that extend over several lattice sites with specified architecture. Furthermore, DiMarzio and Gibbs [1963] define trans and gauche conformations based on rotational states of three successive bonds, whereas LCT describes these

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Chapter 3 Lattice Theories for Polymer –Small-Molecule Mixtures

conformations in terms of relative orientations of two successive bonds. This difference is insignificant for long-chain polymers. The classic GD theory treats only the leading order contribution in both the van der Waals interaction and trans –gauche conformational energies, so the GD density of states is extracted from the partition function as the coefficient of the Boltzmann factor. The GD density of states differs from the LCT zeroth-order density of states. First, DiMarzio and Gibbs employ a Huggins – Guggenhein-type counting model for conformational (i.e., combinatorial) entropy, while the zeroth-order LCT effectively generates the less-accurate Flory counting with immediate self-reversals permitted. However, the nonrandom mixing LCT contributions correct for this counting difference and for the removal of immediate self-reversals, so this formal difference between GD and zeroth-order LCT is inconsequential. Second, DiMarzio and Gibbs use a “maximum-term approximation” to evaluate the number of unoccupied lattice sites, which is related to excess free volume, whereas this excess free volume is determined within LCT from the equation of state as a function of temperature, pressure, and composition. More explicitly, lattice cluster analysis contains coordination number z1 for the second site occupied by the additive, instead of z2 that appears in the GD expression for conformational entropy when the number of sites available for the third segment of the additive is considered from a combinatorial viewpoint. Consequently, the present LCT computation of conformational entropy represents the appropriate extension of the GD theory to structured monomer –polymer systems whose nonrandom mixing contributions to the free energy and conformational entropy cannot be represented in terms of independent partition functions for individual polymer chains. LCT expressions for conformational entropy and free energy are rather complex, but they merely involve polynomials in volume fractions. Hence, they can be evaluated numerically. Most of the comments in this section can be found in the following references on lattice cluster theory: Dudowicz et al. [1990, 2002], Dudowicz and Freed [1991], Freed [2003], and Freed and Dudowicz [1998, 2005].

3.9 SANCHEZ –LACOMBE STATISTICAL THERMODYNAMIC LATTICE FLUID THEORY OF POLYMER –SOLVENT MIXTURES The Flory – Huggins lattice theory, presented earlier in this chapter, does not consider equation-of-state properties of the pure components and cannot predict phase separation of polymer solutions above the upper critical solution temperature (UCST). Sanchez and Lacombe [1978] employed a three-dimensional lattice to describe the thermodynamic properties of polymer solutions, based on the multiplicity of states of high-molecular-weight chains that require r-interconnected lattice sites. Random mixing is assumed, and singlet probabilities are evaluated in terms of species volume fractions. Lattice fluid theory describes disordered fluids, not the ordered crystalline state. Hence, the “close-packed” state is disordered, with more resemblance to the glassy state than the crystalline state. Random close packing of spheres

3.9 Sanchez– Lacombe Statistical Thermodynamic Lattice Fluid Theory

127

corresponds to a solid volume fraction of 64%, whereas hexagonal-close-packed and face-centered cubic-close-packed arrangements of spheres are described by solid volume fractions of 74%. Each CH2 group contributes equally to the molecular close-packed volume of normal alkanes. When the close-packed mass density of alkanes is correlated with reciprocal chain length, the extrapolated intercept for infinite molecular weight yields r  0.934 g/cm3, and the close-packed molar volume is 15.0 cm3/mol. The extrapolated parameters from vapor pressure data on normal alkanes agree favorably with actual density data for linear polyethylene: r  0.904 g/cm3. In general, lattice fluid theory (i) accounts for differences in equation of state properties of the pure components and (ii) includes a thermodynamically unfavorable entropic contribution to the chemical potential of either component in the mixture. This affects diffusional stability via the spinodal inequality, where this entropic term from the equation of state contains nonzero pure-component parameter differences, except for hypothetical mixtures at absolute zero. These temperatureindependent pure-component parameter differences destabilize polymer solutions and cause incipient phase separation. This unfavorable entropic term is small and relatively unimportant at low temperatures, but it becomes large and dominant near the critical point of the mixture. In polymer solutions and low-molecular-weight regular solutions, this unfavorable entropic term is similar in magnitude, but the favorable combinatorial entropic contribution toward stability is much smaller, and almost insignificant, for polymer solutions. Hence, vanishingly small combinatorial entropy in high-molecular-weight polymer solutions might cause these mixtures to be more susceptible to phase separation at any temperature. Heats of mixing at infinite dilution relate directly to energetic interaction parameters for hydrocarbon polymers in hydrocarbon solvents. Nonpolar solutions typically exhibit endothermic enthalpies of mixing which, in some cases, become exothermic at high temperatures. The lattice fluid theory attributes exothermic energetics to the tendency of solvent molecules to experience volume contraction when low concentrations of polymer are introduced, where the magnitude of the contraction is proportional to the solvent’s isothermal compressibility. Macroscopic (i.e., classical) thermodynamic justification for this association between exothermic heats of mixing and volume contraction is provided in the next section. This process is energetically favorable because more intermolecular interactions are operative among solvent molecules when volume contraction occurs. In dilute polymer solutions with exothermic energetics, solvent molecules experience a slightly denser environment than in the pure state. In contrast, entropy decreases in higher-density media, yielding a thermodynamically unfavorable lower entropy of the solvent. Better equation-of-state parameters for the pure components will increase the predictive power of the lattice fluid theory, and re-evaluation of the mixing rules could increase its flexibility. Finally, in the dilute regime, where little overlap occurs between polymer molecules and a nonuniform distribution of polymer segments exists, the mean-field approximations should be revisited and modified because the lattice fluid theory employs uniform segment distribution.

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Chapter 3 Lattice Theories for Polymer –Small-Molecule Mixtures

APPENDIX: THE CONNECTION BETWEEN EXOTHERMIC ENERGETICS AND VOLUME CONTRACTION OF THE MIXTURE This analysis employs the pressure dependence of chemical potentials to define condensed phase activity coefficients and their relations to enthalpy changes upon mixing in nonideal solutions. By definition, the enthalpy of mixing vanishes for ideal solutions, so exothermic or endothermic energetics associated with the mixing process represent nonideal effects. Begin with the total differential of the extensive Gibbs free energy G of a mixture that contains r components. For single-phase behavior, there are r þ 1 degrees of freedom, so r þ 2 independent variables such as temperature T, pressure p, and mole numbers of each component in the mixture Ni are required for complete specification of an extensive thermodynamic property like G in the absence of strong external fields. The functional dependence of G and its total differential is GðT, p, Ni[1ir] Þ dG ¼ S dT þ V dp þ

r X

mi dNi

i¼1

From the previous expression, one employs Maxwell relations to obtain the pressure dependence of the chemical potential mi of species i in the mixture, because second mixed partial derivatives of exact differentials like G do not depend on the order in which differentiation is performed. Hence,     @ mi @V ¼ ¼ Vi @p T,composition @Ni p, T, N j[1 j  r; j=i where V is the extensive volume of the mixture and the quantity on the far right side of the previous equation is the partial molar volume of species i in the mixture. Integration of partial molar volumes with respect to pressure yields the effect of pressure on the chemical potential of species i in the mixture, whereas integration of ideal gas molar volumes of pure materials (i.e., RT/p) with respect to pressure yields the effect of low-pressure conditions on pure-component chemical potentials. The overall objective is to construct the following chemical potential difference at nonideal pressure p, which is required for the extensive Gibbs free energy of mixing DGmixing and subsequent evaluation of the enthalpy of mixing:

mi,mixture ( p)  mi,pure ( p) DGmixing ¼

r X

  Ni mi,mixture ( p)  mi,pure ( p)

i¼1

This chemical potential difference, mi,mixture( p) 2 mi,pure( p), is formulated by conveniently considering species i in the mixture, and as a pure component, at both

Appendix

129

nonideal pressure p and low pressure p , where p corresponds to ideal conditions:   mi,mixture ( p)  mi,pure ( p) ¼ mi,mixture ( p)  mi,mixture ( p ) 0)   þ mi,mixture ( p ) 0)  mi,pure (xi p ) 0)   þ mi,pure (xi p ) 0)  mi,pure ( p ) 0)    mi,pure ( p)  mi,pure ( p ) 0) where xi represents the mole fraction of species i in the mixture at partial pressure xi p . The first term on the right side of the previous equation is obtained by integrating the partial molar volume of species i with respect to pressure from ideal gas conditions at pressure p to nonideal pressure p. The second term vanishes because the partial pressure of species i in the mixture (i.e., xi p ) matches the total pressure of pure component i under ideal gas conditions. The third term is evaluated by integrating the ideal gas molar volume of pure-component i (i.e., RT/p) with respect to pressure between the low-pressure limits of p and xi p . The fourth term is evaluated by integrating the pure-component molar volume of species i, vi, with respect to pressure from ideal gas conditions at p to nonideal conditions at p. Since vi is not equivalent to RT/p over the entire range of operating pressures, the desired result for the chemical potential difference of species i in the mixture versus the undiluted state at nonideal pressure p is

mi,mixture ( p)  mi,pure ( p) ¼ RT ln xi þ

ðp

(V i  vi ) dp

p )0

Nonideal effects of mixing are contained in the second term on the right side of the previous equation. This contribution vanishes for ideal solutions because the partial molar volume of species i in the mixture is equivalent to its pure-component molar volume at all operating pressures, provided that ideal conditions exists from p to p. Condensed phase activity coefficients gi summarize nonideal mixing effects as follows: RT ln gi ¼

ðp

(V i  vi ) dp

p )0

This definition is consistent with the fact that gi ) 1 describes ideal solution behavior because the partial molar volume of mixing (i.e., difference between partial molar volume and pure-component molar volume) vanishes for each component. Focusing on the extensive Gibbs free energy of mixing, dividing the expression by the total number of moles of all species present, and identifying the mole fraction of species i as xi ¼ Ni/NTotal, one arrives at the molar Gibbs free energy of mixing for nonideal solutions: r   DGmixing X ¼ xi mi,mixture ( p)  mi,pure ( p) Dgmixing ¼ NTotal i¼1 ¼

r r X X   xi Gi  gi ¼ RT xi fln xi þ lngi g i¼1

i¼1

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Chapter 3 Lattice Theories for Polymer –Small-Molecule Mixtures

Terms that contain gi represent the nonideal contribution to Dgmixing. The molar enthalpy of mixing for these nonideal solutions is obtained by adding T Dsmixing to the previous expression for Dgmixing. Based on the total differential of the extensive Gibbs free energy of a multicomponent mixture, Maxwell’s relations reveal that the temperature dependence of the chemical potential of species i yields the partial molar entropy of species i:     @ mi @S ¼ ¼ Si @T p,composition @Ni T, p, N j[1 j r, j=i] Analogously, the temperature dependence of the chemical potential of pure component i yields its molar entropy. Hence, the molar entropy of mixing Dsmixing in nonideal solutions can be obtained by taking the temperature derivative of Dgmixing at constant pressure and mixture composition. Since activity coefficients depend on temperature, pressure, and composition, the final result for Dsmixing is 

 @Dgmixing @T p,composition (   ) r X @ mi,pure @ mi,mixture ¼ xi  @T @T p p,composition i¼1 ( )   r r X X   @ lngi ¼ xi Si  si ¼ R xi ln xi þ lngi þ T @T p,composition i¼1 i¼1 ( )   r X @[T ln gi ] ¼ R xi ln xi þ @T p,composition i¼1

Dsmixing ¼ 

Activity coefficient terms in the previous equation represent the nonideal contribution to Dsmixing. For example, if ln gi varies inversely with temperature for each component in the mixture, then the nonideal contribution to the entropy of mixing vanishes and all nonideal effects are enthalpic, as illustrated below. Now, it should be obvious that ideal solutions are athermal because the ideal contributions to Dgmixing and – T Dsmixing are identical, yielding fDhmixinggIdeal ¼ 0. The molar enthalpy of mixing Dhmixing in nonideal solutions is calculated by adding Dgmixing and T Dsmixing. One obtains the following result:   r X @ ln gi Dhmixing ¼ RT 2 xi @T p,composition i¼1 If the activity coefficient for each component is temperature independent, then all nonideal solution effects are entropic in origin. Similar conclusions about the temperature dependence of the Flory – Huggins intermolecular interaction parameter x in polymer – solvent mixtures, based on its enthalpic versus entropic origin, are provided

References

131

in Sections 3.4.4 and 5.3.2. Of particular importance here, one estimates the molar enthalpy of mixing in nonideal solutions when the partial molar volume of mixing for each component depends on composition, but not temperature and pressure. Under these conditions, there is a straightforward relation between Dhmixing and the molar volume change upon mixing, as illustrated below: ðp

RT ln gi ¼

(V i  vi ) dp  p(V i  vi )

p )0

p (V i  vi ) RT   r X @ ln gi 2 ¼ RT xi @T p,composition i¼1

ln gi  Dhmixing

¼p

r X

xi (V i  vi ) ¼ p Dvmixing

i¼1

At the macroscopic level of description, one concludes that mixtures experience volume contraction relative to linear additivity when heats of solution are exothermic. The lattice fluid theory of Sanchez and Lacombe [1978] is consistent with this conclusion from classical thermodynamic analyses of nonideal mixtures.

REFERENCES BATES FS, FREDRICKSON GH. Block copolymers—designer soft materials. Physics Today February:32–38 (1999). BELFIORE LA. Transport Phenomena for Chemical Reactor Design. Wiley, Hoboken, NJ, 2003, Chap. 25, p. 707. CAHN JW. Phase separation by spinodal decomposition in isotropic systems. Journal of Chemical Physics 42(1):93– 99 (1965). DEBENEDETTI PG, SHELL MS. Thermodynamics and the glass transition in model energy landscapes. Physical Review E 69(5):051102 (2004). DIMARZIO EA, GIBBS JH. Molecular interpretation of glass temperature depression by plasticizers. Journal of Polymer Science 1A:1417– 1428 (1963). DUDOWICZ J, FREED KF. Effect of monomer structure and compressibility on the properties of multicomponent polymer blends and solutions; lattice cluster theory of compressible systems. Macromolecules 24:5076–5095 (1991). DUDOWICZ J, FREED KF, MADDEN WG. Role of molecular structure on the thermodynamic properties of melts, blends, and concentrated polymer solutions; comparison of Monte Carlo simulations with cluster theory for the lattice model. Macromolecules 23:4803– 4819 (1990). DUDOWICZ J, FREED KF, DOUGLAS JF. Beyond Flory–Huggins theory; new classes of blend miscibility associated with monomer structural asymmetry. Physical Review Letters 88(9):5503 (2002). EUBANK PT, BARRUFET MA. Simple algorithms for the calculation of phase separation. Chemical Engineering Education winter:36– 41 (1988). FREED KF. Influence of monomer molecular structure on the glass transition in polymers; lattice cluster theory for the configurational entropy. Journal of Chemical Physics 119(11):5730 –5739 (2003).

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Chapter 3 Lattice Theories for Polymer –Small-Molecule Mixtures

FREED KF, DUDOWICZ J. Lattice cluster theory for pedestrians; the incompressible limit and the miscibility of polyolefin blends. Macromolecules 31:6681–6690 (1998). FREED KF, DUDOWICZ J. Influence of monomer molecular structure on the miscibility of polymer blends. Advances in Polymer Science 183:63–126 (2005). KAUZMANN W. The nature of the glassy state and the behaviour of liquids at low temperatures. Chemical Reviews 43(2):219– 256 (1948). OLABISI O, ROBESON LM, SHAW MT. Polymer–Polymer Miscibility. Academic Press, San Diego, CA, 1979, p. 21. SANCHEZ IC, LACOMBE RH. Statistical thermodynamics of polymer solutions. Macromolecules 11(6):1145– 1156 (1978). TANFORD C. Physical Chemistry of Macromolecules. Wiley, Hoboken, NJ, 1961, p. 248. TYLER CA, MORSE DC. Orthorhombic Fddd network in triblock and diblock copolymer melts. Physical Review Letters 94:208302, May 27 (2005). VAN DER PUT PJ. The Inorganic Chemistry of Materials; How to Make Things out of Elements. Plenum Press, New York, 1998, p. 348. ZENI D. Surface Functionalization by Smart Polymers, PhD thesis. University of Trento, Trento, Italy, 2010.

PROBLEMS 3.1. Statistical analysis of athermal binary mixtures that contain trimers and hexamers yields the following expression for the number of distinguishable ways (i.e., the multiplicity of states) to place molecules on a lattice of fixed size. Trimers occupy three adjacent lattice sites and hexamers require six adjacent sites. The multiplicity of states is ln V ¼ (3Ntrimer þ 6Nhexamer ) lnf3Ntrimer þ 6Nhexamer g  5Nhexamer (1  25 ln z)  2Ntrimer z(z  1) where V describes the number of distinguishable ways of arranging these molecules on the lattice, Ntrimer is the number of trimers, Nhexamer is the number of hexamers, and z is the lattice coordination number (i.e., the number of nearest neighbor sites that surround each lattice site). Hint: The total number of lattice sites is 3Ntrimer þ 6Nhexamer. (a) Obtain an expression for the volume fraction of trimers wtrimer in terms of Ntrimer and Nhexamer. Answer

wtrimer ¼

3Ntrimer 3Ntrimer þ 6Nhexamer

(b) Obtain an expression for the volume fraction of hexamers whexamer in terms of Ntrimer and Nhexamer. Answer

whexamer ¼

6Nhexamer 3Ntrimer þ 6Nhexamer

Problems

133

(c) Obtain an expression for the conformational entropy of mixing per mole of lattice sites. Answer  DSmixing ¼ k ln V(Ntrimer , Nhexamer )  ln V(Ntrimer ¼ 0, Nhexamer )  ln V(Ntrimer , Nhexamer ¼ 0)g  ¼ k (3Ntrimer þ 6Nhexamer ) lnf3Ntrimer þ 6Nhexamer g   6Nhexamer lnf6Nhexamer g  3Ntrimer lnf3Ntrimer g 



 3Ntrimer 6Nhexamer þ 6Nhexamer ln ¼ k 3Ntrimer ln 3Ntrimer þ 6Nhexamer 3Ntrimer þ 6Nhexamer DSmixing (NAvogadro ) ¼ Rfwtrimer ln wtrimer þ whexamer ln whexamer g 3Ntrimer þ 6Nhexamer All terms in the multiplicity expression that contain lattice coordination number z affect the combinatorial entropy via k ln V, similar to the Flory –Huggins lattice, but the final expression for the entropy of mixing does not depend on any lattice structural parameters that are required to simulate the mixing process. 3.2. For infinitely dilute solutions that contain extremely high-molecular-weight polymer molecules, the Flory–Huggins thermodynamic interaction parameter, per solvent molecule or lattice site, exhibits the following temperature dependence:

x ¼ 0:25 þ

250 T(K)

(a) At what temperature is phase separation initiated? Answer The spinodal point corresponds to x ¼ 0.5 for the conditions described in the problem statement (see Section 3.5.4 when x  1, wPolymer ) 0). Homogeneous solutions that do not exhibit spinodal decomposition require x to be less than 0.5, whereas phase separation will occur if x is greater than 0.5. Hence, the temperature range where phase separation occurs is described by

x ¼ 0:25 þ T(K) 

250 0:5 T(K)

250 ¼ 333 K 0:75

(b) Does the temperature, calculated in part (a), represent an upper or lower critical solution temperature? Answer Since phase separation occurs below 333 K, it is described as an upper critical solution temperature.

134

Chapter 3 Lattice Theories for Polymer –Small-Molecule Mixtures (c) Consider each term in the temperature-dependent expression for x and determine if it arises from enthalpic interactions, entropic interactions, or some combination thereof.

Answer The interaction free energy of mixing, per total moles of both components in solution, is proportional to the product of x and T, as well as some composition-dependent factors. As illustrated in Section 3.4.4, standard thermodynamic formalism yields

Dsmixing,interaction

Dhmixing,interaction

  @Dgmixing,interaction ¼ @T p,composition

 @(T x) ¼ RySolvent wPolymer @T p,composition 1 ¼ RySolvent wPolymer 4

  @ Dgmixing,interaction ¼ T 2 @T T p,composition

 @ x ¼ RT 2 ySolvent wPolymer @T p,composition ¼ 250 RySolvent wPolymer

Hence, the factor of –0.25, which is dimensionless, contributes favorably to the entropy of mixing. The temperature-dependent term that contains a coefficient of 250, with dimensions of Kelvin units, contributes to the enthalpy of mixing. The mixing process is endothermic because the enthalpy of mixing is always positive, except at the pure-component limits where either the solvent mole fraction or the polymer volume fraction vanishes. If one describes the mixing process by a chemical reaction, with pure components as reactants on the left side of the reaction, and the homogeneous mixture as the product on the right side, then the equilibrium constant for this process increases at higher temperature, as expected for endothermic reactions. This is consistent with the fact that the proposed temperature-dependent expression for x describes endothermic mixing in which the solvation power of the solvent increases at higher temperature until one achieves a homogeneous solution above the UCST (i.e., T 333 K). (d) Repeat the first three parts of this problem if the temperature-dependence of the Flory– Huggins interaction parameter is given by

x ¼ 0:75 

200 T(K)

Answer The nature of the polymer –solvent interactions and the type of solution phase behavior have significant effects on the sign of both coefficients in the proposed temperature-dependent expression for x. Now, homogeneous solutions exist below 800 K, which is a lower critical solution temperature because phase separation occurs at higher temperatures. Polymer–solvent

Problems

135

interactions have an adverse effect on the entropy of mixing, based on the positive temperatureindependent term (i.e., 0.75) in the expression for x. Mixing occurs exothermically at all temperatures, as described by the temperature-dependent term, but the solvation power of the solvent decreases at higher temperature. This is apparent from the proposed expression, because at higher temperature (i) the enthalpic contribution to x becomes less significant, (ii) x increases, and (iii) x exceeds the critical value of 0.5 at 800 K where the first evidence of phase separation occurs.

Chapter

4

dc Electric Field Effects on First- and Second-Order Phase Transitions in Pure Materials and Binary Mixtures A heart gives, a heart takes, in the quivering hope of transformation. —Michael Berardi

C

lassical thermodynamics is employed to analyze electric field effects on phase transitions. Analogous to the Ehrenfest equations/inequality, volume and entropy continuity at Tg are invoked to describe field effects on second-order phase transitions. The modified Prigogine – Defay [1954] ratio contains dielectric properties of the material, and their dependence on temperature and pressure. Computational strategies for nonideal binary mixtures are discussed to identify binodal and spinodal phase boundaries graphically as a function of field strength. Most importantly, one identifies the sign of the “dielectric susceptibility of mixing” as the critical thermodynamic property of mixtures that determines whether strong electric fields will drive systems toward homogeneity or phase separation.

4.1 ELECTRIC-FIELD-INDUCED ALIGNMENT AND PHASE SEPARATION Polar molecules respond to electric fields such that their electric dipole moments align with the field. In some respects, electric fields and mechanical tensile stress can be viewed similarly in an effort to produce molecular alignment. Polymer mobility above the glass transition opposes chain alignment, but electric-field ordering above Tg can be “frozen-in” upon lowering the temperature to produce glassy electrets that exhibit some similarity to piezoelectric devices which convert mechanical stress into Physical Properties of Macromolecules. By Laurence A. Belfiore Copyright # 2010 John Wiley & Sons, Inc.

137

138

Chapter 4 dc Electric Field Effects on First- and Second-Order Phase Transitions

electrical signals. Whereas mechanical stress is a macroscopic stimulus that affects all species in a mixture similarly, electric fields select the polar components such that it might be possible to design novel separation processes. Hence, electric-field-induced phase separation of nonideal binary mixtures that exhibit negative deviations from linear additivity could be an attractive alternative to generate two extremely pure phases when other types of separation processes are not feasible, provided that the field strengths required do not exceed the saturation threshold for dielectric breakdown.

4.2

OVERVIEW

There have been numerous studies of the effects of dc and alternating electric fields on phase transitions in small-molecule mixtures and high-molecular-weight polymers. Furthermore, textbook examples discuss modifications in the melting temperature of pure materials that are subjected to uniform dc fields. This thermodynamic analysis extends some of these predictions to (i) second-order phase transitions and (ii) binodal and spinodal phase boundaries in nonideal binary mixtures. Upon invoking both volume and entropy continuity via the integral approach to phase equilibrium at second-order transitions, electric field effects on the glass transition are developed that parallel the Ehrenfest [1933] equations for the pressure dependence of Tg. Both Tg – field equations predict small changes in the glass transition temperature that scale as the square of the electric field strength. If one equates the dependence of Tg on the magnitude of the electric field via (i) volume continuity and (ii) entropy continuity, it is possible to obtain the electric field analog of the Prigogine – Defay [1954] equality, in which thermophysical properties and discontinuous observables at the zero-field and field-dependent second-order phase transition temperatures are related. When the temperature and pressure dependencies of the relative electric permittivity (i.e., dielectric constant) are neglected in the absence of external fields, one recovers the classic Prigogine – Defay equality (i.e., the lower limit of the Prigogine – Defay [1954] ratio) that was developed from a consideration of volume and entropy continuity for the pressure dependence of Tg, by invoking the differential approach to phase equilibrium. In addition to electric field strength, zero-field activity coefficients and the mixture’s dielectric constant affect binodal and spinodal phase boundaries. When the field-independent interaction (i.e., excess) Gibbs free energy of mixing is described by the two-parameter van Laar model, simulation results are presented graphically to illustrate how electric fields affect the phase behavior of nonideal binary mixtures that do not experience electric saturation.

4.3 ELECTRIC FIELD EFFECTS ON LOWMOLECULAR-WEIGHT MOLECULES AND THEIR MIXTURES Phase diagrams, phase transitions, and solid state morphologies of low-molecularweight liquid crystals, homopolymers, copolymers, and polymer blends have been addressed in the presence of electric fields. Folkins et al. [1991] presented a detailed theoretical analysis to predict the phase diagram of betaine calcium chloride dihydrate

4.4 Electric Field Effects on Polymers and Their Mixtures

139

in externally applied electric fields. The strategy involved formulating an expression for the electric-field-dependent free energy via coupling ionic displacements to crystallographic polarization and, subsequently, identifying the interaction between polarization and the electric field. Equilibrium phase diagrams were calculated by minimizing the free energy with respect to all components of the crystal’s polarization vector. Kroupa [1991] measured dielectric constants and optical birefringence in betaine phosphates and arsenates. Low-temperature phase transitions in the vicinity of 80 –90 K for betaine phosphate and 135– 140 K for deuterated betaine phosphate split when electric fields on the order of 10 – 20 kV/cm induce the formation of a ferroelectric phase. Ye et al. [1991] measured a first-order phase transition in chromium chloride boracite at 264 K induced by mechanical stress. At slightly lower temperature, an electric-field-induced phase transition is observed at 250 K in chromium chloride boracite when the field strength exceeds 85 kV/cm. Pershin and Konoplev [1990] studied orientational order due to electric fields from a theoretical viewpoint. Generic field – temperature phase diagrams were constructed for plastics and liquid crystalline materials. Kuczynski et al. [1991] studied electric field effects on the smectic-C to smectic-A phase transition at 30.5 8C in 4-octyloxy-4-[(2-methyl butyloxy) carbonyl] phenylbenzoate. When a 50-volt dc bias is applied across a planaroriented sample having a thickness of 30 mm, the liquid crystalline phase transition is broadened severely, as detected via light modulation measurements. However, the critical temperature of the smectic-C to smectic-A phase transition is independent of electric field strength [Kuczynski et al., 1991]. Coles and Gleeson [1989] observed electric-field-induced phase transitions in mixtures of cyano-biphenyls with a variety of chiral esters. These blends exhibit mesophases (i.e., blue phases) that occur over a 2– 3 8C temperature range between the chiral nematic liquid crystalline phase and the isotropic molten state. In the presence of ac electric fields, these blue phases transform to (1) chiral nematic phases and (2) homeotropic nematic phases. Characteristic time constants for these field-induced phase transitions are on the order of 100– 200 ms. Kitzerow et al. [1990] studied electric-field-induced phase transitions in liquid crystalline mixtures containing 30 wt % of the chiral compound 1-methyl-heptyloxybenzoyl-4-hexyloxybenzoate. In the absence of external fields, three distinct blue phases are observed sequentially over a 1.1 8C temperature range as the mixture is heated from the cholesteric state at 43.5 8C to the isotropic molten state at 44.6 8C. The boundary between the cholesteric phase and the first blue mesophase exhibits positive temperature dependence on the electric field – temperature projection of the phase diagram for rms field strengths between 40 and 80 V applied across a 12-mm thick sample. Electric fields also affect the phase behavior of lipids in biological membranes. Antonov et al. [1990] demonstrated that a 150-mV bias increases the phase transition temperature of phosphatidic acids by 8– 12 8C.

4.4 ELECTRIC FIELD EFFECTS ON POLYMERS AND THEIR MIXTURES Amundson et al. [1991, 1992, 1993] used small-angle X-ray scattering and optical birefringence measurements via laser-beam technology to demonstrate that a uniform

140

Chapter 4 dc Electric Field Effects on First- and Second-Order Phase Transitions

external electric field can induce macroscopic alignment of block copolymer microstructures. When polystyrene – poly(methylmethacrylate) diblock copolymers are cooled below the molecular-weight-dependent order – disorder transition at 251 8C (MW  37 kDa), stationary plane wave composition patterns develop that mimic lamellae. In an effort to minimize the anisotropic electrostatic contribution to the free energy associated with plane wave patterns, these lamellar surfaces orient predominantly orthogonal to the electrode surfaces. Binary mixtures of polystyrene with poly(ethylene oxide) (PEO) [Wnek et al., 1990] and ternary blends of polystyrene, poly(ethylene oxide), and a polystyrene – poly(ethylene oxide) diblock copolymer [Serpico et al., 1991, 1992] exhibit phase orientation of the minor component (PEO) in the presence of either ac or dc electric fields. The PEO morphology is best described as “pearl chains” oriented in the direction of the field due to the dielectric-constant mismatch between dissimilar phases. Column-like structures form when the applied field is larger than 2 kV/cm. In the ternary polymer – copolymer blends mentioned above, the PS– PEO diblock copolymer (i) reduces interfacial surface tension between immiscible polystyrene and poly(ethylene oxide) phases and (ii) allows the formation and stabilization of an elongated phase as viewed in the optical microscope. Hsu and Lu [1986] investigated isothermal crystallization of undiluted poly(vinylidene fluoride) upon cooling from the molten state in the presence of relatively weak electric fields. Whereas the helical a phase is favored under zero-field conditions, annealing temperatures in the vicinity of 100– 170 8C and moderate field strengths on the order of 70 kV/cm induce a solid – solid phase transformation from the a form to the g form [Hsu and Lu, 1986; Marand and Stein, 1989]. The latter polymorph (i.e., g form of poly(vinylidene fluoride)) is characterized by a large degree of trans sequences as detected by infrared spectroscopy. Koga et al. [1990] observed electric-field-induced phase transformations in copolymers of vinylidene fluoride with trifluoroethylene. At ambient pressure, copolymers with a high content of vinylidene fluoride (i.e., 82– 90 mol %) crystallize as a mixture of three different modifications, identified as a, b, and g, which are isomorphic to the crystallographic modifications of poly(vinylidene fluoride) homopolymers. In the presence of strong ac electric fields on the order of 1600 kV/cm cycling at 1 kHz, mixed-phase copolymer crystals transform completely into the ferroelectric b polymorph as detected by X-ray diffraction, differential scanning calorimetry, and infrared spectroscopy. Tadokoro et al. [1984] studied the ferroelectric-(trans)/paraelectric-(gauche) phase transition at 125 8C in copolymers of vinylidene fluoride with trifluoroethylene subjected to strong ac electric fields, focusing on low-frequency response of the polarization inversion current over a wide temperature range in copolymers that contain a lower content of vinylidene fluoride (i.e., 75 mol % VDF). Electric dipole response to the polarization inversion experiment was interpretted in terms of temperaturedependent motional models for trans – gauche conformational rearrangements of the copolymer segments. Reynolds et al. [1989] also studied microstructural changes that occur in copolymers of vinylidene fluoride (75 mol % VDF) with trifluoroethylene using variable-temperature infrared spectroscopy. Films were subjected to uniaxial tensile deformation with a draw ratio of 1.6, and strong electric fields on the order of 2.2 MV/cm. Infrared data reveal that the crystallites are sensitive to mechanical stress and align with the stretching direction. Chains within the crystallites rotate in

4.6 Theoretical Considerations

141

response to the electric field. Strong temperature-dependent hysteresis of infrared absorption intensity is observed below the Curie temperature when the electric field is cycled between either +1 MV/cm or +2.2 MV/cm. In some respects, the application of an electric field disrupts the orientation that was generated by mechanical deformation. Yitzchaik et al. [1990] employed electric fields on the order of 10– 50 kV/cm to induce asymmetric ordering of dye aggregates in polymeric matrices containing side-chain nematic liquid crystalline moieties. Second harmonic generation coefficients reveal that substituted stilbene derivatives develop orientation both parallel and perpendicular to the dc electric field vectors. Previous investigations of electric-field-induced phase behavior that can be described by the developments in this chapter were published by Reich and Gordon [1979], who applied dc electric fields across thin films of polystyrene (MWPS ¼ 3104) and poly(vinyl methyl ether) MWPVME ¼ 1.4104, and detected cloud points (i.e., on the binodal curve) via laser light scattering techniques. Electric-field-induced phase separation in this classic blend, which exhibits LCST behavior, occurs at 82 8C when the field strength is 272 kV/cm. This observation represents a decrease of 54 8C in the cloud point at the highest field reported, relative to a cloud point of 136 8C in the absence of electric fields.

4.5 MOTIVATION FOR ANALYSIS OF ELECTRIC FIELD EFFECTS ON PHASE TRANSITIONS As indicated in the previous two sections, there have been numerous investigations of electric field effects on phase transitions in a variety of materials, but theoretical and experimental analyses of the glass transition have not occurred. This is a difficult problem, experimentally, because very strong electric fields are required that might exceed the threshold for dielectric breakdown, which is on the order of 30 kV/cm for air and 180 kV/cm for polyethylene. The energy required to orient electric dipoles (i.e., on the order of 10E 2, with dimensions of energy/volume) must compete with the thermal energy associated with stochastic Brownian motion (i.e., NRT/V ) that tends to disrupt dipolar alignment when dielectrics are poled in the molten state. Furthermore, heating an electret in the absence of the field to measure Tg might cause thermally induced misalignment of electric dipoles below the phase transition such that artifacts obscure the true effect of the applied field. Suffice it to say that experimental data are rare and theoretical predictions have not appeared in the literature. First-order phase transition temperatures are much more sensitive to electric fields than secondorder phase transitions. This chapter focuses on theoretical predictions of Tg modification and phase behavior in the presence of dc electric fields.

4.6 4.6.1

THEORETICAL CONSIDERATIONS Electrostatic Preliminaries

The macroscopic formalism to predict phase boundaries, critical solution temperatures, and phase transition temperatures for dielectrics in the presence of static uniform

142

Chapter 4 dc Electric Field Effects on First- and Second-Order Phase Transitions

electric fields begins with the energy representation of the first law of thermodynamics [Kirkwood and Oppenheim, 1961; Ambrosone and Fontana, 2005]. Dielectric materials alter the magnitude of the effective field everywhere within the solid, and the fact that electric fields do not vanish within dielectrics, because they do not support a steady flow of current, has a significant effect on the material’s thermodynamic properties [Landau et al., 1984]. A classical electromagnetic description of internal field distortions due to the accumulation of charge at curved interfacial boundaries that might appear when phase separation occurs is beyond the scope of this chapter. It should not be surprising that rigorous thermodynamic analysis of electric fields is not as simple as the corresponding study of gravitational and centrifugal fields. By invoking a simplification of Faraday’s law for steady state “electrostatic” cases, which states that the curl of the electric field should vanish [Jackson, 1975], the electric field vector E is written as the gradient of a scalar potential. This potential is better known as the electric potential, and it is synonymous with the position-dependent applied voltage, V(r), where r is the position vector with respect to a stationary frame of reference. If the electric potential V is an exact differential, as expected, then it satisfies the following equation: r  E ¼ r  rV ¼ 0 and the specific electric potential for any particular problem is obtained by solving Gauss’ law based on the charge enclosed within the control volume. The force qE experienced by an electric dipole due to the electric field E contributes a work term to the internal energy of the mixture. For dielectrics in electrostatic equilibrium, this work term is directly related to the fact that the surface charge distribution is modified by the electric field. The appropriate work term is given by the scalar (dot) product of the electric field vector E with the differential change in the total dipole polarization vector d P, via the classical mechanics analog of







F dr ¼ qE dr ¼ E q dr where q dr represents a differential change in the dipole moment as individual charge q experiences displacement dr in the field. The dipole polarization vector P of dielectric materials, with dimensions of charge-cm, is defined by the ensemble average of microscopic dipole moments [Landau et al., 1984]. If dielectric materials contain point charges qi at position ri in a laboratory-fixed coordinate system, then the volume-average of the electric dipole moments yields the dipole polarization density vector p(v) with dimensions of charge-cm/cm3, and the total dipole polarization vector is given by the product of p(v) with system volume V: *X + q r (v) i i P ¼ Vp ¼ point charges

If dielectric materials are polarized in the presence of an electric field, then P is nonzero. Furthermore, the average charge density, with dimensions of charge per volume, and the ensemble average of all point charges can be obtained from the divergence of

4.6 Theoretical Considerations

143

the total dipole polarization vector. For example, *X + 1 qi ¼ r p(v) V point



charges

*X point charges

qi

+

  ¼ V r p(v) ¼ r P





Blends of polar polymers or systems that contain liquid crystalline molecules should respond linearly to an external dc electric field if the field strength is not exceedingly large. For isotropic dielectrics that respond linearly to externally applied fields, the total dipole polarization vector P and the electric displacement vector D are parallel to the electric field vector everywhere within the solid medium [Jackson, 1975; Landau et al., 1984]. These relations represent the first-order term for Taylor series expansions of P and D in terms of powers of E, and subsequent truncation due to the fact that internal molecular fields are much stronger than the external field. The scalar proportionality constant that relates electric displacement D to the electric field E is called the dielectric permeability, or the permittivity 1, with dimensions of charge per volt-cm (i.e., D ¼ 1E). The scalar proportionality constant that relates the total dipole polarization P to the electric field E is called the polarization coefficient or the dielectric susceptibility k, defined by [Jackson, 1975] k ¼ V {1  10 } 4p with dimensions of charge-cm2 per volt (i.e., P ¼ kE). k is positive for all materials with volume V, and 10 is the permittivity of free space (i.e., in a vacuum). The previous equation is obtained from the definition of electric displacement D in polarized media [Jackson, 1975], in terms of the electric field E and the polarization density P/V: D ¼ 1E ¼ 10 E þ 4p P V The dielectric constant of the medium, or the relative electric permittivity, is defined as the ratio of 1 to 10. Typical values of the dielectric constant for common materials are (i) 1 for a vacuum, (ii) slightly greater than 1 for gases, (iii) approximately 3 for plastics, (iv) approximately 5 for glass (i.e., silicon dioxide), and (v) approximately 78 for water at 25 8C. The dielectric susceptibility k can also be described as the first derivative of the total dipole polarization with respect to the external electric field in the zero-field limit, where scalars are employed to characterize the total polarization and the external field.

4.6.2 Classical Thermodynamics in the Presence of External Fields If one adds the work term



F dr ¼ E



*X point charges

qi dri

+

144

Chapter 4 dc Electric Field Effects on First- and Second-Order Phase Transitions

experienced by electric dipoles in the presence of a uniform field E to the classic pV– work, then the differential form of the first law of thermodynamics for N-component mixtures in the presence of an electric field [Landau et al., 1984; Ambrosone and Fontana, 2005], dU ¼ T dS  p dV þ

N X



mi dNi þ E dP

i¼1

reveals that the extensive internal energy U(S, V, all Ni, P) is a function of entropy S, volume V, mole numbers Ni, and the total polarization P. Work must be performed on the system (i.e., positive contribution to dU ) in the presence of a uniform field to increase the polarization vector by d P due to differential displacement dri of charge qi. All independent natural variables of the internal energy are extensive. A multivariable Legendre transformation from the internal energy U to the energetic state function C [Damjanovic, 1998],



C(T, p, all Ni , E) ¼ U þ pV  TS  E P generates the following differential expression for C in which temperature T, pressure p, mole numbers Ni, and electric field E now represent the independent natural variables: dC ¼ S dT þ V dp þ

N X



mi dNi  P dE

i¼1

Since T, p, and E are intensive natural variables for the thermodynamic potential C, Euler’s theorem for thermodynamic functions that are homogeneous to the firstdegree with respect to the extensive independent variables [Belfiore, 2003] yields the following expression for C: N X Ni mi C¼ i¼1

proving that the chemical potential of species i, mi, is a partial molar property of C, which should be interpreted as the Gibbs free energy in the presence of uniform electric fields [Damjanovic, 1998]. Several Maxwell relations based on second-mixed partial derivatives of the thermodynamic state function C, defined above via Legendre transformation of the internal energy U, are useful to construct the total differential of the chemical potential of species i. For example,     @ mi ¼  @S ¼ Si @T p,E,composition @Ni T,p,E,all N j½ j=i     @ mi @V ¼ ¼ Vi @Ni T,p,E,all N j½ j=i @p T,E,composition     @ mi ¼  @P ¼ Pi @E T,p,composition @Ni T,p,E,all N j½ j=i

4.6 Theoretical Considerations

145

These relations identify the partial molar entropy, volume, and polarization of species i via temperature, pressure, and electric field dependence of mi, respectively. There are N þ 1 degrees of freedom for homogeneous mixtures of N components in the absence of external fields, and one additional intensive variable, like E, is required to characterize each field. Since the chemical potential of species i is an intensive exact differential, the functional dependence and total differential of mi are given by

mi (T, p, E, x1 , x2 , . . . , xN1 )  N 1  X @ mi dmi ¼ Si dT þ Vi dp  Pi dE þ dxj @xj T,p,E,all xk[k=j,N] j¼1 where xj is the mole fraction of species j in the mixture, and one additional intensive independent variable is required to characterize the external electric field. Useful Maxwell relations based on second-mixed partial derivatives of mi are presented below. The overall objective here is to evaluate the electric-field dependence of (i) chemical potentials, (ii) partial molar entropies, (iii) partial molar enthalpies, and (iv) partial molar volumes that can be used to quantify the requirements for first- and second-order phase equilibrium, and chemical stability. Hence,     @Si @Pi ¼ @E T,p,composition @T p,E,composition     @Vi @Pi ¼ @E T,p,composition @p T,E,composition The total polarization P of isotropic mixtures that do not exhibit ferroelectric behavior can be written as P ¼ kmixture E and the field dependence of the mixture’s dielectric susceptibility kmixture is typically expanded in a Taylor series about its zero-field value. Except when electric saturation occurs in strong external fields, all terms in the expansion for kmixture beyond the zerothorder contribution are typically truncated. Consequently, the leading field-dependent terms in the expression for mi and all of the other partial molar properties (i.e., entropy, enthalpy, and volume) scale as E 2 (i.e., the square of the magnitude of the electric field strength). This is illustrated below for each partial molar property of interest for phase behavior consideration. Chemical potential       @ mi @ kmixture @P ¼ ¼ E @E T,p,composition @Ni T,p,E,all N j½ j=i @Ni T,p,E,all N j½ j=i

mi (T, p, E, composition) ¼ mi (T, p, composition; E ¼ 0)  12 E2 ki þ   

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Chapter 4 dc Electric Field Effects on First- and Second-Order Phase Transitions

The partial molar dielectric susceptibility of species i in a mixture, defined by     @ kmixture 1 @V(1mixture  10 ) ki ¼ ¼ @Ni @Ni T,p,E,all N j[ j=i] 4p T,p,E,all N j[ j=i]   @1mixture 4pki ¼ V þ (1mixture  10 )Vi @Ni T,p,N j[ j=i] ,E strongly influences electric field effects on the chemical potential of species i. Partial molar entropy 

@Si @E

#   @ @P ¼ @T @Ni T,p,E,all N j[ j=i] T,p,composition p,E,composition "  #  @ kmixture @ ¼E @Ni T,p,E,all N j[ j=i] @T



"

p,E,composition

¼E

n

@ ki @T

o p,E,composition

n o @ ki Si (T, p, E, composition) ¼ Si (T, p, composition; E ¼ 0) þ 1 E 2 þ @T p,E,composition 2 Temperature dependence of the partial molar dielectric susceptibility strongly influences electric field effects on the partial molar entropy of species i. Partial molar volume "   #   @Vi @ @P ¼  @p @Ni T,p,E,all N j[j=i] @E T,p,composition T,E,composition "  #  @ @ kmixture ¼E @p @Ni T,p,E,all N j[j=i] T,E,composition   @ ki ¼E @p T,E,composition   1 2 @ ki Vi (T, p, E, composition) ¼ Vi (T, p, composition; E ¼ 0)  E þ @p T,E,composition 2 Pressure dependence of the partial molar dielectric susceptibility strongly influences electric field effects on the partial molar volume of species i. The partial molar enthalpy of species i is obtained from the definition of thermodynamic state function C via Legendre transformation of the internal energy U. Upon taking the partial

4.6 Theoretical Considerations

147

derivative of C with respect to mole numbers Ni at constant temperature, pressure, field strength, and mole numbers of all other species in the mixture, one obtains the following equation for partial molar enthalpy:





C(T, p, all Ni , E) ¼ U þ pV  TS  E P ¼ H  TS  E P       mi (T, p, E, composition) ¼ @C ¼ @H  T @S @Ni T,p,E,N j[ j=i] @Ni T,p,E,N j[ j=i] @Ni T,p,E,N j[ j=i]   @P E @Ni T,p,E,N j[ j=i]



The electric-field-dependent classical thermodynamic developments in this chapter allow one to simplify the previous equation:   Hi (T, p, E, composition) ¼ @H @Ni T,p,E,N j[ j=i]



¼ mi þ TSi þ E Pi  mi (T, p, composition; E ¼ 0)  12 E 2 ki   n o 1 2 @ ki þ T Si (T, p, composition; E ¼ 0) þ 2 E þ E 2 ki @T p,E,composition   n o @ ki þ  ¼ Hi (T, p, composition; E ¼ 0) þ 12 E 2 ki þ T @T p,E,composition n o @(T ki ) þ ¼ Hi (T, p, composition; E ¼ 0) þ 12 E 2 @T p,E,composition All of these electric field effects on thermodynamic properties (1) are consistent with those in the classic treatise on “Electrostatics of Dielectrics” by Landau et al. [1984], and (2) directly parallel the development from Chemical Thermodynamics by Kirkwood and Oppenheim [1961]. The thermodynamic properties of polarizable molecules should not be affected by electric fields if mixtures are chosen judiciously such that the dielectric susceptibility of the medium is composition independent. Whereas Kirkwood and Oppenheim [1961] discuss the effect of uniform dc electric fields on first-order phase transition temperatures for two-phase equilibrium of a pure material, the following sections are applicable to first- and second-order phase transitions that describe two-phase equilibrium in binary and multicomponent mixtures.

4.6.3 First-Order Transition Temperatures Via Phase Equilibrium on the Binodal Curve The binodal phase boundary on the temperature – field diagram at constant composition x, where x represents the only independent mole fraction in binary mixtures,

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Chapter 4 dc Electric Field Effects on First- and Second-Order Phase Transitions

is generated by invoking the integral form of the chemical requirement for two-phase equilibrium between phases a and b. Hence, for each component in binary mixtures (i.e., j ¼ 1, 2), n n

mj (T(E), p, x; E ¼ 0)

mj (T(E), p, E, x)

o a

o a

n o ¼ mj (T(E), p, E, x)

b

n

 12 (kj )a E 2 þ    ¼ mj (T(E), p, x; E ¼ 0) 

1 2 2 (k j )b E

o b

þ 

It is important to emphasize that temperature T(E) in the previous equation represents the a ) b phase transition temperature in the presence of the field E. When the field is absent, the phase transition temperature is T0 and the zero-field chemical potential of species j (i.e., mj (T0, p, x; E ¼ 0)) must be equivalent in each phase. The following strategy is employed in the remainder of this section to predict electric field effects on first-order phase transition temperatures in pure materials and mixtures that correspond to chemical equilibrium on the binodal curve: Step 1:

Use classical thermodynamics in the absence of external fields to predict the temperature dependence of zero-field chemical potentials.

Step 2: Evaluate the difference between zero-field chemical potentials for species j in phases a and b at temperatures T and T0. This chemical potential difference at temperature T0 vanishes because T0 corresponds to the a ) b phase transition in the absence of the electric field. Step 3: Use the previous equation for each component in binary mixtures to predict two-phase equilibrium on the binodal curve via electric field effects on firstorder transition temperatures. Step 4: Average the prediction of first-order phase transition temperatures from Step (3) for each component in the mixture. Under zero-field conditions (i.e., E ¼ 0), the chemical potential of species j is a partial molar property of the Gibbs free energy, and the temperature dependence of mj (T, p, composition) is described by zero-field partial molar entropies. The following expression is similar to the one employed for prediction of equilibrium constants at various temperatures:     n o @(mj =T) @ mj mj (T, p, x; E ¼ 0) 1 1 m þ TS ¼  ¼  j j E¼0 @T T2 T @T p,x,E¼0 T2 p,x,E¼0 ¼  12 Hj (T, p, x; E ¼ 0) T When zero-field partial molar enthalpies are not strong functions of temperature, simple integration of the previous equation from T0 to T provides a useful

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149

approximation for the temperature dependence of chemical potentials in the absence of the field:   mj (T(E), p, x; E ¼ 0) mj (T0 , p, x; E ¼ 0) 1 1   Hj (E ¼ 0)  T(E) T0 T(E) T0   T(E) T(E) mj (T(E), p, x; E ¼ 0)  mj (T0 , p, x; E ¼ 0) þ 1  Hj (E ¼ 0) T0 T0 Now, one invokes the integral representation of chemical equilibrium on the binodal curve where the a ) b phase transition temperature is T(E) in the presence of the external electric field, realizing that mj (T0, p, x; E ¼ 0) must be equivalent in both phases for species j: {mj (T(E), p, x; E ¼ 0)}a  12 (kj )a E 2  {mj (T(E), p, x; E ¼ 0)}b  12 (kj )b E 2 {mj (T0 , p, x; E ¼ 0)}a ¼ {mj (T0 , p, x; E ¼ 0)}b     Ta)b (E) Ta)b (E) [Hj (E ¼ 0)]a  12 (kj )a E 2  1 [Hj (E ¼ 0)]b  12 (kj )b E2 1 T0 T0 (k j )b  (k j )a Ta)b (E)  1  12 E 2 Ta)b (E ¼ 0) [Hj (E ¼ 0)]b  [Hj (E ¼ 0)]a Hence, electric field effects on first-order phase transition temperatures scale as E 2. Reich and Gordon [1979] have demonstrated this phenomenon experimentally for the lower critical solution temperature (LCST) in binary mixtures of polystyrene and poly(vinylmethylether). More importantly, the ratio of the difference between field-dependent partial molar dielectric susceptibilities (i.e., phase b relative to phase a) to the difference between zero-field partial molar enthalpies in phase b relative to phase a governs whether the phase transition shifts to higher or lower temperature at stronger fields. Since phase transition temperatures for mixtures should be the same for all components (i.e., independent of subscript j on the right side of the previous equation) and the previous equation for T(E) was obtained from the integral approach to phase equilibrium by equating the chemical potential of species j in both phases, it is suggested that the important ratio of partial molar properties that governs the sign of the field-dependent temperature shift should be averaged over all N species in the mixture. Hence, at constant pressure and mixture composition, N 2 X (kj )b  (kj )a Ta)b (E) 1 E Ta)b (E ¼ 0) 2N j¼1 [Hj (E ¼ 0)]b  [Hj (E ¼ 0)]a

For pure materials (i.e., N ¼ 1), partial molar properties reduce to pure-component molar properties, and the denominator of the important ratio in the previous equation that governs the sign of the temperature shift will be positive if phase b is thermodynamically favored at higher temperature, because it represents the zero-field molar

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Chapter 4 dc Electric Field Effects on First- and Second-Order Phase Transitions

enthalpy change for endothermic phase transitions. The previous equation is written explicitly for melting of pure materials with dielectric constants that are greater than unity (i.e., 1 . 10), where the crystalline solid is phase a, the liquid is phase b, and v represents the field-dependent molar volume which is discontinuous at Tmelt: 2 {v(1  10 )}Liquid  {v(1  10 )}Crystal Tmelt (E) 1E 8p Dhmelting,E¼0 Tmelt (E ¼ 0)

The permittivity of the medium is 1 (i.e., scalar ratio of electric displacement to electric field for isotropic dielectrics), and v(1 2 10)/4p is the molar dielectric susceptibility of pure materials. This is the electric field analog of the classic Clapeyron equation that describes the pressure dependence of melting transitions, as discussed in Chapter 1. Hence, if first-order melting transitions in pure materials shift to higher temperature in the presence of stronger electric fields due to dipolar ordering, then the molar dielectric susceptibility of the liquid must be smaller than its counterpart in the crystalline solid, even though most materials exhibit a discontinuous increment in molar volume at Tmelt, because the addition of thermal energy to induce melting invariably disrupts the alignment of these electric dipoles. It must be emphasized that severely restricted mobility in the crystalline state hinders the development of polarization in the presence of dc electric fields below Tmelt, so one might conclude that the dielectric susceptibility should be larger in a state of greater mobility. However, electrets are prepared by heating materials into a state of significant mobility, such that dipoles can respond to strong electric fields. Then, the total polarization that develops is frozen-in via cooling below the melting temperature. Next, the crystalline solid is heated in a calorimeter to detect the effect of frozen-in dipolar alignment, prepared in the presence of strong electric fields above Tmelt, on the first-order phase transition temperature Tmelt (E), even though the calorimetric experiments can be performed under zero-field conditions.

4.6.4 Second-Order Phase Transitions in Pure Materials External electric or magnetic fields induce order in molten amorphous materials as electric or magnetic dipoles align in response to the field. Upon forming electrets by lowering the temperature below the glass transition and freezing-in this dipolar alignment, it seems reasonable that higher temperatures should be required to disrupt this frozen-in alignment and induce large-scale translational motion of the chain backbone, as well as reptation, above Tg. Since enthalpy is continuous at secondorder phase transitions for both pure materials and mixtures, the methodology required to analyze first-order phase transitions must be modified for second-order transitions to include discontinuous observables that can be measured or predicted. It is not superficially obvious that l’Hoˆpital’s rule can be applied to the previous equation, in quest of the electric field dependence of Tg because, even though volume and enthalpy are continuous at second-order phase transitions, continuity of the dielectric permeability of the liquid and glass at Tg might not be a valid assumption

4.6 Theoretical Considerations

151

(see the discussion below). Hence, the analyses in this section parallel the Ehrenfest approach by invoking the integral representation of phase equilibrium via volume and entropy continuity at Tg in the presence of external fields. It should be emphasized that Tg is a kinetic transition via actual experimental measurements, not a true equilibrium second-order phase transition. In light of this fact, one expects that electric field effects on both the kinetic and equilibrium glass transition temperatures should be similar.

Entropy Continuity Let’s consider pure materials and develop the analog of the previous equation for second-order phase transitions by invoking entropy continuity (i) at temperature Tg(E) in the presence of a uniform electric field and (ii) at Tg(E ¼ 0) under zerofield conditions. Begin with the field-dependent expression for partial molar entropy that was developed previously in this chapter, because partial molar properties reduce to molar properties for pure materials. Then, adopt the integral approach to phase equilibrium for second-order transitions and equate the molar entropy s in the glassy state (i.e., phase a) and the liquid state (i.e., phase b): {s(Tg (E), p, E)}a ¼ {s(Tg (E), p, E)}b 2

s(Tg (E), p, E) ¼ s(Tg (E), p; E ¼ 0) þ E 8p   2 @[v(1  1 )] 0 a {s(Tg (E), p; E ¼ 0)}a þ E  8p @T p,E   2 @[v(1  10 )]b {s(Tg (E), p; E ¼ 0)}b þ E 8p @T p,E

n

@[v(1  10 )] @T

o

þ

p,E

where the partial molar dielectric susceptibility reduces to the molar dielectric susceptibility of the appropriate phase, v represents molar volume, and Tg in the previous equation corresponds to the second-order phase transition temperature in the presence of the field (i.e., Tg(E)). It is important to emphasize that zero-field entropies of the glass and liquid are equivalent at Tg(E ¼ 0), but not at Tg(E), because the phase transition is affected by the field. Hence, it is necessary to analyze the temperature dependence of zero-field entropy from Tg(E ¼ 0), where s[Tg(E ¼ 0), p; E ¼ 0] is the same in both phases, to Tg(E) where s[Tg(E), p, E] is the same in both phases, but s[Tg(E), p; E ¼ 0] differs between phases a and b. Temperature dependence of zero-field molar entropy at constant pressure is described by the following equations when zero-field molar heat capacities (i.e., Cp,E¼0) are either (i) temperature-averaged or (ii) very weak functions of temperature such that they can be treated as constants: n o C p,E¼0 @s ¼ T @T p,E¼0   Tg (E) s[Tg (E), p; E ¼ 0] ¼ s[Tg (E ¼ 0), p; E ¼ 0] þ C p,E¼0 ln Tg (E ¼ 0)

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Chapter 4 dc Electric Field Effects on First- and Second-Order Phase Transitions

This analysis agrees with the zero-field total differential of pure-component molar enthalpy at constant pressure (i.e., dh ¼ T ds þ v dp ) Cp dT). The electric field analog of the Ehrenfest [1933] equation for second-order phase transitions, based on entropy continuity, is 

   2 @[v(1  1 )] Tg (E) 0 a þE 8p @T Tg (E ¼ 0) p,E     2 @[v(1  10 )]b Tg (E) E þ  {s[Tg (E ¼ 0), p; E ¼ 0]}b þ {Cp,E¼0}b ln 8p @T Tg (E ¼ 0) p,E     @[v(1  10 )]Liquid @[v(1  10 )]Glass    2 @T @T Tg (E) p,E p,E E ln 8p {Cp,E¼0}Liquid  {Cp,E¼0 }Glass Tg (E ¼ 0)

{s[Tg (E ¼ 0), p; E ¼ 0]}a þ {Cp,E¼0}a ln

There are no known exceptions to the fact that the denominator is greater than zero in the important ratio of discontinuous observables at Tg, which governs the sign of the temperature shift in the previous equation. Hence, if electric field ordering due to dipolar alignment increases the glass transition temperature, then the temperature dependence of the molar dielectric susceptibility must be smaller in the liquid state relative to the glassy state, even though field-dependent thermal expansion coefficients, which are directly related to (@v/@T )p,E, exhibit a discontinuous increment upon heating through the second-order phase transition. Hindsight reveals that application of l’Hoˆpital’s rule to the effect of electric fields on first-order phase transitions (i.e., Tmelt), via differentiation of numerator and denominator with respect to temperature, yields the previous equation for field effects on second-order phase transitions if Tg(E) is not significantly different from Tg(E ¼ 0) and ln(1 þ x)  x 2    is expanded in a Taylor series, followed by truncation after the linear term. The previous equation can be simplified if x ¼ {Tg(E)2Tg(E ¼ 0)}/Tg(E ¼ 0): 

Tg (E) ln Tg (E ¼ 0)





 Tg (E)  Tg (E ¼ 0) Tg (E) ¼ ln 1 þ  1 Tg (E ¼ 0) Tg (E ¼ 0)     @[v(1  10 )]Liquid @[v(1  10 )]Glass  2 @T @T p,E p,E E 8p {Cp,E¼0}Liquid  {Cp,E¼0}Glass

The static dielectric permeability is continuous at Tg [Dissado and Hill, 1983] as evidenced by its nonmonotonic temperature dependence in the dipole-glass state of ferroelectric relaxors, such as metal oxide alloys that contain various combinations of lead, magnesium, niobium, scandium, tantalum, titanium, zirconium, and lanthanum [Glinchuk and Stephanovich, 1998]. Experiments reveal that the nonlinear dielectric susceptibility of the electric dipole glass KCl : OH does not diverge at the freezing temperature, which is defined as the temperature of maximum dielectric

4.6 Theoretical Considerations

153

susceptibility [Saint-Paul and Gilchrist, 1986]. One concludes that the static dielectric permeability and the static dielectric susceptibility are continuous at secondorder phase transitions [Dissado and Hill, 1983], but their temperature derivatives are discontinuous, unless electric fields have no effect on the glass transition temperature. Hence, application of l’Hoˆpital’s rule to Tmelt (E) yields the previous equation for Tg(E) via differentiation (i.e., numerator and denominator separately) of the electric field analog of the Clapeyron equation with respect to temperature at constant pressure, composition (i.e., for mixtures), and field strength. Several III– V and II– VI semiconductors exhibit positive temperature dependence of their dielectric constants (i.e., 1/10) in the solid state at 300 K and 1 atmosphere, {@ ln(1/10)/@T}p  1.5  1024 K21, when 1/10  10 [Samara, 1983]. This corresponds to a 3 K change in Tg via the previous equation for very strong static fields (i.e., E  10 MV/cm) when the zero-field glass transition is 423 K (i.e., 150 8C). For comparison, the dielectric constant of H2O at 1 atmosphere is 78 at 25 8C and 100 at 240 8C, so {@1/@T}p , 0 upon passing through the solid – liquid first-order phase transition.

Volume Continuity Now, the integral approach to second-order thermodynamic phase transitions is analyzed by invoking volume continuity (i) at temperature Tg(E) in the presence of a uniform electric field and (ii) at Tg(E ¼ 0) under zero-field conditions. One should obtain the same expression for Tg(E), discussed below, by applying l’Hoˆpital’s rule to Tmelt (E) via differentiation (i.e., numerator and denominator separately) of the electric field analog of the Clapeyron equation with respect to pressure at constant temperature, composition (i.e., for mixtures), and field strength. The fielddependent expression for partial molar volume is simplified for pure materials by replacing partial molar dielectric susceptibility with molar susceptibility, where the latter contains molar volume. Upon equating molar volume v in the glassy state (i.e., phase a) and the liquid state (i.e., phase b), one obtains 

v(Tg (E), p, E)

 a

  ¼ v(Tg (E), p, E) b

  2 @[v(1  1 )] E 0 v(Tg (E), p, E) ¼ v(Tg (E), p; E ¼ 0)  þ 8p @p T,E   2 @[v(1  1 )]   E 0 a  v(Tg (E), p; E ¼ 0) a  8p @p T,E     E 2 @[v(1  10 )]b v(Tg (E), p; E ¼ 0) b  8p @p T,E Now, one evaluates zero-field molar volume of each phase (i.e., a and b) as a function of temperature from Tg(E ¼ 0), where va ¼ vb in the absence of the field, to Tg(E) where the zero-field molar volume of each phase is not the same. Zero-field thermal expansion coefficients, designated by aE¼0, are employed to accomplish this task.

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Chapter 4 dc Electric Field Effects on First- and Second-Order Phase Transitions

Hence, n o aE¼0 ¼ @ ln v @T p,E¼0 v(Tg (E), p; E ¼ 0) ¼ v(Tg (E ¼ 0), p; E ¼ 0) exp

8 T (E) > < gð > :

Tg (E¼0)

aE¼0 dT

9 > = > ;

   v(Tg (E ¼ 0), p; E ¼ 0) exp aE¼0 [Tg (E)  Tg (E ¼ 0)]    v(Tg (E ¼ 0), p; E ¼ 0) 1 þ aE¼0 [Tg (E)  Tg (E ¼ 0)] þ    In the previous expressions for the temperature dependence of zero-field molar volume, if thermal expansion coefficients in the absence of the electric field are assumed to be weak functions of temperature, and Tg(E) is not significantly different from Tg(E ¼ 0), then the exponential is expanded in a Taylor series and truncated after the linear term. Volume continuity yields the following result: 

  ¼ v(Tg (E ¼ 0), p; E ¼ 0) b   va (Tg (E ¼ 0), p; E ¼ 0) 1 þ aE¼0,phase(a) [Tg (E)  Tg (E ¼ 0)]   2 @[v(1  1 )] 0 a E 8p @p T,E    vb (Tg (E ¼ 0), p; E ¼ 0) 1 þ aE¼0,phase(b) [Tg (E)  Tg (E ¼ 0)]   E 2 @[v(1  10 )]b  8p @p T,E v(Tg (E ¼ 0), p; E ¼ 0)



a

The integral approach to phase equilibrium at second-order transitions, based on volume continuity, predicts that uniform dc electric fields affect the glass transition temperature according to     @[v(1  10 )]Liquid @[v(1  10 )]Glass  2 @p @p T,E   T,E Tg (E)  Tg (E ¼ 0)  E 8p v(Tg (E ¼ 0), p; E ¼ 0) aLiquid  aGlass E¼0   @[v(1  10 )] D 2 @p E T,E   ¼ 8p v Tg (E ¼ 0), p; E ¼ 0 DaE¼0 where D implies a difference between electro-thermodynamic properties in the liquid and glassy states at either the field-dependent glass transition (i.e., numerator of the previous equation) or the zero-field Tg (i.e., thermal expansion coefficients in the denominator of the previous equation). There are no known exceptions to the fact that the discontinuity in zero-field thermal expansion coefficients is greater than

4.6 Theoretical Considerations

155

zero in the denominator of the previous equation. Hence, if electric field ordering due to dipolar alignment during the formation of electrets produces an increase in the glass transition temperature, then the previous electric field analog of the Ehrenfest equation via volume continuity suggests that the pressure dependence of the molar dielectric susceptibility must be greater in the liquid state relative to the glass. This claim is supported by the fact that field-dependent isothermal compressibilities, which are directly related to (@v/@p)T,E in the numerator of the previous equation, experience a discontinuous increment as materials are heated through Tg. Several III– V and II – VI semiconductors exhibit negative pressure dependence of their dielectric constants (i.e., 1/10) in the solid state at 300 K and pressures ranging from 1 to 4000 atm, {@ ln(1/10)/@p}T  21.2  1022 GPa21 when 1/10  10 [Samara, 1983]. This corresponds to a 1 K change in Tg via the previous equation for very strong static fields (i.e., E  10 MV/cm), independent of the zero-field glass transition temperature. Electric Field Analog of the Prigogine – Defay Ratio The previous two subsections of this chapter describe electric field effects on the glass transition temperature of pure materials via an integral representation of phase equilibrium that invokes volume and entropy continuity. Similar methodology was discussed in Chapter 1 for the pressure dependence of Tg under zero-field conditions, yielding the classic Ehrenfest equations based on the differential approach to phase equilibrium, once again invoking volume and entropy continuity. Division of the pressure dependence of Tg from volume continuity (i.e., @Tg/@p ¼ Db/Da) by @Tg/@p ¼ Tgv Da/DCp based on entropy continuity yields the Prigogine – Defay [1954] ratio l under zero-field conditions:

l¼

DCp Db  1 Tg v(@Tg ){Da}2

where v is specific volume at the glass transition temperature and D represents discontinuous increments in (i) specific heat Cp, (ii) thermal expansion a, and (iii) isothermal compressibility b that occur as materials are heated through the second-order phase transition. The equality applies to the previous expression if volume and entropy continuity provide adequate descriptions of the pressure dependence of Tg. In many cases, experimental data reveal that the inequality is obeyed, because volume continuity predicts that @Tg/@p is somewhere between two-fold and five-fold greater than @Tg/@p from entropy continuity [Goldstein, 1973, 1975; Nieuwenhuizen, 1997]. It is generally accepted that predictions of @Tg/@p from entropy continuity are more accurate for most, but not all, amorphous polymers, because the discontinuity in isothermal compressibility Db is strongly pressure dependent and it has not been measured for as many materials, relative to Da and DCp, which are easier to obtain experimentally [O’Reilly, 1962]. Due to the lack of accurate Tg measurements for amorphous and semicrystalline polymers in the presence of dc electric fields,

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Chapter 4 dc Electric Field Effects on First- and Second-Order Phase Transitions

predictions from the previous two subsections of this chapter are considered without bias, in an effort to develop the electric field analog of the Prigogine – Defay equality. Upon equating Tg(E) based on volume and entropy continuity, one obtains   @[v(1  10 )] D @p E2  T,E  Tg (E ¼ 0) þ 8p v Tg (E ¼ 0), p; E ¼ 0 DaE¼0 n o @[v(1  10 )] T (E ¼ 0)D 2 g @T p,E  Tg (E ¼ 0)  E 8p DC p,E¼0 Now, the appropriate ratio of thermophysical properties and discontinuous observables at the second-order phase transition temperature is constructed as follows:   @[v(1  10 )] DCp,E¼0 D @p T,E n o  1   @[v(1  10 )] Tg (E ¼ 0)v Tg (E ¼ 0), p; E ¼ 0 DaE¼0 D @T p,E where v is molar volume and Cp,E¼0 is the zero-field molar heat capacity. It is important to emphasize that discontinuous temperature and pressure derivatives of the molar dielectric susceptibility are evaluated at Tg(E) in the presence of the electric field, whereas all of the other thermophysical properties in the previous equation are evaluated at Tg(E ¼ 0) under zero-field conditions. Application of the product rule to the temperature and pressure derivatives of the molar dielectric susceptibility in the previous equation yields the following form for the electric field analog of the Prigogine– Defay equality. The pressure derivative of molar volume introduces the field-dependent coefficient of isothermal compressibility bE=0, with a negative sign, and the temperature derivative of molar volume yields the field-dependent coefficient of thermal expansion aE=0: "   # @1 DCp,E¼0 D  (1  10 )DbE=0 @p T,E  n o   1 @1 Tg (E ¼ 0)v{Tg (E ¼ 0), p; E ¼ 0}DaE¼0 D þ (1  10 )DaE=0 @T p,E In the absence of any external fields, the additional work term experienced by electric dipoles is not required in the first law of thermodynamics, and it is not necessary to consider the dielectric permeability of the medium. Hence, upon neglecting the temperature and pressure derivatives of the dielectric permeability 1, the previous equation reduces to the zero-field Prigogine – Defay equality. In summary, one arrives at this equality from (i) the Ehrenfest equations, based on the differential approach to phase equilibrium, for the zero-field pressure dependence of Tg, and (ii) the integral approach to describe electric field effects on Tg at constant pressure and composition (for mixtures), because both methodologies consider volume and entropy continuity at second-order phase transitions.

4.6 Theoretical Considerations

157

4.6.5 Chemical Stability Limits in Binary Mixtures via the Spinodal Curve Consideration of field-dependent chemical potentials in Section 4.6.2 and the analysis of chemical stability in Section 3.5 indicate that the effect of composition on the chemical potential of species A in binary mixtures, via zero-field activity coefficients gA, total volume V, and a model for the dielectric permeability 1mixture,

mA (T, p, xA , E) ¼ mA,pure (T, p; E ¼ 0) þ RT lnfxA gA (T, p, xA ; E ¼ 0)g   2 @V(1 mixture  10 ) E 8p @NA T,p,NB ,E is required to identify the spinodal condition via inflection points on a graph of C ¼ G – E P per mole of mixture versus mole fraction xA (or a graph of C per mole of lattice sites versus volume fraction wA) at constant temperature, pressure, and electric field strength. It is important to emphasize that the mixing process is constructed as follows; Ni moles of pure component i (i.e., i ¼ A, B), in the presence of a strong electric field, are combined in the presence of the field to yield the appropriate binary mixture. The chemical potential of pure-component i under zero-field conditions, mi,pure(T, p, E ¼ 0), is a molar property of its extensive Gibbs free energy Gi,pure. In the presence of electric fields, mi (T, p, composition, E) is a partial molar property of the energetic state function C ¼ G – E P, as mentioned in Section 4.6.2 via Legendre transformation of the internal energy. It is reasonable to interpret C as the extensive Gibbs free energy of the mixture in the presence of uniform electric fields [Damjanovic, 1998]. Mixtures are stable with respect to spontaneous phase separation when the molar equivalent of C for the mixing process (i.e., Dcmixing) is negative with positive curvature (i.e., with respect to mole fraction xA). These restrictions are summarized mathematically below for binary mixtures [Belfiore, 2003]:





cmixture ¼

B X

xi mi (T, p, xA , E)

i¼A

Dcmixing ¼

B n o X xi mi (T, p, xA , E)  mi,pure (T, p, E) , 0 i¼A

 2   2    @ Dcmixing @ mA @ cmixture ¼ . 0; or .0 @xA T,p,E @x2A @x2A T,p,E T,p,E

4.6.6 Classical Thermodynamic Analysis of Chemical Stability in the Presence of Uniform Electric Fields, Based on a Reference State in Which Both Pure Components Experience the Field The previous equation for Dcmixing is employed to identify binodal and spinodal phase boundaries for nonideal binary mixtures in the presence of external dc electric fields.

158

Chapter 4 dc Electric Field Effects on First- and Second-Order Phase Transitions

The field-dependent contribution to the chemical potential of component i in the mixture is species specific due to contributions from partial molar dielectric susceptibilities, which can be expressed in terms of partial molar dielectric permeabilities and electric-field-dependent partial molar volumes. The field-dependent chemical potential of pure species i is written in terms of its molar dielectric susceptibility, which is essentially the product of its dielectric constant and field-dependent molar volume vi: 2

mi,pure (T, p, E) ¼ mi,pure (T, p; E ¼ 0)  E fvi (1i  10 )g 8p   2 @ fv (1  1 )g i i 0 vi,pure (T, p, E) ¼ vi,pure (T, p; E ¼ 0)  E 8p @p T,E  vi,pure (T, p, E ¼ 0) þ

E2 (1i  10 )bi,E¼0 vi,pure (T, p, E) 8p

Electric field effects on pure-component molar volumes are adopted from previous results in Section 4.6.2, upon replacing the pressure dependence of the partial molar dielectric susceptibility by the pressure dependence of the pure-component molar dielectric susceptibility. Due to the lack of accurate models or experimental data for some of these electrothermodynamic properties, the pressure dependence of purecomponent dielectric susceptibilities is calculated via zero-field coefficients of isothermal compressibility and field-dependent molar volume, neglecting any pressure dependence of the dielectric permeability. The thermodynamics of mixing is described by the following set of equations: Dcmixing ¼ RT

B X

xi lnfxi gi ðT,p,xA ;E ¼0Þg

i¼A

" #  B 2X @V ð1mixture 10 Þ E  xi vi ð1i 10 Þ @Ni 8p i¼A T,p,N j[j=i] ,E     @V ð1mixture 10 Þ @1mixture ¼V þ ð1mixture 10 ÞVi @Ni @Ni T,p,N j[j=i] ,E T,p,N j[j=i] ,E   B B X X V¼ Ni @V ¼ Ni Vi @Ni T,p,Nj[j=i] ,E i¼A i¼A   @ ki Vi ðT,p,xA ,E ÞVi ðT,p,xA ;E ¼0Þ 1 E2 @p T,E,composition 2 According to the previous equation for Dcmixing, the important property of mixtures that governs whether the application of an electric field favors miscibility or phase separation is the sign of the dielectric susceptibility of mixing (per mole of mixture), or the difference between the molar dielectric susceptibility of the mixture and a

4.6 Theoretical Considerations

159

mole-fraction-weighted sum of pure-component molar dielectric susceptibilities: Dielectric susceptibility of mixing " #  X @V(1mixture  10 ) 1 ¼ vi (1i  10 ) xi 4p all species i @Ni T,p,N j[ j=i] ,E For example, if the dielectric susceptibility of mixing is greater than zero, primarily due to positive deviations from ideal mixing such that zero-field activity coefficients exceed unity, then stronger electric fields exhibit a favorable influence on the fielddependent Gibbs free energy of mixing, which enhances the mixing process. A molecular interpretation of positive deviations in the dielectric susceptibility of mixing should be based on the natural tendency of electric fields to induce dipole polarization in multicomponent media that contain polarizable components with different dielectric constants. Simulations that analyze electric field effects on binodal and spinodal phase boundaries employ the following approximations prior to inspecting graphs of Dcmixing versus mole fraction xA at constant temperature, pressure, and electric field strength: (i) Compositional dependence of the dielectric permeability 1mixture for homogeneous mixtures is given by the Landau – Lifshitz expression [Landau et al., 1984] where 1i and wi represent the static permittivity and volume fraction, respectively, of species i: B p ffiffiffiffiffiffiffiffiffiffiffiffiffi X pffiffiffiffi 3 wi 3 1i 1mixture  i¼A

In terms of the two-parameter (i.e., h, s) van Laar activity coefficient model for nonideal binary mixtures of different-sized molecules, described in (iii) below, the following relation between volume fraction wi, mole fraction xi, and mole numbers Ni is employed:

wA ¼

hNA hxA ¼ ; wB ¼ 1  wA hNA þ sNB hxA þ sxB

Based on the Landau – Lifshitz expression for the static dielectric permeability of homogeneous binary mixtures, together with the van Laar relations between volume fractions and mole numbers provided above, the appropriate pressure-independent partial molar dielectric permeability of species A is 

@1mixture @NA



n o 2=3 1=3 1=3 wA wB ¼ 31mixture 1A  1B NA T,p,E,NB

One obtains the partial molar dielectric permeability of species B by interchanging all of the subscripts (i.e., A ) B and B ) A) in the previous

160

Chapter 4 dc Electric Field Effects on First- and Second-Order Phase Transitions

equation. Both expressions for partial molar dielectric permeability satisfy Euler’s equation for homogeneous thermodynamic functions [Belfiore, 2003] because the dielectric permeability of the mixture is an intensive property that is homogeneous to the zeroth degree (i.e., n ¼ 0) with respect to system mass. Hence,   X @(1mixture ) xi ¼0 @Ni T,p,N j[ j=i] ,E all species i The Gibbs free energy change per total moles of the binary mixture in the presence of a uniform field reduces to the following expression that is consistent with the Landau – Lifshitz model for static dielectric permeability: Dcmixing ¼ RT

B X

xi lnfxi gi (T, p, xA ; E ¼ 0)g

i¼A B 2 X   xi (1mixture  10 )Vi (T, p, xA , E)  (1i  10 )vi E 8p i¼A   1 2 @ ki Vi (T, p, xA , E)  Vi (T, p, xA ; E ¼ 0)  E @p T,E,composition 2

It is relatively straightforward to extend the previous expression for Dcmixing to multicomponent mixtures by summing over all species, provided that adequate models are available for 1mixture and zero-field activity coefficients. One possibility is an extension of the Landau – Lifshitz binary model described in (i) above for 1mixture to N components. (ii) Electric field contributions to partial molar volumes require the pressure dependence of partial molar dielectric susceptibilities. Once again, due to the lack of accurate models or experimental data for these electrothermodynamic properties, they are approximated by considering the pressure dependence of the mixture’s total volume V, which yields a field-dependent coefficient of isothermal compressibility that is estimated under zero-field conditions, bE¼0, even though the total mixture volume is evaluated in the presence of the field. Pressure dependencies of the mixture’s dielectric permeability, partial molar volume, and partial molar dielectric permeability are neglected. Hence,   @ ki Vi (T, p, xA , E) ¼ Vi (T, p, xA ; E ¼ 0)  1 E 2 þ @p T,E,composition 2     E 2 @1mixture @V  Vi (T, p, xA ; E ¼ 0)  8p @Ni T,p,E,all N j[ j=i] @p T,E,composition   E 2 V(T, p, xA , E)bE¼0 @1mixture  Vi (T, p, xA ; E ¼ 0) þ 8p @Ni T,p,E,all N j[ j=i]

4.6 Theoretical Considerations

161

where the field-dependent total volume of the mixture V(T, p, xA, E) in the second term on the right side of the last equation is expanded using partial molar properties via Euler’s integral theorem for thermodynamic functions that are homogeneous to the first degree: V(T, p, xA , E) ¼

B X

Ni Vi (T, p, xA , E)

i¼A

(iii) Zero-field activity coefficients for nonideal binary mixtures of different-sized molecules are based on the van Laar model. The Flory –Huggins expression for the nonideal interaction Gibbs free energy of mixing is analogous to the two-parameter van Laar model [Belfiore, 2003], where the van Laar parameters h and s represent a dimensionless interaction Gibbs free energy parameter s and the ratio of molar volume of component A to that of component B (i.e., h/s). Both van Laar parameters adopt equivalent numerical values for regular solutions of equisized molecules. Under these conditions, the van Laar model reduces to the symmetric one-parameter Margules model. The effect of composition on Dgmixing,interaction (i.e., per mole of mixture, not per mole of lattice sites) requires the volume fraction of component A, wA, and the mole fraction of component B, xB, in the first equation below. It is not incorrect to identify h as the dimensionless interaction Gibbs free energy parameter (instead of s) if one employs the mole fraction of component A, xA, and the volume fraction of component B, wB. The van Laar interaction Gibbs free energy of mixing, per mole of lattice sites, is symmetric with respect to the volume fraction of either species (i.e., wAwB), and both parameters h and s are consumed by these composition variables. The appropriate expressions for the nonideal interaction Gibbs free energy of mixing, per mole of mixture (where a mole of mixture is composition dependent for constant-volume lattices), and both activity coefficients gi are [Belfiore, 2003] 



B X

RT hsxA xB ¼ RT swA xB ¼ RT hxA wB hxA þ sxB i¼A h s ln gA ¼ ¼ hw2B ; ln gB ¼ ¼ sw2A f1 þ hxA =sxB g2 f1 þ sxB =hxA g2

Dgmixing

interaction

¼ RT

xi ln gi ¼

(iv) Zero-field partial molar volumes can be estimated from the van Laar activity coefficient model for the nonideal energetics of mixing via the following expression (see the Appendix in Chapter 3): RT ln gi,E¼0 (T, p, xA ) ¼

ðp



 Vi (T, p, xA ; E ¼ 0)  vi,pure (T, p; E ¼ 0) dp

0

Vi (T, p, xA ; E ¼ 0)  vi,pure (T, p; E ¼ 0) þ RT ln gi,E¼0 (T, p, xA ) p

162

Chapter 4 dc Electric Field Effects on First- and Second-Order Phase Transitions

where the first term on the right side of the previous equation represents the zero-field pure-component molar volume of species i. If necessary, the pressure dependence of partial molar volume can be estimated via the second term on the right side of the previous equation.

4.6.7 Numerical Evaluation of the Compositional Dependence of Dcmixing in Nonionic Binary Liquid Mixtures Parameter Declaration Initially, it is necessary to assign numerical values for several parameters that remain constant throughout each simulation. The cgs and mks systems of units are used interchangeably because electric fields are expressed in terms of kV/cm and the permittivity of free space has units of coulomb2/( joule-meter) (i.e., 1 joule ¼ 1 coulombvolt). Consequently, a factor of 1022 is required when 10E 2 appears in the electric field contribution to (i) partial molar volumes with units of cm3/(g-mol), (ii) chemical potentials with units of J/(g-mol), and (iii) the Gibbs free energy of mixing with units of J/(g-mol). The parameters are as follows: Methyl Acetate and Water at 40 8 C and 1 atm (Fig. 4.1) Permittivity of free space (i.e., 8.854  10212 coulomb2/(newton-meter2)) Static dielectric constant of species A, methyl acetate (i.e., 1A/10 ¼ 6.68 @ 25 8C)

10 1A/10

Gibbs Free Energy of Mixing (J/mol)

0

Electric Field Strength E = 0 kV/cm E = 200 kV/cm

–200 –400 –600 –800

E = 300 kV/cm E = 400 kV/cm E = 500 kV/cm E = 600 kV/cm E = 700 kV/cm E = 800 kV/cm E = 900 kV/cm

–1000 –1200 –1400 –1600 –1800 0.0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 Methyl Acetate Mole Fraction

0.8

0.9

1.0

Figure 4.1 Effect of an external dc electric field on the Gibbs free energy of mixing for methyl acetate and water at 40 8C and 1 atmosphere. This system exhibits composition-dependent miscibility (i.e., with a large miscibility gap) under zero-field conditions, that persists as the field is applied because the binodal and spinodal points shift to the right, toward higher concentrations of methyl acetate. Complete homogeneity is not achieved for fields as large as 900 kV/cm. Electric field strength increases from the uppermost curve to the lowermost curve.

4.6 Theoretical Considerations 1B/10 h

s E R T p bE¼0 vA,pure vB,pure TA,boil TB,boil TAzeotrope

163

Static dielectric constant of species B, water (i.e., 1B/10 ¼ 78.54 @ 25 8C) van Laar parameter for species A in binary mixtures with species B (i.e., h ¼ 2.99) van Laar parameter for species B in binary mixtures with species A (i.e., s ¼ 1.89) dc electric field strength, 0–900 kV/cm Universal gas constant, 82.057 cm3-atm/(mol-K) or 8.314 J/(mol-K) Absolute temperature, 40 8C (below the normal boiling point of methyl acetate) Absolute pressure, 1 atmosphere Zero-field coefficient of isothermal compressibility for the binary mixture, 1025 atm21 (must be converted to cm3/J) Zero-field pure-component molar volume of species A at T, p: 79.3 cm3/(g-mol) Zero-field pure-component molar volume of species B at T, p: 18.1 cm3/(g-mol) Normal boiling point of species A, methyl acetate: 330 K Normal boiling point of species B, water: 373 K Minimum-boiling homogeneous azeotrope at xMeAc ¼ 0.95: 329 K

Methanol and Water at 40 8 C and 1 atm (Fig. 4.2) Static dielectric constant of species A, methanol (i.e., 1A/10  30 @ 25 8C) Static dielectric constant of species B, water (i.e, 1B/10 ¼ 78.54 @ 25 8C) van Laar parameter for species A in binary mixtures with species B (i.e., h ¼ 0.55) van Laar parameter for species B in binary mixtures with species A (i.e., s ¼ 0.45) Zero-field pure-component molar volume of species A at T, p: 40.4 cm3/(g-mol) Zero-field pure-component molar volume of species B at T, p: 18.1 cm3/(g-mol) Normal boiling point of species A, methanol: 338 K Normal boiling point of species B, water: 373 K

1A/10 1B/10 h

s vA,pure vB,pure TA,boil TB,boil

Gibbs Free Energy of Mixing (J/mol)

0 –200

Electric Field Strength E = 0 kV/cm E = 500 kV/cm E = 700 kV/cm E = 900 kV/cm

–400 –600 –800 –1000 –1200 –1400 –1600 –1800 –2000 0.0

0.1

0.2

0.3

0.4 0.5 0.6 0.7 Methanol Mole Fraction

0.8

0.9

1.0

Figure 4.2 Effect of an external dc electric field on the Gibbs free energy of mixing for methanol and water at 40 8C and 1 atmosphere, revealing homogeneous single-phase behavior at all field strengths. Electric field strength increases from the uppermost curve to the lowermost curve.

164

Chapter 4 dc Electric Field Effects on First- and Second-Order Phase Transitions

Carbon Disulfide and Acetone at 40 8 C and 1 atm (Fig. 4.3) Static dielectric constant of species A, carbon disulfide (i.e., 1A/10 ¼ 2.641 @ 20 8C) Static dielectric constant of species B, acetone (i.e, 1B/10 ¼ 20.7 @ 25 8C) van Laar parameter for species A in binary mixtures with species B (i.e., h ¼ 1.28) van Laar parameter for species B in binary mixtures with species A (i.e., s ¼ 1.79) Zero-field pure-component molar volume of species A at T, p: 58.9 cm3/(g-mol) Zero-field pure-component molar volume of species B at T, p: 73.5 cm3/(g-mol) Normal boiling point of species A, carbon disulfide: 319 K Normal boiling point of species B, acetone: 329 K

1A/10 1B/10 h

s vA,pure vB,pure TA,boil TB,boil

Methanol– water and carbon disulfide– acetone exhibit homogeneous one-phase behavior in the absence of electric fields, even though these systems exhibit positive deviations from ideality with activity coefficients that are greater than unity (but they are not significantly greater than unity). The compositional dependence of the molar Gibbs free energy of mixing in Figures 4.2 and 4.3 reveals that electric fields cannot induce phase separation because the dielectric susceptibility of mixing is positive.

Gibbs Free Energy of Mixing (J/mol)

0 Electric Field Strength E = 0 kV/cm E = 500 kV/cm E = 700 kV/cm E = 900 kV/cm

–200 –400 –600 –800 –1000 –1200 0.0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 Carbon Disulfide Mole Fraction

0.8

0.9

1.0

Figure 4.3 Effect of an external dc electric field on the Gibbs free energy of mixing for carbon disulfide and acetone at 40 8C and 1 atmosphere, revealing homogeneous single-phase behavior at all field strengths. These mixtures are significantly less sensitive to an applied electric field because the dielectric permeability (i.e., dielectric constant) of the mixture and the mismatch between purecomponent dielectric permeabilities are smaller than those for mixtures of methyl acetate and water in Figure 4.1. Electric field strength increases from the uppermost curve to the lowermost curve.

4.6 Theoretical Considerations

165

Trends Using a minimum of five or six physical properties, such as pure-component dielectric constants and molar volumes, as well as one or two activity coefficient parameters for the Margules or van Laar models, respectively, electrothermodynamic predictions can be generated for low-molecular-weight binary liquid mixtures (i.e., regular solutions). Ambient pressure calculations should be performed below the purecomponent boiling points to address liquid – liquid phase separation (i.e., miscibility gaps). Additional restrictions on temperature are required if the system is classified as a minimum-boiling azeotrope. As mentioned above, the sign of the dielectric susceptibility of mixing determines whether the application of strong fields can drive systems toward homogeneity or phase separation. The previous equations and graphs that summarize and provide visualization of electric field effects on thermodynamic phase behavior of nonideal binary liquid mixtures reveal the following trends. Positive Deviations from Ideal Mixing Let’s begin with classic nonideal mixtures that exhibit positive deviations from linear additivity, or the additive rule of mixtures. In this frequently occurring situation, activity coefficients are greater than unity, field-independent partial molar volumes exceed pure-component molar volumes for each component, the total mixture volume is greater than a mole-fractionweighted sum of pure-component molar volumes, and mixing is endothermic. If activity coefficients are significantly greater than unity, due primarily to rather large positive values of key parameters in the activity coefficient model, then zero-field miscibility gaps (i.e., two-phase regions) exist on the temperature – composition phase diagram because the requirement for chemical stability is violated over a restricted range of mixture compositions. Interestingly enough, these mixtures exhibit positive dielectric susceptibilities of mixing such that the application of strong electric fields will reduce the breadth of the miscibility gap, and shift it, on the composition axis. Aqueous mixtures of methyl acetate reside in this category, adequately described as “electric field modification of phase behaviour.” However, it is not possible to develop complete homogeneity in the methyl acetate – water binary system before one exceeds the threshold for dielectric breakdown. Negative Deviations from Ideal Mixing These strongly interacting nonionic liquids represent the exception, in which homogeneous mixing is described by activity coefficients that are less than unity, but greater than zero, due to negative parameters in the appropriate model (i.e., Margules, van Laar, etc.). Zero-field volumes of mixing are negative and enthalpies of mixing are exothermic, characteristic of energetically attractive interactions between dissimilar components. There are no miscibility gaps in the temperature – composition phase diagram because the criterion for chemical stability is satisfied for all mixture compositions. However, the dielectric susceptibility of mixing is negative, which indicates that the application of electric fields could generate miscibility gaps and drive the system toward phase separation. Hence, the lower critical solution temperature should decrease at higher electric field strengths.

166

Chapter 4 dc Electric Field Effects on First- and Second-Order Phase Transitions

Other Comments If electrostriction is negligible, because the pressure dependence of partial molar dielectric susceptibilities does not allow electric fields to induce significant decreases in partial molar volume and total system volume, then it might not be possible to subject partially miscible systems (that exhibit positive zero-field deviations from ideal mixing) to strong electric fields and induce the breadth of a miscibility gap under zero-field conditions to increase in the presence of the field. In nonionic mixtures, electric fields on the order of 200 –500 kV/cm are typically required for the fielddependent terms that scale as 10E 2 to compete with the field-independent contributions to chemical potentials that scale as RT. Mixtures are less sensitive to an applied field, and stronger fields are required to influence phase boundaries, when the dielectric constant of the mixture decreases and the mismatch between pure-component dielectric permeabilities is smaller. Magnetic-field-induced homogeneity and phase separation might be more attractive than electric-field-induced phase separation because there is no magnetic analog of dielectric breakdown.

4.7

SUMMARY

The integral approach to phase equilibrium for first- and second-order thermodynamic transitions is employed to analyze the effects of uniform dc electric fields on melting temperatures Tmelt and glass transitions Tg of pure materials. A work term due to external fields is added to the classical internal energy, which reveals that most thermodynamic properties of interest scale as the square of the magnitude of the field when systems do not exhibit electric saturation. Upon equating chemical potentials of the solid and liquid at Tmelt, the electric field dependence of first-order phase transitions scales linearly with the discontinuity in (i) molar dielectric susceptibility of pure materials or (ii) partial molar dielectric susceptibilities of all species in multicomponent mixtures. Volume and entropy continuity at second-order phase transitions suggest that discontinuous changes in (i) pressure dependence and (ii) temperature dependence, respectively, of the molar dielectric susceptibility, or partial molar dielectric susceptibility for mixtures, govern the sign of the shift in the glass transition temperature as a function of electric field strength. Both Tg –field equations predict small changes in the glass transition temperature that scale as the square of the electric field strength. These two Ehrenfest-like relations at second-order phase transitions are combined to yield the electric field analog of the Prigogine – Defay equality, which reduces to the correct zero-field ratio of discontinuous thermodynamic properties when the field is removed. Binodal and spinodal phase boundaries for nonionic binary liquid mixtures that are described adequately by van Laar activity coefficients can be predicted via the compositional dependence of the field-dependent Gibbs free energy of mixing. Immiscible mixtures that exhibit strong positive deviations from ideality can be partially homogenized by exposing them to electric fields, because the dielectric susceptibility of mixing is positive. Electric-field-induced phase separation of homogeneous binary mixtures that exhibit negative deviations from ideality and exothermic energetics of mixing might be possible if electric saturation of the medium and dielectric breakdown do not inhibit the demixing process.

Appendix: Nomenclature

167

APPENDIX: NOMENCLATURE Cp,E¼0 D dP dr E F G h N Ni P Pi p q r S Si s s[Tg(E ¼ 0), p; E ¼ 0]

s[Tg(E), p, E] s[Tg(E), p; E ¼ 0]

T Tg(E ¼ 0) Tg(E) Tmelt (E ¼ 0) Tmelt (E) V Vi v v[Tg(E ¼ 0), p; E ¼ 0]

v[Tg(E), p, E] v[Tg(E), p; E ¼ 0]

U xi

Molar heat capacity in the absence of an electric field Electric displacement vector, charge/cm2 Differential Ppolarization vector, given by a sum over all point charges i qi dri Differential displacement vector experienced by charge q in an electric field Electric field vector, volts/cm Force experience by charge q in an electric field Field-independent extensive Gibbs free energy Molar enthalpy of a pure material Total moles, or total number of components in the mixture Moles of component i in a mixture Dielectric polarization vector, charge-cm Partial molar dielectric polarization of species i Pressure Charge, coulombs Displacement vector experienced by charge q in an electric field Extensive entropy Partial molar entropy of species i Molar entropy of a pure material Molar entropy of a pure material in the absence of a dc electric field, evaluated at its zero-field glass transition temperature Molar entropy of a pure material at its glass transition in an electric field Molar entropy of a pure material in the absence of an electric field, evaluated at a temperature that corresponds to its glass transition temperature in the presence of a dc electric field Temperature Glass transition temperature under zero-field conditions Glass transition temperature in the presence of a dc electric field Melting temperature under zero-field conditions Melting temperature in the presence of a dc electric field Extensive volume Partial molar volume of species i Molar volume of a pure material Molar volume of a pure material in the absence of a dc electric field, evaluated at its zero-field glass transition temperature Molar volume of a pure material at its glass transition in an electric field Molar volume of a pure material in the absence of an electric field, evaluated at a temperature that corresponds to its glass transition temperature in the presence of a dc electric field Extensive internal energy Mole fraction of component i in a mixture

168

Chapter 4 dc Electric Field Effects on First- and Second-Order Phase Transitions

Greek Symbols

aE¼0 b D Dhmelting,E¼0 10 1 wi k ki l mi C

c

Thermal expansion coefficient under zero-field conditions Isothermal compressibility Difference between thermodynamic properties in liquid state and glassy state Molar enthalpy change for melting of pure materials under zero-field conditions Permittivity of free space, 8.854  10212 coulomb2/(newton-meter2) Dielectric permeability, charge/(volt-cm), or same dimensions as 10 Volume fraction of component i in a mixture Dielectric susceptibility, V(1 2 10)/4p, charge-cm2/volt Partial molar dielectric susceptibility of species i Prigogine –Defay ratio Chemical potential of component i in a mixture Thermodynamic state function, modified Gibbs free energy in the presence of an electric field Intensive thermodynamic state function, modified Gibbs free energy per mole of mixture in the presence of an electric field

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JACKSON JD. Classical Electrodynamics, 2nd edition. Wiley, Hoboken, NJ, 1975, pp. 33, 34, 144– 146, 210–213. KIRKWOOD JG, OPPENHEIM I. Chemical Thermodynamics, McGraw-Hill, New York, 1961, Chap. 14. KITZEROW HS, CROOKER PP, KWOK SL, HEPPKE G. A blue phase with negative dielectric anisotropy in an electric field; statics and dynamics. Journal de Physique 51(12):1303 (1990). KOGA K, NAKANO N, HATTORI T, OHIGASHI H. Crystallization, field-induced phase transformation, thermally induced phase transition, and piezoelectric activity in poly(vinylidene fluoride/trifluoroethylene) copolymers with high molar content of vinylidene fluoride. Journal of Applied Physics 67(2):965 (1990). KROUPA J. Electric field effects on the phase transitions in betaine phosphates and arsenates. Phase Transitions 36:209 (1991). KUCZYNSKI W, HOFFMAN J, STRYLA B, MALECKI J. Electric field effects on smectic-A/smectic-C phase transitions. Ferroelectrics 114:319 (1991). LANDAU LD, LIFSHITZ EM, PITAEVSKII LP. Electrodynamics of Continuous Media, 2nd edition, Volume 8 in Courses in Theoretical Physics. Pergamon Press, New York, 1984, chap. 2. MARAND H, STEIN RS. Isothermal crystallization of poly(vinylidene fluoride) in the presence of high static electric fields; effect of crystallization temperature and electric field strength on the crystal phase content and morphology. Journal of Polymer Science, Polymer Physics Edition 27(5):1089 (1989). NIEUWENHUIZEN TH M. Ehrenfest relations at the glass transition; solution to an old paradox. Physical Review Letters 79(7):1317–1320(1997). O’REILLY JM. The effect of pressure on the glass transition temperature and dielectric relaxation time of poly(vinyl acetate). Journal of Polymer Science 57:429– 444 (1962). PERSHIN VK, KONOPLEV VA. Critical points in the field–temperature phase diagram of partially ordered systems with isostructural transformations. Solid State Communications 76(9):1131 (1990). PRIGOGINE I, DEFAY R. Chemical Thermodynamics. Longmans Green, New York, 1954, Chap. 19. REICH S, GORDON JM. Electric field dependence of lower critical phase behaviour in polymer–polymer mixtures. Journal of Polymer Science; Polymer Physics Edition 17:371 (1979). REYNOLDS NM, KIM KJ, CHANG C, HSU SL. Spectroscopic analysis of electric field induced structural changes in vinylidene fluoride/tri-fluoroethylene copolymers. Macromolecules 22:1092 (1989). SAINT-PAUL M, GILCHRIST JG. Nonlinear dielectric susceptibility of KCl:OH dipole glass. Journal of Physics: Part C, Solid State Physics 19:2091–2101 (1986). SAMARA GA. Temperature and pressure dependences of the dielectric constants of semiconductors. Physical Review B 27(6):3494–3505 (1983). SERPICO JM, WNEK GE, KRAUSE S, SMITH TW, LUCA DJ, VANLAEKEN A. Effect of block copolymers on the electric field induced morphology of polymer blends. Macromolecules 24:6879 (1991). SERPICO JM, WNEK GE, KRAUSE S, SMITH TW, LUCA DJ, VAN LAEKEN A. Electric field induced morphologies in diblock copolymer-containing polymer blends. Polymer Preprints 33(1):463 (1992); Macromolecules 25:6373 (1992). TADOKORO H, TASHIRO K, NAKAMURA M, KOBAYASHI M, CHATANI Y. Polarization inversion current and ferroelectric phase transition of vinylidene fluoride/trifluoroethylene copolymers. Macromolecules 17: 1452 (1984). WNEK GE, VENUGOPAL G, KRAUSE S. The effects of electric fields on polymer blend morphologies. Polymer Preprints 31(1):377 (1990). YE ZG, RIVERA JP, SCHMID H. Electric field and stress induced phase transitions in chromium-chloride boracite. Ferroelectrics 115:251 (1991). YITZCHAIK S, BERKOVIC G, KRONGAUZ V. Polymers and blends exhibiting two-dimensional asymmetry and optical nonlinearity upon poling by an electric field. Macromolecules 23:3539 (1990).

Chapter

5

Order Parameters for Glasses: Pressure and Compositional Dependence of the Glass Transition Temperature We churn and burn, and dream of becoming something else. —Michael Berardi

Volume and entropy continuity at second-order phase transition temperatures are discussed within the framework of classical thermodynamics using two order parameters to identify structural characteristics of glasses that yield the Ehrenfest inequality for the pressure dependence of Tg. Then, the “order-parameter approach” is combined with statistical lattice models to describe the compositional dependence of Tg for plasticized polymer – diluent blends. The temperature derivative of the Flory – Huggins thermodynamic interaction parameter is discontinuous at Tg.

5.1

THERMODYNAMIC ORDER PARAMETERS

The local structure of glasses depends on the rate at which materials are cooled below Tg from the molten state. Invariably, experimental conditions yield nonequilibrium glasses that might not be described appropriately using equilibrium thermodynamics. Order parameters provide a crutch that introduces additional degrees of freedom for glasses, based on their local structure, and allows one to identify the liquid and glassy states as one moves through the phase transition. Analysis of the glass – liquid second-order phase transition benefits enormously from the establishment of order parameters that characterize glassy state structure qualitatively as a function of experimental cooling rate.

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

171

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Chapter 5 Order Parameters for Glasses

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 The pressure dependence of the glass transition temperature was discussed in Section 1.6 by invoking volume and entropy continuity at an equilibrium secondorder phase transition. Two Ehrenfest equations were obtained when the phase boundary between the liquid and glass exhibits equivalent fluctuations in volume (i.e., volume continuity) or equivalent fluctuations in entropy (i.e., entropy continuity) that can be specified uniquely by pressure p and temperature T. Goldstein [1963, 1975] approached the same problem formulated by Ehrenfest [1933] via the definition of excess thermodynamic state functions that describe changes in volume, entropy, and enthalpy. In each case, Goldstein assumed that temperature and pressure changes on the transition line between the equilibrium liquid and glass occur at constant values of these excess functions. When one internal order parameter z is employed, the intensive Gibbs free energy g is a function of temperature T, pressure p, and z. For the equilibrium liquid, z ¼ z(T, p) such that g exhibits a minimum with respect to z. The glass is described by a constant value of z. Volume and entropy continuity within the context of a single order parameter yield the same Ehrenfest relations that were described in Section 1.6 for the pressure dependence of the glass transition temperature. Now, consider a pure material that requires two independent internal order parameters, y and z, in addition to temperature T and pressure p for a complete description of the intensive (i.e., molar) Gibbs free energy. Hence, g ¼ g(T, p, z, y) and dg ¼ s dT þ v dp  A dz  B dy where molar entropy s, molar volume v, and affinities A and B are state functions of temperature T, pressure p, and internal order parameters z and y. Consistent with the phase rule for single-phase behavior of a pure material that exhibits two degrees of freedom, the equilibrium liquid is defined uniquely by p and T because both dimensionless order parameters adopt the functional form z ¼ z(T, p) and y ¼ y(T, p) to minimize the molar Gibbs free energy. Hence, the equilibrium liquid state is described by dg ¼ s dT þ v dp  2      @g @A @ g A¼ ¼ 0; ¼ ,0 @z T,p,y @z T,p,y @z2 T,p,y  2      @g @B @ g ¼ 0; ¼ ,0 B¼ @y T,p,z @y T,p,z @y2 T,p,z Order parameters y and z are constants for a given rigid glassy structure, but these constants depend on the cooling rate employed to achieve the glassy state. Since

5.2 Ehrenfest Inequalities

173

molar volume v ¼ v(T, p, z, y), its total differential is         @v @v @v @v dv ¼ dT þ dp þ dz þ dy @T p,z,y @p T,z,y @z T,p,y @y T,p,z     @v @v ¼ vaGlass ; ¼ vbGlass @T p,z,y @p T,z,y where the appropriate definitions of glassy state properties such as thermal expansion coefficient aGlass and compressibility factor bGlass are based on temperature and pressure derivatives, respectively, of molar volume at constant z and y. Temperature and pressure derivatives of the total differential expression for molar volume can be evaluated for the equilibrium liquid with affinities A ¼ 0 and B ¼ 0 to calculate aLiquid and bLiquid, respectively:       @v @v @z ¼ vaLiquid ¼ vaGlass þ @T p,A¼0,B¼0 @z T,p,y @T p,A¼0,B¼0     @v @y þ @y T,p,z @T p,A¼0,B¼0       @v @v @z ¼ vbLiquid ¼ vbGlass þ @p T,A¼0,B¼0 @z T,p,y @p T,A¼0,B¼0     @v @y þ @y T,p,z @p T,A¼0,B¼0 Condensed notation is employed for temperature and pressure derivatives of molar volume such that the previous two differential expressions are rewritten rather compactly in terms of the discontinuities in thermal expansion and isothermal compressibility at the glass transition temperature:     @z @y þ vy v Da ¼ vz @T p,A¼0,B¼0 @T p,A¼0,B¼0     @z @y v Db ¼ vz þ vy @p T,A¼0,B¼0 @p T,A¼0,B¼0     @v @v vz ¼ ; vy ¼ @z T,p,y @y T,p,z where molar volume v is evaluated at Tg. With knowledge of the fact that the pressure dependence of Tg via volume continuity is given by the ratio of Db to Da, the previous set of equations reveals that the temperature and pressure dependence of both internal order parameters in the equilibrium liquid state (i.e., A ¼ 0 and B ¼ 0) govern this classic result obtained by Ehrenfest [1933] more than 70 years ago. Next, the same sequence of calculations is performed using the total differential of molar entropy s, which is a function of T, p, z, and y. Properties of the glassy state are obtained at constant z and y, whereas those of the equilibrium liquid correspond to A ¼ 0 and B ¼ 0.

174

Chapter 5 Order Parameters for Glasses

At constant pressure, the temperature derivative of molar entropy is given by the ratio of the molar heat capacity and temperature. Isothermally, the pressure derivative of molar entropy is given by the product of molar volume and thermal expansion coefficient, based on a Maxwell relation that originates from the total differential of the Gibbs free energy. The order parameter development based on molar entropy reveals that the second Ehrenfest equation is influenced strongly by the temperature and pressure dependence of z and y for the equilibrium liquid. Justification for these claims is quantified below, beginning with the total differential of molar entropy and the definitions of glassy state properties:         @s @s @s @s dT þ dp þ dz þ dy ds ¼ @T p,z,y @p T,z,y @z T,p,y @y T,p,z     @s 1 @s ¼ Cp,Glass ; ¼ vaGlass @T p,z,y T @p T,z,y The following liquid state properties can be derived from temperature and pressure derivatives of molar entropy when both affinities vanish (i.e., A ¼ 0 and B ¼ 0). Two equations reveal that Cp,Liquid 2 Cp,Glass ¼ DCp and aLiquid 2 aGlass ¼ Da can be calculated from the temperature and pressure dependence of z and y, as well as the dependence of molar entropy on z and y:       @s 1 1 @s @z ¼ Cp,Liquid ¼ Cp,Glass þ @T p,A¼0,B¼0 T T @z T,p,y @T p,A¼0,B¼0     @s @y þ @y T,p,z @T p,A¼0,B¼0       @s @s @z ¼ vaLiquid ¼ vaGlass þ @p T,A¼0,B¼0 @z T,p,y @p T,A¼0,B¼0     @s @y þ @y T,p,z @p T,A¼0,B¼0 In condensed notation, the ratio of the following two equations yields the pressure dependence of the glass transition temperature, as prescribed by the second Ehrenfest equation:     1 @z @y DCp ¼ sz þ sy T @T p,A¼0,B¼0 @T p,A¼0,B¼0     @z @y v Da ¼ sz þ sy @p T,A¼0,B¼0 @p T,A¼0,B¼0     @s @s sz ¼ ; sy ¼ @z T,p,y @y T,p,z where both temperature T and molar volume v must be evaluated at Tg. Total differentials of the affinities, A and B, are used to calculate temperature and pressure

5.2 Ehrenfest Inequalities

175

derivatives of the internal order parameters, z and y, for the equilibrium liquid where A ¼ 0 and B ¼ 0. Since the Gibbs free energy is an exact differential, it is acceptable to reverse the order of second mixed partial differentiation without affecting the final result. For example, if g ¼ g(T, p, z, y), then   @g ¼ gz ¼ A(T, p, z, y) A¼  @z T,p,y         @A @A @A @A dT þ dp þ dz þ dy dA ¼ @T p,z,y @p T,z,y @z T,p,y @y T,p,z dA ¼ sz dT  vz dp  gzz dz  gyz dy ( "  #)  2  @ g @ @g gzz ¼ . 0; gyz ¼ @z2 T,p,y @y @z T,p,y

T,p,z

For the equilibrium liquid, where both affinities A and B vanish and the order parameters depend only on T and p, the temperature and pressure dependence of z are obtained by setting dA ¼ 0 at constant y. Hence,     @z @z sz ¼ ¼ @T p,A¼0,B¼0 @T p,y,A¼0,B¼0 gzz     @z @z vz ¼ ¼ gzz @p T,A¼0,B¼0 @p T,y,A¼0,B¼0 Analogously, the temperature and pressure dependence of order parameter y for the equilibrium liquid are obtained by requiring that the total differential of affinity B must vanish at constant z. The appropriate set of equations is summarized below.   @g ¼ gy ¼ B(T, p, z, y) B¼ @y T,p,z         @B @B @B @B dT þ dp þ dz þ dy dB ¼ @T p,z,y @p T,z,y @z T,p,y @y T,p,z dB ¼ sy dT  vy dp  gzy dz  gyy dy (   )  2  @ g @ @g gyy ¼ . 0; gzy ¼ 2 @y T,p,z @z @y T,p,z T,p,y     @y @y sy ¼ ¼ @T p,A¼0,B¼0 @T p,z,A¼0,B¼0 gyy     @y @y vy ¼ ¼ gyy @p T,A¼0,B¼0 @p T,z,A¼0,B¼0 This analysis of temperature – pressure relations at an equilibrium second-order phase transition has generated three important expressions for the following discontinuous

176

Chapter 5 Order Parameters for Glasses

observable quantities at Tg: Da, Db, and DCp, where D represents the difference between thermodynamic properties slightly above and slightly below the glass transition temperature. These expressions and the corresponding substitutions for the temperature and pressure dependence of order parameters z and y are summarized below.     @z @y v Da ¼ vz þ vy @T p,A¼0,B¼0 @T p,A¼0,B¼0     @z @y vz sz vy sy ¼ sz  sy ¼ þ .0 gyy @p T,A¼0,B¼0 @p T,A¼0,B¼0 gzz     v2y v2 @z @y  vy ¼ z þ .0 v Db ¼ vz @p T,A¼0,B¼0 @p T,A¼0,B¼0 gzz gyy     s2y s2 1 @z @y DCp ¼ sz þ sy ¼ z þ .0 Tg @T p,A¼0,B¼0 @T p,A¼0,B¼0 gzz gyy The inequality in each case has been introduced solely based on experimental observations. However, the second and third inequalities are obvious based on the thermodynamics and mathematics of this order-parameter analysis. It is important to realize that the temperature and pressure dependence of order parameters z and y are characteristic of the equilibrium liquid (i.e., A ¼ 0 and B ¼ 0), whereas vz , vy , sz, and sy are characteristic of either volume or entropy differences between nonequilibrium glasses that are produced at different cooling rates. Inspection of the previous set of three relations suggests that the difference between (i) the product of the second and the third and (ii) the square of the first provides a comparison between the two Ehrenfest relations. For example,   v DCp Da 2 2 Db DCp  (v Da) ¼ v Da Db  Tg v Da Db Tg ( )( )   v2y s2y v2z s2z vz sz vy sy 2  þ þ þ ¼ gzz gyy gzz gyy gzz gyy ¼

(vz sy  vy sz )2 0 gzz gyy

As presented in Section 1.6, prior to the introduction of order parameters to characterize the solid state morphology of glasses, both Ehrenfest equations for the pressure dependence of the glass transition temperature yield the following equality:   @Tg Db Tg v Da ¼ ¼ @p composition Da DCp which corresponds to a Prigogine – Defay ratio of unity [Prigogine and Defay, 1954]. When the Prigogine – Defay equality is satisfied, vz sy 2 vy sz ¼ 0. This is a special case where volume and entropy differences between nonequilibrium glasses that are produced at different cooling rates require only one order parameter to characterize

5.3 Compositional Dependence of the Glass Transition Temperature

177

glassy structures. In other words, z and y monitor the same process as fluctuations in entropy and volume occur during vitrification (i.e., glass formation), such that equivalent results are obtained when z is replaced by y. Justification for this claim is provided below:     @s @s     @z T,p,y @y T,p,z sz sy @s @s ¼ )   ¼  ) ¼ @v @v vz vy @v T,p,y @v T,p,z @z T,p,y @y T,p,z If two order parameters are required to describe the morphological structure of a glass, then any one of the following inequalities implies that the others must also be true [Yourtee, 1973]:     @s @s Db Tg v Da = ) . v z sy = v y s z ) Da DCp @v T,p,y @v T,p,z These relations are known as the Ehrenfest inequalities, and they provide accurate predictions for nonpolar materials [Wu, 1999]. Experimentally, one typically observes that the pressure dependence of the glass transition temperature via volume continuity (i.e., dTg/dp ¼ Db/Da) is between two-fold and five-fold larger than that based on entropy continuity (i.e., dTg/dp ¼ Tgv Da/DCp), [Nieuwenhuizen, 1997]. Using bisphenol-A polycarbonate as a practical example of a “polar” polymer to examine the Ehrenfest predictions, typical values for its physical properties (i.e., Tg ¼ 423 K, r  1.176 g/cm3) and discontinuities in thermal expansion (i.e., Da  3.78  1024 K21), isothermal compressibility (i.e., Db  1  1025 atm21), and specific heat (i.e., DCp  0.29 J/(g-K)) yield the following results: Db/Da  2.65  1022 K/atm, Tgv Da/DCp  4.75  1022 K/atm. The reported experimental value of dTg/dp is 4.4  1022 K/atm [O’Reilly, 1962]. It is generally accepted that predictions of dTg/dp from entropy continuity are more accurate for most, but not all, amorphous polymers, because the discontinuity in isothermal compressibility Db is strongly pressure dependent and it has not been measured for as many materials, relative to Da and DCp, which are easier to obtain experimentally. The polar nature of the carbonate functional group in bisphenol-A polycarbonate and the relatively high glass transition temperature of this unique high-impact glassy polymer most likely preclude any verification of the Ehrenfest inequalities.

5.3 COMPOSITIONAL DEPENDENCE OF THE GLASS TRANSITION TEMPERATURE 5.3.1

Calorimetric Results and Objectives

It is well known that the glass transition temperature of an amorphous polymer is depressed by the addition of low-molecular-weight miscible plasticizers. This effect is illustrated in Figure 5.1 for four additives that decrease the Tg of bisphenol-A polycarbonate. The diluents and their respective glass transition temperatures are dibutyl

178

Chapter 5 Order Parameters for Glasses 160 Dinitrobiphenyl Diphenylphthalate Dibutylphthalate Dibutylsuccinate

Glass Transition Temperature (°C)

140

120

100

80

60

40

0

5

10

15

20

25

30

Diluent Mass Fraction (%)

Figure 5.1 Effect of four structurally different plasticizers on the glass transition temperature of bisphenol-A polycarbonate in miscible binary mixtures [Belfiore, 1982].

succinate (Tg,diluent ¼2105 8C), dibutyl phthalate (Tg,diluent ¼ 291 8C), dinitrobiphenyl (Tg,diluent ¼ 223 8C), and diphenyl phthalate (Tg,diluent ¼ 215 8C). At constant composition of the polymer – diluent blend, the rate of decrease of Tg with respect to diluent concentration is larger for the lower-Tg diluents, dibutyl succinate and dibutyl phthalate, relative to those blends containing the higher-Tg diluents, dinitrobiphenyl and diphenyl phthalate. Furthermore, the slope of Tg versus additive concentration is greatest for each binary mixture when the diluent concentration approaches zero. The major objective of this section is to develop a relation between the compositional dependence of Tg and changes in the energetic mixing characteristics of polymer – diluent blends when they are cooled below Tg from the molten state [Belfiore and Cooper, 1983]. The methodology follows the order parameter formalism used to develop the classic Ehrenfest equations that describe the pressure dependence of Tg in terms of discontinuous thermodynamic observables in the vicinity of a secondorder phase transition.

5.3.2

Assumptions of the Theory

(a) In amorphous polymeric materials, there is an equilibrium glass transition temperature, Tg,equilibrium, that can be attained only in experiments conducted on infinite time scales. According to the conformational entropy theory of DiMarzio and Gibbs [1963] discussed in Section 3.7, Tg,equilibrium corresponds to the temperature at which the conformational entropy of the

5.3 Compositional Dependence of the Glass Transition Temperature

179

macromolecule initially vanishes upon cooling. Under most experimental conditions, the measured glass transition temperature contains a kinetic contribution that is attributed to some type of rate effect (i.e., heating or cooling rate, probe frequency, etc.). In all cases, the experimental glass transition temperature is higher than Tg,equilibrium. (b) In polymer – diluent blends, the hypothetical diluent concentration dependence of Tg,equilibrium can be superposed onto the experimental curve of Tg versus diluent concentration. DiMarzio and Gibbs [1963] suggest that only the slopes and shapes of the two curves are important, not the absolute magnitude of the phase transition temperatures. (c) Below Tg, the total differential of the molar Gibbs free energy g of a binary mixture is a function of temperature T, pressure p, mole fraction of one of the components (i.e., diluent) yDiluent, and a single dimensionless intensive order parameter z, which can be expressed as follows: dg ¼ s dT þ v dp þ (mDiluent  mPolymer )dyDiluent  A dz where the coefficients s (molar entropy), v (molar volume), mi (chemical potential), and A (affinity) are state functions. Order parameter z is constant for a reasonably low-energy amorphous structure below Tg. Hence, z depends on the experimental cooling rate employed to produce the glassy material. (d) Above the glass transition temperature, order parameter z in the equilibrium liquid state has the characteristics of a state function (i.e., z ¼ z(T, p, yDiluent)), and z is independent of sample history. At constant temperature T, pressure p, and diluent mole fraction yDiluent, the order parameter adjusts itself to minimize the molar Gibbs free energy of the binary mixture:  2    @g @ g ¼ 0; .0 @z T,p,yDiluent @z2 T,p,yDiluent The previous statement implies that the affinity A vanishes in the equilibrium liquid state. (e) The Flory –Huggins lattice theory was discussed in Section 3.4.6 for the extensive conformational entropy of mixing in concentrated polymer solutions. The equation for DSmixing, presented below, is not useful for numerical calculations because the summation extends over the total number of polymer molecules in the mixture. The most important aspect of the following expression [Belfiore and Cooper, 1983] "   DSmixing ¼ k (x  1)NPolymer ln wPolymer  xNPolymer ln(xNPolymer )  1

þx

NPolymer X1 i¼0



ln NDiluent þ x(NPolymer  i)



#

180

Chapter 5 Order Parameters for Glasses

is the fact that temperature does not appear implicitly or explicitly. NDiluent and NPolymer are the number of molecules of diluent and polymer, respectively, 0 0 wPolymer is the polymer volume fraction, x ¼ VPolymer =VDiluent is the ratio of pure-component molar volumes of polymer and diluent, respectively, or the number of segments in a polymer chain, and k is Boltzmann’s constant. This lattice theory prediction of DSmixing is not restricted to dilute solutions. The development below considers that the extensive conformational entropy of mixing for concentrated polymer solutions exhibits the following generic dependence on blend composition: DSmixing ¼ f (NPolymer , NDiluent ; x) where the generic function f (NPolymer, NDiluent; x) is independent of temperature. (f) The temperature-dependent nonideal extensive Gibbs free energy of mixing due to polymer – diluent interactions is based on the Flory – Huggins lattice theory, as discussed in Section 3.4.4, with an interaction parameter x that does not depend on the composition of the binary system. This expression for DGInteraction is adopted from Flory’s enthalpy of mixing in nonideal binary systems. In the most general case, it is assumed that there are both enthalpic and entropic contributions to the nonideal interaction free energy of mixing. Hence, DGInteraction ¼ kT xNDiluent wPolymer The Flory – Huggins interaction parameter x is temperature dependent, but composition independent. Thermodynamic analysis of the previous expression for DGInteraction yields the following results for the nonideal extensive enthalpy and entropy of mixing due to polymer – diluent interactions:   @ DGInteraction T @T p,composition   @x ¼ kT 2 NDiluent wPolymer @T p,composition   @(DGInteraction) ¼ @T p,composition   @(T x) ¼ kNDiluent wPolymer @T p,composition

DHInteraction ¼ T 2

DSInteraction



5.4 Diluent Concentration Dependence of the Glass Transition Temperature

181

For example, if the interaction parameter x scales inversely with temperature such that a1 x (T) ¼ a0 þ T then the zeroth-order coefficient a0 yields an interaction entropy of mixing, and the coefficient of T 21 (i.e., a1) is enthalpic, in origin, as illustrated below: DSInteraction ¼ ka0 NDiluent wPolymer DHInteraction ¼ ka1 NDiluent wPolymer (g) Interaction parameters that are independent of blend composition agree, to some extent, with the experimental data of Eichinger and Flory [1968], who demonstrated that x depends very weakly on the concentration of polyisobutylene in cyclohexane at 25 8C. (h) The conformational characteristics of the pure components and their mixtures are described hypothetically by a single lattice that simply adjusts its size to accommodate the appropriate number of molecules. This implies that neither the size of the lattice sites nor the ratio of pure-component 0 0 =VDiluent ) depends on temperature. Furthermore, molar volumes (i.e., VPolymer there is no deviation of the fractional free volume of the mixtures from simple additivity.

5.4 DILUENT CONCENTRATION DEPENDENCE OF THE GLASS TRANSITION TEMPERATURE VIA CLASSICAL THERMODYNAMICS Based on the total differential of the specific Gibbs free energy dg of a binary mixture in terms of its natural independent variables, as outlined above in Assumption (c), the molar entropy of the polymer – diluent blend is also a function of temperature T, pressure p, diluent mole fraction yDiluent, and order parameter z, because   @g s(T, p, yDiluent , z) ¼  @T p,yDiluent ,z Now, the total differential of molar entropy is     @s @s dT þ dp ds ¼ @T p,yDiluent ,z @p T,yDiluent ,z     @s @s dyDiluent þ dz þ @yDiluent T,p,z @z T,p,yDiluent Thermodynamic properties of the equilibrium liquid are evaluated at A ¼ 0, and those in the glassy state correspond to a constant value of the order parameter z. On the

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Chapter 5 Order Parameters for Glasses

transition line between the liquid and glassy states, one evaluates properties simultaneously at A ¼ 0 and z ¼ constant. Based on the previous expression for the total differential of molar entropy, the mole fraction derivatives of s at constant T and p are         @s @s @s @z  ¼ @yDiluent T,p,A¼0 @yDiluent T,p,z @z T,p,yDiluent @yDiluent T,p,A¼0 Analogously, the temperature derivatives of s at constant pressure and composition are         @s @s @s @z  ¼ @T p,yDiluent ,A¼0 @T p,yDiluent ,z @z T,p,yDiluent @T p,yDiluent ,A¼0 The differences between partial derivatives on the left sides of the previous two equations represent discontinuous second derivatives of the Gibbs free energy at a second-order phase transition, like Tg. The ratio of these two equations yields the diluent concentration dependence of temperature along the transition line that separates the equilibrium liquid and glassy states. According to the triple-product rule,   @z   @yDiluent @T   T,p,A¼0 ¼  @z @yDiluent p,z,A¼0 @T p,yDiluent ,A¼0     @s @s  @yDiluent T,p,A¼0 @yDiluent T,p,z   ¼   @s @s  @T p,yDiluent ,A¼0 @T p,yDiluent ,z Differences between the temperature dependence of molar entropy in the denominator of the previous equation correspond to the product of T 21 and the discontinuity in molar heat capacities in the liquid and glassy states at composition yDiluent. In other words,      @s @s 1  ¼ Cp,Liquid  Cp,Glass @T p,yDiluent ,A¼0 @T p,yDiluent ,z T Based on the total differential of the molar Gibbs free energy of a binary mixture, as illustrated in Assumption (c) above, the following Maxwell relation allows one to re-express the compositional dependence of molar entropy in terms of the temperature derivative of chemical potentials:     @s @ (m ¼  mPolymer ) @yDiluent T,p,A¼0 @T Diluent p,yDiluent ,Liquid     @s @ (m ¼  mPolymer ) @yDiluent T,p,z @T Diluent p,yDiluent ,Glass Now, the diluent concentration dependence of the glass transition temperature at constant pressure is given by the following ratio of discontinuous observables at Tg, where D represents the difference between thermodynamic properties evaluated in

5.5 Compositional Dependence of the Glass Transition Temperature

183

the liquid and glassy states:     @Tg,Mixture Tg,Mixture @ (mDiluent  mPolymer ) ¼ D @yDiluent p DCp(molar) @T p,yDiluent The previous equation contains macroscopic quantities from classical thermodynamics that are discontinuous at Tg. Order parameter z and affinity A were instrumental in the identification of glasses and liquids, respectively, but neither z nor A appears in the final result from classical thermodynamics, as illustrated by the previous equation.

5.5 COMPOSITIONAL DEPENDENCE OF THE GLASS TRANSITION TEMPERATURE VIA LATTICE THEORY MODELS The diluent concentration dependence of Tg contains macroscopic quantities from classical thermodynamics. In this section, the combinatorial statistics of, and energetic interactions between, polymer segments and diluent molecules are incorporated into the macroscopic formulation via the Flory – Huggins lattice theory. Even though the Flory – Huggins conformational entropy of mixing does not depend on detailed intramolecular characteristics of polymer chains, the interaction free energy of mixing contains intermolecular effects via the x parameter. Expressions in Sections 3.4.4 and 3.4.6 are employed to generate the extensive Gibbs free energy of mixing for trace amounts of plasticizer in an amorphous polymer DGMixing ¼ kT xNDiluent wPolymer  Tf (NPolymer , NDiluent ; x) ¼ RT xnDiluent wPolymer  Tf (NPolymer , NDiluent ; x) where the plasticizer is a diluent that assumes the role of a solvent, and the number of diluent molecules NDiluent is re-expressed in terms of moles (i.e., nDiluent) and Avogadro’s number. Relative to the standard states, defined by pure diluent and perfectly ordered polymer, the chemical potentials of diluent and polymer are calculated from DGMixing as follows:   @DGMixing 0 mDiluent  mDiluent (T) ¼ @nDiluent T,p,nPolymer   @f ¼ RT xw2Polymer  T @nDiluent T,p,nPolymer   @DGMixing mPolymer  m0Polymer (T) ¼ @nPolymer T,p,nDiluent   @f 2 ¼ RT xxwDiluent  T @nPolymer T,p,nDiluent where volume fraction is denoted by w. The temperature dependence of mDiluent 2 mPolymer at constant pressure and composition is calculated from these expressions,

184

Chapter 5 Order Parameters for Glasses

where x is temperature dependent, but f is not. One obtains the following result that must be evaluated slightly above and slightly below the glass transition temperature:   @ d  0 (mDiluent  mPolymer ) ¼ mDiluent  m0Polymer @T dT p,yDiluent

mDiluent  m0Diluent mPolymer  m0Polymer  T T   @ x  þ RT w2Polymer  xw2Diluent @T p þ

At an equilibrium second-order phase transition for polymer – diluent blends, the chemical potential of the diluent is equivalent in the liquid and glassy states, and the same is true for mPolymer. Furthermore, the temperature dependence of purecomponent chemical potentials (i.e., pure-component molar entropy) is not discontinuous at a second-order phase transition. Hence, only the last term on the right side of the previous equation exhibits a discontinuity at Tg. When trace amounts of diluent are present and the magnitude of the concentration dependence of the glass transition temperature is largest, results from this section and the previous one yield  lim

yDiluent )0

@Tg,Mixture @yDiluent

 p

2 RTg,Polymer ¼ DCp,Polymer(molar)

(

@x @T





@x  @T p,Glass



)

p,Liquid

Consistent with Tg depression by plasticizers and the fact that the discontinuity in specific heat is greater than zero at the glass transition temperature, the previous equation predicts that the temperature dependence of the Flory – Huggins intermolecular interaction parameter is larger for glasses than for liquids. However, one must realize that the most common temperature dependence for x is such that it decreases at higher temperature (see Section 3.4.4), so x versus T should exhibit a steeper negative slope for liquids relative to glasses. Hence, f@ x/@Tgp,Glass . f@ x/ @Tgp,Liquid, but the magnitude of this slope is greater above the glass transition temperature if Tg depression is induced by low-molecular-weight additives. This inequality must be reversed to accommodate Tg enhancement by transition-metal and lanthanide complexes (see Chapter 6).

5.6

COMPARISON WITH OTHER THEORIES

In this section, the previous equation for the compositional dependence of Tg in the limit of trace amounts of diluent is compared with other well-known results. Chow [1980] obtained an expression for the effect of diluent volume fraction wDiluent on the glass transition temperature Tg,Mixture of a miscible amorphous polymer – diluent 0 blend via equilibrium statistical thermodynamics. If Tg,Polymer is the glass transition temperature of the undiluted polymer, DCp(specific) the discontinuity in specific heat 0 the repeat unit molecular of the mixture at its glass transition temperature, MWPolymer

5.6 Comparison with Other Theories

185

weight of the polymer, j the lattice coordination number, and R the gas constant, then Chow’s [1980] derivation yields the following result: ( ) Tg,Mixture ¼ b[(1  Q) ln(1  Q) þ Q ln Q] ln 0 Tg,Polymer   y 0Polymer jR wDiluent b¼ ; Q ¼ 0 DCp(specific) jy 0Diluent 1  wDiluent MWPolymer where yi0 is the pure-component molar volume of species i. Very good agreement has been demonstrated between the previous equation and experimental data for several polystyrene – diluent mixtures. However, one disadvantage of Chow’s approach is its failure to distinguish the effects of structurally different plasticizers and antiplasticizers on their ability to decrease Tg in the limit of trace amounts of diluent. Based on the previous equation, the slope of the mixture’s Tg versus diluent volume fraction wDiluent is  2   y 0Polymer dTg,Mixture RTg,Mixture 1 1Q  ¼ ln 0 dwDiluent Q MWPolymer DCp(specific) y 0Diluent 1  wDiluent The approximations in Chow’s statistical thermodynamic theory require that both Q and wDiluent should be much less than unity. The previous equation predicts that plasticizer efficiency, defined by 2dTg,Mixture/dwDiluent in the limit of vanishing diluent concentration, is infinitely large and independent of the specific polymer – diluent pair. Kelley and Bueche [1961] and Fujita and Kishimoto [1961] invoked continuity of total system volume and assumed linear additivity of fractional free volume to obtain the following compositional dependence of Tg,Mixture that represents the volume continuity analog of the well-known Gordon – Taylor equation (see Section 1.13). Tg,Mixture ¼

wDiluent DaDiluent Tg,Diluent þ (1  wDiluent )DaPolymer Tg,Polymer wDiluent DaDiluent þ (1  wDiluent )DaPolymer

When polymer – diluent blends contain trace amounts of additive, the previous expression reduces to (see Problem 1.9 in Chapter 1)    @Tg,Mixture DaDiluent  ¼ Tg,Polymer  Tg,Diluent lim wDiluent )0 @ wDiluent p DaPolymer where Dai represents the discontinuity in thermal expansion coefficient for component i at its pure-component glass transition temperature, Tg,Polymer or Tg,Diluent. This prediction from linear free-volume theory is consistent with the Tg data presented earlier in this chapter for bisphenol-A polycarbonate blends with diluents that exhibit different chemical structure and glass transition temperatures. The initial slope of the mixture’s glass transition temperature with respect to diluent volume fraction wDiluent is steeper for the lower-Tg diluents. Couchman and Karasz [1978] used classical thermodynamics and continuity of total system entropy to develop the compositional dependence of Tg,Mixture for miscible binary mixtures via the following expression which is

186

Chapter 5 Order Parameters for Glasses

known “classically” as the Gordon – Taylor equation: Tg,Mixture ¼

vDiluent DCp,Diluent Tg,Diluent þ (1  vDiluent )DCp,Polymer Tg,Polymer vDiluent DCp,Diluent þ (1  vDiluent )DCp,Polymer

where vDiluent represents diluent mass fraction in the mixture and DCp,i is the discontinuity in specific heat for species i at its pure-component glass transition temperature. As discussed in Section 1.7 when the glass transition process in miscible blends is analyzed from the viewpoint of entropy continuity, five assumptions must be satisfied to arrive at the previous Tg – composition relation in binary systems. (a) Entropy continuity: Si,Liquid (Tgi ) ¼ Si,Glass (Tgi ) Stotal,Liquid (Tg,Mixture ) ¼ Stotal,Glass (Tg,Mixture ) DSmixing,Liquid (Tg,Mixture ) ¼ DSmixing,Glass (Tg,Mixture ) (b) Ideal mixtures: vi,Glass ¼ vi,Liquid (c) Upon heating, Tg occurs at constant pressure. (d) Temperature-independent specific heats: Cpi,Liquid = f (T) and Cpi,Glass =f (T):   (e) ln Tgi =Tgk  Tgi =Tgk  1 þ    : The Couchman – Karasz equation employs mass fraction vi as the important concentration variable because their analysis is based on specific entropy of each pure component in the mixture. In the limit of trace amounts of diluent, one obtains (see the answer to Problem 1.11b);  lim

vDiluent )0

  @Tg,Mixture DCp,Diluent  ¼ Tg,Polymer  Tg,Diluent @ vDiluent p DCp,Polymer

Gordon et al. [1977] obtained the previous equation via a macroscopic formulation of the Gibbs– DiMarzio conformational entropy description of the glass transition temperature, as discussed in Chapter 3.

5.7

MODEL CALCULATIONS

The combined classical and statistical thermodynamic analysis of glass transition temperature depression in miscible amorphous polymer –diluent blends predicts that (@ x/@T )p is discontinuous at an equilibrium second-order phase transition. This conclusion is consistent with the order parameter formalism that yields both of the Ehrenfest equations, together with the Flory – Huggins chemical potentials of polymer and diluent when the dimensionless free-energy interaction parameter x depends on temperature but not composition in binary mixtures. The discontinuity D(@ x/@T )p at Tg, where D signifies property differences above and below the glass transition temperature (i.e., liquid– glass), is estimated for polycarbonate blends that contain two chemically dissimilar diluents, dinitrobiphenyl and dibutylsuccinate, using the

5.7 Model Calculations

187

initial slope of the Tg – concentration data presented earlier in this chapter (see Figure 5.1). One estimates differences between the temperature dependence of x above and below Tg at constant pressure as follows: (

@x @T





@x  @T p,Liquid

 p,Glass

)



@x ¼D @T

 p

" #   DCp,Polymer(molar) MWDiluent @Tg,Mixture lim ¼ 2 0 vDiluent )0 @ vDiluent p RTg,Polymer MWPolymer where DCp,Polymer(molar) represents the discontinuity in molar heat capacity of the 0 ). The undiluted polymer (i.e., based on its repeat unit molecular weight, MWPolymer ratio of molecular weights of diluent and polymeric repeat unit is required when mass fraction vi replaces mole fraction yi to describe diluent concentration in binary mixtures. Since the ratio of DCp,Polymer(molar) to the polymer’s repeat unit molecular weight corresponds to the discontinuity in specific heat for the undiluted polymer at its own pure-component Tg, the previous equation reduces to     DCp,Polymer(specific) @Tg,Mixture @x ¼ MWDiluent lim D 2 vDiluent )0 @ vDiluent p @T p RTg,Polymer Numerical evaluation of various quantities in the previous equation yields the results in Table 5.1 for bisphenol-A polycarbonate blends that contain two structurally dissimilar diluents, 2,20 -dinitrobiphenyl and dibutylsuccinate. Initial slopes of Tg,Mixture versus diluent mass fraction were calculated via linear interpolation between 0 and 5 wt % diluent. The error introduced by this approximation underestimates the actual initial slope and provides a lower limit on the absolute value of D(@ x/@T )p. Steeper initial slopes of Tg,Mixture versus diluent mass fraction are consistent with greater plasticizer efficiency, and the previous equation suggests that these polymer – diluent blends exhibit larger discontinuities in (@ x/@T )p. (See Table 5.2.)

Table 5.1 Discontinuities in the Temperature Dependence of the Flory–Huggins Dimensionless Free-Energy Interaction Parameter for Bisphenol-A Polycarbonate –Diluent Blends Thermophysical property (in the limit of pure polymer) (@Tg,Mixture/@ vDiluent)p; vDiluent ) 0 MWDiluent (daltons) Tg,Polymer (K) DCp,Polymer(specific) (J/g-K) D(@ x/@T )p @ Tg,Polymer (K21)

Polycarbonate– dinitrobiphenyl

Polycarbonate– dibutylsuccinate

2500 K 244 423 0.29 22.4  1022

2900 K 230 423 0.29 24.0  1022

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Chapter 5 Order Parameters for Glasses

Table 5.2 Discontinuities in the Temperature Dependence of the Flory– Huggins Dimensionless Free-Energy Interaction Parameter for Polymer–Diluent Blends that Contain Bisphenol-A Polycarbonate or Polystyrene a Polymer–diluent blend Polycarbonate – p-pentyloxycinnamic acid Polystyrene –terephthal-bis-4n-butylaniline Polystyrene –n(ethoxybenzylidine)-p-nbutylaniline

(@Tg,Mixture/@ vDiluent)p vDiluent ) 0

D(@ x/@T )p @ Tg,Polymer  1022 (K21)

2480 K 2104 K

22.2 [Belfiore and Patwardhan 1988] 21.0 [Weiss et al., 1983]

2400 K

22.7 [Weiss et al., 1983]

a

Df@ x/@Tgp ¼ f@ x/@Tgp,Liquid 2 {@ x/@T}p,Glass.

5.8

LIMITATIONS OF THE THEORY

Polymeric glasses invariably exist in a nonequilibrium state, which depends on the cooling rate and formation pressure that polymers experience in the molten state. Upon cooling below Tg, the sluggishness of the material’s viscoelastic response precludes an exact interpretation of glass transition phenomena in terms of equilibrium thermodynamics. In this respect, it is instructive to discuss the validity of the equilibrium order parameter formalism presented in this chapter for the pressure and diluent concentration dependence of Tg. Although equilibrium second-order transitions are well understood theoretically, most experimental measurements of Tg should be interpreted using nonequilibrium thermodynamics to explain phenomena governed mostly by kinetic processes. According to DiMarzio and Gibbs [1963], the absolute magnitudes of the equilibrium and experimentally measured glass transition temperatures are not the same, but the diluent concentration dependence of both of these Tg values in miscible polymer – diluent blends should have the same slopes and shapes.

REFERENCES BELFIORE LA. Molecular Dynamics of Polycarbonate – Diluent Systems, PhD thesis. University of Wisconsin-Madison, September 1982. BELFIORE LA, COOPER SL. Bisphenol-A polycarbonate –diluent interactions. Journal of Polymer Science; Polymer Physics Edition 21:2135–2157 (1983). BELFIORE LA, PATWARDHAN AA. Thermodynamic miscibility of polymer/liquid-crystal blends. Polymer Engineering and Science 28(14):916–925 (1988). CHOW TS. Molecular interpretation of the glass transition temperature of polymer– diluent systems. Macromolecules 13(2):362– 364 (1980). COUCHMAN PR, KARASZ FE. Classical thermodynamic discussion of the effect of composition on glass transition temperatures. Macromolecules 11(1):117 –119 (1978). DIMARZIO EA, GIBBS JH. Molecular interpretation of glass temperature depression by plasticizers. Journal of Polymer Science 1A:1417–1428 (1963).

Problem

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EHRENFEST P. Phase changes in the ordinary and extended sense, classified according to the corresponding singularities of the thermodynamic potentials. Proceedings of the Academy of Sciences (Amsterdam) 36:153 (1933). EICHINGER BE, FLORY PJ. Thermodynamics of polymer solutions: polyisobutylene and cyclohexane. Transactions of the Faraday Society 64:2061 (1968). FUJITA H, KISHIMOTO A. Interpretation of viscosity data for concentrated polymer solutions. Journal of Chemical Physics 34(2):393 (1961). GOLDSTEIN M. Some thermodynamic aspects of the glass transition: free volume, entropy, and enthalpy theories. Journal of Chemical Physics 39:3369 (1963). GOLDSTEIN M. Validity of the Ehrenfest equations for systems with more than one ordering parameter. Journal of Applied Physics 46(10):4153 –4156 (1975). GORDON JM, ROUSE GB, GIBBS JH, RISEN WM. Compositional dependence of glass-transition properties. Journal of Chemical Physics 66(11):4971 –4976 (1977). KELLEY FN, BUECHE F. Viscosity and glass transition temperature relations for polymer– diluent systems. Journal of Polymer Science 50:549 (1961). NIEUWENHUIZEN THM. Ehrenfest relations at the glass transition; solution to an old paradox. Physical Review Letters 79(7):1317–1320 (1997). O’REILLY JM. The effect of pressure on the glass transition temperature and dielectric relaxation time of poly(vinyl acetate). Journal of Polymer Science 57:429– 444 (1962). PRIGOGINE I, DEFAY R. Chemical Thermodynamics. Longmans Green, New York, 1954, Chap. 19. WEISS RA, HUH WS, NICOLAIS L. Thermal and rheological properties of blends of polystyrene and thermotropic liquid crystals. Polymer Engineering and Science 23(14):779– 783 (1983). WU J. The glassy state, ideal glass transition, and second-order phase transition. Journal of Applied Polymer Science 71:143–150 (1999). YOURTEE JP. The Mechanical Behavior of Densified Polystyrene, PhD thesis. University of Wisconsin– Madison, 1973.

PROBLEM 5.1. A 50 : 50 random copolymer of D-lactic acid and L-lactic acid (i.e., PDLLA), with a glass transition temperature of 588C, is processed in supercritical carbon dioxide at 458C and 100 bar total pressure. Discuss the competing effects of pressure and diluent concentration on Tg. If possible, include the concept of reduced variables (i.e., temperature and pressure) with respect to the critical constants, and consider the enhanced diffusional flux of CO2 in the supercritical state relative to the liquid state. Then, in an effort to mimic spongy trabecular or cancellous bone, calculate the mass fraction of solubilized CO2 that is required to generate a flexible porous PDLLA matrix at the processing conditions indicated above. Estimate the time required for CO2 diffusion to equilibrate throughout PDLLA, whose film thickness is L. (Hint: equilibration should be achieved after approximately five diffusion time constants, according to the analyses in Sections 2.6 and 16.8.)

Chapter

6

Macromolecule –Metal Complexes: Ligand Field Stabilization and Glass Transition Temperature Enhancement Fierce grinning spawn of a molten mother: an obsidian soul. —Michael Berardi

Inorganic models of transition-metal coordination between d-block salts and functional polymers are introduced and analyzed to explain increases in Tg via coordination crosslinks. d-Electron configurations and stabilization of metal d-electrons due to the surrounding ligands in the first-shell coordination sphere are correlated with thermal synergy in several macromolecule –metal complexes. These energetic models that exhibit reduced local symmetry about the metal center in the molten state, relative to the glassy state, can be extended to binary systems for which glass transition data are not available.

6.1

LIGAND FIELD STABILIZATION

Many transition-metal elements and ions have outermost, or frontier, electrons that occupy d-orbitals, and all five of these d-orbitals exhibit the same energy for a “free” ion. When functional side groups in a polymer chain aggregate in the vicinity

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

191

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Chapter 6 Macromolecule–Metal Complexes: Ligand Field Stabilization

(i.e., first-shell coordination sphere) of a transition-metal ion and form a complex, the energies of the d-orbitals are perturbed—some move to lower energy and others move to higher energy such that the average energy of all five d-orbitals is essentially the same. The number of d-electrons in a complex is a function of the atomic number and valence of the transition-metal ion. These electrons occupy the perturbed d-orbitals of the transition-metal complex, beginning with the orbitals at lowest energy, subjected to the Pauli exclusion principle that, at most, two electrons with opposite spin angular momentum can occupy each orbital. Ligand field stabilization describes the difference between (i) the energies of all d-electrons that occupy perturbed d-orbitals in a complex and (ii) the energies of the same d-electrons when they occupy unperturbed degenerate orbitals of the “free” metal ion, in the absence of the surrounding ligands. Stabilization of metal d-electrons occurs when they occupy orbitals at lower energy in a complex relative to the d-orbital energies of the free ion.

6.2

OVERVIEW

When transition-metal cations from the d-block of the Periodic Table coordinate to ligands in the side group of a polymer and modify the thermal response of a macromolecular complex, the enhancement in Tg can be explained by focusing on ligand field stabilization of the metal d-electrons [Belfiore et al., 1992]. The methodology to identify attractive coordination complexes and predict relative increases in Tg is described in terms of the local symmetry of the complex, molecular orbital energies, and the d-electron configuration [Belfiore et al., 2001a]. Interelectronic repulsion is considered for pseudo-octahedral d6 and d7 complexes in the glassy state when there is ambiguity in the order in which the d-orbitals are populated. Ligand field stabilization energies are calculated for simple octahedral geometries and 5-coordinate complexes with reduced symmetry, such as square pyramidal, trigonal bipyramidal, and pentagonal planar, for example, in molybdenum hexacarbonyl complexes with poly(vinylamine) above and below the glass transition temperature. If pseudooctahedral transition-metal complexes bridge two different macromolecules in the glassy state via coordination crosslinks, then 5-coordinate complexes with one surviving metal – polymer bond above Tg represent reasonable geometries in the molten state. This model of thermochemical synergy in macromolecule – metal complexes with no adjustable parameters considers the glass transition as an endothermic process in which sufficient thermal energy must be supplied to dissociate intermolecular bridges or coordination crosslinks and produce coordinatively unsaturated molten state complexes. The enhancement in Tg correlates well with the difference between ligand field stabilization energies in the glassy and molten states for Ru2þ (d6), Co2þ (d7), and Ni2þ (d8) complexes with either poly(4-vinylpyridine) or poly(L-histidine). Larger relative increases in Tg are measured in complexes with the synthetic poly(a-amino acid) relative to those with poly(4-vinylpyridine). Poly(vinylamine) complexes with cobalt chloride hexahydrate [Belfiore et al., 1997] and several lanthanide trichloride

6.3 Methodology of Transition-Metal Coordination in Polymeric Complexes

193

hydrates [Das et al., 2000] exhibit some of the largest increases in the glass transition temperature that have been measured to date.

6.3 METHODOLOGY OF TRANSITION-METAL COORDINATION IN POLYMERIC COMPLEXES 6.3.1 Polymeric Coordination Complexes with d-Block Salts that Exhibit an Increase in Tg It is well known that organic plasticizers decrease a polymer’s glass transition temperature [Kelley and Bueche, 1961], as described by previous researchers via entropy continuity, volume continuity, free volume concepts [Fujita and Kishimoto, 1961], and the conformational entropy description of Tg [DiMarzio and Gibbs, 1963] when flexible diluents are employed. The glass transition temperature is depressed more at higher diluent concentrations until phase separation occurs. Hence, inexpensive brittle polymers can be used in applications that require more flexible and compliant materials if miscible plasticizers are available to lower the glass transition temperature. When additives increase a polymer’s Tg, explanations are based on the existence of specific interactions and the formation of molecular complexes or nanoclusters, because one does not typically employ diluents with glass transition temperatures that are higher than that of the undiluted polymer. Complexation between amorphous polymers and transition metal salts are operative in organic – inorganic hybrids that exhibit enhanced glass transition temperatures relative to Tg of the undiluted polymer [Belfiore et al., 1993]. Chain mobility is hindered when transition metals coordinate to favorable ligands in the polymer’s side group via acid – base interactions. Coordination pendant groups form when p-orbitals of the ligand with comparable energy and the same symmetry properties as d-orbitals of the metal form s-bonds. If one functional side group in the polymer occupies a vacant site in the first-shell coordination sphere of the metal center, then Tg increases by approximately 10– 30 8C relative to the undiluted polymer. This occurs, for example, in poly(4-vinylpyridine) complexes with zinc acetate dihydrate [Belfiore and McCurdie, 1995]. Coordination crosslinking occurs when the transition metal forms s-bonds with at least two functional side groups on different polymer chains. These interactions should produce mobility-restricting nanoclusters in the polymeric matrix. Figure 6.1 illustrates two modes of complexation between macromolecules and metal cations, denoted by M: intrachain coordination (upper left) versus interchain coordination (lower left). Mixed-mode coordination is illustrated on the right side of Figure 6.1. It seems reasonable that the glass transition temperature should experience larger enhancements when a single metal center coordinates to more functional side groups in several different polymer chains, analogous to multifunctional crosslinking agents. The overall objective of this chapter is to (i) estimate differences between electronic energies of dn configurations for macromolecule – metal complexes in the glassy

194

Chapter 6 Macromolecule–Metal Complexes: Ligand Field Stabilization

Figure 6.1 Schematic representation of intrachain versus interchain coordination of metal centers to functional side groups in linear polymers. (Courtesy of E. Y. Chen, Department of Chemistry, Colorado State University.)

and molten states and (ii) correlate these d-electron energy differences with enhancements in the glass transition temperature.

6.3.2 Chemical Bonding, Coordination, and Transition-Metal Compatibilization Divalent late transition metals like cobalt (d7), nickel (d8), and copper (d 9) in the first row of the d-block can utilize five 3d orbitals, one 4s orbital, and three 4p orbitals to form 4-, 5-, or 6-coordinate complexes [Shriver et al., 1990]. As a general rule, if there are N ligands in the first-shell coordination sphere of a transition-metal complex, then there should be N bonding molecular orbitals, N antibonding molecular orbitals, and 92N nonbonding molecular orbitals. Exceptions to this rule occur in some squareplanar complexes when three orbitals with the same symmetry properties overlap and form chemical bonds [Belfiore et al., 1995b]. Usually, some coordination sites in the first-shell of the metal center are occupied by neutral ligands such as waters of hydration, acetonitrile, benzonitrile, or carbon monoxide (i.e., CuO). Anionic ligands in rather close proximity to the metal cation are required for charge neutrality. Pyridine ligands in poly(4-vinylpyridine), and copolymers that contain 4-vinylpyridine repeat units, coordinate to divalent zinc, copper, nickel, cobalt and ruthenium [Belfiore et al., 1993; Belfiore and McCurdie, 1995]. Alkene ligands in the main chain or side group of diene polymers, such as polybutadiene and polyisoprene, coordinate to palladium(II) and platinum(II), but not nickel(II) [Bosse´ et al., 1995, 1996; Belfiore et al., 1996; Belfiore and Das, 2004]. The imidazole ring in the histidine side group of the synthetic poly(a-amino acid), poly(L-histidine), coordinates to divalent cobalt, nickel, copper, ruthenium, and palladium [McCurdie and Belfiore, 1999a]. One of the most attractive applications of this technology is transition-metal compatibilization of polymers that are immiscible in the absence of the inorganic component. Complexation will induce miscibility if the transition-metal center acts as a bridge between two dissimilar chains by coordinating to appropriate ligands in

6.3 Methodology of Transition-Metal Coordination in Polymeric Complexes

195

the side group of both polymers. This has been demonstrated for copolymer blends of styrene – 4-vinylpyridine and 4-vinylpyridine – butylmethacrylate [Belfiore and McCurdie, 1995]. The proposed structure of this miscible ternary system is illustrated in Figure 6.2. Nickel acetate tetrahydrate and cobalt chloride hexahydrate function as transition-metal compatibilizers and produce miscible 4-vinylpyridine copolymer blends. Dichlorobis(acetonitrile) palladium(II) compatibilizes diene polymer blends, such as (i) atactic 1,2-polybutadiene with atactic 3,4-polyisoprene [Bosse´ et al., 1996] and (ii) 1,2-polybutadiene with cis-polybutadiene [Belfiore et al., 1999] via high-temperature palladium-catalyzed chemical crosslinking. Palladium(II) also compatibilizes 3,4-polyisoprene and (i) lightly sulfonated polystyrene, with or without Zn2þ neutralization of the sulfonic acid groups, and (ii) random copolymers of ethylene and methacrylic acid [Das et al., 2001a]. Tetrakis(triphenylphosphine)-palladium(0) compatibilizes 1,2-polybutadiene with poly(4-bromostyrene) [Das et al., 2001b] via a macromolecular analog of the Heck

Figure 6.2 Molecular model that illustrates the concept of transition-metal compatibilization of two immiscible vinylpyridine copolymers. The octahedral bonding characteristics of nickel acetate tetrahydrate, together with its previously published crystal structure [VanNiekerk and Schoening, 1953; Downie et al., 1971], are used to postulate the structure of the amorphous polymeric coordination complex. It is proposed that the divalent metal salt sheds two hard-base waters of hydration and coordinates to pyridine side groups in copolymers of styrene with 4-vinylpyridine (i.e., on the left) and butylmethacrylate with 4-vinylpyridine (i.e., on the right).

196

Chapter 6 Macromolecule–Metal Complexes: Ligand Field Stabilization

reaction (i.e., oxidative addition followed by olefin coordination, migratory insertion, b-hydrogen elimination, etc.) [Hegedus, 1999].

6.3.3 Ligand Field Stabilization Energy Description of the Enhancement in Tg for Polymeric Complexes with Transition Metals When transition metals coordinate to ligands in the main chain or side group of amorphous polymers and modify the thermal response of a macromolecular complex, the enhancement in Tg is based on the well-known correlation of lattice enthalpies of hexaaqua transition-metal complexes from the first row of the d-block with octahedral ligand field stabilization energies if these complexes exhibit high-spin weak-field electronic configurations [Shriver et al., 1990; Figgis and Hitchman, 2000]. In complexes with octahedral or tetrahedral symmetry, if the energy difference between two nondegenerate metal-based d-orbitals is smaller than the repulsive energy that electrons experience when they are paired with opposite spin in the same orbital, then the electronic configuration is described as high spin in a weak ligand field. If one considers the increasingly exothermic enthalpy of formation of divalent hexaaqua transition-metal complexes (i.e., M2þ(H2O)6) as a function of the number of d-electrons from calcium (d0) to zinc (d10), then the additional exothermic effect relative to linear trends from Ca2þ to Mn2þ and Mn2þ to Zn2þ, illustrated in Figure 6.3, correlates with the stabilization of metal d-electrons for octahedral

Exothermic Hydration Enthalpy (MJ/mol)

0

1

2

3

4

5

6

7

8

9

10

3.0

3.0

2.9

2.9

2.8

2.8

2.7

2.7

2.6

2.6

2.5

2.5

2.4

2.4 Ca(II) 0

Zn(II)

Mn(II) 1

2

3

4 5 6 7 Number of d-Electrons

8

9

10

Figure 6.3 Hydration enthalpies of divalent hexa-aqua metal complexes from the first row of the d-block as a function of the number of d-electrons from Ca2þ to Zn2þ. Both dashed lines illustrate linear trends when ligand field stabilization energies, appropriate for a weak ligand field, are subtracted from the experimental hydration enthalpies.

6.3 Methodology of Transition-Metal Coordination in Polymeric Complexes

197

complexes in the first row of the d-block that exhibit weak-field electronic configurations. These empirical correlations between thermodynamic properties and delectron energies of octahedral complexes provide support for analyzing Tg enhancements in macromolecule – metal complexes via ligand-field-induced stabilization of metal d-electrons. Furthermore, this stabilization must be larger for complexes in the glassy state relative to the corresponding molten state complexes to realize metal-induced increases in the glass transition temperature with respect to Tg of the undiluted polymer. The approach described below deviates considerably from wellknown free-volume and conformational entropy models of the glass transition. Energetic stabilization of metal d-electrons due to the presence of the ligands is invoked to explain relative increases in Tg when transition-metal complexes coordinate to polar side groups in amorphous polymers. The absolute magnitude of Tg and the discontinuity in specific heat DCp at the glass transition temperature are not predicted by these energetic ligand field stabilization models.

6.3.4 Energetic Ligand Field Models and the Methodology of Transition-Metal Coordination The methodology to identify attractive coordination complexes and predict increases in Tg is described in terms of the local symmetry of the complex, energies of the five metal-based atomic d-orbitals when ligands surround the metal center, and the d-electron configuration. The glass transition occurs when sufficient thermal energy is supplied to dissociate a coordination crosslink or bridge between two different polymer chains. If nearby low-molecular-weight neutral ligands, such as waters of hydration (i.e., lattice waters), acetonitrile, benzonitrile, or carbon monoxide, do not occupy the vacant site in the coordination sphere of the transition metal when a polar side group in the polymer is removed from the first shell, then the molten state complex exhibits reduced symmetry relative to the complex below Tg. Detailed calculations are considered for octahedral complexes in the glassy state and 5-coordinate complexes in the molten state. Differences between electronic energies of a dn configuration for polymeric complexes in the glassy and molten states are used to predict relative increases in Tg. Then, these predictions based on ligand field stabilization energies are compared with experimental results in an attempt to establish universal trends. In general, one associates the coordination numbers to complexes that exhibit point group symmetries, as summarized in Table 6.1. Well-Defined Low-Molecular-Weight Transition-Metal Complexes that Increase Tg The X-ray crystallography literature is useful to locate previously published crystal structures of attractive low-molecular-weight d-block metal complexes. Information about coordination numbers and the molecular point groups, summarized in Table 6.1, is useful to (i) postulate possible ligand substitution schemes that involve weakly bound lattice waters, acetonitrile, or carbon monoxide; (ii) adopt the same symmetry for the complex that forms between polymer and metal center in the

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Table 6.1 Summary of Molecular Point Groups and Coordination Numbers for Transition-Metal Complexes (M ¼ Metal, L ¼ Ligand, X ¼ Anionic Ligand) Coordination number 3, ML3 4, ML4 or MX2L2 5, ML5 6, ML6 7, ML7 8, ML8 10, ML10 12, ML12

Point group symmetry (structural formula; acronym) Facial trivacant (C3v ), trigonal planar (D3h), T-shaped (C2v ) Tetrahedral (ML4; Td), trigonal pyramid (ML4; C3v ), square planar (ML4; C4v ), trans-square-planar (MX2L2; D2h), cis-square-planar (MX2L2; C2v ) Square pyramid (C4v ), trigonal bipyramid (D3h), pentagonal planar (D5h) Octahedral (Oh), hexagonal planar (D6h), pentagonal pyramid (C5v ) Heptagonal planar (D7h), pentagonal bipyramid (D5h), hexagonal pyramid (C6v ) Cubic (Oh), square antiprism (D4d), octagonal planar (D8h), hexagonal bipyramid (D6h) Pentagonal antiprism (D5d) Icosahedral (Ih)

glassy state; and (iii) calculate ligand field splittings and the corresponding ligand field stabilization energies below Tg. Table 6.2 summarizes molecular point groups, coordination numbers, and the number of weakly bound neutral ligands that can be displaced for several late transition-metal complexes on the right side of the d-block that are useful for Tg-enhancement of functional polymers. Attractive Polymeric Ligands Polymers with functional groups that coordinate to transition-metal complexes represent an important design criterion for glass transition temperature enhancement. Table 6.2 Transition-Metal Complexes that Increase the Glass Transition Temperature of Functional Polymers Metal complex Cobalt chloride Nickel chloride Copper chloride Zinc chloride Nickel acetate Zinc acetate Dichlorotricarbonyl ruthenium(II) Dichlorobis(acetonitrile)palladium(II) Dichlorobis(benzonitrile)platinum(II)

Structural formula, including neutral ligands in first shell

Molecular point group (pseudo)

Coordination number

Number of weakly bound neutral ligands 4 4 2

CoCl2(H2O)6 NiCl2(H2O)6 CuCl2(H2O)2 ZnCl2 Ni(OOCH3)2(H2O)4 Zn(OOCH3)2(H2O)2 [RuCl2(CO)3]2

Octahedral Octahedral Tetrahedral Polymeric Octahedral Tetrahedral Octahedral

6 6 4

PdCl2(CH3CN)2

Square planar

4

PtCl2(C6H5CN)2

Square planar

4

6 4 6

4 2 3 and a vacant site 2 2

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“Reactive blending” is based on the fact that selected functional groups in the main chain or side group of the polymer are stronger bases than the original weakly bound neutral ligands in the coordination sphere of the transition metal. Hence, these ligands in the polymer will displace neutral ligands like (i) waters of hydration, (ii) carbonyls (i.e., CuO), (iii) acetonitriles, and (iv) benzonitriles. In most cases, weak-base neutral ligands are displaced by strong-base polymeric ligands that are also neutral. In less frequent situations, polymeric ligands cleave the dihalide bridge in a dimeric transition-metal complex and coordinate to the vacant site after cleavage [Benedetti et al., 1972]. The hard-and-soft acid – base theory [Pearson, 1969, 1973] is useful for selecting proper combinations of polymeric ligands and transition metals that have an affinity for each other. Identifying Attractive Interactions via Hard-and-Soft Acids and Bases One possible set of guidelines for acid – base interactions follows concepts from hard-and-soft acid – base theory to identify the hardness of the metal center as an acid and the hardness of the ligands, including important functional groups in the polymer, as bases. Frontier orbital energy differences between the highest occupied and lowest unoccupied molecular orbitals are small and perturbations in the electronic distribution occur rather easily, yielding covalent bonds for soft acid – base pairs. In contrast, when the frontier molecular orbital energy differences are large, it is difficult to perturb the electronic distribution, and ionic bonding dominates in hard acid – base pairs. There is an affinity between acids and bases with the same classification. If there is a mismatch in hardness between the metal center and a neutral ligand, then this metal – ligand bond should be the focus of a potential displacement reaction. Basic functional groups in the polymer with the same hardness classification as the metal could displace a ligand of dissimilar hardness. Waters of hydration (i.e., lattice waters) are neutral hard bases. Hence, if the low-molecularweight metal complex contains lattice waters in the first-shell coordination sphere of a soft or borderline acidic metal center, then soft or borderline basic ligands in the side group of the polymer might displace these hard bases. Hardness classifications for various acids and bases [Pearson, 1969, 1973] are summarized in Table 6.3.

Table 6.3 Hard, Soft, and Borderline Classifications for Acids and Bases Acids:

Hard Borderline Soft

Hþ, Liþ, Naþ, Kþ, Mg2þ, Ca2þ, Mn2þ, Cr3þ, Fe3þ, Co3þ Fe2þ, Co2þ, Ni2þ, Cu2þ, Zn2þ, Rh3þ, Ir3þ, Ru2þ, Os2þ Cuþ, Agþ, Auþ, Pd2þ, Cd2þ, Pt2þ, zero-valent metal atoms

Bases:

Hard Borderline Soft

22 F2, Cl2, OH2, H2O, NH3, RNH2, NO2 3, O 2 2 NO2 , Br , N2, C5H5N, C6H5NH2 H2, I2, CN2, CuO, C6H6, PR3, P(OR)3 R represents an alkyl group, like CH3 or C2H5

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Chapter 6 Macromolecule–Metal Complexes: Ligand Field Stabilization

Displacement of Weak Neutral Bases in the First-Shell Coordination Sphere by Stronger Bases One should consider all of the basic ligands in the coordination sphere of the lowmolecular-weight metal complex, as well as potential basic ligands in the main chain or side group of the polymer, and rank the strengths of these bases using the pKA/pKB scale. The pKA scale, which summarizes the strengths of acids and bases, is completely independent of the hardness classification discussed in the previous section via Table 6.3. At 25 8C, the equilibrium constant for the reaction HA(aqueous) þ H2 O(liquid) , H3 Oþ þ A is defined by KA: KA ¼

[H3 Oþ ][A ] [HA][H2 O]

and pKA ¼ log KA ¼ log{KA1 }. Stronger acids HA have a greater tendency to donate Hþ to H2O and generate H3Oþ and A2; they exhibit larger values of KA, smaller values of KA1 , and smaller values of pKA. The acidity constant for H2O is 10214 at 25 8C, which corresponds to the equilibrium constant for the previous reaction when HA(aqueous) is replaced by H2O. Hence, H2 O(aqueous) þ H2 O(liquid) , H3 Oþ þ OH with KA ¼ KW ¼

[H3 Oþ ][OH ] ¼ 1014 [H2 O]2

If reactants and products are reversed in the first reaction, H3 Oþ þ A , HA(aqueous) þ H2 O(liquid) with 1 [HA][H2 O] ¼ KA [H3 Oþ ][A ] and one adds the previous two reactions, then the net reaction is H2 O(liquid) þ A , HA(aqueous) þ OH where the equilibrium constant for this reaction is defined by KB, and pKB ¼ 2log KB.

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Hence,

KB ¼

[HA][OH ] [HA][H2 O] [H3 Oþ ][OH ] 1014 ¼  ¼ KA [H2 O][A ] [H3 Oþ ][A ] [H2 O]2

Stronger bases A2 can extract Hþ from H2O more readily to generate HA and OH2; they exhibit larger values of KB, smaller values of KB1 , and smaller values of pKB. Furthermore, log KB ¼ 14  log KA pKA þ pKB ¼ 14 The pKA of water as an acid is 14, and its conjugate base (the hydroxyl anion, OH2) has a pKB of 0. The pKB of water as a base is 14, and its conjugate acid (the hydronium cation, H3Oþ) has a pKA of 0. pKA and pKB sum to 14 for an acid/conjugate-base pair. Acidity and basicity increase, respectively, when pKA and pKB decrease. When an acid is stronger than the hydronium cation, pKA is negative. When a base is stronger than the hydroxyl anion, pKB is negative. Acidity constants for several acid/conjugate-base pairs in aqueous solution at 25 8C are provided in Table 6.4 [March, 1985]. If the polymer contains functional groups that are stronger bases than some of the neutral ligands chemically bound to the metal center, then these weak basic ligands that occupy sites in the coordination sphere are susceptible to displacement reactions. When weak bases are displaced by stronger bases, the metal – ligand s-bonds that form are stronger than those that are dissociated, and this type of ligand exchange Table 6.4 Acidity Constants for Acid (HA)/Conjugate-Base (A2) Pairs in Aqueous Solution at 25 8C Acid Hydriodic Hydrobromic Hydrochloric Phenol superacid Ether superacids Hydronium Hydrofluoric Carboxylic Pyridinium Cyanide Ammonium Tertiary ammonium Secondary ammonium Primary ammonium

HA

A2

HI HBr HCl C6H5OHþ 2 R1R2OHþ H3Oþ HF RCOOH C5H5NþH HCN N þ H4 R3NþH R2NþH2 RNþH3

I2 Br2 Cl2 C6H5OH R1OR2 H2 O F2 RCOO2 C5H5N CN2 NH3 NR3 R2NH RNH2

pKA 211 29 27 26.4 23.5 0 3.45 425 5.25 9.3 9.25 10– 11 11 10– 11

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corresponds to an exothermic reaction. Consequently, some of the ligand – based bonding molecular orbitals are stabilized (i.e., lowered in energy) after ligand displacement occurs, and the metal – ligand s-antibonding eg orbitals are destabilized in systems with local octahedral symmetry. This increases the ligand field splitting.

Anionic Ligands Are the Last Ones that Should Be Displaced to the Second Shell Weak basic neutral ligands with a different hardness classification than the metal center are most susceptible to displacement reactions. If all of the lattice waters (i.e., hard bases) and some of the weak-base chloride anions (i.e., hard bases) in a divalent metal chloride hydrate are displaced by strong-base polymeric ligands, then the following ligand exchange reaction is possible: M2þ Cl2 (H2 O)4 þ (4 þ y)[Polymer] ) M2þ Cl2y [Polymer]4þy þ ( y)Cl þ 4H2 O where 0 , y  2, and the displaced anionic ligands reside in the second shell. One should not propose a scheme that displaces anionic ligands in the coordination sphere of a cationic metal center unless (i) all of the neutral basic ligands have already been displaced and (ii) the displacing ligand in the polymer is a very strong base. If anionic ligands are displaced, then they must reside in the second shell. For example, cobalt chloride hexahydrate, CoCl2(H2O)6, forms complexes with poly (vinylamine) when the lone pair of electrons on the amino nitrogen displaces all four waters of hydration in the first shell [Belfiore and Indra, 2000]. It is also possible that amino nitrogens displace one or both of the anionic chlorides to the second shell [House, 1987; Constable, 1990; Pomogailo, 1996]. Hence, cobalt(II) acts as a multifunctional bridge between several amino side groups. This structure, which forms via self-assembly, is postulated to explain the unusually large increase in Tg for these cobalt(II) complexes (i.e., 45 8C per mol % salt, up to 3 mol % Co2þ) relative to the undiluted polymer [Belfiore et al., 1997]. Contrary to some of the results discussed later in this chapter for cobalt complexes with poly(4-vinylpyridine) and poly(L-histidine), in which the ligands in the side groups of the polymers are much weaker bases (i.e., by four or five orders of magnitude since pKA is a logarithmic scale) than poly(vinylamine), Co2þ performs exceptionally well with poly(vinylamine) due to its ability to coordinate several amino groups, as described by the previous ligand substitution reaction. This is not possible for dichlorotricarbonyl– ruthenium(II), fRuCl2(CuO)3g2, because it is increasingly difficult for amino side groups in poly(vinylamine) to displace more than one carbonyl ligand in the ruthenium complex after cleaving the dichloride bridge [Benedetti et al., 1972; Stephenson and Wilkinson, 1966]. Hence, CoCl2(H2O)6 is superior to fRuCl2(CuO)3g2 from the viewpoint of enhancing the glass transition temperature of poly(vinylamine) [Belfiore and Indra, 2000], based on a reduction in chain mobility and the formation of nanoscale clusters when NH2 side groups occupy sites in the coordination sphere of Co2þ versus Ru2þ, where Co2þ contains weak-base lattice waters and Ru2þ contains CuO ligands.

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203

Complexes with the Same Local Symmetry Above and Below the Glass Transition If Tg of the polymeric complex is higher than Tg of the undiluted polymer, then one postulates that the glass transition occurs when sufficient thermal energy is provided to remove N or N21 ligands in the side group of the polymer from the first-shell coordination sphere of the transition-metal center. Hence, one ligand in the polymer could survive this dissociation process if only N21 ligands are removed. Now, the transition-metal complex represents a bulky coordination pendant group. The N21 vacant sites in the coordination sphere of the metal could be occupied by neutral basic ligands that were displaced originally by polymeric ligands, if this process seems reasonable. In other words, displaced CuO ligands that bubble out of solution or sublime should not be used to fill vacant sites, but waters of hydration that have not volatilized are available. Complexes with Reduced Symmetry Above Tg The previous displacement reaction preserves ligand arrangements and geometries of macromolecule – metal complexes above and below the second-order phase transition. If ligands that were displaced originally are not available to occupy vacant sites when metal – polymer s-bonds are dissociated, then it might seem reasonable to postulate a decrease in coordination number for coordinatively unsaturated complexes that survive above Tg [Belfiore et al., 2001a]. This approach is adopted for zero-valent molybdenum hexacarbonyl complexes as described below, because displaced CuO ligands are not available to occupy vacant sites above Tg. When geometric perturbations occur as macromolecule– metal complexes transform from glasses to highly viscous liquids, ligand field stabilization energies (LFSEs) for the state of lower symmetry above the glass transition temperature require more complex methods of analysis in comparison with LFSE calculations for octahedral and tetrahedral geometries. LFSE calculations for 5-coordinate transition-metal complexes in the molten state are discussed below in significant detail. All of the possible 5-coordinate geometries in Table 6.1 should be considered if the glass transition corresponds to the dissociation of one polymeric side group from the coordination sphere of a macromolecule – metal complex with pseudo-octahedral symmetry in the glassy state. Similarly, all possible 3-coordinate geometries should be considered when pseudo-tetrahedral macromolecule – metal complexes are heated into the molten state, and Tg is described by the formation of a 3-coordinate complex as one ligand in the side group of a functional polymer is removed from the coordination sphere of the metal center. When the geometry of the complex is not pseudo-octahedral or pseudo-tetrahedral, the five d-orbitals of the metal do not split into a triply degenerate set and a doubly degenerate set, according to molecular symmetry [Burdett, 1980]. Hence, the ligand field splitting does not represent the energy difference between any two orbitals at different energy. If D0 ¼ 10Dq represents the octahedral ligand field splitting, or the energy difference between eg and t2g metal-based molecular orbitals in complexes with local octahedral symmetry, then there are several examples where the energy difference between two adjacent nondegenerate d-orbitals in ML5 complexes is much less than 10Dq.

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Consequently, it is not uncommon for the pairing energy that characterizes interelectronic repulsion to be larger than the energy difference between two adjacent nondegenerate d-orbitals. This scenario produces a high-spin population of metal d-orbitals, because electrons would rather populate vacant orbitals at slightly higher energy instead of occupying orbitals at lower energy that already contain one electron.

6.3.5 Consideration of Interelectronic Repulsion and Ligand Field Splitting When There Is Ambiguity in the d-Electron Configuration Quantum-chemical group contribution methods and tabulated parameters are available to estimate the ligand field splitting (i.e., D0 or DT) for transition-metal complexes with local octahedral Oh or tetrahedral Td symmetry [Burdett, 1980; Figgis and Hitchman, 2000]. Detailed calculations of the energy difference between metal-based molecular orbitals for ML6 and ML4 complexes are summarized below. It is also possible to estimate the Racah interelectronic repulsion energy, B ¼ B0(12hk), in the presence of an octahedral arrangement of ligands if B0 is known for the free metal ion (i.e., see Table 6.5). The parameters h and k are characteristic of the ligand and metal, respectively. The Racah interelectronic repulsion energy B is inversely proportional to the

Table 6.5 Jørgensen’s Parameters for Ligand Field Splittings and Interelectronic Repulsion in Octahedral ML6 Complexes Metal ion 2þ

Mn Ni2þ Co2þ Fe2þ Cu2þ Fe3þ Cr3þ Co3þ Ru2þ Ti3þ Mn3þ Mn4þ Mo3þ Rh3þ Ir3þ Pt4þ 1

g 8.0–8.5 8.7–8.9 9.0–9.3 10.0 12.0 14.0 17.0–17.4 18.2–19.0 20 20.3 21 23 24.0–24.6 27.0 32 36

k 0.07 0.12 0.24

0.24 0.21 0.35

B0 (cm21) 960 1080 1120 1060 1240 1030 620 1140

0.5 0.15 0.3 0.3 0.5

See Figure 6.7 and the discussion in Section 6.6.6.

Ligands 2

Br Cl2 F2 (NH2)2CO CH3COO2 H2 O CH3NH2 CH3CN C5H5N NH3 Histidine1 CN2 CuO

f

h

0.72 –0.76 0.78 –0.80 0.90 0.91 –0.92 0.94 –0.96 1.00 1.17 1.22 1.23 –1.25 1.25 1.32 1.7 5–8

2.3 2.0 0.8 1.2 1.0

1.4 2.0

6.3 Methodology of Transition-Metal Coordination in Polymeric Complexes

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effective radius of the d-electron cloud on the metal. When ligands interact with the metal and form a complex, expansion of the d-electron cloud occurs, which is known as the nephelauxetic (i.e., cloud expansion) effect. As a consequence of the delocalization of metal d-electrons over the ligands, interelectronic repulsion is weaker and the Racah B parameter is reduced in magnitude relative to the free metal ion. These considerations are important when there is ambiguity in the electronic configuration and the ligand field stabilization energy for dn complexes. For example, octahedral d7 complexes exhibit a low-spin ft2gg6fegg1 ground state (i.e., see Table 6.13), with six paired electrons occupying the triply degenerate t2g orbitals at lower energy and one electron occupying a higher energy doubly degenerate eg orbital, when   D0 D0 . B B critical A high-spin ft2gg5fegg2 ground state is preferred (i.e., see Table 6.13), with five electrons occupying the triply degenerate orbitals at lower energy and two unpaired electrons occupying the doubly degenerate higher energy orbitals, when D0 , B

  D0 B critical

In other words, in the presence of ligands that are classified as strong bases with large pKA values or small pKB values, and strong p-acceptors at the top of the spectrochemical series, heavy-metal octahedral complexes exhibit large energy differences between t2g and eg metal-based molecular orbitals. Hence, the ligand field splitting is large, metal d-electron density is delocalized significantly over these basic ligands such that the Racah interelectronic repulsion energy is reduced, and the low-spin electronic configuration is favored, especially for complexes with heavy-metal centers from the second and third rows of the d-block that contain CuO ligands. For transition-metal complexes with tetrahedral symmetry, the five degenerate d-orbitals of the metal split into a doubly degenerate set with e-symmetry at lower energy and a triply degenerate set with t2-symmetry at higher energy [Figgis and Hitchman, 2000]. The energy difference between these electronic orbitals is approximately two-fold smaller for tetrahedral complexes relative to octahedral complexes. Now, for d7 complexes with seven d-electrons and tetrahedral symmetry, there is no ambiguity in the electronic configuration because four paired electrons populate the doubly degenerate e-orbitals at lower energy and three unpaired electrons populate the triply degenerate t2 orbitals at higher energy, feg4ft2g3, irrespective of the ligand field strength. For octahedral complexes, one should use Jørgensen’s “group contribution” methodology [Burdett, 1980; Figgis and Hitchman, 2000] to estimate the ratio of the ligand field splitting D0 (i.e., the energy difference between eg and t2g metal-based molecular orbitals) to the reduced Racah interelectronic repulsion energy B, as discussed below, and inspect the appropriate Tanabe – Sugano diagram [Shriver et al., 1990], which summarizes electronic states and d –d transitions for dn complexes with octahedral symmetry. A summary of the critcal values of D0/B on

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Chapter 6 Macromolecule–Metal Complexes: Ligand Field Stabilization

the horizontal axes of the Tanabe – Sugano diagrams for dn complexes is provided below, where n ¼ 4, 5, 6, 7. A crossover occurs in the electronic ground state from high-spin to low-spin when the ligand field splitting D0 matches and, subsequently, exceeds the pairing energy for interelectronic repulsion. The pairing energy is defined by BfD0/Bgcritical when two d-electrons with opposite spin occupy the same molecular orbital. Number of d-Electrons

{D0/B}critical

4 5 6 7

27 28 20 22

As mentioned earlier in this chapter, pseudo-octahedral d7 cobalt(II) chloride complexes with poly(vinylamine) induce significant increases in the polymer’s glass transition temperature. Quantum-chemical group contribution predictions for 6-coordinate Co2þ complexes with anionic chloride ligands, lattice waters, and methyl amine as a model ligand for the amino side group of poly(vinylamine) yield estimates of D0/B between 11 and 14 [Belfiore et al., 1997]. Hence, the d-electron configuration in these pseudo-octahedral d7 complexes is ft2gg5fegg2 for a weak ligand field because D0/B is less than the critical value of 22, as expected for complexes from the first row of the d-block that do not contain CuO ligands. If a complex exhibits tetrahedral symmetry, one should proceed with calculations based on tabulated parameters for octahedral geometries and reduce the octahedral ligand field splitting, D0, by a factor of 4/9 (i.e., DT  0.45D0) [Shriver et al., 1990]. It seems reasonable that tetrahedral complexes with four ligands in the first-shell coordination sphere of the transition metal will not expand the d-electron cloud as much as six ligands with octahedral symmetry. Consequently, tetrahedral complexes should experience a smaller reduction in the Racah interelectronic repulsion energy B from its value for the free transition-metal ion. For example, divalent cobalt tetrachloride [CoCl4]22 exhibits an experimental Racah parameter B ¼ 730 cm21, based on an analysis of its electronic spectrum [Figgis, 1966]. Quantum-chemical group contribution predictions of the corresponding octahedral complex [CoCl6]42 suggest that B  580 cm21. Since the free-ion interelectronic repulsion energy B0 for Co2þ is 1120 cm21 [Figgis, 1966], B ¼ 730 cm21 for the tetrahedral complex [CoCl4]22 represents a 72% reduction in B0 relative to the octahedral complex. In general, estimates of B in the presence of ligands for complexes with Td symmetry are not as straightforward as estimates of the tetrahedral ligand field splitting DT (i.e., the energy difference between t2 and e metal-based molecular orbitals): DT ¼ 49 D0 The Tanabe – Sugano diagram for a dn complex with tetrahedral symmetry is equivalent to the Tanabe– Sugano diagram for a d102n complex with octahedral symmetry. The important questions that must be considered are: (i) What molecular orbital does the fourth metal d-electron populate in a complex with octahedral symmetry? and (ii) What orbital does the third metal d-electron populate in a complex with

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207

tetrahedral symmetry? The answers are: (i) the d-orbital at lower energy is populated, and electron pairing occurs, when   D0 D0 . B B critical which corresponds to the strong-field, low-spin situation; and (ii) the d-orbital at higher energy is populated, with spin-correlation for all of these unpaired electrons, when   D0 D0 , B B critical which corresponds to the weak-field, high-spin case. Jørgensen’s Parametric Representation of Ligand Field Splitting and Interelectronic Repulsion For several ML6 complexes with true octahedral symmetry, it is possible to estimate electronic properties empirically that agree quite well with data from electronic spectroscopy, based on the information in Table 6.5 [Figgis, 1966; Burdett, 1980; Figgis and Hitchman, 2000]. One predicts the octahedral ligand field splitting (i.e., D0 ¼ 10Dq), or the energy difference between the triply degenerate t2g metal-based molecular orbitals at lower energy and the higher energy doubly degenerate eg antibonding metal-based molecular orbitals, and the Racah interelectronic repulsion energy B, for ML6 complexes with six identical monodentate ligands as follows: D0 ¼ 10Dq ¼ fg[103 cm1 ] B ¼ B0 (1  hk) where g and k are characteristic of the metal, f and h are unique to the six ligands, and B0 describes the interelectronic repulsion energy of the free metal ion in the absence of the ligand field. The metal-based g-factor provides the strongest influence on D0. As described below, the rule of average environments is invoked to predict electronic properties of mixed-ligand complexes with pseudo-octahedral symmetry [Figgis and Hitchman, 2000]. The concept of the magnitude of a cubic ligand field is appropriate for complexes that exhibit approximate cubic symmetry. For example, if xk represents a normalized weighting factor for the fraction of Lk-type ligands in a pseudo-octahedral complex, then the octahedral ligand field splitting of these mixed-ligand complexes is predicted as follows: X D0 (MLA LB LC ) ¼ xk D0 (ML6 )k k¼A,B,C

where D0(ML6)k represents the octahedral ligand field splitting of an ML6 complex with six identical monodentate Lk-type ligands that can be predicted using the parameters in Table 6.5. For mixed-ligand complexes with pseudo-tetrahedral symmetry,

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Chapter 6 Macromolecule–Metal Complexes: Ligand Field Stabilization

(i) predict octahedral ligand field splittings for ML6 complexes with six identical Lk-type ligands using parameters in Table 6.5, (ii) scale each of these predictions for D0(ML6)k by 0.45 to obtain DT(ML4)k for complexes with true tetrahedral symmetry, and (iii) use the following equation to predict pseudo-tetrahedral ligand field splittings for mixed-ligand complexes: 4 X x D0 (ML6 )n DT (MLA LB LC ) ¼ 9 n¼A,B,C n where xn represents a normalized weighting factor for the fraction of Ln-type ligands in these pseudo-tetrahedral complexes. Polymeric Complexes with Enhanced Glass Transition Temperatures Inorganic models of the glass transition process, based on a consideration of ligand field splittings and ligand field stabilization energies for dn complexes with specific geometries and coordination numbers, predict an increase in Tg if metal d-electrons are stabilized when complexes form in solution and persist in the glassy state. If there is no change in local symmetry of metal complexes in the glassy and molten states, simply a change in ligand environment when ligand exchange occurs to simulate the glass transition process, then metal d-electrons experience destabilization when N or N21 polymeric ligands are removed from the coordination sphere of the transition metal via thermal energy and the original ligands of weaker basicity occupy these vacant sites in the first shell. Hence, these ligand displacement reactions that simulate the glass transition process are endothermic because stronger metal – ligand s-bonds are dissociated and weaker ones reform. If the molten state is described better by a reduction in symmetry of the metal complex relative to the glassy state, then dissociation of at least one ligand in the side group of the polymer from the first-shell coordination sphere of the metal center is also an endothermic process, based on metal – ligand bond energies or destabilization of metal d-electrons in the molten state complex of reduced symmetry. All of these models, described qualitatively above and investigated quantitatively below, are consistent with the fact that Tg is enhanced by most transition-metal salts investigated to date. Polymeric Complexes with Reduced Glass Transition Temperatures Ligand field models predict a decrease in Tg if metal d-electrons are stabilized when N or N21 polymeric ligands of similar hardness are removed from the coordination sphere of the transition metal via the addition of thermal energy and the original ligands of stronger basicity, but different hardness, occupy these vacant sites in the molten state. Under these conditions, ligand displacement reactions that simulate the glass transition process are “exothermic” because weaker metal – ligand s-bonds are dissociated and stronger ones reform. Transition-metal salts can decrease the glass transition temperature, and stabilization of metal d-electrons in the molten state provides an energetic explanation for this phenomenon, even though the glass

6.4 Pseudo-Octahedral d8 Nickel Complexes with Poly(4-vinylpyridine)

209

transition process is unequivocally endothermic. For example, Co2þ, Ni2þ, Cu2þ, and Zn2þ chlorides decrease the glass transition temperature of poly(L-lysine)hydrobromide [Belfiore and McCurdie, 2000], because the presence of bromide counterions that neutralize quaternary lysine side groups (i.e., (CH2)4NþH3) at neutral pH values precludes any complexation between these divalent metal cations and the amino nitrogen lone pair. The definition of the acidity equilibrium constant KA in Section 6.3.4 is useful to estimate the relative fractions of free base (A2) and protonated base (HA) in terms of the relative magnitudes of pH vs. pKA. Other Considerations It is helpful to use the 18-electron rule [Hegedus, 1999] as a standard for estimating the rates of ligand substitution reactions in organometallic complexes with carboncontaining ligands, like CuO, acetate, and acetonitrile. Complexes that are coordinatively saturated with a total of 18 metal d-electrons and ligand electrons in the frontier orbitals usually follow a dissociative mechanism of ligand exchange that is inherently slow. Transition states are described by lower coordination number, relative to reactants or products, when substitution proceeds by a dissociative mechanism. Coordinatively unsaturated complexes with less than 18 electrons are labile, and the associative mechanism of ligand substitution is much faster, relative to the dissociative mechanism of ligand exchange. Transition states are described by higher coordination numbers, relative to reactants or products, when substitution proceeds by an associative mechanism.

6.4 PSEUDO-OCTAHEDRAL d8 NICKEL COMPLEXES WITH POLY(4-VINYLPYRIDINE) 6.4.1

Ligand Field Stabilization Energies

Nickel acetate tetrahydrate forms coordination complexes with poly(4-vinylpyridine) (P4VP) and increases the glass transition temperature of this amorphous polymer by 102 8C when the Ni2þ concentration is 36 mol %. The complete compositional dependence of the effect of Ni2þ on P4VP’s Tg is summarized in Table 6.6 [Belfiore et al., 1992]. The formalism outlined above is employed to analyze Tg enhancement when the local symmetry about Ni2þ does not change as complexes are heated through the second-order phase transition. The energy difference between the t2g and eg metalbased molecular orbitals in octahedral ML6 complexes with six identical monodentate ligands can be predicted via Jørgensen’s parameters listed in Table 6.5 [Figgis and Hitchman, 2000]. Pyridine is a physically realistic model that captures the electronic characteristics of pyridine side groups in the polymer, but it cannot reproduce the correct steric hindrance due to the chain backbone. Analysis of Tg enhancement based on structurally simple model compounds is consistent with the energetic description of the glass transition process in this chapter, whereas an entropic model of Tg modification that does not consider steric hindrance as ligands in the polymer’s side group coordinate to the metal center is not acceptable. One predicts ligand field splittings

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Chapter 6 Macromolecule–Metal Complexes: Ligand Field Stabilization Table 6.6 Effect of Nickel Acetate Tetrahydrate on the Glass Transition Temperature of Poly(4-vinylpyridine) Mole fraction nickel acetate

Glass transition temperature (8C)

Tg,Complex 2Tg,P4VP (8C)

149 175 201 227 251 189 171

0 26 52 78 102 40 22

0.000 0.055 0.110 0.219 0.359 0.528 0.771

for the 6-coordinate nickel acetate anion [Ni(CH3COO)6]42, the hexa-aqua nickel cation [Ni(H2O)6]2þ, and the 6-coordinate nickel pyridine cation [Ni(C5H5N)6]2þ, some of which agree with spectroscopic data: Ligand Field Splitting (cm21)

Ni21 Complex 42

[Ni(CH3COO)6] [Ni(H2O)6]2þ [Ni(C5H5N)6]2þ

8544 8900 11125

The “rule of average environments” [Figgis and Hitchman, 2000] is subsequently invoked, which states that “when all ligands coordinate to a metal center in monodentate fashion, the ligand field splitting for a pseudo-octahedral mixed-ligand complex is obtained from a weighted average of the splittings calculated for each of the mono-ligand 6-coordinate complexes separately.” Hence, “group-contribution” quantum-chemical empiricism is employed to predict the octahedral ligand field splitting for nickel acetate tetrahydrate, nickel acetate trihydrate coordinated to one pyridine side group, and nickel acetate dihydrate coordinated to two pyridine side groups. The latter complex [i.e., Ni(CH3COO)2(H2O)2(C5H5N)2] represents a “coordination crosslink” where nickel acetate forms metal– ligand s-bonds with pyridine side groups on two different macromolecular chains. One estimates the pseudo-octahedral ligand field splitting D0 for these three mixed-ligand complexes via the parameters in Table 6.5: Ni21 Complex Ni(CH3COO)2(H2O)4 Ni(CH3COO)2(H2O)3(C5H5N) Ni(CH3COO)2(H2O)2(C5H5N)2

Ligand Field Splitting (cm21) 8781 9152 9523

Ligand field stabilization energies (LFSEs) are based on the electronic energy level separation between the t2g and eg metal-based molecular orbitals and the d-electron configuration of a transition-metal complex. When metal d-electrons are influenced by the Coulombic ligand field potential appropriate to an octahedral distribution of electron donors around the metal center, the five metal d-orbitals split into a triply

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211

degenerate set of molecular orbitals, denoted by dxy, dyz, and dzx with t2g symmetry, that are at lower energy relative to the doubly degenerate pair denoted by dx2 y2 and dz2 with eg symmetry. These molecular orbitals with eg symmetry (i) have lobes that are directed along the metal – ligand bond axis and (ii) are of the correct symmetry to participate in metal – ligand s-bonding. The three degenerate orbitals with t2g symmetry at lower energy (i) have lobes that are directed to each side of the metal –ligand bond axis and (ii) are of the correct symmetry to participate in metal – ligand p-bonds. The ground state electronic configuration of a d8 octahedral complex is ft2gg6fegg2, which does not depend on the strength of the ligand field or the ratio of the ligand field splitting D0 to the Racah interelectronic repulsion energy. Quantum mechanical calculations based on zeroth-order perturbation theory allow one to determine the energies of the t2g and eg metal-based molecular orbitals. Relative to the five degenerate atomic d-orbitals of the free metal cation, the t2g orbitals are 0.4D0 lower in energy, and the eg orbitals are 0.6D0 higher in energy [Shriver et al., 1990]. Hence, the ligand field stabilization energy is 120% of D0 when six metal d-electrons are spin-paired in the t2g molecular orbitals and the remaining two unpaired electrons populate the eg orbitals for an octahedral d8 complex that does not distort to tetragonal or square-planar geometries. Since 1 cm21 (i.e., wavenumber) corresponds to 11.963 J/mol, calculations of the ligand field splittings given above yield an LFSE for Ni(CH3COO)2(H2O)2(C5H5N)2 that is 5.3 kJ/mol larger than the LFSE of Ni(CH3COO)2(H2O)3(C5H5N).

6.4.2 Coordination Crosslinks Versus Coordination Pendant Groups Empirical quantum-chemical predictions summarized above from inorganic chemistry and ligand field theory are correlated with macroscopic enhancements in the glass transition temperature for Ni(II) complexes with poly(4-vinylpyridine). The 102 8C enhancement in Tg of P4VP occurs when the metal/pyridine-ligand concentration ratio is approximately 1 : 2 on a molar basis. It is postulated that thermal synergy (i.e., the enhancement in Tg) is a consequence of coordination crosslinking where the nickel cation forms metal – ligand bonds with two pyridine nitrogen lone pairs on different macromolecular chains. This coordination complex is modeled by Ni(CH3COO)2(H2O)2(C5H5N)2 and the proposed molecular structure is illustrated in Figure 6.4, based on the following facts: (i) The crystal structure of undiluted nickel acetate tetrahydrate is pseudooctahedral [VanNiekerk and Schoening, 1953; Downie et al., 1971]. (ii) The 6-coordinate d8 nickel complex is strongly favored from an equilibrium viewpoint when good donor ligands such as pyridine are present [Cotton and Wilkinson, 1972], even though the macromolecule– metal complex is completely amorphous. (iii) Nickel(II) and pyridine are classified as a borderline acid – base pair, whereas lattice waters in the first shell of nickel acetate tetrahydrate are hard bases.

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Figure 6.4 Molecular model of nickel acetate dihydrate coordinated to two pyridine side groups in P4VP, illustrating the concept of coordination crosslinks. This model is adapted from the geometry of nickel acetate tetrahydrate, based on its crystal structure [VanNiekerk and Schoening, 1953; Downie et al., 1971]. It is proposed that pyridine side groups in the polymer displace weak-base waters of hydration in the coordination sphere of the divalent nickel cation.

(iv) Stronger metal – ligand s-bonding occurs when pyridines replace weak-base waters of hydration in the coordination sphere of Ni(II). (v) Coordination crosslinks were proposed, but not necessarily defined, by Agnew [1974, 1976] for transition-metal complexes of nickel(II) chloride with poly(4-vinylpyridine). The onset of translational and rotational motion of the chain backbone in P4VP is severely restricted until enough thermal energy is supplied to (i) dissociate one nickel– pyridine bond for each metal center that is coordinated to two pyridine ligands from different macromolecular chains and (ii) induce the glass – rubber transition. The structure of the proposed pseudo-octahedral model compound in the molten state is Ni(CH3COO)2(H2O)3(C5H5N), because coordination of each metal center to one pyridine nitrogen lone pair only increases the size of the side group in P4VP, similar to a para-substituent on the phenyl ring of polystyrene. Interestingly enough, a t-butyl (ZC(CH3)3) side group in the para-position of the styrene ring increases the glass transition temperature of polystyrene by approximately 30 8C [Brandrup and Immergut, 1975; Malhotra et al., 1981]. This is comparable to the Tg enhancement of P4VP by nickel acetate for the “equimolar” complex, identified in Table 6.6 at a Ni2þ mole fraction of 0.528, where each nickel cation hypothetically coordinates to one pyridine ligand and the thermal synergy is 40 8C. The thermal energy required to dissociate coordination crosslinks

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213

and induce the glass – rubber transition is estimated by RTg,Complex, where R is the gas constant and Tg,Complex depends on composition. Relative to the undiluted polymer or the polymer attached to a bulky nickel acetate trihydrate pendant group, the thermal energy required to remove one pyridine ligand from the coordination sphere of the nickel cation on a molar basis and disrupt coordination crosslinks is estimated by   R Tg,Complex  Tg,Undiluted Polymer No effort is made to account for the concentration-dependent effect on the glass transition temperature of P4VP due to nickel acetate trihydrate pendant groups coordinated to the lone pair of electrons on nitrogen in the pyridine ring. This “bulky side group effect” becomes more important when the nickel concentration exceeds 33 mol % and approaches 50 mol %.

6.4.3 Ligand Field Model of the Glass Transition in Macromolecule –Metal Complexes A simple coordination-interaction model is formulated that accounts for the disruption of coordination crosslinks and includes ligand field stabilization energies for model complexes in the glassy and molten states. Both metal complexes have the same local symmetry (i.e., pseudo-octahedral) above and below the glass transition temperature: Ni(CH3 COO)2 (H2 O)2 (C5 H5 N)2 þ H2 O ) Ni(CH3 COO)2 (H2 O)3 (C5 H5 N) þ C 5 H5 N The energetics of this ligand substitution scheme are endothermic because (i) pyridine is a stronger base than water by approximately five orders of magnitude (see Table 6.4), and (ii) the metal complex on the left has an estimated LFSE that is 5.3 kJ/mol larger than that for the complex on the right of the above reaction (i.e., D(LFSE) ¼ LFSEGlass 2LFSELiquid  5.3 kJ/mol). Free water and pyridine are included for completeness in the ligand exchange process, but they are excluded from energetic considerations. When the disruption of metal– ligand bonds, instead of the stabilization of metal d-electrons, is correlated with Tg enhancement, the energetics of each species in the proposed ligand displacement reaction is considered. The endothermic nature of the ligand exchange process, illustrated above, is consistent with the fact that (i) energy must be supplied to disrupt coordination crosslinks, and (ii) the glass transition temperature of P4VP– Ni2þ complexes is enhanced relative to Tg of undiluted P4VP. The energy input required to remove a pyridine ligand from the coordination sphere of the nickel cation and achieve the rubbery state is comparable to the enhancement in Tg. Pyridine and P4VP are electronically similar when nickel(II) coordination to the nitrogen lone pair is considered, but there is a significant

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Chapter 6 Macromolecule–Metal Complexes: Ligand Field Stabilization

amount of steric hindrance due to the chain backbone that is not captured by pyridine when coordination occurs. The compositional dependence of coordination interactions is adopted from the Flory – Huggins lattice theory [Flory, 1953] for nonideal mixing energetics of polymer – small-molecule blends with (i) RTx replaced by D(LFSE) and (ii) the polymer segment (i.e., volume) fraction replaced by the mole fraction of the repeat unit (see Section 3.4.4). The difference between the use of mole fraction versus volume fraction for the compositional dependence of nonideal mixing energetics is equivalent to the difference between the Margules and van Laar models for the excess free energy of mixing. In this respect, the proposed model matches the characteristics of the Margules formulation. Hence, with the aid of ligand field stabilization, Tg enhancement via metal complexation is estimated from the following energetic equality:   R Tg,Complex  Tg,Undiluted Polymer ¼ b[D(LFSE)]z(1  z ) where z represents mole fraction of the metal complex. The empirical parameter b is included in the previous equation to account for at least three possible scenarios that have been overlooked by the simple energetic model. (i) Ligand field splitting calculations appropriate to small-molecule crystalline coordination complexes have been adopted to predict stabilization energies for amorphous polymer – metal-salt blends. It is not possible for nickel acetate to coordinate to an amorphous polymer, like P4VP, with long-range crystallographic order and true octahedral symmetry. Distortions to tetragonal and square-planar geometries are not uncommon in d8 complexes to lower the energy of the electronic configuration. Hence, b accounts for amorphous imperfections and the possibility that distortions to lower symmetry might occur. (ii) When nickel coordinates to two pyridine ligands, there is no guarantee that these ligands reside on different macromolecular chains producing “effective” crosslinks. Intramolecular loops form if both ligands originate from the same chain, and Tg should not increase much, if at all, due to “ineffective” crosslinks. Hence, b accounts for the fraction of effective intermolecular coordination crosslinks. (iii) Nickel acetate trihydrate pendant groups coordinated to nitrogen in the pyridine ring could affect the glass transition temperature of P4VP. If this type of coordination occurs for nickel acetate concentrations below 35 mol %, then this effect is contained in the parameter b. The linear least squares calculation of b is 0.7 for P4VP – Ni2þ complexes with pseudo-octahedral symmetry and D(LFSE)  5.3 kJ/mol. This suggests that the experimentally measured enhancement in Tg represents 70% of predictions based on the octahedral ligand field model. If nickel complexes with poly(4-vinylpyridine) exhibit tetrahedral coordination above and below Tg, then (i) the electronic configuration of eight metal d-electrons in the molecular orbitals of the complex is feg4ft2g4,

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215

(ii) tetrahedral ligand field splittings are 4/9th as large as the corresponding octahedral ligand field splittings, (iii) LFSE for the electronic configuration feg4ft2g4 is 80% of the tetrahedral ligand field splitting, and (iv) D(LFSE) is 2.4 kJ/mol for the disruption of coordination crosslinks without a reduction in local symmetry above Tg. If tetrahedral parameters for nickel acetate coordinated to either one or two pyridine ligands are employed in the energetic model of Tg enhancement, then the linear least squares calculation of b is 1.6 to generate agreement between prediction and experimental glass transition temperature data. This is physically unrealistic because crystal field coordination parameters should not underestimate bonding in the amorphous phase by approximately 60%.

6.4.4 Linear Least Squares Analysis of D(LFSE) via the Compositional Dependence of Tg in P4VP – Ni21 Complexes Subject to the Constraint that b  1 In light of the glass transition data for nickel acetate tetrahydrate and P4VP, summarized in Table 6.6, linear least squares analysis of the energetic ligand field model of Tg enhancement provides insight about the required magnitude of the ligand field stabilization energy parameter (i.e., D(LFSE)) for macromolecule– metal complexes of arbitrary geometry. Begin by constructing the square of the difference between the actual Tg enhancement data in Table 6.6 and predictions using the energetic expression in the previous section that includes D(LFSE). Using the first five data points in Table 6.6, one obtains Minimization function ¼

5  X

R(DTg )i  b[D(LFSE)]zi (1  zi )

2

i¼1

Since the maximum glass transition temperature of P4VP occurs at a nickel acetate mole fraction of 0.36, whereas the Margules compositional dependence of Tg in the ligand field model is symmetric and predicts maximum enhancement at z ¼ 0.50, the optimum value of b in the previous equation was based on the first five data points in Table 6.6. Optimization proceeds as follows: d (Minimization function) db ¼ 2

5  X

 R(DTg )i  b[D(LFSE)]zi (1  zi ) [D(LFSE)]zi (1  zi ) ¼ 0

i¼1 5  X

 R(DTg )i  b[D(LFSE)]zi (1  zi ) zi (1  zi ) ¼ 0

i¼1

When ligand field stabilization energies for glassy and molten state complexes are predicted using the methodologies described in this chapter, one calculates the best value

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Chapter 6 Macromolecule–Metal Complexes: Ligand Field Stabilization

of b to match experimental data: X5

b¼

R(DTg )i zi (1  zi ) X5 fzi (1  zi )g2 D(LFSE) i¼1 i¼1

As mentioned above, it seems reasonable that b should not exceed unity if crystal field coordination parameters do not underestimate metal – ligand bonding in the amorphous phase. This restriction on b allows one to identify the minimum value of D(LFSE) for complexes with arbitrary geometry and coordination number. Based solely on experimental data and the proposed compositional dependence of glass transition temperature enhancement, X5 D(LFSE) 

i¼1 X 5

R(DTg )i zi (1  zi )

i¼1

fzi (1  zi )g2

Tg enhancement data for nickel acetate complexes with poly(4-vinylpyridine) require that ligand field stabilization energy differences above and below Tg, D(LFSE), must be greater than 3.8 kJ/mol for physically realistic predictions via the energetic ligand field model. This linear least squares analysis precludes tetrahedral coordination of Ni2þ to pyridine side groups in P4VP, where D(LFSE) was predicted to be 2.4 kJ/mol.

6.5 d6 MOLYBDENUM CARBONYL COMPLEXES WITH POLY(VINYLAMINE) THAT EXHIBIT REDUCED SYMMETRY ABOVE THE GLASS TRANSITION TEMPERATURE 6.5.1

Experimental Results

Chromium (3d6), molybdenum (4d6), and tungsten (5d6) hexacarbonyls form coordination complexes with acrylonitrile and poly(acrylonitrile) (PAN) [Ross et al., 1963]. The nitrogen lone pair in the acrylonitrile side group (i.e., ZCuN:) displaces a carbonyl ligand and occupies an octahedral site in the coordination sphere of the transition metal. Metal(CuO)5(CH2CHCuN:) and Metal(CuO)4(CH2CHCuN:)2 have been prepared from the corresponding acetonitrile complexes, Metal(CuO)5(CH3CuN:) and Metal(CuO)4(CH3CuN:)2. Formation of these acetonitrile complexes from zero-valent coordinatively saturated d6 hexacarbonyls was accelerated photochemically (i.e., to eject carbon monoxide ligands) because displacement of CuO by acetonitrile proceeds via a dissociative mechanism where the intermediate complex is 5-coordinate. In light of these results, the analysis below focuses on poly(vinylamine) (PVAm) complexes with d6 molybdenum hexacarbonyl to stimulate experimental

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217

investigations of these materials that have not been prepared in the laboratory to date. If two CuO ligands are displaced from the coordination sphere of molybdenum and the metal bridges amino side groups on two different PVAm chains via the lone pair on nitrogen, then Mo(CuO)4(PVAm)2 represents a coordinatively crosslinked pseudooctahedral complex with considerably less mobility than the undiluted polymer. This glassy structure is postulated below Tg, and it is analyzed via ligand field models that exhibit reduced symmetry in the molten state.

6.5.2 Ligand Field Splitting Parameters for Molybdenum Hexacarbonyl Low-spin d6 octahedral complexes of molybdenum exhibit ligand field stabilization energies that are 240% of the corresponding octahedral ligand field splitting [Shriver et al., 1990]. The strong-field nature of heavy-metal centers from the second row of the d-block and p-acceptor carbonyl ligands at the top of the spectrochemical series force all six metal d-electrons to populate the lower energy triply degenerate t2g molecular orbitals in complexes with local octahedral symmetry. The experimental ligand field splitting for molybdenum hexacarbonyl is 32,150 cm21 [Beach and Gray, 1968; Pruchnik, 1990]. This data point allows one to bracket an acceptable range of Jørgensen’s g-factor for molybdenum in the zero-valent oxidation state, because fCuO gMo(0) ¼ 32,150 cm1 based on quantum-chemical group contribution estimates of the octahedral ligand field splitting when all six monodentate ligands are identical. Since CuO is the strongest p-acceptor in the spectrochemical series of ligands, its f-factor must be larger than that for cyanide anions, CN2, which has a value of 1.7, as listed in Table 6.5. Hence, the dimensionless g-factor for Mo(0) is gMo(0) ¼

32:150 , 18:9 fCuO

Further consideration of the octahedral ligand field splitting for manganese hexacarbonyl in the þ1 oxidation state (i.e., D0 ¼ 41,650 cm21) [Beach and Gray, 1968; Pruchnik, 1990] and Jørgensen’s g-factors for Mn2þ, Mn3þ, and Mn4þ (i.e., 8.5, 21, 23, respectively, Table 6.5) allow one to estimate the f-factor for CuO between 5 and 8. The g-factor for Mn1þ must be less than 8.5, yielding a reasonable estimate of the g-factor for Mo(0) between 4 and 6.5 to reproduce experimental ligand field splittings for Mo(CuO)6 and fMn(CuO)6g1þ. These estimates of ligand field splitting parameters (i.e., f- and g-factors), not included in Jørgensen’s data base, are consistent with the fact that D0 is smaller when the metal center is in a lower oxidation state and the effective nuclear charge experienced by the ligands (i.e., þZe) is smaller.

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Chapter 6 Macromolecule–Metal Complexes: Ligand Field Stabilization

6.5.3 Ligand Field Stabilization for Complexes of Molybdenum Hexacarbonyl and Poly(vinylamine) in the Glassy State Six metal d-electrons in zero-valent molybdenum complexes with local octahedral symmetry adopt a strong-field electronic configuration given by ft2gg6. This corresponds to an energetic stabilization of 2.4D0 because each t2g metal-based molecular orbital is stabilized by 0.4D0 relative to the five degenerate d-atomic orbitals of Mo(0). Methylamine, with an f-factor of 1.17 (i.e., see Table 6.5), is employed in model calculations as a “small-molecule” analog of the polymer that provides a reasonable estimate of electronic interactions between Mo(0) and poly(vinylamine), but it cannot capture the true steric hindrance imposed by the chain backbone when coordination occurs. Hence, one should not replace the polymer by methylamine and expect to interpret Tg enhancement via an entropic model, because entropy and steric hindrance strongly influence the number of amino side groups that occupy sites in the first-shell coordination sphere of Mo(0). The following complexes are of interest when poly(vinylamine) displaces carbonyl ligands, possibly with photochemical assistance, in the coordination sphere of molybdenum; Mo(CuO)x(CH3NH2)62x, where x ¼ 4, 5, 6. Based on the “rule of average environments” discussed in Section 6.4.1 [Figgis and Hitchman, 2000], the ligand field stabilization energy for pseudo-octahedral complexes with x CuO ligands and (62x) methylamine ligands is h nx o xi LFSE(T , Tg ) ¼ 2:4gMo(0) fCuO þ 1  fCH3 NH2 (11:96) kJ=mol 6 6

6.5.4 Quantum Mechanical Model Parameters and Trigonal Bipyramid 5-Coordinate d6 Complexes of Molybdenum Hexacarbonyl and Poly(vinylamine) with D3h Symmetry Above Tg Models are required to estimate the relative energies of the five d-orbitals in the molten state before ligand field stabilization energies can be calculated for 5-coordinate complexes. These energies are expressed in terms of the parameters Dq and Cp [Burdett, 1980], where 10Dq represents the corresponding octahedral ligand field splitting D0. In terms of atomic parameters for octahedral complexes where six point charges, each one of magnitude 2ze, are placed a distance L from the metal center with effective nuclear charge þZe, Dq and Cp are defined as follows [Burdett, 1980]: Dq ¼

Zze2 kr4 l ; 6L5

7Cp L2 kr 2 l ¼ 12Dq kr 4 l

In these expressions, kr nl represents the average (i.e., expectation value) of r n with respect to the radial part of the d-electron wavefunctions, r is the radius of the electron cloud about the metal center, and L is the metal– ligand bond distance. There are five 3d-orbital wavefunctions, and each one exhibits a different spherical harmonic

6.5 d6 Molybdenum Carbonyl Complexes with Poly(vinylamine)

219

expression in terms of polar angle Q and azimuthal angle w in spherical coordinates. However, the radial part of each 3d-orbital wavefunction r(r) is the same, as given by [Shriver et al., 1990]  3=2 2   1 Z 2Zr Zr r(r) ¼ pffiffiffiffiffi exp  3a0 3a0 9 30 a0 ˚ is the Bohr radius and Z represents the atomic number. The where a0 ¼ 0.529 A second and fourth moments (i.e., expectation values) of the radial part of the 3d-orbital wavefunctions are defined by ð1 rn fr(r)g2 r 2 dr n r¼0 kr l ¼ ð 1 fr(r)g2 r 2 dr r¼0 n

with n ¼ 2, 4. Hence, kr l increases (i.e., n . 0) when the d-electron cloud experiences more delocalization due to the ligands as a consequence of the nephelauxetic effect. A similar conclusion is based on the radial part of the 4d-orbital wavefunctions. The following d-orbital energy levels are available for trigonal bipyramid complexes when 7Cp/12Dq ¼ 1 [Burdett, 1980]: d-Orbital z2 xy x 2 2y 2 xz yz

Energy (Dq) þ6.21 þ0.035 þ0.035 23.14 23.14

When six metal d-electrons populate these orbitals, z 2 at highest energy remains vacant because the energy difference between z 2 and xz or yz is greater than 93% of the corresponding octahedral ligand field splitting, which should exceed the pairing energy that characterizes interelectronic repulsion. Hence, pairing occurs in xz and yz, while z 2 is unoccupied. This is reasonable for heavy metals like Mo with p-acceptor carbonyl ligands and large D0 that produce a strong-field electronic configuration given by  1 [d6 ]: fxzg2 fyzg2 fxyg1 x2  y2

D3h symmetry with 7Cp=12Dq ¼ 1

Stabilization of the metal d-electrons is predicted to be, LFSE  1.25D0. The following d-orbital energy levels are available for trigonal bipyramid complexes when 7Cp/12Dq ¼ 2 [Burdett, 1980], corresponding to less d-electron delocalization by the surrounding ligands: d-Orbital 2

z xy x 2 2y 2 xz yz

Energy (Dq) þ7.07 20.82 20.82 22.72 22.72

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Chapter 6 Macromolecule–Metal Complexes: Ligand Field Stabilization

When six metal d-electrons populate these orbitals, z 2 at highest energy remains vacant because the energy difference between z 2 and xz or yz is greater than 97% of the corresponding octahedral ligand field splitting. Once again, spin-pairing occurs in xz and yz, while z 2 is unoccupied. The strong-field electronic configuration is  1 [d6 ]: fxzg2 fyzg2 fxyg1 x2  y2

D3h symmetry with 7Cp=12Dq ¼ 2

which also corresponds to a stability factor given by LFSE  1.25D0.

6.5.5 Square Pyramid 5-Coordinate d6 Complexes of Molybdenum Hexacarbonyl and Poly(vinylamine) with C4v Symmetry Above Tg The methodology of identifying d-electron configurations and ligand field stabilization energies in the previous section is repeated here for complexes with square pyramid geometry. Now, there could be ambiguity in the order that six metal d-electrons populate the molecular orbitals of the complex. Low-spin and high-spin configurations are considered. The following d-orbital energy levels are available when 7Cp/12Dq ¼ 1 [Burdett, 1980]: Energy (Dq)

d-Orbital x2 – y2 z2 xy xz yz

þ7.43 þ2.57 22.57 23.715 23.715

The x 2 2y 2 d-orbital at highest energy remains vacant because the energy difference between x 2 2y 2 and xz or yz is greater than 110% of the corresponding octahedral ligand field splitting. Hence, electrons are spin-paired in xz and yz. The energy difference between z 2 and xy is 51% of D0, so z 2 and xy each contain one electron for a high-spin configuration, whereas z 2 is vacant at low spin. The low-spin electronic configuration is given by Low-spin [d6 ]: fxzg2 fyzg2 fxyg2

C4v symmetry with 7Cp=12Dq ¼ 1

with a ligand field stabilization energy of LFSE ¼ 2.0D0. If the electronic configuration is high spin with x 2 2y 2 vacant, then one obtains;  1 High-spin [d6 ]: fxzg2 fyzg2 fxyg1 z2

C4v symmetry with 7Cp=12Dq ¼ 1

This corresponds to LFSE ¼ 1.49D0. d-Orbital energy levels are provided below for square pyramid complexes when 7Cp/12Dq ¼ 2 [Burdett, 1980], which corresponds to less electron delocalization by the surrounding ligands such that the fourth moment of the radial part of the

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221

d-electron wavefunctions decreases more rapidly than the second moment (i.e., relative to 7Cp/12Dq ¼ 1): d-Orbital x 2 2y 2 z2 xy xz yz

Energy (Dq) þ9.14 þ0.86 20.86 24.57 24.57

Now, the x 2 2y 2 d-orbital at highest energy undoubtedly remains vacant because the energy difference between x 2 2y 2 and xz or yz is greater than 137% of the corresponding octahedral ligand field splitting. The energy difference between z 2 and xy is less than 20% of D0, so each of these d-orbitals should contain one electron. Hence, the electronic configuration is  1 C4v symmetry with 7Cp=12Dq ¼ 2 [d6 ]: fxzg2 fyzg2 fxyg1 z2 with a stabilization energy of LFSE ¼ 1.83D0.

6.5.6 Pentagonal Planar 5-Coordinate d6 Complexes of Molybdenum Hexacarbonyl and Poly(vinylamine) with D5h Symmetry Above Tg The third possible 5-coordinate geometry in the molten state is pentagonal planar, and the d-orbital energy levels are provided below when 7Cp/12Dq ¼ 1 [Burdett, 1980]: d-Orbital x 2 2y 2 xy z2 xz yz

Energy (Dq) þ4.825 þ4.825 21.07 24.29 24.29

The energy difference between xy or x 2 2y 2 and xz or yz is slightly greater than 90% of the corresponding octahedral ligand field splitting, so spin-pairing in xz and yz is considered. Since xy or x 2 2y 2 and z 2 are separated by 59% of D0, spin-pairing in z 2 is considered also. The strong-field d-electron configuration is Low-spin [d6 ]: {xz}2 {yz}2 {z2 }2

D5h symmetry with 7Cp=12Dq ¼ 1

with electronic stabilization given by LFSE ¼ 1.93D0. d-Orbital energy levels are summarized below for pentagonal planar complexes when 7Cp/12Dq ¼ 2 [Burdett, 1980]: d-Orbital 2

x 2y xy z2 xz yz

2

Energy (Dq) þ9.10 þ9.10 25.35 26.42 26.42

222

Chapter 6 Macromolecule–Metal Complexes: Ligand Field Stabilization

Once again, spin-pairing should occur in yz, xz, and z 2. The highest energy d-orbitals (i.e., x 2 2y 2 and xy) remain vacant because they are 1.55D0 above the lowest energy d-orbitals. Most importantly, they are 1.45D0 above z 2. The strong-field configuration is Low-spin [d6 ]: {xz}2 {yz}2 {z2 }2

D5h symmetry with 7Cp=12Dq ¼ 2

and the largest ligand field stabilization energy of all possible scenarios discussed above is given by LFSE ¼ 3.64D0.

6.5.7 Ligand Field Stabilization of 5-Coordinate d6 Complexes of Molybdenum Hexacarbonyl and Poly(vinylamine) Above Tg When sufficient thermal energy is supplied to dissociate at least one metal – nitrogen chemical bond in the glassy complex Mo(CuO)4(CH3NH2)2, this coordinatively saturated crosslinked structure reverts to a 5-coordinate complex above the glass transition temperature. The ligand dissociation reaction is Mo(CuO)4 (CH3 NH2 )2 ) Mo(CuO)4 (CH3 NH2 ) þ CH3 NH2 Force field calculations favor a square pyramid geometry in the molten state, where the lone remaining amino ligand could occupy a site in the equatorial plane or at the apical position of the square pyramid. In either case, the “rule of average environments” is invoked to predict 10Dq for the 5-coordinate complex above Tg, Mo(CuO)4(CH3NH2): D0 ¼ 10Dq ¼ gMo(0)

h o xi fCuO þ 1  fCH3 NH2 (11:96) kJ=mol 5 5

nx

where x ¼ 4. The previous three sections of this chapter have summarized d-electron configurations and ligand field stabilization energies for 5-coordinate heavy metal d6 complexes with D3h, C4v , and D5h symmetry. In each case, LFSE is expressed in terms of Dq, and Jørgensen’s quantum-chemical group contribution method of estimating this parameter for mixed-ligand complexes is useful. Even though force field calculations favor a square pyramid geometry in the molten state after one amino functional group in the side chain of poly(vinylamine) is removed from the first shell of molybdenum, LFSE is estimated by considering all possible 5-coordinate geometries and d6 electronic configurations described above. Equal weighting factors for complexes with different symmetry and “spin state” are employed to calculate an “average” LFSE above the glass transition temperature, as summarized in Table 6.7. The difference between ligand field stabilization energies above and below the glass transition temperature, in which coordinatively unsaturated molten state

6.5 d6 Molybdenum Carbonyl Complexes with Poly(vinylamine)

223

Table 6.7 Ligand Field Stabilization Energies for 5-Coordinate d6 Complexes in the Molten State 5-Coordinate geometry d6 electronic structure

7Cp/12Dq

LFSE

D3h, trigonal bipyramid: fxzg2 fyzg2 fxyg1 fx 2 2y 2g1

1

1.25D0

D3h, trigonal bipyramid: fxzg2 fyzg2 fxyg1 fx 2 2y 2g1

2

1.25D0

C4v , square pyramid, low-spin: fxzg2 fyzg2 fxyg2

1

2.00D0

C4v , square pyramid, high-spin: fxzg2 fyzg2 fxyg1 fz 2g1

1

1.49D0

C4v , square pyramid, high-spin: fxzg2 fyzg2 fxyg1 fz 2g1

2

1.83D0

D5h, pentagonal planar, low-spin: fxzg2 fyzg2 fz 2g2

1

1.93D0

D5h, pentagonal planar, low-spin: fxzg2 fyzg2 fz 2g2

2

3.64D0 Average LFSE(T . Tg) ¼ 1.9D0

complexes exhibit reduced symmetry relative to 6-coordinate complexes in the glassy state, is calculated as follows:  h nx o x D(LFSE) ¼ LFSE(T , Tg )  LFSE(T . Tg ) ¼ gMo(0) 2:4 fCuO þ 1  fCH3 NH2 6 6  nx oi x 1:9 fCuO þ 1  fCH3 NH2 (11:96) kJ=mol 5 5 with x ¼ 4 when zero-valent heavy-metal centers, such as Mo(0) coordinate to four CuO ligands and two amino side groups in poly(vinylamine) below the glass transition temperature. One predicts that D(LFSE)  58 kJ/mol when the g-factor of Mo(0) is 5, and the f-factors for CuO and CH3NH2 are 6 and 1.17, respectively. Hence, the difference between ligand field stabilization energies above and below Tg for molybdenum hexacarbonyl complexes with poly(vinylamine) is predicted to be an order of magnitude larger than D(LFSE)  5.3 kJ/mol for pseudo-octahedral nickel acetate complexes with poly(4-vinylpyridine). Since the stabilization of metal d-electrons for Mo(CuO)4(CH3NH2)2 is significantly larger than Mo(CuO)4(CH3NH2), this should translate into dramatic enhancements in the glass transition temperature of heavy-metal carbonyl complexes with amorphous polymers that contain strongly basic functional side groups, but experimental data are not available for Mo(CuO)6 and poly(vinylamine) to verify these predictions. Experimental results described below for ruthenium carbonyl complexes with either poly(4-vinylpyridine) or poly(L-histidine) are consistent with the proposed ligand field stabilization models with reduced symmetry above the glass transition temperature.

224

Chapter 6 Macromolecule–Metal Complexes: Ligand Field Stabilization

6.6 COBALT, NICKEL, AND RUTHENIUM COMPLEXES WITH POLY(4-VINYLPYRIDINE) AND POLY(L-HISTIDINE) THAT EXHIBIT REDUCED SYMMETRY IN THE MOLTEN STATE 6.6.1 Polymeric Coordination Complexes with d-Block Salts Unlike the well-known phenomenon of plasticization [Kelley and Bueche, 1961], transition-metal salts typically increase the glass transition temperature of polymers that contain attractive ligands in the side group. A plausible mechanism discussed previously in this chapter involves acid – base interactions between the metal center and appropriate functional groups in the polymer via ligand exchange. Whereas plasticizers interact weakly with the polymer via van der Waals forces and enhance the fractional free volume of the binary mixture [Fujita and Kishimoto, 1961], metal – ligand s-bonds form between transition metals and favorable functional groups in the macromolecule. Since coordination numbers between 4 and 6 are most common in d-block complexes [Cotton and Wilkinson, 1972], as summarized in Table 6.2, opportunities exist for basic ligands in the side group of the polymer to occupy sites in the first-shell coordination sphere of an acidic metal center. The concept of coordination crosslinks is realized when functional groups from more than one chain occupy sites in the first shell of a single metal center. This type of structure, illustrated schematically in Figure 6.1, exhibits reduced mobility in the vicinity of these thermoreversible crosslinks, which is consistent with an increase in the glass transition temperature. Multifunctional metal centers that coordinate to basic ligands in several different chains could be responsible for the formation of nanoclusters with significant reduction in chain mobility and dramatic increases in Tg. This has been observed recently in polymeric complexes with several lanthanide trichloride hydrates from lanthanum to lutetium in the first row of the f-block [Das et al., 2000; Belfiore et al., 2001b], but ligand field splitting energies of lanthanide complexes (i.e., 100 cm21, 1.2 kJ/mol) [Ashcroft and Mortimer, 1970] are much too small to provide the dominant contribution to Tg enhancement. In addition to exhibiting increased glass transition temperatures, macromolecule – metal complexes could form gels during preparation in dilute solution [Bosse´ et al., 1997]. When gelation occurs in aqueous media, applications for water purification, drug release, and artificial muscles become attractive [Tanaka et al., 1995]. If gels are sensitive to variations in pH, temperature, or electric field strength, then it might be possible to exploit these “molecular gates” and use them for controlled release of encapsulated molecules with a specific target. Of particular interest in this chapter, the methodology for producing and analyzing macromolecule – metal complexes with significantly enhanced glass transition temperatures is described from an energetic viewpoint that considers the stabilization of metal d-electrons. A systematic study of Tg enhancement in poly(4-vinylpyridine) and poly(L-histidine) via ruthenium(II), cobalt(II), and nickel(II) is employed to (i) extend the methodology outlined above for molybdenum hexacarbonyl complexes

225

6.6 Cobalt, Nickel, and Ruthenium Complexes with Poly(4-vinylpyridine) 0

Tg(complex) – Tg(polymer) (°C)

100 80

5

10

15

20

25

Dichlorotricarbonylruthenium(II) Nickel acetate tetrahydrate Cobalt chloride hexahydrate

100 80

60

60

40

40

20

20

0

0 0

5

10 15 Mole % Transition Metal Ion

20

25

Figure 6.5 Effect of cobalt chloride hexahydrate, nickel acetate tetrahydrate, and dichlorotricarbonylruthenium(II) on the glass transition temperature of poly(4-vinylpyridine). The polymer’s molecular weight is 2  105 daltons.

with poly(vinylamine) and (ii) compare predictions with experimental data. Poly(4vinylpyridine) and poly(L-histidine) contain nitrogen lone pairs in either the pyridine side group or the imidazole ring of the histidine side group that form s-bonds with appropriate d-orbitals of the transition-metal cation. Ruthenium(II), cobalt(II), and nickel(II) enhance the glass transition temperatures of these polymers, as illustrated in Figures 6.5 and 6.6. Following the same theme from previous sections of this chapter, transition-metal-induced enhancements of Tg in selected amorphous polymers are correlated with ligand field stabilization energy differences between complexes in the glassy and molten states. The ligand field model outlined above for molybdenum hexacarbonyl complexes with poly(vinylamine) considers a reduction in symmetry and a decrease in coordination number of the metal center in the molten state above Tg, due to dissociation of a ligand in the polymer’s side group from the first-shell coordination sphere. Since the glassy state is described by complexes with local tetrahedral or octahedral symmetry, geometric distortions of 3-coordinate and 5-coordinate polymer – metal complexes in the molten state are considered in the analyses below.

6.6.2

Ruthenium d6 Complexes

Ruthenium(II) is an attractive d-block metal cation for Tg enhancement because strong-field low-spin d6 metal centers with pseudo-octahedral symmetry exhibit very large ligand field splittings and stabilization energies [Shriver et al., 1990]. Ru2þ is classified as a borderline acid [Pearson, 1969, 1973] that exhibits an affinity for borderline bases, like pyridine ligands in the side group of poly(4-vinylpyridine).

226

Chapter 6 Macromolecule–Metal Complexes: Ligand Field Stabilization 0

1

2

3

4

5

6

7

8

9

10 120

Tg(complex) – Tg(polymer) (°C)

120 Dichlorotricarbonylruthenium(II), d6 Nickel chloride hexahydrate, d8 Cobalt chloride hexahydrate, d7

100

100

80

80

60

60

40

40

20

20

0

0 0

1

2

3 4 5 6 7 Mole % Transition-Metal Ion

8

9

10

Figure 6.6 Effect of CoCl2(H2O)6, NiCl2(H2O)6, and fRuCl2(CO)3g2 on the glass transition temperature of poly(L-histidine). The polymer’s molecular weight is (1.5 –5.0)  104 daltons.

Reactions of a particular ruthenium dimer (i.e., fRuCl2(CuO)3g2) with both stoichiometric and excess amounts of pyridine are well documented [Benedetti et al., 1972; Stephenson and Wilkinson, 1966]. In both cases, the dichloride bridge is cleaved and either one or two pyridine ligands coordinate to each metal center forming complexes with pseudo-octahedral symmetry. The first pyridine ligand occupies the vacant site generated from cleavage of the dichloride bridge. The second pyridine ligand displaces carbon monoxide in the coordination sphere of the metal. Vibrational spectroscopic studies of [RuCl2(CuO)3]2 in the vicinity of 1900 – 2200 cm21 fingerprint the infrared absorptions of CuO that are sensitive to s-donation and p back-donation [Bruce and Stone, 1967; Cleare and Griffith, 1969]. Electron-rich metal centers backbond to p-acceptor ligands like CuO and shift the vibrational absorption frequencies of carbon monoxide to lower energy. This process is described by the Dewar – Chatt model of chemical bonding in transition-metal complexes [Shriver et al., 1990], where metal-based t2g molecular orbitals in systems with local octahedral symmetry donate electron density to the antibonding p orbitals of carbon monoxide. These antibonding orbitals have larger amplitude on the less electronegative atom of CuO, and this carbon atom participates in s-bonding with the metal center. Hence, the orbital energy levels are similar, proximity is satisfied, and the wavefunctions of interest have the correct symmetry for t2g – p molecular orbital overlap to achieve metal-to-ligand flow of electron density into the antibonding orbital of CuO that weakens its infrared (i.e., triple-bond) stretching frequency. Solid state carbon-13 NMR spectroscopic data reveal that heteronuclear spin diffusion between 1H in

6.6 Cobalt, Nickel, and Ruthenium Complexes with Poly(4-vinylpyridine)

227

poly(4-vinylpyridine) and the carbonyl 13C nuclei of [RuCl2(CuO)3]2 is operative [Belfiore et al., 1993]. This observation of intermolecular polarization transfer in Figure 16.31, after the establishment of a magnetization gradient, is consistent with molecular dispersion at the nanoscale level (i.e., dipolar distances between 1H and 13 C that correspond to tenths of a nanometer, or a few angstroms) and subsequent complexation of two dissimilar components. More recently, ruthenium(II) complexes have been used to induce supramolecular assembly of dissimilar copolymer blocks, specifically when polystyrene and poly(ethylene oxide) are functionalized with terpyridine end groups that coordinate to the pseudo-octahedral metal center [Fustin et al., 2005].

6.6.3

Cobalt d7 Complexes

Cobalt chloride hexahydrate forms transition-metal complexes with amino, pyridine, and imidazole ligands in the side group of poly(vinylamine), poly(4-vinylpyridine), and poly(L-histidine), respectively. Two independent X-ray crystallographic studies [Young, 1960; Mizuno et al., 1959] have deduced a pseudo-octahedral geometry for CoCl2(H2O)6 with two chloride anions and four equatorial lattice waters in the first-shell coordination sphere of Co2þ. The two remaining waters of hydration are “free,” but they reside near the apical chlorides and form hydrogen bonds with these anions [Nicholls, 1973]. Several 6-coordinate Co2þ complexes with multiple nitrogen-containing ligands have been prepared and characterized [House, 1987; Constable, 1990; Pomogailo, 1996]. These coordination compounds support the concept that multiple ligands in the side group of nitrogen-containing polymers could displace lattice waters and occupy sites in the coordination sphere of Co2þ, which is a borderline acid [Pearson, 1969, 1973].

6.6.4

Nickel d8 Complexes

Nickel(II) complexes are useful to induce synergistic Tg response in amorphous polymers with nitrogen-containing ligands in the side group. The hexahydrates of nickel chloride and cobalt chloride adopt the same coordination number and ligand arrangement [Wells, 1984]. Nickel acetate tetrahydrate exhibits a pseudo-octahedral geometry in the solid state with four equatorial lattice waters and two apical monodentate acetate ligands [VanNiekerk and Schoening, 1953; Downie et al., 1971]. In the most favorable situation, pseudo-octahedral Ni2þ forms metal– ligand s-bonds with nitrogen lone pairs in two different macromolecular chains. The 6-coordinate d8 nickel complexes are strongly favored from an equilibrium viewpoint when good donor ligands are present [Cotton and Wilkinson, 1972]. The 5-coordinate square pyramid and trigonal bipyramid complexes, and 4-coordinate tetrahedral and square planar complexes are also common. Physically realistic mechanisms by which nickel acetate tetrahydrate and nickel chloride hexahydrate enhance the glass transition temperatures of poly(4vinylpyridine) and poly(L-histidine), respectively, are discussed below.

228

Chapter 6 Macromolecule–Metal Complexes: Ligand Field Stabilization

6.6.5 5-Coordinate Complexes with Reduced Symmetry in the Molten State d-Orbital Energies for 5-Coordinate Complexes Above Tg Trigonal Bipyramid dn Complexes with D3h Symmetry Electronic energies relative to the five degenerate d-orbitals of the free metal ion, dn configurations (i.e., n ¼ 6, 7, 8), and ligand field stabilization energies are summarized in Table 6.8 for trigonal bipyramid complexes when Cp/Dq is either 1.71 or 3.43 [Burdett, 1980]. If each d-orbital contains one electron, then the total electronic energy is exactly the same as that for the free metal ion, and there is no net stabilization due to the ligand field. One arrives at the same conclusion for d10 complexes that contain divalent zinc, because each d-orbital contains a pair of electrons with opposite spin. Stabilization is prevalent when there is a larger population of electrons in lower energy orbitals. In a weak ligand field, electrons occupy vacant orbitals whenever possible, instead of pairing with opposite spin in lower energy orbitals. This produces less ligand field stabilization. In a strong ligand field, it is more probable that two electrons with opposite spin will be paired at lower energy, instead of occupying vacant higher energy orbitals. In general, larger ligand field stabilization

Table 6.8 Electronic Energy Calculations for 5-Coordinate Trigonal Bipyramid Complexes with D3h Symmetry d-Orbital energies (units of Dq) 7Cp/12Dq 1 2

dxz

dyz

dxy

dx2 y2

dz2

23.14 22.72

23.14 22.72

þ0.035 20.815

þ0.035 20.815

þ6.21 þ7.07

d-Electron configurations and ligand field stabilization energies Number of d-electrons

7Cp/12Dq

Ligand field strength

d-Electron configuration

LFSE (Dq)

6 6 6 6

1 2 1 2

Weak Weak Strong Strong

fxzg2 fyzg1 fxyg1 fx 2 2y 2g1 fz 2g1 fxzg2 fyzg1 fxyg1 fx 2 2y 2g1 fz 2g1 fxzg2 fyzg2 fxyg1 fx 2 2y 2g1 fxzg2 fyzg2 fxyg1 fx 2 2y 2g1

3.14 2.72 12.51 12.51

7 7 7 7

1 2 1 2

Weak Weak Strong Strong

fxzg2 fyzg2 fxyg1 fx 2 2y 2g1 fz 2g1 fxzg2 fyzg2 fxyg1 fx 2 2y 2g1 fz 2g1 fxzg2 fyzg2 fxyg2 fx 2 2y 2g1 fxzg2 fyzg2 fxyg2 fx 2 2y 2g1

6.28 5.44 12.46 13.33

8 8 8 8

1 2 1 2

Weak Weak Strong Strong

fxzg2 fyzg2 fxyg2 fx 2 2y 2g1 fz 2g1 fxzg2 fyzg2 fxyg2 fx 2 2y 2g1 fz 2g1 fxzg2 fyzg2 fxyg2 fx 2 2y 2g2 fxzg2 fyzg2 fxyg2 fx 2 2y 2g2

6.25 6.26 12.42 14.14

6.6 Cobalt, Nickel, and Ruthenium Complexes with Poly(4-vinylpyridine)

229

energies are possible in a strong ligand field. The z 2 orbital in trigonal bipyramid complexes with 6, 7, or 8 metal d-electrons remains vacant in a strong ligand field because the energy difference between z 2 and xz or yz is 93% (i.e., Cp/Dq ¼ 1.71) to 97% (i.e., Cp/Dq ¼ 3.43) of the octahedral ligand field splitting. Electronic configurations of d6, d7, and d8 complexes are provided in Table 6.8 for weak and strong ligand fields. Once Dq is measured or predicted, the information in Table 6.8 is useful to estimate the energetic stabilization of metal d-electrons for coordinatively unsaturated trigonal bipyramid complexes in the molten state. Square Pyramid d n Complexes with C4v Symmetry When all bond angles are 908, energies of the five d-orbitals for square pyramid complexes [Burdett, 1980] are summarized in Table 6.9, where an energy of zero is assigned to the degenerate orbitals of the free metal ion. The x 2 2y 2 d-orbital remains vacant for all complexes that contain eight electrons or less, when the ligand field is strong. For d6 complexes with 7Cp/12Dq ¼ 1 in a strong ligand field, z 2 is vacant because the energy difference between z 2 and xy is more than 51% of the octahedral ligand field splitting. The d-electron configurations in Table 6.9 consider 5-coordinate complexes with 6, 7, or 8 metal d-electrons. LFSE

Table 6.9 Electronic Energy Calculations for 5-Coordinate Square Pyramid Complexes with C4v Symmetry d-Orbital energies (units of Dq) 7Cp/12Dq 1 2

dxz

dyz

dxy

23.715 24.57

23.715 24.57

22.57 20.86

dz2

dx2 y2

þ2.57 þ7.43 þ0.86 þ9.14

d-Electron configurations and ligand field stabilization energies Number of d-electrons

7Cp/12Dq

Ligand field strength

d-Electron configuration

LFSE (Dq)

6 6 6 6

1 2 1 2

Weak Weak Strong Strong

fxzg2 fyzg1 fxyg1 fz 2g1 fx 2 2y 2g1 fxzg2 fyzg1 fxyg1 fz 2g1 fx 2 2y 2g1 fxzg2 fyzg2 fxyg2 fxzg2 fyzg2 fxyg1 fz 2g1

3.72 4.57 20.00 18.28

7 7 7 7

1 2 1 2

Weak Weak Strong Strong

fxzg2 fyzg2 fxyg1 fz 2g1 fx 2 2y 2g1 fxzg2 fyzg2 fxyg1 fz 2g1 fx 2 2y 2g1 fxzg2 fyzg2 fxyg2fz 2g1 fxzg2 fyzg2 fxyg2 fz 2g1

7.43 9.14 17.43 19.14

8 8 8 8

1 2 1 2

Weak Weak Strong Strong

fxzg2 fyzg2 fxyg2 fz 2g1 fx 2 2y 2g1 fxzg2 fyzg2 fxyg2 fz 2g1 fx 2 2y 2g1 fxzg2 fyzg2 fxyg2 fz 2g2 fxzg2 fyzg2 fxyg2 fz 2g2

10.00 10.00 14.86 18.28

230

Chapter 6 Macromolecule–Metal Complexes: Ligand Field Stabilization

predictions in the far right column of this table are useful to analyze molten state complexes with reduced symmetry. Pentagonal Planar d n Complexes with D5h Symmetry These 5-coordinate complexes exhibit d-orbital energies [Burdett, 1980] that are summarized in Table 6.10. In the presence of a strong ligand field, d6 complexes do not populate xy or x 2 2y 2. There is no difference between weak field and strong field d8 complexes because the two orbitals at highest energy are degenerate. LFSE predictions for these 5-coordinate complexes are summarized in the far right column of Table 6.10. Summary of LFSE Calculations for 5-Coordinate dn Complexes Ligand field stabilization energies for all possible 5-coordinate geometries of a dn complex in either weak or strong fields are averaged with equal weighting factors. These results are summarized in Table 6.11. For d6, d7, and d8 complexes, the dependence of LFSE on the number of d-electrons follows opposite trends for weak and strong ligand fields. In other words, LFSE increases from d6 to d8 for weak-field configurations, whereas LFSE decreases from d6 to d8 for strong-field configurations. All calculations are presented in units of Dq, where the corresponding octahedral ligand field splitting is given by 10Dq. Parametric estimates of Dq are summarized in the following section for (i) pseudo-octahedral mixed-ligand complexes with two Table 6.10 Electronic Energy Calculations for 5-Coordinate Pentagonal Planar Complexes with D5h Symmetry d-Orbital energies (units of Dq) 7Cp/12Dq 1 2

dxz

dyz

dz2

dxy

dx2 y2

24.29 26.42

24.29 26.42

21.07 25.35

þ4.825 þ9.10

þ4.825 þ9.10

d-Electron configurations and ligand field stabilization energies Number of d-electrons

7Cp/12Dq

Ligand field strength

d-Electron configuration

LFSE (Dq)

6 6 6 6

1 2 1 2

Weak Weak Strong Strong

fxzg2 fyzg1 fxzg2 fyzg1 fxzg2 fyzg2 fxzg2 fyzg2

fz 2g1 fxyg1 fx 2 2y 2g1 fz 2g1 fxyg1 fx 2 2y 2g1 fz 2g2 fz 2g2

4.29 6.42 19.30 36.38

7 7 7 7

1 2 1 2

Weak Weak Strong Strong

fxzg2 fyzg2 fxzg2 fyzg2 fxzg2 fyzg2 fxzg2 fyzg2

fz 2g1 fxyg1 fx 2 2y 2g1 fz 2g1 fxyg1 fx 2 2y 2g1 fz 2g2 fxyg1 fz 2g2 fxyg1

8.58 12.84 14.48 27.28

8 8

1 2

Weak, Strong Weak, Strong

fxzg2 fyzg2 fz 2g2 fxyg1 fx 2 2y 2g1 fxzg2 fyzg2 fz 2g2 fxyg1 fx 2 2y 2g1

9.65 18.19

6.6 Cobalt, Nickel, and Ruthenium Complexes with Poly(4-vinylpyridine)

231

Table 6.11 Averaged Ligand Field Stabilization Energies for 5-Coordinate dn Complexes Number of d-electrons

Ligand field strength

LFSE (Dq)

d6 d6

Weak Strong

4.14 19.83

d7 d7

Weak Strong

8.29 17.35

d8 d8

Weak Strong

10.06 14.59

polymeric ligands in the glassy state and (ii) 5-coordinate mixed-ligand complexes with one polymeric ligand in the molten state, above the glass transition temperature.

6.6.6 Stabilization of Metal d-Electrons in Mixed-Ligand Complexes The following generic ligand dissociation reaction is proposed to analyze the onset of Tg in macromolecule – metal complexes with enhanced glass transition temperatures relative to the undiluted polymers: M2þ (LA )2 (LB )2 (LC )2 ) M2þ (LA )2 (LB )2 LC þ LC where M2þ is either Ru2þ, Co2þ, or Ni2þ; LA is either Cl2 or CH3COO2; LB is either H2O or CuO; and LC is the nitrogen lone pair in the side group of either poly(4vinylpyridine) or poly(L-histidine). Reactive blending in dilute solution places two side groups from these polar polymers in the coordination sphere of a single metal center. After solvent evaporation, 6-coordinate glassy complexes are modeled via the structure on the left side of the previous dissociation reaction. Ligand field stabilization energies, based on the information in Table 6.11 and Table 6.13, are computed in Table 6.12 for the appropriate 6-coordinate and 5-coordinate complexes with 6, 7, or 8 d-electrons that simulate the glassy and molten states. Bold numbers with asterisks in the far right column of Table 6.12 identify the most probable LFSE, based on strength of the ligand field and the number of d-electrons. The “rule of average environments” [Figgis, 1966; Figgis and Hitchman, 2000] was invoked to calculate 10Dq for mixed-ligand complexes that are 6-coordinate below Tg and 5-coordinate above Tg. The averaging procedure was performed as follows. For 6-coordinate complexes with three different types of ligands, denoted by M2þ(LA)2(LB)2(LC)2, one predicts 10Dq (kJ/mol) via Jørgensen’s parameters in Table 6.5: f10Dqg6-Coordinate ¼

11:963 gM2þ f2f (LA ) þ 2f (LB ) þ 2f (LC )g 6

232

Chapter 6 Macromolecule–Metal Complexes: Ligand Field Stabilization

Table 6.12 Ligand Field Stabilization Energies for Macromolecule–Metal Complexes Above (5-Coordinate) and Below (6-Coordinate) the Glass Transition Temperature Complex d6 Complexes RuCl2(CO)2[P4VP]2

10Dq (kJ/mol)

Ligand field strength

LFSE (kJ/mol)

640

Weak Strong Weak Strong Weak Strong Weak Strong

4Dq ¼ 256 24Dq ¼ 1536 4.14Dq ¼ 294 19.83Dq ¼ 1406 4Dq ¼ 259 24Dq ¼ 1553 4.14Dq ¼ 295 19.83Dq ¼ 1414

Weak Strong Weak Strong Weak Strong Weak Strong

8Dq ¼ 88.9 18Dq ¼ 200 8.29Dq ¼ 88.0 17.35Dq ¼ 184 8Dq ¼ 91.3 18Dq ¼ 205 8.29Dq ¼ 89.4 17.35Dq ¼ 187

Weak Strong Weak Strong Weak Strong Weak Strong

12Dq ¼ 134 12Dq ¼ 134 10.06Dq ¼ 109 14.59Dq ¼ 158 12Dq ¼ 131 12Dq ¼ 131 10.06Dq ¼ 104 14.59Dq ¼ 150

RuCl2(CO)2[P4VP]

709

RuCl2(CO)2[PHIS]2

647

RuCl2(CO)2[PHIS]

713

d7 Complexes CoCl2(H2O)2[P4VP]2

111

CoCl2(H2O)2[P4VP]

106

CoCl2(H2O)2[PHIS]2

114

CoCl2(H2O)2[PHIS]

108

d8 Complexes Ni(CH3COO)2(H2O)2[P4VP]2

112

Ni(CH3COO)2(H2O)2[P4VP]

108

NiCl2(H2O)2[PHIS]2

109

NiCl2(H2O)2[PHIS]

103

For 5-coordinate complexes with three different types of ligands, denoted by M2þ(LA)2(LB)2(LC), Jørgensen’s prediction of 10Dq (kJ/mol) is f10Dqg5-Coordinate ¼

11:963 gM2þ f2f (LA ) þ 2f (LB ) þ f (LC )g 5

where anionic, neutral, and polymeric ligands are denoted by LA, LB, and LC, respectively. An average Jørgensen f-factor of 6.0 was employed for CuO in fRuCl2(CuO)3g2, which is consistent with the fact that CuO is the strongest p-acceptor in the spectrochemical series [Shriver et al., 1990]. The f-factor for imidazole in poly(L-histidine) was determined from an empirical correlation between Brønsted ionization equilibrium constants in Table 6.4 (i.e., pKB ¼ 142pKA) and

Jørgensen’s f-parameter for Ligand Field Splittings

6.6 Cobalt, Nickel, and Ruthenium Complexes with Poly(4-vinylpyridine) 5

10

15

20

1.8 Jørgensen’s f-values Third-order polynomial

1.6

233

25 1.8 1.6

1.4

1.4

1.2

1.2

1.0

1.0

0.8

0.8

0.6 5

20 10 15 Brønsted Ionization Equilibrium Constant, pKB

0.6 25

Figure 6.7 Empirical correlation between Jørgensen f-factors for prediction of octahedral ligand field splittings and Brønsted ionization equilibrium constants (i.e., pKB) in aqueous solution at 25 8C.

Jørgensen’s f-factors in Table 6.5 for three anionic ligands (i.e., Br2, Cl2, CN2) and two neutral ligands (i.e., H2O and C5H5N). The following third-order polynomial was used to (i) match these five data pairs for f versus pKB with a correlation coefficient better than 0.999 and (ii) estimate the Jørgensen f-factor for six monodentate imidazole ligands, which is within the range of the data set: f ¼ 2:61  2:51  101 pKB þ 1:31  102 pKB2  2:52  104 pKB3 Agreement between this empirical correlation and the five data points for three anionic and two neutral ligands, mentioned above, is illustrated graphically in Figure 6.7. Since the Brønsted ionization equilibrium constant for the imidazole ring in histidine is pKB ¼8.0 [Bohinski, 1983], the Jørgensen f-factor for imidazole is estimated to be 1.32, as indicated in Table 6.5.

6.6.7 Consideration of Interelectronic Repulsion and D0 When There Is Ambiguity in the d-Electron Configuration for Complexes with Pseudo-Octahedral Symmetry There is ambiguity in the electronic configuration for d6 and d7 octahedral complexes, due to the strength of the ligand field. Table 6.13 summarizes information about d-orbital energies, B0 for Ru2þ, Co2þ, and Ni2þ, crossover (i.e., electron pairing) energies for weak and strong ligand fields in terms of fD0/Bgcritical, weak-field and strong-field electronic configurations, and the corresponding LFSEs for d6, d7, and d8 complexes with local octahedral symmetry. For Ru2þ in the second row of the dblock, the large g-factor (i.e., 20), the extremely large f-factor for CuO, and the

234

Chapter 6 Macromolecule–Metal Complexes: Ligand Field Stabilization

Table 6.13 Racah Interelectronic Repulsion Energies for Free Metal Cations, Weak/StrongField Crossover Energies, d-Electron Configurations, and LFSEs for Pseudo-Octahedral dn Complexes Complex

B0 (cm21)

fD0/Bg

d-Electron configuration

LFSE (Dq)

d6 Complexes Ru2þ

620

,20 .20

ft2gg4fegg2 ft2gg6

4 24

d7 Complexes Co2þ

1120

,22 .22

ft2gg5fegg2 ft2gg6fegg1

8 18

d8 Complexes Ni2þ

1080

—

ft2gg6fegg2

12

Note: xy, yz, and xz d-orbitals (denoted by t2g) are degenerate at 24Dq; while z 2 and x 2 2y 2 d-orbitals (denoted by eg) are degenerate at þ6Dq.

small value of B0 (620 cm21) [Figgis, 1966; Figgis and Hitchman, 2000] suggest that strong-field electronic configurations are most probable for these heavy-metal d6 complexes because D0/B is invariably greater than the weak-field/strong-field crossover at fD0/Bgcritical  20. Hence, spin pairing occurs in dxy, dyz, and dxz, whereas the z 2 and x 2 2y 2 d-orbitals are vacant, corresponding to LFSE ¼ 24Dq. For Co2þ in the first row of the d-block, a much smaller g-factor (i.e., 9.2) and a very large value of B0 (i.e., 1120 cm21) [Figgis, 1966; Figgis and Hitchman, 2000], with no carbonyl ligands, argue in favor of weak-field electronic configurations with D0/B less than the crossover at 22. Now, the higher energy d-orbitals (i.e., z 2 and x 2 2y 2) contain one electron each, and LFSE is 8Dq. There is no ambiguity in electronic configuration for d8 Ni2þ complexes with local octahedral symmetry. However, ligand field strength, or D0/B, influences the electronic configuration for 5-coordinate d8 complexes. Since B0 and Jørgensen’s g-factor for Ni2þ are similar to those for Co2þ (see Table 6.5), and no carbonyl ligands occupy sites in the first shell of Ni2þ for the complexes of interest in this chapter, it is reasonable to adopt weak-field electronic configurations for Ni(CH3COO)2(H2O)2[P4VP] and NiCl2(H2O)2[PHIS] in the molten state.

6.6.8 Correlation Between Tg Enhancement and the Difference Between Ligand Field Stabilization Energies in the Glassy and Molten States Ligand field stabilization energies are calculated in Table 6.12 for macromolecule – metal complexes in the glassy and molten states. Based on consideration of interelectronic repulsion in the previous section, asterisks and bold print in Table 6.12 identify the most probable LFSEs for 5- and 6-coordinate complexes of Ru2þ (i.e., strong field), Co2þ (i.e., weak field), and Ni2þ (weak field). When the appropriate ligand field strength is considered in Table 6.12, LFSEs are larger for glassy 6coordinate complexes than they are for 5-coordinate complexes above Tg. More stabilization of metal d-electrons due to geometry and the surrounding ligands in

235

6.6 Cobalt, Nickel, and Ruthenium Complexes with Poly(4-vinylpyridine)

the glassy state is consistent with the fact that these complexes exhibit thermochemical synergy with respect to Tg, upon removal of one ligand in the polymer’s side group from the first-shell coordination sphere of the metal center. This is analogous to the fact that larger LFSEs for fM(H2O)6g2þ yield more exothermic hydration enthalpies relative to linear trends from Ca2þ to Mn2þ to Zn2þ [Shriver et al., 1990; Figgis and Hitchman, 2000] for divalent hexa-aqua metal complexes from the first row of the dblock (see Fig. 6.3). Furthermore, when one lattice water is removed from these hexaaqua complexes, the logarithm of the kinetic rate constant for this process, or the free energy of activation from the 6-coordinate complex to the transition state, is correlated empirically with the difference between LFSEs (i.e., units of Dq) of octahedral ML6 and square pyramidal ML5, without any geometric perturbations of the 5-coordinate complex [Burdett, 1980]. In this analysis of Tg enhancement, Dq is estimated for 5and 6-coordinate mixed-ligand complexes above and below the glass transition temperature, respectively, when geometric perturbations are considered for 5-coordinate complexes. Then, the difference between LFSEs (i.e., DLFSE, units of kJ/mol) in the glassy and molten states is correlated with the increase in Tg for poly(4-vinylpyridine) and poly(L-histidine) complexes that contain 1 mol % Ru2þ, Co2þ, and Ni2þ. These results are presented in Figure 6.8. The reduction in chain mobility and the increase in Tg is more pronounced when the first trace of metal cation is present. There is further enhancement of Tg at higher concentrations of metal cations, but the “relative” increase in Tg is not as significant as the initial effect. Most Tg – composition behavior, experimental or theoretical, is nonlinear. Ligand field analysis is correlated with the initial slope of Tg versus composition, in the range from 0 to 1 mol % metal cation. Without adjustable parameters that might account for differences in complexation efficiency between pyridine and imidazole, it

Tg(complex, 1 mol %) – Tg(polymer) (°C)

80

0

70

20

40

60

80

100

120

140

80 70

Poly(L-histidine)/Co,Ni,Ru Poly(4-vinylpyridine)/Co,Ni,Ru

60

60

50

50

40

40

30

30

20

20

10

10 0

20 40 60 80 100 120 LFSE(glassy state) – LFSE(molten state) (kJ/mol)

140

Figure 6.8 Correlation between Tg enhancement at 1 mol % metal cation and differences between ligand field stabilization energies in the glassy and molten states for Co2þ, Ni2þ, and Ru2þ complexes with poly(4-vinylpyridine) and poly(L-histidine).

236

Chapter 6 Macromolecule–Metal Complexes: Ligand Field Stabilization

is not possible to generate a universal correlation for Tg enhancement, based on the six complexes that were analyzed in this section. Nevertheless, a priori calculations of this nature, together with a reasonable model for the onset of Tg, are useful to identify macromolecule – metal complexes that exhibit thermochemical synergy.

6.6.9 Tetrahedral Co21 Complexes Below Tg and 3-Coordinate Complexes in the Molten State An alternate viewpoint of macromolecule – metal complexes with Co2þ is presented in this section, when two ligands in the side group of the polymer occupy sites in the firstshell coordination sphere of the metal center below Tg. CoCl2(H2O)6 is pink and the anhydrous salt is blue [Young, 1960; Nicholls, 1973]. Cobalt chloride adopts a tetrahedral geometry in ethanol with a characteristic blue color [Nicholls, 1973]. X-ray diffraction data on dark blue crystals of dichlorobis(4-vinylpyridine)cobalt(II) suggest that the structure of this 4-coordinate pseudo-tetrahedral complex contains two 4-vinylpyridine ligands and no waters of hydration [Nyholm et al., 1961; Agnew and Larkworthy, 1965; Admiraal and Gafner, 1968]. These studies are significant because they demonstrate that the borderline acid Co2þ sheds its four hard-base lattice waters in favor of two borderline-base pyridine ligands. Tetrahedral symmetry of the metal center is a common occurrence for d7 Co2þ complexes [Cotton and Wilkinson, 1972]. If geometric perturbations occur during preparation of complexes with poly(4-vinylpyridine) in ethanol, then pseudo-octahedral CoCl2(H2O)6 might revert to tetrahedral coordination with two pyridine ligands and two anionic chloride ligands. If this 4-coordinate structure persists in the glassy state, then the onset of Tg might occur when one pyridine ligand in the side group of the polymer is removed from the coordination sphere of Co2þ due to the addition of thermal energy. Now, the coordinatively unsaturated molten state complex above Tg is 3-coordinate, and the possibilities range from facial trivacant (i.e., nonplanar), where all bond angles are 908, to trigonal planar, where all bond angles are 1208 (see Table 6.1). The following scheme represents a model for the glass transition process in Co2þ complexes with poly(4-vinylpyridine) that have been prepared from ethanol: CoCl2 (P4VP)2 ) CoCl2 (P4VP) þ P4VP Stabilization energies for metal d-electrons above and below Tg, based on this ligand dissociation reaction are presented in Table 6.14. Weak-field electronic configurations are favored for 4-coordinate and 3-coordinate d7 metal complexes from the first row of the d-block, with no carbonyl ligands. One predicts that LFSE for 4-coordinate tetrahedral Co2þ complexes with poly(4-vinylpyridine) below Tg is larger than LFSE for the corresponding 3-coordinate complex in the molten state by 5.8 kJ/mol. Since these complexes were prepared in ethanol, the horizontal coordinate (i.e., D(LFSE)) of the empirical correlation for P4VP –Co2þ complexes in Figure 6.8 was changed from 0.9 kJ/mol (i.e., Table 6.12) to 5.8 kJ/mol (i.e., Table 6.14). Previous analysis of P4VP – Co2þ complexes that were assumed to be pseudo-tetrahedral above and below Tg [Belfiore et al., 1993; Belfiore and McCurdie, 1995] yielded LFSEs that

6.6 Cobalt, Nickel, and Ruthenium Complexes with Poly(4-vinylpyridine)

237

Table 6.14 Electronic Energy Calculations for 3-Coordinate (C3v and D3h) and 4-Coordinate (Td) d7 Co2þ Complexes d-Orbital energies (Dq) for facial trivacant complexes with C3v symmetry (all bond angles are 908) 7Cp/12Dq 1, 2

dxz

dyz

dxy

dx2 y2

dz2

22.00

22.00

22.00

þ3.00

þ3.00

d-Orbital energies (Dq) for trigonal planar complexes with D3h symmetry 7Cp/12Dq 1 2

dxz

dyz

dz2

dxy

dx2 y2

22.57 23.85

22.57 23.85

20.65 23.21

þ2.895 þ5.46

þ2.895 þ5.46

d-Orbital energies (Dq) for tetrahedral complexes with Td symmetry 7Cp/12Dq

dx2 y2

dz2

dxy

1, 2

22.67

22.67

þ1.78

dyz

dxz

þ1.78

þ1.78

7

Ligand field stabilization energies for 3- and 4-coordinate d complexes Symmetry

7Cp/12Dq

Ligand field strength

C3v

1, 2

Weak

D3h

1

Weak

D3h

2

Weak

d-Electron configuration

LFSE (Dq)

fxzg2 fyzg2 fxyg1 fx 2 2y 2g1 fz 2g1 fxzg2 fyzg2 fz 2g1 fxyg1 fx 2 2y 2g1 fxzg2 fyzg2 fz 2g1 fxyg1 fx 2 2y 2g1

4.00 5.14 7.69

Average LFSE for 3-coordinate complexes Td

2

2 2

2 2

1, 2

Weak

fx 2y g fz g fxyg1 fyzg1 fxzg1

d7 Complexes

State

10Dq (kJ/mol)

CoCl2[P4VP]2 CoCl2[P4VP] CoCl2[PHIS]2 CoCl2[PHIS]

Glass Molten Glass Molten

111.7 103.5 116.1 106.4

5.21 5.34

LFSE (kJ/mol) 5.34Dq ¼ 59.7 5.21Dq ¼ 53.9 5.34Dq ¼ 62.0 5.21Dq ¼ 55.4

are 3.7 kJ/mol larger in the glassy state (i.e., CoCl2(P4VP)2) relative to the molten state (i.e., CoCl2(P4VP)(H2O)). Predictions for Co2þ complexes with poly(L-histidine), based on the ligand dissociation reaction in this section, reveal that LFSE for pseudo-tetrahedral complexes in the glassy state is 6.6 kJ/mol larger than that for 3-coordinate complexes in the molten state. However, these poly(L-histidine) – metal complexes were prepared in aqueous solution [McCurdie and Belfiore, 1999a], suggesting that at least two lattice

238

Chapter 6 Macromolecule–Metal Complexes: Ligand Field Stabilization

waters should be retained in a 6-coordinate glassy state structure. Pseudo-octahedral glassy complexes that revert to 5-coordinate complexes via the onset of Tg represent a better model, so D(LFSE) of 1.9 kJ/mol from Table 6.12 is employed on the horizontal axis of Figure 6.8 for poly(L-histidine) –Co2þ instead of 6.6 kJ/mol calculated in Table 6.14.

6.7 TOTAL ENERGETIC REQUIREMENTS TO INDUCE THE GLASS TRANSITION VIA CONSIDERATION OF THE FIRST-SHELL COORDINATION SPHERE IN TRANSITION METAL AND LANTHANIDE COMPLEXES 6.7.1 Density Functional Estimates of Metal –Ligand Bond Dissociation Energies Instead of focusing on the strength of one metal –ligand bond and the corresponding ligand field stabilization energy that represents only a small fraction of the total bond energy, modifications in the glass transition temperature can be correlated with the difference between the total energetics of all reactants and products in a proposed ligand dissociation scheme. Molecular engineering design focuses on all of the metal– ligand bonds that involve basic functional groups in the polymer, because these “weak links” determine whether the material can withstand larger forces before failure occurs and higher temperatures prior to viscous flow or thermal degradation. Density functional methods [Pople et al., 1986; Wimmer, 1991; Ziegler and Tschinke, 1991] are useful to simulate the energetics of the glass transition process. Even though most schemes in this chapter focus on the dissociation of one metal –ligand chemical bond, the energetics of all reactants and products consider molecular orbital overlap that is sensitive to coordination number and geometry of the complex. A plausible strategy is described below for zero-valent d6 transition-metal hexacarbonyl complexes (i.e., M(CuO)6) with poly(vinylamine), where amino side groups in the polymer displace CuO in the first-shell coordination sphere of the metal via assistance from UV radiation. Heating these hybrid organic – inorganic materials above the glass transition temperature dissociates metal/polymer – ligand s-bonds and produces coordinatively unsaturated complexes of lower symmetry in the molten state. Initially, one estimates the first CuO single bond dissociation energy for transition-metal hexacarbonyls M(CuO)6 with true octahedral symmetry. Since all of the ligands are equivalent, there is no difference between removing equatorial versus apical ligands, as there is with 5-coordinate square pyramid complexes. One CuO ligand can be removed from a hexacarbonyl complex by restricting one of the metal – ˚ . The total electronic energy of the original hexacarcarbon bond distances to be 10 A bonyl complex is compared with that of the distorted complex in which the stretched ˚ . The difference between the total electronic metal – carbon bond length is 10 A energy of the original and distorted complexes is the first metal – carbon single bond dissociation energy. Since these calculations force the 5-coordinate (i.e., distorted) complex to be square pyramidal, it should be instructive to compare the total electronic

6.7 Total Energetic Requirements to Induce the Glass Transition

239

energies of the complexes in (i), (iii), and (iv) below to establish the best methodology for quantitative predictions of the first metal – CuO single bond dissociation energy. (i) M(CuO)6 versus square pyramid M(CuO)5 with one CuO ligand occupying ˚. the sixth octahedral site at M – CuO single bond distances of 8, 10, and 12 A (ii) Square pyramid versus trigonal bipyramid M(CuO)5, for application in (iii) and (iv) below. (iii) M(CuO)6 versus square pyramid M(CuO)5 and CuO, where the energies of M(CuO)5 and CuO are calculated separately. (iv) M(CuO)6 versus trigonal bipyramid M(CuO)5 and CuO, where the energies of M(CuO)5 and CuO are calculated separately. Results from (i), (iii), and (iv) above should be compared with experimental CuO single bond dissociation energies [Ziegler and Tschinke, 1991] for zero-valent d6 metal hexacarbonyls (i.e., M(CuO)6). Zero-Valent d6 Hexacarbonyl

First CuO Bond Dissociation Energy

Cr(CuO)6 Mo(CuO)6 W(CuO)6

162 kJ/mol 126 kJ/mol 166 kJ/mol

The method of choice that provides the best match with experimental data is employed in the following section.

6.7.2 The Energetics of Ligand Dissociation Reactions in Model Systems: Comparison with Experimental Tg Enhancements for d-Block and f-Block Complexes Energy-minimized conformations from molecular mechanics are useful to estimate the energetics of metal – ligand bond dissociation. A plausible thermal dissociation reaction for generic zero-valent metal carbonyl complexes that corresponds to the onset of the glass transition is M(CuO)4 [Polymer]2 ) M(CuO)4 [Polymer] þ Polymer The complex on the left side of the previous reaction simulates coordination crosslinks, where ligands in the side group of the polymer occupy apical or equatorial sites in the first shell, and the complex on the right side represents a coordination pendant group. Typical models for the polymer in the previous dissociation reaction are: Polymer (Acronym) Poly(vinylamine) (PVAm) Poly(4-vinylpyridine) (P4VP) Poly(L-histidine) (PHIS)

Model Compounds Methyl amine, CH3NH2 Pyridine, C5H5N Imidazole, C3H4N2

These model compounds provide reasonable estimates of the ligands’ electronic characteristics, as required for an energetic description of the glass transition process. At the present time, any connection between the density functional calculations summarized above

240

Chapter 6 Macromolecule–Metal Complexes: Ligand Field Stabilization

and experimental Tg enhancement data is empirical. It is necessary to establish a quantitative link between the formation of mobility-restricting nanoclusters in the coordination sphere of the metal center and the efficiency of Tg enhancement, where the latter is summarized in Table 6.15 for several polymer/metal-complex combinations. Table 6.15 Enhancement in Tg (8C) per mol % Metal Cation (Between 0 and 1 mol %) for Several Macromolecule–Metal Complexes a d-Block complexes

P4VP

Mg(CH3COO)2(H2O)4 Ca(CH3COO)2(H2O) Ni(CH3COO)2(H2O)4 Cu(CH3COO)2(H2O)2 Zn(CH3COO)2(H2O)2 Zn(CH3[CH2]10COO)2 CoCl2(H2O)6 NiCl2(H2O)6 CuCl2(H2O)2 ZnCl2 fRuCl2(CuO)3g2 PdCl2(CH3CuN)2

0 0 5.2 1.0 1.3 26.2 3.4

10

Lanthanide complexes LaCl3(H2O)6 CeCl3(H2O)x PrCl3(H2O)7 NdCl3(H2O)6 SmCl3(H2O)6 EuCl3(H2O)6 GdCl3(H2O)x TbCl3(H2O)6 DyCl3(H2O)6 HoCl3(H2O)6 ErCl3(H2O)x TmCl3(H2O)7 YbCl3(H2O)6 LuCl3(H2O)6

[Xe]4f0 [Xe]4f1 [Xe]4f2 [Xe]4f3 [Xe]4f5 [Xe]4f6 [Xe]4f7 [Xe]4f8 [Xe]4f9 [Xe]4f10 [Xe]4f11 [Xe]4f12 [Xe]4f13 [Xe]4f14

Polymers (acronym) Poly(4-vinylpyridine) (P4VP) Poly(vinylamine) (PVAm) Poly(L-histidine) (PHIS) Poly(L-lysine)hydrobromide (PLYS) a

PVAm

PHIS

PLYS

15 30 2.8 3 69 17

24.6 25.4 21.1 26.2

17

45

25

Tg (@ 0.5 mol % Ln3þ with PVAm) (8C) 104 102 102 103 74 79 60 110 107 105 74 112 106 112 MW (daltons) 5

2  10 2.3  104 (1.5–5.0)  104 2  105

The asterisks identify polymer– metal-salt mixtures that exhibit a decrease in Tg.

Tg (8C), undiluted 145 57 169 178

6.9 Epilogue

6.8

241

SUMMARY

Stabilization of metal d-electrons has been employed previously to explain thermodynamic [Shriver et al., 1990; Figgis and Hitchman, 2000] and kinetic [Burdett, 1980] data for 6-coordinate hexa-aqua divalent transition-metal complexes from the first row of the d-block. Kinetic data for the dissociation of one lattice water from M2þ(H2O)6 were analyzed by postulating a 5-coordinate square pyramidal product [i.e., M2þ(H2O)5] that was not allowed to distort. The methodology described in this chapter allows for geometric distortions in the molten state to model the glass transition process in macromolecule – metal complexes with enhanced Tg values. By focusing on weakly basic ligands with a different hardness classification than the metal center, ligand exchange in the first-shell coordination sphere of d-block cations was invoked to couple at least two different chains via coordination crosslinks. Molybdenum hexacarbonyl complexes with poly(vinylamine) have been analyzed using ligand field models, with hope that experimental data will follow. For complexes based on poly(4-vinylpyridine) and dichlorotricarbonylruthenium(II) at low metal cation concentrations, group theory analysis of infrared data [McCurdie and Belfiore, 1999b] yields a glassy state structure where two pyridine side groups from the polymer occupy sites in the first shell of Ru2þ. Dissociation of one of these metal – polymer chemical bonds at high temperature produces a 5-coordinate complex with reduced symmetry in the molten state. Ligand field stabilization energy differences between 6-coordinate glassy complexes and 5-coordinate molten complexes have been correlated with the enhancement in Tg for Ru2þ, Co2þ, and Ni2þ complexes with poly(L-histidine). For similar complexes with poly(4-vinylpyridine), Ru2þ and Ni2þ are considered to be pseudo-octahedral below Tg and 5-coordinate above Tg, but the corresponding Co2þ complexes prepared from ethanol are considered to be pseudo-tetrahedral below Tg and 3-coordinate above Tg. At 1 mol % of the d-block metal cations, there is much more enhancement in the glass transition of poly(L-histidine), relative to that for poly(4-vinylpyridine). Adjustable parameters were not introduced to develop a correlation between Tg,Complex 2Tg,Undiluted Polymer, due to self-assembled nanoclusters, and the difference between ligand field stabilization energies below and above Tg. The model predicts thermochemical synergy for six classes of macromolecule – metal complexes. However, universality of the correlation has not been demonstrated for these six complexes, even though the basicity of the important functional group in the polymer’s side chain influences predictions of ligand field stabilization energies via Jørgensen’s quantum-chemical group contribution method.

6.9

EPILOGUE

The major postulate in this chapter, that the enhancement in Tg should correlate with ligand field stabilization energy differences between the glass and the liquid, mimics Linus Pauling’s development of the electronegativity scale in which the partial ionic character of covalent bonds between dissimilar atoms, due to the ionic resonance

242

Chapter 6 Macromolecule–Metal Complexes: Ligand Field Stabilization

energy, was correlated with the square of the difference between electronegativities of the two elements. Pauling [1960] introduced the electronegativity scale to correlate the difference between (i) AB bond energies, where A and B represent dissimilar atoms, and (ii) the arithmetic or geometric average of the AA and BB bond energies. Pauling [1960] states that “the electronegativity scale systematizes the field of inorganic thermochemistry” in which glass transition temperatures of macromolecule – metal complexes are a subset. If one envisions that the arithmetic average of AA and BB represents the additive rule of mixtures when thermophysical properties, like the glass transition temperature, are considered for binary systems, then synergistic enhancement of bond energies or Tg with respect to the average has been described by differences between (i) electronegativities of the dissimilar atoms or (ii) ligand field stabilization energies of the two states (i.e., below and above Tg), respectively. It should be emphasized that synergistic enhancement of the glass transition in this chapter has been analyzed with respect to Tg of the undiluted polymer, not the additive rule of mixtures prediction of Tg for polymer and low-molecular-weight metal complex. Glass transition temperatures of completely crystalline transition-metal and lanthanide salts are extremely difficult, if not impossible, to measure experimentally because, in the process of heating these inorganic compounds into the molten state, thermal decomposition is highly probable. The classic Gordon – Taylor equation for the compositional dependence of the glass transition temperature in binary mixtures with component mass fractions vi (see Sections 1.7 and 1.9, and Fig. 1.1), Tg,mixture ¼

    v1 DCp,1@Tg,1 Tg,1 þ v 2 DCp,2@Tg,2 Tg,2 v1 DCp,1@Tg,1 þ v 2 DCp,2@Tg,2

predicts that Tg,mixture lies between the pure-component glass transitions, Tg,1 and Tg,2. The additive rule of mixtures (i.e., linear weight-fraction-weighted sum of purecomponent glass transition temperatures) is obtained from the Gordon– Taylor equation when the discontinuous increment in specific heat DCp is the same for each component in these binary mixtures. Kwei et al. [1984, 1989] proposed an empirical modification of the Gordon – Taylor equation to account for inflection points in the compositional dependence of Tg,mixture and synergistic enhancement of the mixture’s glass transition temperature with respect to the additive rule of mixtures (i.e., when h ¼ 1) by including a quadratic mass-fraction term as follows: Tg,mixture ¼

h¼

v1 Tg,1 þ hv2 Tg,2 þ cv1 v 2 v1 þ hv2 DCp,2@Tg,2 DCp,1@Tg,1

The Kwei equation reduces to the additive rule of mixtures for Tg,mixture when h ¼ 1 and c ¼ 0. Positive values of the quadratic coefficient c (i.e., with h ¼ 1) yield positive deviations of the mixture’s Tg with respect to the Gordon– Taylor equation, with negative curvature and no inflection point.

References

243

APPENDIX: PHYSICAL INTERPRETATION OF THE PARAMETERS IN THE KWEI EQUATION FOR SYNERGISTIC ENHANCEMENT OF THE GLASS TRANSITION TEMPERATURE IN BINARY MIXTURES Let’s begin by analyzing the shape of the Tg – composition relation (i.e., Tg,mixture vs. v2) when h ¼ 1 to investigate the effect of c on the slope and curvature. One obtains the following expressions:   dTg,mixture ¼ Tg,2  Tg,1 þ c(1  2v2 ) d v2 @h¼1  2  d Tg,mixture ¼ 2c dv22 @h¼1 It should be obvious that the quadratic coefficient c governs the curvature (i.e., second derivative) of Tg,mixture versus v2 when h ¼ 1. Furthermore, the magnitude and sign of c, relative to the difference between pure-component glass transition temperatures, determine whether the maximum and/or minimum in Tg,mixture versus composition occurs in binary mixtures or at the pure-component limits. For example, if the following equation does not yield a mass fraction of component 2 between 0 and 1,

v2,@Tg, max=min ¼

Tg,2  Tg,1 þ c 2c

then Tg,mixture versus composition exhibits extrema at Tg,1 and Tg,2. From the viewpoint of backbone stabilization and the Margules model for excess thermodynamic state functions in binary mixtures, the quantity kBoltzmanncv1v2 represents the excess chain backbone stabilization energy in the blend relative to a weighted average of the backbone stabilization energy in each pure material. If this excess energy is positive (i.e., c . 0), then blends exhibit higher glass transition temperatures relative to the additive rule of mixtures.

REFERENCES ADMIRAAL LJ, GAFNER G. Chemical Communications 1221 (1968). AGNEW NH. Journal of Polymer Science, Polymer Chemistry Edition 14:2819 (1976). AGNEW NH, LARKWORTHY LF. Journal of the Chemical Society 4669 (1965). AGNEW NH, COLLIN RJ, LARKWORTHY LF. Journal of the Chemical Society, Dalton Transactions 3:272 (1974). ASHCROFT SJ, MORTIMER CT. Thermochemistry of Transition Metal Complexes. Academic Press, New York, 1970, p. 402. BEACH NA, GRAY HB. Journal of the American Chemical Society 90:5713 (1968). BELFIORE LA, DAS PK. Journal of Polymer Science, Polymer Physics Edition 42:2270 (2004). BELFIORE LA, INDRA EM. Journal of Polymer Science, Polymer Physics Edition 38:552 (2000). BELFIORE LA, MCCURDIE MP. Journal of Polymer Science, Polymer Physics Edition 33:105 (1995). BELFIORE LA, GRAHAM HRG, UEDA E. Macromolecules 25:1411 & 2935 (1992). BELFIORE LA, MCCURDIE MP, UEDA E. Macromolecules, 26:6908 (1993).

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BELFIORE LA, BOSSE´ F, DAS PK. Polymer International 36:165 (1995). BELFIORE LA, DAS PK, BOSSE´ F. Journal of Polymer Science, Polymer Physics Edition 34:2675 (1996). BELFIORE LA, INDRA EM, DAS PK. Macromolecular Symposia, Polymer– Solvent Complexes 114:35 (1997). BELFIORE LA, SUN X, DAS PK, LEE JY. Polymer 40:5583 (1999). BELFIORE LA, MCCURDIE MP. Polymer Engineering and Science 40:738 (2000). BELFIORE LA, MCCURDIE MP, DAS PK. Polymer 42:9995 (2001a). BELFIORE LA, RUZMAIKINA IY, DAS PK. Polymer Engineering and Science 41:1196 (2001b). BENEDETTI E, BRACA G, SBRANA G, SALVETTI F, GRASSI B. Journal of Organometallic Chemistry 37:361 (1972). BOHINSKI RC. Modern Concepts in Biochemistry, 4th edition. Allyn & Bacon, Boston, 1983, p. 61. BOSSE´ F, DAS PK, BELFIORE LA. Macromolecules 28:6993 (1995). BOSSE´ F, DAS PK, BELFIORE LA. Journal of Polymer Science, Polymer Physics Edition 34:909 (1996). BOSSE´ F, DAS PK, BELFIORE LA. Polymer Gels and Networks 5:387 (1997). BRANDRUP J, IMMERGUT EH. Polymer Handbook, 2nd edition. Wiley-Interscience, Hoboken, NJ, 1975, Chap. 3, p. 152. BRUCE MI, STONE FGA. Journal of the Chemical Society A:1238 (1967). BURDETT JK. Molecular Shapes: Theoretical Models of Inorganic Stereochemistry. Wiley-Interscience, Hoboken, NJ, 1980, pp. 124– 133, 138, 139, 167–169, 189. CLEARE MJ, GRIFFITH QP. Journal of the Chemical Society A:372 (1969). CONSTABLE EC. Metals and Ligand Reactivity. Ellis Horwood, New York, 1990, p. 115. COTTON FA, WILKINSON G. Advanced Inorganic Chemistry: A Comprehensive Text, 3rd edition. WileyInterscience, Hoboken, NJ, 1972, Chap. 25, pp. 875, 890– 897. DAS PK, RUZMAIKINA IY, BELFIORE LA. Journal of Polymer Science, Polymer Physics Edition 38:1931 (2000). DAS PK, LEE JK, RUZMAIKINA IY, BELFIORE LA. Polymer 42:8873 (2001a). DAS PK, LEE JK, BELFIORE LA. Journal of Polymer Science, Polymer Physics Edition 39:677 (2001b). DIMARZIO EA, GIBBS JH. Journal of Polymer Science, Part A 1:1417 (1963). DOWNIE TC, HARRISON W, RAPER ES, HEPWORTH MA. Acta Crystallographica, Section B, Structural Science B27:706 (1971). FIGGIS BN. An Introduction to Ligand Fields. Wiley-Interscience, Hoboken, NJ, 1966, pp. 17, 37, 52, 236–237, 242–245. FIGGIS BN, HITCHMAN MA. Ligand Field Theory and Its Applications. Wiley-VCH, Hoboken, NJ, 2000, pp. 18, 105, 139, 140, 167 –171, 194–195, 218–221. FLORY PJ. Principles of Polymer Chemistry. Cornell University Press, Ithaca, NY, 1953, pp. 507–511. FUJITA H, KISHIMOTO A. Journal of Polymer Science 28:547 (1958). FUJITA H, KISHIMOTO A. Journal of Chemical Physics 34:393 (1961). FUSTIN CA, LOHMEIJER BGG, DUWEZ AS, JONAS AM, SCHUBERT US, GOHY JF. Nanoporous thin films from self-assembled metallo-supramolecular block copolymers. Advanced Materials 17:1162–1165 (2005). HEGEDUS LS. Transition Metals in the Synthesis of Complex Organic Molecules, 2nd edition. University Science Books, Sausalito, CA, 1999, pp. 4, 13– 15. HOUSE DA. Ammonia and amines, in Comprehensive Coordination Chemistry: Synthesis, Reactions, Properties and Applications of Coordination Compounds, Wilkinson G, Gillard RD, and McCleverty JA, editors. Pergamon Press, Oxford, UK, 1987, Vol. 2, Chap. 13.1, pp. 23–43. KELLEY FN, BUECHE F. Journal of Polymer Science 50:549 (1961). KWEI TK. Journal of Polymer Science, Polymer Letters Edition 22:307 (1984). KWEI TK, LIN AA, REISER A. Macromolecules 22:4112 (1989). MALHOTRA SL, LESSARD P, BLANCHARD LP. Journal of Macromolecular Science-Chemistry A15:121 (1981). MARCH J. Advanced Organic Chemistry: Reactions, Mechanisms, and Structure, 3rd edition. Wiley, Hoboken, NJ, 1985, Chap. 8. MCCURDIE MP, BELFIORE LA. Journal of Polymer Science, Polymer Physics Edition 37:301 (1999a). MCCURDIE MP, BELFIORE LA. Polymer 40:2889 (1999b). MIZUNO J, UKEI K, SUGAWARA T. Journal of the Physical Society of Japan 14:383 (1959).

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NICHOLLS D. Cobalt, in Comprehensive Inorganic Chemistry, Volume 3, Bailar JC, Emele´us HJ, Nyholm RS, and Trotman-Dickenson AF, editors. Pergamon Press, Oxford, UK, 1973. NYHOLM RS, GILL NS, BARCLAY GA, CHRISTIE TI, PAULING PJ. Journal of Inorganic and Nuclear Chemistry 18:88 (1961). PAULING L. The Nature of the Chemical Bond, and the Structure of Molecules and Crystals, 3rd edition. Cornell University Press, Ithaca, NY, 1960, Chap. 3. PEARSON RG. In Survey of Progress in Chemistry: Volume 6, Scott A, editor. Academic Press, New York, 1969, Chap. 1, pp. 12, 13. PEARSON RG, editor. Hard and Soft Acids and Bases, Benchmark Papers in Inorganic Chemistry. Dowden, Hutchinson, and Ross, Stroudsburg, PA, 1973. POMOGAILO AD. In Macromolecule– Metal Complexes, Ciardelli F, Tsuchida E, and Wo¨hrle D, editors. Springer-Verlag, Berlin, 1996, p. 52. POPLE JA, HEHRE WJ, RADOM L, SCHLEYER PVR. Ab Initio Molecular Orbital Theory. Wiley-Interscience, Hoboken, NJ, 1986. PRUCHNIK FP. Organometallic Chemistry of the Transition Elements, Duraj SA, translator. Plenum Press, New York, 1990, p. 45. ROSS BL, GRASSELLI JG, RITCHEY WM, KAESZ HD. Inorganic Chemistry 2:1023 (1963). SHRIVER DF, ATKINS PW, LANGFORD CH. Inorganic Chemistry. WH Freeman, New York, 1990, pp. 17, 150, 206–211, 214, 445, 504– 506, 683. STEPHENSON TA, WILKINSON G. Journal of Inorganic and Nuclear Chemistry 28:945 (1966). TANAKA T, WANG C, KING K. Faraday Discussions 101:201 (1995). VANNIEKERK JN, SCHOENING FRL. Acta Crystallographica 6:609 (1953). WELLS AF. Structural Inorganic Chemistry, 5th edition. Oxford University Press, New York, 1984, pp. 674– 675. WIMMER E. Density functional theory for solids, surfaces and molecules: from energy bands to molecular bonds, in Density Functional Methods in Chemistry, Labanowski JK and Andzelm JW, editors. Springer-Verlag, Heidelberg, 1991. YOUNG RS, editor. Cobalt: Its Chemistry, Metallurgy, and Uses. Reinhold Publishing, New York, 1960, p. 76. ZIEGLER T, TSCHINKE V. Density functional theory as a practical tool in organometallic energetics and dynamics, in Density Functional Methods in Chemistry, Labanowski JK and Andzelm JW, editors. Springer-Verlag, Heidelberg, 1991.

Part Two

Semicrystalline Polymers and Melting Transitions

Chapter

7

Basic Concepts and Molecular Optical Anisotropy in Semicrystalline Polymers The iron law of nature says: “creatures that fart and burp cannot take themselves seriously.” —Michael Berardi

Introductory crystallization topics are followed by a discussion of semicrystalline polymers that form spherulites. Then, optical anisotropy is analyzed from the interaction of birefringent materials with polarized light in spherical coordinates. Segment orientation distribution functions are developed via entropy maximization with constraints. This Langevin distribution is employed to predict the molecular optical anisotropy of random-coil and rigid-rod polymers. The connection between the morphology of semicrystalline polymers and their mechanical properties is discussed qualitatively.

7.1

SPHERULITIC SUPERSTRUCTURE

Polymer chains undergo crystallization by organizing into flat plate-like layers known as lamellae. In an effort to minimize unfavorable surface contact between crystalline and amorphous regions, further organization occurs when plate-like lamellae twist, branch, and form spherical space-filling crystalline regions known as spherulites. These highly organized structures interact uniquely with polarized light such that the field of view in an optical microscope with cross polars resembles a Maltese cross, which is characteristic of a spherulitic superstructure. From small-scale to large-scale order that summarizes the crystallization process, (i) polymer chains pack within a unit cell, (ii) unit cells stack within lamellae with chain folding in the vicinity of the lamellar surface, and (iii) lamellae form space-filling birefringent spherulites that can be viewed in an optical microscope. Polymer crystallization always corresponds to (i) and (ii), but the formation of spherulitic superstructures may or may not occur. Physical Properties of Macromolecules. By Laurence A. Belfiore Copyright # 2010 John Wiley & Sons, Inc.

249

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7.2

COMMENTS ABOUT CRYSTALLIZATION

The requirements for macromolecules to crystallize are discussed next, together with some characteristics of semicrystalline polymers.

7.2.1

Tacticity

For vinyl polymers with the following repeat unit chemical structure, (CH2CHR), where R represents a side group significantly different in size relative to H, stereoregularity is required because a center of chirality exists. The carbon atom in the chain backbone, attached to side group R, is a chiral center because four different groups are connected tetrahedrally to this carbon atom. The left and right sides of the chain backbone with respect to this chiral carbon are not identical from the viewpoint of stereochemistry. Hence, isotactic and syndiotactic configurational isomers, with R groups on the same side and opposite sides, respectively, of an extended chain, are capable of crystallizing, but the atactic isomer with random placement of R groups should be almost completely amorphous. It is not possible to interconvert from one tactic isomer to another by rotation about carbon – carbon single bonds in the chain backbone. Complete stereoregularity is not required for crystallization. Stereoregular polymerization of vinyl monomers requires the use of heterogeneous Ziegler – Natta catalysts that direct migration of the growing alkyl polymer chain from the metal to the less-substituted terminus of the adjacent metal-p-coordinated alkene monomer. This creates an extended alkyl polymer and a vacant site in the first-shell coordination sphere of the metal center that will be occupied by the next alkene monomer. Actually, migration of the growing alkyl polymer chain can occur at either the more-substituted or the less-substituted terminus of the h2 metalp-coordinated alkene monomer. “Olefin-coordination/migratory-insertion” mechanisms are common throughout transition-metal organometallic chemistry [Hegedus, 1999]. This propagation mechanism for stereospecific polymerization is very similar to the classic Heck reaction. The monomer’s alkene functional group coordinates to a transition-metal catalyst, like titanium in the þ4 oxidation state, by occupying a vacant site in the first shell, followed by migratory insertion of the growing polymer chain s-alkyl intermediate in head-to-tail fashion.

7.2.2

Helical Conformations

When isotactic vinyl polymers crystallize, the chain adopts a 3/1 helical conformation where three repeat units are required for one complete revolution of the helix. This conformation results when carbon – carbon single bonds in the chain backbone alternate between trans and gauche rotational isomers. Hence, there must be sufficient mobility for backbone bonds to adopt the correct rotational isomeric state, which is consistent with experimental observations that crystallites are thicker with fewer defects when crystallization occurs isothermally at temperatures closer to the melting point. This process is directly related to the rate of crystal growth. The rather complex

7.2 Comments about Crystallization

251

three-dimensional potential energy surface for polymer conformations exhibits local minima as a function of adjacent Ramachandran backbone bond rotation angles w and c that correspond to trans (1808), gaucheþ (608), and gauche – (3008). The energy barriers between these local minima represent Arrhenius-type activation energies for interconversion between two different chain conformations. More flexible chain segments are characterized by smaller activation energies that separate trans and gauche rotational isomers on the potential energy surface. The cis or “eclipsed” conformation at w ¼ 08 is highly unfavorable, as evidenced by the fact that it corresponds to the highest energy state. All of these rotational states for carbon –carbon single bonds are illustrated best with the aid of Newman projections in organic chemistry. As a general rule, polymers crystallize in low-energy conformations that correspond to local minima on the potential energy surface. Chain conformations in the disordered amorphous state correspond to much higher energies than any crystalline conformation of a given macromolecule. This is analogous to the energetics of protein folding in biological macromolecules where the native or folded conformation is at lowest energy in the “folding funnel” hierarchy relative to molten globules and denatured (i.e., unfolded) proteins. The planar zigzag “extended” crystalline conformation of polyethylene, with repeat unit chemical structure (CH2CH2)x, no centers of chirality, and orthorhombic space group symmetry of the unit cell, is obtained when all carbon – carbon backbone bonds exist in trans rotational states. Poly(ethylene oxide), with repeat unit chemical structure (CH2CH2O)x, no centers of chirality, and monoclinic symmetry of the unit cell, crystallizes as a 7/2 helix (i.e., 7 repeat units are required for 2 complete revolutions) where carbon – carbon bonds are in a gauche (i.e., g) rotational state and both carbon – oxygen bonds are trans (i.e., t) [Tadokoro 1979]. Isotactic polypropylene, with repeat unit chemical structure [CH2C H(CH3)]x and chiral center denoted by C , crystallizes as a 3/1 helix where the carbon – carbon backbone single bonds alternate between trans and gauche. This is the a-form of isotactic polypropylene with a monoclinic unit cell. Poly(oxymethylene), with repeat unit chemical structure (CH2O)x, no chiral centers, and a trigonal unit cell, crystallizes as a 9/5 helix in which all backbone bonds are essentially gauche. The crystalline conformation of poly(oxymethylene) does not contain any backbone bonds in a trans rotational state. These four examples of crystalline chain conformations suggest that a higher percentage of gauche rotational states for backbone single bonds yields a more compact helix with higher crystal density in which fewer repeat units are required for one complete revolution. For example, the 9/5 helical conformation of poly(oxymethylene) exhibits a crystal density of 1.49 g/cm3, whereas poly(ethylene oxide)’s 7/2 helix exhibits a crystal density of 1.228 g/cm3. This information is summarized in Table 7.1 for four different semicrystalline polymers.

7.2.3

Polymorphism

This phrase describes the tendency of a specific polymer to adopt more than one crystalline structure. One classic example is the three crystalline forms exhibited by

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Chapter 7 Basic Concepts and Molecular Optical Anisotropy

Table 7.1 Chain Conformations of Selected Crystalline Polymers Chemical structure

Polymer Polyethylene (PE) Poly(ethylene oxide) (PEO) Isotactic poly(propylene) (iPP) Poly(oxymethylene) (POM)

Rotational states

Helical pitch

Percentage gauche

(CH2CH2) (CH2CH2O)

All trans ttg

Planar zigzag 7/2 or 3.5/1

0 33

[CH2CH(CH3)]

tg

3/1

50

(CH2O)

All gauche

9/5 or 1.8/1

100

Table 7.2 Physical Properties of Three Crystalline Polymorphs of Isotactic Poly(1-butene)

Crystalline polymorph Form 1 Form 2 Form 3

Crystalline density g/cm3

Melting temperature (8C)

Enthalpy of fusion (cal/g)

0.951 0.920 0.897

133 118 100

34 29 28

Helical pitch

Backbone bond rotational states

Unit cell symmetry

3/1 11/3 4/1

t180g60 t163g77 t159g83

Trigonal Tetragonal Orthorhombic

Source: Belfiore et al. [1984]. The density of completely amorphous poly(1-butene) is 0.864 g/cm3.

isotactic poly(1-butene), whose repeat unit chemical structure is (CH2CHR)x, where the stereoregular side group R is CH2CH3. As mentioned above, isotactic vinyl polymers, such as poly(1-butene), crystallize in a 3/1 helical conformation where the backbone bond rotational states alternate between trans (1808, or t180) and gauche (608, or g60). There are two additional helical conformers of this vinyl polymer in which the backbone bond rotation angles deviate slightly from pure trans and pure gauche. Notice how the physical properties of these polymorphs depend on helical pitch in Table 7.2, such that the crystal density and melting temperature decrease when more repeat units are required for one complete revolution of the helix.

7.2.4

Predicting the Melting Temperature

Tmelt can be predicted, either qualitatively or quantitatively, from the following thermodynamic relation: Dgmelting ¼ Dhfusion  T Dsfusion where g is free energy, h is enthalpy, s is entropy, and D represents a difference between thermodynamic state functions in the disordered amorphous phase versus the crystalline regions. At the equilibrium melting temperature, Tmelt, the free

7.2 Comments about Crystallization

253

energy of the crystallites is the same as the free energy of the amorphous molten material. Hence, one estimates Tmelt from Dhfusion Tmelt ¼ Dsfusion This expression is useful for analysis of first-order phase transitions, like melting, but not second-order glass transitions (see Problem 1.5 in Chapter 1). Factors that affect the enthalpy change upon melting are polarity, intermolecular interactions, and the degree of crystallinity. Factors that affect the entropy change upon melting are related to chain flexibility and the number of conformations available to the polymer in the disordered amorphous phase. In general, Tmelt is higher for polar polymers with larger dipole moments, as a consequence of larger Dhfusion. Conversely, Tmelt is lower for polymers with more flexible linkages in the chain backbone, due to larger Dsfusion. One should exercise caution in estimating Tmelt qualitatively when competing factors are present.

7.2.5

Single Crystals, Lamellae, and Chain Folding

When a potentially crystallizable polymer is dissolved at very low concentrations in an appropriate low volatile solvent, most likely at elevated temperature, and the temperature is lowered to a favorable range below the melting point of the polymer where crystallization can occur, flat plate-like lamellar structures develop. These lamellae grown from dilute solution are stacked upon each other and contain polymer chains in which the c-axis of each crystallographic unit cell is oriented in the same direction. An integer number of repeat units in each chain and an integer number of revolutions of a helical conformation are required for periodicity along the c-axis of the unit cell. If backbone bond lengths and bond angles are similar, then the c-axis of the unit cell is longer when more backbone bonds are required to achieve periodicity of a helical ˚ for the 3/1 helix of isotactic polypropylene conformation. For example, c ¼ 6.50 A ˚ for the 9/5 helix of (i.e., 6 carbon– carbon bonds in 3 repeat units), c ¼ 17.39 A poly(oxymethylene) (i.e., 18 carbon – oxygen bonds in 9 repeat units), and c ¼ ˚ for the 7/2 helix of poly(ethylene oxide) (i.e., 7 carbon – carbon bonds and 19.48 A 14 carbon – oxygen bonds in 7 repeat units). X-ray and electron diffraction experiments performed on these single crystals yield dimensions and angles of the crystallographic unit cell which allow one to calculate crystal densities based on an integer number N of polymer chains in each cell. For example, if there are 4 chains of isotactic polypropylene in each unit cell and 3 repeat units per chain are required for periodic behavior of the 3/1 helix, then 12 repeat units occupy each unit cell. More importantly, diffraction experiments on single crystals reveal that the chain backbone, or the c-axis of a helical conformation, is normal to the large plate-like surfaces of the lamellae. Lamellar thick˚ , and this thickness can be increased by nesses are typically on the order of 100 – 200 A annealing crystals between Tg and Tmelt. Detailed analysis of the thermodynamics of chain folding is required to determine the optimum lamellar thickness (see Section 8.11). This occurs when the Gibbs free energy change for crystallization is minimized, based on contributions from chain ends, chain folds, and intermolecular

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Chapter 7 Basic Concepts and Molecular Optical Anisotropy

interactions between the molten amorphous phase and chain segments on the fold surface. High-molecular-weight polymers exhibit contour lengths and chain lengths along the c-axis that are much larger than the thickness of a lamella. Hence, polymer chains in these single crystals must fold back on themselves after passing through the thinnest dimension of a lamella so that they can reenter the same lamella in a pattern described by regular or irregular adjacent reentry. The phenomenon of chain folding was proposed initially by K.H. Storks in 1938 via electron diffraction studies of natural rubber (i.e., gutta-percha), which is primarily trans-1,4-polyisoprene, and this publication [Storks, 1938] has received more than 70 citations. Original comments from the classic 1938 publication are included, here: “It is surprising that most of the crystallites are oriented with their fiber axis directions normal to the plane of a thin film whose thickness is much smaller than the contour length of gutta-percha. Hence, macromolecular chains may possibly fold by a mechanism of rotation around single bonds. Chains fold back and forth upon themselves such that adjacent stems remain parallel. The chemical repeat unit is short, so relatively few folds per chain are required. It is conceivable that infrequent folding should not affect the crystal structure of the polymer.” An important factor that contributes to the phenomenon of chain folding is the reduction in density experienced by chain segments in the vicinity of the fold surface, because this allows for easier intramolecular rotation about backbone single bonds to assist the folding process. Stems describe segments of one polymer chain that traverse a single lamella. For example, an N-stem crystal layer is defined as a polymer chain that folds {N 2 1} times as it adds to the lateral surface of a growing lamella. As lamellae grow, chain folding of crystal stems on the lamellar surface is consistent with a reduction of the overall free energy for crystal growth. A higher concentration of sequential gauche rotational states is required for chains within the fold regions to undergo reentry. The regularity of reentry into the same lamella can be described by the probability that chain segments near the fold surface immediately reenter the crystallite. Hence, crystallographic conformations in the vicinity of the fold surfaces do not follow the regular pattern of trans and gauche rotational states described above. When the crystallization conditions correspond to stable lamellar growth, chain folding from a nucleation center occurs three dimensionally, with defect structures near the chain ends and on the fold surface. Stereo-irregular chain segments (i.e., atactic) and headto-head instead of head-to-tail arrangements of two adjacent repeat units also contribute to defects that reduce the melting temperature. Irregular nonadjacent reentry describes a two-phase morphology where polymer chains fold within one lamella, and then traverse the amorphous region before entering different lamellae. These interlamellar “tie molecules” represent links between different crystalline regions, and they provide mechanical integrity to two-phase semicrystalline polymers that are above their glass transition temperature but below the melting point. These “switchboard” models of a two-phase morphology of semicrystalline polymers are consistent with the following optical properties of solid materials with respect to visible light. If the sample is completely amorphous and homogeneous, then it should be transparent. If a two-phase morphology of crystalline and amorphous regions exists, then the material is translucent or opaque if the refractive indicies of both regions are different. This is the most common situation, but exceptions exist. For example, if a material is translucent or opaque, then it is phase-separated with a different refractive index for at least two of

7.3 Spherulitic Superstructures that Exhibit Molecular Optical Anisotropy

255

the phases. Transparent materials are, most likely, amorphous and homogeneous, but they could exist as two separate phases that have the same refractive index. The opacity or transparency of a two-phase material with respect to visible light depends on the presence or absence, respectively, of a mismatch between refractive indicies of the two phases.

7.2.6

Molecular Optical Anisotropy

In the presence of an electric field, it is much easier to delocalize electrons or distort the electron density distribution along the chain backbone due to p-overlap of p-orbitals that do not participate directly in s-bonding which is characteristic of covalent carbon– carbon single bonds. Hence, the polarizability of each polymer chain is much larger along its backbone, or optic axis, relative to the ability to distort the electron density distribution normal to the chain backbone. Single-chain calculations of optical anisotropy, averaged over the distribution of segment orientations with respect to the end-to-end chain vector, are discussed at the end of this chapter for freely jointed and rigid rod-like polymers. These differences in polarizability, parallel versus perpendicular to the chain backbone, are obscured by the disordered nature of an ensemble of random-coil polymer chains in the amorphous state. Hence, amorphous materials are optically isotropic. However, if (i) chains adopt a particular conformation, as required for crystallization, (ii) all chains are aligned along the thinnest dimension of each lamella, and (iii) a group of lamellae is not disorganized, then differences in polarizability parallel and perpendicular to the chain backbone yield differences in polarizability along two mutually orthogonal axes of a coordinate system which is fixed in the laboratory frame of reference. Hence, single crystals are optically anisotropic, by virtue of chain organization within each lamella and the fact that the c-axes of all crystallographic unit cells are oriented in the same direction. A birefringent material exhibits optical anisotropy with respect to a fixed laboratory frame of reference. If lamellae are disorganized, then the optical anisotropy due to chain ordering within each lamella is obscured because there is no large-scale superstructure. Hence, disordered crystallites and amorphous materials are both optically isotropic because it is not possible to identify at least two orthogonal coordinate directions in a fixed laboratory reference frame that exhibit different polarizabilities.

7.3 SPHERULITIC SUPERSTRUCTURES THAT EXHIBIT MOLECULAR OPTICAL ANISOTROPY When polymers crystallize from the molten state upon cooling below the melting temperature, they form either ordered or disordered lamellae. This section focuses on the formation of three-dimensional space-filling spherulites that result when ordered lamellae grow radially outward from a nucleation center. This phenomenon is illustrated below in Figure 7.1 (on the left side) for undiluted poly(ethylene oxide) PEO, via polarized optical microscopy. Crystallites do not necessarily evolve in spherically symmetric fashion from their inception. In some cases, spherulitic superstructures develop from much simpler

256

Chapter 7 Basic Concepts and Molecular Optical Anisotropy

Figure 7.1 Optical micrographs of the crystalline regions in poly(ethylene oxide), PEO, using polarized light in which the polarizer is perpendicular to the analyzer (i.e., cross polars). In the polymer– polymer blend, poly(2-vinylpyridine), P2VP, is completely amorphous, introducing imperfections into the spherulitic superstructure of PEO.

dendritic chain-folded crystal precursors. Hence, the absence of spherulites in a semicrystalline polymer does not imply that crystallization proceeds via a mechanism that is fundamentally different from the process that yields spherulites. If dendritic precursors impinge upon each other during the early stages of growth and inhibit the formation of spherulites, then the birefringent nature of the crystallites is severely distorted. This is illustrated in Figure 7.1 (right side) when poly(ethylene oxide) is blended with an amorphous polymer, poly(2-vinylpyridine) (P2VP), such that P2VP interferes with the crystallization process in PEO as a consequence of hydrogen-bonding interactions between two dissimilar chains in the amorphous phase. If impingement of dendritic precursors does not occur, then it might be difficult to envision how flat plate-like structures grow radially to form a space-filling aggregate. Twisting and branching of lamellae occur to accomplish this task. An imbalance between surface forces on the upper and lower lamellar surfaces, where chain folding occurs, is responsible for the fact that lamellae twist within spherulites. Since the lamellae are ordered, these crystalline regions are birefringent, which implies that the polarizability of a spherulite exhibits directionality in the laboratory frame of reference. In mathematical terms, polarizability is a second-rank tensor that contains nine scalars aij arranged in a 3  3 matrix. The ability to distort the electron density distribution in the ith-coordinate direction due to application of an electric field in the jth-coordinate direction is given by aij. The 3  3 matrix of scalars representing the polarizability tensor contains nonzero entries only on the main diagonal when the molecular axis along the chain backbone coincides with one of the axes in the coordinate system of the fixed laboratory reference frame. Under these conditions, the molecular axes within the polymer chain represent the principal axes of the polarizability tensor. Since the spherical coordinate system best exploits the symmetry of a spherulite, the polarizability tensor is represented as follows:

a¼

3 X i, j¼1

di dj aij dij ¼

3 X i¼1

spherical

di di aii ) dr dr arr þ dQ dQ aQQ þ df df aff coordinates

7.3 Spherulitic Superstructures that Exhibit Molecular Optical Anisotropy

257

where di is a unit vector in the ith-coordinate direction, didj is an ordered pair of unit vectors, known as a unit dyad for representation of a second-rank tensor, and dij is the Kroneker delta, which assumes a value of unity when the two subscripts are the same but is zero otherwise. When an electric field (i.e., E) interacts with a birefringent material and distorts the electron density distribution, as prescribed by the polarizability tensor, a dipole moment is induced which can be calculated as follows:



pinduced ¼ a E ¼

3 X

di di aii

i¼1



3 X

d j Ej

j¼1

where Ej is the component of the electric field along the jth-coordinate direction. Vector – tensor mathematics in the previous expression can be manipulated to obtain a vector equation for the induced dipole moment. This is accomplished using summation notation and the definition of the scalar “dot” product of unit vectors:



di dj ¼ dij Hence, pinduced ¼

3 X



di (di dj )aii Ej ¼

i, j¼1

3 X

di (dij )aii Ej

i, j¼1

The summations in the previous expression contain nine terms. However, six of these terms vanish because di is orthogonal to dj. Summation notation and the Kroneker delta account for the orthonormal property of unit vectors in spherical coordinates by reducing the nine-term sum to a three-term sum. This is accomplished by removing the summation over index j and replacing subscript j by subscript i in all scalar components. Hence, pinduced ¼

3 X i¼1

spherical

di aii Ei ) dr arr Er þ dQ aQQ EQ þ df aff Ef coordinates

This is the starting point to analyze the angular dependence of transmitted light for semicrystalline polymers in an optical microscope. In the equatorial plane of a spherulite at polar angle Q ¼ p/2 in spherical coordinates, lamellae grow radially outward from the nucleation center and chain folding occurs within each lamella. Hence, the chain backbone and the primary direction in which the electron density distribution can be distorted (i.e., the optic axis) are exclusively in the f-direction. The relative magnitudes of the principal components of the polarizability tensor in spherical coordinates are

aff  arr  aQQ and the induced oscillating dipole moment that results when a light source impinges on this birefringent material is essentially “along the chain backbone”: pinduced  df aff Ef

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Chapter 7 Basic Concepts and Molecular Optical Anisotropy

where



Ef ¼ d f E With respect to the direction of light propagation (i.e., z-direction in rectangular coordinates), electric field vectors oscillate in the transverse (i.e., x-y) plane.

7.4 INTERACTION OF A BIREFRINGENT SPHERULITE WITH POLARIZED LIGHT Polarized optical microscopy, or optical microscopy with cross polars, is the experimental technique used to detect spherulitic superstructures because transmitted light exhibits angular dependence in the focal plane of the microscope. Electromagnetic radiation propagates in the z-direction and a polarizer selects components of the electric field that oscillate in the x-direction. Hence, E ¼ dx E Polarized light impinges on a birefringent spherulite and induces an oscillating dipole moment, given by



pinduced  df (df dx )aff E When cross polars are placed in the optical path, this oscillating dipole moment is sampled in the y-direction because the analyzer is perpendicular to the polarizer. Notice that the direction of light propagation, its electric field components, and the component of the induced dipole moment that is sampled by the analyzer are described best in rectangular coordinates. However, the polarizability tensor and pinduced are presented in spherical coordinates to exploit the symmetry of a spherulite. The azimuthal angular direction (i.e., 0 f 2p) is the symmetry variable in spherical coordinates. In the focal plane of the microscope at constant Q ¼ p/2 (i.e., x-y plane), the f-direction is related to the orientation of the polarizer (i.e., x-direction) and analyzer (i.e., y-direction) by trigonometry: df ¼ dx sin f þ dy cos f The component of the induced oscillating dipole moment along the analyzer direction is







dy pinduced  (dy df )(df dx )aff E ¼ aff E sin f cos f In the language of spectroscopy and quantum mechanics, transmitted light intensity is given by the square of the expectation value of the electric dipole moment operator. In agreement with this formalism, when a birefringent spherulite interacts with

7.4 Interaction of a Birefringent Spherulite with Polarized Light

259

cross-polarized light, transmitted light intensity in the binocular field of view of an optical microscope is given by



Transmitted light intensity ¼ {dy pinduced}2  {aff E sin f cos f}2 ¼

2 1 4 {aff E}

sin2 2f

The angular dependence of transmitted light exhibits extremum conditions (i.e., maxima or minima) when the derivative of the previous expression with respect to azimuthal angle f vanishes. One obtains the following result: d {Transmitted light intensity}  14 {aff E}2 (2 sin 2f)(2 cos 2f) df ¼ 12 {aff E}2 sin 4f ¼ 0 The previous equation is satisfied when angle 4f is a multiple of p. Hence,

f¼

np ; 0 n 7 4

Maxima occur at odd values of n (i.e., 1, 3, 5, 7) and minima occur at even values of n (i.e., 0, 2, 4, 6). The pattern that one observes in the field of view is known classically as the Maltese cross. It is simulated in Figure 7.2 using polar coordinates with a gray-scale gradient in the radial direction, and compared with a realistic spherulite of undiluted semicrystalline poly(ethylene oxide). This birefringent pattern is an unambiguous indicator that polymeric crystallites exhibit order and that a spherulitic superstructure exists.

Figure 7.2 Prediction and observation of transmitted light intensity (i.e., Maltese cross) that results when a birefringent spherulite interacts with polarized light such that the polarizer is perpendicular to the analyzer (i.e., cross polars).

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7.5 INTERACTION OF DISORDERED LAMELLAE WITH POLARIZED LIGHT The starting point for analysis of the angular dependence of transmitted light in the optical microscope is the induced dipole moment in spherical coordinates:

pinduced ¼

3 X

spherical

di aii Ei ) dr arr Er þ dQ aQQ EQ þ df aff Ef coordinates

i¼1

This expression was derived above and simplified for a birefringent spherulite. If lamellae are not organized into a spherulitic superstructure, then the crystalline regions are not birefringent, even though chain folding occurs within each lamella and polarizability is much greater along the chain backbone than perpendicular to the chain backbone. Under these conditions, all three principal components of the polarizability tensor are comparable in magnitude because it is not possible to orient one coordinate direction in the fixed laboratory frame of reference along the chain backbone. Consequently, it is not easier to distort the electron density distribution along any particular axis in the fixed laboratory frame of reference. Hence,

arr  aQQ  aff Once again, electromagnetic radiation propagates in the z-direction and the oscillating electric field vectors are polarized in the x-direction, which implies that E ¼ dx E In spherical coordinates, the r-, Q-, and f-components of the polarized electric field are

    ¼ d  E ¼ E{d  d }

Er ¼ dr E ¼ E{dr dx} EQ ¼ dQ E ¼ E{dQ dx} Ef

f

f

x

If a represents each principal component of the polarizability tensor, then the induced dipole moment in the disordered crystalline regions is written as follows:







pinduced ¼ aE{dr (dr dx ) þ dQ (dQ dx ) þ df (df dx )} For cross-polarized light, the analyzer samples the y-component of p induced, which is calculated from the previous expression:















dy pinduced ¼ aE{(dy dr )(dr dx ) þ (dy dQ )(dQ dx ) þ (dy df )(df dx )}

7.6 Interaction of Disordered Lamellae with Unpolarized Light

261

This result is simplified by employing trigonometry to relate unit vectors in rectangular and spherical coordinates. One obtains the following vector relations: dr ¼ dx sin Q cos f þ dy sin Q sin f þ dz cos Q dQ ¼ dx cos Q cos f þ dy cos Q sin f  dz sin Q df ¼ dx sin f þ dy cos f Now, the following scalar “dot” products of unit vectors can be evaluated:

 

 

 

dy dr ¼ sin Q sin f; dy dQ ¼ cos Q sin f; dy df ¼ cos f dr dx ¼ sin Q cos f; dQ dx ¼ cos Q cos f; df dx ¼ sin f Explicit evaluation of the y-component of p induced, which is sampled by the analyzer, yields



dy pinduced ¼ aE{sin2 Q sin f cos f þ cos2 Q sin f cos f  sin f cos f} ¼ 0 Hence, there is no transmitted light in the binocular field of view of the optical microscope when the polarizer is perpendicular to the analyzer. Electric-fieldinduced oscillating dipole moments vanish along the direction of the analyzer when lamellae are disorganized and the crystalline regions are not birefringent. One concludes that isotropic materials, including unoriented amorphous polymers, water and air, produce a completely dark field of view in an optical microscope when the polarizer and analyzer are oriented perpendicular to each other (i.e., cross-polarized light). Hence, it is rather straightforward to distinguish birefringent spherulites from disordered isotropic regions of a semicrystalline polymer via optical microscopy with cross polars. However, if a spherulitic superstructure does not develop, then the crystalline regions cannot be identified by polarized optical microscopy. Under these conditions, differential scanning calorimetry is the preferred analytical technique to detect the presence of crystallites via measurements of melting transitions.

7.6 INTERACTION OF DISORDERED LAMELLAE WITH UNPOLARIZED LIGHT Electromagnetic radiation propagates in the z-direction and the electric field vectors oscillate in the transverse plane, such that E ¼ E{dx þ dy} which is a simplified representation of unpolarized light. Since a spherulitic superstructure does not develop and the crystalline regions are not birefringent, there is no preferred direction in which the electron density distribution can be distorted in the fixed laboratory frame. Under these conditions, the electric-field-induced

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Chapter 7 Basic Concepts and Molecular Optical Anisotropy

oscillating dipole moment is;

pinduced ¼

3 X

spherical

di aii Ei ) dr arr Er þ dQ aQQ EQ þ df aff Ef coordinates

i¼1

where

arr  aQQ  aff Now, the r-, Q-, and f-components of the electric field vector in spherical coordinates for this simplified representation of unpolarized light are

  



Er ¼ dr E ¼ E{dr (dx þ dy )}

 

EQ ¼ dQ E ¼ E{dQ (dx þ dy )} Ef ¼ df E ¼ E{df (dx þ dy )} Even though the incident light is not polarized, the analyzer samples either the x-component or the y-component of p induced, the latter of which is calculated as follows:  dy pinduced ¼ aE (dy dr )[dr (dx þ dy )] þ (dy dQ )[dQ (dx þ dy )]  þ (dy df )[df (dx þ dy )]















which can be simplified greatly with assistance from trigonometry, as described in the previous section.



dy pinduced ¼ aE{sin Q sin f[sin Q cos f þ sin Q sin f] þ cos Q sin f  [cos Q cos f þ cos Q sin f] þ cos f[sin f þ cos f]} ¼ aE where a represents each principal component of the polarizability tensor. Transmitted light intensity in the binocular field of view is



Transmitted light intensity ¼ {dy pinduced}2  {aE}2 ¼ constant Hence, there is no angular dependence of transmitted light intensity in the optical microscope when disorganized lamellae are viewed with unpolarized light. The field of view is not dark, as it is when polarized light is employed and the analyzer is perpendicular to the polarizer. Isotropic materials are not birefringent and do not produce angular dependence of transmitted light intensity under any conditions. The field of view is either completely bright if the incident light is

7.7 Molecular Optical Anisotropy of Random Coils and Rigid Rod-Like Polymers

263

unpolarized, or completely dark if the polarizer and analyzer are crossed at right angles.

7.7 MOLECULAR OPTICAL ANISOTROPY OF RANDOM COILS AND RIGID ROD-LIKE POLYMERS 7.7.1 Segment Orientation with Respect to the End-to-End Chain Vector: Inverse Langevin Distribution Function Consider a freely jointed chain with no bond angle or valance cone restrictions. There are N segments, each of length l, and the vector from one end of the chain to the other (i.e., the end-to-end chain vector) is r. Of particular interest in this section is the development of the segmental orientation distribution function fr(Q), where fr(Q) sin Q dQ (i.e., spherical coordinates) represents the probability that a randomly chosen chain segment is oriented in the range from Q to Q þ dQ with respect to a fixed end-to-end chain vector. For cylindrically symmetric carbon – carbon single bonds in the chain backbone that are oriented at angle Q with respect to r, all nine scalar components of the polarizability tensor in a laboratory-fixed rectangular Cartesian coordinate system are averaged with respect to the distribution function fr(Q). The molecular optical anisotropy of a single chain becomes significant when flexible macromolecules are stretched. Chain orientation due to an external force field produces macroscopic birefringence that represents a visual image of the state of stress in polymers. The following strategy is employed to derive the segmental orientation distribution function fr(Q): Step 1:

Begin with a single chain that contains a total of N segments, each of length l, and align the end-to-end chain vector along the positive z-axis of a rectangular Cartesian coordinate system, where polar angle Q ¼ 0 in spherical coordinates.

Step 2:

If there are ni segments oriented at angle Qi with respect to the end-to-end chain vector, then one calculates the total number of distinguishable ways V that these ni segments can be distributed among the total number of segments N. It is necessary to consider segments oriented at all possible angles with respect to r. In other words, polar angle Q ranges from 0 to p.

Step 3:

The most probable distribution for fr(Q) corresponds to chain conformations with maximum entropy. The final result maximizes the multiplicity of states identified by V in Step 2 via Boltzmann’s entropy equation (i.e., S ¼ k ln V), where k is Boltzmann’s constant. One must incorporate two constraints in the maximization problem via the method of Lagrange multipliers. Since V contains a product of several factorial quantities, the preferred approach maximizes the natural logarithm of V, as prescribed

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by Boltzmann’s equation, after employing Stirling’s approximation for ln(ni!). The constraints are X ni ¼ N

l

X

i

ni cos Qi ¼ r

i

where cos Qi accounts for the projection of segment ni along the z-axis and r is the magnitude of the end-to-end chain vector, which is constant. Kuhn and Gru¨n [1942] considered the projection of each segment vector along the axis of the end-to-end chain vector r, whereas Flory [1969] considered projections along an arbitrary axis. One evaluates two Lagrange multipliers (i.e., a and b) via continuous expressions for ni (Qi) and both constraints as follows: ðp n(Q) sin Q dQ ¼ N 0

ðp

l n(Q) cos Q sin Q dQ ¼ r 0

The appropriate volume element in spherical coordinates requires the “solid angle,” which accounts for the factor of sin Q in these integral constraints. If the energy of each chain conformation were an important consideration, then the calculation described below would be much more complex. However, freely jointed chains exhibit no bond angle or valence cone restrictions, and each conformation is “equally likely,” exhibiting the same energy. The system seeks a state of maximum entropy via the following “counting problem.” The multiplicity V of distinguishable conformations of N chain segments, each of length l, with ni segments oriented at angle Qi with respect to a fixed end-to-end chain vector r is given by N!

V¼Y i

{ni (Qi )}!

Division by each ni! is required so that identical chain conformations are not counted multiple times. The previous expression for V invariably yields an integer for this “counting problem,” but the results below for ni as a function of Qi are not restricted to whole numbers, particularly when the discrete solution ni (Qi) is extended to the continuous limit, yielding n(Q). The strategy focuses on maximizing ln V after using Stirling’s approximation for the factorial of a large argument (i.e., n  1):   pffiffiffiffiffiffiffiffiffi 1 1 þ     n! ¼ nn 2p n exp (n) 1 þ 12n 288n2   ln n!  12 ln(2p) þ n þ 12 ln n  n

7.7 Molecular Optical Anisotropy of Random Coils and Rigid Rod-Like Polymers

265

The objective function that must be maximized is ln V ¼ ln(N!) 

X

ln(ni !)

i

X      1 1 ¼ 12 ln(2p) þ N þ 12 ln N  N  2 ln(2p) þ ni þ 2 ln ni  ni 

X

 1

¼ 12 ln(2p) þ N þ 2 ln N 

i 1 2 ln(2p)

   þ ni þ 12 ln ni

i

The method of Lagrange multipliers accounts for the two constraints mentioned above and yields the following modification of the discrete objective function: ( G ¼ ln V þ a N 

X

) ni

( þb rl

X

i

) ni cos Qi

i

where a and b represent generic Lagrange multipliers. The extremum conditions are

@G @nj

N,a,b,nk[k=j]



@G @a @G @b

  1 ¼ 1þ þ ln nj  a  bl cos Qj ¼ 0 2nj



¼N

X

N,b,ni



¼rl N,a,ni

ni ¼ 0

i

X

ni cos Qi ¼ 0

i

In the first extremum condition, it is not rigorously possible to vary a certain group of segments nj oriented at angle Qj with respect to the end-to-end chain vector, while the total number of segments N and all other groups of segments nk at other orientation angles remain constant, because the first constraint is not satisfied. However, for very large numbers of segments, N is approximately constant when nj varies slightly. Furthermore, since the number of segments nj oriented at angle Qj with respect to the end-to-end chain vector is, in general, quite large also (i.e., nj  1), the first extremum condition given above simplifies to

  @G 1 ¼ 1þ þ ln nj  a  bl cos Qj @nj N,a,b,nk[k=j] 2nj  ln nj  a  bl cos Qj ¼ 0 which provides an estimate for each discrete nj in terms of Qj. In the continuous limit, the number of chain segments n(Q) oriented at angle Q with respect to the end-to-end chain vector r is given by the following function: n(Q)  exp{a  bl cos Q}

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Chapter 7 Basic Concepts and Molecular Optical Anisotropy

Lagrange multipliers a and b are determined from continuous representations of the two constraints mentioned above: X

ni )

i

ðp

ðp

n(Q) sin Q dQ ¼ exp(a) exp(bl cos Q) sin Q dQ

0

0

2 sinh(bl) exp(a) ¼ N bl ðp X l ni cos Qi ) l n(Q) cos Q sin Q dQ ¼

i

0

ðp

¼ l exp( a) exp(bl cos Q) cos Q sin Q dQ 0

  2 sinh(bl) ¼ exp(a)  cosh(bl) ¼ r b bl where the second constraint is evaluated via integration by parts. One obtains the following solution for the Lagrange multipliers a and b (i.e., replace b by 2b in all of the equations above without affecting any of the results), and the distribution function fr(Q): exp(a) ¼ N

bl 2 sinh(bl)

r 1 ¼ coth(bl)  ¼ L(bl) bl Nl fr (Q) ¼

n(Q) bl exp{bl cos Q} ¼ 2 sinh(bl) N

The requirement that the projections of all segments along the z-axis must sum to the magnitude of the end-to-end chain vector r, together with the fact that the total number of chain segments N is constant, yields a relation between r, N, l, and the Lagrange multiplier b known as the Langevin function (i.e., L(x) ¼ coth(x) – x 21). For this reason, fr(Q) is called the Langevin or inverse Langevin distribution function. The parameter of interest in fr(Q) is bl, which can be calculated from the inverse Langevin function of r/Nl (i.e., bl ¼ L 21(r/Nl )). The development in this section yields a normalized distribution function with respect to the “solid angle” in spherical coordinates (i.e., sin Q dQ), as illustrated below: ðp

ðp bl fr (Q) sin Q dQ ¼ exp{bl cos Q} sin Q dQ ¼ 1 2 sinh(bl)

0

0

In general, there is a higher probability of finding segments that are oriented in the direction of r, but this discrimination becomes much more pronounced, and the

7.7 Molecular Optical Anisotropy of Random Coils and Rigid Rod-Like Polymers

267

breadth of the distribution decreases significantly, when the end-to-end chain length r approaches its rigid-rod limit of Nl. There is a direct connection between the inverse Langevin distribution function fr(Q) for individual segment orientations with respect to the end-to-end chain vector and the Kratky – Porod persistence length aP, which qualitatively describes length along the chain backbone that one must travel before the next segment no longer correlates with the end-to-end vector (i.e., Q ¼ 908). The persistence length aP and the end-to-end chain length r approach the upper limit of Nl for rigid rods and thermoreversible gels with fiber-like morphology. Furthermore, aP increases if fr(Q ¼ 0) is larger, as illustrated in Figure 7.3 when the end-to-end chain length r corresponds to a larger fraction of the rigid-rod limit. The results presented in this section are summarized as follows. One obtains an exponential segment orientation distribution function fr(Q) by applying the method of Lagrange multipliers with two constraints to optimize a multiplicity function V. The logarithmic form of the multiplicity function conforms to Boltzmann’s entropy expression (i.e., S ¼ k ln V) because all orientations of the segments of a freely jointed chain have the same energy. With assistance from Stirling’s approximation for the factorial of a large argument, one arrives at the most probable distribution by seeking a solution that maximizes the entropy (i.e., via ln V) of a single chain. The independent variable Q in the distribution function corresponds to the polar angle in spherical coordinates, because one seeks the projection of each chain segment along the direction of the end-to-end chain vector r that is coincident with the z-axis of a rectangular Cartesian coordinate system. Since the distribution function is exponential, and continuous expressions for the two constraints yield integrals of fr(Q) over the complete range of Q (i.e., finite limits from 0 ) p), the two Lagrange multipliers that appear in fr(Q) are evaluated in terms of hyperbolic functions. Similar hyperbolic functions do not appear in the free volume formulation of small molecules diffusing

Orientation Distribution Function

10 90% of rigid limit 85% of rigid limit 80% of rigid limit 75% of rigid limit 70% of rigid limit 60% of rigid limit 50% of rigid limit 30% of rigid limit

8 6 4 2 0

0

30

60 90 120 Polar Angle, Q (degrees)

150

180

Figure 7.3 Effect of the end-to-end chain length as a fraction (i.e., 0.3 to 0.9) of the rigid-rod limit (i.e., Nl ) on the Langevin orientation distribution function fr(Q). The length of the end-to-end vector increases from the lowermost curve to the uppermost curve.

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Chapter 7 Basic Concepts and Molecular Optical Anisotropy

through polymers on a lattice (i.e., see Section 2.3 in Chapter 2) because the free volume per lattice cell was considered to have a physically unrealistic infinite upper limit.

7.7.2 Rectangular Cartesian Components of the Polarizability Tensor for Cylindrically Symmetric Bonds in the Chain Backbone For simplicity, let’s assume that the polarizability of carbon– carbon backbone single bonds in linear hydrocarbon chains is cylindrically symmetric about the bond axis, which is oriented in the radial direction of the spherical coordinate system. Hence, the backbone bond under investigation is oriented at polar angle Q with respect to the z-axis, and azimuthal angle f with respect to the x-axis in the xy-plane. As discussed in the previous section, the end-to-end vector of a single chain is coincident with the z-axis, but the orientation of each segment with respect to the z-axis is described by the Langevin distribution function fr(Q). Focus on one segment of a linear polymer chain that is oriented in the radial direction (spherical coordinates). The bond polarizability tensor contains only diagonal entries in spherical coordinates because the cylindrically symmetric bond axis exhibits flexibility with respect to Q and f such that carbon – carbon bonds always coincide with the radial direction of the fixed laboratory reference frame. Under these conditions, the molecular axes within cylindrically symmetric backbone single bonds represent the principal axes of the bond polarizability tensor a. The same alignment between the molecular axes and the laboratory reference frame is not possible using cylindrical coordinates because bond vectors exhibit flexibility with respect to the z-axis, such that preferential alignment of each bond along one coordinate direction does not occur. Hence, it should not be surprising that individual bond polarizability tensors for cylindrically symmetric carbon –carbon single bonds contain nonzero off-diagonal elements in cylindrical coordinates. Analogous to the discussion in Section 7.3 for chain polarizability tensors in a spherulite, a contains no off-diagonal entries in spherical coordinates, and one writes:

a ¼ dr dr arr þ dQ dQ aQQ þ df df aff where arr corresponds to the principal component of the polarizability tensor along the bond axis (i.e., hereafter referred to as a1) and aQQ ¼ aff represent principal values of a along the two principal directions of the polarizability ellipsoid that are transverse to the bond axis (i.e., hereafter referred to as a2). The cylindrically symmetric nature of carbon –carbon single bonds in the chain backbone is responsible for the fact that both principal values of a in the transverse plane are equal (i.e., aQQ ¼ aff ). It is important to emphasize that the two distinctly different principal values of the bond polarizability tensor, a1 and a2 (where a1  a2), characterize each segment of the polymer chain and are described best in spherical coordinates. However, when one focuses on the entire chain composed of N freely jointed segments, in which the end-to-end vector is oriented along the z-axis, the rectangular Cartesian coordinate system is chosen to evaluate aParallel and aPerpendicular via ensemble averaging of azz and axx or ayy,

7.7 Molecular Optical Anisotropy of Random Coils and Rigid Rod-Like Polymers

269

respectively, with respect to the Langevin segment orientation distribution function fr(Q). Hence, all nine scalar components of the symmetric bond polarizability tensor are calculated in rectangular coordinates, where a contains off-diagonal entries because each chain segment is not necessarily oriented along any of the three fixed coordinate directions (i.e., x, y, or z) in the fixed laboratory reference frame. However, none of the nonzero off-diagonal entries survives ensemble averaging with respect to the Langevin segment orientation distribution function, yielding principal axes and principal values of the polarizability ellipsoid for the entire chain in rectangular coordinates. With assistance from trigonometric relations between unit vectors in rectangular and spherical coordinates, repeated here for convenience, dr ¼ dx sin Q cos f þ dy sin Q sin f þ dz cos Q dQ ¼ dx cos Q cos f þ dy cos Q sin f  dz sin Q df ¼ dx sin f þ dy cos f it is possible to obtain all nine scalar components of a in rectangular coordinates. For example, the xx-component of the bond polarizability tensor is

  ¼ d  {(d  d )d a þ (d  d )d a þ (d  d )d a } ¼ d  {d a sin Q cos f þ d a cos Q cos f  d a sin f}

axx ¼ dx (dx a ) x x

x

r

r rr

x

f

x

Q QQ

2

2

f ff

f 2

Q 2

r 1

2

Q

2

2

¼ a1 sin Q cos f þ a2 cos Q cos f þ a2 sin f ¼ a1 sin2 Q cos2 f þ a2 cos2 Q cos2 f þ a2 (1cos2 f) ¼ a1 sin2 Q cos2 f  a2 (1cos2 Q) cos2 f þ a2 ¼ (a1  a2 )sin2 Q cos2 f þ a2 In general, the jk-component of a in any orthonormal coordinate system is obtained by constructing the scalar “dot” product of unit vectors dj and dk with the second-rank polarizability tensor as follows: a jk ¼ dk (dj a)

 

The yy-component of the bond polarizability tensor is calculated below:

 

 





ayy ¼ dy (dy a) ¼ dy {(dy dr )dr arr þ (dy dQ )dQ aQQ þ (dy df )df aff } 2

2

2

2

¼ a1 sin Q sin f þ a2 cos Q sin f þ a2 cos2 f ¼ a1 sin2 Q sin2 f þ a2 cos2 Q sin2 f þ a2 (1  sin2 f) ¼ a1 sin2 Q sin2 f  a2 (1  cos2 Q)sin2 f þ a2 ¼ (a1  a2 )sin2 Q sin2 f þ a2 The zz-component of a is significant because it yields aParallel per chain segment when it is averaged with respect to the Langevin segment orientation distribution function

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Chapter 7 Basic Concepts and Molecular Optical Anisotropy

fr(Q). One calculates azz as follows:

 

 





azz ¼ dz (dz a) ¼ dz {(dz dr )dr arr þ (dz dQ )dQ aQQ þ (dz df )df aff} 2

2

2

2

¼ a1 cos Q þ a2 sin Q ¼ a1 cos Q þ a2 (1cos Q) ¼ (a1  a2 )cos2 Q þ a2 The off-diagonal elements of the bond polarizability tensor are nonzero because each segment is oriented randomly with respect to any of the rectangular coordinate axes, even though the end-to-end chain vector is coincident with the z-axis. Hence, the molecular axes for each chain segment cannot, in general, be aligned with the rectangular Cartesian coordinate axes in the fixed laboratory frame of reference. Since a is a symmetric second-rank tensor, ajk ¼ akj as illustrated below. The xy-component of a is

 

    ¼ d  {d a sin Q cos f þ d a cos Q cos f  d a sin f}

axy ¼ dy (dx a) ¼ dy {(dx dr )dr arr þ (dx dQ )dQ aQQ þ (dx df )df aff } y

f 2

Q 2

r 1

2

2

¼ a1 sin Q sin f cos f þ a2 cos Q sin f cos f  a2 sin f cos f ¼ a1 sin2 Q sin f cos f þ a2 (1sin2 Q)sin f cos f  a2 sin f cos f ¼ (a1  a2 )sin2 Q sin f cos f One obtains the same result for the yx-component of a, as illustrated below:

 

   ¼ d  {d a sin Q sin f þ d a



ayx ¼ dx (dy a) ¼ dx {(dy dr )dr arr þ (dy dQ )dQ aQQ þ (dy df )df aff} x

r 1

Q 2

cos Q sin f þ df a2 cos f}

¼ a1 sin2 Q sin f cos f þ a2 cos2 Q sin f cos f  a2 sin f cos f ¼ (a1  a2 )sin2 Q sin f cos f The final two independent scalar components of the bond polarizability tensor are axz,

 

   ¼ d  {d a sin Q cos f þ d a



axz ¼ dz (dx a) ¼ dz {(dx dr )dr arr þ (dx dQ )dQ aQQ þ (dx df )df aff} z

r 1

Q 2

cos Q cos f  df a2 sin f}

¼ a1 sin Q cos Q cos f  a2 sin Q cos Q cos f ¼ (a1  a2 )sin Q cos Q cos f and ayz,

 

   ¼ d  {d a sin Q sin f þ d a



ayz ¼ dz (dy a) ¼ dz {(dy dr )dr arr þ (dy dQ )dQ aQQ þ (dy df )df aff } z

r 1

Q 2

cos Q sin f þ df a2 cos f}

¼ a1 sin Q cos Q sin f  a2 sin Q cos Q sin f ¼ (a1  a2 )sin Q cos Q sin f

7.7 Molecular Optical Anisotropy of Random Coils and Rigid Rod-Like Polymers

271

All six independent scalar components of the symmetric bond polarizability tensor in rectangular coordinates are summarized below and evaluated when a cylindrically symmetric carbon – carbon single bond is aligned with the z-axis (i.e., Q ¼ 0, any value of f ), such that azz ¼ a1 and axx ¼ ayy ¼ a2, with no nonzero off-diagonal elements.

axx ¼ (a1  a2 )sin2 Q cos2 f þ a 2 ayy ¼ (a1  a 2 )sin2 Q sin2 f þ a 2 azz ¼ (a1  a 2 )cos2 Q þ a 2 axy ¼ (a1  a 2 )sin2 Q sin f cos f axz ¼ (a1  a 2 )sin Q cos Q cos f ayz ¼ (a1  a 2 )sin Q cos Q sin f When a cylindrically symmetric bond vector is oriented in the r-direction (spherical coordinates), the previous results for all scalar components of the bond polarizability tensor in rectangular coordinates reduce to





a jk ¼ (a1  a2 )(dj dr )(dk dr ) þ a2 d jk d jk ¼ Kronecker delta,

) j, k ¼ x, y, z

7.7.3 Ensemble Averaging of ajk in Rectangular Coordinates with Respect to the Langevin Segment Orientation Distribution Function As discussed earlier in this chapter, the Langevin distribution fr(Q) depends on polar angle Q, but not azimuthal angle f, in spherical coordinates. Normalization of fr(Q) was illustrated in Section 7.7.1 with respect to the complete range of polar angles (i.e., 0 Q p). An equivalent statement of normalization for fr(Q) that considers the complete range of both orientation angles in spherical coordinates is 1 2p

2ðp

ðp

ðp bl fr (Q)sin Q dQ ¼ exp{bl cos Q}sin Q dQ ¼ 1 2 sinh(bl)

0

0

df 0

It is important to include integration with respect to azimuthal angle f in the normalization statement and the averaging process because several scalar components of the bond polarizability tensor in rectangular coordinates exhibit dependence on f. In general, averaging the jk-component of a with respect to the segment

272

Chapter 7 Basic Concepts and Molecular Optical Anisotropy

orientation distribution function corresponds to 1 hajk i ¼ 2p

2ðp ð p

a jk (Q, f) fr (Q) sin Q dQ df 0 0

bl ¼ 4p sinh(bl)

2ðp ð p

a jk (Q, f) exp{bl cos Q} sin Q dQ d f 0 0

Now, inspection of the cos f and/or sin f dependence of each ajk, in which j = k, reveals that all off-diagonal scalar components of a do not survive the averaging process, because hsin fi ¼ hcos fi ¼ hsin f cos fi ¼ 0 However, axx and ayy contain cos2 f and sin2 f, respectively, which yield nonzero averages that are calculated as follows:

1 hcos fi ¼ 2p 2

hsin2 fi ¼

2ðp

1 2p

1 (cos f) df ¼ 4p 2

2ðp

0

0

2ðp

2ðp

0

(sin2 f) d f ¼

1 4p

(1 þ cos 2f) df ¼

1 2

(1cos 2f) df ¼

1 2

0

The Langevin distribution which depends on polar angle Q is not required to average a function of azimuthal angle f. The most significant calculations in the averaging process that require fr(Q) involve sin2 Q and cos2 Q, which appear in axx, ayy, and azz. One must consider the following definite integral expression to evaluate ksin2 Ql and kcos2 Ql. If x ¼ bl cos Q, then ðp

1 bl (cos Q) expfbl cos Qg sin Q dQ ¼ (bl)2 2

0

bl ð

x 2 e x dx

bl

¼ 2 sinh(bl)  þ

4 cosh(bl) bl

4 sinh(bl) (bl)2

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273

via two consecutive integrations by parts. Division of the previous equation by 2 sinh(bl ) yields the ensemble average of cos2 Q with respect to fr(Q). Hence, ðp bl hcos Qi ¼ (cos2 Q) exp{bl cos Q} sin Q dQ 2 sinh(bl) 2

0

  2 2 2 1 coth(bl) þ ¼1 coth(bl)  ¼1 bl bl bl (bl)2 n o 2 2 r L(bl) ¼ 1  ¼1 bl bl N l where the constraints in Section 7.7.1 for the Lagrange multiplier optimization of the segment orientation distribution function reveal that the Langevin function of argument bl (i.e., L(bl )) is equivalent to the ratio of the magnitude of the end-to-end chain vector r to the product of the total number of segments N and the length of each segment l. Similar evaluation of the ensemble average of sin2 Q with respect to fr(Q) yields ðp bl hsin Qi ¼ (sin2 Q) exp{bl cos Q} sin Q dQ 2 sinh(bl) 2

0

ðp bl ¼ (1  cos2 Q) exp{bl cos Q} sin Q dQ 2 sinh(bl) 0

¼

ðp

bl exp{bl cos Q} sin Q dQ  hcos2 Qi 2 sinh(bl) 

0

2 n r o 2 n r o ¼1 1 ¼ bl N l bl N l

Let’s summarize the results of ensemble averaging of all scalar components of the bond polarizability tensor in rectangular coordinates with respect to the Langevin segment orientation distribution function. Remember that the end-to-end vector of the entire chain, which contains N segments each of length l, is aligned with the z-axis of the fixed laboratory reference frame. On a per-segment basis, kazz l is identified as aParallel, and both kaxx l and kayy l correspond to aPerpendicular. The molecular optical anisotropy of the entire chain is given by the product of N segments and the difference

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between aParallel and aPerpendicular. One obtains the following summary: 1 2 n r o haxx i ¼ aPerpendicular ¼ (a1  a2 )hsin2 Qihcos2 fi þ a2 ¼ (a1  a2 ) þ a2 bl N l 2 1 2 n r o hayy i ¼ aPerpendicular ¼ (a1  a2 )hsin2 Qihsin2 fi þ a2 ¼ (a1  a2 ) þ a2 bl N l 2  2 n r o þ a2 hazz i ¼ aParallel ¼ (a1  a2 )hcos2 Qi þ a2 ¼ (a1  a2 ) 1  bl N l ha jk i ¼ 0; j = k Now, construct the difference between aParallel and aPerpendicular and multiply the result by N segments to obtain the optical anisotropy of the entire chain:  3 n r o N(aParallel  aPerpendicular ) ¼ N(a1  a2 ) 1  bl N l

7.7.4 Optical Anisotropy of Rigid Rod-Like Polymers in the Completely Extended Conformation Consider this unique minimum-entropy conformation of a freely jointed chain in which all N interconnected segments are aligned with the z-axis. This corresponds to the full-extension limit where the magnitude of the end-to-end vector r is equivalent to the contour length of the chain, Nl. Hence, the Langevin function of argument bl equals unity, or bl is evaluated as the inverse Langevin function when the argument is unity. Taylor series expansions of the Langevin function and its inverse are employed for numerical evaluation of the previous equation in the rigid-rod limit, where y ¼ r/Nl ) 1 and x ¼ bl: 1 x x3 2x5 x7  þ  y ¼ L(x) ¼ coth(x)    þ x 3 45 945 4725 9 297 5 1539 7 y þ y þ  x ¼ L1 ( y)  3y þ y3 þ 5 175 875 In this example, the second series expansion is applicable to evaluate bl from L 21(1). Since the series does not converge, one concludes that b ) 1, which implies that an infinitely large value of x is required for coth(x) 2 1/x to approach unity. The molecular optical anisotropy of rigid rod-like polymers is calculated from the final equation of the previous section when r ¼ Nl and b ) 1: N(aParallel  aPerpendicular ) ¼ N(a1  a2 ) Polarized optical microscopy is useful to measure the left side of the previous equation, which is directly related to birefringence as described below. Upon division by the degree of polymerization, one obtains the difference between the principal

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275

components of the bond polarizability tensor in spherical coordinates for cylindrically symmetric single bonds in the chain backbone of highly oriented polymers.

7.7.5 Birefringence Calculations via Optical Microscopy with Cross Polars Step 1: Step 2:

Place a birefringent sample on the rotating optical microscope stage using a glass slide and/or cover slip. Focus on a birefringent region of the sample. Insert the polarizer – analyzer combination into the optical path. The polarizer should be perpendicular to the analyzer.

Rotate the optical stage and find the extinction angle for the birefringent material which produces maximum darkness in the binocular field of view. Step 4: Insert the first-order waveplate into the optical path and observe transmitted light intensity in the binocular field of view. Step 5: Without rotating the optical stage, the color associated with the transmitted light in the presence of the first-order waveplate at the extinction angle from Step 3 is noted and translated into a primary wavelength using a polarization color chart, which can be found in Wood [1977] and Robinson and Bradbury [1992]. Step 3:

Step 6: Remove the first-order waveplate. Step 7: Rotate the optical stage until the maximum transmitted light intensity is observed in the binocular field of view. Insert the first-order waveplate and identify the primary wavelength of the transmitted light in the binocular field of view from Step 7 using the same polarization color chart from Step 5. Step 9: The optical path difference is calculated by subtracting these transmitted light wavelengths from Step 5 and Step 8.

Step 8:

Divide the optical path difference from Step 9 by the sample thickness to obtain the dimensionless birefringence, which is equivalent to the difference between two principal components of the refractive index tensor, nParallel – nPerpendicular. Step 11: Use the Lorentz – Lorenz equation, derived in 1880, to relate principal components of the refractive index tensor, nParallel or nPerpendicular, and the polarizability tensor, aParallel or aPerpendicular.   n2Parallel  1 4p ¼ raParallel 310 n2Parallel þ 2   n2Perpendicular  1 4p ¼ raPerpendicular 310 n2Perpendicular þ 2 Step 10:

where r is the number of molecules per unit volume, or the number density. In general, ni increases when ai is larger. To facilitate the connection

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between birefringence measurements in Step 10 and predictions of aParallel 2 aPerpendicular, expand the left side of both equations, above, about a refractive index of unity, truncate the expansions after the linear terms, and subtract the equation for nPerpendicular from that for nParallel. Simplified Analysis of the Lorentz –Lorenz Equation In the presence of an external electric field Eexternal, each molecule experiences an internal contribution Einternal from its neighbors in the polarized medium, in addition to the external field Eexternal. The Clausius – Mossotti equation, derived independently by Mossotti in 1850 and Clausius in 1879 based on an average continuum approximation [Jackson, 1975; Aklonis and MacKnight, 1983], describes the effective field within a spherical shell occupied by several molecules as Eeffective ¼ Eexternal þ Einternal ¼ Eexternal þ

4p (v) p 310

where p(v) is the polarization density of the medium with dimensions of charge per cm2, and 10 is the permittivity of free space with dimensions of charge per volt-cm. The spherical shell of interest must be sufficiently small such that p(v) is constant throughout this element of volume. The polarization density p(v) is defined as the product of the ensemble average of microscopic dipole moments per molecule with the number density of the medium r. The molecular polarizability of the medium a, with dimensions of charge-cm2/volt, allows one to calculate the ensemble average of microscopic dipole moments per molecule in the presence of an effective electric field Eeffective as follows: aEeffective. Hence, the polarization density p(v) is given by   4p (v) (v) p p ¼ raEeffective ¼ ra Eexternal þ 310   310 ra (v) Eexternal p ¼ 310  4pra Polarization density is also related to the external field via the dielectric permeability 1, where the ratio of 1 to 10 is defined as the dielectric constant of the medium (see Section 4.6.1). Hence, the Lorentz – Lorenz equation connects the polarizability a to the dielectric permeability 1 as follows:   1  10 310 ra Eexternal ¼ Eexternal p(v) ¼ 4p 310 4pra 1  10 310 ra ¼ 4p 310 4pra This relation between a and 1 can be inverted to yield 4p 10 (1  10 ) ra ¼ 3 1 þ 210 Division of each term on the right side of the previous equation by the permittivity of free space 10 introduces the dielectric constant (i.e., 1/10) of the medium, which is

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277

equivalent to the square of the refractive index n. Hence, the final form of the Lorentz – Lorenz equation relates polarizability a to the refractive index n via   4p n2  1 2  (n  1)     ra ¼ 2 310 n þ2 3

7.7.6 Optical Anisotropy of Random-Coil Polymers that Exhibit Gaussian Chain Statistics These calculations for a single chain cannot be verified experimentally unless the endto-end vector of each chain experiences the same orientation along the z-axis in the fixed laboratory frame of reference. If electrons can be delocalized more easily along an individual bond axis rather than perpendicular to this axis, such that a1 . a2, then the optical anisotropy of the entire chain, NfaParallel 2 aPerpendicularg, with respect to the z-axis of a fixed laboratory reference frame will be greater than zero, even though each freely jointed chain segment exhibits random orientation with respect to the z-axis. For Gaussian coils, the magnitude of the end-to-end vector r is significantly smaller than the contour length of the chain Nl. Hence, y ¼ r/Nl 1 and x ¼ bl ¼ L 21( y). The series expansion for the inverse Langevin function is applicable for small arguments and the series can be truncated after a few (i.e., 2) terms. The molecular optical anisotropy of random coils that obey Gaussian statistics is calculated below when the mean-square end-to-end chain distance ,r 2 . is given by Nl2 for freely jointed chains that contain N segments, and l represents the length of each segment:  3y N(aParallel  aPerpendicular ) ¼ N(a1  a2 ) 1  1 L (y) 9 297 5 1539 7 L1 ( y)  3y þ y3 þ y þ y þ  5 175 875 L1 ( y) 3 99 4 513 6  1 þ y2 þ y þ y þ  3y 5 175 875 Upon inverting L 21( y)/3y, subtracting the result from unity, and retaining only the leading term in the series, one obtains an expression for the optical anisotropy of random coils that is positive and independent of the number chain segments N when NfaParallel 2 aPerpendicularg is averaged with respect to the Gaussian distribution function, yielding the final result; 0.6 (a1 2 a2). 3y 3 36 4 108 6  1  y2  y  y þ  L1 (y) 5 175 875  3 2 36 4 108 6 y þ y þ  N(aParallel  aPerpendicular )  N(a1  a2 ) y þ 5 175 875  2   2  3 r 3 r ¼ (a1  a2 ) N(aParallel  aPerpendicular )  (a1  a2 ) 2 r2 i 5 5 h Nl

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Chapter 7 Basic Concepts and Molecular Optical Anisotropy

7.8 BIREFRINGENCE OF RUBBERY POLYMERS SUBJECTED TO EXTERNAL FORCE FIELDS Birefringence of an anisotropic material is defined as the phenomenon that occurs when the refractive index tensor exhibits directionality and the wavelengths of monochromatic light waves oscillating along two mutually perpendicular coordinate directions in the transverse plane, relative to the direction of light propagation, are different. Birefringent materials have the unique ability to depolarize electromagnetic radiation. In other words, if the incident radiation is designed to be plane-polarized, then the emerging electric field vectors in the transverse plane, perpendicular to the direction of light propagation, are elliptically polarized.

7.8.1 The Connection Between the Refractive Index Tensor in Anisotropic Media and Depolarization The physical properties of isotropic media do not exhibit directionality, which means that the properties are uniform throughout the medium and they do not adopt values that are “coordinate-direction specific.” The index of refraction describes the speed at which light travels through the medium relative to the speed of light in vacuum. Correctly speaking, the refractive index of the medium is the ratio of the speed of light in vacuum to the speed of light in the medium. In isotropic media, the wavelength of monochromatic light, although less than its wavelength in vacuum, is independent of the direction of propagation, and there is no effect on the amplitude or frequency of the oscillating electric field vectors that characterize electromagnetic radiation. By definition, refractive index is a second-rank tensor, not a scalar, in anisotropic (i.e., birefringent) media such that light travels at different speeds along different coordinate directions. Birefringence is calculated from the difference between any two principal values of the refractive index tensor, which exhibits no nonzero offdiagonal entries when the molecular axes of the material are aligned with the coordinate directions of a fixed laboratory frame of reference. For example, if monochromatic electromagnetic radiation propagates in the z-direction through a birefringent material in which the index of refraction is larger in the x-direction relative to the y-direction, then oscillating electric field vectors in the transverse plane have a shorter wavelength along the x-axis relative to their wavelength along the y-axis. Upon emerging from this birefringent material, the oscillating electric field vectors in the plane that is transverse to the direction of light propagation will be “out-of-phase” and experience destructive interference. This “phase shift” is responsible for depolarization of the incident radiation. It does not occur in isotropic media.

7.8.2

Intrinsic Birefringence

This type of anisotropic behavior occurs naturally in semicrystalline polymers that form a spherulitic superstructure, as described earlier in this chapter, as well as

7.9 Mechanical Properties of Semicrystalline Polymers

279

low-molecular-weight liquid crystalline materials that exhibit order in less than three dimensions. Optical microscopy with cross polars detects angular dependence of transmitted light in both cases.

7.8.3

Orientation Birefringence

There is no net birefringence in unstretched amorphous polymers. Birefringence develops when these materials are subjected to tensile, compressive, or shear stresses. The stress-optical properties of amorphous polymers are interesting because orientation birefringence is induced mechanically and the state of stress is measured via optical techniques. Stretched polymers are birefringent because randomly oriented chains untangle and align with the direction of the applied stress. At the molecular level, the fraction of chemical bonds in the chain backbone that is oriented preferentially in the stretch direction increases at higher strain. The Langevin segment orientation distribution function fr(Q) discussed earlier in this chapter illustrates this concept for a single chain when the magnitude of the end-to-end chain vector r corresponds to a larger fraction of the completely extended rigid-rod limit (i.e., Nl ). In other words, fr(Q ¼ 0) increases continuously as r ) Nl (see Fig. 7.3).

7.8.4

Optical Properties of Crosslinked Polymers

If one is interested in the optical properties of long-chain molecules that have been chemically crosslinked, then it is necessary to determine how the three principal components of the polarizability tensor depend on the state of strain. Two important assumptions are invoked to correlate mechanical stress and orientation birefringence in crosslinked amorphous rubbery networks. (a) Network strands between crosslink junctions are approximated by a sequence of statistically equivalent segments, where the length of each segment is known as the “Kuhn statistical segment length.” (b) The conformation of each network strand is described by a Gaussian distribution, before and after stretching, provided that the elongation ratios do not exceed 200%.

7.9 CHAIN FOLDING, INTERSPHERULITIC CONNECTIVITY, AND MECHANICAL PROPERTIES OF SEMICRYSTALLINE POLYMERS Consider the following questions that have been addressed by members of the American Chemical Society’s Division of Polymer Chemistry discussion list. The responses have been edited for clarity and to insure anonymity.

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Chapter 7 Basic Concepts and Molecular Optical Anisotropy

Question #1: As chains fold within lamellae that grow radially from the nucleation center of spherulite A, is it possible for some of these chains to undergo irregular nonadjacent reentry and enter lamellae in nearby spherulite B which impinges upon spherulite A? Question #2: If the answer to Question #1 is yes, then how do the mechanical properties of a semicrystalline polymer differ if a spherulitic superstructure forms where, in case C, chains undergo irregular nonadjacent re-entry from spherulite A to spherulite B but, in case D, chains are not shared among adjacent spherulites? Question #3: If everything else is the same except morphology, then how do the mechanical properties of a semicrystalline polymer differ where, in case E, a spherulitic superstructure forms and, in case F, lamellae are disorganized?

Eight detailed responses follow. RESPONSE #1: These questions were well-researched in the 1960s. Under all realistic conditions, chains are always shared among adjacent lamellae and among adjacent spherulites. There is direct electron microscopic evidence for this phenomenon. If chains did not connect spherulites or, during the early stages of crystallization, form entanglements with other chains in the amorphous phase, then semicrystalline polymers would have no mechanical strength and they would be entirely brittle (in fact, powdery). The references in Science and Journal of Polymer Science are extensive and superb (e.g., Keith et al. [1965, 1966]). There are other references as well (e.g., Keith et al. [1971, 1980]), but the ones that should not be missed are the first two references. RESPONSE #2: Irregular, nonadjacent re-entry folding of chains into the same lamellae or into a neighboring lamella is most common, especially in relatively stiff chain polymers (PET, PEEK, PBT, PPS, etc.), in random copolymers with noncrystallizable comonomers (ethylene-octene, ethylene-butene, ethylene-vinyl acetate, etc.), and in fairly high molar mass homopolymers. The linking of two spherulites by a single chain or by multiple chains, although it must happen, will be in the end a fairly rare event. Indeed, only a small fraction of chains find themselves at the boundary between spherulites. As mentioned below, a significant amount of noncrystalline material often gets segregated at interspherulitic boundaries. It is well known, for example, that cracks propagate in semicrystalline polymers by travelling through interspherulitic boundaries. Also, diluents, plasticizers and low molar mass or low tacticity materials present in semicrystalline polymers are often rejected between spherulites during their growth. This phenomenon certainly accounts for the fact that spherulite boundaries are the usual location where cracks propagate. Hence, everything else being equal, one expects smaller spherulites to produce better mechanical properties. This is “one” of the reasons why nucleating agents are often useful, beyond their role in increasing crystallization rate and decreasing the scattering of light (i.e., increasing transparency). In reference to Question #3, the latter morphology should give rise to better mechanical properties. In stretched semicrystalline polymer films, there are no spherulites and the mechanical properties are excellent but, in this case, lamellae are not randomly oriented. RESPONSE #3: There must be tie chains between lamellae, otherwise the mechanical properties of semicrystalline polymers would be poor. When polyolefins are stretched, chains are literally pulled out of the lamellae. There is a schematic illustration of this phenomenon in publications by Peterlin and Meinel [1971a, b]. RESPONSE #4: Polymer chain re-entry need not be adjacent, as evidenced by the “switchboard model” of crystallization. Hence, when two spherulites impinge upon each other, the

7.9 Mechanical Properties of Semicrystalline Polymers

281

last chains in the melt between them might have a good chance of connecting these spherulites. The mechanical properties such as fracture energy should be improved by these bonds. Wool [1995] discusses the role of chain entanglements in fracture energy calculations, and there is a chapter on the mechanical behavior of crystalline polymer blends. RESPONSE #5: Crystal growth in polyolefins, specifically polypropylene, depends on shearinduced effects that are different from static crystal growth seen under a microscope using hot stage techniques. The shape of crystal growth in polypropylene influences physical properties such as tensile strength, elongation, impact (i.e., Charpy, Izod, notched and unnotched, flexural modulus, etc.). The specific answers are fabrication and application dependent. Shear-induced forces and molecular weight distribution of the polymer affect morphological changes. RESPONSE #6: There are indeed tie chains that connect spherulites and form entanglements with other chains in the amorphous phase. A structure in which the chains are confined to a single spherulite will have no mechanical integrity, in the extreme case. Such a structure in an ideal case would behave like an assembly of independent particles with weak dispersion and/or electrostatic interactions holding them together. These materials will not have any strength. Tie chains between spherulites are necessary to impart strength to the material. Mechanical strength increases when there is a higher density of tie chains. A spherutic structure is brittle. A structure composed of disorganized lamellae is ductile. RESPONSE #7: The answer to Question #1 is yes, but it is not very probable. The regularity of chain folding within a particular lamella decreases with increasing crystallization rate (i.e., increasing supercooling). Hence, at high rates of crystallization, there is more irregular reentry and increasing numbers of chains that go from one lamella to another (i.e., tie chains). However, these lamellae are usually within the same spherulite because, in the course of spherulitic growth, the distance between spherulites is large compared with chain dimensions. Toward the end of the growth period, as spherulites impinge, some interspherulitic connectivity could occur, but it would only be for a very small fraction of the chains. In response to the second question, for mechanical properties such as modulus, morphological differences associated with the sharing versus confinement of a single chain are not very important. They may be important for fracture and tensile strength, where failure sometimes occurs at weak spherulite boundaries, the strength of which depends on the number of interconnecting chains. An important factor could be impurities that might accumulate at and weaken the interface. Experiments have been performed where spherulite size was purposely changed, for example, through use of nucleating agents, but the average properties such as modulus were not affected very much. If everything else is the same, then the discussion above is applicable. However, if in one case, spherulites are formed and in another, they are not, as mentioned in Question #3, then everything else is not the same. Something must be different to account for this. In most cases, it is the presence of nucleating agents, often in a not very well-controlled manner. Of course, differences in mechanical properties also depend on molecular weight and its distribution, branching, supercooling, and so on, but then “everything else” is not the same. RESPONSE #8: Lustiger et al. [1998] address some of these questions; (1) There is no reason that chains from lamellae in spherulite A cannot enter a lamella in spherulite B. This is really the basis of the above-mentioned publication that focuses on spherulite boundary strengthening as a concept to enhance toughness. (2) If chains are confined to one spherulite, then the results are catastrophic. During stress-strain testing, such a material undergoes brittle failure, interspherulitically. This is demonstrated in a series of photographs in the above-mentioned publication. (3) A random lamellar morphology displays better mechanical properties than a spherulitic morphology. As spherulites grow, they have a tendency to reject non-crystallizable chain segments,

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and this unwanted amorphous material accumulates at the spherulite boundaries acting as a locus of failure. In a random lamellar morphology, this large-scale segregation cannot take place. For example, Lustiger [1983] provides a comparison between different polyethylenes with and without spherulites.

Biographical Sketch Paolo Corradini, Professor of Chemistry at the University of Naples “Federico II,” passed away on March 1, 2006 at age 75. Corradini was truly a pioneer in polymer science and one of the “fathers” of polymer crystallography. His work on stereospecific polymerization and elucidation of the structures of the first stereoregular polymers with Giulio Natta is legendary. His determination of the 3/1 helical conformation of isotactic polypropylene in 1956 was an unprecedented tour de force because, at that time, there were no other helical synthetic polymers: the only helical structures known were the alpha helix in proteins and the double helix in DNA. Over the next five decades, Paolo continued his studies of the molecular conformation and chain packing of synthetic polymers and provided our central understanding of their crystal structures. The materials whose structures he solved, such as the polypropylenes, polystyrenes, and polybutadienes, are now among the top industrial polymers in production and use. But Corradini did not only resolve structures, he provided much of the terminology and established many of the basic principles that define how polymer chains assume specific conformations and pack in their crystalline lattices. Paolo Corradini was a wonderful and kind gentleman, always gracious, helpful, and understanding. He was full of enthusiasm, generous and inspiring in his interactions with young colleagues, and a dedicated and patient teacher. An excellent Tribute to Paolo Corradini was published by Prof. Claudio De Rosa in Macromolecules 35:7167 (2002). This tribute was written by Andrew J. Lovinger, March 3, 2006.

REFERENCES AKLONIS JJ, MACKNIGHT WJ. Introduction to Polymer Viscoelasticity, 2nd edition. Wiley, Hoboken, NJ, 1983, pp. 196– 203. BELFIORE LA, SCHILLING FC, TONELLI AE, LOVINGER AJ, BOVEY FA. Magic-angle-spinning carbon-13 NMR spectroscopy of three crystalline forms of isotactic poly(1-butene). Macromolecules 17:2561– 2565 (1984). FLORY PJ. Statistical Mechanics of Chain Molecules. Wiley, Hoboken, NJ, 1969, pp. 316–325. HEGEDUS LS. Transition Metals in the Synthesis of Complex Organic Molecules, 2nd edition. University Science Books, Sausalito, CA, 1999. JACKSON JD. Classical Electrodynamics, 2nd edition. Wiley, Hoboken, NJ, 1975, pp. 152–155. KEITH HD, PADDEN FJ, VADIMSKY RG. Intercrystalline links in bulk polyethylene. Science 150:1027 (1965). KEITH HD, PADDEN FJ, VADIMSKY RG. Intercrystalline links in polyethylene crystallized from the molten state. Journal of Polymer Science, Physics Edition (A-2) 4:267 (1966). KEITH HD, PADDEN FJ, VADIMSKY RG. Intercrystalline links: a critical evaluation. Journal of Applied Physics 42(12):4585 (1971). KEITH HD, PADDEN FG, VADIMSKY RG. The origin of inter-crystalline links. Journal of Polymer Science, Physics Edition B 18(11):2307– 2309 (1980).

Problems

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KUHN W, GRU¨N F. Relations between chain flexibility and birefringence of biaxially oriented polymeric materials via the inverse Langevin distribution [in German]. Kolloid-Zeitscnft 101:248– 271 (1942). LUSTIGER A, MARKHAM RL. The importance of tie molecules in preventing polyethylene fracture under long-term loading conditions. Polymer 24(12):1647– 1654 (1983). LUSTIGER A, MARZINSKY CN, MUELLER RR. Spherulite boundary strengthening concept for toughening polypropylene. Journal of Polymer Science, Physics Edition 36(12):2047– 2056 (1998). PETERLIN A, MEINEL G. Plastic deformation of polyethylene: mechanical properties and morphology of drawn low density polyethylene. European Polymer Journal 7(6):657– 670 (1971a). PETERLIN A, MEINEL G. Plastic deformation of polyethylene: change of mechanical properties during drawing. Journal of Polymer Science (A-2) 9(1):67–83 (1971b). ROBINSON PC, BRADBURY S. Qualitative polarized-light microscopy, in Microscopy Handbook, Volume 9. Oxford University Press, Oxford, UK, 1992, Chap. 6. STORKS KH. Electron diffraction examination of some linear high polymers. Journal of the American Chemical Society 60(8):1753–1761 (1938): courtesy of communications with Prof. AS Vaughan, University of Southampton, England. TADOKORO H. Structure of Crystalline Polymers. Wiley, NY, 1979, p. 334. WOOD AE. Crystals and Light, 2nd edition. Dover Pubications, New York, 1977, Chap. 10. WOOL RP. Polymer Interfaces: Structure & Strength. Hanser-Gardner, New York, 1995.

PROBLEMS 7.1. An isotactic vinyl polymer with repeat unit (CH2CHR) exhibits a glass transition at 80 8C and a melting transition at 200 8C. Thermal degradation does not occur below 350 8C. When this polymer is cooled from the molten state to temperatures between Tm and Tg, the time required for recrystallization is approximately 8 hours. There is not enough chain mobility for the polymer to crystallize below Tg. (a) Sketch specific volume versus temperature from 25 8C to 250 8C during the first heating trace at a rate of 25 8C per minute. The polymer is semicrystalline. (b) Sketch the response of a differential scanning calorimeter when this polymer is heated from 25 8C to 250 8C during the first heating trace at a rate of 25 8C per minute. The polymer is semicrystalline. (c) Sketch specific volume versus temperature during the first cooling trace from 250 8C to 25 8C at a rate of 50 8C per minute. After cooling from 250 8C to 25 8C at a rate of 50 8C per minute, the polymer is held isothermally at 25 8C for approximately 1 hour. Then, POM and DSC are performed. (d) Describe the field of view in a polarized optical microscope at 25 8C when the analyzer is perpendicular to the polarizer (i.e., cross-polarized light). (e) Sketch the response of a differential scanning calorimeter when this polymer is heated from 25 8C to 250 8C during the second heating trace at a rate of 25 8C per minute. After cooling from 250 8C to 25 8C at a rate of 50 8C per minute, the polymer is held isothermally at 25 8C for approximately 8 hours. Then, POM and dilatometry are performed. (f) Describe the field of view in a polarized optical microscope at 25 8C when the analyzer is perpendicular to the polarizer (i.e., cross-polarized light). (g) Sketch specific volume versus temperature when this polymer is heated from 25 8C to 250 8C during the third heating trace at a rate of 25 8C per minute. Now, a strong dc (i.e., direct current) electric field is applied to this polymer in the molten state at 250 8C. Consequently, all of the chains are oriented in the same direction in

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Chapter 7 Basic Concepts and Molecular Optical Anisotropy response to the electric field. Then, the polymer is cooled from 250 8C to 25 8C at a rate of 50 8C/min in the presence of the field. The electric field is removed 1 hour after the polymer thermally equilibrates at 25 8C. Then, POM is performed. (h) The polymer is viewed in a polarized optical microscope at 25 8C when the optic axis is aligned with the direction of the polarizer. Under the influence of an electric field, it is easiest to delocalize electrons along the optic axis. Describe the field of view in an optical microscope with cross polars when the analyzer is perpendicular to the polarizer (i.e., cross-polarized light). (i) Now the polymer is viewed in the polarized optical microscope at 25 8C when the optic axis is perpendicular to the direction of the polarizer. Describe the field of view in an optical microscope with cross polars when the analyzer is perpendicular to the polarizer (i.e., cross-polarized light). ( j) What angle of orientation of the optic axis with respect to the direction of the polarizer produces maximum transmitted light intensity for these electric-field-aligned polymer chains in an optical microscope with cross polars when the analyzer is perpendicular to the polarizer?

7.2. Derive the angular dependence of transmitted light intensity in an optical microscope when birefringent spherulites interact with unpolarized light. Electromagnetic radiation propagates in the z-direction. Electric field vectors that oscillate in the transverse plane have components of equal magnitude in the x- and y-directions. Use azimuthal angle f in spherical coordinates to describe the angular dependence of transmitted light intensity in the equatorial plane of a single spherulite. Present your results in graphical form. Answer Use the formalism developed in this chapter for birefringent crystallites that result when ordered lamellae generate a spherulitic superstructure. The interaction of electromagnetic radiation with the polarizability tensor of crystallizable chain molecules, in which the optic axis is aligned with the f-direction in spherical coordinates (i.e., aff  arr  aQQ), reduces to the following expression for the induced dipole moment: pinduced ¼

3 X

  spherical di aii Ei ) df aff Ef ¼ df aff df E



coordinates arr aQQ aff

i¼1

With respect to the direction of propagation of electromagnetic radiation (i.e., z-direction in rectangular coordinates), electric field vectors oscillate in the transverse plane, such that   E ¼ E dx þ dy This is a simplified representation of unpolarized light when the polarizer is removed from the optical path. To be consistent with previous calculations in this chapter, the analyzer samples the y-component of pinduced, which is calculated as follows:   dy pinduced  aff E dy df [(df dx ) þ (df dy )]









Assistance from the trigonometric relation; df ¼2d x sin f þ d y cos f, allows one to evaluate the required “dot” products of unit vectors in rectangular and spherical coordinates in the previous expression. Hence



dy pinduced  aff E cos ffsin f þ cos fg

Problems

285

Figure 7.4 Graphical answer to Problem 7.2: prediction of transmitted light intensity that results when a birefringent spherulite interacts with unpolarized light. Notice that there are minor lobes in the first and third quadrants (i.e., at right angles with respect to the major lobes). The amplitudes of these minor lobes are approximately 3– 4% of the amplitudes of the major lobes. The angular dependence of transmitted light intensity in the binocular field of view is obtained by squaring the magnitude of {dy pinduced}:



 2 Transmitted light intensity ¼ dy pinduced  (aff E)2 cos2 ffcos f  sin fg2



This function of azimuthal angle f in the focal plane of an isolated spherulite is illustrated in Figure 7.4 using polar coordinates with a gray-scale gradient in the radial direction. 7.3. (a) One very large lamella, grown from dilute solution with dimensions in the micron-size range, is removed from the solvent and observed in an optical microscope with cross polars. Electric field vectors that are characteristic of polarized light oscillate parallel to the axis at Q ¼ 0 in the transverse plane, and the analyzer selects the component of the induced dipole moment along the axis at Q ¼ 908. The experiment begins by aligning the major lamellar surface, which contains the folds, at Q ¼ 0. Then, the microscope stage is rotated slowly and smoothly over a range of Q from 08 to 3608. Sketch the transmitted light intensity in the binocular field of view as a function of the angle of rotation of the microscope stage. Be quantitative on the horizontal axis that tracks the angle of rotation. (b) In the second experiment that employs the same orientations of the polarizer and analyzer, a liquid crystalline material that contains rigid-rod molecules is placed in the optical microscope. All molecules in this liquid crystal are aligned such that their major optic axis is coincident with Q ¼ 0, initially. The optic axis is defined as the major coordinate direction in which electron delocalization occurs in response to an electric field. During the experiment, the microscope stage is rotated slowly and smoothly over a range of Q from 08 to 3608. Sketch the transmitted light intensity in the binocular field of view as a function of the angle of rotation of the microscope

286

Chapter 7 Basic Concepts and Molecular Optical Anisotropy stage. Be quantitative on the horizontal axis which tracks the angle of rotation. Provide a graph of experimental results for the lamella and the liquid crystalline material on the same set of axes and label the curves.

7.4. Qualitatively describe the field of view in an optical microscope when the following two experiments are performed. The sample is a semicrystalline polymer containing disorganized lamellae with random orientations that do not form a spherulitic superstructure. The microscope stage is not rotated. In both cases, light propagates in the z-direction. (a) Electromagnetic radiation is polarized along the x-direction and the component of the induced oscillating dipoles is analyzed along the y-direction. (b) Electromagnetic radiation is not polarized, the E-field vectors oscillate along the x- and y-directions, and the component of the induced oscillating dipoles is analyzed along the y-direction. 7.5. Isotactic poly(methylmethacrylate) and isotactic polystyrene are cooled below their melting temperatures, but above their glass transition temperatures, to induce crystallinity at their respective optimum annealing temperatures. In each case, a wide-angle X-ray powder diffraction photograph is obtained to investigate the crystal structure that develops. Explain why some of the powder diffraction rings for isotactic poly(methylmethacrylate) have a smaller diameter, or smaller scattering angle for coherent reflections, than those for isotactic polystyrene. Hint: Consider the “pitch,” or number of repeat units per one revolution, of the helical conformation in each polymer. Consult Tadokoro [1979] in the References Section. 7.6. Use your knowledge of semicrystalline polymers to explain the lyrics in the first verse of Polythene Pam, written by John Lennon and Paul McCartney: Well, you should see Polythene Pam She’s so good looking but she looks like a man Well, you should see her in drag dressed in her polythene bag Yes, you should see Polythene Pam 7.7. You have just completed an awesome day of bike riding in the Dolomiti mountains of northeast Italia, returning to Trento via Val di Cembra. Now, it is time to enjoy some fabulous pizza at Alessandro’s in Piedicastello, via Ponte San Lorenzo that crosses the Adige Fiume, and a bottle of Moscato Rosa Dolce from Cantina Fratelli Pellegrini in Lavis. Describe in as much morphological detail as possible, the rather expensive “crystal” wine “glass” that allows you to drink this amazing vino. Answer Crystals exhibit long-range three-dimensional order, known as crystallographic order. Glasses are amorphous solids (i.e., amorphous silicon dioxide, for example), exhibiting, at most, short-range order. Expensive crystal glasses are composed of silicon dioxide (i.e., quartz), but now the word “glass” refers to the container, which is not necessarily an amorphous solid. Silicon dioxide exists in more than 10 different crystallographic forms (alpha-quartz, beta-quartz, etc.), as well as an amorphous solid (i.e., cheap window glass and cheap wine glasses). These inexpensive forms of amorphous silicon dioxide could be described as supercooled liquids that have been quenched to temperatures where the rate of crystallization vanishes, and this temperature range is below the glass transition, so the terms “glass” and “amorphous solid” are applicable, here.

Chapter

8

Crystallization Kinetics via Spherulitic Growth From bleeding wounds, swimming visions crystallize. —Michael Berardi

Q

uantitative models of nucleation and growth rates for impinging and nonimpinging spherulites are discussed within the framework of classical thermodynamics that include effects of supercooling and fractional free volume. Mathematical analysis of the Avrami equation is useful to parameterize the kinetics of crystallization and identify actual experimental annealing temperatures where maximum rates should yield significant fractions of crystallinity. Thermodynamic descriptions of chain folding in semicrystalline-amorphous polymer – polymer blends identify optimum lamellar thicknesses when hydrogen bonding between dissimilar chains competes with the crystallization process. Flory – Huggins lattice models are employed to describe melting-point depression in binary systems. Linear least squares analysis of Tm-depression data yields quantitative information about the enthalpy of fusion and the interaction free energy of mixing (i.e., Flory – Huggins x parameter).

8.1

NUCLEATION AND GROWTH

Crystallization involves two processes that occur sequentially and spontaneously. Stable nuclei must form, such that their size is large enough to overcome a formidable barrier whose height decreases at crystallization temperatures that are significantly below Tm. The growth of these stable nuclei requires mobility that increases at higher temperature as chains in the amorphous phase reorganize and add to growing plate-like structures, known as lamellae. Since the rate of nucleation increases at lower temperature and the rate of growth increases at higher temperature, the overall process exhibits a maximum rate somewhere between the glass transition and the melting point. Crystallites are smaller and exhibit more defects in their structure when crystallization occurs at lower temperature. Physical Properties of Macromolecules. By Laurence A. Belfiore Copyright # 2010 John Wiley & Sons, Inc.

287

288

Chapter 8 Crystallization Kinetics via Spherulitic Growth

8.2 HETEROGENEOUS NUCLEATION AND GROWTH PRIOR TO IMPINGEMENT When stereoregular polymers crystallize below the melting temperature and a spherulitic superstructure develops, it is possible to measure time-dependent growth of these three-dimensional birefringent aggregates via polarized optical microscopy. Heterogeneous nucleation represents a crystallization mechanism where all “seeds” are “planted” at time t0 and lamellae grow radially outward from these nucleation centers. Hence, all spherulites should be of similar size if this mechanism dominates the crystallization process. One requirement of polarized optical microscopy is that the focal plane of the microscope must be coincident with the equatorial plane of an isolated spherulite to measure accurate growth rates. This condition will be satisfied if the spherulite of interest in the field of view is identified during its early stages of growth when it appears as a “point” at 50 to 200 magnification. The temperaturedependent growth rate G is defined by the time rate of change of the radius r of a single spherulite: G(T) ¼

dr dt

When the growth rate G(T ) is essentially constant during the early stages of isothermal crystallization, one calculates the spherulite radius as follows: ðr

ðt

r(t) ¼ dr ¼ G(T) dt0 ¼ G(T){t  t0 } 0

0

t0

The volume of one space-filling three-dimensional spherulite is V1-Spherulite ¼ 43p{r(t)}3 ¼ 43p [G(T)]3 {t  t0 }3 Since all of the nucleation centers were planted at t ¼ t0, the number density of nucleation centers NS is the same as the number of spherulites per unit volume of polymer. Hence, the volume fraction of spherulites, which represents the total volume of spherulites per volume of polymer, is synonymous with the volume fraction of crystallinity, XV(t). The simple model described above yields XV (t; T) ¼ 43pNS G3 (T){t  t0 }3 which is an accurate representation of the crystalline volume fraction during the early stages of spherulitic growth before impingement occurs. The scaling law for this process at early times (i.e., t  t0) is XV (t)  {t  t0 }n where n is the dimensionality of crystal growth. In this example, n ¼ 3. Timedependent patterned crystal growth in circular or rectangular fashion on a catalytic surface or inert substrate exhibits a scaling exponent of n ¼ 2. These simple models

8.3 Avrami Equation for Heterogeneous Nucleation

289

break down when the growth of a single spherulite is affected by neighboring spherulites, which causes growth rates to decrease.

8.3 AVRAMI EQUATION FOR HETEROGENEOUS NUCLEATION THAT ACCOUNTS FOR IMPINGEMENT OF SPHERULITES The primary objective of this section is to predict the time dependence of the volume fraction of crystallinity when spherulitic growth rates are reduced by the presence of neighboring spherulites, as illustrated in Figure 8.1 via polarized optical micrographs for poly(ethylene oxide) with a quarter-waveplate in the field of view. Let’s consider the time rate of change of the volume of a single isolated spherulite prior to impingement, as discussed in the previous section. Differentiation of V1-Spherulite yields   dV dr ¼ 4p{r(t)}2 ¼ (External surface area)(Growth rate) dt 1-Spherulite dt The following strategy is adopted after impingement occurs: Step 1:

Multiply the previous expression by the number density of spherulites NS which have been nucleated heterogeneously at time t0. This provides one with a rate equation for the volume fraction of crystallinity, because XV (t; T) ¼ NS V1-Spherulite (t; T)   dXV dV ¼ NS dt dt 1-Spherulite

Step 2:

Calculate the external surface area of a single spherulite by assuming constant growth rate during isothermal crystallization: External surface area ¼ 4p{r(t)}2 ¼ 4pG2 (T){t  t0 }2

Figure 8.1 Optical micrographs with cross polars that illustrate the impingement of spherulites in poly(ethylene oxide). Spatially dependent gray-scale intensity in both micrographs is attributed to optically anisotropic spherulites and the presence of a quarter-waveplate in the field of view.

290

Chapter 8 Crystallization Kinetics via Spherulitic Growth

Step 3: Introduce a truncation factor f, which represents the probability that impingement does not occur, to account for the fact that (i) spherulitic growth rates decrease after impingement and (ii) the external surface area of a single spherulite is not 4pfr(t)g2. The truncation factor is proportional to the volume fraction of amorphous material. Justification for this claim is based on geometrical considerations of intersecting spheres and circles. Hence, f (t)  1  XV (t) This is reasonable because the probability of impingement increases at higher crystalline volume fractions. These considerations yield the following expression for the time rate of change of the volume of a single spherulite after impingement occurs:   dV dr ¼ 4p{r(t)}2 f ðtÞ ¼ 4p[G(T)]2 (t  t0 )2 G(T)f (t) dt 1-Spherulite dt ¼ 4p[G(T)]3 (t  t0 )2 {1  XV (t)} One calculates the time dependence of the volume fraction of crystallinity from   dXV dV ¼ NS ¼ 4pNS [G(T)]3 (t  t0 )2 {1  XV (t)} dt dt 1-Spherulite Separation of variables yields the Avrami equation for heterogeneous nucleation, or the time dependence of the spherulite volume fraction during isothermal crystallization, XV(t; T ). The initial condition is XV ¼ 0 at t ¼ t0: XVð(t)

XV (t0 )

dXV ¼ 4pNS [G(T)]3 1  XV

t0ð ¼t

(t 0  t0 )2 dt 0

t0 ¼t0

ln{1  XV (t; T)} ¼ 43pNS [G(T)]3 (t  t0 )3   XV (t; T) ¼ 1  exp 43pNS [G(T)]3 (t  t0 )3 This result is consistent with predictions of XV(t; T ) prior to impingement, as discussed in the previous section. If one expands the exponential function in the Avrami equation when t  t0 during the early stages of spherulitic growth and truncates terms that scale as (t 2 t0)6 or higher, then XV (t; T)  43pNS G3 (T){t  t0 }3 þ   

Volume Fraction of Crystallinity

8.3 Avrami Equation for Heterogeneous Nucleation

291

0.8 0.7 0.6

Prior to Impingement After Impingement

0.5 Rate Constant = 10–4 minute–3 0.4 0.3 0.2 0.1 0.0 0

5 10 15 Crystallization Time (minutes)

20

Figure 8.2 Comparison of the predicted time dependence of the volume fraction of crystallinity for heterogeneous nucleation at time t ¼ 0 in the absence (i.e., triangles) and presence (i.e., circles) of spherulitic impingement when the crystallization rate constant is 1024 minute23. Measurable deviation between the two curves suggests that impingement occurs after approximately 14 minutes.

If the crystallization mechanism proceeds via heterogeneous nucleation and spherulites are observed in the optical microscope fitted with cross polars, then one predicts that impingement occurs when there is a significant difference between XV(t; T ) calculated from the previous two equations, as illustrated in Figure 8.2. If temperature-dependent crystallization rate constants, with dimensions of (time)23, are defined by KCrystallization (T) ¼ 43 pNS G3 (T) then impingement occurs earlier when KCrystallization is larger. For example, the time required for impingement to occur for two different crystallization rate constants is summarized in Table 8.1. Since macromolecules do not exhibit 100% crystallinity due to (i) the presence of stereo-irregular units along the chain when centers of chirality exist, (ii) head-to-head versus head-to-tail addition of monomer units during free-radical polymerization, (iii) irregularities in the vicinity of chain ends and the fold regions, etc., it is necessary

Table 8.1 Empirical Correlation Between Crystallization Rate Constants and the Time Required for Spherulitic Impingement to Occur KCrystallization (minute)23 1023 1024

tImpingement (minutes) 7 14

292

Chapter 8 Crystallization Kinetics via Spherulitic Growth

to modify the Avrami equation such that XV , 1 as t ) 1. Hence,   XV (t; T) ¼ 1  exp  43pNS [G(T)]3 (t  t0 )3 XV (t ) 1)

8.4 CRYSTALLIZATION KINETICS AND THE AVRAMI EQUATION FOR HOMOGENEOUS NUCLEATION OF SPHERULITES This mechanism proceeds via continuous nucleation and growth of spherulites throughout the entire crystallization process. Nucleation occurs in the amorphous molten material below the melting temperature. During the time interval from l to l þ dl, the number of spherulites nucleated in the amorphous phase per volume of polymer is given by dNS. The temperature-dependent nucleation rate is proportional to the volume fraction of amorphous material, and this rate process is assumed to proceed via first-order kinetics, as illustrated below, where the appropriate concentration variable is the amorphous volume fraction: dNS ¼ KHomogeneous (T){1  XV} dl KHomogeneous(T ) is a temperature-dependent rate constant for homogeneous nucleation, with dimensions of (volume-time)21, that decreases at higher temperature according to the developments in Sections 8.8 and 8.9. If spherulites are nucleated at time l and exhibit constant growth rate G(T ) during isothermal crystallization, then their radius at time t is r(t; l) ¼

r(t) ð r(l)¼0

dr ¼ G(T)

t0ð ¼t

dt0 ¼ G(T){t  l}

t0 ¼l

The volume of this single isolated spherulite at time t, which was nucleated homogeneously at time l, is V1-Spherulite (t; l) ¼ 43p{r(t; l)}3 ¼ 43p G3 (T){t  l}3 The incremental volume fraction of crystallinity at time t, dXV, due to spherulites that were nucleated between time l and l þ dl, is given by the product of the incremental number density of spherulites which occurred between l and l þ dl and the volume of these spherulites at time t. Hence, dXV ¼ V1-Spherulite (t; l) dNS ¼ 43p G3 (T)KHomogeneous (T){1  XV }{t  l}3 d l Now, consider all possible nucleation times from l ¼ t0 to l ¼ t. This allows one to determine the total volume fraction of spherulites during isothermal crystallization

8.5 Linear Least Squares Analysis of the Kinetics of Crystallization

293

between times t0 and t, XV(t; T ), via separation of variables with an initial condition of XV ¼ 0 at t ¼ t0: XVð(t) XV (t0 )

lмt dXV ¼ 43p [G(T)]3 KHomogeneous (T) (t  l)3 d l 1  XV l¼t0

ln{1XV (t; T)} ¼ 13p [G(T)]3 KHomogeneous (T){t  t0 }4   XV (t; T) ¼ 1  exp  13p [G(T)]3 KHomogeneous (T)(t  t0 )4 Once again, this form of the Avrami equation must be modified to account for the fact that macromolecules do not exhibit 100% crystallinity. This is accomplished as follows:   XV (t; T) ¼ 1  exp  13 p[G(T)]3 KHomogeneous (T)(t  t0 )4 XV (t ) 1)

8.5 LINEAR LEAST SQUARES ANALYSIS OF THE KINETICS OF CRYSTALLIZATION VIA THE GENERALIZED AVRAMI EQUATION Results from the previous two sections are summarized below for the time dependence of the volume fraction of spherulites during isothermal crystallization: XV (t; T) ¼ 1  exp{kn (T)t n } XV (t ) 1) where t0 ¼ 0 without loss of generality, n is the generalized Avrami exponent, and kn(T ) is the temperature-dependent crystallization rate constant with dimensions of (time)2n. The analyses above suggest that the values and expressions in Table 8.2 are appropriate for two different crystallization mechanisms. When the Avrami exponent n is greater than unity, the time dependence of the volume fraction of crystallinity follows a sigmoidal increase from zero at the inception of the process to the upper limit given by XV(t ) 1). Larger values of the Avrami Table 8.2 Comparison Between Avrami Exponents and Crystallization Rate Constants for Homogeneous and Heterogeneous Nucleation Mechanisms Crystallization mechanism

Avrami exponent

Rate constant (time)2n

Heterogeneous nucleation

n¼3

kn (T) ¼ 43p NS G3 (T)

Homogeneous nucleation

n¼4

kn (T) ¼ 13p G3 (T)KHomogeneous (T)

294

Chapter 8 Crystallization Kinetics via Spherulitic Growth

exponent and smaller rate constants correspond to a longer induction period, or time delay, before crystallization occurs isothermally at temperatures below the melting point. Nucleating agents operate via the heterogeneous mechanism and provide a foreign solid surface in the molten amorphous polymer upon which spherulites grow. Relative to homogeneously nucleated crystals, the presence of nucleating agents is consistent with smaller Avrami exponents, larger crystallization rate constants, and higher temperatures between Tg and Tm, where the maximum rate of crystallization occurs. Natural aerosols in the air originate from sea spray, dust storms, wildfires, and volcanoes. Pollution-based aerosols, primarily due to diesel exhaust and coal-burning power plants, are concentrated above industrialized cities and, together with natural aerosols, they provide nucleation sites within storm clouds where water vapor condenses into liquid droplets, subsequently forming ice crystals during winter. Hence, snowflakes that form in polluted storm clouds are generated at a relatively faster rate via this heterogeneous nucleation process, and the rate of crystallization scales linearly with the number density of aerosol particles. However, these heterogeneously nucleated snowflakes that form around pollution-based aerosols are smaller in size, undergo less frequent collisions with other water droplets in the same storm clouds and, ultimately, contain less moisture, supporting the theory that air pollution contributes to reduced snowpack in the mountains and lower spring runoff [Borys et al., 2000]. At the same temperature below the melting point of a given polymer, nucleating agents accelerate the rate of crystallization relative to the formation of homogeneously nucleated crystals. Hence, a larger volume fraction of crystallinity is expected after the process occurs for a given time t in the presence of a nucleating agent relative to XV when the nucleating agent is absent, but the upper limit of crystallinity is not affected much. Polarized optical micrographs, illustrated in Figure 8.3, reveal the effect of the nucleating agent, lithium perchlorate (i.e., LiClO4), on the spherulitic morphology and increased rate of crystallization in poly(ethylene oxide). Crystallization time and temperature (i.e., ambient) are invariant in both micrographs.

Figure 8.3 Optical micrographs of spherulites in poly(ethylene oxide), viewed with cross polars, illustrating the effect of a nucleating agent (i.e., LiClO4) on the development of several smaller spherulites at primary interspherulitic boundaries.

8.5 Linear Least Squares Analysis of the Kinetics of Crystallization

295

Experimentally, one measures the rate at which thermal energy is liberated during isothermal crystallization via DSC when the calorimeter operates in constanttemperature mode. Hence, quantitative information about XV(t; T ) is available from differential scanning calorimetry (DSC), regardless of whether semicrystalline polymers form spherulites or disordered lamellae. When the calorimeter operates isothermally, it provides information about the rate at which thermal energy must be added to or removed from the sample to maintain constant temperature. Since the formation of crystallites from the amorphous molten material is an exothermic process, the calorimeter removes thermal energy from the sample at the same rate at which it is generated by the crystallization process. The sample behaves similar to a constant-volume batch reactor. Detailed kinetic analysis in the next chapter reveals that the time rate of change of XV is proportional to the rate at which thermal energy must be removed during isothermal operation. Knowledge of the heat of fusion for the semicrystalline polymer under investigation, and integration of the DSC output curve versus time yield XV(t; T ). Experimental determination of the Avrami exponent n and the crystallization rate constant kn(T ) is achieved via linear least squares analysis of the generalized Avrami equation after performing the following algebraic rearrangement: XV (t; T) ¼ exp{kn (T)t n } XV (t ) 1)   XV (t; T) ¼ kn (T)t n ln 1  XV (t ) 1)    XV (t; T) ¼ ln{kn (T)} þ n ln t ln  ln 1  XV (t ) 1) 1

The experimental DSC data from isothermal crystallization at temperature T are processed as follows, excluding the initial data point at XV(t ¼ 0; T ) ¼ 0 Step 1: Integrate the isothermal DSC output curve at several different crystallization times t and divide the integrated results by the tabulated heat of fusion, as well as the initial mass of amorphous material, to determine XV(t; T ). It is necessary to account for the fact that calorimetry provides a measure of the mass fraction of crystallinity, whereas the Avrami equation was developed in terms of the volume fraction of spherulites. Hence, crystalline and amorphous densities are also necessary to obtain accurate values of XV(t; T ) from the isothermal DSC output curve. Perform linear least squares analysis of XV(t; T ) using a first-order polynomial. In other words, the model is y(x) ¼ a0 þ a1x. Step 3: The independent variable is x ¼ ln t, where crystallization time t appears on the horizontal axis of the DSC output curve. Step 4: The dependent variable is    XV (t; T) y ¼ ln ln 1  XV (t ) 1)

Step 2:

296

Chapter 8 Crystallization Kinetics via Spherulitic Growth

Step 5:

The first-order coefficient a1 (i.e., slope) in the model is the generalized Avrami exponent n.

Step 6:

The zeroth-order coefficient a0 (i.e., intercept) in the model is related to the crystallization rate constant kn(T ) as follows: a0 ¼ ln{kn (T)}

8.6 HALF-TIME ANALYSIS OF CRYSTALLIZATION ISOTHERMS Once the generalized Avrami exponent is known from linear least squares analysis of XV(t; T ), as described above, one should calculate the crystallization rate constant from half-time analysis and compare the result with Step 6 in the previous section. The half-time, t1/2, is defined as the time required to achieve 50% of the maximum crystallinity during an isothermal experiment. Hence, XV (t ¼ t1=2 ) ¼ 12 XV (t ) 1) Rearrangement of the generalized Avrami equation allows one to calculate kn(T ) if the exponent n is known: XV (t ¼ t1=2 ) 1 n ¼ ¼ 1  exp{kn (T)t1=2 } 2 XV (t ) 1) kn (T) ¼

ln 2 n t1=2

The Avrami exponent n can be verified by calculating the slope of a normalized crystallization isotherm at t1/2. The development proceeds as follows: XV (t; T) ¼ 1  exp{kn (T)t n } XV (t ) 1)    d XV (t; T) n1 n ¼ nkn (T)t1=2 exp{kn (T)t1=2 } dt XV (t ) 1) t¼t1=2 The previous expression for the rate of crystallization is simplified via the definition of t1/2:    d XV (t; T) n ln 2 ¼ dt XV (t ) 1) t¼t1=2 2t1=2 Hence, n¼

   2t1=2 d XV (t; T) ln 2 dt XV (t ) 1) t¼t1=2

8.7 Maximum Rate of Isothermal Crystallization

297

Half-time analysis and linear least squares analysis should yield consistent results and allow one to quantify isothermal rates of crystallization via the generalized Avrami equation.

8.7 MAXIMUM RATE OF ISOTHERMAL CRYSTALLIZATION Based on the generalized Avrami equation for the time dependence of the volume fraction of crystallinity, one calculates the isothermal rate of crystallization as follows:   d XV (t; T) ¼ nkn (T)t n1 exp{kn (T)t n } dt XV (t ) 1) For a crystallization mechanism in which the exponent n . 1, the crystallization rate approaches zero at the beginning of the process (i.e., t )0) and when the process nears completion (i.e., t ) 1). The maximum rate of isothermal crystallization occurs at time tmax, which corresponds to the inflection point of the sigmoidal-shaped curve of XV versus time, where the second derivative of XV(t; T ) vanishes. Hence, tmax is calculated from the following equation:   d2 XV (t; T) ¼ nkn (T)t n2 [n1 nkn (T)t n ] exp{kn (T)t n } ¼ 0 dt2 XV (t ) 1) The rate of isothermal crystallization proceeds quickest when   n1 1=n tmax ¼ nkn (T) The half-time t1/2 is coincident with the inflection point of XV versus time when the Avrami exponent n ¼ (1 2 ln 2)21 ¼ 3.26, which implies that t1/2 ¼ tmax. The half-time is greater than tmax when n , 3.26, and t1/2 occurs prior to the inflection point of XV versus time when n . 3.26, but differences between t1/2 and tmax could be difficult to resolve for reasonable values of n between 3 and 4. These Table 8.3 Effect of the Avrami Exponent n on the Half-Time and tmax when kn ¼ 1023 (minute)2n Avrami exponent, n 2 3 3.26 4 5

Half-time (minutes)

tmax/t1/2

26 8.8 7.4 5.1 3.7

0.85 0.99 1.00 1.02 1.03

298

Chapter 8 Crystallization Kinetics via Spherulitic Growth

trends are summarized in Table 8.3. The maximum rate of crystallization is evaluated when t ¼ tmax:    d XV (t; T) n1 n ¼ nkn (T)tmax exp{kn (T)tmax } dt XV (t ) 1) t¼tmax ¼ w(n){kn (T)}1=n where

w(n) ¼ (n1)

(n1)=n 1=n

n

  n1 exp  n

and w ¼ 1 when n ¼ 1. One indentifies fkn(T )g1/n as the crystallization rate constant with dimensions of (time)21. Temperature dependence of this rate constant allows one to predict the optimum crystallization temperature (i.e., TC) where the kinetics of crystallization proceed most rapidly. The following sections address the temperature dependence of kn(T ). Since half-time measurements provide a route to calculate kn(T ), the important results of this section are summarized in terms of t1/2:   n1 1=n t1=2 tmax ¼ n ln 2 and the maximum rate of crystallization is

1.00

0.05

0.80

0.04 Volume Fraction of Crystallinity Rate of Crystallization

0.60 0.40

0.02 0.01

0.20 0.00

0.03

0

16

32 48 64 Crystallization Time (minutes)

Rate of Crystallization (min–1)

Volume Fraction of Crystallinity

   d XV (t; T) w(n){ln 2}1=n ¼ w(n){kn (T)}1=n ¼ t1=2 dt XV (t ) 1) t¼tmax

0.00 80

Figure 8.4 Predictions from the Avrami equation for heterogeneous nucleation followed by two-dimensional crystal growth (i.e., n ¼ 2) when the crystallization rate constant is 1023 minute22. The normalized volume fraction of crystallinity is evaluated with respect to the maximum possible crystalline volume fraction that is achieved at infinite time. The rate of crystallization has dimensions of inverse minutes. The crystallization half-time is t1/2  26 minutes, and the maximum rate of crystallization occurs at 85% of t1/2.

8.8 Thermodynamics and Kinetics of Homogeneous Nucleation

Volume Fraction of Crystallinity Rate of Crystallization

0.80

0.50 0.40

0.60

0.30

0.40

0.20

0.20

0.10

0.00

0

2

4 6 8 Crystallization Time (minutes)

Rate of Crystallization (min–1)

Volume Fraction of Crystallinity

1.00

299

0.00 10

Figure 8.5 Predictions from the Avrami equation for homogeneous nucleation (i.e., n ¼ 4) when the crystallization rate constant is 1023 minute24. The normalized volume fraction of crystallinity is evaluated with respect to the maximum possible crystalline volume fraction that is achieved at infinite time. The rate of crystallization has dimensions of inverse minutes. The crystallization half-time is t1/2  5.1 minutes, and the maximum rate of crystallization occurs at 102% of t1/2.

Sigmoidal-shaped crystallization isotherms are illustrated in Figures 8.4 and 8.5 for two values of the Avrami exponent (i.e., n ¼ 2 and n ¼ 4), together with a comparison between the half-time and the time at which the rate of crystallization proceeds the fastest when the crystallization rate constant is 1023 (minute)2n. (See Table 8.3.) Larger relative differences between t1/2 and tmax exist, for n ¼ 2, when the crystallization isotherm exhibits significant asymmetry about the inflection point where t ¼ tmax.

8.8 THERMODYNAMICS AND KINETICS OF HOMOGENEOUS NUCLEATION Consider nucleation and spontaneous growth of a spherulite of radius r that accompany the following “reaction”: Molten amorphous polymer () Spherulite There are at least two processes that contribute to the extensive total free energy change, DGSpherulite, for the formation of a spherulite. (i) A favorable volumetric process given by 4 3 3p r Dg

¼ 43p r3 {gSpherulite gAmorphous}

300

Chapter 8 Crystallization Kinetics via Spherulitic Growth Table 8.4 Effect of Temperature on the Volumetric Free Energy Density Difference Between Spherulites and Molten Amorphous Material Dg ¼ gSpherulite 2 gAmorphous

Temperature T ¼ Tmelt T , Tmelt T . Tmelt

0 ,0 .0

In the previous equation, g represents a free energy density (i.e., energy/ volume) for each specific phase in the above-mentioned “reaction.” Since the favored phase corresponds to the one with lower free energy, the qualitative temperature dependence of Dg given in Table 8.4 is appropriate. For typical crystallization temperatures below the melting point, Dg is negative and this volumetric process represents a favorable contribution to DGSpherulite. (ii) An unfavorable surface-related process given by 4p r 2 s where s is the surface free energy change (i.e., energy/area) that accompanies the formation of “foreign” spherulitic surfaces in the molten amorphous phase. This process is unfavorable because s . 0. Now, it is possible to predict the total free energy change for the formation of a spherulite of radius r. However, it should be emphasized that this model for the Gibbs free energy change that accompanies spherulite formation, and the next model that considers the addition of an N-stem crystal layer to a growing lamella, do not capture effects of (i) chain microstructure (i.e., configurational isomers), (ii) packing of left- and right-handed helices in the unit cell, or (iii) chain branching: DGSpherulite ¼

3 4 3pr Dg

  rDg þs þ 4p r s ¼ 4p r 3 2

2

For crystallinity to develop in the molten amorphous phase below Tmelt, nucleation and subsequent growth of spherulites should occur spontaneously. The corresponding mathematical statement from thermodynamics is 

@ DGSpherulite @r

 ,0 T

such that radial growth occurs in a regime where the total free energy decreases continuously. There is a thermodynamic driving force to minimize the total free energy change that accompanies spherulite formation, and this driving force is assisted by the fact that spheres exhibit maximum volume-to-surface ratio, relative to flat plates

8.8 Thermodynamics and Kinetics of Homogeneous Nucleation

301

or long cylinders. When crystallization occurs below Tmelt where Dg , 0, DGSpherulite versus r exhibits a double root at r ¼ 0, and another root at rroot ¼

3s Dg

During isothermal crystallization, the slope of DG versus r is given by   @ DGSpherulite ¼ 4p r(r Dg þ 2s) @r T and the second derivative of DG versus r is  2  @ DG ¼ 8p (r Dg þ s) Spherulite @r 2 T Hence, zero slopes occur at r ¼ 0, and rcritical ¼

2s Dg

The zero slope at r ¼ 0 represents a local minimum in DGSpherulite. The extremely important zero slope at rcritical corresponds to a maximum in DGSpherulite, denoted by DGcritical. Hence, random thermal fluctuations in the amorphous phase below Tmelt must provide sufficient thermal energy to overcome the barrier represented by DGcritical such that spontaneous spherulitic growth can occur. The magnitude of this energy barrier is 3 2 Dg þ 4prcritical s¼ DGcritical ¼ DGSpherulite (r ¼ rcritical ) ¼ 43prcritical

16ps 3 3(Dg)2

Hence, random thermal fluctuations due to Brownian motion in the molten amorphous phase below Tmelt must overcome DGcritical for chain segments to (i) rearrange conformationally, (ii) adopt the appropriate sequence of rotational isomeric states within the unit cell, (iii) undergo chain folding in a lamellar structure, and (iv) exhibit spontaneous spherulitic growth via homogeneous nucleation. The temperature dependence of this energy barrier is obtained by considering the free energy density difference between a spherulite and the amorphous molten polymer below the melting point: Dg ¼ gSpherulite  gAmorphous ¼ Dh  T Ds where the enthalpy change (i.e., Dh) and the entropy change (i.e., Ds) are negative because the isothermal crystallization process is exothermic and chains are more ordered in the crystallites relative to the disordered amorphous phase. Since Dg vanishes at the melting temperature, the ratio of (2Dh) to (2Ds) is given by   Dh ¼ Tmelt Ds @Tmelt

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Chapter 8 Crystallization Kinetics via Spherulitic Growth

This ratio is assumed to remain the same at other temperatures where Dg does not vanish. Hence, the temperature dependence of Dg is approximated as follows:         Ds T Tmelt  T Dg ¼ Dh 1  T ¼ Dh ¼ Dh 1  Dh Tmelt Tmelt This temperature dependence of Dg allows one to predict the energy barrier that must be surpassed and the critical spherulite size above which spontaneous growth occurs: DGcritical ¼ rcritical ¼

2 16ps 3 Tmelt 2 3(Dh) {Tmelt  T}2

2sTmelt (Dh){Tmelt  T}

The “degree of supercooling” is defined as Tmelt 2 T, when crystallization occurs at temperature T. Now, it should be obvious that homogeneous nucleation of spherulites at the melting temperature is highly unfavorable because the energy barrier is infinitely high and the critical spherulite size is infinitely large. This is synonymous with the fact that the rate of homogeneous nucleation is vanishingly small. Nucleation rates increase at lower crystallization temperatures because the barrier height decreases and the critical spherulite size is smaller. However, the final spherulite size is smaller when crystallization proceeds at lower temperatures, due to increased surface free energy, and the melting temperature of these smaller “imperfect” spherulites is lower than that of spherulites that form at higher temperature. The crystallite imperfection content is greater and Tmelt is lower when crystallization occurs more rapidly at lower temperature.

8.9 TEMPERATURE DEPENDENCE OF THE CRYSTALLIZATION RATE CONSTANT The maximum rate of isothermal crystallization is proportional to the crystallization rate constant with dimensions of inverse time, fkn(T )g1/n, as described in Section 8.7. Two different models are proposed to simulate the effect of temperature on rates of crystallization, in agreement with the following qualitative trends. (i) The rate constant for homogeneous nucleation KHomogeneous increases at lower temperature. Hence, d KHomogeneous , 0 dT (ii) The energy barrier that must be surpassed to allow spontaneous growth of homogeneously nucleated spherulites increases at higher temperature, as one approaches Tm. Hence, d DGcritical . 0 dT

8.9 Temperature Dependence of the Crystallization Rate Constant

303

(iii) Spherulitic growth rates, defined by G ¼ dr/dt, increase at higher temperature. This growth process requires sufficient mobility for disordered chain segments in the amorphous phase to reorganize conformationally and adopt the appropriate rotational isomeric states in the crystal. An increase in fractional free volume at higher temperature allows faster growth rates above the glass transition. Hence, dG .0 dT Rates of crystallization are growth-rate controlled at low temperatures and nucleation-rate controlled at higher temperatures. A window of plausible temperatures to induce crystallization exists between Tg and Tmelt, and one expects vanishingly small rates at both of these phase transition temperatures. The first model combines all of the above-mentioned effects into one temperature-dependent exponential function. For temperatures between Tg and Tmelt, where a maximum in the rate of crystallization exists,   DGcritical (T) {kn (T)}1=n  exp z(T Tg )a The exponent a is positive and the constant z has dimensions of energy/K a. This functional form for the crystallization rate constant yields a slower rate of crystallization when the energy barrier for spontaneous nucleation increases as one approaches the melting temperature, and when mobility decreases at lower temperature. It is not true that conformational rearrangements are “frozen” and chain mobility ceases completely at the glass transition temperature, but the concept of inducing crystallization in macromolecules below Tg is somewhat contradictory because glasses exhibit exceedingly large motional time constants. The second model accounts for temperature-dependent nucleation and growth rates separately. The rate of homogeneous nucleation is modeled as an activated process with a temperature-dependent energy barrier given by DGcritical(T ). Hence,   DGcritical (T) 1=n {KHomogeneous (T)}  exp RT where R is the gas constant. Temperature-dependent growth rates are modeled analogous to rate processes like diffusion and molecular motion in the vicinity of the glass transition, resulting from a redistribution of useful free volume (i.e., not interstitial free volume) that is easy to manipulate without an activation barrier: {G3 (T)}1=n ¼

 3=n   dr f   exp dt f (T)  f (Tg )

where f(T ) represents fractional free volume and f  is the critical fractional free volume required for conformational reorganization via rotation about single bonds

304

Chapter 8 Crystallization Kinetics via Spherulitic Growth

in the backbone as chains in the amorphous phase diffuse, reptate, fold, and adopt the correct conformation for crystallization within lamellae. A reasonable value for f  is 0.025, which corresponds to 2.5% relative empty space between chains at the glass transition. Hence, {kn (T)}1=n  {G3 (T)KHomogeneous (T)}1=n     DGcritical (T) f  exp  exp f (T)  f (Tg ) RT

8.10 OPTIMUM CRYSTALLIZATION TEMPERATURES: COMPARISON BETWEEN THEORY AND EXPERIMENT Based on two different models for temperature-dependent crystallization rate constants in the previous section and the maximum rate of isothermal crystallization at t ¼ tmax in Section 8.7 via the generalized Avrami equation, it is possible to predict the optimum annealing temperature TC where crystallization proceeds fastest. This occurs between Tg and Tmelt. One calculates TC from the following extremum condition:    d d XV (t; T) d ¼ w(n) [{kn (T)}1=n ] ¼ 0 dT dt XV (t ) 1) @tmax dT The first model from the previous page yields " # az(TC Tg )a1 (Tmelt TC )2 2z(TC Tg )a (Tmelt TC ) 1=n 2 ¼0 w(n){kn (TC )} gTmelt {z(TC Tg )a (Tmelt TC )2 }2

g¼

16ps 3 3(Dh)2

The parameter g has dimensions of energy. For a . 1, the previous equation is satisfied when TC is either Tg or Tmelt, which corresponds to the absolute minimum rate of crystallization. The maximum rate of crystallization occurs at annealing temperature TC when the following algebraic equation is satisfied: az(TC Tg )a1 (Tmelt TC )2 2z(TC Tg )a (Tmelt TC ) ¼ 0 This yields an optimum crystallization temperature given by TC ¼ Tg þ

a (Tmelt Tg ) 2þa

which agrees with empirical observations when a ¼ 4. Hence, the optimum annealing temperature is TC ¼ Tg þ 23 (Tmelt Tg )

8.10 Optimum Crystallization Temperatures

305

The second model that accounts for temperature-dependent nucleation and growth separately yields   2 d f  gTmelt  w(n){kn (TC )}1=n ¼0 dT Da(T Tg ) RT(Tmelt T)2 One calculates the optimum annealing temperature TC from   f R(Tmelt TC )2  2RTC (Tmelt TC ) 2 ¼0 þ gTmelt Da(TC Tg )2 [RTC (Tmelt TC )2 ]2 which simplifies to the following nonlinear algebraic equation for TC: f 2 g 2 T (Tmelt TC )3 ¼ Tmelt (3TC Tmelt )(TC Tg )2 Da C R where the discontinuous increment in the coefficient of thermal expansion at the glass transition is Da  4.8  1024 K21. To illustrate the correspondence between Tg, TC, and the free energy parameter g for optimum crystallization kinetics, if Tmelt ¼ 523 K, Tg varies from 173 K to 423 K, and TC assumes its optimum value based on the first model with a ¼ 4, then the previous equation, which represents the extremum condition for the second model, reveals that g/R decreases linearly as Tg and TC increase. See Table 8.5. To illustrate the relation between Tmelt, TC, and g for optimum crystallization kinetics, if Tg ¼ 223 K, Tmelt varies from 273 K to 523 K, and TC assumes its optimum value based on the first model with a ¼ 4, then the previous equation reveals that g/R increases as Tmelt and TC increase, but with slightly negative curvature. (See Table 8.6.) Perhaps the best application of the previous equation uses experimental values for Tg, Tmelt, and g for semicrystalline polymers to identify the optimum crystallization Table 8.5 Relation Between the Free Energy Parameter g for Spherulite Formation and the Glass Transition Temperature of a Semicrystalline Polymer with Tmelt ¼ 523 K and Optimum Crystallization Temperature Given by TC ¼ Tg þ 23 [Tmelt  Tg ]

g/R (K) 1.32 1.21 1.11 1.00 0.89 0.78 0.67 0.56 0.40

Tg (K)

TC (K)

173 203 233 263 293 323 353 383 423

406 416 426 436 446 456 466 476 490

306

Chapter 8 Crystallization Kinetics via Spherulitic Growth Table 8.6 Relation Between the Free Energy Parameter g for Spherulite Formation and the Melting Temperature of a Semicrystalline Polymer with Tg ¼ 223 K and Optimum Crystallization Temperature Given by TC ¼ Tg þ 23 [Tmelt  Tg ]

g/R (K) 0.39 0.55 0.68 0.79 0.88 0.95 1.02 1.08 1.14

Tmelt (K)

TC (K)

273 303 333 363 393 423 453 483 523

256 276 296 316 336 356 376 396 423

temperature. For example, consider a semicrystalline polymer in which Tg  373 K and Tmelt  523 K. The optimum annealing temperature, where the kinetics of crystallization proceed most rapidly, is 473 K based on the first model with a ¼ 4. The second model reveals that this optimum crystallization temperature decreases nonlinearly (i.e., with positive curvature) as the energy parameter g increases. This effect is illustrated in Figure 8.6. Notice that the second model predicts a range of crystallization temperatures, above and below TC,optimum ¼ 473 K based on the first model, for reasonable values of the energy parameter g. If there is no surface free energy change

Crystallization Temperature (Kelvin)

530 520 510 500 490 480 470 460 450 0.00

0.35

0.70 1.05 1.40 Energy Parameter, g /R (Kelvin)

1.75

2.10

Figure 8.6 Effect of the surface free energy parameter s (and g) on the optimum crystallization temperature for a semicrystalline polymer (i.e., isotactic polystyrene) in which Tg ¼ 373 K and Tmelt ¼ 523 K.

8.11 The Energetics of Chain Folding in Semicrystalline Polymer –Polymer Blends

307

(i.e., penalty) that accompanies the formation of spherulites in the amorphous phase (i.e., both s and g vanish), then the energy barrier DGcritical ) O and the previous simulations reveal that the optimum crystallization temperature coincides with Tmelt. Obviously, this is an ideal situation that doesn’t occur in practice. In-depth discussions of the Lauritzen –Hoffman theory of cystallization growth rates can be found in the following references: Lauritzen and Hoffman [1960], Davis et al. [1976], and Pillai et al. [2000].

8.11 THE ENERGETICS OF CHAIN FOLDING IN SEMICRYSTALLINE POLYMER –POLYMER BLENDS THAT EXHIBIT MULTIPLE MELTING ENDOTHERMS 8.11.1

Overview

The melting and crystallization behavior of semicrystalline homopolymers has received much attention both in academic and industrial research because the morphological aspects of polymer crystals and spherulites influence mechanical, optical, and other macroscopic properties. Changes in crystallization or melting may cause dramatic variations in the properties of engineering materials from an application and processing viewpoint. An interesting feature of melting behavior, the presence of multiple endotherms, has been observed in some semicrystalline homopolymers such as polyethylene, poly(ethylene oxide), poly(butylene terephthalate), and poly(ether ether ketone). It has been suggested that multiple melting endotherms are caused by the following: (i) recrystallization and subsequent melting of as-formed unstable crystals that melt initially at relatively low temperatures; (ii) melting of crystals that have different lamellar thicknesses and/or different spherulitic superstructures; (iii) melting of quasi-stable crystals that have been produced from different crystallization conditions and thermal histories, with subsequent recrystallization to a more stable form; and (iv) melting of crystals that have an inequivalent distribution of internal defects and/or surface defects. Different tie-molecule segment lengths between lamellae could affect the mobility of the intercrystalline component and contribute to the conformational entropic stabilization or destabilization of the crystallites. Localized surface free energy differences also produce multiple melting, and stereo-irregular defects produce different melting temperatures. Tmelt associated with the segments that contain a few stereoirregular units can be increased by thermal treatment. This is a concept that has been verified experimentally for segmented polyurethane block copolymers. Suggestions (i) and (ii) mentioned above represent the most probable origins of multiple melting behavior. Kinetic models have also been proposed to explain multiple melting. A systematic study of multiple melting in partially compatible polymer – polymer blends has not been documented sufficiently, despite the fact that semicrystalline

308

Chapter 8 Crystallization Kinetics via Spherulitic Growth

blends of two macromolecules represent an important research area—namely, the chemical design of multicomponent systems. The thermodynamic driving force for compatibility in many blends results from the exothermic energetics of specific interactions, such as hydrogen bonding, interpolymer complexation, and transition-metal coordination because the entropic contribution to chemical stability is extremely small when the component molecular weights are large. Strong interactions between dissimilar functional groups in the amorphous phase of a crystalline/amorphous polymer – polymer blend could produce changes in crystalline morphology and surface free energy on the periphery of the lamellae (see Section 8.11.3). Consequently, the melting transitions of the semicrystalline polymer could be distorted due to the presence of the energetically attractive amorphous component in the mixtures.

8.11.2 Dependence of Multiple Melting Endotherms on the Chemical Structure of the Crystallizable Component in Polymer –Polymer Blends Multiple melting behavior has been observed recently for a variety of strongly interacting polymer blends in which the crystallizable component is either poly(ethylene oxide), poly(vinylidene fluoride), or main-chain polyesters. It was postulated that hydrogen-bond strength and specificity might have a strong influence on crystallization and melting, leading to complex heating traces in the differential scanning calorimeter (Fig. 8.7). Polyesters with different [COO]/[CH2] molar ratios have been blended with the hydrogen-bond donor poly(vinylphenol), PVPh, to investigate multiple melting and the effects attributed to hydrogen bonding. The crystalline morphology of main-chain polyesters is sensitive to hydrogen-bond strength and composition. This sensitivity, coupled with the fact that, in all of the main-chain polyesters chosen, the carbonyl group is equally accessible to the hydroxyl functional group of PVPh, allows for a systematic study of the effect of hydrogen-bond concentration on multiple melting phenomena. Blends of PVPh with a variety of main-chain polyesters reveal unique melting behavior in each case. High-resolution solid state carbon-13 NMR and FTIR spectroscopic data have identified strong interactions between the hydroxyl functional group of PVPh and carbonyl oxygens in the polyester repeat unit. Association between PVPh hydroxyl groups and the polyester carbonyl moiety is rather strong, although weaker than the self-association of hydroxyl groups in undiluted PVPh. This attraction between functional groups in dissimilar chain segments can be detected indirectly via highresolution NMR spectroscopy of the carbon-13 site that is only one bond removed from the lone pairs of electrons on oxygen. Figure 8.8 reveals NMR chemical shift lineshapes for the carbonyl carbon (i.e., between 170 and 180 ppm) in high-resolution solid state spectra of semicrystalline/amorphous polymer – polymer blends of poly (vinylphenol) with (a) poly(ethylene adipate), PEA; (b) poly(1,4-butylene adipate), PBA; and (c) polycaprolactone, PCL. The carbonyl 13C NMR signal in the crystalline domains exhibits a full-width-athalf-height of 1 – 2 ppm when the glass transition temperature of the blends is below the temperature of the NMR experiment (i.e., indicated by solid triangles in the spectra

8.11 The Energetics of Chain Folding in Semicrystalline Polymer –Polymer Blends (a)

(b)

309

Heating rate 10 °C/min 50

80 wt % PEA

55 60

70 wt % PHMS

90 wt % PBA

65 Endotherm

Endotherm

70 wt % PCL

70 75 80 85 90

60 wt % PES 95 100 PES wt % 10

30

50 70 90 Temperature (°C)

110

20

40 60 80 100 120 Temperature (°C)

Figure 8.7 (a) DSC thermograms illustrating multiple melting behavior in solid state binary mixtures of poly(vinylphenol), MW ¼ 30 kDa, with (i) 60 wt % poly(ethylene succinate), PES, MW ¼ 11 kDa; (ii) 90 wt % poly(1,4-butylene adipate), PBA; (iii) 70 wt % poly(hexamethylene sebacate), PHMS, MW ¼ 62 kDa; (iv) 70 wt % polycaprolactone, PCL, MW ¼ 32 kDa; and (v) 80 wt % poly(ethylene adipate), PEA, MW ¼ 11 kDa. (b) Multiple melting behavior in binary solid state mixtures of poly(vinylphenol) and poly(ethylene succinate), at various blend compositions indicated at the left of each endotherm [Qin and Belfiore, 1990].

Figure 8.8 Dipolar decoupled carbon-13 NMR spectra of (a) poly(ethylene adipate), MW ¼ 11 kDa, (b) poly(1,4-butylene adipate), and (c) polycaprolactone, MW ¼ 32 kDa in the carbonyl carbon chemical shift region via cross-polarization and magic-angle spinning, in blends with poly(vinylphenol), MW ¼ 30 kDa. High-resolution spectra with triangles correspond to isotropic carbonyl 13C NMR lineshapes that reflect polyester chain mobility at experimental temperatures that are above the glass transition temperature, rather than the presence or absence of a polyester-rich crystalline phase [Belfiore et al., 1993].

310

Chapter 8 Crystallization Kinetics via Spherulitic Growth

in Fig. 8.8). In all cases, a single composition-dependent glass transition temperature is measured by DSC, which increases monotonically from below ambient for polyesterrich blends to well above ambient for blends that are rich in poly(vinylphenol). When the concentration of the amorphous proton donor, PVPh, is sufficient to thwart crystallization of the polyester and increase the glass transition temperature of the blends above the temperature of the NMR experiment (i.e., 15 8C), the linewidth of the carbonyl resonance increases three- to fourfold (i.e., 5 –6 ppm). When the blends are completely amorphous and the glass transition temperature is above 15 8C, the polyester carbonyl 13C lineshape reveals at least two morphologically inequivalent microenvironments. A partially resolved carbonyl signal in rigid amorphous blends is (i) identified at higher chemical shift relative to the crystalline resonance and (ii) attributed to hydrogen bonding in the amorphous phase. This interaction-sensitive hydrogen-bonded carbonyl signal, which is rather broad, accounts for an increasing fraction of the overall NMR absorption envelope of the carbonyl carbon site when the polyester is saturated with PVPh. It has been suggested that surface energetics due to hydrogen bonds between semicrystalline and amorphous blend components in the vicinity of the fold surface, alter the crystallization, melting, and recrystallization processes of the polyester. The existence of hydrogen bonds on the lamellar surface favorably decreases the surface free energy of the crystallites. This interaction at the interface between crystalline and amorphous domains stabilizes the formation of rather thin lamellae with more chain folds. Consequently, thinner lamellae are consistent with (i) more surface area per unit volume between dissimilar segments of the crystalline and amorphous components and (ii) higher probability that the interacting functional groups will form hydrogen bonds and provide energetic stabilization for partial mixing of two high-molecular-weight polymers.

8.11.3 Theoretical Considerations that Account for Strong Interactions on the Periphery of the Crystallites With the aid of Hoffman’s theory that explains homopolymer crystallization [Lauritzen and Hoffman, 1960; Davis et al., 1976; Pillai et al., 2000], a simple modification is proposed to illustrate how multiple melting endotherms might be generated for the semicrystalline component in strongly interacting blends. The formalism begins by postulating the functional dependence of the Gibbs free energy of formation for an N-stem crystal layer, DGN-stem, upon cooling below Tmelt from the molten state. An N-stem crystal layer is defined as a polymer chain that folds (N21) times as it adds to the lateral surface of a growing lamella. As lamellae grow, chain folding of crystal stems on the lamellar surface is consistent with a reduction of the overall free energy for crystal growth. Another factor that contributes to chain folding is a reduction in packing density of the chain in the vicinity of the fold surface. This reduction in density allows for easy rotation about carbon – carbon backbone bonds, which is required for chain folding. If it can be demonstrated that this free energy function exhibits local kinetic minima and/or a global thermodynamic minimum corresponding to a preferred lamellar thickness, then the concept of multiple melting will be justified by

8.11 The Energetics of Chain Folding in Semicrystalline Polymer –Polymer Blends

311

calculating a relatively small crystal stem height (i.e., the lamellar thickness) from the extremum condition. Thermodynamically or kinetically favored lamellar thicknesses should decrease when stronger intermolecular interactions exist (see Fig. 8.11). These rather thin lamellae, that are stabilized by surface free energy effects on the periphery of the crystallites, melt below the primary transition temperature. As mentioned above, multiple endotherms result when these chains recrystallize to a more stable form during the heating trace in the calorimeter and subsequently melt again at higher temperature. The appropriate extensive free energy function for the addition of an N-stem crystal layer to a growing lamella is represented by the following equation: DGN -stem ¼ Nab{sends=folds þ sHydrogen Bonds } þ (Na þ 2b)LsLateral þ (DGHydrogen Bonds )Fold Surface þ (DGHydrogen Bonds )Amorphous Phase  NabL(DhMelting ) Geometric Considerations a is the width of one crystal stem; b is the depth of one stem; L is the height of each stem or, more appropriately, the lamellar thickness; and N represents the number of stems that add to a growing lamella. Hence, Nab represents either the upper or lower surface area of the N-stem layer. Half of the folds reside on the upper surface and the other half protrude from the lower surface. The lateral surface area of the aggregation of N stems that are exposed to the amorphous material can be represented by (Na þ 2b)L  NaL. The total volume of this newly formed N-stem crystal layer is given by NabL. Energetic Considerations for Chain Folds, Chain Ends, and the Presence of Lateral Surfaces

send is the surface free energy density for one chain end on the lamellar surface; sfold is the surface free energy density for one chain fold on the lamellar surface; and sLateral represents the surface free energy density on the lateral periphery of a lamella. These surface free energy densities, with dimensions of energy per area, allow one to construct contributions to the extensive Gibbs free energy for the formation of an N-stem segment of one lamella. The unfavorable nonspecific surface free energy density due to two chain ends and N21 chain folds on the upper and lower lamellar surfaces is sends=folds ¼ 2send þ (N 1)sfold Nonspecific energetics due to the presence of a foreign lamellar surface in the amorphous molten phase, as described by sLateral, occur across NaL where the N-stem layer contacts amorphous material only on one side (i.e., the other side of the stem contacts the growing lamella) and across 2bL which accounts for stem segments at both ends of the chain.

312

Chapter 8 Crystallization Kinetics via Spherulitic Growth

Melting DhMelting is the endothermic (i.e., .0) enthalpy of melting per volume of the crystalline phase, which contributes favorably to the extensive free energy of formation of an N-stem crystal. Hydrogen Bonding in the Amorphous Phase and on the Periphery of the Lamellae

wAmorphous is the volume fraction of the amorphous phase, and 1Hydrogen Bond represents the volumetric energy density for hydrogen-bond formation between dissimilar blend components in the amorphous phase. When hydrogen bonds form between dissimilar blend components on the fold surface of the crystallites, there is a favorable contribution to the extensive Gibbs free energy. The exothermic surface free energy density (i.e., per unit area) due to these interactions is expressed as follows:   wAmorphous NabL sHydrogen Bonds ¼  1Hydrogen Bond 1  wAmorphous Nab wAmorphous L1Hydrogen Bond ¼ 1  wAmorphous The ratio of amorphous to crystalline volume fractions corresponds to the ratio of total amorphous volume to total crystalline volume, NabL represents the volume of the newly formed N-stem segment of one lamella, and Nab is the total surface area for contact between functional groups on the fold surface and those in the amorphous phase that interact via hydrogen bonds. An unknown multiplication factor is required to relate NabL to the total volume of the crystalline phase, but this same factor relates Nab to the total surface area for contact between hydrogen bonding partners, where one functional group is part of the chemical structure of the completely amorphous component and the complementary functional group belongs to the semicrystalline component on the fold surface of the lamellae. The extensive Gibbs free energy change that accounts for dissociation of hydrogen bonds in the amorphous phase represents an unfavorable contribution to the formation of an N-stem crystal layer. Energy is required to disrupt these interactions and provide motional freedom to those chain segments of the semicrystalline polymer that participate in hydrogen bonding in the amorphous phase: {DGHydrogen Bonds}Amorphous Phase ¼

wAmorphous 1Hydrogen Bond (NabL) 1  wAmorphous

where NabL represents the volume of chain segments that previously interacted with the completely amorphous component via hydrogen bonds in the amorphous phase. Now, these segments diffuse, reptate, and fold into an N-stem layer within one lamella. Hence, these segments of the semicrystalline polymer in the amorphous phase undergo the appropriate conformational rearrangements that are necessary for crystallization, participate in the formation of the N-stem crystal layer, and interact via

8.11 The Energetics of Chain Folding in Semicrystalline Polymer –Polymer Blends

313

hydrogen bonds on the fold surface. The chemical structure of the semicrystalline polymer does not contain complementary functional groups that interact strongly via hydrogen bonds or any other energetic attractions between similar molecules. Hence, hydrogen bonding is an exothermic process that does not occur within a lamella, but only on its fold surface where contacts are possible with the completely amorphous component. The favorable contribution to the overall free energy change is given by {DGHydrogen Bonds}Fold Surface ¼ 

wAmorphous 1Hydrogen Bond ab (LContour  NL) 1  wAmorphous

where LContour is the contour length of the semicrystalline chain that traverses a lamella and NL represents the length of the same chain in the N-stem crystal. If the N-stem crystalline segment follows a square-tooth pattern and all segments of the polymer chain reside in one lamella or on its surface, then the following relation is valid: LContour ¼ NL þ (N 1)a If the folds involve a few additional chain segments that protrude slightly into the amorphous phase and the remaining segments of the polymer chain reside in one lamella, then LContour ¼ NL þ (N  1)ka

k1

When the contour length of a single chain is constant, this relation between the number of crystal stems N in one lamella and the lamellar thickness L provides a constraint in the optimization problem, as described below.

Total Free Energy Function for the Formation of an N-Stem Crystal Layer The free energy change for hydrogen bond formation on the fold surface excludes any possibility that strong association between similar or dissimilar chain segments occurs within the bulk of the lamellae, due to the chemical structure of the semicrystalline polymer and the fact that the totally amorphous component cannot reside in the lamellar regions. The extensive Gibbs free energy change associated with the addition of an N-stem crystal layer to a growing lamella reduces to the following expression: ( ) wAmorphous L1Hydrogen Bond DGN -stem ¼ Nab 2send þ (N 1)sfold  1 wAmorphous þ (Na þ 2b)LsLateral þ

LContour

wAmorphous 1Hydrogen Bond [NabL(N 1)ka2 b] 1 wAmorphous

NabL(DhMelting ) ¼ NL þ (N 1)ka ¼ constant

k1

Hoffman’s crystallization theory has been modified for strongly interacting polymer blends by the terms in the previous equation that contain the volumetric energy density

314

Chapter 8 Crystallization Kinetics via Spherulitic Growth

for hydrogen-bond formation between dissimilar blend components, 1Hydrogen Bond. If all of the important phenomena that contribute to the formation of an N-stem crystal layer have been considered and modeled correctly, then thin unstable lamellae with a larger number of chain folds should form more rapidly and experience stable growth in blends that experience strong interactions on the lamellar surface. This description of the formation of thin lamellae, coupled with crystallization kinetics that could dictate a distribution of stem heights L, supports the concept and observation of multiple melting endotherms. As mentioned earlier in this chapter, this model does not capture effects of (i) chain microstructure (i.e., configurational isomers), (ii) packing of leftand right-handed helices in the unit cell, or (iii) chain branching, but the trends illustrated below represent reasonable conclusions when strong intermolecular interactions occur on the periphery of the crystallites. Lamellae with rather small stem heights will melt invariably at temperatures below the primary transition that is characteristic of larger crystals. Upon melting or high-temperature annealing, the crystallizable segments of these thin lamellae will undergo chain unfolding and form more stable crystals with larger stem heights L. Experimental conditions such as heating rate and annealing temperature, and physical properties such as the glass transition temperature and melt viscosity also govern the stability and melting temperature of the newly formed crystals.

Optimum Lamellar Thickness Consider the following calculations, illustrated in Figures 8.9 and 8.10, based on developments in the previous section that yield an optimum lamellar thickness at the free energy minimum for a specific set of conditions. This is the numerical analog of the “method of Lagrange multipliers,” where optimization is performed with one constraint (i.e., the contour length of the chain is constant). Lengths are given in nanometers (i.e., a, b, L, and LContour), and energies are provided in microjoules (i.e., DGN-stem), mJ/nm2 (i.e., send, sfold, and sLateral ), or mJ/nm3 (i.e., 1Hydrogen Bond and DhMelting). Numerical calculations displayed in Figures 8.9– 8.12

wAmorphous ¼ 0:80 1Hydrogen Bond ¼ 100

8.11.4

a ¼ 1:5 send ¼ 1

b ¼ 1:3 k¼5 sfold ¼ 2 sLateral ¼ 10

LContour ¼ 10000 DhMelting ¼ 150

Summary of Multiple Melting Behavior

The consequences of strong interactions between dissimilar components of a polymer – polymer blend in which one component is semicrystalline are (i) a larger decrease in the Gibbs free energy change for crystal formation when the hydrogen bond strength increases, and (ii) several sequential melting events that occur below the primary transition temperature. In addition to the effect of stem height and number of folds on DGN-stem, the energetic balance between DhMelting and 1Hydrogen Bond is an

Free Energy of Formation (mJ)

8.11 The Energetics of Chain Folding in Semicrystalline Polymer –Polymer Blends

315

–1.6 × 106 –2.0 × 106 Optimum Lamellar Thickness - 13 nm –2.4 × 106 –2.8 × 106 –3.2 × 106 –3.6 × 106 –4.0 × 106

0

40

80 120 160 Lamellar Thickness (nm)

200

Figure 8.9 Dependence of the free energy of formation of an N-stem crystal layer on lamellar thickness in binary blends of semicrystalline and amorphous polymers that interact strongly via hydrogen bonding in the vicinity of the fold surface. Simulations were performed using the parametric values defined on the previous page. The optimum lamellar thickness is obtained numerically by free energy minimization subject to one constraint (i.e., constant contour length of a single chain that folds N 2 1 times within one lamella).

important factor that governs the optimum lamellar thickness (i.e., see Figs. 8.11 and 8.12). Due to the exothermic nature of the crystallization process, DhMelting contributes favorably to the free energy of formation of an N-stem crystal layer below Tmelt. The magnitude of this enthalpic crystallization term is proportional to lamellar volume (i.e., NabL) and the defect-free nature of the crystals that are formed. The volumetric

Free Energy of Formation (mJ)

–1.6 × 106 –2.0 × 106 –2.4 × 106 –2.8 × 106 –3.2 × 106 –3.6 × 106 –4.0 × 106 0 Figure 8.10

Optimum Number of Stems - 490

200

400 600 800 1000 Number of Stems in a Lamella

1200

Dependence of the free energy of formation of an N-stem crystal layer on the number of stems in blends of semicrystalline and amorphous polymers that interact strongly via hydrogen bonding in the vicinity of the fold surface. Simulations were performed using the parametric values defined on the previous page. The optimum lamellar thickness and number of stems are obtained numerically by free energy minimization subject to one constraint (i.e., constant contour length of a single chain).

316

Chapter 8 Crystallization Kinetics via Spherulitic Growth 90 Lamellar Thickness (nm)

80 70 60 50 40 30 20 10 0 50

70

90 110 130 Hydrogen Bond Energy (mJ/nm3)

150

Figure 8.11 Effect of hydrogen bond strength in the vicinity of the fold surface between dissimilar species in binary blends of semicrystalline and amorphous polymers on the optimum lamellar thickness. Free energy minimization for the formation of an N-stem crystal layer was performed using the parametric values defined in the previous section.

energy density for hydrogen-bond formation, 1Hydrogen Bond, is responsible for competing factors when crystallization occurs in hydrogen-bonded blends. For example, in addition to cooling below Tmelt, it is necessary to disrupt hydrogen bonds between chain segments of the semicrystalline polymer and the totally amorphous polymer in the amorphous phase so that the former can undergo diffusion or reptation

Free Energy of Formation (mJ)

–2.5 × 106 –3.0 × 106 –3.5 × 106 –4.0 × 106 –4.5 × 106 –5.0 × 106 –5.5 × 106 –6.0 × 106 50

70

90

110

Hydrogen Bond Energy

130

150

(mJ/nm3)

Figure 8.12 Effect of hydrogen bond strength in the vicinity of the fold surface between dissimilar species in binary blends of semicrystalline and amorphous polymers on the optimum free energy of formation of an N-stem crystal layer.

8.12 Melting Point Depression in Polymer–Polymer and Polymer– Diluent Blends

317

into the lamellae. However, only a small fraction of these hydrogen bonds reform on the fold surface. Consequently, the free energy change for hydrogen bond disruption and formation exhibits competing effects on the overall process. Some of these effects favor thinner lamellae and others favor thicker lamellae. In the absence of kinetic factors, the crystallization process will search for a thermodynamic minimum in the free energy function that corresponds to an optimal value of L, or an optimal distribution of lamellar thicknesses, subject to the constraint that the contour length of a single chain that traverses one lamella N times is constant. When hydrogen bonds are stronger and the value of 1Hydrogen Bond is larger, the crystallization equilibrium shifts in favor of shorter stem heights or lamellar thicknesses. This trend is illustrated in Figure 8.11. Furthermore, the minimum value of the Gibbs free energy for the formation of an N-stem crystal layer from the amorphous phase, which corresponds to the optimum lamellar thickness, decreases when the hydrogen bond strength increases, as illustrated in Figure 8.12. These calculations were performed using the parametric values defined in the previous section, except for the fact that hydrogen bond strength was varied over a reasonable range, with an upper limit that matches the enthalpy of melting.

8.12 MELTING POINT DEPRESSION IN POLYMER – POLYMER AND POLYMER –DILUENT BLENDS THAT CONTAIN A HIGH-MOLECULAR-WEIGHT SEMICRYSTALLINE COMPONENT When a semicrystalline polymer above its melting point is diluted with a miscible low-molecular-weight component in the liquid state and cooled to a temperature range where crystallization occurs, the measured melting temperature upon heating is lower than Tmelt of the pure polymer. Depression of the melting temperature is not due to differences in cooling rate, crystallization temperature, or subsequent heating rate required to measure Tmelt, but to the presence of an impurity (i.e., diluent) that interferes with the crystallization process, producing more imperfect crystals composed of thinner lamellae. Thermodynamic analysis of the effect of low-molecularweight miscible impurities on the equilibrium melting temperature of a semicrystalline polymer begins by equating the polymer’s chemical potential in both phases. The liquid phase binary mixture contains a semicrystalline polymer above its melting point and a miscible diluent. Based on the Flory – Huggins lattice model, the extensive Gibbs free energy of mixing was derived in Section 3.4 as DGmixing ¼ RT{nDiluent ln wDiluent þ nPolymer ln wPolymer þ xnDiluent wPolymer} where ni represents mole numbers, wi corresponds to volume fraction, and x is the dimensionless polymer –diluent energetic interaction parameter. The liquid

318

Chapter 8 Crystallization Kinetics via Spherulitic Growth

phase chemical potential of the polymer that is consistent with the Flory – Huggins mean-field model is   @DGmixing 0 mPolymer (T, p, wDiluent )  mPure Liquid Polymer (T, p) ¼ ¼ @nPolymer T,p,nDiluent (     nPolymer @ wPolymer nDiluent @ wDiluent þ RT ln(1 wDiluent ) þ wDiluent @nPolymer T,p,nDiluent wPolymer @nPolymer T,p,nDiluent )   @ wPolymer þ x nDiluent @nPolymer T,p,nDiluent   ¼ RT ln(1  wDiluent )  (x  1)wDiluent þ xxw2Diluent where x is the ratio of the molar volume of polymer to that of the diluent, or the polymer’s degree of polymerization, and the standard state m0Pure Liquid Polymer is defined as pure liquid polymer. One equates this expression for the chemical potential of the polymer in a single-phase binary liquid phase mixture to the chemical potential of pure crystalline polymer at the depressed melting temperature Tm,Depressed due to the presence of the diluent. Hence,

mPolymer (Tm,Depressed , p, wDiluent ) ¼ m0Pure Crystalline Polymer (Tm,Depressed , p) Subtract the standard state chemical potential of pure liquid polymer from both sides of the previous statement of chemical equilibrium and use the Flory – Huggins lattice results for the activity of the polymer in the liquid phase, which is valid for concentrated polymer solutions:

mPolymer (Tm,Depressed , p, wDiluent )  m0Pure Liquid Polymer (Tm,Depressed , p) ¼ RTm,Depressed {ln(1 wDiluent )  (x1)wDiluent þ xxw2Diluent } ¼ m0Pure Crystalline Polymer (Tm,Depressed , p)  m0Pure Liquid Polymer (Tm,Depressed , p) The difference between the chemical potentials of pure crystalline polymer and pure liquid polymer at temperature Tm,Depressed is equivalent to the molar Gibbs free energy of crystallization of the undiluted polymer, which vanishes only at the melting temperature of the undiluted polymer. In terms of the enthalpy and entropy of crystallization, one writes

m0Pure Crystalline Polymer (Tm,Depressed , p)  m0Pure Liquid Polymer (Tm,Depressed , p) ¼ DgCrystallization ¼ DhCrystallization  Tm,Depressed DsCrystallization    DsCrystallization ¼ DhCrystallization 1  Tm,Depressed DhCrystallization   Tm,Depressed ¼ DhCrystallization 1  Tm,Pure

8.12 Melting Point Depression in Polymer–Polymer and Polymer– Diluent Blends

319

where DhCrystallization is the molar enthalpy change upon solidification, based on the molar mass of the entire polymer chain. Since the molar Gibbs free energy of crystallization of the undiluted polymer vanishes at the pure-component melting temperature Tm,Pure, the ratio of DhCrystallization to DsCrystallization is given by Tm,Pure, and this ratio is assumed to be temperature-independent between the pure-component and depressed melting temperatures. The statement of chemical equilibrium for the semicrystalline polymer in the binary liquid mixture and the undiluted crystalline phase at Tm,Depressed yields   RTm,Depressed ln(1  wDiluent )  (x  1)wDiluent þ xxw2Diluent   Tm,Depressed ¼ DhCrystallization 1  Tm,Pure The first two terms on the left side of the previous equation represent entropic contributions to the polymer’s activity in binary liquid mixtures, whereas the term that contains the Flory – Huggins interaction parameter x represents an energetic contribution. The entropic terms can be neglected when melting point depression in semicrystalline-amorphous polymer – polymer blends is considered because the high-molecular-weight nature of both species minimizes the combinatorial aspects of mixing. Begin with the previous equation for polymer – diluent blends, replace the enthalpy change for crystallization, per mole of the entire chain, by – DhFusion, expand the logarithmic term because the diluent volume fraction is exceedingly small (i.e., ln(1  wDiluent )  wDiluent  12w2Diluent    ), and algebraically rearrange the prediction for diluent-induced melting point depression: 1 Tm,Depressed



1 Tm,Pure

¼

    xR 1  x w2Diluent wDiluent þ DhFusion 2x

Linear least squares analysis of the melting point depression equation is performed when one has data pairs that correspond to Tm,Depressed versus wDiluent at low diluent concentrations. The data point at wDiluent ¼ 0 yields Tm,Pure, which is used to force the zeroth-order coefficient of the polynomial, as illustrated below. Choose a second-order polynomial, y(z) ¼ a0 þ a1z þ a2z 2, and make the following association between the polynomial and the physical model: (i) Independent variable: (ii) Dependent variable:

z ¼ wDiluent y¼

a0 ¼

(iii) Zeroth-order coefficient: wDiluent ¼ 0) (iv) First-order coefficient:

1 Tm,Depressed 1 Tm,Pure

(forced, based on the data point at

a1 ¼ xR=DhFusion

(v) Second-order coefficient:

a2  2xxR/DhFusion, when x 1

320

Chapter 8 Crystallization Kinetics via Spherulitic Growth

The first-order coefficient a1 yields the enthalpy of fusion per mole of repeat units (i.e., DhFusion/x) for a hypothetical 100% crystalline polymer and the Flory – Huggins dimensionless polymer – diluent interaction parameter x, on a repeat-unit basis, is calculated from the second-order coefficient. If the molar volume of the polymeric repeat unit y Repeat Unit is different from the molar volume of diluent y Diluent, then this ratio (i.e., y Repeat Unit/y Diluent) must be included as a multiplicative factor in both the first-order and second-order coefficients of the linear least squares analysis. The Flory – Huggins lattice model assumes that diluent molecules and polymer segments each occupy one site. Hence, the degree of polymerization x was identified as the ratio of molar volumes of the entire polymer chain to that of the diluent, excluding the possibility that y Repeat Unit might be different from y Diluent. Another linear least squares procedure for the melting point depression equation of high-molecularweight materials (i.e., x 1), in the presence of low-molecular-weight miscible impurities, is based on the following algebraic rearrangement when the first data point at wDiluent ¼ 0 and Tm,Pure is used to identify the melting temperature of the pure polymer, but this data point is excluded from the analysis: 1



1

wDiluent Tm,Depressed



1 Tm,Pure



xR {1  xwDiluent } DhFusion

Now, a first-order polynomial is employed, y(z) ¼ a0 þ a1z, and the following association between the polynomial and physical model yields: (i) Independent variable: (ii) Dependent variable:

z ¼ wDiluent  1 1 1 y¼  wDiluent Tm,Depressed Tm,Pure

(iii) Zeroth-order coefficient: (iv) First-order coefficient:

a0 ¼ xR/DhFusion a1  2xxR/DhFusion, when x 1

The intercept and slope yield numerical values for the enthalpy of fusion and the interaction parameter, per mole of polymeric repeat units (i.e., DhFusion/x and x, respectively). Once again, a multiplicative factor of y Repeat Unit/y Diluent must appear in a0 and a1 if this molar volume ratio is not unity. Analysis of melting point depression in semicrystalline-amorphous polymer – polymer blends is performed by adopting the previous equations and neglecting the entropic contribution to the semicrystalline polymer’s activity in miscible binary liquid mixtures because high-molecular-weight chains have negligible combinatorial freedom when they are placed on a lattice. Let polymeric variables and parameters in the previous analysis correspond to those of the semicrystalline polymer, and diluent (i.e., impurity) properties are replaced by those of the amorphous polymer. Hence, upon (i) replacing wDiluent by the volume fraction of the amorphous polymer in miscible molten-state blends, (ii) retaining the repeat-unit molar volume ratio of semicrystalline polymer to amorphous polymer, and (iii) associating the degree of

8.12 Melting Point Depression in Polymer–Polymer and Polymer– Diluent Blends

321

polymerization x with that of the semicrystalline polymer, the melting point depression equation reduces to xSemicrystalline Polymer R 1 1  ¼ Tm,Depressed Tm,Pure DhFusion   V Semicrystalline Polymer  xw2Amorphous Polymer V Amorphous Polymer Now, linear least squares analysis of Tm,Depressed versus wAmorphous, with knowledge of the pure-component melting temperature of the semicrystalline polymer from the initial data point at wAmorphous ¼ 0, requires a first-order polynomial; y(z) ¼ a0 þ a1z, such that: (i) Independent variable: (ii) Dependent variable:

z ¼ w2Amorphous Polymer y¼

(iii) Zeroth-order coefficient:

1 1  Tm,Depressed Tm,Pure a0 ¼ 0 (forced)

  xSemicrystalline Polymer R V Semicrystalline Polymer x (iv) First-order coefficient: a1 ¼  DhFusion V Amorphous Polymer Knowledge of the repeat-unit molar volume of each polymer, as well as the enthalpy of fusion per repeat unit of the semicrystalline polymer, DhFusion/ xSemicrystalline Polymer, allows one to calculate the dimensionless binary energetic interaction parameter x (per repeat unit of the semicrystalline polymer) from the slope of the polynomial model. This procedure invariably yields a negative x-parameter when melting point depression occurs. Exothermic energetic interactions in the molten state are required for two high-molecular-weight polymers to exhibit miscibility. Melting point depression of the semicrystalline polymer is a consequence of this compatibilization in the molten state, due to specific interactions such as hydrogen bonding, ionic attractions of charged functional groups, or metal complexation. If energetic interactions in the molten state are endothermic and unfavorable, then the blend is immiscible because favorable entropic contributions to the Gibbs free energy of mixing are negligible for high-molar-mass species. Consequently, the x-parameter is greater than zero and the melting point depression equation incorrectly predicts an increase in the melting temperature of the semicrystalline polymer. The predictions are incorrect because the lattice model assumes a homogeneous liquid phase that is not applicable for immiscible binary mixtures of two high-molecular-weight polymers in the molten state. When polymer – polymer blends are immiscible in the molten state and one component is potentially crystallizable, negligible melting point depression is observed. This insensitivity of the semicrystalline polymer’s melting temperature to blend composition at low concentrations of the impurity (i.e., the diluent or amorphous component) is a reasonable diagnostic probe of the incompatible nature of these blends in the molten state.

322

Chapter 8 Crystallization Kinetics via Spherulitic Growth

8.12.1

Hoffman –Weeks Analysis

As a final note about melting point depression, the equations developed in the previous section are based on the statement of chemical equilibrium via the polymer’s chemical potential in each phase. Hence, equilibrium melting temperatures (i) for pure polymer Tm,Pure and (ii) in the presence of an impurity Tm,Depressed are required in these equations. Linear least squares analysis of actual experimental data employs melting temperatures TExperimental that are lower than the equilibrium melting points, because when crystallization occurs at temperatures below Tm,Pure or Tm,Depressed, relatively fast rates of crystallization produce imperfect crystals that contain rather thin lamellae. As initially developed by Hoffman and Weeks, it is necessary to extrapolate experimentally measured melting temperatures TExperimental versus crystallization temperatures TC until these two temperatures asymptotically approach each other. For example, TExperimental increases when crystallization occurs at higher temperatures TC below the melting point, with TExperimental . TC. Equilibrium melting temperatures Tm,Pure or Tm,Depressed are obtained by extrapolating TExperimental versus TC until TExperimental ¼ TC. This corresponds to vanishingly slow rates of crystallization at the melting point, which yields the equilibrium melting temperature that is almost impossible to achieve under realistic experimental conditions because the driving force for nucleation vanishes at the melting point.

REFERENCES BELFIORE LA, QIN C, UEDA E, PIRES ATN. Carbon-13 solid state NMR detection of the isotropic carbonyl lineshape in blends of poly(vinylphenol) with main-chain polyesters. Journal of Polymer Science; Polymer Physics Edition 31(4):409– 418 (1993). BORYS RD, LOWENTHAL DH, MITCHELL DL. Atmospheric Environment 34:2593–2602 (2000). DAVIS GT, HOFFMAN JD, LAURITZEN JI. The rate of crystallization of linear polymers with chain folding, in Treatise on Solid State Chemistry, Volume 3. Plenum Press, New York, 1976, pp. 497 –614. LAURITZEN JI, HOFFMAN JD. Theory of formation of polymer crystals with folded chains in dilute solution. Journal of Research of the National Bureau of Standards A—Physics & Chemistry 64A(1):73–102 (1960). PILLAI KM, ADVANI SG, BENARD A, JACOB KI. Numerical simulation of crystallization in high-density polyethylene extrudates. Polymer Engineering and Science 40(11):2356 (2000). QIN C, BELFIORE LA. Thermal investigation of hydrogen-bonding interactions and their effect on the melting/crystallization phenomena in polymer– polymer blends. ACS Polymer Preprints 31(1): 263 (1990).

PROBLEMS 8.1. Qualitatively sketch the temperature dependence of the density of semicrystalline isotactic polystyrene from ambient temperature (i.e., 25 8C) to 300 8C, and be extremely quantitative on the horizontal temperature axis. The glass transition temperature Tg is 105 8C, and the actual melting temperature Tmelt is 220 8C.

Problems

323

8.2. Isotactic polystyrene (i.e., 99% isotactic) is cooled rapidly (i.e., quenched) from 300 8C (state #1) to room temperature (state #2). The sample is heated to 175 8C and maintained at that temperature for a length of time given by 3tCrystallization (state #3), where the time constant for crystallization is

tCrystallization ¼

1 {kn (T)}1=n

Then, it is cooled slowly to room temperature (state #4). (a) Sketch the temperature dependence of specific enthalpy for this cool –heat –cool scheme. Be quantitative on the temperature axis, and identify states #1 through #4 on the graph. (b) Sketch the isothermal response of this sample in a differential scanning calorimeter while the temperature is held at 175 8C for a length of time given by 3tCrystallization. Put time on the horizontal axis. On the vertical axis, endothermic events are represented by upward deflections and exothermic events correspond to downward deflections. (c) The sample in state #4 is heated at 20 8C per minute via DSC from 25 8C to 130 8C. Sketch the DSC heating trace from ambient to 130 8C. 8.3. A differential scanning calorimeter, operating in isothermal mode, provides quantitative information about the time dependence of the volume fraction of crystallinity, XV(t; T ) versus t, for a semicrystalline polymer that crystallizes at temperature T between Tg and Tmelt. (a) Obtain an expression for the Avrami exponent n during isothermal crystallization at temperature T in terms of the slope of XV(t; T ) versus time t at the half-time t1/2 when this semicrystalline polymer achieves 50% of its maximum volume fraction of crystallinity. Express your answer in terms of the slope @ t ¼ t1/2:    d XV (t; T) dt XV (t ) 1) @t¼t1=2 and the half-time, only. (b) Calculate a numerical value for the Avrami exponent n if the half-time is 3 hours at temperature T and the slope of the crystallization isotherm @ t1/2 is 0.4 (hour)21. In other words,    d XV (t; T) ¼ 0:4 h1 dt XV (t ) 1) @t¼t1=2 ¼3 h 8.4. At the optimum annealing temperature TC, your analysis of the crystallization isotherm, XV(t; TC) versus t, in Problem 8.3 yields an Avrami exponent of n ¼ 4 when the halftime t1/2 is 2 hours. How long should you anneal this polymer at TC to induce a significant amount of crystallinity? A numerical answer is required here. 8.5. Isotactic polypropylene with repeat unit fCH2CH(CH3)g exhibits a glass transition at 210 8C and a melting transition at 185 8C. Thermal degradation does not occur below 300 8C. This polymer crystallizes in a 3/1 helical conformation where the backbone bond rotation angles alternate between trans and gauche.

324

Chapter 8 Crystallization Kinetics via Spherulitic Growth (a) What annealing temperature should be used to induce crystallization in the 3/1 helical conformation most rapidly under vacuum? The major coherent Bragg reflection from wide-angle X-ray powder diffraction using Cu ˚ occurs at scattering angle of 2Q ¼ 13.68. K-a radiation with a wavelength of 1.541 A The d-spacing via Bragg’s law corresponds to the c-dimension of a monoclinic unit cell, where two of the unit cell angles are 908 and the third unit cell angle is 998. The other ˚ and b ¼ 20.96 A ˚ , where 1 A ˚ ¼ 1028 cm and unit cell dimensions are a ¼ 6.65 A 23 Avogadro’s number is 6.02214  10 . There are 4 chains in each unit cell, and 3 repeat units per chain are required for periodic behavior of the 3/1 helix. Hence, there are 12 repeat units in each unit cell. (b) Calculate the density of isotactic polypropylene that is 100% crystalline. Pychnometry measurements for an actual sample of isotactic polypropylene, with methanol as the nonsolvent, are tabulated below at 20 8C. The density of methanol at 20 8C is 0.7914 g/cm3. Mass of the pychnometer: 10.003 g Mass of the pychnometer, filled to the meniscus with methanol: 13.926 g Mass of the semicrystalline sample of isotactic polypropylene: 1.566 g Mass of the pychnometer, polypropylene, and methanol: 14.116 g (c) Calculate the weight fraction of crystallinity in the actual sample at 20 8C if the amorphous polymer density is 0.854 g/cm3. (d) Calculate the volume fraction of crystallinity in the actual sample at 20 8C if the amorphous polymer density is 0.854 g/cm3. (e) Sketch the density of the actual sample of isotactic polypropylene from 230 8C to 200 8C, during the first heating scan at 20 8C per minute. Be careful. The handbook value for the heat of fusion of 100% crystalline isotactic polypropylene is (Dhfusion)100% crystal ¼ 2100 calories per mole of repeat units. (f) Predict the experimental heat of fusion in calories for isotactic polypropylene if the DSC experiment is performed on 20 milligrams of the actual polymer.

Numerical Calculations that Accompany the Solution to Problem 8.5 (see Table 8.7): Calculations for isotactic polypropylene that crystallizes as a 3/1 helix Tg ¼ 210 8C, glass transition temperature Tm ¼ 185 8C, melting temperature TC,optimum ¼ Tg þ 0.67fTm – Tgg, optimum crystallization temperature WAXD parameters for a monoclinic unit cell where two of the angles are 908 and the third unit cell angle is 998 ˚ a ¼ 6.65 A ˚ b ¼ 20.96 A a ¼ 908 b ¼ 908 g ¼ 998

Problems

325

Table 8.7 Numerical Solution to Problem 8.5 for isotactic polypropylene a ¼ 6.650 a ¼ 90.000 Avogadro ¼ 6.02eþ23 b ¼ 90.000 Crystal density ¼ 0.934 Dhfusion,expt ¼ 30.183 g ¼ 99.000 Mass fraction ¼ 0.604 Mass total ¼ 14.116 MWrepeat ¼ 42.000 Polymer density ¼ 0.901 Pychnometer þ Solvent Mass ¼ 13.926 Tg ¼ 210.000 Tm ¼ 185.000 Volume of unit cell ¼ 8.96e222

(angstroms) (degrees) (number/mol) (degrees) (g/cm3) (cal/g) (degrees) (%Crystallinity) (grams) (g/mol) (g/cm3) (grams) (8C) (8C) (cm3)

Actual DSC mass ¼ 20.000 Amorphous density ¼ 0.854 b ¼ 20.960 c ¼ 6.507 Dhfusion,100 ¼ 2100.000 Ð DSC area Cp dT ¼ 0.604 l ¼ 1.541 Mass Polymer ¼ 1.566 MeOH density ¼ 0.791 Number in unit cell ¼ 12.000 Pychnometer mass ¼ 10.003 Tc,opt ¼ 120.000 u ¼ 6.800 Volume fraction ¼ 0.582

(mg) (g/cm3) (angstroms) (angstroms) (cal/mol) (calories) (angstroms) (grams) (g/cm3) (#repeat) (grams) (8C) (degrees) (%Crystallinity)

˚ , wavelength of copper K-a radiation l ¼ 1.541 A c-Dimension of the unit cell from coherent Bragg scattering at 2Q: 2Q ¼ 13.68 n ¼ 1, for first-order reflections nl ¼ 2c sin Q VUnit Cell ¼ (abc sin a sin b sin g)  10224, cm3 MWrepeat ¼ 42 daltons, mass of 1 mole of repeat units NAvogadro ¼ 6.02214  1023 molecules/mol Number of repeat units in one unit cell: there are 4 chains per unit cell, and 3 repeat units per chain are required for periodic behavior of the 3/1 helix NUnit Cell ¼ 12 Density of the 100% crystalline polymer, g/cm3:

r100%Crystalline Polymer ¼

NUnit Cell MWrepeat VUnit Cell NAvogadro

Pychnometry measurements for the density of semicrystalline isotactic polypropylene: massPychnometer ¼ 10.003 grams, mass of the pychnometer Mass of the pychnometer and the nonsolvent, methanol, filled to the meniscus: massPychnometerþMethanol ¼ 13.926 grams rMethanol ¼ 0.7914 g/cm3, density of methanol at 20 8C Mass of semicrystalline isotactic polypropylene: massPolymer ¼ 1.566 grams Mass of the pychnometer, the polymer, and MeOH filled to the meniscus: massPychnometer,Polymer,Methanol ¼ 14.116 grams Density of semicrystalline isotactic polypropylene via pychnometry:

rSemicrystalline Polymer ¼

rMethanol massPolymer massPychnometerþMethanol þ massPolymer  massPychnometer,Polymer,Methanol

326

Chapter 8 Crystallization Kinetics via Spherulitic Growth Density of completely amorphous polypropylene: rAmorphous Polymer ¼ 0.854 g/cm3 Volume fraction of crystallinity:

wCrystallinity ¼

rSemicrystalline Polymer  rAmorphous Polymer r100%Crystalline Polymer  rAmorphous Polymer

Mass fraction of crystallinity:

v Crystallinity ¼

r100%Crystalline Polymer w rSemicrystalline Polymer Crystallinity

Enthalpy of fusion for hypothetical 100% crystalline isotactic polypropylene: fDhfusiong100%Crystalline ¼ 2100 calories per mole of repeat units Experimental enthalpy of fusion for the actual DSC heating trace, cal/g:

v Crystallinity ¼

MWrepeat {Dhfusion }Experimental {Dhfusion }100%Crystalline

Mass of polypropylene tested via DSC: massPolymerDSC ¼ 20 mg Integration of Cp versus T in the vicinity of Tm for the actual DSC sample of polypropylene: EnthalpyDSC ¼

massPolymerDSC {Dhfusion }Experimental 1000

8.6. An isotactic vinyl polymer (i.e., 90% isotactic), like poly(propylene) [CH2CH(CH3)], exhibits a glass transition at 210 8C and an actual melting transition at 170 8C. This polymer crystallizes in a 3/1 helical conformation where the backbone bond rotation angles alternate between trans and gauche. The semicrystalline polymer is heated under vacuum from ambient to 225 8C at a rate of 10 8C/min. Then, it is removed from the vacuum oven and immersed in liquid nitrogen. The crystallization time constant for isotactic poly(propylene) at its optimum annealing temperature is 4 hours. It is reasonable to assume that the crystallization time constant is much longer than 4 hours at all other temperatures. The polymer’s molecular weight is 5  105 daltons. (a) Calculate the optimum annealing temperature to induce a significant amount of crystallinity in isotactic poly(propylene), i-PP, in the 3/1 helical conformation. (b) The sample of i-PP, described above, is recovered from the liquid nitrogen dewar and allowed to equilibrate at ambient temperature for 10 minutes. Now, the sample is placed in a pychnometer that contains a nonsolvent, like methanol, and its density is measured. Estimate the density of this sample of isotactic poly(propylene) at 25 8C. A numerical answer is required here. 8.7. As a continuation of Problem 8.6, 20 different dog-bone-shaped samples of i-PP, quenched from the molten state in liquid nitrogen, are removed from the liquid nitrogen dewar and annealed under vacuum at the optimum crystallization temperature. One sample is removed from the vacuum oven every 15 minutes for ambient-temperature stress –strain testing at a strain rate of 1 inch per minute. Remember that Tg is below ambient, but Tm is above ambient. The modulus of elasticity is calculated from the initial slope of the stress –strain curve.

Problems

327

(a) Sketch the ambient-temperature stress –strain curve for the sample of i-PP that was removed from the vacuum oven after 15 minutes of annealing at the optimum crystallization temperature. Put stress on the vertical axis and strain on the horizontal axis. (b) Sketch the ambient-temperature stress –strain curve for the sample of i-PP that was removed from the vacuum oven after 5 hours of annealing at the optimum crystallization temperature. Put stress on the vertical axis and strain on the horizontal axis. Be semiquantitative with respect to your sketch in part (a). (c) Sketch ambient-temperature elastic modulus versus annealing time in the vacuum oven at the optimum crystallization temperature for all 20 samples of i-PP. Put modulus on the vertical axis and annealing time on the horizontal axis. (d) Estimate the ambient-temperature density of the 20th (i.e., last) sample of isotactic poly(propylene) that was removed from the vacuum oven after 5 hours of annealing at the optimum crystallization temperature. 8.8. An isotactic vinyl polymer with repeat unit (CH2CHR) exhibits a glass transition at 80 8C and a melting transition at 200 8C. Thermal degradation does not occur below 350 8C. When this polymer is cooled from the molten state to temperatures between Tm and Tg, the time required for recrystallization is approximately 8 hours at 140 8C. There is not enough chain mobility for the polymer to crystallize below Tg. This polymer is heated above the melting temperature at a rate of 25 8C per minute. Then, it is cooled from 250 8C to 140 8C at a rate of 25 8C per minute, and a spherulitic superstructure develops during isothermal crystallization at 140 8C. (a) Sketch the total free energy change DGSpherulite for spherulite formation as a function of the radius of the spherulite. DGSpherulite accounts for volumetric and surface-related thermodynamic phenomena as spherulites nucleate and grow. Include two curves on one set of axes—one for isothermal crystallization at 140 8C, and another curve that corresponds to isothermal crystallization at 175 8C. (b) During isothermal crystallization at 140 8C, a spherulitic superstructure develops via heterogeneous nucleation at time t ¼ 0. The spherulite number density NS and growth rate G produce the following crystallization rate constant for spherulitic growth: 3 4 3pNS G

¼ 104 h3

Do the spherulites impinge upon each other after 8 hours of growth at 140 8C? 8.9. Use your knowledge of (i) the temperature dependence of crystallite growth rates, (ii) pressure effects on melting, (iii) the “physics of snow”, and (iv) the analogous resistance posed by short carbon nanotube “whiskers” to explain why improperly waxed skate skis glide very poorly over freshly groomed snow at 5 8F that has not experienced sunlight or temperatures in the 25–30 8F range. In comparison, skate skiing on freshly groomed snow in brilliant sunlight at 27 8F is the equivalent of “blue velvet”.

Chapter

9

Experimental Analysis of Semicrystalline Polymers Turning, I heard the stars shatter—what laughter! —Michael Berardi

Isothermal crystallization kinetic data from calorimetry are analyzed using the formalism of coupled heat and mass transfer in constant-volume batch reactors. Rate constants and scaling exponents in the generalized Avrami equation are obtained from linear least squares analysis of crystallization exotherms. Liquid crystalline phase transitions are discussed briefly from an experimental viewpoint using differential scanning calorimetry. Simple pychnometry measurements are discussed to calculate densities of semicrystalline polymers and thermal expansion coefficients of liquids and solids.

9.1

SEMICRYSTALLINITY

Unlike small molecules that have an opportunity to crystallize completely, the fraction of crystallinity in high-molecular-weight polymers is, in general, significantly less than 100%. Hence, the morphology (i.e., solid state phase behavior) of semicrystalline polymers must consider crystallites in an amorphous matrix. Several physical variables, including mass density, enthalpy of fusion, crystallographic scattering of X-rays, and absorptions from infrared spectroscopy or magnetic resonance can be measured to estimate the fraction of crystallinity in polymeric materials based on realistic physical models. When ambient temperature is above Tg and below Tm, the ambient-temperature mechanical response of semicrystalline polymers depends strongly on the fraction of crystallinity.

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

329

330

Chapter 9 Experimental Analysis of Semicrystalline Polymers

9.2 DIFFERENTIAL SCANNING CALORIMETRY: THERMOGRAMS OF SMALL MOLECULES THAT EXHIBIT LIQUID CRYSTALLINE PHASE TRANSITIONS BELOW THE MELTING POINT Milligram-quantity samples are heated at a constant rate (i.e., typically 5 8C/min or 10 8C/min) in a calorimeter that monitors the power differential between sample and reference (i.e., empty pan), as required to maintain both at the same temperature. The vertical axis of the output curve, with dimensions of power, represents the product of heat capacity Cp and heating rate. Typically, the endothermic direction is identified on the vertical axis. Temperature is displayed on the horizontal axis. One distinguishes between first-order and second-order phase transitions via the behavior of Cp in the vicinity of Tm and Tg. Specific heat is discontinuous at first-order phase transitions (i.e., Tm and transitions to a liquid crystalline state with conformation disorder in at

(a)

(b) 47.2 J/g 70.1 °C

60.0 J/g 103.8 °C

HO

13.3 J/g 149.6 °C

C

O

CH2 CH3 6

O

HO C

O CH2 CH3 5

O Solid–Nematic

Smectic–Nematic

Endotherm

73.2 °C Solid–Solid

Endotherm

Nematic–Isotropic 152.4 °C Nematic–Isotropic

Solid–Smectic

106.4 °C 40

60

80

100 120 140 160 180 200 220 240

70

80

90

Temperature (°C)

100 110 120 130 140 150 160 170 Temperature (°C)

(c) 18.1 J/g 80.1 J/g 114.5 °C 144.7 °C 15.2 J/g 180.0 °C

182.0 °C Nematic–Isotropic

H C C

C HO

O

O CH2

4

Endotherm

119.1 °C Solid–Solid H

CH3 Solid–Nematic 147.1 °C

40

60

80

100 120 140 160 180 200 220 240 Temperature (°C)

Figure 9.1 Thermograms from differential scanning calorimetry for small molecules that exhibit liquid crystalline behavior below the melting temperature. (a) p-Hexyloxybenzoic acid, solid–solid phase transition at 73 8C, solid– nematic transition at 106 8C, nematic –isotropic (i.e., melting) transition at 152 8C. (b) p-Heptyloxybenzoic acid, solid– smectic phase transition at 91 8C, smectic–nematic transition at 94 8C, nematic – isotropic (i.e., melting) transition at 146 8C. (c) p-Pentyloxycinnamic acid, solid–solid phase transition at 119 8C, solid–nematic transition at 147 8C, nematic– isotropic (i.e., melting) transition at 182 8C. Transition temperatures are reported at the endothermic peak, not the onset or endpoint, and endothermic response is downward. Taken from the MS thesis of AA Patwardhan (1986).

9.3 Isothermal Analysis of Crystallization Exotherms

331

least one dimension), whereas the temperature derivative of Cp exhibits a discontinuous increment at second-order phase transitions, like Tg. DSC heating traces are illustrated in Figure 9.1 for (i) solid – solid conformational rearrangements below Tm, (ii) melting transitions to the isotropic state, and (iii) other first-order transitions from a state of three-dimensional order to a liquid crystalline state (i.e., nematic or smectic phases).

9.3 ISOTHERMAL ANALYSIS OF CRYSTALLIZATION EXOTHERMS VIA DIFFERENTIAL SCANNING CALORIMETRY If digital control is implemented to maintain isothermal operation of a crystallizable polymer and an external sink of thermal energy is available to remove the heat liberated when crystallization occurs, then it is possible to analyze the kinetics of crystallization via coupled heat and mass transfer in a constant-volume diathermal system. For this exothermic process, the rate at which thermal energy is generated must be equivalent to the rate at which thermal energy is removed by heat transfer across the wall of the aluminum sample pan. The calorimeter operates similarly to an isothermal constant-volume batch reactor, and one seeks a prediction for the transient mass fraction of crystallinity, which is analogous to fractional conversion in reactive systems, from the thermal history required to maintain constant temperature. The calorimeter functions as a digital controller by monitoring the rate at which thermal energy must be removed from the system. Analysis begins by writing an unsteady state total energy balance for a constant-volume batch reactor with no exchange of mass between the system and the surroundings. The most general form of the extensive total energy balance for a closed system that performs no mechanical work on the surroundings, with dimensions of energy per time, is   dE dQ ¼ dt dt Input where total energy E is the sum of kinetic, potential, and internal energies, and fdQ/dtgInput is the rate of heat exchange between the surroundings (i.e., the calorimeter equipped with digital control) and the crystallizable polymer (i.e., the system). This is essentially the first law of thermodynamics in differential form. By convention, the rate of heat exchange is positive when the system receives heat from the surroundings, so the heat transfer term on the right side of the previous equation is negative. Since the kinetic and potential energies of the system do not change with time, the unsteady state total energy balance reduces to the following thermal energy balance:   dU dQ ¼ dt dt Output where U is the extensive internal energy of the system. Standard thermodynamic formalism is employed to express the total differential of internal energy in terms of temperature T, pressure p, and mass numbers MAmorphous and MCrystalline for the

332

Chapter 9 Experimental Analysis of Semicrystalline Polymers

amorphous and crystalline fractions of the sample, respectively. Hence, the system is analyzed as a binary mixture that converts amorphous material to crystallites as crystallization occurs, with subsequent liberation of thermal energy that must be removed to maintain isothermal operation in the calorimeter. The total differential of U is     @U @U dT þ dp dU ¼ @T p, MAmorphous , MCrystalline @p T, MAmorphous , MCrystalline     @U @U dMAmorphous þ dMCrystalline þ @MAmorphous T, p, MCrystalline @MCrystalline T, p, MAmorphous The time derivative of the previous expression yields Eq. (9.1):     dU @U dT @U dp ¼ þ dt @T p, MAmorphous , MCrystalline dt @p T, MAmorphous , MCrystalline dt   dMAmorphous @U þ dt @MAmorphous T, p, MCrystalline     dMCrystalline @U dQ ¼ þ dt @MCrystalline T, p, MAmorphous dt Output

(9:1)

which is appropriate for this analysis of crystallization kinetics. The coefficients of dT/dt and dp/dt in Eq. (9.1) are evaluated from the total differential expression for the extensive internal energy of multicomponent systems in terms of its natural variables S (i.e., extensive entropy) and V (i.e., extensive volume) at constant composition: dU ¼ T dS  p dV This differential form of the first law is used in conjunction with a Maxwell relation and the definition of thermophysical properties like heat capacity Cp, thermal expansion coefficient a, and isothermal compressibility k to calculate the temperature and pressure coefficients of the extensive internal energy. For example,       @U @S @V ¼T p ¼ Cp paV @T p, composition @T p, composition @T p, composition       @U @S @V ¼T p ¼ V{kp aT} @p T, composition @p T, composition @p T, composition     @S @V ¼ @p T, composition @T p, composition where Cp is an extensive heat capacity with units of energy per Kelvin,     @H @S ¼T Cp ¼ @T p, composition @T p, composition

9.3 Isothermal Analysis of Crystallization Exotherms

333

a is the coefficient of thermal expansion (i.e., 1024 K21 for liquids, 1/T for ideal gases),   @ ln V a¼ @T p, composition and k is the coefficient of isothermal compressibility (i.e., 1026 atm21 for liquids, 1/p for ideal gases),   @ ln V k¼ @p T, composition The coefficients of dMAmorphous/dt and dMCrystalline/dt in Eq. (9.1) are defined as partial specific internal energies of species i, because differentiation with respect to mass numbers of component i is performed at constant T, p, and mass numbers of the other phase in semicrystalline polymers. Unsteady state mass balances for the crystalline and amorphous phases are provided below, where each phase is interpreted as a separate component in binary mixtures with one type of transformation (i.e., crystallization) between the two species. Hence, if RCrystallization represents the rate of crystallization with dimensions of mass per amorphous volume per time, wAmorphous is the decreasing volume fraction of amorphous material, and V is the total system volume, then the temporal behavior of MAmorphous and MCrystalline is given by d MAmorphous ¼ wAmorphous VRCrystallization dt d MCrystalline ¼ wAmorphous VRCrystallization dt where wAmorphous is a decreasing function of time during the crystallization process. Now, Eq. (9.1) adopts the following form when the previous mass and thermal energy balances are combined: dU dT dp ¼ {Cp paV} þ V{k p aT} þ wAmorphous VRCrystallization dt dt dt ( )      @U @U dQ ¼   @MCrystalline T, p, MAmorphous @MAmorphous T, p, MCrystalline dt Output The difference between partial specific internal energies of the crystalline and amorphous phases can be interpreted as the specific internal energy change (i.e., negative) for the crystallization process, DUCrystallization. For reactive systems, when the product of stoichiometric coefficient y i (i.e., þ1 for crystallites, 21 for amorphous material, 0 for inerts) and the partial molar internal energy of species i is summed over all components in the mixture, one obtains an exact expression for the molar internal energy change for the reaction [Tester and Modell, 1986]. In other words, X  @U  yi DUReaction ¼ @Ni T, p, all Nj [ j=i] all reactive species

334

Chapter 9 Experimental Analysis of Semicrystalline Polymers

even though pure-component molar internal energies are typically employed in practice to calculate DUReaction. Generalized predictions of nonisothermal crystallization in constant-volume diathermal systems that operate at constant pressure can be obtained from   dT dQ þ wAmorphous VRCrystallization DUCrystallization ¼  {Cp paV} dt dt Output For example, analysis of crystallization kinetics under nonisothermal conditions at a constant cooling rate from the molten state to temperatures below the melting point requires consideration of the first term on the left side of the previous equation, where the time-rate-of-change of system temperature is given by the experimental cooling rate (i.e., negative). For isothermal operation, kinetic analysis is based on the following closed system thermal energy balance:   dQ wAmorphous VRCrystallization (DUCrystallization ) ¼ dt Output where fdQ/dtgOutput represents the rate at which thermal energy must be removed from the sample via heat transfer across the wall of the aluminum sample pan, as specified by the calorimeter to maintain isothermal operation. Circular data points for the time dependence of the rate of isothermal crystallization in Figures 8.4 and 8.5 of Section 8.7 represent generic calorimeter response profiles for fdQ/dtgOutput. The primary objective of this analysis is to evaluate the Avrami parameters and the mass fraction of crystallinity from isothermal measurements of fdQ/dtgOutput. The previous equation indicates that this rate of heat exchange must be balanced by the rate of thermal energy generation due to exothermic crystallization. The important thermodynamic quantity of interest is the internal energy change DUCrystallization, instead of enthalpy, because the first law was constructed for closed systems with no contributions from convective transport. One typically requires enthalpy changes when pV– work across the inlet and outlet planes for fluid flow is combined with internal energy in the first law for open systems. The unsteady state mass balance for the amorphous phase of a semicrystalline polymer, at temperatures below the melting point, is employed to introduce the mass fraction of crystallinity vCrystalline(t) into the previous thermal energy balance. For example, d MAmorphous ¼ wAmorphous VRCrystallization dt MAmorphous (t) v Crystalline (t) ¼ 1  MAmorphous (t ¼ 0) MAmorphous (t ¼ 0)

(9:2)

d v Crystalline ¼ wAmorphous VRCrystallization dt

This methodology is consistent with the formalism for reactive systems in which only one chemical reaction occurs, and fractional conversion is defined in terms of an important (i.e., key) reactant. Hence, both fractional conversion and the mass fraction of crystallinity are identically zero at t ¼ 0. The previous definition of vCrystalline in

9.4 Kinetic Analysis of the Mass Fraction of Crystallinity

335

terms of the residual mass of the amorphous phase (i.e., less residual amorphous mass corresponds to a higher mass fraction of crystallinity) is reasonable because the only options for amorphous material are to (i) adopt the correct rotational state for backbone bonds and add to growing lamellae, (ii) remain in the amorphous phase, or (iii) exist as interlamellar or interspherulitic impurities. Case (i) is the only situation that contributes to the crystalline mass fraction of a semicrystallizable polymer. When the steady state thermal energy balance is combined with the unsteady state species mass balance, the time dependence of the mass fraction of crystallinity (i.e., dvCrystalline/dt) can be calculated from the calorimeter’s digital controller response, which monitors the rate of thermal energy removal across the wall of the aluminum sample pan during isothermal crystallization. The relevant equation is   dvCrystalline dQ ¼ (9:3) MAmorphous (t ¼ 0){DUCrystallization} dt Output dt The rate of thermal energy removal on the right side of the previous equation is measured experimentally, and the product of the initial mass of amorphous material MAmorphous(t ¼ 0) with the specific internal energy change for crystallization DUCrystallization represents a hypothetical upper limit on the amount of thermal energy that should be liberated if 100% crystallization is achieved. For solids (i.e., crystallites) or highly viscous liquids (i.e., molten amorphous material), DUCrystallization defined by  DUCrystallization ¼

@U @MCrystalline



 

T, p, MAmorphous

@U @MAmorphous

 T, p, MCrystalline

is essentially the same as the specific enthalpy change for crystallization, corresponding to a hypothetical 100% crystalline polymer. In other words, the contribution from pV– work is negligible, so DUCrystallization  DHCrystallization.

9.4 KINETIC ANALYSIS OF THE MASS FRACTION OF CRYSTALLINITY VIA THE GENERALIZED AVRAMI EQUATION A model for the rate of crystallization RCrystallization with dimensions of mass per amorphous volume per time is required to extract kinetic parameters for the crystallization process via analysis of Eqs. (9.2) and (9.3). This was discussed in Section 8.7 for homogeneous and heterogeneous nucleation via the generalized Avrami equation. At temperatures below the melting point, the appropriate rate expression is ( ) MAmorphous (t ¼ 0) [v Crystalline (t ) 1)  v Crystalline (t ¼ 0)] RCrystallization  wAmorphous V  nkn (T)t n1 exp{kn (T)t n}

336

Chapter 9 Experimental Analysis of Semicrystalline Polymers

where n is the Avrami exponent, kn(T) is a temperature-dependent crystallization rate constant, with dimensions of (time)2n, and vCrystalline(t ) 1)2 vCrystalline(t ¼ 0) represents an upper limit on the fraction of crystallinity that samples can achieve during isothermal kinetic experiments due to imperfections in chain microstructure, endgroups on low-molecular-weight chains, the presence of impurities, and so on. It should be emphasized that crystallization models in Chapter 8 were based on the volume fraction of crystallinity, prior to and after impingement of adjacent spherulites. In contrast, the calorimetric data of interest in this chapter are sensitive to the mass fraction of crystallinity. When the previous expression for RCrystallization is combined with the unsteady state mass balance, given by Eq. (9.2), the mass fraction of crystallinity must satisfy the following equation: d v Crystalline  [v Crystalline (t ) 1)  v Crystalline (t ¼ 0)]nkn (T)t n1 exp{kn (T)t n} dt which can be integrated rather easily to yield the integral form of the generalized Avrami equation, subject to the initial condition that the mass fraction of crystallinity is vCrystalline(t ¼ 0). The Avrami prediction for vCrystalline(t) is

v Crystalline (t)  v Crystalline (t ¼ 0)  1  exp{kn (T)t n} v Crystalline (t ) 1)  v Crystalline (t ¼ 0)

(9:4)

Comparison of the previous two equations with those in Sections 8.5 and 8.7 reveals that no distinction is made between mass fraction of crystallinity and volume fraction of crystallinity. Crystal and amorphous densities are required to relate these two crystalline fractions, as illustrated in Section 9.6. Experimentally, one uses differential and integral forms of the steady state thermal energy balance, together with the calorimeter’s measurement of the rate of thermal energy removal, to estimate the Avrami exponent n, the temperature-dependent crystallization rate constant kn(T ), and the mass fraction of crystallinity during isothermal crystallization at temperature T. Rearrangement and integration of Eq. (9.3) yield   dQ dt Output dv Crystalline ¼ dt MAmorphous (t ¼ 0){DUCrystallization} ð j¼t   dQ dj dj Output j¼0 v Crystalline (t) ¼ v Crystalline (t ¼ 0) þ MAmorphous (t ¼ 0){DUCrystallization} The initial mass fraction of crystallinity vCrystalline(t ¼ 0) will be zero if the sample is heated above the melting temperature and soaked in the molten state for sufficient time to melt all of the crystallites that exist prior to this kinetic study below Tm. It is not recommended to initiate isothermal crystallization kinetics below the melting temperature with samples that are not completely amorphous. When vCrystalline(t ¼ 0) ¼ 0, it must be emphasized that integration of the calorimeter’s rate of thermal energy removal across the wall of the aluminum sample pan, from t ¼ 0 to t ) 1, will always be

9.5 Measurements of Crystallinity via Differential Scanning Calorimetry

337

less than the product of MAmorphous(t ¼ 0) and DUCrystallization, because polymers do not achieve 100% crystallinity. Substitution of the previous expression for vCrystalline into Eq. (9.4) for samples that exhibit no crystallinity prior to kinetic studies suggests the following linear least squares analysis of the data. Upon taking the natural logarithm of rearranged Eq. (9.4) twice with vCrystalline(t ¼ 0) ¼ 0, one obtains 8 2ð j )1   39 dQ > > > dj > > >    = < 6 d j Output 7 v Crystalline (t) 6 j¼t 7 ¼ ln ln6 ð j)1   ln ln 1 7 > 4 5> v Crystalline (t ) 1) dQ > > > dj > ; : j d j¼0 Output ¼ ln{kn (T)} þ n ln(t) A linear polynomial model [i.e., y(x) ¼ a0 þ a1x] is required to match the physical model described by the previous equation. The independent variable x for linear least squares analysis is the log of the crystallization time, ln t. The dependent variable y is given by 8 2ð j )1   39 dQ > > > > d j > > 6 < 7= j d Output 7 6 j¼t y ¼ ln ln6ð j )1   7 > 4 5> dQ > > > dj > ; : j d j¼0 Output It is necessary to exclude the initial data point at t ¼ 0 from the regression analysis. The first-order coefficient in the polynomial model a1, or the slope, corresponds to the Avrami exponent n. The temperature-dependent crystallization rate constant, with dimensions of (time)2n, is obtained from the zeroth-order coefficient a0: kn (T) ¼ exp(a0 ) If isothermal crystallization experiments are performed at several different temperatures between the glass and melting transitions, and linear least squares analysis of the data yields an Avrami exponent n and crystallization rate constant kn at each temperature, then one identifies the optimum crystallization temperature TC where fkn(T )g1/n, with dimensions of inverse time, exhibits a maximum. In-depth theoretical analysis of optimum crystallization temperatures was discussed in Sections 8.9 and 8.10.

9.5 MEASUREMENTS OF CRYSTALLINITY VIA DIFFERENTIAL SCANNING CALORIMETRY The primary objective of this section is to obtain an expression for the mass fraction of crystallinity from calorimetric data in the vicinity of the melting transition. This analysis of first-order thermodynamic phase transitions focuses on (i) the discontinuity in enthalpy at Tmelt, (ii) the temperature derivative of specific enthalpy at constant pressure, or specific heat, which is described by a delta function at the melting

338

Chapter 9 Experimental Analysis of Semicrystalline Polymers

temperature under ideal conditions, and (iii) the fact that DSC thermograms provide a snapshot of specific heat versus temperature (actually, the product of specific heat and heating rate vs. temperature). Deviations from ideality that induce broadening in Cp versus temperature near Tmelt can be attributed to the following: (i) Heat Transfer Limitations. Not all regions of the sample experience the same temperature simultaneously, particularly when the surface-to-volume ratio is small for bulk materials relative to high surface-to-volume ratio powders. (ii) Range of Melting Temperatures. Crystallites that contain thicker lamellae melt at higher temperature. Also, higher molecular weight chains reduce the probability that imperfections in the vicinity of the chain ends will depress Tmelt. (iii) Crystallite Imperfections. Less stable crystals invariably melt at lower temperature. If there is a sufficient driving force for nucleation in this molten material at temperatures below the primary melting point, then recrystallization might occur with subsequent melting of more perfect crystals at higher temperature. Experimental methods to measure crystallinity can be developed by constructing reasonable models for the temperature and mass fraction dependence of the specific heat of semicrystalline polymers (i) in regions where no thermal transitions occur, such that the crystalline mass fraction remains constant, and (ii) in the vicinity of the melting temperature. When the nonzero crystalline mass fraction is independent of temperature below Tmelt, one defines the baseline specific heat as a linear weighted sum of the specific heats of the completely crystalline and completely amorphous materials, where mass fraction of crystallinity vC is employed to construct the appropriate weighting factors. Hence, {Cp (T)}baseline ¼ vC {Cp (T)}100%Crystalline þ (1 vC ){Cp (T)}100%Amorphous In the vicinity of the melting transition, the actual specific heat of semicrystalline materials is much larger than that given by the previous equation as a consequence of the discontinuity in specific enthalpy and its temperature derivative at Tmelt. Under ideal conditions, specific heat is described by a delta-function increment, and the change in crystalline mass fraction with respect to temperature (i.e., dvC/dT) is represented by a comparable delta-function decrement. Both of these delta functions are broadened over a finite temperature range in the vicinity of Tmelt, as mentioned above. When the crystalline mass fraction decreases with respect to temperature during a DSC heating trace, the following specific heat model is reasonable: {Cp (T)}actual ¼ vC {Cp (T)}100%Crystalline þ (1 vC ){Cp (T)}100%Amorphous þ Dhfusion,

100% Crystalline

  d vC  dT

Rearrangement of the previous equation and integration with respect to temperature in the vicinity of Tmelt yields the mass fraction of crystallinity of semicrystalline materials

9.6 Analysis of Crystallinity via Density Measurements

339

in the morphological state below Tmelt. Hence, one subtracts the baseline specific heat from the actual data in the vicinity of the melting transition and integrates the broadened delta-function DSC response from below Tmelt, where the crystalline mass fraction is vC, to above Tmelt, where there is no longer any crystalline material. Manipulation of the previous equation is performed when the heat of fusion of hypothetical 100% crystalline materials is not temperature dependent: aboveð T melt

ð0

below T melt

100% Crystalline vC

[{Cp (T)}actual {Cp (T)}baseline ] dT ¼ Dhfusion,

d vC

The mass fraction of crystallinity in the morphological state below Tmelt is defined as the experimental heat of fusion for the actual semicrystalline polymer, given by the left side of the previous equation, relative to the heat of fusion for a hypothetical material that is 100% crystalline, where the latter quantity is tabulated for many polymers in handbooks and the refereed journal literature. The appropriate working equation that allows one to estimate the mass fraction of crystallinity vC in the morphological state below Tmelt is ð above T melt

vC 

[{Cp (T)}actual  {Cp (T)}baseline ] dT

below T melt

Dhfusion, 100%Crystalline

9.6 ANALYSIS OF CRYSTALLINITY VIA DENSITY MEASUREMENTS The primary objective of this section is similar to the previous section, but the methodology is based on densities instead of specific heat versus temperature. Except for water and a few other materials that contract upon melting, the density of a 100% crystalline material is greater than the corresponding density of its 100% amorphous counterpart. Furthermore, the density of a semicrystalline polymer ( ractual ) lies somewhere between the densities of its 100% crystalline ( rCrystal ) and 100% amorphous ( rAmorphous) counterparts. The first approach assumes that the crystalline and amorphous volumes of a semicrystalline material are additive: Total mass Crystalline mass Amorphous mass ¼ þ ractual rCrystal rAmorphous where the following expression for total mass and definition of vC are appropriate: Total mass ¼ Crystalline mass þ Amorphous mass

vC ¼

Crystalline mass Total mass

340

Chapter 9 Experimental Analysis of Semicrystalline Polymers

Algebraic rearrangement of the volume additivity statement yields an expression for the mass fraction of crystallinity in terms of a ratio of differences of inverse densities. After multiplication of numerator and denominator of this ratio by the product of rCrystal and rAmorphous, further manipulation yields ( ) rCrystal ractual  rAmorphous vC ¼ ractual rCrystal  rAmorphous The second approach employs volumes and densities to invoke linear additivity of crystalline and amorphous masses: {Total volume}ractual ¼{Crystalline volume}rCrystal þ {Amorphous volume}rAmorphous The following expressions for total volume and definition of the volume fraction of crystallinity wC are appropriate: Total volume ¼ Crystalline volume þ Amorphous volume

wC ¼

Crystalline volume Total volume

Algebraic rearrangement of the mass additivity statement yields an expression for the volume fraction of crystallinity, which is invariably less than the mass fraction of crystallinity for semicrystalline polymers:

wC ¼

ractual  rAmorphous rCrystal  rAmorphous

9.7 PYCHNOMETRY: DENSITY AND THERMAL EXPANSION COEFFICIENT MEASUREMENTS OF LIQUIDS AND SOLIDS Equations are presented in the previous section to quantify the mass fraction vC and volume fraction wC of crystallinity in terms of densities. Unit cell dimensions (i.e., a, b, c) and angles (i.e., a, b, g) from wide-angle X-ray diffraction experiments characterize the crystallographic unit cell volume. For example, the volume of a crystallographic unit cell is given by Unit cell volume ¼ abc(sin a)(sin b)(sin g) An integer number of repeat units is required for periodicity along the chain backbone and an integer number of chains per unit cell is postulated to calculate a reasonable crystal density. As an example, there are 4 chains in each unit cell and 3 repeat units per chain are required for periodic behavior of the 3/1 helix in isotactic poly(propylene), Z[CH2CH(CH3)]Z, yielding 12 repeat units in each unit cell. If one postulates (i) 3 chains or (ii) 5 chains per unit cell, which might not be consistent with the symmetry

9.7 Pychnometry: Density and Thermal Expansion Coefficient Measurements

341

of the crystallographic lattice, then calculations of the density of a hypothetical 100% crystalline poly(propylene) with either (i) 9 or (ii) 15 repeat units in each unit cell will be either (i) too low relative to rAmorphous or (ii) too high for hydrocarbons. Pychnometry measurements yield accurate volumes of liquids or mixtures of liquids and solids that allow one to calculate rAmorphous and ractual, as required to predict vC and wC from equations provided in the previous section. Initially, one identifies a nonsolvent for a solid semicrystalline polymer and locates the nonsolvent density to at least four significant figures at the appropriate temperature. The nonsolvent density rNonsolvent must be less than the density of the actual semicrystalline polymer, ractual, to ensure that the solid does not float in the liquid. If mNonsolvent is the mass of liquid required to completely fill a nominal 5-mL pychnometer to its meniscus, then mNonsolvent/rNonsolvent represents a more accurate measure of the volume of the pychnometer that is used in subsequent calculations. Next, mPolymer is the mass of the solid semicrystalline polymer that is added to the empty pychnometer, which is then filled to the meniscus with the same nonsolvent. If mTotal is the mass of solid polymer and nonsolvent in the pychnometer, then volume additivity of both components yields the following expression, where mTotal is greater than mNonsolvent: mNonsolvent mPolymer mTotal  mPolymer ¼ þ rNonsolvent ractual rNonsolvent Rearrangement of the previous equation provides the desired relation to calculate the density of the actual solid semicrystalline polymer via pychnometry:

ractual

  mTotal  mNonsolvent 1 ¼ rNonsolvent 1  mPolymer

If mTotal is expressed as the mass of polymer mPolymer plus the mass of nonsolvent in the presence of polymer mNonsolvent/Polymer required to fill the pychnometer to its meniscus, then ractual reduces to 

ractual ¼ rNonsolvent

9.7.1

mPolymer mNonsolvent  mNonsolvent=Polymer



Thermal Expansion Coefficients

It is possible to modify the pychnometer described in the previous section by adding a highly accurate open-ended graduated pipette above the neck of the nominal 5-mL flask and submerging the entire apparatus into a well-controlled temperature bath. As described in this section, volume (v) versus temperature (T) measurements in the modified pychnometer are analyzed by invoking volume additivity to calculate volumetric coefficients of thermal expansion for liquids and solids via dynamic cooling. Liquid within the pychnometer rises into the graduated region of the pipette when temperature increases, and measurements of total volume versus bath temperature are

342

Chapter 9 Experimental Analysis of Semicrystalline Polymers

recorded as temperature decreases toward ambient. If cooling rates are not extremely slow, then sample temperatures might not be uniform and measurements could lack sufficient accuracy. Based on the definition of thermal expansion a,     1 @v @ ln v a¼ ¼ v @T p,composition @T p,composition one calculates a from the slope of lnfvg versus temperature, which can be performed for pure liquids with low volatility (i.e., the nonsolvent) and a mixture of the solid polymer with the nonsolvent, as illustrated in Figure 9.2 for (i) glycerol (upper graph) and (ii) polystyrene in glycerol (lower graph) between 30 8C and 60 8C. These two experiments yield direct calculations of aNonsolvent and aMixture, respectively, via linear least squares analysis. Next, nonlinear least squares analysis of volume (i.e., vMixture) versus temperature data for the mixture of the solid polymer and the nonsolvent provides an estimate of the thermal expansion coefficient for the solid polymer. Upon invoking volume additivity of the solid polymer and the nonsolvent, and taking the partial derivative of this simple relation with respect to temperature at constant pressure and composition, one obtains 

@vMixture @T

vMixture ¼ vPolymer þ vNonsolvent     @vPolymer @vNonsolvent ¼ þ @T @T p, composition p, composition p, composition



2.325 2.320 2.315

In{u}

2.310 2.305 2.300 2.295 2.290 2.285 2.280 25

30

35

40

45

50

55

60

65

Temperature (°C)

Figure 9.2 Dilatometric data for (i) glycerol (upper graph) and (ii) polystyrene in glycerol (lower graph) in a modified 10-mL pychnometer, where volume v is expressed in mL (i.e., millilitres) on the vertical axis. The calculated volumetric coefficients of thermal expansion, from the slope of lnfvg versus temperature, are 4.9992  1024 K21 for glycerol, and 4.7636  1024 K21 for the mixture of polystyrene in glycerol. Literature references for the thermal expansion coefficient of glycerol are 4.99  1024 K21 [Ohanian, 1994] and 5.200  1024 K21 [Lide, 1996].

9.7 Pychnometry: Density and Thermal Expansion Coefficient Measurements

343

It should be emphasized that vMixture at each temperature is obtained from experimental data for the mixture of the solid polymer and the nonsolvent, aMixture is calculated from the slope of ln{vMixture} versus temperature, aNonsolvent is calculated from the slope of ln{vNonsolvent} versus temperature in a separate pychnometry experiment that excludes the polymer, vNonsolvent in the presence of the polymer is predicted at any temperature via aNonsolvent, as illustrated below, and aPolymer and vPolymer are chosen to satisfy the following equation:

aMixture vMixture ¼ aPolymer vPolymer þ aNonsolvent vNonsolvent via nonlinear least squares minimization with one degree of freedom, aPolymer. For temperature-insensitive coefficients of thermal expansion, as suggested by pychnometry data mentioned above for aMixture and aNonsolvent, one assumes that aPolymer is also independent of temperature and employs the following expressions for vPolymer and vNonsolvent via integration of the defining equation for a from ambient temperature to temperatures slightly above ambient where volatilization of completely degassed high-boiling-point solvents is insignificant:  vPolymer (T) ¼  vNonsolvent (T) ¼

mPolymer ractual

 exp{aPolymer (T  Tambient )} @Tambient

vMixture 

mPolymer ractual

 exp{aNonsolvent (T  Tambient )} @Tambient

9.7.2 Nonlinear Least Squares Analysis of Volume –Temperature Data for Mixtures The last three equations from the previous subsection are used in conjunction with N data points for vMixture versus temperature to construct an objective function to be minimized with one degree of freedom: Objective function ¼

N X

{aMixture vMixture (Ti )  aPolymer vPolymer (Ti )

i¼1

 aNonsolvent vNonsolvent (Ti )}2 To reiterate, (i) vMixture(Ti) is obtained directly from pychnometry data for a mixture of the solid polymer and the nonvolatile solvent, (ii) aMixture is calculated from linear least squares analysis of this same data set, specifically the slope of lnfvMixtureg versus temperature, (iii) aNonsolvent is calculated from linear least squares analysis of volume – temperature data for the pure nonsolvent in an independent set of pychnometry experiments that excludes the solid polymer, and (iv) vPolymer(Ti) and vNonsolvent (Ti) are calculated at each temperature via the last two equations in the previous subsection in conjunction with ambient-temperature pychnometry data for the density of the solid polymer (i.e., ractual at Tambient) and the volume of the solid polymer – nonsolvent

344

Chapter 9 Experimental Analysis of Semicrystalline Polymers

mixture prior to increasing the bath temperature. Since v in all of these equations represents extensive volume, not specific or molar volume, it is necessary to predict vNonsolvent (Ti) via the last equation of the previous subsection because volume – temperature data for the pure nonsolvent yield an accurate aNonsolvent that is not influenced by the polymer, but nonsolvent volumes in pychnometer experiments that contain the solid polymer cannot be obtained directly from the data set for the pure nonsolvent. Nonlinear least squares minimization of the objective function with respect to aPolymer is implemented via a numerical analog of the following equation:   @(Objective function) ¼0 @ aPolymer aMixture ,aNonsolvent One constraint is that aMixture should be less than aNonsolvent to yield volumetric thermal expansion coefficients for solids that are greater than zero. Thermal shrinkage of anisotropic materials at higher temperature is consistent with negative coefficients of linear expansion in one particular coordinate direction, but the sum of all three coefficients of linear expansion yields the coefficient of volumetric expansion discussed in this chapter, which should be positive for uncrosslinked materials, as well as crosslinked solids that are not subjected to strains beyond the thermoelastic inversion point. Experimental analysis of volumetric coefficients of thermal expansion is pursued in greater detail using actual pychnometry data in Problems 9.5 and 9.6 for amorphous poly(vinylamine).

REFERENCES HAMMOND CR. The elements, in CRC Handbook of Chemistry and Physics, 55th edition, Weast RC, editor. CRC Press, Boca Raton, FL, 1974, p. B-85. LIDE DR. Fluid properties, in CRC Handbook of Chemistry and Physics, 77th edition, Lide DR and Frederikse HPR, editors. CRC Press, Boca Raton, FL, 1996. OHANIAN HC. Principles of Physics. WW Northon, New York, 1994. PATWARDHAN AA. Predicting Miscibility in Polymer Systems: A Group Contribution Approach, MS thesis, Colorado State University, 1986, Chapter 4. TESTER JW, MODELL M. Thermodynamics and Its Applications, 3rd edition. Prentice-Hall, Englewood Cliffs, NJ, 1997, pp. 769– 770.

PROBLEMS 9.1. A melt-quenched film of isotactic poly(1-butene) is annealed at ambient temperature to induce a considerable fraction of crystallinity via the 3/1 helical conformer. The following experiment is performed to quantify the crystalline content of the annealed material. A nominal 5-milliliter flask is used as a standard density-measuring device. Methanol, with a density of 0.7914 g/cm3 at 20 8C is chosen as the nonsolvent for the experiment. One finds that 3.934 grams of methanol completely fill the flask to the meniscus in the absence of polymer. Next, a strip of the annealed poly(1-butene) film with a mass of 1.560 grams is inserted in the flask and methanol is then added to fill the flask to the same volume as in the previous trial (in the absence of polymer). Due to density differences

Problems

345

between isotactic poly(1-butene) and methanol, the total mass of material in the flask is now 4.140 grams. Using the same pychnometer with methanol as the nonsolvent, the density of completely amorphous poly(1-butene) was estimated to be 0.864 g/cm3. (a) (b) (c) (d)

Calculate the density of semicrystalline poly(1-butene) at ambient temperature. Calculate the volume fraction of crystallinity. Calculate the mass fraction of crystallinity. Why was methanol, a nonsolvent for isotactic poly(1-butene), used instead of a good solvent? 9.2. A liquid (i.e., nonsolvent) of known density is used to fill a pychnometer to its meniscus in an effort to determine the exact volume of the pychnometer. Then, two separate and independent experiments are performed. In the first experiment, 1.358 grams of isotactic poly(1-butene) in the 3/1 helical polymorph with hexagonal crystal symmetry is introduced into the pychnometer and then mNonsolvent;3/1 is added to the meniscus level. In the second experiment, 1.358 grams of isotactic poly(1-butene) in the 11/3 helical polymorph with tetragonal crystal symmetry is introduced into the pychnometer and then mNonsolvent;11/3 is added to the meniscus level. Compare the nonsolvent masses, mNonsolvent;3/1 and mNonsolvent;11/3, that are required to fill the pychnometer to the meniscus when each of the different crystalline polymorphs is present. Are these masses equal, or is one of them larger? Both polymorphs of isotactic poly(1-butene) exhibit the same mass fraction of crystallinity—35%. 9.3. Design a series of simple experiments in logical order that allow you to determine the correct compression molding temperature for a polymer that is obtained from the commercial distributor as small pellets. The desired task is to thermoform these pellets into a uniform thin film (thickness ¼ 0.010 inch) with no defects from which samples can be cut in the shape of a dog bone for mechanical testing. The analytical department of your research and development laboratory has the capability of performing pychnometry, wide-angle X-ray diffraction (WAXD), carbon-13 solid state NMR spectroscopy (13C NMR), hotstage polarized optical microscopy (POM), differential scanning calorimetry (DSC), and thermogravimetric analysis (TGA). The TGA experiment measures weight loss of a sample in air or dry nitrogen as a function of temperature. The temperature range where large weight losses occur usually signifies the onset of thermal decomposition. You are not required to take advantage of all of these analytical capabilities. If necessary, a particular experiment can be performed multiple times. The following charges per sample will be billed to your account to perform the experiments. WAXD costs $10/sample. 13C solid state NMR costs $30/sample. Pychnometry costs $5/sample. DSC costs $25/sample. TGA costs $15/sample. Twelve POM photographs during one heating/cooling cycle cost $40/sample. Each attempt to compression mold the pellets requires operator intervention at $25 per trial. If the same experiment is performed twice, then the costs listed above must be doubled. The research director suggests that you should be somewhat conservative because there are limited funds in the budget for this research project. Provide a logical sequence of experiments to thermoform these polymeric pellets into a uniform 10-mil-thick film and report your total cost to achieve this task. 9.4. Reformulate the objective function for nonlinear least squares analysis of nonisothermal pychnometry data (i.e., vMixture vs. temperature) to predict the volumetric coefficient of thermal expansion of a solid that is submerged in a nonsolvent when one does not take the partial derivative with respect to temperature of the simple volume additivity relation for a two-component mixture, as discussed in Sections 9.7.1 and 9.7.2.

346

Chapter 9 Experimental Analysis of Semicrystalline Polymers Table 9.1 Volume–Temperature Data for Poly(vinylamine) in Benzaldehyde Temperature (8C)

Volume (mL)

37.0 36.0 35.0 34.0 33.0 32.0 31.0 29.8 29.0 28.0

10.2433 10.2349 10.2273 10.2183 10.2094 10.2014 10.1923 10.1818 10.1741 10.1636

9.5. Analyze the volume–temperature data in Table 9.1 for glassy poly(vinylamine), with Tg ¼ 56 8C, in benzaldehyde (i.e., Tboil ¼ 178 8C, aNonsolvent  8.6682  1024 K21) and predict the polymer’s volumetric coefficient of thermal expansion below the glass transition temperature. Independent ambient-temperature pychnometry data using toluene as the nonsolvent (i.e., Tboil ¼ 110.6 8C, rNonsolvent ¼ 0.8669 g/cm3) reveal that the solid density of poly(vinylamine) is 1.212 g/cm3. In the current variable-temperature experiments, 0.1019 g of this polymer was introduced into the modified pychnometer. Then, benzaldehyde was degassed and added to the pychnometer to yield a total volume of 10.0791 mL at 24 8C. Answer Enter the number of volume–temperature data points for the current experiment: N ¼ 10 Enter temperature in Kelvin and volume in mL for the solid –polymer –nonsolvent mixture from Table 9.1 (i.e., better known as a Lookup Table in Engineering Equation Solver): Duplicate i = 1,N Ti = lookup(i,1) + 273 fvMixturegi = lookup(i,2) End

Use linear least-squares analysis to calculate the volumetric coefficient of thermal expansion for the solid-polymer–nonsolvent mixture from the slope of lnfvMixtureg versus absolute temperature (see Appendix B in Chapter 14 to calculate the slope a1 ¼ aMixture for a first-order polynomial): sum# 1 ¼

N X i¼1

aMixture ¼

Ti ; sum#2 ¼

N X i¼1

Ti2 ; sum#3 ¼

N X

ln {(vMixture )i}; sum#4 ¼

i¼1

N(sum#4)  (sum#1)(sum#3) ¼ 8:5965  104 K1 N(sum#2)  (sum#1)2

N X i¼1

Ti ln {(vMixture )i }

Problems

347

2.594 2.593 2.592

In{u }

2.591 2.590 2.589 2.588 2.587 2.586 302 303 304 305 306 307 308 309 310 Temperature (K)

Figure 9.3 Dilatometric data for benzaldehyde in a modified 10-mL pychnometer, where volume v is expressed in mL (i.e., millilitres) on the vertical axis. The calculated volumetric coefficient of thermal expansion, from the slope of lnfvg versus temperature, is 8.6682  1024 K21. Enter the volumetric coefficient of thermal expansion for the nonsolvent, benzaldehyde, from linear least squares analysis of independent volume– temperature data illustrated in Figure 9.3 in the absence of the solid polymer (exactly analogous to the calculation outlined above for aMixture):

aNonsolvent ¼ 8:6682  104 K1 Enter the actual density of the solid polymer via ambient-temperature pychnometry experiments in toluene:

rPolymer ¼ ractual ¼ 1:212 g=cm3 Enter the mass of the solid polymer in the variable-temperature pychnometry experiments: mPolymer ¼ 0:1019 g Enter the total volume of the solid-polymer–nonsolvent mixture in the modified pychnometer at ambient temperature: {vMixture}ambient ¼ 10:0791 mL Enter ambient temperature in Kelvin: Tambient ¼ 24 þ 273 Construct the objective function that should be minimized with respect to aPolymer by invoking volume additivity. Then take the partial derivative of this simple additivity relation

348

Chapter 9 Experimental Analysis of Semicrystalline Polymers

with respect to temperature: Objective function ¼

N X

[aMixture {vMixture}i  aPolymer {vPolymer}i  aNonsolvent {vNonsolvent}i ]2

i¼1

{vPolymer}i ¼ vPolymer (Ti ) ¼

mPolymer exp{aPolymer (Ti  Tambient )} rPolymer

{vNonsolvent}i ¼ vNonsolvent (Ti ) "

# mPolymer exp{aNonsolvent (Ti  Tambient )} ¼ {vMixture}ambient  rPolymer

Thus, aPolymer  5.28  1025 K21 via “quadratic approximations” minimization of the Objective function ¼ 5.41  10212 (mL/K)2. 9.6. Use the methodology developed in Problem 9.4 to analyze the volume–temperature data in Table 9.2 for glassy poly(vinylamine) with 1 mol % cobalt chloride hexahydrate (i.e., Tg ¼ 92 8C) in benzaldehyde (i.e., Tboil ¼ 178 8C, aNonsolvent  8.6682  1024 K21) and predict the volumetric coefficient of thermal expansion for this macromolecule– metal complex below the glass transition temperature. Independent ambient-temperature pychnometry data using toluene as the nonsolvent (i.e., Tboil ¼ 110.6 8C, rNonsolvent ¼ 0.8669 g/cm3) reveal the following solid densities: 1.212 g/cm3 for poly(vinylamine); 1.956 g/cm3 for CoCl2(H2O)6, which compares well with the literature value of 1.924 g/cm3 [Hammond, 1974]; and 1.213 g/cm3 for poly(vinylamine) with 1 mol % CoCl2(H2O)6, which compares well with either mass-fraction-weighted or volumefraction-weighted theoretical predictions of 1.237 g/cm3. In the current variabletemperature experiments, 0.1026 g of this macromolecule–metal complex with 1 mol % CoCl2(H2O)6 was introduced into the modified pychnometer. Then, benzaldehyde was degassed and added to the pychnometer to yield a total volume of 13.2146 mL at 24 8C.

Table 9.2 Volume–Temperature Data for a Solid Complex of Poly(vinylamine) with 1 mol % CoCl2(H2O)6 in Benzaldehyde Temperature (8C)

Volume (mL)

37.5 36.5 35.5 34.5 33.5 32.5 31.5 30.5 29.5 28.5

13.3702 13.3593 13.3466 13.3351 13.3240 13.3123 13.3004 13.2882 13.2760 13.2647

Problems

349

Answer Enter the number of volume–temperature data points for the current experiment: N ¼ 10 Enter temperature in Kelvin and volume in mL for the solid-polymer-complex–nonsolvent mixture from Table 9.2 (i.e., better known as a Lookup Table in Engineering Equation Solver): Duplicate i = 1,N Ti = lookup(i,1)+273 fvMixturegi = lookup(i,2) End

Enter the volumetric coefficient of thermal expansion for the nonsolvent, benzaldehyde, from linear least squares analysis of independent volume–temperature data in the absence of the solid macromolecule– metal complex:

aNonsolvent ¼ 8:6682  104 K1 Enter the actual density of the solid macromolecule–metal complex via ambient-temperature pychnometry data using toluene as the nonsolvent:

rPolymer ¼ ractual ¼ 1:213 g=cm3 Enter the mass of the solid macromolecule–metal complex in the variable-temperature pychnometry experiments: mPolymer ¼ 0:1026 g Enter the total volume of the solid-polymer-complex–nonsolvent mixture in the modified pychnometer at ambient temperature: {vMixture}ambient ¼ 13:2146 mL Enter ambient temperature in Kelvin Tambient ¼ 24 þ 273 Invoke volume additivity and construct the objective function that should be minimized with respect to aPolymer. Do not perform partial differentiation with respect to temperature: Objective function ¼

N X

[{vMixture}i  {vPolymer }i  {vNonsolvent}i ]2

i¼1

{vPolymer}i ¼ vPolymer (Ti ) ¼

mPolymer exp{aPolymer (Ti  Tambient )} rPolymer

{vNonsolvent}i ¼ vNonsolvent (Ti )  mPolymer  ¼ {vMixture}ambient  exp{aNonsolvent (Ti  Tambient )} rPolymer Thus, aPolymer  5.61  1025 K21 via “quadratic approximations” minimization of the Objective function ¼ 6.96  1026 mL2. 9.7. Develop mass-fraction-weighted and volume-fraction-weighted predictions for the density of multicomponent mixtures. Then, compare these theoretical predictions with the pychnometry data in Table 9.3 for glassy poly(vinylamine) complexes that contain cobalt chloride hexahydrate, CoCl2(H2O)6.

350

Chapter 9 Experimental Analysis of Semicrystalline Polymers

Table 9.3 Thermophysical Properties, Including Measured and Theoretical Densities, of Poly(vinylamine) Complexes with Cobalt Chloride Hexahydrate Macromolecule –metal complex

Glass transition temperature (8C)

Measured density (g/cm3)

Theoretical density (g/cm3)

56 92

1.212 1.213

— 1.237

143

1.236

1.261

210

1.246

1.305

—

1.956

1.924 a

Poly(vinylamine) PVA/1 mol % CoCl2(H2O)6 PVA/2 mol % CoCl2(H2O)6 PVA/4 mol % CoCl2(H2O)6 CoCl2(H2O)6 a

Literature value [Hammond, 1974].

Answer The first approach assumes that the total volume of a multicomponent mixture can be represented by additive contributions from the volume of each component: X Mass of species i Total mass ¼ rtheoretical rpure component i all species i which yields the following inverted mass-fraction-weighted prediction for the mixture’s density: 1

¼

r theoretical

vi weighted

(

X all species i

vi

)

rpure component i

The second approach employs volumes and densities to invoke additivity of the mass of each component in multicomponent mixtures, yielding a linear volume-fraction-weighted prediction for the mixture’s density: {Total volume}rtheoretical ¼

X

{Volume of species i}rpure component i

all species i

r theoretical

wi weighted

¼

X

{wi rpure component i}

all species i

The mass-fraction-weighted prediction for the mixture’s density is employed to illustrate the procedure used to obtain theoretical densities of macromolecule–metal complexes in Table 9.3. The conversion between mole fraction and mass fraction requires molar masses of the individual components—MWPVA repeat unit ¼ 43 daltons and MWCobalt chloride hexahydrate ¼ 238 daltons. Since the relation between mass fraction vi and mole fraction yi is yi MWi vi ¼ P yj MWj j

Problems

351

the theoretical inverted mass-fraction-weighted prediction of the mixture’s density is given by 1 rtheoretical

¼

r theoretical

¼

v PVA v CoCl2 (H2 O)6 þ r PVA r CoCl2 (H2 O)6

v i weighted

v i weighted

yPVA MWPVA þ yCoCl2 (H2 O)6 MWCoCl2 (H2 O)6 1 yPVA MWPVA þ yCoCl2 (H2 O)6 MWCoCl2 (H2 O)6 rPVA rCoCl2 (H2 O)6 X yi MWi 1

extrapolated to

all species i

) X v weighted multicomponent mixtures

r theoretical

1 yj MWj r all species j j

i

Theoretical densities for poly(vinylamine) complexes with cobalt chloride hexahydrate were predicted using this equation together with the experimental pure-component density of CoCl2(H2O)6, calculated as 1.956 g/cm3. Next, the linear volume-fraction-weighted prediction of the density of multicomponent mixtures requires interconversion between mole fraction and volume fraction, based on pure-component molar volumes y i of individual components; y i ¼ MWi/ri. Begin with the relation between volume fraction wi and mole fraction yi: MWi r wi ¼ ¼ X i MWj y j yj yj rj all species j all species j yi

yy Xi i

and substitute this expression for wi into the linear volume-fraction-weighted prediction for the density of multicomponent mixtures: X yi MWi X X all species i rtheoretical ¼ {wi rpure component i} ¼ {wi ri} ¼ X MWj wi weighted all species i all species i yj rj all species j Hence, the linear volume-fraction-weighted theoretical prediction of the density of multicomponent mixtures is equivalent to the inverted mass-fraction-weighted prediction. It is instructive to compare the functional form of both expressions for the density of multicomponent mixtures with the discrete functional form (not the continuous functional form) of two equivalent expressions for the number-average molecular weight Mn of mixtures that contain several components (see Problem 12.1 in Chapter 12): X X vi  1 rtheoretical ¼ {wi ri }; ¼ r theoretical ri wi weighted all species i all species i vi weighted

Mn ¼

X all species i

{yi MWi };

X  vi  1 ¼ Mn all species i MWi

In terms of the average density expressions, the linear volume-fraction-weighted sum is based on additivity of the mass of each component in the mixture, whereas the inverted mass-fractionweighted sum invokes volume additivity. In terms of number-average molecular weight, the linear mole-fraction-weighted sum is based on additivity of the mass of each component in

352

Chapter 9 Experimental Analysis of Semicrystalline Polymers

the mixture, whereas the inverted mass-fraction-weighted sum is based on additivity of the moles of each species in the mixture. 9.8. Search the research literature, handbooks, and textbooks for thermoplastic polymers that exhibit the following characteristics: (i) optical transparency on the order of 80% and (ii) volumetric coefficient of thermal expansion less than 4.0  1025 K21 in the solid state. Answer Poly(ether ether ketones), PEEK, exhibit approximately 50% transparency, due to the presence of crystallites. Their solid state coefficients of thermal expansion are typically  5  1025 K21. Quenching from the molten state, above the melting temperature, should eliminate most of the crystalline phase and increase transparency above 50%. However, elimination of the crystalline phase in PEEK might increase its solid state thermal expansion coefficient above 5  1025 K21. 9.9. Qualitatively explain why transient batch-reactor material balances for amorphous and crystalline mass (i.e., see expressions in Section 9.3) require that the rate of crystallization RCrystallization must be multiplied by the product of amorphous volume fraction wAmorphous and total system volume V. What are the dimensions of RCrystallization? Where do “reactants” reside for the crystallization process? How should both unsteady state material balances be modified if crystallization is modeled as a heterogeneous “reaction” where chains from the amorphous phase add to the lateral periphery (not the fold surface) of growing lamellae, such that RCrystallization has dimensions of mass per surface area per time.

Part Three

Mechanical Properties of Linear and Crosslinked Polymers

Chapter

10

Mechanical Properties of Viscoelastic Materials: Basic Concepts in Linear Viscoelasticity Lonely birds bear food from the land of the dead. —Michael Berardi

Linear viscoelasticity is introduced and Maxwell’s constitutive equation is developed by combining Hooke’s law of elasticity and Newton’s law of viscosity. Time dependence of stress and strain is analyzed for creep, stress relaxation, dynamic mechanical testing, and the torsion pendulum. The concept of the dimensionless Deborah number is used to introduce the principle of time – temperature superposition. The Boltzmann superposition integral for linear viscoelasticity connects dynamic properties and stress relaxation moduli via Fourier transformation. The irreversible degradation of mechanical energy to thermal energy in forced-vibration and freevibration experiments is analyzed quantitatively and related qualitatively to the fluctuation – dissipation theorem in statistical physics.

10.1 MATHEMATICAL MODELS OF LINEAR VISCOELASTICITY Macroscopic mechanical response of high-molecular-weight polymers subjected to time-dependent stress and strain is an extremely important topic with significant industrial relevance. Linear response is achieved at small deformations. Hooke’s law is appropriate for metallic solids and Newton’s law describes the behavior of low-molecular-weight viscous liquids. These laws are combined to describe the

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

355

356

Chapter 10 Mechanical Properties of Viscoelastic Materials

flow behavior of memory fluids, as well as the mechanical properties of polymeric solids. Memory fluids store elastic energy, whereas viscous fluids dissipate mechanical energy and provide mechanisms for impact resistance and damping. All of these characteristics of polymeric materials are captured mathematically with assistance from springs and dashpots, or capacitors and resistors via electrical analogs. Of particular importance, oscillatory experiments and the corresponding analysis of in-phase and out-of-phase response allow one to bridge the gap between molecular structure and macroscopic material properties.

10.2 OBJECTIVES The primary purpose of this chapter is to introduce simple laws of mechanics for elastic solids and viscous liquids, present analogies between mechanical and electrical concepts, and develop simple constitutive equations that describe the mechanical response of viscoelastic solids and memory fluids. Four well-defined mechanical experiments are described in which the time dependence of stress and strain is predicted and related to important material properties.

10.3 SIMPLE DEFINITIONS OF STRESS, STRAIN, AND POISSON’S RATIO A rectangular-shaped or dog-bone-shaped sample has initial length Linitial and cross-sectional area Ainitial that are measured using a caliper prior to deformation. When force F is applied in tensile mode, the deformed sample length L increases. Almost all materials will exhibit a decrease in cross-sectional area Ainstantaneous because Poisson’s ratio is greater than zero. Poisson’s ratio y characterizes lateral contraction upon extension. When y is between zero and 0.5, materials contract laterally such that their volume increases due to uniaxial tensile deformation. Incompressible liquids and chemically crosslinked rubber-like solids are described by y ¼ 0.5, because no volume change occurs when these materials deform. Cork is useful as a stopper because it is characterized by y ¼ 0, which implies that the initial cross-sectional area is not affected by tensile or compressive forces. Lateral expansion and volume increase due to uniaxial tensile deformation, as well as lateral contraction and volume shrinkage due to compression, are described by negative values of Poisson’s ratio (i.e., with stable values as large as 24), but these auxetic materials represent the exception, not the rule. The following references introduce some cellular foams and fibrous materials with negative Poisson’s ratios that expand laterally upon extension: Alderson et al. [2002], Almgren [1985], Brandel and Lakes [2001], Evans et al. [2004], Lakes [1987], Rovati [2003], Ruzzene et al. [2002, 2003], Scarpa et al. [2002, 2004], Wang et al. [2001], Webber et al. [2000], and Yang et al. [2003]. Lateral contraction and volume decrease due to uniaxial tensile deformation would be described hypothetically by y . 0.5, but this effect is not observed in any known materials. Hence, the most common situation corresponds to L . Linitial and Ainstantaneous , Ainitial after tensile deformation, such that total volume of the sample

10.4 Stress Tensor

357

increases (i.e., 0 , y , 0.5). True and engineering stresses, with units of force per unit area, are defined as follows:

strue ¼

Force Ainstantaneous

;

sengineering ¼

Force Ainitial

where Ainstantaneous is the instantaneous cross-sectional area of the sample when its deformed length is L. Unless Poisson’s ratio is negative, true tensile stress is always larger than engineering tensile stress because materials contract laterally upon extension, but strue , sengineering when materials are compressed and y . 0. There are several definitions of strain, all of which do not yield the same magnitude. Engineering strain is defined relative to an undeformed frame of reference, such as the initial sample length. Hence, L  Linitial gengineering ¼ ¼d1 Linitial where elongation d is given by the ratio of L to Linitial. An undeformed sample is characterized by an elongation of 1, with zero strain. True strain is defined with respect to a continuously deforming frame of reference, based on the sum of infinitesimal changes in sample length relative to the instantaneous length x of the sample. In mathematical terms,   ðL dx L ¼ ln gtrue ¼ x Linitial Linitial

When samples are stretched and their length increases, engineering strain is always larger than true strain. However, for very small deformation where L is only slightly larger than Linitial, Taylor series expansion of gtrue yields an alternating series in which the leading first-order term is synonymous with engineering strain: 

gtrue ¼ ln

L

Linitial



  L  Linitial L  Linitial     ¼ gengineering     ¼ ln 1 þ  Linitial Linitial

The discussion below does not distinguish between true and engineering properties, identifying s as stress and g as strain. However, material failure occurs when the applied force produces internal stresses that exceed the upper limit of true stress. In other words, there is significant stress intensity in regions where the cross-sectional area has decreased considerably. This effect is captured by monitoring true stress, not engineering stress. It should be mentioned that stress and strain are actually secondrank tensors, as described below.

10.4 STRESS TENSOR A second-rank tensor, like s, contains nine scalars that completely describe the state of stress in a material. These scalars are identified via two subscripts on s, where each

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Chapter 10 Mechanical Properties of Viscoelastic Materials

subscript represents a particular coordinate direction in an orthogonal coordinate system (i.e., rectangular, cylindrical, or spherical coordinates). For example, sij represents the jth-component of a vector force per unit area which acts across a surface that is perpendicular to the ith-coordinate direction. In other words, the simple surface is oriented such that the unit normal vector everywhere on the surface points in the ith-coordinate direction. Consider the simple surface in rectangular coordinates that is perpendicular to the x-direction and intersects the x-axis at x ¼ b. On this surface, y and z vary, but x remains constant. This is consistent with the fact that the unit normal vector on the surface is oriented in the x-direction. The vector force per unit area that acts across this surface due to the stress tensor has components in the x-, y-, and z-directions, given by sxx, sxy, and sxz, respectively. The nine scalars that characterize s can be presented in 3  3 matrix form, where the elements in the first, second, and third rows contain all three components (i.e., x-, y-, and z-components) of the vector forces per unit area that act across three mutually perpendicular simple surfaces with unit normal vectors in the x-, y-, and z-directions, respectively. The three scalars on the main diagonal from upper left to lower right have two identical subscripts, sii, and they are classified as normal stresses because the surface across which the stress acts is perpendicular to the direction in which the force is applied. These normal stresses can act in tension or compression. The definition of stress in the previous section is based on one of these three normal components in tension. There are six off-diagonal components of s in which the subscripts are different, sij, where i = j. These are classified as shear stresses because the surface across which the stress acts is parallel to the direction in which the force is applied. This is analogous to stating that shear forces act in the plane of the surface. The stress tensor is symmetric, which implies that sij ¼ sji. This translates into the fact that the matrix representation of s is symmetric about the main diagonal from upper left to lower right. Hence, there are only three independent shear stress components because s is a symmetric second-rank tensor. There are a total of six independent scalars that completely describe the state of stress in a material; three shear components and three normal components.

10.5 STRAIN AND RATE-OF-STRAIN TENSORS Consider a material displacement vector u that describes the state of deformation at time t relative to a reference state at time t0. In rectangular coordinates, where the x-, y-, and z-directions are identified by subscripts 1, 2, and 3, respectively, the displacement vector for a differential volume element is written in terms of unit vectors di in the ith-coordinate direction and the corresponding scalar displacements ui: u(t0 ) t) ¼ d1 u1 (x1 , x2 , x3 , t; t0 ) þ d2 u2 (x1 , x2 , x3 , t; t0 ) þ d3 u3 (x1 , x2 , x3 , t; t0 ) The components of the second-rank strain tensor g are defined by   1 @uj @ui gij ¼ þ 2 @xi @xj

10.6 Hooke’s Law of Elasticity

359

This tensor is symmetric because gij ¼ gji. The symmetric second-rank rate-of-strain tensor is obtained by taking the partial time derivative of g, realizing that each scalar component of the displacement vector is an exact differential. This implies that the order of mixed second partial differentiation can be reversed without affecting the final result. Hence,      @ gij 1 @ @uj @ @ui ¼ þ @t 2 @xi @t @xj @t Since the time rate of change of each component of the material displacement vector is synonymous with the same component of the velocity vector, vk ¼

@uk @t

each scalar component of the rate-of-strain tensor can be written as   @ gij 1 @vj @vi ¼ þ @t 2 @xi @xj which is expressed in tensor form as @g 1 ¼ g_ ¼ {rv þ (rv)T } @t 2 where (rv)T is the transpose of the velocity gradient tensor. All of the second-rank tensors presented in this section and the previous section are symmetric. The next two sections discuss well-known fundamental laws of mechanics, which provide relations between these symmetric tensors.

10.6 HOOKE’S LAW OF ELASTICITY The constitutive relation between stress and strain for perfectly elastic isotropic solids is given by Hooke’s law:

s ¼ Eg where the modulus of elasticity E is a proportionality constant that measures the resistance to deformation. This symmetric second-rank tensor relation implies that nine scalar equations must be satisfied, but only six of these equations are independent. In other words,

sij ¼ E gij For anisotropic solids in which material properties such as elastic modulus exhibit directionality, E is a fourth-rank tensor with 81 components that describes how Eijkl couples sij to glk. Obviously, the laws of mechanics are simplified for isotropic materials, when the elastic modulus is a scalar instead of a higher-order tensor. Hooke’s law is modeled mechanically by a spring and electronically by a capacitor. These elements store either mechanical energy when the spring deforms, or electrical energy when electric charge accumulates on the plates of the capacitor. There is

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Chapter 10 Mechanical Properties of Viscoelastic Materials

complete memory in both cases because all of the mechanical or electrical energy can be recovered. Springs release mechanical energy upon removal of the applied stress when they return to their equilibrium dimensions. Capacitors release electrical energy or power when transient current flows upon removal of the external voltage drop across its plates. Real isotropic materials exhibit deviations from purely elastic response when the linearity between s and g is no longer obeyed at strains beyond 5 –10%. Hence, the elastic modulus is measured experimentally from the initial slope of the stress –strain curve, as indicated below:   ds E ¼ lim g )0 d g

10.7 NEWTON’S LAW OF VISCOSITY The constitutive relation between stress and rate-of-strain is given by Newton’s law for viscous liquids that are incompressible, Newtonian, and isotropic: @g @t where the proportionality constant h is the viscosity, which measures resistance to flow. Typically in fluid mechanics, Newton’s law contains a negative sign, by convention, to indicate the direction in which momentum is transferred and to emphasize analogies between the molecular fluxes for heat, mass, and momentum transport. Once again, this relation between symmetric second-rank tensors implies that nine scalar equations must be satisfied, but only six of these equations are independent. For anisotropic liquids, viscosity is a fourth-rank tensor, requiring a maximum of 36 scalars to characterize flow resistance because the constitutive equation is symmetric. Viscous characteristics of a viscoelastic solid or liquid are modeled mechanically by a plunger or piston in a container of a Newtonian fluid, and electronically by a resistor with real impedance R. In each case, there is no memory of past history, and mechanical or electrical energy is completely dissipated into thermal energy.

s¼h

10.8 SIMPLE ANALOGIES BETWEEN MECHANICAL AND ELECTRICAL RESPONSE As described by the previous two laws of mechanics for perfectly elastic isotropic solids and purely viscous liquids, the important mechanical quantities of interest are stress s, strain g, and rate-of-strain @ g/@t. The analogous quantities in circuit theory are voltage drop V, charge q, and current i ¼ dq/dt, respectively. Completely elastic response in the linear regime is modeled mechanically by a spring with static modulus E, such that

s ¼ Eg

10.9 Phase Angle Difference between Stress and Strain and Voltage and Current

361

The electrical analog of Hooke’s law is given by the relation between voltage drop and charge stored on the plates of a capacitor with capacitance C via Farad’s law: V¼

q C

Hence, the static compliance J of the spring (i.e., J ¼ 1/E) is analogous to C. Materials subjected to the same stress will exhibit larger strain if they are more compliant with a lower elastic modulus. Analogously, capacitors with larger capacitance store more charge on their plates when the same voltage drop is imposed across them. The electrical analog of Newton’s law of viscosity is Ohm’s law, which relates voltage drop across a resistor to the current flowing through the circuit. Hence,

s¼h

@g ; @t

V ¼ iR ¼ R

dq dt

where viscosity h and resistance R are analogous.

10.9 PHASE ANGLE DIFFERENCE BETWEEN STRESS AND STRAIN AND VOLTAGE AND CURRENT IN DYNAMIC MECHANICAL AND DIELECTRIC EXPERIMENTS This is an important topic that will be discussed in more detail after the properties of viscoelastic solids are introduced. However, it is instructive to analyze the harmonic response of elastic solids and viscous liquids separately when they are subjected to oscillatory forcing functions. In other words, one imposes sinusoidal stress on an elastic solid and an analogous sinusoidal voltage drop across a capacitor. Measurements of the induced strain and current reveal that the responses are linear because g and i ¼ dq/dt oscillate at the same frequency as the forcing functions that are imposed on the elastic solid and the capacitor. Higher order harmonic response at 2v, 3v, and so on, is characteristic of materials that exhibit nonlinear behavior at large strain. In linear dynamic experiments, where the property of interest is the phase angle difference between the forcing function and the induced response, one proceeds as follows if the sample always experiences tensile stress (i.e., sdc  s0):

s (t; v) ¼ sdc þ s0 sin(vt) Hooke’s law describes how a perfectly elastic isotropic solid responds to the application of harmonic tensile stress:

g (t; v) ¼ J{sdc þ s0 sin(vt)} where J ¼ 1/E is the static compliance of the spring. Hence, stress and strain oscillate in-phase at the same frequency for an elastic solid. Analogously, voltage and charge oscillate in-phase at the same frequency when a harmonic voltage drop is applied

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Chapter 10 Mechanical Properties of Viscoelastic Materials

across a capacitor. In dynamic dielectric experiments, the phase angle difference between voltage and current is most important. In this case, V(t; v) ¼ V0 sin(vt) i(t; v) ¼

dq dV ¼C ¼ CV0 v cos(vt) dt dt

Voltage and current oscillate 908 out-of-phase at the same frequency in the electrical analog of an elastic solid, and current leads the voltage via ELI the ICE man. The same set of dynamic experiments is performed hypothetically on a viscous liquid by agitating or wiggling the container, and the electrical analog of a viscous liquid. Newton’s law applied to a viscous element with sdc ¼ 0 yields an expression for the rate-of-strain, which must be integrated to calculate the harmonic strain: @g 1 s0 ¼ s (t; v) ¼ sin(vt) @t h h Hence,

g (t; v) ¼ Constant 

s0 cos(vt) vh

Now, stress and strain oscillate 908 out-of-phase at the same frequency for a viscous liquid. The phase angle difference between s and g lies somewhere between 08 and 908 for viscoelastic solids and liquids that are subjected to dynamic mechanical testing. Analogously, when a harmonic voltage drop is applied across a resistor, Ohm’s law reveals that the induced current oscillates in-phase with V at the same frequency. These results are summarized below for both types of dynamic experiments. Phase angle difference between s and g Phase angle difference between V and i

Elastic Solids

Viscous Liquids

08 908

908 08

The temperature and frequency dependence of the phase angle difference between (i) s and g in dynamic mechanical testing and (ii) V and i for dynamic dielectric spectroscopy allows one to identify molecular motion and viscoelastic phase transitions in polymeric solids. By convention, larger phase angle differences are characteristics of materials that can dissipate mechanical or electrical energy into thermal energy via their viscous component. Hence, the dynamic dielectric experiment actually reports 908 minus the phase angle difference between voltage and current.

10.10 MAXWELL’S VISCOELASTIC CONSTITUTIVE EQUATION Maxwell [1867] proposed the following relation between stress and strain to describe the behavior of viscoelastic materials that exhibit the combined characteristics of elastic solids and viscous liquids. These materials reveal “fading memory,” which is a superposition of the excellent memory of elastic solids and the terrible memory

10.10 Maxwell’s Viscoelastic Constitutive Equation

363

of viscous liquids. Maxwell’s model contains only one material response time or relaxation time constant l, which is temperature dependent:

s (t) þ l(T)

@s @g ¼h @t @t

where the viscosity h is not shear-rate dependent, and l ¼ h/E. Strain additivity was invoked to obtain the previous equation that could describe nonlinear viscoelastic response if material properties (i.e., h, E, l ) exhibit dependence on strain or rate of strain. The effect of strain on viscoelastic relaxation times is discussed phenomenologically in Chapter 11. Partial derivatives are employed in Maxwell’s model, instead of total derivatives, because stress and strain depend on position and time. Only the time dependence is discussed in this chapter on the mechanical properties of viscoelastic materials in the linear regime. The validity of Maxwell’s model is revealed by considering two different limiting cases, as described below. Case 1: When stress s varies very slowly over an experimental time scale given by texpt, then order-of-magnitude estimates of the two terms on the left side of the previous equation suggest that the first term is more important. In other words,   @s s s l l(T) @t texpt Hence, when the experimental time scale is much longer than the material response time (i.e., texpt  l ), viscoelastic materials exhibit liquid-like behavior and follow Newton’s law of viscosity. Observation of the flow characteristics of (i) silly putty during a period of several days or (ii) colored glass windows of a medieval church for a few centuries represent examples of these relative time scales. Case 2: At the other extreme, when viscoelastic materials are subjected to rapidly changing stress over experimental time scales that are much shorter than the material response time (i.e., texpt  l ), the second term on the left side of the Maxwell model dominates:   @s s s l l(T) @t texpt Now, one recovers Hooke’s law upon integration of the remaining terms in Maxwell’s equation:

l(T)

@s @g h ; @t @t

s ¼ E g þ Constant

The integration constant must vanish if the material recovers its original dimensions when the stress is removed. Hence, viscoelastic materials exhibit solid-like behavior and follow Hooke’s law of elasticity when

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Chapter 10 Mechanical Properties of Viscoelastic Materials

texpt  l(T ). Silly putty behaves in solid-like fashion when it abruptly impacts a solid surface, and a “belly flop” dive into a swimming pool can be perceived as an impact with “solid” water.

10.11 INTEGRAL FORMS OF MAXWELL’S VISCOELASTIC CONSTITUTIVE EQUATION The one-time-constant model in differential form, presented in the previous section, can be integrated for any strain-rate history after (i) division by l and (ii) multiplication by the integrating factor, which is exp(t/l ). The result is t 1  t  h @g t @s exp exp þ s (t) exp ¼ @t l l l l l @t Now, the left side of the previous equation can be combined into a single term:  t o h @ g  t  @n exp s exp ¼ l l @t l @t Integration is performed with respect to time variable Q over all past history, from the distant past, Q )21, to the present time, Q ¼ t, subject to the condition of finite stress as Q )21. Hence,

s (t) exp

nto

l

¼

Q¼t ð

  h Q @g dQ exp l @Q l

Q)1

When material property l depends only on temperature, the final result from linear viscoelasticity for time-dependent stress in memory fluids and solids that experience strain-rate history given by @ g/@Q is

s (t) ¼

Q¼t ð

  h t  Q @g dQ exp  l @Q l

Q)1

It reveals that as the time interval t 2 Q increases, forcing functions imposed on viscoelastic materials at time Q have increasingly diminishing contributions to the present state of stress at time t. The exponential weighting factor indicates that viscoelastic materials exhibit excellent memory about strain rates imposed at time Q which is close to t, but recollection of strain rates imposed in the distant past is fuzzy (i.e., Q much earlier than t, such that t 2 Q is large). As illustrated later in this chapter,     h tQ tQ ¼ ER (t  Q; T) exp  ¼ E exp  l l(T) l represents the time-dependent relaxation modulus at time t for the one-time-constant Maxwell model when a “jump” strain is imposed on the material at time Q. Hence,

10.11 Integral Forms of Maxwell’s Viscoelastic Constitutive Equation

365

integration of Maxwell’s viscoelastic constitutive equation and identification of the time-dependent relaxation modulus yield a generic expression for the time dependence of stress that is consistent with the Boltzmann superposition integral for linear viscoelastic response: Q¼t ð

s (t) ¼

ER (t  Q; T)

@g dQ @Q

Q)1

The present state of stress s (t) depends on the complete strain-rate history, via @ g/@Q, where time variable Q ranges from 21 to present time t. The previous equation also illustrates a concept from linear response theory, where the total stress is obtained by linear superposition of effects due to imposing an infinite number of sequential infinitesimal strains that define the complete strain history experienced by a memory fluid or solid. Relaxation moduli and the Boltzmann superposition integral for linear viscoelasticity are discussed in more detail later in this chapter. Now, integrate the previous expression by parts when present time t does not change. Let u ¼ ER (t  Q; T); @g dQ; v ¼ dv ¼ @Q

du ¼ wð¼t

@ER (t  Q; T) (dQ) @(t  Q)

@g dw ¼ g (Q ) t) @w

w¼Q

Hence,

s (t) ¼ [g (Q ) t)ER (t  Q;

T)]Q¼t Q)1

þ

Q¼t ð

  @ER (t  Q; T) g (Q ) t) dQ @(t  Q)

Q)1

The first term on the right side of the previous expression vanishes at the upper and lower limits because: (i) g (Q ) t) ¼ 0 for Q ¼ t, since there is no deformation in state t relative to state Q if both states are identical. (ii) ER(t 2 Q; T ) ¼ 0 for Q )21, since there is no memory about the strain history in the extreme past unless crosslinks are present. An equivalent statement of the Maxwell model for the time-dependent stress experienced by uncrosslinked materials in the linear regime is

s (t) ¼

Q¼t ð

  @ER (t  Q; T) g (Q ) t) dQ @(t  Q)

Q)1

where @ER/@(t 2 Q) represents a memory function. Hence, the following three expressions are equivalent for the time dependence of stress in viscoelastic materials

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Chapter 10 Mechanical Properties of Viscoelastic Materials

with one temperature-dependent relaxation time constant l:

s (t) þ l(T)

@s @g ¼h @t @t

s (t) ¼

Q¼t ð

ER (t  Q; T)

@g dQ @Q

Q)1

s (t) ¼

Q¼t ð



 @ER (t  Q; T) g (Q ) t) dQ @(t  Q)

Q)1

This constitutes the Maxwell model, which is equivalent to an elastic element (i.e., spring) in series with a viscous element (i.e., dashpot).

10.12 MECHANICAL MODEL OF MAXWELL’S VISCOELASTIC CONSTITUTIVE EQUATION The objective of this section is to consider the generic mechanical response of viscoelastic materials that are modeled as an elastic element in series with a viscous element, and demonstrate that stress and strain obey Maxwell’s equation discussed in the previous two sections. When time-dependent stress is applied to a spring and dashpot in series at time t0, the following rules are invoked: (i) The same stress is experienced by each element in series. In more rigorous terms, the same force is experience by each element in series. (ii) Total strain is obtained by adding the contribution to strain from each element. Likewise, total strain rate is obtained by adding the contribution to @ g/@t from each element. (iii) Unrestricted (i.e., free) springs respond instantaneously and reach their elastic limit if they are not hindered by dashpots in parallel. (iv) Dashpots do not respond instantaneously. However, once they begin to flow, their response will continue indefinitely unless they are restricted by springs in parallel. Statements (i) and (ii) are employed to analyze the response of this mechanical model with one viscoelastic time constant l ¼ h/E. If s (t) is imposed on each element, then the following strains and strain rates occur: Strain, g(t)

Rate of Strain, @ g/@ @t

Elastic element

s (t)=E

Viscous element

ðt

1 @s E @t 1 s (t) h

t0

1 s (t0 ) dt0 h

10.13 Four Well-Defined Mechanical Experiments

367

If @ g/@t represents the total rate of strain, then @g ¼ @t

    @g @g 1 @s 1 þ s þ ¼ @t Elastic @t Viscous E @t h

Multiplication by viscosity h yields the differential form of Maxwell’s constitutive equation for viscoelastic response with one temperature-dependent relaxation time, l ¼ h/E: @s @g ¼h s (t) þ l(T) @t @t The electrical analog of the Maxwell model consists of a parallel arrangement of a resistor and a capacitor. In this case, each circuit element experiences the same voltage drop and the total current is obtained by summing the current through each element. This corresponds to the fact that elastic and viscous elements in series experience the same stress, whereas the total rate of strain is obtained by summing the strain rate in each element.

10.13 FOUR WELL-DEFINED MECHANICAL EXPERIMENTS The following laboratory experiments allow one to measure the viscoelastic response of polymeric materials when the forcing function (i.e., s or g) is well defined. Materials are designed to withstand realistic conditions based on their performance in these laboratory tests. 1. Creep and Creep Recovery. Materials are subjected to a “jump” stress by placing them under a constant load. Creep response is obtained by measuring the time-dependent strain while engineering stress remains constant. Creep recovery measures the material’s ability to recover its original dimensions after the stress is removed. These results are useful because most structural materials are subjected to constant stress for extended times, and it is important to minimize creep, or the time-dependent creep compliance, for these applications. 2. Stress Relaxation. Materials are subjected to a “jump” strain by stretching them instantaneously to a small but constant strain. The induced stress decreases as time evolves. For uncrosslinked polymers, stress decays exponentially to zero at long time. The asymptotic response at long times yields nonzero stress in crosslinked solids because viscous flow is restricted by the crosslink junctions. When materials exhibit large strains at failure, obvious questions arise that address potential energy dissipation mechanisms associated with the deformation process. Stress relaxation measurements that focus on viscous flow at constant strain provide a well-controlled macroscopic probe of the viscoelastic time constants that are associated with energy dissipation processes.

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Chapter 10 Mechanical Properties of Viscoelastic Materials

3. Dynamic Mechanical Testing. Materials are stretched so that they always experience tension. Then, strain oscillates harmonically about this dc offset. Since the response is linear, the induced tensile stress oscillates at the same frequency with no evidence of overtones. One of the most important properties in these dynamic tests is the phase angle difference between stress and strain, which is measured as a function of frequency and temperature. The results are useful to detect viscoelastic relaxation phenomena, such as the glass transition and micro-Brownian molecular motion below Tg, where mechanical energy is dissipated irreversibly to thermal energy. Viscoelastic materials damp mechanical vibrations when the average response time of the material at a particular temperature matches the inverse of the vibrational frequency in radians per second. 4. Fracture Testing. Materials are subjected to a constant rate of strain until cracks propagate catastrophically to cause mechanical fracture at large enough strain. The most important properties in this experiment are the elastic modulus at very small strain and ultimate stress and strain at failure. In many cases, fracture occurs in the nonlinear regime. The rate of strain is an important parameter in these tests, and it must be mentioned together with the measurement temperature when elastic moduli and ultimate properties are reported.

10.14 LINEAR RESPONSE OF THE MAXWELL MODEL DURING CREEP EXPERIMENTS Constant engineering stress s0 is applied instantaneously to a viscoelastic material at time t0, and the stress is removed at time t1. In practice, this “jump” stress is accomplished using a servohydraulic mechanical testing system in load-control mode. Creep response occurs during the time interval from t0 to t1, where predictions and measurements of g (t) represent the desired result. The Maxwell model

s (t) þ l(T)

@s @g ¼h @t @t

reduces to

s0 ¼ h

@g @t

It should be obvious from this simplified equation of motion that the viscous element will exhibit liquid-like flow. This is reasonable because, after the initial step change in stress, the viscoelastic material is not subjected to any abruptly changing stress. The time dependence of total strain is given by   s0 t þ Constant g (t) ¼ h The integration constant is determined from the instantaneous response of the spring when a “jump” stress is applied at time t0. There is no immediate response of the

10.15 Creep Recovery of the Maxwell Model

369

viscous element. Motion of the spring is not restricted by any viscous element in parallel. Hence, s0 g (t ¼ t0 ) ¼ E and the complete creep response of the Maxwell model is   1 t  t0 þ g (t  t0 ) ¼ s0 h E This result is used to construct an intrinsic property of viscoelastic materials, known as the creep compliance JC(t 2 t0) via the following definition: JC (t  t0 ) ¼

g (t  t0 ) 1 t  t0 ¼ þ s0 h E

The leading term (i.e., 1/E) is the static compliance of the spring. A summary of the generic characteristics of JC(t) is provided below: (i) JC(t) is a time-dependent intrinsic material property that does not depend on the magnitude of the “jump” stress s0 when response occurs in the linear regime. If the response is nonlinear at larger strain and the spring is stretched beyond its elastic limit, then the static compliance is much larger than 1/E and the material will not recover its original dimensions when the stress is removed. (ii) JC(t) measures the ease of deformation. Materials that are easier to deform exhibit larger creep compliances. (iii) JC(t) is measured directly from g (t) in a creep experiment, and it is dominated by the contribution from the amorphous component in semicrystalline polymers.

10.15

CREEP RECOVERY OF THE MAXWELL MODEL

At time t1 2t0 after application of a constant stress s0, materials with one viscoelastic relaxation time l that obey the Maxwell model exhibit the following strain:   1 t1  t0 þ g (t1 t0 ) ¼ s 0 h E Now, the stress is removed, the spring recovers instantaneously, and the time dependence of strain during creep recovery is obtained by neglecting s (t) and @ s/@t in the equation of motion for the Maxwell model. The result is

h

@g ¼0 @t

which implies that total strain is independent of time after the spring recovers at time t1. Since the instantaneous recovery of the spring at time t1 cancels its instantaneous

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Chapter 10 Mechanical Properties of Viscoelastic Materials

deformation at time t0, the time-independent strain of the Maxwell model during creep recovery (i.e., t . t1) is s0 s0 g (t) ¼ g (t1  t0 )  ¼ (t1  t0 ) E h This represents irrecoverable deformation, or “permanent set,” due to viscous flow that occurred during creep response from the time interval t0 ) t1. The creep response and creep recovery of the Maxwell model are characteristic of linear amorphous polymers without chemical crosslinks, because these materials deform and recover instantaneously, and they exhibit permanent set due to viscous flow or chain – chain slippage.

10.16 LINEAR RESPONSE OF THE MAXWELL MODEL DURING STRESS RELAXATION A viscoelastic material is subjected to a “jump” strain g0 at time t0, and the time dependence of the induced stress s (t) is measured experimentally and predicted via the Maxwell model. From a practical viewpoint, it is not possible to impose a finite strain instantaneously upon a sample with mass, due to inertial restrictions. However, one must realize that most viscoelastic models discussed in this chapter are “massless” in which the inertial component is neglected. With the aid of servohydraulic mechanical testing systems, very large rates of strain are possible, and g0 can be achieved realistically in a few tens of milliseconds. If a rapid strain rate is imposed on a sample at time t0 and strain g0 is achieved at time t0 þ 1, then one should not measure s (t) with confidence prior to t0 þ 51. The rate of strain dg/dt vanishes after time t0 þ 1, and predictions from the Maxwell model yield

s (t) þ l(T)

@s ¼0 @t

Separation of variables allows simple integration of the previous equation: ds dt ¼ d ln s ¼  s l(T) Hence,

  t s (t) ¼ (Constant) exp  l(T)

The realistic boundary condition is employed, where strain g0 is achieved almost instantaneously at time t0 þ 1 and the spring deforms immediately. Hence, the stress in both elements at time t0 þ 1 is Eg0, even though the viscous element has not responded yet. The spring retracts when the dashpot begins to flow so that the total strain is always g0. At any time t  t0 þ 1, stress within each element of the Maxwell model is given by the product of the spring force constant (i.e., elastic modulus E) and the instantaneous deformation of the elastic element. As illustrated by the previous equation, {g (t)}Elastic is an exponentially decreasing function of time. The

10.16 Linear Response of the Maxwell Model during Stress Relaxation

371

integration constant in the previous equation is evaluated at time t0 þ 1. The complete time-dependent stress response is   t  t0 s (t  t0 ) ¼ E g0 exp  l(T) where t0 ¼ t0 þ 1. Stress relaxation is the most appropriate experiment to measure or predict the relaxation modulus ER(t 2 t0 ), which is an intrinsic material property defined by   s (t  t0 ) t  t0 ER (t  t0 ) ¼ ¼ E exp  g0 l(T) Analogous to the time-dependent creep compliance JC(t) obtained from creep experiments, a summary of the generic characteristics of relaxation moduli are provided below: (i) ER(t) is a time-dependent intrinsic material property that does not depend on the magnitude of the “jump” strain g0 when response occurs in the linear regime. If the response is nonlinear at larger strain and the spring is stretched beyond its elastic limit, then its static modulus is much less than E. Viscoelastic time constants, such as l(T ), increase at larger jump strains in the nonlinear regime [Das and Belfiore, 2004]. (ii) ER(t) measures the resistance to deformation. Materials that are easier to deform exhibit smaller relaxation moduli, typically with shorter relaxation times. (iii) ER(t) is measured directly from s (t) in a stress relaxation experiment, and the decay of stress is dominated by the contribution from the amorphous component in semicrystalline polymers. (iv) Even though the static modulus of the spring is the inverse of its static compliance, previous results for the Maxwell model subjected to creep and stress relaxation experiments reveal that ER(t) and JC(t) are not related inversely. In other words, ER (t) =

1 JC (t)

In general, the product of the Laplace transforms of ER(t) and JC(t) is given by 1/p 2, where p is the Laplace variable. Detailed analyses of ER and JC via Boltzmann superposition integrals are illustrated in Appendix A.3 of this chapter. The result is L{ER (t)}t)p L{JC (t)}t)p ¼ ER ( p)JC ( p) ¼

1 ¼ {L(t)}t)p p2

where L represents the Laplace operator. If one performs a Laplace inversion of the previous equation via the convolution theorem (i.e., see Section 10.22.3), then the

372

Chapter 10 Mechanical Properties of Viscoelastic Materials

convolution integral of ER(t) and JC(t) is given by 1

L {ER ( p)JC ( p)} ¼

x¼t ð

ER (x)JC (t  x) dx ¼

x¼0

where

x¼t ð

ER (t  x)JC (x) dx ¼ t

x¼0

  t ER (t) ¼ E exp  l(T) JC (t) ¼

1 t þ E h

for the Maxwell model, and t0 ¼ 0 without loss of generality.

10.17 TEMPERATURE DEPENDENCE OF THE STRESS RELAXATION MODULUS AND DEFINITION OF THE DEBORAH NUMBER The response of viscoelastic materials is governed by the ratio of two competing time scales. The experimental time scale texpt, or a characteristic time for the relevant deformation process, is given by t 2 t0 , where stress or strain is applied almost instantaneously to a sample at time t0 . The response time, or relaxation time, of the material is given by l(T ), which represents the ratio of viscosity h to modulus E for the onetime-constant Maxwell model. If material response is described by a parallel configuration of N Maxwell elements with several time constants li (1 i N ), then one could compare each li with texpt, or compare the average time constant of the material kll with texpt. Material response is faster at higher temperature. Hence, dl/dT is negative and l is shorter at higher temperature. Let’s consider the stress relaxation modulus for the one-time-constant Maxwell model, given by   t  t0 ER (t  t0 ; T) ¼ E exp  l(T) When the time scale for the relevant deformation process is much shorter than the material response time, for example, at low temperature, t  t0  l(T) ER  E Under these conditions, materials exhibit solid-like behavior with maximum resistance to deformation because stress relaxation has not occurred during the time scale of the experiment, t 2 t0 . Now, either increase texpt or increase temperature, such that t  t0  l(T) ER ) 0

10.18 Other Combinations of Springs and Dashpots

373

The resistance to deformation is minimal because uncrosslinked materials exhibit liquid-like behavior and most of the stress has already relaxed at much earlier times. Notice that material response at long times and high temperature is similar. In fact, there is an equivalence between time and temperature, or frequency and temperature in the analysis of viscoelastic properties. For a material with one viscoelastic time constant, most of the stress relaxes when the experimental time scale is comparable to l. All of this behavior is captured in the Deborah number De, defined by De ¼

l(T) texpt

The scaling of time in polymer viscoelasticity is achieved by this dimensionless parameter. Materials are solid-like at high Deborah numbers, and liquid-like at low Deborah numbers. Since dl/dT is negative, solid-like behavior occurs at low temperature or short texpt. Materials behave like liquids at high temperature or long texpt. Viscoelastic relaxation occurs when the Deborah number is on the order of unity. In the fifth chapter of the book of Judges in the Old Testament, Deborah declared that the mountains flowed like rivers before the Lord. The religious concept is that God’s time scale is infinite and everything will flow if one waits long enough, even the mountains. On the basis of this reference, Marcus Reiner named the ratio of two important time scales the Deborah number, which characterizes the behavior of viscoelastic materials. (This connection between the scaling of time in rheology, the Deborah number, and the Old Testament is not meant to discriminate among nor endorse any religious documents.)

10.18 OTHER COMBINATIONS OF SPRINGS AND DASHPOTS Viscoelastic models of the human chest have been developed to predict compression during high-speed impact. Leg muscles can be modeled with springs and dashpots to simulate the motion of skiers as they negotiate a slalom or giant slalom course. The Maxwell model discussed above contains an elastic element in series with a viscous element. The Voigt model contains the same two elements in parallel. Creep response and creep recovery of the Voigt model are analyzed below. A few other combinations are useful to describe polymer chain dynamics [Mansfield, 1983] and realistic viscoelastic response. (i) A Maxwell element in series with a Voigt element, which contains two springs and two dashpots. (ii) The Maxwell– Wiechert model contains N Maxwell elements in parallel. This discrete model of time-dependent stress and the relaxation modulus is useful to develop the concept of a continuous distribution of viscoelastic relaxation times when the number N of Maxwell elements is infinitely large.

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(iii) The Voigt – Kelvin model contains N Voigt elements in series. As N approaches infinity, this discrete model for time-dependent strain and the creep compliance yields the continuous distribution of viscoelastic retardation times.

10.19 EQUATION OF MOTION FOR THE VOIGT MODEL As mentioned above, this model contains an elastic element in parallel with a viscous element. Hence, the spring is not “free.” When time-dependent stress s (t) is applied to the Voigt configuration, the following rules apply: (i) Stresses are additive. Hence, the total stress s (t) is given by the sum of the stress in each leg of the parallel assembly. (ii) The strain in each leg is the same, and equals the total strain. (iii) The viscous element restricts the elastic element from responding instantaneously. Hence, there is no immediate deformation of the complete assembly when a “jump” stress is applied to initiate creep response. It is essentially impossible to impose a “jump” strain, as required for stress relaxation. (iv) The elastic element restricts deformation within the viscous element after the spring has reached its equilibrium displacement. Statements (i) and (ii) yield the desired relation between stress and strain. From the additivity of stress in each leg of the model, {s (t)}Total ¼ {s (t)}Elastic þ {s (t)}Viscous Hooke’s law and Newton’s law provide expressions for the stress in each element: {s (t)}Elastic ¼ E{g (t)}Elastic   @g {s (t)}Viscous ¼ h @t Viscous Since gElastic ¼ gViscous, the final result is {s (t)}Total ¼ s (t) ¼ E g (t) þ h

@g @t

Once again, the single response time l of the material is given by h/E. The Voigt model does not exhibit stress relaxation, as illustrated by the previous equation of motion when g (t) ¼ g0. The elastic element deforms, although not instantaneously, to this constant level of strain such that the stress is Eg0, due solely to the spring. This induced stress does not decrease with time. Furthermore, one calculates the stress relaxation modulus via division of the induced stress by the constant level of strain g0. Hence, the stress relaxation modulus of the Voigt model is essentially the static modulus E of the elastic element, which reveals no time dependence. The

10.19 Equation of Motion for the Voigt Model

375

electrical analog of the Voigt model corresponds to a capacitor and resistor in series. Circuit elements in series experience the same current, analogous to the fact that mechanical elements in parallel experience the same strain and rates of strain. The total voltage drop across circuit elements in series is obtained by summing the individual voltage drops. This is analogous to the fact that the total stress experienced by a parallel configuration of mechanical elements is obtained by summing the stress in each element.

10.19.1 General Solution to the Equation of Motion for the Voigt Model The objective of this section is to integrate the previous equation when viscoelastic materials are subjected to an arbitrary time-dependent stress s (t) at time t0. Integrating factor methodology requires a coefficient of unity for the first-derivative term (i.e., @ g/@t) and identifies the integrating factor as exp(t/l ). After division by viscosity h, the equation of motion for the Voigt model is multiplied by exp(t/l ) to obtain the following result: et=l

@ g 1 t=l @ 1 þ e g (t) ¼ {g (t)et=l } ¼ et=l s (t) @t l h @t

Since motion of the spring is restricted by the viscous element in parallel, the Voigt model does not exhibit instantaneous strain when stress s is applied initially at t0. Hence, g ¼ 0 at t ¼ t0, and integration of the previous equation yields gð(t)

t=l

t=l

d{g (t)e } ¼ g (t)e

g (t0 )¼0

1 ¼ h

ðt

eQ=l s (Q) dQ

Q¼t0

where integration variable Q is employed on the right side of the previous equation, instead of t, to distinguish between present time t and past time Q, which ranges from time t0, when stress s is applied initially, to present time t. The general solution for time-dependent strain g (t) is 1 g (t) ¼ h

ðt

  (t  Q) dQ s (Q) exp  l(T)

Q¼t0

which should be compared with the Boltzmann superposition integral for g (t) in Appendix A.2 at the end of this chapter. The experimental time scale for the relevant deformation process is identified as t 2 Q, which ranges from t 2 t0 to 0, and the exponential term in the integrand of the previous equation represents a time derivative of the creep compliance JC(t 2 Q) with respect to past time Q.

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Chapter 10 Mechanical Properties of Viscoelastic Materials

10.20 LINEAR RESPONSE OF THE VOIGT MODEL IN CREEP EXPERIMENTS At time t0, constant stress s0 is applied instantaneously to a viscoelastic material that obeys the Voigt model with one temperature-dependent time constant l(T ). The time dependence of strain g (t) can be predicted from the general solution in the previous section. Hence, s (Q) ¼ s0 for times t  t0, and one obtains the following result:

s0 g (t) ¼ h

ðt

     (t  Q) s0 (t  t0 ) dQ ¼ 1  exp  exp  l(T) E l(T)

Q¼t0

As the time scale for the relevant deformation process increases significantly (i.e., t 2 t0 ) 1), the asymptotic limit for g (t) corresponds to the equilibrium response of the Voigt model during creep, which is identical to the equilibrium extension of the elastic element. The elastic response of the spring is retarded by the viscous element in parallel, but the retardation or time delay is much shorter at higher temperature because l decreases as temperature increases. Since creep experiments are designed specifically to measure time-dependent strain and calculate intrinsic material response via the creep compliance, JC(t) for the Voigt model is    g (t  t0 ) 1 (t  t0 ) 1  exp  ¼ s0 l(T) E   1 1  EJC (t  t0 ) ¼ exp  De JC (t  t0 ) ¼

Solid-like behavior persists at very large Deborah numbers where the creep compliance is negligible. At higher temperature, De is much smaller for viscoelastic materials that respond like liquids, and JC(t 2 t0) asymptotically approaches its maximum value, which corresponds to the static compliance of the spring, 1/E.

10.21

CREEP RECOVERY OF THE VOIGT MODEL

A viscoelastic material with chemical crosslinks has experienced constant stress s0 during the time interval t1 2t0, and its strain asymptotically approaches the equilibrium deformation of the elastic element. Hence,    s0 (t1  t0 ) 1  exp  g (t1  t0 ) ¼ E l(T) Now, the stress is removed at time t1 and strain recovery occurs. The equation of motion of the Voigt model describes the time dependence of strain during creep

10.22 Creep and Stress Relaxation for Maxwell and Voigt Elements

377

recovery: @g 1 þ g (t) ¼ 0 @t l(T) Separation of variables yields the following solution for g versus t:   (t  t1 ) g (t) ¼ g (t1  t0 ) exp  l(T) Notice that strain is continuous because the viscous element in parallel with the elastic element restricts any instantaneous response of the spring when a “jump” stress is applied at t0 and removed at t1. All of the strain that occurs during creep response is completely recovered during the recovery phase. Hence, the Voigt model exhibits no irrecoverable deformation. This model captures some characteristics of crosslinked polymers. For example, there is no permanent set and stress relaxation does not occur. If instantaneous response is required for an accurate description of crosslinked materials, then it is necessary to include an unrestricted spring in series with the Voigt element. Unrestricted dashpots, which simulate viscous flow and chain – chain slippage, are not consistent with the mechanical response of crosslinked polymers.

10.22 CREEP AND STRESS RELAXATION FOR A SERIES COMBINATION OF MAXWELL AND VOIGT ELEMENTS An interesting viscoelastic model is based on the following elements in series: 1. An unrestricted spring with static modulus E1. 2. A second unrestricted spring with static modulus E2. 3. A Voigt element with spring E3 and dashpot h3 in parallel. This element contains a single viscoelastic time constant: l3 ¼ h3/E3. 4. An unrestricted dashpot with viscosity h4.

10.22.1

Creep Response

When a constant “jump” stress s0 is applied to the entire configuration at time t0, each of the four elements described above experiences this constant stress s0 because they are aligned in series. The total strain is additive. Hence,

g Total ¼ g Elastic Element(1) þ g Elastic Element(2) þ g Voigt þ g Viscous Element(4)

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Chapter 10 Mechanical Properties of Viscoelastic Materials

Each unrestricted elastic element deforms instantaneously and exhibits constant strain that is inversely proportional to its elastic modulus. In other words,

s0 E1 s0 ¼ E2

g Elastic Element(1) ¼ g Elastic Element(2)

When the Voigt element is subjected to constant stress s0 at time t0, the timedependent strain during creep was developed in Section 10.20 of this chapter:    s0 t  t0 1  exp  {g (t  t0 )}Voigt ¼ E3 l3 (T) The unrestricted Viscous Element (4) is sluggish and experiences no deformation at time t0, but it responds linearly with a time-dependent strain that is inversely proportional to its viscosity h4. The complete creep response of this viscous element is {g (t  t0 )}Viscous Element(4) ¼

s0 (t  t0 ) h4

The total creep response of the entire configuration is obtained from a linear superposition of strain in all four elements, as dictated by the fact that they are in series. Hence,  {g (t  t0 )}Total ¼ s0

    1 1 t  t0 1 t  t0 þ þ þ 1  exp  h4 l3 (T) E1 E2 E3

Irrecoverable deformation occurs in unrestricted Viscous Element (4), but the strain induced in the other three elements is completely reversible. The two unrestricted elastic elements recover immediately, and the Voigt element recovers sluggishly, as dictated by time constant l3.

10.22.2

Stress Relaxation

Now, a “jump” strain g0 is applied to the entire configuration at time t ¼ 0, and one seeks the time dependence of stress s (t) which is experienced equally by each element. Since the total strain and the total rate of strain are additive, all of the individual rates of strain must sum to zero because the total strain is constant. Hence,         @g @g @g @g þ þ þ ¼0 @t Elastic Element(1) @t Elastic Element(2) @t Voigt @t Elastic Element(4) Each element is analyzed separately to determine its rate of strain.

10.22 Creep and Stress Relaxation for Maxwell and Voigt Elements

379

1. Hooke’s law is employed to calculate the deformation rate in the first unrestricted spring:

s (t) ¼ E1 {g}Elastic Element(1)   @g 1 @s ¼ @t Elastic Element(1) E1 @t 2. A similar result is obtained via Hooke’s law for the rate of strain in the second unrestricted spring: 

 @g 1 @s ¼ @t Elastic Element(2) E2 @t

3. One employs the equation of motion for the Voigt model, with time-dependent stress s (t), to calculate the deformation in each leg of this parallel assembly of spring E3 and dashpot h3. Since the elastic and viscous stresses must sum to s (t), one obtains  E3 {g (t)}Voigt þ h3

@g @t

 ¼ s (t) Voigt

Division by viscosity h3 yields 

 @g 1 1 þ {g (t)}Voigt ¼ s (t) @t Voigt l3 h3

Laplace transform methodology is employed to map {g (t)}Voigt into the Laplace domain. The transformation from g (t) to g ( p) via operator L and transformed variable p, together with a theorem for the Laplace transform of the first derivative of g (t), yields the following result, based on the previous equation:

L

  @g 1 1 þ L{g (t)}Voigt,t)p ¼ L{s (t)}t)p @t Voigt,t)p l3 h3   @g L ¼ p{g ( p)}Voigt  {g (t ¼ 0)}Voigt @t Voigt,t)p

The Voigt element experiences no deformation at time t ¼ 0 because the spring is restricted by the viscous component in parallel. Hence, one solves for {g ( p)}Voigt

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Chapter 10 Mechanical Properties of Viscoelastic Materials

in the Laplace domain from the transformed equation of motion: 

 1 1 pþ {g ( p)}Voigt ¼ s ( p) l3 h3

h3 {g ( p)}Voigt ¼

10.22.3

s ( p) p þ 1=l3

Convolution Integrals

This theorem from the theory of Laplace transforms is useful here because the previous equation reveals that, in the Laplace domain, deformation of the Voigt element is given by the product of two transformed functions. In other words, L{s (t)}t)p ¼ s ( p)    t 1 ¼ L exp  l3 t)p p þ 1=l3 In general, the product of two separate transformed functions in the Laplace domain L{ f (t)}L{g(t)} where 1 ð

L{ f (t)}t)p ¼

f (t) exp(pt) dt t¼0 1 ð

L{g(t)}t)p ¼

g(t) exp(pt) dt t¼0

can be manipulated via the definition of this integral transformation. For example, using u and v as integration variables,

L{f (t)}t)p L{g(t)}t)p ¼

1 ð

f (u) exp(pu) du u¼0

¼

1 ð

1 ð v¼0

81 <ð :

u¼0

g(v) exp(pv) dv v¼0

9 = f (u)g(v) exp{p(u þ v)} du dv ;

10.22 Creep and Stress Relaxation for Maxwell and Voigt Elements

381

Now, let w ¼ u þ v, and eliminate u as an integration variable. Since the range of u is from 0 ) 1, the corresponding range of w is from v ) 1. The product of Laplace transforms is 9 8 1 ð ð < 1 = f (w  v)g(v) exp{pw} dw dv L{ f (t)}t)p L{g(t)}t)p ¼ ; : v¼0

w¼v

Upon reversing the order of integration (i.e., integrate first with respect to v, and then with respect to w) and accounting for the integration limits properly (i.e., v ranges from 0 ) w, and w ranges from 0 ) 1), one obtains 8 w 9 8 w 9 1 ð <ð = <ð = f ( p)g( p) ¼ exp{pw} f (w  v)g(v) dv dw ¼ L f (w  v)g(v) dv : ; : ; v¼0

w¼0

v¼0

w)p

This is the convolution theorem, also known as Borel’s theorem, which allows one to perform a rather quick Laplace inversion when an expression in the Laplace domain is written as the product of two transformed functions. In other words, letting w ¼ t and v ¼ x, ðt ðt f (t  x)g(x) dx ¼ f (x)g(t  x) dx L1 { f ( p)g( p)} ¼ x¼0

x¼0

Application of the convolution theorem to the transformed equation of motion for the Voigt element yields the following integral expression for time-dependent strain in this parallel configuration of a spring and dashpot:     ðt ðt x (t  x) dx ¼ dx h3 {g (t)}Voigt ¼ s (t  x) exp  s (x) exp  l3 l3 x¼0

x¼0

This result from the theory of Laplace transforms is identical to the general solution for the equation of motion of the Voigt model in Section 10.19.1 when t0 ¼ 0.

10.22.4 Leibnitz Rule for Differentiating One-Dimensional Integrals When the Limits of Integration Are Not Constant The previous equation for the deformation in each leg of the Voigt element must be differentiated with respect to time to calculate the rate of strain. This operation is performed with assistance from the Leibnitz rule. For example, if

G(t) ¼

w(t) ð

f (x, t) dx x¼u(t)

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Chapter 10 Mechanical Properties of Viscoelastic Materials

then, in general, G(t) ¼ G{t, u(t), w(t)} ¼ G(t, u, w) The hierarchy is that G depends directly on t, u, and w; whereas u and w both depend on t. Hence, the total differential of G is  dG ¼

@G @t



 dt þ

u,w

   @G @G du þ dw @u t,w @w t,u

Therefore, dG ¼ dt



@G @t



 þ

u,w

   @G du @G dw þ @u t,w dt @w t,u dt

The first term on the right side of the previous equation is obtained by taking the partial derivative of f (x, t) with respect to t “inside the integral,” because the limits of integration are treated as constants. Now, it is necessary to obtain expressions for (@G/@u)t,w and (@G/@w)t,u. This is achieved by introducing a new function, J(x, t), which represents the indefinite integral of f (x, t) with respect to x. In other words, J(x, t) ¼

G(t, u, w) ¼

ð f (x, t) dx w(t) ð

f (x, t) dx ¼ J(w, t)  J(u, t)

x¼u(t)

The “fundamental theorem of calculus” states that   @J ¼ f (x, t) @x t Therefore, if one replaces x by either u or w, then     @J @J ¼ f (u, t); ¼ f (w, t) @u t @w t The relation between G and J above yields the following partial derivatives of interest: 

   @G @J ¼ ¼  f (u, t) @u t,w @u t     @G @J ¼þ ¼ þ f (w, t) @w t,u @w t

10.22 Creep and Stress Relaxation for Maxwell and Voigt Elements

383

The final result is dG ¼ dt

w(t) ð x¼u(t)

  @f (x, t) dw du  f (u, t) dx þ f (w, t) @t dt dt x

[upon returning to the analysis of the individual rates of strain] Application of this integration rule to {g (t)}Voigt provides the desired result for the rate of strain in the Voigt element. The third term on the right side of the previous equation vanishes when the lower integration limit u(t) ¼ 0 is constant. Hence, the Leibnitz rule is applied to the convolution integral on the extreme right side of the expression for {g (t)}Voigt, where stress s is evaluated at argument x, not t 2 x:

h3 {g (t)}Voigt ¼

ðt

  (t  x) dx s (x) exp  l3

x¼0



h3

   ðt @g 1 (t  x) ¼ s (t)  s (x) exp  dx @t Voigt l3 l3 x¼0

4. Newton’s law of viscosity is employed to calculate the rate of strain in the unrestricted dashpot, which experiences time-dependent stress s (t):   @g 1 ¼ s (t) @t Viscous Element(4) h4 Additivity of the rates of strain in all four series elements yields an integro-differential equation for the time dependence of stress s (t) during stress relaxation when the entire configuration is subjected to a constant “jump” strain g0 at time t ¼ 0: 

     ðt 1 1 @s 1 1 1 (t  x) dx ¼ 0 þ þ þ s (t)  s (x) exp  h3 h4 h3 l3 l3 E1 E2 @t x¼0

For simplicity, let 1 1 1 ¼ þ ; E E1 E2

1 1 1 ¼ þ h h3 h4

Now, apply the Leibnitz rule for differentiating an integral with variable limits and differentiate the integro-differential equation to obtain a second-order ordinary differential equation (ODE) for s (t):       2 ðt 1 @ s 1 @s 1 1 (t  x) dx ¼ 0  þ s (t) þ s (x) exp  h @t h3 l3 l3 E @t 2 h3 l23 x¼0

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Chapter 10 Mechanical Properties of Viscoelastic Materials

Employ the following substitution from the original integro-differential equation for s (t) to eliminate the integral expression in the previous equation: 1 h3 l3

ðt

      (t  x) 1 @s 1 þ s (x) exp  s (t) dx ¼ l3 h E @t

x¼0

Hence, one calculates the time dependence of stress during stress relaxation from the following second-order ODE:   2     1 @ s 1 1 @s 1 1 1 þ þ þ  s (t) ¼ 0 h E l3 @t l3 h h3 E @t 2 which simplifies to the final result: @2s @s þ cs (t) ¼ 0 þb @t 2 @t E 1 E a ¼ 1; b ¼ þ ; c ¼ h l3 l3 h4 a

10.22.5 Boundary Conditions and Qualitative Characteristics of s (t) During Stress Relaxation Two conditions on s (t) are required to obtain a unique solution to the previous equation. Realize that the Voigt element and the unrestricted dashpot do not respond immediately at t ¼ 0 when the entire configuration is subjected to “jump” strain g0. Hence, at t ¼ 0,

g0 ¼ {g (t ¼ 0)}Elastic Element(1) þ {g (t ¼ 0)}Elastic Element(2)   1 1 1 ¼ þ s (t ¼ 0) ¼ s (t ¼ 0) E1 E2 E This leads to the first condition:

s (t ¼ 0) ¼ E g0 where 1/E is the equivalent static compliance (i.e., J1 þ J2) of two unrestricted (i.e., free) springs in series, analogous to the equivalent capacitance (i.e., C1 þ C2) of two capacitors in parallel. Due to the presence of unrestricted dashpot h4, the stress decays to zero at long times, and it should do so with zero slope. Hence, 

s (t ) 1) ¼ 0;

 @s ) 0 @t t)1

10.24 Stress Relaxation via the Equivalence Between Time and Temperature

385

In general, stress relaxation for this series configuration of Maxwell and Voigt elements begins at Eg0 and decays to zero. The response can be characterized as underdamped, overdamped, or critically damped when the discriminant of the characteristic equation based on the linear second-order ODE for s(t) (i.e., b 2 24ac) is negative, positive, or zero, respectively. Since all five parameters for this model (i.e., E1, E2, E3, h3, h4) must be positive in a physically realistic situation, actual calculations reveal that the discriminant is invariably positive, the characteristic equation exhibits two real negative roots, and the system is overdamped. Hence, the residual stress decays exponentially to zero, with no overshoot.

10.23 THE PRINCIPLE OF TIME –TEMPERATURE SUPERPOSITION As mentioned earlier, there is an equivalence between time and temperature when the viscoelastic properties of amorphous polymers are under investigation. If one wants a snapshot of the complete time dependence of the stress relaxation modulus at one particular temperature, it is necessary to make measurements over 10 or 15 decades on the log time axis. From a practical viewpoint, this is nearly impossible. Viscoelastic response at very short times is limited by the restriction that one should wait about five times the duration required to achieve a “jump” strain prior to making accurate stress relaxation measurements. At the other extreme, computerized data acquisition files will become overloaded and instrumental stability could drift when measurements are required on a time scale of tens or hundreds of hours. The solution to this dilemma is summarized as follows: (i) measure stress relaxation data isothermally over a reasonable time scale that is physically realistic; (ii) repeat these measurements at several different temperatures; and (iii) maintain equivalence of the Deborah number when stress relaxation data are measured at different temperatures and shift the data horizontally on the log time axis to obtain a continuous curve for the complete response at one reference temperature Treference. These concepts are illustrated empirically and theoretically in the following sections.

10.24 STRESS RELAXATION VIA THE EQUIVALENCE BETWEEN TIME AND TEMPERATURE Consider stress relaxation measurements at two different temperatures such that one obtains the same relaxation modulus. Since material response times decrease at higher temperature (i.e., dl/dT , 0), the observation time for stress decay must be shorter when experiments are performed at higher temperature to achieve the following equivalence: ER (t  t0 ; T) ¼ ER (treference  t0 ; Treference ) The observation times at temperatures T and Treference, to achieve the same relaxation modulus, are t 2 t0 and treference 2 t0, respectively. For example,

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Chapter 10 Mechanical Properties of Viscoelastic Materials

(a) if T . Treference, then t 2 t0 , treference 2 t0; (b) if T , Treference, then t 2 t0 . treference 2 t0. The ratio of observation times at temperatures T and Treference is defined as the time – temperature shift factor: aT (T; Treference ) ¼

(t  t0 )@T (treference  t0 )@Treference

If the time axis is linear, then aT represents a multiplicative factor to relate observation times at two different temperatures. For the most common case of a logarithmic time axis, aT is an additive factor. Furthermore, (a) if T . Treference, then aT (T; Treference) , 1; (b) if T , Treference, then aT (T; Treference) . 1. Stress relaxation data are shifted horizontally along the time axis such that the Deborah number is the same at temperatures T and Treference. In other words, stress relaxation moduli measured at these two different temperatures will be equivalent when De(Treference ) ¼ De(T)

l(Treference ) l(T) ¼ treference  t0 t  t0 This provides another expression for the time – temperature shift factor, which is consistent with the equivalence between time and temperature because dl/dT , 0: aT (T; Treference ) ¼

(t  t0 )@T l(T) ¼ (treference  t0 )@Treference l(Treference )

Since relaxation moduli are measured at temperature T during observation times {t – t0}, it is necessary to identify the corresponding observation times {treference – t0} for isothermal measurements at Treference to achieve the objective of creating log ER versus log{treference – t0}. The relation between observation times, and the appropriate additive shift on the log time axis, are achieved via the definition of aT. Hence, log{treference  t0 }@Treference ¼ log{t  t0 }@T  log{aT (T; Treference )} The complete strategy is summarized below: (i) Measure stress relaxation data isothermally at temperature T over a reasonable time scale (i.e., t 2 t0) that will yield accurate data. (ii) Calculate stress relaxation moduli versus t 2 t0 at temperature T via division of s (t) by the constant level of “jump” strain g0, which yields linear response such that ER does not depend on g0. (iii) Maintain equivalence of the Deborah number and shift log ER versus log{t 2 t0} horizontally to obtain the same relaxation moduli when the observation times are log{treference 2 t0}.

10.24 Stress Relaxation via the Equivalence Between Time and Temperature

387

Measurements are obtained at temperature T, and it is desired to create the complete behavior for log ER versus log(treference – t0) at temperature Treference. Hence, the data should be shifted horizontally as described below. If vertical shifting is required as well, then the viscoelastic material does not exhibit rheologically simple behavior. (i) If T . Treference, then aT , 1 and the data must be shifted forward on the log time axis by an additive factor. The relaxation moduli measured during observation times t 2 t0 at temperature T require much longer observation times to generate the complete picture at lower temperature. (ii) If T , Treference, then aT . 1 and the data must be shifted backward on the log time axis by an additive factor. The relaxation moduli measured during observation times t 2 t0 at temperature T require much shorter observation times to generate the complete picture at higher temperature. Experimental stress relaxation data were correlated as follows in the 1950s for the temperature dependence of aT: 1 a1 ¼ a0 þ T  Treference log aT (T; Treference ) a0 ¼

1 ; C1 (Treference )

a1 ¼

C2 (Treference ) C1 (Treference )

Linear least squares analysis can be performed using a first-order polynomial: y ¼ a0 þ a1 x where 1 ; T  Treference 1 ; (ii) the dependent variable is y ¼ log aT (T; Treference ) (iii) the intercept is a0 and the slope is a1. (i) the independent variable is x ¼

Williams, Landel, and Ferry (WLF) correlated their dynamic compliance data (i.e., J0 and J00 ) using a slightly modified procedure, relative to the one described above. For example, (T  Treference ) ¼ a0 (T  Treference ) þ a1 log aT (T; Treference ) where a0 and a1 are defined above in terms of C1 and C2. Once again, linear least squares analysis can be performed using a first-order polynomial, where (i) the independent variable is x ¼ T  Treference ; (T  Treference ) ; (ii) the dependent variable is y ¼ log aT (T; Treference ) (iii) the intercept is a1 and the slope is a0.

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Chapter 10 Mechanical Properties of Viscoelastic Materials

Rearrangement of either linear model yields the following expression for the WLF shift factor (i.e., known universally as the WLF equation); log aT (T; Treference ) ¼

C1 (T  Treference ) C2 þ (T  Treference )

where C1 and C2 depend on the reference temperature, but their product (i.e., C1C2  900 K) is independent of Treference. When the reference temperature is chosen as Tg, the universal WLF constants are C1 (Treference ¼ Tg )  17:44 C2 (Treference ¼ Tg )  51:6 K When the reference temperature is Treference ¼ Tg þ 50 K, the WLF constants are C1  8:86 C2  101:6 K Both sets of WLF parameters suggest that polymer chain motion ceases 51.6 K below the glass transition temperature because the time – temperature shift factor is infinitely large when T ¼ Tg 251.6 K. In other words, if it is desired to create the complete behavior of log ER versus log{treference – t0} at Treference ¼ Tg and stress relaxation experiments are performed at a temperature of 51.6 K below Tg, then an infinite observation time would be required to achieve a finite decrease in relaxation modulus for an uncrosslinked polymer. In light of these results from stress relaxation, one might consider a modification of the rate of growth during crystallization upon cooling from the molten state (see Sections 8.9 and 8.10). In other words, free-volume-dependent growth rates associated with diffusion and reptation of amorphous chains into the lamellar regions as they adopt the correct crystalline conformation should not vanish at the glass transition temperature, but rather 51.6 K below Tg. Adam and Gibbs [1965] employed kinetic theory based on the conformational entropy of glasses to test for universality of the WLF equation and its parameters. For example, the temperature derivative of log aT is d C1 C2 {log aT (T; Treference )} ¼ dT (C2 þ T  Treference )2 which reduces to the following result at the glass transition temperature when Tg is not the reference temperature:  d C1 C2 {log aT (T; Treference )} ¼ 2 dT (C þ T 2 g  Treference ) T¼Tg The previous result is much simpler when Treference ¼ Tg:  d C1 {log aT (T; Tg )} ¼ ¼ 0:34 K1 C2 dT T¼Tg

10.25 Semi Theoretical Justification for the Empirical form of the WLF Shift Factor

389

One obtains the same numerical result for the temperature derivative of log aT at Tg, as indicated in the previous equation, when the reference temperature is Tg or 50 K above the glass transition temperature. In other words, the appropriate choices for C1 and C2 when Treference ¼ Tg (i.e., C1 ¼ 17.44, C2 ¼ 51.6 K) and Treference ¼ Tg þ 50 K (i.e., C1 ¼ 8.86, C2 ¼ 101.6 K) yield the same result in the previous two equations, which is independent of the reference temperature. The fact that the temperature derivative of log aT at the glass transition temperature is not a function of Treference suggests that the WLF equation exhibits universality. This concept is employed below to develop a scaling law for the apparent activation energy at Tg for molecular motion and viscous transport. The following sections describe the rationale for correlating the temperature dependence of aT via the linear models described above.

10.25 SEMI THEORETICAL JUSTIFICATION FOR THE EMPIRICAL FORM OF THE WLF SHIFT FACTOR aT (T; Treference) Molecular mobility near Tg and the concept of cooperative segmental motions in longchain molecules have been combined within the framework of statistical thermodynamics to arrive at the WLF equation [Beuche, 1956]. When one segment of a polymer chain moves as a result of thermal fluctuations, several nearby segments must cooperate and move or conformationally rearrange as well to allow the overall process to occur. However, the WLF equation suggests that the complex temperature behavior of molten polymers is not peculiar to long-chain molecules, because small molecules exhibit similar effects near the glass transition [Bueche, 1959]. Along these lines, it is instructive to consider the stress relaxation modulus for the one-timeconstant Maxwell model and integrate the “area under the curve” of ER(t 2 t0; T ) from time t0 when the “jump” strain is applied to infinite time. If the viscoelastic material is crosslinked beyond the percolation threshold such that there is a continuous path throughout the network via intrachain and interchain chemical bonds, then the relaxation modulus will not decay to zero and the area under the curve approaches infinity. For linear polymers, as well as materials that are crosslinked below the percolation threshold, the relaxation modulus will decay to zero and the area under the curve is finite. Furthermore, higher temperature response causes more rapid decay of the relaxation modulus and a reduction in the area under the curve. The quantity of interest is 1 ð t¼t0

ER (t  t0 ; T) dt ¼

1 ð

  (t  t0 ) E exp  dt l(T)

t¼t0



  (t  t0 ) 1 ¼ E l(T) ¼ h(T) ¼ E l(T) exp  l(T) t¼t0 Even though the result of this integration yields the zero-shear-rate viscosity of the polymer h(T ) when the Maxwell model for the relaxation modulus is employed,

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Chapter 10 Mechanical Properties of Viscoelastic Materials

the same result is obtained for any functional form of ER(t 2 t0; T ). This trend is reasonable because, at higher temperature, the relaxation modulus decreases faster, the area under the curve is smaller, and viscosities of highly viscous liquids decrease. Now, consider the equivalence of relaxation moduli measured at temperatures T and Treference when the Deborah number is the same: ER (t  t0 ; T) ¼ ER (treference  t0 ; Treference ) such that the observation times at each temperature are related via aT (T; Treference). Differential changes in the observation time at temperature T (i.e., dt) are related to similar changes in observation times at Treference (i.e., dtreference) via the following equation: dt ¼ aT (T; Treference ) dtreference because the shift factor depends on temperature, but not time. This information is employed to recalculate the zero-shear-rate polymer viscosity via integration of the stress relaxation modulus:

h(T) ¼

1 ð

1 ð

ER (t  t0 ; T) dt ¼ aT (T; Treference )

ER (treference  t0 ; Treference ) dtreference

treference ¼t0

t¼t0

¼ aT (T; Treference )h(Treference ) In the previous equation, the zero-shear-rate viscosity of a viscoelastic material at the reference temperature (i.e., 1012 – 13 Pascal-second when the reference temperature is Tg) is calculated by integrating the stress relaxation modulus versus treference from the time when a jump strain is applied at temperature Treference until the stress decays to a negligible fraction of its initial value for uncrosslinked materials. Hence, three equivalent definitions of the time – temperature shift factor are aT (T; Treference ) ¼

(t  t0 )@T l(T) h(T) ¼ ¼ (treference  t0 )@Treference l(Treference ) h(Treference )

10.26 TEMPERATURE DEPENDENCE OF THE ZERO-SHEAR-RATE POLYMER VISCOSITY VIA FRACTIONAL FREE VOLUME AND THE DOOLITTLE EQUATION Since the previous equation reveals that the time – temperature shift factor is related to viscosity, the temperature dependence of viscosity should yield an expression for aT in terms of T and Treference. The Doolittle [1951] equation is employed to accomplish this task. At temperatures that are sufficiently removed from the freezing point, Doolittle [1951] correlated the viscosities of liquid normal paraffins in terms of inverse fractional free volume instead of inverse temperature: ln h(T) ¼ A þ

B f (T)

10.26 Temperature Dependence of the Zero-Shear-Rate Polymer Viscosity

391

This equation predicts accurate viscosities of the homologous series of n-alkanes with molecular weights from 72 daltons (i.e., C5) to 898 daltons (i.e., C64). The trend is reasonable because, at higher temperature, fractional free volume f (T ) increases and viscosities decrease because the constant B is greater than zero. Based on transport in hard-sphere liquids via redistribution of useful free volume and the formation of holes or vacancies, B represents a critical fractional free volume that must be overcome for molecular “jumps” to occur, most likely due to Brownian motion. Experimentally, it has been found that B is close to unity. The choice of B ¼ 1 for T .Tg is consistent with the fact that the temperature coefficient of fractional free volume is given by Da at Tg, instead of aLiquid. Now, the time – temperature WLF shift factor correlates inversely with fractional free volume: log{aT (T; Treference )} ¼ log{h(T)}  log{h(Treference )}   1 1 1 B  ¼ 2:303 f (T) f (Treference )   B f (Treference )  f (T) ¼ 2:303f (Treference ) f (T) The temperature dependence of fractional free volume is expanded in a Taylor series about Treference, only the zeroth-order and first-order terms are retained, and Tg is chosen as the reference temperature. As illustrated in a Chapter 1 (i.e., Section 1.11), f (T)  f (Tg ) þ Da(T  Tg ) þ    The expression for aT is structurally similar to the empirical form that was used for data analysis to calculate the WLF constants C1 and C2, as described in Section 10.24: log{aT (T; Tg )} ¼

B 2:303f (Tg )



Da(T  Tg ) f (Tg ) þ Da(T  Tg )



Experimental values for C1 and C2 in the relation between aT, T, and Treference, based on actual stress relaxation data, can be used to evaluate important free-volume parameters. Since C1 (Treference ¼ Tg ) ¼ 17:44 ¼

B 2:303f (Tg )

C2 (Treference ¼ Tg ) ¼ 51:6 K ¼

f (Tg ) Da

and B  1, the principle of time – temperature superposition, together with Doolittle’s equation for h(T ) and the temperature dependence of fractional free volume, yield f (Tg )  0:025 Da(T ¼ Tg )  4:8  104 K1

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Chapter 10 Mechanical Properties of Viscoelastic Materials

These parameters are universal for many polymers, suggesting that the glass transition is an iso-free-volume state consisting of 2.5% void space.

10.27 APPARENT ACTIVATION ENERGY FOR aT AND THE ZERO-SHEAR-RATE POLYMER VISCOSITY The previous discussion suggests that the temperature dependence of aT and the zero-shear-rate polymer viscosity follows the WLF equation, not the Arrhenius equation. In fact, this statement is true for the temperature dependence of all rate processes in the vicinity of the glass transition temperature. In other words, the log of a temperature-dependent parameter that describes rate processes, like viscous transport, diffusion, or molecular motion, should be correlated inversely with fractional free volume, not inversely with temperature. Even though these rate processes occur via the useful empty space between molecules that redistributes itself with little or no energy change, it is possible to define an apparent activation energy for these processes which is consistent with the Arrhenius equation. For rate constants, like viscosity, material response time, and aT, that decrease at higher temperature, the apparent activation energy is defined as d d ln aT Eapparent ¼ R   {ln aT (T; Treference )} ¼ RT 2 1 dT d T Results from Section 10.24 reveal “WLF universality” for the temperature derivative of ln aT at the glass transition temperature, for any choice of Treference. Hence, the apparent activation energy at the glass transition temperature scales as the square of Tg (see Section 2.5). It is possible to express the apparent temperature-dependent activation energy in terms of the WLF parameters C1 and C2: Eapparent (T) ¼ 2:303RT 2

C1 C2 (C2 þ T  Treference )2

Evaluation of the previous equation at T ¼ Treference yields 2 Eapparent (Treference ) ¼ 2:303RTreference

C1 C2

The ratio of WLF constants C2 and C1 (i.e., C2/C1) represents the linear least squares slope of 1 1 versus log aT (T; Treference ) T  Treference which exhibits dependence on Treference, but it is not a function of Doolittle’s parameter B in the expression for zero-shear-rate polymer viscosity versus inverse fractional free volume.

10.28 Comparison of the WLF Shift Factor aT at Different Reference Temperatures

393

10.28 COMPARISON OF THE WLF SHIFT FACTOR aT AT DIFFERENT REFERENCE TEMPERATURES The WLF parameters C1 and C2, which depend on Treference, can be related to their counterparts when the reference temperature changes. In other words, if C1A and C2A are tabulated or have been calculated at TrefA, then a simple procedure exists to calculate C1B and C2B at TrefB. Begin with the definition of aT, using TrefA as the reference temperature: aT (T; TrefA ) ¼

l(T) l(T) l(TrefB ) ¼ ¼ {aT (T; TrefB )}{aT (TrefB ; TrefA )} l(TrefA ) l(TrefB ) l(TrefA )

WLF expressions are employed for each shift factor in the previous equation. For example, log aT (T; TrefA ) ¼

C1A (T  TrefA ) C2A þ (T  TrefA )

log aT (T; TrefB ) ¼

C1B (T  TrefB ) C2B þ (T  TrefB )

log aT (TrefB ; TrefA ) ¼

C1A (TrefB  TrefA ) C2A þ (TrefB  TrefA )

Rearrange the relations among these three shift factors, insert the corresponding WLF equations, and combine terms that are referenced to TrefA using a common denominator: log aT (T; TrefA )  log aT (TrefB ; TrefA ) ¼ log aT (T; TrefB ) C1A (T  TrefA ) C1A (TrefB  TrefA ) C1A C2A (T  TrefB ) þ ¼ C2A þ (T  TrefA ) C2A þ (TrefB  TrefA ) C{T þ (C2A  TrefA )} ¼

C1B (T  TrefB ) T þ (C2B  TrefB )

where C ¼ C2A þ TrefB 2 TrefA. The following relations among WLF parameters, C1 and C2, referenced to two different temperatures, should be obvious from the previous equation: C2A  TrefA ¼ C2B  TrefB C1A C2A ¼ C1B C ¼ C1B {C2A þ TrefB  TrefA } ¼ C1B C2B These correspondences reveal universality of the WLF parameters, because the product of C1 and C2 is the same for all reference temperatures. Furthermore, this product is 900 K.

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Chapter 10 Mechanical Properties of Viscoelastic Materials

10.29 VOGEL’S EQUATION FOR THE TIME– TEMPERATURE SHIFT FACTOR An alternate three-parameter expression for aT proposed by Vogel is log aT (T; Treference ) ¼

1  K2 K1 (T  T0 )

The Vogel parameters K1, K2, and T0 can be related to the WLF parameters C1, C2, and Treference by equating the previous expression to the WLF shift factor: log aT (T; Treference ) ¼

C1 (T  Treference ) 1 ¼  K2 C2 þ (T  Treference ) K1 (T  T0 )

1  K2 (T  T0 ) K ¼ 1 T  T0 The following correspondences between the two sets of parameters are obvious: C1 ¼ K2 Treference  C2 ¼ T0 C1 Treference ¼

1 þ K2 T0 K1

1 ¼ C1 Treference  K2 T0 K1 ¼ C1 {C2 þ T0 }  C1 T0 ¼ C1 C2 ¼ 900 K Hence, the time – temperature shift factor can be presented using Vogel’s format with the classic WLF parameters: C1 C2 log aT (T; Treference ) ¼  C1 C2 þ (T  Treference )

10.30 EFFECT OF DILUENT CONCENTRATION ON THE WLF SHIFT FACTOR aC IN CONCENTRATED POLYMER SOLUTIONS Zero-shear-rate viscosities of concentrated polymer solutions h(T, wDiluent) are dominated by the macromolecule, not the solvent, particularly for high-molecular-weight chains that are above the threshold for entanglement formation (see Section 10.33). Following the development of Fujita and Kishimoto [1961], the solvent contribution to the solution viscosity is neglected and one writes

h(T, wDiluent )  NPolymer z(T, wDiluent )  NPure Polymer (1  wDiluent )z(T, wDiluent ) where NPure Polymer and NPolymer are the number densities of polymeric repeat units in the undiluted state and in concentrated solutions, respectively, wDiluent is the

10.30 Effect of Diluent Concentration on the WLF Shift Factor

395

diluent or solvent volume fraction, and z is the monomeric friction coefficient that exhibits functional dependence on fractional free volume, as described by Doolittle’s equation. Under isothermal conditions at temperature T, the concentration-dependent WLF shift factor is defined as follows:   h(T, wDiluent ) z(T, wDiluent ) 1  wDiluent ¼ aC (wDiluent ; T, wreference ) ¼  h(T, wreference ) z(T, wreference ) 1  wreference where wreference represents the diluent volume fraction in the reference mixture whose average material response time is kllreference. Hence, the concentration-dependent shift factor aC(wDiluent; wreference) is constructed from a ratio of monomeric friction coefficients for two polymer solutions. By comparison, temperature-dependent shift factors aT (T; Treference) are given by the ratio of zero-shear-rate viscosities at temperatures T and Treference for a viscoelastic material at constant composition. Since polymer concentration is the same at temperatures T and Treference in the construction of aT, the ratio of zero-shear-rate viscosities translates directly into the corresponding ratio of monomeric friction coefficients. In general, aT represent a ratio of material response times at temperatures T and Treference, whereas aC corresponds to a ratio of viscoelastic time constants for two solutions at diluent compositions wDiluent and wreference. If monomeric friction coefficients are described by Doolittle’s equation, then b1 ln{z(T, wDiluent )} ¼ b0 þ fMixture (T, wDiluent ) and the concentration-dependent WLF shift factor is correlated in terms of temperature and diluent composition via fractional free volume of the mixture, fMixture(T, wDiluent), as follows: ln{aC (wDiluent ; T, wreference )} ¼ ln{z(T, wDiluent )}  ln{z(T, wreference )} ¼

b1 b1  fMixure (T, wDiluent ) fMixture (T, wreference )

Ferry [1980] describes how aC decreases at higher diluent volume fractions. This trend is reasonable because viscoelastic relaxation times decrease at higher dilution in lower-viscosity solutions. Further analysis of the compositional dependence of aC requires knowledge of the effect of temperature and diluent concentration on the fractional free volume of polymer solutions. As discussed in Sections 1.11 and 1.12, linear free-volume theory yields the following relations: fMixture (T, wDiluent ) ¼ fPolymer (T) þ wDiluent G(T) fPolymer (T) ¼ fPolymer (Tg,Polymer ) þ [aLiquid,Polymer  {1  fPolymer (Tg,Polymer )}  aGlass,Polymer ](T  Tg,Polymer ) G(T) ¼ fDiluent (T)  fPolymer (T) where a represents the coefficient of thermal expansion, either in the liquid or glassy states, fractional free volume at the glass transition temperature is approximately 2.5%

396

Chapter 10 Mechanical Properties of Viscoelastic Materials

(i.e., 0.025) for any material, and G(T ) is the plasticizer efficiency parameter, defined as the difference between fractional free volume of pure diluent and pure polymer at temperature T. Algebraic manipulation, which employs fMixture(T, wDiluent) ¼ fMixture(T, wreference) þ G(T ){wDiluent 2 wreference}, provides the following concentration dependence of the WLF shift factor: 2 1 fMixture (T, wreference ) fMixture (T, wreference ) ¼ þ ln{aC (wDiluent ; T, wreference )} b1 b1 G(T)   1  wDiluent  wreference

Isothermal dynamic viscoelastic measurements on polymer solutions yield average relaxation times kll, or terminal relaxation times as described below. These viscoelastic time constants exhibit dependence on diluent concentration such that kll decreases when the diluent volume fraction increases. The previous equation suggests that free-volume parameters can be obtained via linear least squares analysis. The procedure is as follows: Step 1:

Choose a “reference” solution with average viscoelastic relaxation time kllreference that is dictated by the diluent volume fraction at which the fractional free volume of the mixture is desired.

Step 2:

Calculate the concentration-dependent WLF shift factor aC for each solution via division of its viscoelastic relaxation time kll by kllreference. It is also possible to calculate aC, based on its definition, from concentrationdependent zero-shear-rate viscosity data in polymer solutions at diluent volume fraction w Diluent. Do not analyze data for the reference solution, where w Diluent ¼ wreference and aC ¼ 1.

Step 3: Step 4:

Use a first-order polynomial model: y(x) ¼ a0 þ a1 x. The independent variable is x¼

Step 5:

The dependent variable is y¼

Step 6:

1 ¼ ln{aC (wDiluent ; T, wreference )}

 ln

fMixture (T, wreference ) b1

The slope, or first-order coefficient, is a1 ¼

1

hl(wDiluent ; T)i hl(wreference ; T)i

The intercept, or zeroth-order coefficient, is a0 ¼

Step 7:

1 wDiluent  wreference

2 fMixture (T, wreference ) b1 G(T)



10.31 Stress Relaxation Moduli via the Distribution of Viscoelastic Time Constants

397

Typically, one sets b1 ¼ 1 and uses the slope and intercept to evaluate the plasticizer efficiency parameter G(T ) and the fractional free volume of the reference mixture at temperature T and diluent volume fraction wreference. Since fMixture (T, wreference ) ¼ fPolymer (T) þ wreference G(T) G(T) ¼ fDiluent (T)  fPolymer (T) it is possible to estimate the fractional free volume of each pure component at temperature T. Then, knowledge of the glass transition temperature of pure polymer Tg,Polymer and pure diluent Tg,Diluent, together with the assumption that Tg is an iso-free-volume state with approximately 2.5% fractional free volume, provides information to evaluate the difference between thermal expansion coefficients in the liquid and glassy states for each pure component as follows: fPolymer (T) ¼ fPolymer (Tg,Polymer ) þ [aLiquid,Polymer  {1 fPolymer (Tg,Polymer )}  aGlass,Polymer ](T  Tg,Polymer )  0:025 þ [aLiquid,Polymer  0:975aGlass,Polymer ](T  Tg,Polymer ) fDiluent (T) ¼ fDiluent (Tg,Diluent ) þ [aLiquid,Diluent  {1 fDiluent (Tg,Diluent )}  aGlass,Diluent ](T  Tg,Diluent )  0:025 þ [aLiquid,Diluent  0:975aGlass,Diluent ](T  Tg,Diluent )

10.31 STRESS RELAXATION MODULI VIA THE DISTRIBUTION OF VISCOELASTIC TIME CONSTANTS The primary objective of this section is to modify the one-time-constant Maxwell model because it is not sufficient to describe the viscoelastic response of real materials. A spring and dashpot in series predict that the relaxation modulus decreases abruptly when the observation time is comparable to the material response time. This abrupt decrease in ER occurs predominantly over one decade on the logarithmic time axis, as dictated by the Maxwell model. Real materials do not respond in this fashion. The principle of time – temperature superposition allows one to construct the actual stress relaxation response of real materials, which require several viscoelastic time constants for an accurate description of mechanical properties. From a modeling viewpoint, real materials that exhibit N relaxation times are described by N Maxwell elements in parallel. This is the Maxwell – Wiechert model, where two parameters are required to define each spring and dashpot. These parameters are the spring modulus Ei and the material response time li ¼ hi/Ei for the ith element. Consider stress relaxation of the Maxwell – Wiechert model, where the entire assembly is subjected to a “jump” strain g0 at time t0. The following rules apply to elements in parallel: Rule 1: Each Maxwell element experiences the same strain g0 ¼ constant, for t  t0.

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Chapter 10 Mechanical Properties of Viscoelastic Materials

Rule 2: Total stress sTotal is obtained by adding the stress in each Maxwell element. Hence, N X sTotal ¼ si i¼1

where si is the stress experienced by the ith Maxwell element. Now, it is necessary to analyze each Maxwell element separately to calculate si during stress relaxation. The result is exactly the same as the previous analysis of stress relaxation for the Maxwell model (see Section 10.16), except that one must include subscripts to indicate that the ith element exhibits static modulus Ei and material response time li. Hence, the time dependence of stress in the ith element is   (t  t0 ) si (t  t0 ) ¼ Ei g0 exp  li (T) Total stress in the Maxwell – Wiechert model is obtained by summing the contribution si from each of the N elements in parallel. Then, the relaxation modulus for this model, obtained directly from a “constant” strain experiment, is calculated via division of sTotal by g0. The result is   N sTotal (t  t0 ) X (t  t0 ) ¼ Ei exp  ER (t  t0 ) ¼ g0 li (T) i¼1 The Maxwell – Wiechert model exhibits a discrete spectrum of viscoelastic time constants. In other words, there are N discrete data pairs: Ei and li. When the number N of data pairs approaches infinity, the discrete spectrum becomes a continuous spectrum of viscoelastic time constants and, in the continuous limit, these data pairs map out a continuous function ED(l ), which is known as the distribution of viscoelastic relaxation times (i.e., ED(l ) is not a normalized distribution function). These concepts are illustrated below, based on the results of the Maxwell – Wiechert model: ER (t  t0 ) ¼

N X i¼1

  (t  t0 ) Ei exp  ) N!1 li (T)

1 ð

  (t  t0 ) ED (l) exp  dl l

l¼0

where ED(l ) dl represents the non-normalized probability that the viscoelastic material has a relaxation time between l and l þ dl. However, if the relaxation modulus is normalized via division of ER(t 2 t0) by ER(0), then evaluation of the previous integral expression when t ¼ t0 yields the following normalized distribution of viscoelastic relaxation times, ED(l )/ER(0): {ER (t  t0 )}t¼t0 ¼1¼ ER (0)

1 ð

l¼0

  ED (l) dl ER (0)

If ED(l ) is known, then one can predict all of the linear viscoelastic properties of a material. For example, if a polymer exhibits two discrete relaxation times t1 and t2

10.31 Stress Relaxation Moduli via the Distribution of Viscoelastic Time Constants

399

with corresponding spring constants E1 and E2, then the distribution function is described by Dirac delta functions: ED (l) ¼ E1d(l  t1 ) þ E2 d(l  t2 ) Normalization of the previous expression can be obtained via division by E1 þ E2. These delta functions imply that the distribution function exhibits spikes at l ¼ t1 and l ¼ t2. The relaxation modulus is calculated as follows: ER (t  t0 ) ¼

  (t  t0 ) dl [E1 d(l  t1 ) þ E2 d(l  t2 )] exp  l

1 ð

l¼0

When the integrand contains a delta function, integration is based on the following theorem: þ1 ð d(x  a) f (x, t) dx ¼ f (a, t) 1

Normalized Storage and Loss Moduli

In other words, if the integration variable x is part of the argument of a delta function, then choose x within the integration limits such that the argument is zero where

Storage Modulus Loss Modulus

0.8

0.6

0.4

0.2

0.0 100 Figure 10.1

101 102 Experimental Test Frequency (radians/second)

103

Frequency dependence of the storage (i.e., elastic) modulus and loss modulus (lower curve) for a viscoelastic model that contains two Maxwell elements in parallel, such that the distribution function, ED(l ), for this two-time-constant model is spiked at 1023 s and 1021 s, with weighting factors of 60% and 40%, respectively.

400

Chapter 10 Mechanical Properties of Viscoelastic Materials

the delta function exhibits a spike, and evaluate the remainder of the integrand at the chosen value of x. For the previous example of a material that exhibits two discrete relaxation times, formalism based on the distribution of viscoelastic relaxation times yields     (t  t0 ) (t  t0 ) þ E2 exp  ER (t  t0 ) ¼ E1 exp  t1 t2 If t1 , t2, then this material exhibits solid-like behavior with an elastic modulus given by E1 þ E2 when t 2 t0  t1. This is essentially the equivalent static modulus of two springs in parallel or, analogously, the equivalent capacitance of two capacitors in parallel. Liquid-like behavior with very little resistance to deformation is achieved when t 2 t0  t2. For observation times that are greater than t1 and less than t2 (i.e., t1 , t 2 t0 , t2) the relaxation modulus exhibits a plateau at ER  E2. This plateau region is broader when t2 (i.e., the molecular-weight-dependent terminal relaxation time) is much larger than t1. A snapshot of ER(t 2 t0) can be obtained from the elastic (i.e., storage) component of the dynamic modulus from oscillatory testing versus test frequency, as discussed in Section 10.38 and illustrated in Figure 10.1, for a distribution function that contains two discrete relaxation times, t1 and t2.

10.32 STRESS RELAXATION MODULI AND TERMINAL RELAXATION TIMES There are two different distribution functions that characterize the spectrum of relaxation times. One of these, ED(l ), was discussed above via an extension of the Maxwell – Wiechert model. The other distribution function, HD(l ), is defined below in terms of the time-dependent relaxation modulus:

ER (t  t0 ) ¼

1 ð

1     ð (t  t0 ) (t  t0 ) ED (l) exp  HD (ln l) exp  dl ¼ d ln l l l

l¼0

l¼0

Hence, HD (ln l) ¼ lED (l) The reason for introducing HD(ln l ) is that one typically approximates HD as a “flat” distribution between the limits of l1 and l2, where l1 is the shortest response time of the material, and l2 represents the terminal relaxation time which depends on molecular weight. In other words, the viscoelastic material exhibits no response times that are greater than l2. The “flat” distribution function is described by HD (ln l) ¼ E0

l1 l l2

E0 l

l1 l l2

ED (l) ¼

10.32 Stress Relaxation Moduli and Terminal Relaxation Times

401

Evaluation of the parameter E0 is possible when the stress relaxation modulus is normalized with respect to its initial value. For example, lð2       {ER (t  t0 )}t¼t0 E0 E0 l2 ¼1¼ d ln l ¼ ln ER (0) ER (0) ER (0) l1 l¼l1

E0 ¼

ER (0) ln(l2 =l1 )

Now, set t0 ¼ 0 and calculate the relaxation modulus via the ED distribution function, realizing that ED ¼ 0 when l , l1 and l . l2: lð2 t  E0 ER (t) ¼ exp dl l l l¼l1

Multiply and divide the integrand by t, “normalize” the relaxation modulus by E0, and let x ¼ t/l: ER (t) ¼ E0

lð2

lð 2 =t   t=ðl1 t  l t 1 1 ¼ ¼ exp(x) dx exp x exp(x) d d l l t x x

l¼l1

l1 =t

t=l2

The result can be expressed in terms of the “exponential integral,” which is defined by 1 ð 1 exp(x) dx Eexp,int (z) ¼ x x¼z

This integral is denoted by E1 in most texts and requires numerical evaluation. It is tabulated in Abramowitz and Stegun [1965] for various values of the dimensionless parameter z. Hence, for some small nonzero constant 1 that is less than both t/l1 and t/l2, ER (t) ¼ E0

t=ðl1

1 exp(x) dx ¼ x

8 > ð <1 1 > : x

8 > ð <1 1 > : x 1

x¼t=l1

exp(x) dx 

1 exp(x) dx x

1 1 ð

exp(x) dx 

1



t=ðl2

1 exp(x) dx  x

1

t=l2

¼

t=ðl1

9 > = 1 exp(x) dx > x ;

1 ð

x¼t=l2

9 > = 1 exp(x) dx > x ;

    t t ¼ Eexp,int  Eexp,int l2 l1

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Chapter 10 Mechanical Properties of Viscoelastic Materials

If the shortest relaxation time l1 is small enough (i.e., millisecond regime) such that t/l1 ) 1 when t . 0, then       ER (t) t t t  Eexp,int ¼ Eexp,int ) 1  Eexp,int l2 (MW) l1 l2 (MW) E0 Justification for the neglect of the exponential integral at relatively large arguments (i.e., t/l1) is provided below. Hence, truncation error in the previous equation contributes insignificantly when t/l1  5. Notice that at a given observation time t, longer terminal relaxation times l2 yield a smaller argument for the exponential integral on the extreme right side of the previous equation, a larger value for Eexp,int (t/l2), and a larger relaxation modulus, which correlates directly with the effects of chain entanglements. Selected Values of the Exponential Integral, Eexp,int (z) z

Eexp,int (z)

0.05 0.1 0.2 0.3 0.4 0.5 0.7 1 2 5 7 10

2.47  100 1.82  100 1.22  100 9.06  1021 7.02  1021 5.60  1021 3.74  1021 2.19  1021 4.89  1022 1.15  1023 1.15  1024 4.16  1026

If the “flat” distribution for HD is reasonable, then it should be possible to (i) measure stress relaxation moduli as a function of time t, (ii) normalize the data via division of ER(t) by ER(0), (iii) neglect the response at very short times because the terminal relaxation time does not influence the early data, and (iv) identify the argument of the exponential integral which increases linearly with time (i.e., z(t) ¼ kt) to match experimental data. Hence, ER (t) 1  Eexp,int (kt) ER (0) ln(l2 =l1 ) If the parameter k is found to reproduce experimental relaxation moduli versus time, then t kt ¼ l2 (MW) and the terminal relaxation time for the viscoelastic material of a given molecular weight is the inverse of k. Now, stress relaxation experiments are performed on various molecular weight fractions of a given polymer, and the procedure described above is

10.34 Zero-Shear-Rate Viscosity via the Distribution of Viscoelastic Relaxation Times

403

employed to identify the parameter k and the terminal relaxation time l2 for each sample with different molecular weight. The results can be summarized in terms of the following molecular weight scaling laws for the terminal relaxation times:

l2  a{MW}1:0 when MW , MWcritical l2  b{MW}3:3 when MW . MWcritical where MWcritical represents the critical molecular weight required for chain entanglements. MWcritical depends on the chemical nature of the polymer and, as illustrated by the values in Section 10.33, MWcritical is typically in the range of a few thousand daltons to a few tens-of-thousand daltons. The scaling laws presented above illustrate how an increase in chain length beyond MWcritical has a significant effect on terminal relaxation times, which extends the plateau modulus to much longer times or, equivalently, to considerably higher temperatures. When polymers exhibit a distribution of chain lengths, the previous terminal relaxation time scaling laws require the weightaverage molecular weight.

10.33 THE CRITICAL MOLECULAR WEIGHT REQUIRED FOR ENTANGLEMENT FORMATION

Polymer Polyethylene Poly(styrene) Poly(propylene) Poly(vinyl alcohol) Poly(vinyl acetate) Poly(vinyl chloride) Poly(methyl methacrylate) Poly(ethylene oxide) Poly(propylene oxide) Polycarbonate

Experimental MWcritical (daltons) 4000 31200 7000 7500 24500 11000 18400 4400 5800 4800

Theoretical MWcritical (daltons) 4200 32000 7600 7000 25000 10700 18000 5000 5000 4300

10.34 ZERO-SHEAR-RATE VISCOSITY VIA THE DISTRIBUTION OF VISCOELASTIC RELAXATION TIMES Let’s return to a previous result in Section 10.25, where the zero-shear-rate viscosity for an uncrosslinked polymer is obtained via the area under the stress relaxation modulus curve from time t0, when a “jump” strain is imposed on the sample, to

404

Chapter 10 Mechanical Properties of Viscoelastic Materials

infinite times: 1 ð

h(T) ¼

ER (t  t0 ; T) dt

t¼t0

Now, express the stress relaxation modulus ER(t 2 t0) in terms of the distribution of viscoelastic relaxation times ED(l ) and reverse the order of integration:

h(T) ¼

1 ð

2 4

l¼0

t¼t0

¼

1 ð

1 ð

3   (t  t0 ) d l5 dt ED (l) exp  l 2

E D (l )4

l¼0

1 ð

3 1   ð (t  t0 ) dt5d l ¼ exp  lED (l) dl ¼ hl(T)i l l¼0

t¼t0

This calculation reveals that the zero-shear-rate polymer viscosity is equivalent to the first moment of the distribution of viscoelastic relaxation times, or the average relaxation time for the material. By definition, the nth moment of the distribution is

n

hl i ¼

1 ð

ln ED (l) d l

l¼0

Realize, of course, that the distribution function ED(l ) is not normalized and that ED(l ) dl has dimensions of modulus. Also, the temperature dependences of h and kll are the same, and they follow WLF formalism via the temperature dependence of the shift factor aT in the vicinity of the glass transition temperature. The molecular weight dependence of zero-shear-rate viscosities is obtained by calculating the first moment of the distribution of viscoelastic relaxation times when the HD distribution is “flat” between l1 and l2. The result is

h(T; MW) ¼

1 ð

l¼0

lED (l) d l ¼

lð2

E0 d l ¼ E0 {l2  l1}  E0 l2 (T; MW)

l¼l1

when the shortest relaxation time l1 can be neglected relative to the terminal relaxation time l2. Hence, viscosities are directly proportional to terminal relaxation times, the latter of which scale as MW 1.0 or MW 3.3 depending on the molecular weight range relative to MWcritical. Weight-average molecular weights are required in these scaling laws for zero-shear-rate viscosity when chains exhibit a distribution of molecular weights. High-shear-rate viscosities for shear-thinning materials exhibit negligible dependence on molecular weight.

10.35 The Boltzmann Superposition Integral for Linear Viscoelastic Response

405

10.35 THE BOLTZMANN SUPERPOSITION INTEGRAL FOR LINEAR VISCOELASTIC RESPONSE This principle was mentioned initially in Section 10.11 when the equation of motion for the Maxwell model was integrated for any type of strain rate history. Consider a viscoelastic material with a stress relaxation modulus given by ER(t) and focus on small strains. In this regime of linear response, the following principles are applicable: 1. Strain, which is a continuous two-state function of time, can be viewed as an infinite sequence of infinitesimal “steps” Dgi, where each step persists for an infinitesimal observation time Dti. As each time interval approaches zero, the sequence of discrete steps simulates the continuous strain history g (t). 2. During each infinitesimal time interval Dti ¼ tiþ1 2 ti, which begins at time ti, stress relaxation is operative in response to the infinitesimal step strain Dgi ¼ gi 2 gi21. In other words, “jump” strain Dgi is imposed upon a viscoelastic material at time ti and it survives for a duration given by Dti before another jump strain is superimposed on the previous one. The induced stress si during this time interval is a linear function of the small step strain:

si (t) ¼ ER (t  ti )Dgi where the relaxation modulus is strain independent in the regime of small strains. The previous equation is exactly the same as the stress relaxation response of a viscoelastic material that can be described by the Maxwell model. At time t1, the material is subjected to “jump” strain Dg1 ¼ g1, because g0 ¼ 0. 3. Total stress s (t) is calculated from a linear superposition of time-dependent stresses described in principle 2. Hence,

s (t) ¼

1 X

si (t) ¼

i¼1

1 X i¼1

 ER (t  ti )

 Dgi Dti Dti

4. As each time interval Dti approaches zero, Dgi/Dti is synonymous with the rate of strain history @ g/@Q, and the infinite sum of stresses is written in integral form:   ðt @g dQ s (t) ¼ ER (t  Q) @Q Q¼t1

The previous result corresponds to zero stress at t ¼ t1, when the material is subjected to the initial step strain. Now, redefine the reference state such that s (t) approaches zero as t ) 21. This change of reference states modifies s (t) by an additive constant, and it is equivalent to replacing t1 in the lower integration limit by 21. Hence, the Boltzmann superposition integral predicts that the induced stress in a viscoelastic material can be calculated for any type of strain rate history, provided that the

406

Chapter 10 Mechanical Properties of Viscoelastic Materials

stress relaxation modulus is known and total strain remains in the linear regime:   ðt @g dQ s (t) ¼ ER (t  Q) @Q Q)1

If the strains are large enough and mechanical response cannot be described by the postulates of linear viscoelasticity, then the previous equation must be modified by including (i) strain dependence of the stress relaxation modulus and (ii) additional integrals in a series expansion that contains higher order terms based on the rate of strain history. As an example of linear response, consider steady state shear flow of a memory fluid between two parallel plates, where one of the plates is stationary and the other one moves with a constant linear velocity from left to right. In this case, the viscoelastic fluid experiences a constant rate of strain, and @ g/@Q ¼ @ g/@t can be removed from the previous integral expression. The Boltzmann superposition integral resembles Newton’s law of viscosity: 9 8 t = @g < ð @g ¼h s (t) ¼ ER (t  Q) dQ ; @t : @t Q)1

where the quantity in brackets in the previous equation represents the zero-shear-rate polymer viscosity because the Boltzmann expression is restricted to small strains and small velocity gradients. Hence, ðt

h(T) ¼

ER (t  Q; T) dQ

Q)1

Now, let the observation time at constant t be replaced by s ¼ t 2 Q, and ds ¼ 2dQ. The previous integral is re-expressed as

h(T) ¼

1 ð

ER (s; T) ds s¼0

which states that the zero-shear-rate polymer viscosity is equivalent to the area under the stress relaxation modulus versus time curve, beginning when the “jump” strain is applied, and extending to infinite time. This result was obtained in Section 10.25 using an expression for the stress relaxation modulus from the Maxwell model.

10.36 ALTERNATE FORMS OF THE BOLTZMANN SUPERPOSITION INTEGRAL FOR s (t ) Begin with the general expression for s (t) from the previous section:

s (t) ¼

ðt Q)1

 ER (t  Q)

 @g dQ @Q

10.37 Linear Viscoelastic Application of the Boltzmann Superposition Principle

407

where the dependence of the state of stress on present time t appears in the upper limit of integration, and in the observation time for the relaxation modulus. Now, change integration variables from Q to observation time s ¼ t 2 Q. Hence, g (Q) becomes g (t 2 s), and @ g (Q)/@Q becomes –{@ g (t 2 s)/@s}t. The result is

s (t) ¼ 

1 ð

s¼0

  @ g (t  s) ER (s) ds @s t

The previous equation is employed to relate stress relaxation moduli and dynamic moduli via complex variable analysis and Fourier transformation in Section 10.40. Now, begin with the first expression for time-dependent stress in this section and integrate by parts:   @ER (t  Q) dQ u ¼ ER (t  Q); du ¼ @Q t   @ g (Q) dQ; v ¼ g (Q) dv ¼ @Q t One obtains

s (t) ¼ g (t)ER (0)  g (1)ER (1) 



ðt

g (Q) Q)1

 @ER (t  Q) dQ @Q t

The second term on the right side of the previous equation vanishes because strain is finite when t )21, but the relaxation modulus decays to zero at infinite observation times (i.e., ER(1) ¼ 0) for linear polymers or materials that are crosslinked below the percolation threshold. A change of variables in the integral of the previous equation, letting s ¼ t 2 Q, yields 1   ð dER (s) ds s (t) ¼ g (t)ER (0) þ g (t  s) ds s¼0

10.37 LINEAR VISCOELASTIC APPLICATION OF THE BOLTZMANN SUPERPOSITION PRINCIPLE: ELASTIC FREE RECOVERY Consider squeezing a viscoelastic material that has the shape of a solid rubber ball through a tube in which the tube diameter is smaller than that of the ball. If the deformation is small enough and within the regime of linear response, then the constitutive equation that relates stress s to the rate of strain @ g/@t is described adequately by the Boltzmann superposition integral. The polymeric material is subjected to a constant rate of strain K for times t less than zero (i.e., –1 , t , 0). Stress is removed at time t ¼ 0 when the constrained material exits the tube and experiences free recovery,

408

Chapter 10 Mechanical Properties of Viscoelastic Materials

based on its memory, toward the undeformed state. The primary objective of this linear viscoelastic problem is to predict the total possible recoverable deformation g1, defined by

g1 ¼

1 ð

 @g dt @t

0

in terms of the generalized stress relaxation modulus ER(t). Since free recovery occurs for times t . 0 in the absence of any stress, but not in the absence of the rate of strain, the original expression for s (t) via the Boltzmann superposition integral yields   d g (Q) dQ ER (t  Q) dQ

ðt

s (t) ¼

Q)1

ð0

¼

ER (t  Q)K dQ þ

Q)1

ðt



 dg (Q) dQ ¼ 0 ER (t  Q) dQ

Q¼0

The strategy to isolate g1 involves (i) changing integration variables, (ii) integrating the previous equation from t ¼ 0 to t ) 1, and (iii) reversing the order of integration. Each of these steps is illustrated below. Begin by changing variables from Q to s, via s ¼ t – Q. One obtains 1 ð

K s¼t

ER (s) ds ¼



ðt ER (s) s¼0

 @ g (t  s) ds @s t

Now, integrate the previous equation with respect to time t, from t ¼ 0 to t ) 1: 8 9 8 9 1 1  = ð ð < ðt ð <1 = @ g (t  s) ER (s) ds dt ¼ ER (s) ds dt K : ; : ; @s t t¼0

s¼t

t¼0

s¼0

The left side of this double integral involves integration with respect to s from s ¼ t to s ) 1, followed by integration with respect to t from t ¼ 0 to t ) 1. Upon reversal, one integrates with respect to t from t ¼ 0 to t ¼ s, followed by integration with respect to s from s ¼ 0 to s ) 1. The left side of the previous equation reduces to 9 9 8 8 s 1 1 1 ð ð ð ð <1 = <ð = ER (s) ds dt ¼ K ER (s) dt ds ¼ K sER (s) ds K ; ; : : t¼0

s¼t

s¼0

t¼0

s¼0

10.37 Linear Viscoelastic Application of the Boltzmann Superposition Principle

409

Now, consider the following double integral: 8 9 1  = ð < ðt @ g (t  s) ER (s) ds dt : ; @s t t¼0

s¼0

where one integrates first with respect to s from s ¼ 0 to s ¼ t, then with respect to t from t ¼ 0 to t ) 1. Upon reversal, integration occurs first with respect to t from t ¼ s to t ) 1, followed by integration with respect to s from s ¼ 0 to s ) 1. One obtains 8 9 81 9 1 1  = = ð ð  ð < ðt < @ g (t  s) @ g (t  s) ER (s) ds dt ¼ ER (s) dt ds : ; : @s @t t s ; t¼0

s¼0

s¼0

t¼s

Integration with respect to t on the right side of the previous equation is recast in terms of variable Q, where Q ¼ t – s, to isolate the total irrecoverable deformation, g1: 1 ð t¼s

 ð1  Q¼ts @ g (t  s) dg (Q) dQ ¼ g1 dt ) @t dQ s Q¼0

The final prediction of elastic recovery from linear viscoelastic analysis is ð1 K sER (s) ds s¼0 ð g1 ¼ 1 ER (s) ds s¼0

where the denominator on the right side of the previous equation represents the zero-shear-rate polymer viscosity that decreases at higher temperature. The effect of constant strain rate K on the magnitude of free recovery is obvious, because g1 scales linearly with K. Hence, an n-fold increase in strain rate produces response that is more elastic-like, so the total elastic recovery also increases n-fold. The effect of temperature on g1 is obtained by invoking the relaxation modulus for the single-time-constant Maxwell model and evaluating both integrals in the previous equation. For example, ER (s) ¼ E exp{s=l(T)} 1 ð

ER (s) ds ¼ E

s¼0 1 ð

s¼0

1 ð

exp{s=l(T)} ds ¼ E l(T)[exp{s=l(T)}]s¼0 s)1 ¼ E l(T)

s¼0

sER (s) ds ¼ E

1 ð

2 s exp{s=l(T)}ds ¼ E l(T)[(s þ l) exp{s=l(T)}]s¼0 s)1 ¼ E l (T)

s¼0

g1 ¼ K l(T)

410

Chapter 10 Mechanical Properties of Viscoelastic Materials

Since uncrosslinked viscoelastic materials exhibit less memory and behave more viscous-like at higher temperature, with shorter relaxation times l, the total elastic recovery decreases as temperature increases. Thought-provoking exercise: Alessandro’s pizza dough in Piedicastello, dopo Ponte San Lorenzo en route from Trento, is a hydrated viscoelastic polysaccharide that exhibits significant elastic free recovery in response to intermittent stress. Without releasing your inner carbohydrate-loading capacity, sketch the diameter of circular-shaped pizza dough vs. time for 24 hours after it is prepared and subjected to short square-wave stress pulses, downward and radially outward, every six hours.

10.38 DYNAMIC MECHANICAL TESTING OF VISCOELASTIC SOLIDS VIA FORCED VIBRATION ANALYSIS OF TIME-DEPENDENT STRESS AND DYNAMIC MODULUS E (t ; v) Results from “forced vibration” experiments are useful to analyze molecular motion and damping characteristics of viscoelastic materials. At the molecular level, one correlates motion of segments of the polymer chain with viscoelastic relaxation. When these relaxation processes occur at lower temperature or higher frequency, the motion that gives rise to this resonance condition exhibits a smaller moment of inertia. Hence, viscoelastic relaxation at lower temperature or higher frequency is associated with localized motion of the polymer chain and only involves a few repeat units, at most. This is characteristic of small-scale micro-Brownian molecular motion below the glass transition temperature. At the macroscopic level, the combination of temperature and frequency that corresponds to the resonance condition, or viscoelastic relaxation, allows one to design materials that can dissipate mechanical energy irreversibly to thermal energy. This is advantageous for vibration damping, but one must realize that thermal energy is generated when damping occurs. The forcing function in these experiments is an oscillatory strain that depends on time t and frequency v. From a practical viewpoint, it is necessary to superimpose this harmonic excitation on a constant dc level of strain to ensure that thin-film solids are always under tension to maintain rectangular geometry. For example, the strain history is

g (t; v) ¼ gdc þ g0 sin vt where gdc . g0. The characteristic observation time in these dynamic experiments is based on one cycle of oscillation, or 2p/v. If the viscoelastic material is described best by a distribution of relaxation times, then the average relaxation time kll is a reasonable representation of the material response time. Hence, the ratio of the material response time to the characteristic time scale of the experiment, or the Deborah number, is given by vhl(T)i De  2p The equivalence between temperature and frequency in these dynamic experiments is obtained from an analysis of the Deborah number. For example, rigid elastic-like solid

10.38 Dynamic Mechanical Testing of Viscoelastic Solids

411

behavior in the glassy state occurs at large values of De. This can be achieved when either (i) kll is large at low temperature or (ii) v is large. In other words, solid-like response occurs at low temperature or high frequency. Liquid-like behavior at small Deborah numbers is achieved at high temperature or low frequency. If these dynamic experiments are performed isothermally as a function of the frequency of excitation, then the corresponding high-temperature response is found at low frequency. Viscoelastic relaxation occurs when the resonance condition is satisfied and the Deborah number is on the order of unity, or vl  2p, where l is the material response time of the viscoelastic process that undergoes relaxation. When dynamic experiments are performed at higher frequency, viscoelastic relaxation occurs at higher temperature because it is necessary to maintain vl  2p at resonance and material response times are shorter at higher temperature. Since both gdc and g0 are small and within the regime of linear response, one calculates the induced stress s (t; v) from the Boltzmann superposition integral with a strain-independent relaxation modulus. Begin with the initial representation of the Boltzmann superposition integral for linear viscoelastic response (i.e., the first equation in Section 10.36):   ðt @g dQ s (t) ¼ ER (t  Q) @Q Q)1

and evaluate the rate-of-strain history:

g (Q; v) ¼ gdc þ g0 sin vQ   @g ¼ vg0 cos vQ @Q v Notice that the dc level of strain required to maintain tension on the sample throughout the experiment does not affect the rate-of-strain history. However, stress relaxation will occur in response to gdc that is not described by the equations below. Now, evaluate the stress relaxation modulus ER(t 2 Q) for a viscoelastic material that exhibits a continuous distribution of material response times l: ð1

ER (t  Q) ¼

  (t  Q) dl ED (l) exp  l

l¼0

The induced stress is ðt

s (t; v) ¼ vg0

Q)1

2 4

1 ð

3  (t  Q) d l5cos(vQ) dQ ED (l) exp  l 

l¼0

Upon reversing the order of integration, one obtains 2 t 3 1   ð ð  t Q cos(vQ) dQ5 d l s (t; v) ¼ vg0 ED (l) exp  4 exp l l l)0

Q)1

412

Chapter 10 Mechanical Properties of Viscoelastic Materials

Mathematics tables allow one to evaluate the first integration with respect to Q. The result is ðt

v

    t  Q (vl)2 1 exp exp cos(vt) sin(vt) þ cos(vQ) dQ ¼ l l vl 1 þ (vl)2

Q)1

Now, the induced stress becomes

s (t; v) ¼ g0 sin(vt)

1 ð

l¼0

(vl)2 E D (l ) dl þ g0 cos(vt) 1 þ (vl)2

1 ð

E D (l ) l¼0

vl dl 1 þ (vl)2

Since the harmonic excitation for the time dependence of strain was expressed as a sine wave, the Boltzmann superposition integral reveals linear response because the induced stress also oscillates at frequency v. There are no overtones in s (t; v) at 2v, 3v, and so on. In other words, frequency doubling and tripling does not occur. Furthermore, one identifies the elastic contribution to the induced stress as the coefficient of sin(vt), which is in-phase with g (t; v). The viscous contribution to s (t; v) is given by the coefficient of cos(vt), or sin(vt þ p/2), which is 908 out-of-phase with g (t; v). The previous result for s (t; v) is divided by the magnitude of the oscillatory part of the strain function (i.e., g0, not gdc) to generate the dynamic modulus E (t; v), defined by E (t; v) ¼

s (t; v) ¼ E 0 (v, T) sin(vt) þ E 00(v, T) cos(vt) g0

The storage modulus E0(v), 0

E (v, T) ¼

1 ð

E D (l ) l¼0

(vl)2 dl 1 þ (vl)2

is associated with mechanical energy stored during each cycle of the dynamic experiment due to the elastic nature of the material. The viscoelastic loss modulus E00 (v), 1 ð vl 00 ED (l) dl E (v, T) ¼ 1 þ (vl)2 l¼0

measures the mechanical energy that is degraded irreversibly to thermal energy during each cycle of the dynamic experiment due to the viscous nature of the material. All of these moduli, E , E0 , and E00 (i.e., dynamic, storage, and loss), are not affected by the dc level of strain required to maintain tension on the sample because, if viscoelastic response occurs in the linear regime, then moduli are independent of strain.

10.39 Phasor Analysis of Dynamic Viscoelastic Experiments via Complex Variables

413

Knowledge of the distribution of viscoelastic relaxation times for a particular material allows one to predict the performance of this material in a dynamic test and calculate the frequency dependence of the storage and loss moduli. To illustrate this concept, consider a viscoelastic material that can be modelled by a spring and dashpot in series with one material response time t. The following analysis is specific to the Maxwell model, where the distribution function is spiked at l ¼ t. Hence, ED (l) ¼ E d(l  t) The storage and loss moduli for this material can be simplified greatly. For example, 0

E (v, T) ¼ E

1 ð

d ( l  t)

(vl)2 (vt)2 d l ¼ E 1 þ (vl)2 1 þ (vt)2

d ( l  t)

vl vt dl ¼ E 1 þ (vl) 1 þ (vt)2

l¼0 00

E (v, T) ¼ E

1 ð

l¼0

The storage modulus achieves its maximum value at very high frequency or, analogously, in the low-temperature limit for a glassy material. E0 decreases at lower frequency as the “rubbery plateau” behavior occurs at higher temperature. When the storage modulus exhibits its steepest decrease with respect to frequency or temperature, the loss modulus exhibits a maximum because molecular motion of chain segments in the backbone or side group is responsible for viscoelastic relaxation with energy-absorbing characteristics when resonance occurs at vt  1. The dominant peak in E 00 and the corresponding largest decrease in E0 (i.e., by about three or four orders of magnitude) occur at the glass transition temperature. This transition is obscured, to some extent, when chemical crosslinks are present. Predictions of storage E0 and loss E00 moduli versus test frequency are illustrated in Figure 10.1 when the distribution function ED(l ) contains two discrete relaxation times, t1 ¼ 1023 s (w1 ¼ 60%) and t2 ¼ 1021 s (w2 ¼ 40%). In other words, the normalized distribution of viscoelastic relaxation times is given by the following expression for two Maxwell elements in parallel: ED (l) ¼ w1 d (l  t1 ) þ w2 d (l  t2 )

10.39 PHASOR ANALYSIS OF DYNAMIC VISCOELASTIC EXPERIMENTS VIA COMPLEX VARIABLES The oscillatory nature of the excitation forcing function at frequency v can be represented in phasor notation, using exp(ivt), where i is the square root of 21. This methodology is convenient because the real and imaginary components of the response function provide decoupled information about the elastic and viscous

414

Chapter 10 Mechanical Properties of Viscoelastic Materials

nature of real materials. Hence, the time dependence of strain in a forced vibration experiment is given by

g (t; v) ¼ gdc þ g0 exp(ivt) ¼ gdc þ g0 {cos vt þ i sin vt} where g0 is the magnitude of the strain wave, and gdc  g0. The corresponding rate of strain is 

 @g ¼ ivg0 exp(ivt) ¼ vg0 {i cos vt  sin vt} @t v

Results from the previous section reveal that the induced stress in a viscoelastic material is not completely in-phase with the harmonic forcing function g. In complex variable notation, this is considered by introducing a phase angle difference d between stress and strain, where d depends on frequency and temperature, but not time. The phasor representation of stress, with magnitude s0, is

s (t; v) ¼ s0 exp{i(vt þ d)} is defined as the ratio of complex stress Now, the complex dynamic modulus E Complex to the oscillatory part of the complex strain phasor:

{E (v, T)}Complex ¼

s (t; v) s0 s0 ¼ exp(id) ¼ {cos d þ i sin d} g0 exp{i v t} g0 g0

For perfectly elastic isotropic solids that store mechanical energy with total memory, stress and strain are completely in-phase (i.e., d ¼ 0). In this case, the complex dynamic modulus is real, and it reduces to the storage modulus E0 . For viscous Newtonian liquids that are totally dissipative with no ability to store mechanical energy, stress and strain are completely out-of-phase (i.e., d ¼ p/2), the complex dynamic modulus is imaginary, and it reduces to the loss modulus E00 . These limiting cases are consistent with the following representation of the complex dynamic modulus: {E (v, T)}Complex ¼ E0 (v, T) þ iE 00 (v, T) ¼

s0 {cos d þ i sin d} g0

Hence, the storage and loss moduli are related to the phase angle difference between stress and strain in dynamic mechanical spectroscopy as follows: s0 E 0 (v, T) ¼ cos d (v; T) g0 s0 E00 (v, T) ¼ sin d (v; T) g0 Finally, the ratio of the loss modulus to the storage modulus is given by the tangent of the phase angle difference between stress and strain in dynamic experiments: E00 (v; T) ¼ tan d (v; T) E 0 (v; T)

10.40 Fourier Transformation of the Stress Relaxation Modulus

415

which is referred to as “tan del” or the loss tangent. An alternate viewpoint of this phasor analysis defines the complex dynamic viscosity h Complex as the ratio of complex stress to the complex rate of strain, where vg0 is the magnitude of the harmonic strain rate. Hence,

s (t; v) s0 s0 exp(id) ¼ {sin d  i cos d} {h (v, T)}Complex ¼   ¼ @g ivg0 vg0 @t v Limiting behavior reveals that the imaginary component survives when stress and strain are completely in-phase for perfectly elastic isotropic materials that store mechanical energy, whereas the real component is favored when d ¼ p/2 for completely dissipative viscous response. These trends are consistent with the following definitions of the storage and loss components of the complex dynamic viscosity: s0 {sin d  i cos d} {h (v, T)}Complex ¼ hLoss (v, T)  ihStorage (v, T) ¼ vg0

hStorage (v, T) ¼

s0 E 0 (v, T) cos d(v, T) ¼ vg0 v

hLoss (v, T) ¼

s0 E 00 (v, T) sin d(v, T) ¼ vg0 v

Both formalisms associate the “loss” component of the complex dynamic modulus or viscosity with irreversible degradation of mechanical energy to thermal energy. Dynamic experiments on viscoelastic solids typically provide the temperature or frequency dependences of E0 and E00 , as well as tan d. When harmonic oscillations are imposed on viscoelastic liquids, dynamic data are presented in terms of hStorage and hLoss.

10.40 FOURIER TRANSFORMATION OF THE STRESS RELAXATION MODULUS YIELDS DYNAMIC MODULI VIA COMPLEX VARIABLE ANALYSIS Dynamic properties of viscoelastic materials can be obtained from stress relaxation data via one-sided Fourier transformation of the relaxation modulus. Hence, one imposes a jump strain in the linear regime, measures the time dependence of the stress decay, and calculates storage and loss moduli as illustrated below. This procedure is similar to the one employed in Fourier transform infrared spectroscopy and pulsed NMR spectroscopy where information in the frequency domain is accessible from Fourier transformation of (i) transmitted light versus mirror position in a Michaelson interferometer and (ii) the free induction decay of magnetization in the rotating frame of reference as nuclear spins return to their equilibrium position along the static magnetic field. Complex variable analysis of dynamic mechanical spectroscopy begins with the second expression for the time dependence of stress via the Boltzmann superposition integral in Section 10.36 and employs phasor notation for harmonic strain. It is important to emphasize that the following equation

416

Chapter 10 Mechanical Properties of Viscoelastic Materials

requires a rate-of-strain history that is unique to dynamic mechanical testing, even though predictions based on the linear superposition theorem contain the stress relaxation modulus ER(s) that is not specific to any combination of springs and dashpots: 1   ð @ g (t  s) s (t) ¼  ER (s) ds @s t s¼0

The time dependence of g and @ g/@t has been addressed in the previous section. Now, one requires a representation of strain in terms of the quantity, t 2 s. Hence,

g (t; v) ¼ gdc þ g0 exp(ivt) g (t  s; v) ¼ gdc þ g0 exp{iv(t  s)}   @ g (t  s; v) ¼ ivg0 exp{iv(t  s)} @s t The complex dynamic modulus is obtained via division of s (t) from the Boltzmann superposition integral by the oscillatory part of the strain phasor, g0 exp(ivt). The real and imaginary components of the following equation identify the storage and loss moduli, respectively:

{E (v, T)}Complex

s (t) ¼ iv ¼ g0 exp(ivt) ¼v

1 ð

1 ð

ER (s) exp(ivs) ds s¼0

ER (s){sin(vs) þ i cos(vs)} ds

s¼0

¼ E 0 (v, T) þ iE 00 (v, T) 0

E (v, T) ¼ v

1 ð

ER (s) sin(vs) ds s¼0

00

E (v, T) ¼ v

1 ð

ER (s) cos(vs) ds s¼0

As stated above in general terms, the Fourier sine transform of the relaxation modulus (i.e., the step response) yields the dynamic storage modulus, whereas the Fourier cosine transform of ER(s) provides a route to calculate the dynamic loss modulus from time-dependent stress relaxation data. The general methodology discussed in this section is also useful to predict dynamic properties of viscoelastic media in which high-resolution video microscopy and particle-tracking software allow one to map the trajectories of nanoparticle motion in complex fluids. This is the field of microrheology, or nanorheology, where high-speed photography provides timeresolved data to calculate the mean-squared displacement of particles injected into viscoelastic fluids. Fourier transformation of the time-dependent mean-squared

10.41 Energy Dissipation and Storage During Forced Vibration

417

displacement of these particles yields the dynamic modulus of the medium, with accuracy that is comparable to bulk fluid rheological techniques.

10.41 ENERGY DISSIPATION AND STORAGE DURING FORCED VIBRATION DYNAMIC MECHANICAL EXPERIMENTS When viscoelastic materials are subjected to oscillatory deformation at frequency v, mechanical energy is converted irreversibly to thermal energy during each cycle, which persists for time t given by 2p/v. The second law of thermodynamics dictates the path by which this irreversible process occurs. In other words, complete conversion of thermal energy to mechanical energy is described as a “perpetual motion machine of the second kind,” and this process violates the second law. Quantitative evaluation of the mechanical energy per unit volume dissipated during each cycle is given by the scalar “double dot” product of the stress tensor and the differential strain tensor, integrated over a complete cycle of oscillation. For the dynamic mechanical experiments of interest in this discussion, one writes 2pð=v   ððð Energy dissipation @g ¼ s dg ¼ s dt @t v Volume t¼0

System Volume

This is completely analogous to calculating the work done on the system due to an external force F that causes a differential displacement d l as ð Work ¼ F d l



In dynamic mechanical spectroscopy, s and g are collinear and they both act in tension. Engineering stress is represented by a tensile force per initial unit cross-sectional area, and engineering strain is calculated from a displacement per initial sample length in the stretch direction. Hence, the connection between F and s, and d l and dg contains an additional factor that represents the initial sample volume which does not change much during the application of a harmonic tensile strain. Typically, dynamic testing is performed on very thin samples with negligible volume so that the amount of thermal energy generated is not too large. Let’s return to the results from dynamic mechanical testing, given by g (t; v) ¼ gdc þ g0 sin vt   @g ¼ vg0 cos vt @t v

s (t; v) ¼ g0 E (t; v) ¼ g0 {E (v, T) sin vt þ E 00 (v, T) cos vt} where the dynamic modulus is defined with respect to the magnitude of the oscillatory strain excitation, and calculate the mechanical energy per unit volume that is dissipated to thermal energy during each cycle of oscillation. This calculation also contains information about energy stored during the first and third quarters of the cycle

418

Chapter 10 Mechanical Properties of Viscoelastic Materials

when the sample is stretched and compressed, respectively, relative to the equilibrium dc level of strain gdc. There is no energy stored during the complete cycle because the sample returns to its equilibrium position twice during each cycle. The dissipation of mechanical energy to thermal energy is Energy dissipation ¼ vg 20 Volume

2pð=v

{E 0 (v, T) sin(vt) þ E 00 (v, T) cos(vt)} cos(vt) dt

t¼0

¼

vg 20 E 0 (v,

2pð=v

sin(vt) cos(vt) dt

T) t¼0

þ

vg 20 E 00 (v,

2pð=v

T)

cos2 (vt) dt

t¼0

As mentioned above, the term on the right side of the previous equation, which contains the storage modulus E0 , vanishes over the complete cycle. The term that contains the loss modulus E00 yields the following result: Energy dissipation ¼ vg 20 E 00 (v, T) Volume

2pð=v

cos2 (vt) dt

t¼0

¼ g 20 E 00 (v, T)

2ðp

(cos2 x) dx ¼ pg 20 E 00 (v, T)

x¼0

The previous equation reveals that energy dissipation in viscoelastic materials is related directly to the imaginary part of the complex dynamic modulus, E00 (v), or the absorption function for the dissipation process, which can be calculated from the Fourier cosine transform of the step response (i.e., stress relaxation modulus), via the final equation in Section 10.40 . Combining all of these concepts in the following expression for energy dissipation during dynamic mechanical testing, one obtains 8 1 9 ð < = Energy dissipation ¼ pg 20 Im iv ER (t) exp(ivt) dt : ; Volume t¼0

where Im selects the imaginary part of the complex dynamic modulus. This viscoelastic relation between energy dissipation and Fourier transformation of the stress relaxation modulus is consistent with the fluctuation –dissipation theorem in statistical physics. In other words, thermally induced micro-Brownian molecular motion, that is random in nature, produces fluctuations in the microscopic stress tensor that can be autocorrelated, to yield the stress relaxation modulus, and Fourier transformed to reveal how these stochastic perturbations are distributed among the natural frequencies of viscoelastic materials. Dissipation of mechanical energy into thermal energy occurs during stress relaxation when resonance conditions are satisfied (i.e., at maxima in E00

10.42 Free Vibration Dynamic Measurements via the Torsion Pendulum

419

where material response (i.e., natural) frequencies match the frequencies of external perturbations that fluctuate periodically with time). If viscoelastic materials are required to provide adequate vibration damping, then the previous two equations reveal that one should focus on designs that maximize the loss modulus at the temperature and frequency of operation. This can be achieved from a molecular engineering viewpoint by (i) correlating the effects of chemical structure on E00 and (ii) choosing polymers for damping applications that contain functional groups with the required moment of inertia to dissipate mechanical energy via resonance at temperature T and frequency v. Now, consider the first and third quarters of each cycle where the sample is either stretched or compressed from its “pseudo-equilibrium” position gdc, to gdc + g0. Even though the sample is compressed relative to gdc during the second half of the cycle, it remains under tension because gdc . g0. Since the mechanical energy stored during stretching or compression is the same if deformation remains within the elastic limit, focus on the work term that contains the storage modulus, consider the first quarter-cycle of harmonic strain, and multiply the result by a factor of 2 to account for the third quarter-cycle: Energy Storage (First and Third quarters) ¼ 2vg 20 E 0 (v, T) Volume

2pð =4v

sin(vt) cos(vt) dt t¼0

¼

2g 20 E 0 (v,

2pð=4

T)

1

2 sin

 2x dx ¼ g 20 E 0 (v, T)

x¼0

The ratio of (i) the energy dissipated during one complete cycle of oscillation to (ii) the energy stored during the first and third quarters is defined as the logarithmic decrement D. Hence, D¼

pg02 E 00 (v, T) ¼ p tan d g02 E 0 (v, T)

This relation between the logarithmic decrement and the loss tangent provides the necessary connection between energy dissipation in free- and forced-vibration dynamic measurements on viscoelastic materials. Forced-vibration experiments have been discussed above via the application of harmonic excitation g (t; v) ¼ gdc þ g0 sin vt, or g (t; v) ¼ g0 exp(ivt). Free-vibration experiments are described hypothetically below via the Voigt element with mass, and practically using an inverted torsion pendulum.

10.42 FREE VIBRATION DYNAMIC MEASUREMENTS VIA THE TORSION PENDULUM The viscoelastic models discussed above are “massless” and contain no acceleration, or inertial, term in the force balance that underlies the governing equation of motion. Consequently, the time dependence of either stress or strain in the Maxwell and Voigt

420

Chapter 10 Mechanical Properties of Viscoelastic Materials

models represents first-order response. First-order systems do not exhibit “overshoot” as the new steady state is approached in response to “step” disturbances. Obviously, realistic materials are not massless, which implies that an inertial force should be included with viscous and elastic forces to analyze the dynamic response of a system that is perturbed from its equilibrium state. The disturbance of interest in a torsion pendulum corresponds to the angle of twist of a sample about a vertical rotation axis. Then, the material returns to its equilibrium position via underdamped oscillatory motion in the absence of any external forces or torques. Gravity plays no role in this experiment because the rotation axis is vertical and oscillatory motion occurs in the horizontal plane. Since mass is added to the sample, usually in the form of flat disks, to reduce the frequency of oscillations, an “inverted” pendulum with additional system mass placed above the sample eliminates the effect of tensile creep. At time t ¼ 02, an external torque is applied to twist the sample such that its strain is g (0). At time t ¼ 0þ, the external torque is removed and the oscillatory strain response decays to zero due to the dissipative viscous component. A perfectly elastic isotropic material would oscillate, or “ring,” indefinitely in undamped fashion, similar to the harmonic oscillator in classical mechanics.

10.42.1

Analysis of the Voigt Element with Mass

The appropriate equation of motion is developed from a macroscopic force balance on a spring and dashpot in parallel that contains additional lumped mass m, which controls the oscillation frequency. Elastic and viscous forces oppose the displacement and velocity, respectively, of the material as it undergoes underdamped oscillatory motion about its equilibrium position. The most general form of the vector force balance on an incompressible, but deformable, viscoelastic solid with density r, viscosity h, and tensile modulus of elasticity E is written in terms of the displacement vector u, velocity vector v, deviatoric stress tensor s, strain tensor g, and the rate-of-strain tensor dg=dt, where all three second-rank tensors are symmetric. The control volume is differentially thick in all coordinate directions and g is the gravitational acceleration vector:

r

@2u ¼ r s þ rg @t 2 @g s ¼ E g  h @t 1 @g 1 @u ¼ {rv þ (rv)T}; v ¼ g ¼ {ru þ (ru)T}; @t 2 2 @t



As a first approximation, the one-dimensional quasi-macroscopic force balance, that should be compared with previous massless viscoelastic models in this chapter, is constructed in terms of the axial (i.e., z) component of the displacement vector in cylindrical coordinates, uz, when gravitational forces are neglected. Hence, the z-component of the previous vector force balance is required, with gz ¼ 0. Now, the control volume is differentially thick only in the z-direction, which coincides with the direction of

10.42 Free Vibration Dynamic Measurements via the Torsion Pendulum

421

oscillatory motion when the sample experiences an initial tensile displacement. Let szz represent the normal restoring force per unit area (i.e., normal stress) that opposes displacement and velocity of the material relative to its equilibrium position. A linear momentum balance (i.e., Newton’s law) on the differential control volume yields the following distributed-mass model for the time and spatial dependence of the zcomponent of the displacement vector, uz(t, z):

r

@ 2 uz @ szz ¼ 2 @t @z

szz ¼ E

  @uz @ @uz h @z @t @z

This reduces to the one-dimensional hyperbolic wave equation with speed of sound given by the square root of the ratio of elastic modulus E to density r in perfectly elastic isotropic solids when the dissipative component vanishes (i.e., h ¼ 0):

szz ¼ E

@uz @z

@ 2 uz E @ 2 uz ¼ @t 2 r @z2 The viscoelastic problem can be simplified by replacing this distributed-mass model for the time and spatial dependence of the displacement vector’s z-component in the rod-like sample, uz(t, z), by a macroscopic lumped-mass model for time-dependent normal strain gzz(t) ¼ @uz/@z  uz/L, using a Voigt element with an attached cylindrical disk of mass m. The actual mass of the viscoelastic material is neglected completely, relative to m. Let L and A represent the original sample length in the direction of oscillatory motion and the corresponding cross-sectional area, respectively. The macroscopic version of Newton’s law for this problem is   d2 uz uz d u z  A m 2 ¼ FRestoring ¼  E þ h dt L dt L   2 L d gzz dg m ¼ E gzz  h zz dt A dt2 The previous distributed-mass and lumped-mass models for tensile and compressive strain in the z-direction are reformulated to simulate twisting motion in the torsion pendulum via an angular momentum balance that includes elastic and viscous torques, generated within the sample of interest, which oppose oscillatory displacement and motion in the tangential direction. In this lumped-mass formulation, one calculates the angle of twist c(t) of the cylindrical disk with mass m, where, once again, the mass of the viscoelastic material is neglected completely relative to the mass of the disk. Both the sample and the disk have cylindrical symmetry, with radii RSample and RDisk, respectively, and LSample is the original length of the viscoelastic material. In each circular cross-sectional plane of the viscoelastic sample with area A ¼ pR 2Sample and local radius r (i.e., 0 r RSample) from the axis of rotation, the total differential restoring force dFRestoring due to elastic and viscous shear stresses,

422

Chapter 10 Mechanical Properties of Viscoelastic Materials

sElastic and sViscous, is additive for the Voigt model: dFRestoring ¼ {sElastic þ sViscous} dA rc LSample   dg d rc r dc  h ¼ 2h ¼ h dt dt LSample LSample dt

sElastic ¼ 2Gg ¼ G sViscous

where G is the shear modulus of elasticity and h is the shear viscosity of the viscoelastic sample. In the limit of small shear strain g, one divides the local arc length at the end of the sample where it is attached to the disk (i.e., rc) by twice the original sample length LSample, according to the definition of the symmetric strain tensor, above. Hence, shear strain g within the viscoelastic sample is related to the angle of twist of the attached disk. The corresponding differential restoring torque dTRestoring is constructed from the cross-product of the local lever arm r, measured from the rotation axis of the viscoelastic sample, and the differential restoring force dFRestoring, which is then integrated over the total cross-sectional shear area of the sample to evaluate the macroscopic restoring torque. This is illustrated below: dTRestoring ¼ r  d FRestoring     rc r dc 1 dc 2 dA ¼  r dA dTRestoring ¼ r G þh Gc þ h LSample dt LSample dt LSample   ð 1 dc J Gc þ h TRestoring ¼ dTRestoring ¼  dt LSample J¼

ðð

2

r dA ¼

A

2ðp RSample ð 0

1 r 2 r dr dQ ¼ pR4Sample 2

0

where J is the moment of inertia of the sample based on its cross-sectional area A, or the area moment of inertia. The angle of twist c satisfies the following angular momentum balance, which requires that the restoring torque within the viscoelastic material must match the product of the moment of inertia of the disk and its angular acceleration, where the latter is given by the second time derivative of c (t). One arrives at a second-order ordinary differential equation for c (t) that is extremely similar to the previous lumped-mass expression for normal strain gzz:   d2 c 1 dc J Gc þ h IDisk 2 ¼ TRestoring ¼  dt dt LSample ð ðð ð IDisk ¼ r2 d m ¼ rDisk r2 dV mDisk

¼ rDisk {2pLDisk}

VDisk Rð Disk

0

4 (r 2 )r dr ¼ 12 rDisk pRDisk LDisk ¼ 12 mR2Disk

10.42 Free Vibration Dynamic Measurements via the Torsion Pendulum

423

where IDisk is the moment of inertia of the disk. Hence, one solves the previous angular momentum balance by postulating an exponential solution for the angle of twist c (t):

c (t)  exp(zt) and calculating z from the characteristic equation:

IDisk z2 þ

hJ GJ zþ ¼0 LSample LSample sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   hJ 2 4IDisk GJ   + LSample LSample LSample z¼ 2IDisk pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J(4IDisk GLSample  h2 J) hJ ¼ +i 2IDisk LSample 2IDisk LSample 

hJ



Actual testing of viscoelastic solids reveals that these systems exhibit underdamped oscillatory response. This occurs when 4IDiskGLSample . Jh2, such that viscoelastic behavior is dominated by the elastic contribution and the roots of the characteristic equation are imaginary. Hence, ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) " t J(4IDisk GLSample  h2 J) hJ t A cos 2IDisk LSample 2IDisk LSample ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi)# t J(4IDisk GLSample  h2 J) þ B sin 2IDisk LSample

 c(t) ¼ exp 

Except for time t, the argument of the trigonometric functions in the preceding equation identifies the frequency (i.e., b) of these damped oscillations, which can be reduced by increasing the mass m, or moment of inertia IDisk, of the disk. This increase in the mass of the system also allows the oscillatory response to continue for longer times, where parameter a characterizes the rate of damping. Notice that completely elastic materials, for which h ¼ 0, exhibit undamped oscillatory response that continues indefinitely. Experimental determination of the oscillation frequency b and the damping parameter a allows one to infer the sample’s viscoelastic parameters, G and h, assuming that inertial effects within the sample are negligible relative to those of the disk. This assumption is valid when the oscillation frequency of the system (i.e., primarily the disk) is small relative to the frequency of shear-wave propagation in the sample, where the latter travels at the speed of sound. The following analysis of c (t) is employed to identify extreme displacements on the same side of the equilibrium position. These successive maxima are used to calculate the logarithmic decrement. Begin with the solution for

424

Chapter 10 Mechanical Properties of Viscoelastic Materials

c (t) and identify time t when the angular velocity of the disk vanishes: @c ¼ a exp(at){A cos(bt) þ B sin(bt)} @t þ b exp(at){A sin(bt) þ B cos(bt)} ¼ 0 (bB  aA) cos(bt) ¼ (bA þ aB) sin(bt) tan(bt) ¼ tan Q ¼

bB  aA bA þ aB

hJ ; b¼ a¼ 2IDisk LSample

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J(4IDisk GLSample  h2 J) 2IDisk LSample

Now, calculate time t when successive maxima occur on the same side of the equilibrium position:   bB  aA Q ¼ bt ¼ arctan bA þ aB 1 tn ¼ (Q þ np); n ¼ 0, 2, 4, . . . b The ratio of the magnitudes of the angle of twist at two successive maxima (i.e., tn and tnþ2) on the same side of the equilibrium position is evaluated using c (tn) and c (tnþ2) as follows:

 exp  ab (Q þ np) [A cos(Q þ np) þ B sin(Q þ np)] c (tn )

 ¼ c (tnþ2 ) exp  ab (Q þ [n þ 2]p) [A cos(Q þ [n þ 2]p) þ B sin(Q þ [n þ 2]p)] cos(Q þ np) ¼ cos Q cos(np)  sin Q sin(np) ¼ (1)n cos Q sin(Q þ np) ¼ sin Q cos(np) þ cos Q sin(np) ¼ (1)n sin Q   c (tn ) 2pa [(1)n A cos Q þ (1)n B sin Q] ¼ exp c (tnþ2 ) b [(1)nþ2 A cos Q þ (1)nþ2 B sin Q]   2pa = f (n) ¼ exp b These results suggest that the ratio of successive maxima for damped oscillatory response does not depend on which two maxima are considered. The logarithmic decrement, defined previously in this chapter as the ratio of (i) energy dissipated during one complete cycle of oscillation to (ii) energy stored during the first and third quarters, is calculated from free-vibration experiments as the natural logarithm of c (tn)/c (tnþ2). Hence,   c (tn ) 2pa 2p ¼ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D ¼ p tan d ¼ ln c (tnþ2 ) b 4IDisk GLSample 1 J h2

Appendix A: Linear Viscoelasticity

425

McCrum et al. [1967] suggest that random error can be reduced in actual measurements of the logarithmic decrement by averaging several (i.e., N ) ratios of the angle of twist at successive maxima on the same side of the equilibrium position. For even values of n, beginning with the first maximum that occurs on the same side of equilibrium after the initial displacement,     2N 1 X c (tn ) 1 c (t2 ) ¼ ln ln D¼ c (tnþ2 ) c (t2Nþ2 ) N n¼2,(even) N As expected, this analysis of the torsion pendulum reveals that (i) materials with no viscous component (i.e., h ¼ 0) do not dissipate mechanical energy irreversibly to thermal energy, (ii) stress and strain, or torque and angle of twist, are completely in-phase (i.e., d ¼ 0), and (iii) the oscillatory amplitude of the angle of twist in free-vibration experiments does not decrease from its initial displacement (i.e., D ¼ 0 and c (tn)  c (tnþ2)). The effect of additional system mass, in the form of thin disks, that reduces the oscillation frequency and allows damped response to continue for longer times, must be removed from the previous result for the logarithmic decrement so that one obtains intrinsic damping properties of viscoelastic materials under observation (see Problem 10.22 at the end of this chapter).

APPENDIX A: LINEAR VISCOELASTICITY A.1 Boltzmann Superposition Principle for Time-Dependent Strain via Continuous Stress History Consider a viscoelastic material with creep compliance given by JC(t) and focus on small “jump” stresses applied in sequence. The following principles apply when the response is linear at small strain; 1. Stress, which is a continuous function of time, can be viewed as an infinite sequence of infinitesimal “steps” Dsi, where each step stress persists for an infinitesimal observation time Dti. As each time interval approaches zero, the sequence of discrete steps simulates the continuous stress history s (t). 2. During each infinitesimal time interval Dti ¼ tiþ1 – ti, which begins at time ti, creep occurs in response to the infinitesimal step stress Dsi ¼ si – si21. In other words, “jump” stress Dsi is imposed upon a viscoelastic material at time ti and it survives for a duration given by Dti before another jump stress is superimposed on the previous one. The induced strain gi during this time interval is a linear function of the small step stress:

gi (t) ¼ JC (t  ti )Dsi The creep compliance JC is only a function of the experimental time scale t – ti when stress and strain are not exceedingly large. The previous equation is

426

Chapter 10 Mechanical Properties of Viscoelastic Materials

exactly the same as the ones in Sections 10.14 and 10.20 for the creep response of viscoelastic materials that can be described by the Maxwell and Voigt models, respectively. At time t0, the material is subjected to “jump” stress Ds0 ¼ s0, because s21 ¼ 0. 3. Total strain g (t) is calculated from a linear superposition of time-dependent strains described in Principle 2 above. Hence,

g (t) ¼

1 X

gi (t) ¼

i¼0

1 X i¼0

  Ds i Dti JC (t  ti ) Dti

4. As each time interval Dti approaches zero, Dsi/Dti is synonymous with the continuous stress history @s/@Q, and the infinite sum of strains is written in integral form:   ðt @s dQ g (t) ¼ JC (t  Q) @Q Q¼t0

The previous result corresponds to zero strain at t ¼ t0, when the material is subjected to the initial jump stress. Now, redefine the reference state such that g (t) approaches zero as t ) 21. This change of reference states modifies g (t) by an additive constant, and it is equivalent to replacing t0 in the lower integration limit by 21. Hence, the Boltzmann superposition integral predicts that the induced strain in viscoelastic materials can be calculated for any type of continuous stress history, provided that the creep compliance is known and total strain remains in the linear regime:   ðt @s dQ g (t) ¼ JC (t  Q) @Q Q)1

If strain becomes large enough and mechanical response cannot be described by the postulates of linear viscoelasticity, then the previous equation must be modified by including (i) dependence of the creep compliance JC on stress or strain and (ii) additional integrals (i.e., kernel functions) in a series expansion that contains higher order terms based on the continuous stress history.

A.2 Alternate Form of the Boltzmann Superposition Integral for Time-Dependent Strain g(t) It is slightly awkward to analyze the creep response of viscoelastic materials subjected to jump stress s0 at time t0 via the previous equation because (i) stress versus time is described by a step function, and (ii) the stress history required for the Boltzmann superposition integral corresponds to a delta function. However, integrands that contain delta functions are relatively straightforward to evaluate, as illustrated below in the

Appendix A: Linear Viscoelasticity

427

prediction of creep response: @s ¼ s0 d(Q  t0 ) @Q   ðt @s dQ g (t) ¼ JC (t  Q) @Q Q)1

¼ s0

ðt

JC (t  Q)d(Q  t0 ) dQ ¼ s0 JC (t  t0 )

Q)1

One evaluates the integrand by choosing integration variable Q such that the argument of the delta function vanishes, because d(0) ¼ 1 and delta functions vanish at all other positive and negative arguments. It should be obvious that the previous result for creep response g (t) conforms to the definition of creep compliance JC when the relevant time scale of observation is t – t0. Another route to analyze creep response begins with the Boltzmann superposition integral for time-dependent strain g (t) and performs integration by parts. For example, at constant time t,  u ¼ JC (t  Q); du ¼

@ JC (t  Q) @Q

 dv ¼

g (t) ¼

 @s dQ; v ¼ s (Q) @Q ðt

 dQ t



 @s dQ JC (t  Q) @Q

Q)1

¼ {s (Q)JC (t 

Q)}Q¼t Q)1





ðt

@ JC (t  Q) s (Q) @Q

Q)1

 dQ t

If the creep compliance is finite at infinite observation time and it is reasonable to assume that stress s approaches zero as Q ) 21, then the previous expression for g (t) reduces to

g (t) ¼ s (t)JC (0) 



ðt

s (Q) Q)1

 @ JC (t  Q) dQ @Q t

As an illustrative example, consider creep response of the one-time-constant Voigt model that contains elastic and viscous elements in parallel. At time t0, one imposes constant jump stress s0 on a viscoelastic material. The induced strain g (t)

428

Chapter 10 Mechanical Properties of Viscoelastic Materials

from the previous equation reduces to 2 3 2 3 Q¼t  ðt  ð @ 6 7 6 7 JC (t  Q) dQ5 ¼ s0 4JC (0)  g (t) ¼ s0 4JC (0)  {dJC (t  Q)}t 5 @Q t Q¼t0

Q¼t0

) s0 JC (t  t0 ) because the stress history vanishes for times t that are less than t0. Since there are no free springs that can respond instantaneously to jump stress s0 at time t0, the Voigt element exhibits no instantaneous strain. Hence, its creep compliance vanishes when the observation time for the relevant deformation is nonexistent (i.e., JC(0) ¼ 0). Furthermore, JC is finite at infinite observation time, as required to validate the previous equation, because there are no unrestricted dashpots. These claims are substantiated via inspection of JC(t – Q) for the Voigt model in Section 10.20. Knowledge of the creep compliance is employed in the Boltzmann superposition integral. One obtains the following results for the Voigt model:    1 (t  Q) 1  exp JC (t  Q) ¼ l(T) E    @ 1 (t  Q) exp JC (t  Q) ¼  l(T) @Q E l(T) t 2 3  ðt  @ 6 7 JC (t  Q) dQ5 g (t) ¼ s0 4JC (0)  @Q t Q¼t0

s0 ) E l(T)

ðt



   (t  Q) s0 (t  t0 ) dQ ¼ 1  exp exp l(T) E l(T)

Q¼t0

A.3 Laplace Transform Analysis of the Relation Between Creep Compliance JC and Stress Relaxation Modulus ER via Boltzmann Superposition Integrals The primary objective of this section is to develop relations between JC and ER in the Laplace domain and in the time domain with assistance from the convolution integral theorem. Begin with the general expression for time-dependent strain g (t) in Section A.2 of this chapter (i.e., using an alternate form of the Boltzmann superposition integral) and replace observation time t – Q for the creep compliance by modified integration variable s. Hence, s ¼ t – Q:

g (t) ¼ s (t)JC (0) þ

1 ð s¼0



 d JC (s) ds s (t  s) ds

Appendix A: Linear Viscoelasticity

429

Now, it is necessary to take the Laplace transform of the previous equation, such that independent variable t is mapped into Laplace variable p, as follows:

L{g (t)}t)p ¼ g (p) ¼ JC (0)L{s (t)}t)p 81 9 1  = ð <ð d JC (s) ds dt þ ept s (t  s) : ; ds t¼0

s¼0

Evaluation of the double integral in the previous equation is obtained by (i) reversing the order of integration and (ii) applying the shift theorem for the Laplace transform of functions that are translated along the time axis (i.e., s (t – s)). For example, 1 ð

t¼0

ept

81 <ð :

s¼0

9 81 9 1 = ð <ð = d d JC (s) ds dt ¼ JC (s) s (t  s) ept s (t  s) dt ds ; : ; ds ds 

s¼0

¼

1 ð

t¼0

d JC (s)[eps L{s (t)}t)p ] ds ds

s¼0

¼ s ( p)

ð1

eps



 d JC (s) ds ds

s¼0

Next, Laplace transform analysis of the derivative of the creep compliance, dJC/ds, is employed to simplify the complete expression for g ( p) in the Laplace domain. The final result is L{g (t)}t)p ¼ g (p) ¼ JC (0)L{s (t)}t)p þ s ( p)

1 ð

eps



 d JC (s) ds ds

s¼0

¼ JC (0)s ( p) þ s ( p)[ pL{JC (s)}s)p  JC (0)] ¼ pJC ( p)s ( p) Similar analysis of time-dependent stress s(t) is performed to obtain an expression for s ( p) in the Laplace domain. Begin with an alternate form of the Boltzmann superposition integral for s (t) in Section 10.36:

s (t) ¼ g (t)ER (0) þ

1 ð

  dER (s) ds g (t  s) ds

s¼0

The appropriate sequence of steps includes (i) taking the Laplace transform of the previous equation to map s (t) into s ( p), (ii) reversing the order of integration, by integrating first with respect to time t and then with respect to time delay s, (iii) applying the shift theorem to evaluate the Laplace transform of functions like g (t – s) that

430

Chapter 10 Mechanical Properties of Viscoelastic Materials

have been translated along the horizontal time axis such that g (t 2 s) ¼ 0 for times t , s, and (iv) using the derivative theorem to obtain the Laplace transform of the derivative of the stress relaxation modulus, dER/ds (i.e., L{dER/ds}s)p ¼ pL{ER(s)}s)p – ER(0)). As a consequence of performing these operations, one obtains the following result: 1   ð ps d ER (s) ds e L{s (t)}t)p ¼ s ( p) ¼ ER (0)L{g (t)}t)p þ g ( p) ds s¼0

¼ pER ( p)g ( p) The equations in this subsection reveal an important relation between JC and ER in the Laplace domain: 1 L{JC (t)}t)p L{ER (t)}t)p ¼ 2 ¼ {L(t)}t)p p The convolution theorem for Laplace transforms, summarized in Section 10.16 and developed in Section 10.22.3, provides the methodology to invert the product of JC(s) and ER(s) to obtain the following relation in the time domain:

L1 {ER ( p)JC (p)} ¼

Q¼t ð

ER (Q)JC (t  Q) dQ

Q¼0

¼

Q¼t ð

ER (t  Q)JC (Q) dQ ¼ t

Q¼0

A.4 Distribution of Viscoelastic Relaxation Times via the Continuous Analog of the Voigt– Kelvin Model Application of the principle of time – temperature superposition to creep or stress relaxation data (i.e., time – temperature shifting), as well as dynamic response via frequency – temperature shifting, reveals that several decades in time or frequency at the reference temperature are required to provide a snapshot of molecular dynamics in viscoelastic materials characterized by several relaxation times. Hence, the objectives of this section are to analyze the creep response of N Voigt elements in series and extend this model to the continuous limit, identifying the distribution of relaxation or retardation times JD(l ). The following rules apply when N Voigt elements are connected in series and the entire configuration is subjected to constant stress s0 at time t ¼ t0: Rule 1:

Each Voigt element experiences the same stress s0.

Appendix A: Linear Viscoelasticity

431

Rule 2: Linear response theory at small strain implies that the total strain response {g(t)}Total is obtained by summing the strain in each Voigt element, gi (t). Hence, {g (t)}Total ¼

N X

gi (t)

i¼1

Rule 3:

Within each Voigt element, strain gi (t) is the same for the spring with modulus Ei and the dashpot with viscosity hi in parallel, but the elastic and viscous stresses add to a constant (i.e., s0). The equation of motion for the ith Voigt element is {si}Viscous þ {si}Elastic ¼ hi

Rule 4:

@ gi þ Ei g i ¼ s 0 @t

Each Voigt element is characterized by a different relaxation time li, defined by the ratio of viscosity hi to the spring modulus Ei. These material properties are independent of strain in the linear regime.

The creep response of each individual Voigt element was described in Section 10.20, yielding the desired solution to the previous equation for gi (t). Now, one (i) constructs the total strain response {g(t)}Total and (ii) divides by s0 to obtain the creep compliance JC(t – t0) of the Voigt – Kelvin model. The desired result is    N N X 1 X 1 (t  t0 ) g (t) ¼ 1  exp  JC (t  t0 ) ¼ s0 i¼1 i li (T) E i¼1 i One simulates the continuous limit by letting the number N of Voigt elements, which is equivalent to the number of discrete relaxation times li, approach infinity. This is implemented when the discrete pairs, Ei and hi or 1/Ei and li (i.e., 1/Ei represents the static compliance Ji of the Hookean spring in the ith Voigt element), map out a continuous function JD(l ) known as the distribution of viscoelastic relaxation times under creep conditions. The following expression for creep compliance is useful in conjunction with the Boltzmann superposition integral to analyze energy storage and loss via the dynamic compliance in forced vibration experiments when real materials require a continuous spectrum of relaxation times:

JC (t  t0 ) ¼

 N X 1 i¼1

Ei

1      ð (t  t0 ) (t  t0 ) 1  exp  ) JD (l) 1  exp  dl N!1 li (T) l

l¼0

The time scale of the relevant deformation is t – t0, and relaxation time l exhibits temperature dependence. If viscoelastic materials are rheologically simple, then

432

Chapter 10 Mechanical Properties of Viscoelastic Materials

each subrelaxation process in the spectrum has the same temperature dependence, or activation energy. This is a requirement if the principle of time – temperature superposition is valid.

A.5 Dynamic Mechanical Spectroscopy of Viscoelastic Liquids via Forced-Vibration Analysis of Time-Dependent Strain and Dynamic Compliance J (t ; v) Analogous solid state experiments were described in Section 10.38 from the viewpoint of time-dependent stress and dynamic modulus. For viscoelastic liquids, samples are placed in rotational rheometers with parallel-disk or cone-and-plate geometry such that the gap spacing between the 1-inch diameter stationary plate and the rotating disk or cone is approximately 1 mm. Whereas thin-film solids require that harmonic excitation must be superposed on a dc level of strain to maintain tensile displacement at all times, shear stresses and the corresponding torques in rotational rheometers oscillate about their equilibrium values with positive and negative deviations. Hence, if the stress history at frequency v is given by a cosine wave, s (t; v) ¼ s0 cos(vt), where s0 is the amplitude of the forcing function, then the appropriate Boltzmann superposition integral for g (t; v) yields,

g (t; v) ¼

ðt Q)1

  @ s (Q; v) JC (t  Q) dQ ¼ vs0 @Q v

ðt

JC (t  Q) sin(vQ) dQ

Q)1

subject to the condition that strain g vanishes as t ) 21. Upon combining the previous two equations and reversing the order of integration, one predicts a strain wave that oscillates at the same frequency v as the applied stress via linear response theory: 8 9    = ðt < 1 ð (t  Q) g (t; v) ¼ vs0 JD (l) 1  exp  d l sin(vQ) dQ : ; l Q)1

¼ s0

1 ð

l¼0

þ s0

ð1 l¼0

l¼0

8 <

JD (l) v :

ðt Q)1

8 < JD (l)et=l v :

9 = sin(vQ) dQ dl ; ðt

Q)1

9 = eQ=l sin(vQ) dQ d l ;

When one integrates sin(vQ) with respect to Q from 21 to time t, it is acceptable to neglect the cosine function at the lower limit of integration because total strain g (t) vanishes when t ) 21. Evaluation of the second integral with respect to Q is obtained with assistance from the following formulas that were extracted from

Appendix A: Linear Viscoelasticity

433

integral tables: ð

eax sin(bx) dx ¼

1 eax {a sin(bx)  b cos(bx)} a2 þ b2

a ¼ l1 ; b ¼ v ðt

v

eQ=l sin(vQ) dQ ¼

  (vl)2 1 t=l e sin( v t)  cos( v t) vl 1 þ (vl)2

Q)1

Three integral expressions for g (t; v) that contain the distribution of viscoelastic relaxation times JD(l ) in the Boltzmann superposition integral are combined to yield the complete induced strain as follows: 9 81 = < ð 1 cos(vt) g (t; v) ¼ s0 JD (l) d l : 1 þ (vl)2 ; l¼0

þ s0

81 <ð :

l¼0

9 = vl sin(vt) JD (l) d l 1 þ (vl)2 ;

Once again, no harmonic overtones at higher frequency (i.e., 2v, 3v, etc.) are predicted by this linear response theory (i.e., the Boltzmann superposition principle) for induced strain. Dynamic compliance J (t; v) is defined via division of the strain wave by the amplitude of the forcing function s0. Hence, 9 81 = < ð g (t; v ) 1 cos(vt) ¼ JD (l) d l J (t; v) ¼ : s0 1 þ (vl)2 ; þ

81 < ð :

l¼0

l¼0

9 = vl J D (l ) dl sin(vt) 2 ; 1 þ (vl)

Since the harmonic excitation was modeled as a cosine wave to describe the stress history imposed on viscoelastic liquids, the coefficient of cos(vt) in the expression for J (t; v) represents the frequency-dependent storage compliance J0 (v), which selects the component of the strain wave that is in-phase with the harmonic forcing function s (t; v). J0 (v) is characteristic of the elastic fraction of viscoelastic materials, and it provides a measure of the ease of deformation. Hence, the storage compliance is essentially nonexistent at low temperatures or high frequencies, characteristic of high-Deborah-number solids. The coefficient of sin(vt) in the previous expression for J (t; v), which selects the out-of-phase component of the induced strain wave with respect to the harmonic forcing function, represents the frequency-dependent loss compliance J 00 (v) that exhibits maxima when resonance occurs. Viscoelastic relaxation at the glass transition satisfies the criterion for resonance, where the experimental time scale during one cycle of oscillation, given by 2p/v, is comparable to

434

Chapter 10 Mechanical Properties of Viscoelastic Materials

the response time of the material. Hence, there is a maximum in J 00 versus temperature or frequency at the glass transition, with a corresponding sigmoidal-shaped increase in the storage compliance J0 on the high-temperature or low-frequency side of this transition. For realistic viscoelastic materials that exhibit a distribution of relaxation times, described by JD(l ), quantitative expressions for the storage and loss compliances are given by 1 ð 1 0 JD (l) dl Storage: J (v) ¼ 1 þ (vl)2 l¼0

00

Loss: J (v) ¼

1 ð

JD (l) l¼0

vl dl 1 þ (vl)2

These integral transformations between the distribution function JD(l ) and dynamic properties are pursued in significant detail in Appendix C from the viewpoint of stress relaxation moduli. Problem 10.18d in this chapter employs complex variable theory to relate storage and loss compliances to storage and loss moduli. It is a rather difficult task to measure dynamic compliance in a rotational rheometer (i.e., J ¼ J0 – iJ 00 ), separate the real J0 and imaginary J 00 components, and extract the distribution of relaxation times JD from these experimental material properties. The methodology of inverse problems is designed to address this aspect of dynamic mechanical spectroscopy. On the other hand, if a reasonable model for the distribution function can be postulated, then analytical or numerical integration of the previous two equations yields predictions of material properties that can be compared with experimental data from rotational rheometers. For example, if mechanical response of viscoelastic materials is described adequately by one relaxation time t, then a delta-function distribution is appropriate for JD(l ) ¼ Jd(l 2 t), where J represents the static compliance of a spring configured in parallel with a dashpot (i.e., the Voigt model). The corresponding storage and loss compliances are obtained by straightforward integration of the previous two equations: 0

Storage: J (v) ¼ J

1 ð

d(l  t)

1 1 dl ¼ J 2 1 þ (vl) 1 þ (vt)2

d(l  t)

vl vt dl ¼ J 2 1 þ (vl) 1 þ (vt)2

l¼0 00

Loss: J (v) ¼ J

1 ð

l¼0

This simple one-relaxation-time model exhibits trends that are observed experimentally. As mentioned above, the storage compliance decreases at higher frequency or lower temperature, characteristic of glassy materials. The low-frequency limit of J0 corresponds to the static compliance of the spring and the high-frequency limit of J0 suggests that glasses exhibit infinite resistance to deformation, which is somewhat exaggerated. Resonance occurs at vt ¼ 1, where J 00 exhibits a maximum due to viscoelastic relaxation, most likely coincident with the glass transition process. When the temperature or frequency dependence of storage and loss compliance is analyzed

Appendix B: Finite Strain Concepts for Elastic Materials

435

together, experimental data for multiple relaxation processes reveal that more intense resonance peaks in J 00 occur in harmony with larger increments or decrements in J0 when viscoelastic relaxation dissipates more mechanical energy irreversibly to heat. The connection between molecular structure and viscoelastic relaxation is discussed further in Sections 15.7 and 15.8.

APPENDIX B: FINITE STRAIN CONCEPTS FOR ELASTIC MATERIALS Elastic materials are introduced from an “engineering mechanics” viewpoint. Stress and strain ellipsoids are defined and evaluated for simple flow fields.

B.1 Ellipsoids Three-dimensional objects with elliptical shape provide a pictorial and mathematical description of the states of stress and strain in elastic materials. Analogous to the major and minor axes of an ellipse in two dimensions, there are three principal axes of an ellipsoid, and the principal components, or lengths of these principal axes, provide information about material deformation and the forces, or stresses, required to achieve this state of deformation.

B.2 Overview The primary objective of the information in this appendix is to provide a framework for the analysis of elastic solids. For homogeneous materials, it is sufficient to analyze these materials via an arbitrarily small volume element that can be placed anywhere within the system. If materials are isotropic, then their physical properties do not exhibit directionality. In other words, physical properties are the same in any coordinate direction and, for isothermal systems, there is no reason to believe that any physical properties should change differently in one direction relative to any other direction. Deformations are classified as homogeneous if planes with the material remain plane. Homogeneous deformations, or homogeneous flows, are useful because all rheological properties of “simple” fluids can be obtained from experiments in which the deformation is homogeneous. In contrast, complex fluids require experiments that subject materials to inhomogeneous deformations before all rheological properties can be measured.

B.3 Deformation and the Strain Ellipsoid Deformation is a “two-state” process. In other words, for any two states defined by t0 and t1, homogeneous deformation in three dimensions can be described by the strain ellipsoid. The principal axes, or directions, of strain are oriented along the major axes of the ellipsoid and the principal values of strain (i.e., elongation ratios—l1, l2, l3) correspond to one-half of the lengths of these principal axes. Hence, elongation

436

Chapter 10 Mechanical Properties of Viscoelastic Materials

ratios are defined with respect to the principal axes of the strain ellipsoid. Each li is constructed from the ratio of one-half of the length of the principal axis in state t1 relative to one-half of the length of the same principal axis in reference state t0, where the latter is unity by convention because the strain ellipsoid is a unit sphere in state t0. By definition, each elongation ratio li is greater than zero. The strain s associated with any state of deformation is a two-state quantity that is determined completely by three elongation ratios along principal axes of the strain ellipsoid and three angles within the material that describe the orientation of each principal axis in state t1 relative to state t0. All three principal axes of strain are mutually orthogonal in each state (i.e., t0 and t1). Consider the following deformational processes from state t0 to state t1: Rigid Translation. All three principal values of strain are unity (i.e., li ¼ 1, i ¼ 1, 2, 3) and each orientation angle is zero. A small strain is defined as one in which each elongation ratio is “close to unity.” Isotropic Dilation or Contraction. Each principal value of strain is the same, given by the cube root of the ratio of sample volume in state t1 to sample volume in state t0. Each orientation angle is zero. Uniaxial Tensile Deformation. Materials are stretched in “direction #1” and isotropic lateral contraction occurs in both directions transverse to stretching such that sample volume remains constant. Hence, l1 .1, l2 ¼ l3 ,1, and the product of all three principal values of strain is unity. Each orientation angle is zero. Strain Ellipsoid for Simple Two-Dimensional Shear There is a family of material planes, perpendicular to “direction #2” that slide in “direction #1” at constant separation without stretching. If 1 is the “angle of shear,” then the magnitude of shear s ¼ tan 1. Invoke the stipulation that simple shear occurs at constant volume (i.e., the product of all three elongation ratios is unity) and that two-dimensional shear implies no change in the principal value of strain along “direction #3” (i.e., l3 ¼ 1). The orientation angle in direction #3 is zero, whereas the orientation angles of the other two principal axes in state t1 have rotated by angle x with respect to reference state t0. Hence, the unit sphere in the undeformed reference state t0 is transformed into an ellipsoid in state t1 after shear s. Mathematical analysis of the strain ellipsoid is necessary to identify the principal axes and principal values of strain after shear deformation via maximization/minimization subject to a constraint. The method of Lagrange multipliers yields the following results:

l1 ¼ cot x l2 ¼ tan x l3 ¼ 1 l1 l2 l3 ¼ 1 s ¼ tan 1 ¼ 2 cot 2x For very small shear strains, the limiting value of x is 458 as the magnitude of the shear s approaches zero. As stated above, two-dimensional shear in the x – y plane

Appendix B: Finite Strain Concepts for Elastic Materials

437

(i.e., directions #1 and #2) implies that sample dimensions in direction #3 remain unchanged in states t0 and t1. Hence, z(t0) ¼ z(t1). The fact that shear planes “slide” at constant separation implies that y(t0) ¼ y(t1) and x(t1) ¼ x(t0) þ sy(t0), where the magnitude of shear s is given by the tangent of shear angle 1. Since the strain ellipsoid in reference state t0 is a unit sphere, one writes {x(t0 )}2 þ {y(t0 )}2 þ {z(t0 )}2 ¼ 1 There are no xy, xz, or yz cross terms in reference state t0 because the x-, y-, and z-axes in this reference state represent principal axes of strain, and all three principal values of strain are unity. When the previous unit sphere expression in state t0 is written in terms of sample dimensions in state t1, {x(t1 )  sy(t1 )}2 þ {y(t1 )}2 þ {z(t1 )}2 ¼ 1 {x(t1 )}2  2sx(t1 )y(t1 ) þ (1 þ s2 ){y(t1 )}2 þ {z(t1 )}2 ¼ 1 the presence of “xy cross-terms” supports the fact that the original x- and y-axes in reference state t0 are no longer principal axes of strain after simple shear deformation, but the absence of xz and yz cross-terms indicates that the z-axis is a principal axis of strain in states t0 and t1. The following analysis identifies the principal axes of the strain ellipsoid in the x– y plane at constant z. Any shear plane perpendicular to the z-axis can be employed because there is no deformation in the z-direction. Hence, for simplicity, one sets z(t1) ¼ 0 and searches for extreme values of x 2 þ y 2 after the material has been subjected to shear s in deformed state t1. If the origin of the xyz-coordinate system is coincident with the center of the strain ellipsoids in states t0 and t1, then the square of the length of position vector r from the origin to any point on the ellipsoid is given by x 2 þ y 2 þ z 2. The major axis of the strain ellipsoid in the xy-plane where z ¼ 0 is coincident with a maximum in x 2 þ y 2, whereas the minor axis of strain corresponds to a minimum in x 2 þ y 2, subject to the constraint that x and y in state t1 must satisfy the following equation of an ellipse: {x(t1 )}2  2sx(t1 )y(t1 ) þ (1 þ s2 ){y(t1 )}2 ¼ 1 The method of Lagrange multipliers is employed to identify the principal axes of strain in state t1. Construct the function C(x, y, j ), where j is the Lagrange multiplier: C(x, y, j ) ¼ x2 þ y2 þ j{x2  2sxy þ (1 þ s2 )y2  1} Extreme values of C must satisfy these three equations:   @C ¼ 2x þ j{2x  2sy} ¼ 0 @x y,j   @C ¼ 2y þ j{2(1 þ s2 )y  2sx} ¼ 0 @y x,j   @C ¼ x2  2sxy þ (1 þ s2 )y2  1 ¼ 0 @ j x,y

438

Chapter 10 Mechanical Properties of Viscoelastic Materials

Elimination of the Lagrange multiplier j from the first two equations in the previous set of three equations yields relations between x and y at extreme points on the strain ellipsoid in state t1, where the ratio of y to x is identified as tan x, and x is the orientation angle of each principal axis of strain in state t1 with respect to the corresponding principal axis of strain in state t0. Hence, values of x and y that maximize or minimize C must satisfy the following relation: x  sy (1 þ s2 )y  sx ¼ x y x y s ¼  ¼ cot x  tan x ¼ 2 cot(2x) y x

B.4 Stress Ellipsoid At any point within an isotropic solid, the stress in state t1 is described by a symmetric second-rank tensor whose scalar components in a 3  3 matrix are referenced to the “stress-free” state t0. If stress is homogeneous and uniform, then each scalar component is independent of position. For any given element of surface dS in a material with outward directed unit normal vector n, the stress tensor identifies a force per unit area, or traction vector f, that acts across this surface, but f is not necessarily parallel to n. However, at any point in a material, it is always possible to rotate the coordinate axes and locate three mutually perpendicular coordinate directions with unit normal vectors n1, n 2, and n 3, such that the traction vector f i is parallel to n i, where n i defines the orientation of a differential surface element across which f i acts. In other words, f 1 ¼ n1 p1 f 2 ¼ n2 p2 f 3 ¼ n3 p3 where n i is a unit vector oriented along the ith principal axis of the stress ellipsoid and pi is the corresponding principal value of stress. This is analogous to placing the nine scalars that describe the complete state of a symmetric second-rank tensor, like the stress tensor, in ordered fashion into a 3  3 matrix and finding the eigenvalues and eigenvectors that diagonalize the matrix via an orthogonal transformation which preserves vector lengths before and after the coordinate axes undergo rotation. Since the 3  3 matrix is symmetric, all three eigenvalues are real and the eigenvectors that correspond to nondegenerate eigenvalues are orthogonal. The eigenvalues are given by pi, the three orthogonal eigenvectors are n i, all forces are classified as normal forces, and there are no nonzero off-diagonal elements of the 3  3 matrix. Hence, six quantities are required to describe the complete state of stress at any point in a material; three principal axes and the corresponding three principal values of stress. Since each pi is positive for tensile stress and negative in compression, the magnitude of pi is identified with one-half of the length of the principal axis oriented in the direction of n i.

Appendix B: Finite Strain Concepts for Elastic Materials

439

If the stress tensor that includes both pressure and viscous forces per unit area is given by p¼

3 X

di dj pij

i, j¼1

where {pij} represents a set of nine scalars that describe the complete state of stress, then the previous results for traction vector f n acting across a differential element of surface with unit normal vector n can be expressed as follows:



fn ¼ n p ¼ n



3 X

di dj pij

i, j¼1

The “dot” operation in the previous expression contracts unit normal vector n with d i, where the latter is the unit vector on the left side of the unit dyad d id j in summation notation for the stress tensor that is closest to the “dot.” If unit normal vector n is oriented along one of the principal axes of the stress ellipsoid and the principal value of stress in that direction is given by p, then the previous expression for f n yields np because f n is parallel to n. Hence, 3 X

fn ¼

i, j¼1



(n di )dj pij ¼ np

Now, take the scalar “dot” product of the previous equation with an arbitrarily chosen unit vector in the kth-coordinate direction (i.e., d k). One obtains



dk f n ¼

3 X



i, j¼1

¼

3 X



(n di )(dk dj )pij ¼

3 X





(n di )dkj pij

i, j¼1



(n di )pik ¼ (dk n)p

i¼1

The scalar dot product of d k with d j yields the Kronecker delta dkj, which is only nonzero and equal to unity when d j equals d k. Hence, it is acceptable to let j ¼ k in the summation and remove the sum over index j. For a given state of stress that is described by {pik}, the three principal values of the stress ellipsoid are calculated from the previous eigenvalue equation. This is illustrated explicitly by the following set of three linear algebraic equations that are written for d k ¼ d 1, d k ¼ d 2, and d k ¼ d 3, respectively:

  ¼ d ; (n  d )p

  þ (n  d )p

  þ (n  d )p

  ¼ (d  n)p

dk ¼ d1 ; (n d1 )p11 þ (n d2 )p21 þ (n d3 )p31 ¼ (d1 n)p dk ¼ d2 ; (n d1 )p12 þ (n d2 )p22 þ (n d3 )p32 ¼ (d2 n)p dk

3

1

13

2

23

3

33

3

440

Chapter 10 Mechanical Properties of Viscoelastic Materials

Upon rearrangement, the system of equations can be written as follows:

  (n  d )p



 

(n d1 ){ p11  p} þ (n d2 )p21 þ (n d3 )p31 ¼ 0 (n d1 )p12 þ (n d2 ){ p22  p} þ (n d3 )p32 ¼ 0 1

13

 



þ (n d2 )p23 þ (n d3 ){p33  p} ¼ 0

Since unit normal vector n is oriented along a principal axis of stress, each scalar “dot” product (n d k) represents the projection of a principal axis of the stress ellipsoid in the kth coordinate direction. If some of these scalar products are not zero (i.e., k ¼ 1, 2, 3), then the following necessary and sufficient condition for nontrivial solutions to the previous system of linear algebraic equations yields a cubic equation for p, which has three real roots because pij ¼ pji. If (n d 1), (n d 2), and (n d 3) represent three unknowns in the system of linear equations, then the necessary and sufficient condition such that not all (n d k) ¼ 0 requires that the determinant of the following coefficient matrix,











2

p11  p 4 p12 p13

p21 p22  p p23

3 p31 5 p32 p33  p

must vanish. Three principal values of stress are calculated from   Det pij  pdij  ¼ 0

B.5 Perfectly Elastic Isotropic Solids At any given temperature, solid materials possess a unique shape in the reference “stress-free” state identified by state t0. Perfectly elastic solids subjected to stress in state t1 attain this unique stress instantaneously, as determined only by states t0 and t1. This picture provides a useful idealization for lightly crosslinked elastomers above their glass transition temperature. Under isothermal conditions and in the limit of small strain, the mechanical properties of perfectly elastic isotropic solids are described completely by two constants (i.e., Young’s modulus of elasticity and Poisson’s ratio), and one scalar function of three independent scalar variables for compressible fluids. This scalar function contains only two independent variables when the fluid is incompressible because the product of all three principal values of strain (i.e., elongation ratios) must be unity when isothermal deformation does not induce any change in volume. The isotropic nature of these materials suggests that the stress and strain ellipsoids should have the same symmetry properties and that the principal axes of both ellipsoids should coincide. Furthermore, each principal value of stress can be constructed in terms of a symmetric function of all three principal values of strain.

Appendix B: Finite Strain Concepts for Elastic Materials

441

B.6 Principal Values of Stress When a Perfectly Elastic Isotropic Solid is Subjected to Simple Two-Dimensional Shear This problem was introduced in Section B.3 of Appendix B in this chapter, from the viewpoint of the strain ellipsoid. The important shear stress in the xy-plane is p12 ¼ p21 ¼ s. There are no shear forces in the z-direction, identified as direction #3. Hence, p13 ¼ p23 ¼ 0. Due to the symmetry of the stress tensor, p31 ¼ p32 ¼ 0, also. Furthermore, d z ¼ d 3 is a principal axis of stress as illustrated below. Principal values of stress are calculated from the determinant of the coefficient matrix: 2

p11  p 4 s 0

3 s 0 p22  p 0 5 0 p33  p

which must vanish. One solves the following cubic equation for three real roots, p1, p2, and p3: ( p33  p)[( p11  p)( p22  p)  s 2 ] ¼ 0 p3 ¼ p33 p2  ( p11 þ p22 )p þ p11 p22  s 2 ¼ 0 p ¼ 12 ( p11 þ p22 ) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + 12 ( p11 þ p22 )2  4( p11 p22  s 2 ) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 12 ( p11 þ p22 ) + 12 ( p11  p22 )2 þ 4s 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p1 ¼ 12 ( p11 þ p22 ) þ 12 N12 þ 4s 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 ¼ 12 ( p11 þ p22 )  12 N12 þ 4s 2 where N1 is the first normal stress difference, defined by N1 ¼ p11 – p22.

B.7 Evaluation of the Principal Axes of the Stress Ellipsoid for Simple Two-Dimensional Shear The origin of an xyz-coordinate system is placed at the center of the stress ellipsoid in reference state t0 and final state t1 after a perfectly elastic isotropic solid has experienced shear stress s ¼ p12 ¼ pxy ¼ pyx in the xy-plane. The square of the length of position vector r from the origin to any point on the ellipsoid is given by x 2 þ y 2 þ z 2. The major axis of stress in the xy-plane where z ¼ 0 is coincident with a maximum in x 2 þ y 2, whereas the minor axis of stress corresponds to a minimum in x 2 þ y 2, subject to the constraint that x and y in state t1 must satisfy the following equation

442

Chapter 10 Mechanical Properties of Viscoelastic Materials

of an ellipse, known as the “stress quadric”: 2 X

pij xi xj ¼ pxx x2 þ 2s xy þ pyy y2 ¼ 1

i, j¼1

The x- and y-axes coincide with the principal directions of stress in reference state t0, but these principal axes have rotated by angle b in state t1 relative to the “stress-free” state t0. The z-axis is a principal direction of stress in both states because simple shear occurs in the xy-plane. Lagrange multiplier optimization is applied to the function F(x, y, z ), where z is the Lagrange multiplier: F(x, y, z ) ¼ x2 þ y2 þ z{pxx x 2 þ 2s xy þ pyy y 2  1} Optimization of F, subject to the elliptic constraint, is performed below to identify extreme values of x and y on the stress ellipsoid and principal axes of stress in the xy-plane in state t1 such that the ratio of y to x on these axes is defined by tan b. The following equations must be satisfied simultaneously:   @F ¼ 2x þ z{2pxx x þ 2s y} ¼ 0 @x y,z   @F ¼ 2y þ z{2pyy y þ 2s x} ¼ 0 @y x,z   @F ¼ pxx x2 þ 2s xy þ pyy y2  1 ¼ 0 @ z x,y Elimination of the Lagrange multiplier z from the first two equations in this set yields the desired relation between x and y along the principal directions of stress: pxx x þ s y pyy y þ s x ¼ x y pxx þ s tan b ¼ pyy þ s cot b pxx  pyy ¼ cot b  tan b ¼ 2 cot(2b) s The first normal stress difference N1 is defined by N1 ¼ pxx – pyy. As a consequence of the isotropic nature of perfectly elastic solids that experience simple two-dimensional shear in the xy-plane, the stress and strain ellipsoids must exhibit the same symmetry properties. This implies that the principal axes of both ellipsoids coincide, and b ¼ x. Hence, Lagrange multiplier analysis of both ellipsoids yields, pxx  pyy N1 ¼ ¼s s s In the limit of infinitesimally small strain s, both ellipsoids rotate by 458 relative to the stress-free reference state t0. Furthermore, shear stress s scales linearly with strain for

Appendix C: Distribution of Linear Viscoelastic Relaxation Times

443

elastic solids, suggesting that the first normal stress difference N1 scales as s 2. The difference between the two principal values of stress in the xy-plane in state t1, when a perfectly elastic isotropic solid experiences two-dimensional simple shear, is p1  p2 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi N12 þ 4s2 ¼ s s2 þ 4

APPENDIX C: DISTRIBUTION OF LINEAR VISCOELASTIC RELAXATION TIMES Advanced mathematical techniques are introduced to analyze relaxation time distributions for linear viscoelastic response. Fourier transforms and double-Laplace transforms are employed to evaluate frequency response functions from time-dependent stress relaxation data.

C.1 Integral Transforms These transformations map a function of one variable into a related function of another variable. Laplace and Fourier transforms are the ones typically employed in the physical sciences. When an infinite spectrum of relaxation times is required to describe material response, the distribution function and the stress relaxation modulus are related by the integral transform given in Eq. (1) below. Furthermore, the stress relaxation modulus and the complex dynamic modulus are related by Fourier transformation. Both of these integral transforms are combined to reveal that the distribution of viscoelastic relaxation times and the complex dynamic modulus are related by double Laplace transformation. Experimental techniques are available to measure stress relaxation moduli and complex dynamic moduli for viscoelastic solids. Hence, numerical predictions of the distribution of viscoelastic relaxation times can be obtained by measuring mechanical properties and inverting the appropriate integral transform.

C.2 Overview All linear viscoelastic properties of polymeric materials can be predicted if the relaxation time distribution function ED(l ) is known. The objective of the calculations in this appendix is to investigate several methods that might allow one to calculate ED(l ). Illustrative examples are provided when the stress relaxation modulus follows exponential decay with one or two time constants because, in these simple cases, ED(l ) is described by either one or two delta functions. Let’s begin with the stress relaxation modulus ER(t) for the Maxwell – Wiechert model, as discussed in Section 10.31, when the number N of Maxwell elements in parallel approaches the continuous limit (i.e., N ) 1): ER (t  t0 ) ¼

1 ð

l¼0

  (t  t0 ) ED (l) exp  dl l

(1)

444

Chapter 10 Mechanical Properties of Viscoelastic Materials

If the jump strain imposed on the sample were large enough and if the linear hypothesis were not justified, then relaxation time l and the distribution function ED could depend on strain in the nonlinear regime. In the linear regime, the previous equation suggests that the relaxation modulus ER(t) and the distribution function ED(l ) are related by an integral transformation. It is relatively straightforward to calculate the relaxation modulus if ED(l ) is known, because the previous integral can be evaluated analytically or numerically. Unfortunately, there are no experiments that provide direct measurement of the distribution of relaxation times. However, ER(t) is measured directly via stress relaxation experiments, and, in principle, one can invert the previous integral transform to obtain the distribution function.

C.3 Inverse Moments of the Distribution of Viscoelastic Relaxation Times Divide the previous equation by the initial value of the stress relaxation modulus, ER(0), and expand the exponential function in a Taylor series about the point t ¼ t0. One obtains the following result: 1   ð ER (t  t0 ) E D (l ) (t  t0 ) 1 (t  t0 )2 1 (t  t0 )3 ¼ 1 þ  þ    dl l ER (0) ER (0) 2! l2 3! l3 l¼0

It should be emphasized that ED(l )/ER(0) is a normalized distribution function j (l ), which is obvious if one evaluates the previous expression at t ¼ t0: E D (l ) ¼ j (l) ER (0) {ER (t  t0 )}t¼t0 ¼1¼ ER (0)

1 ð

j (l ) d l l¼0

Definition of the nth-inverse moment of the normalized distribution of viscoelastic relaxation times, 1   ð 1 1 ¼ j (l ) d l n l ln l¼0

allows one to rewrite the normalized stress relaxation modulus as a polynomial with independent variable t 2 t0, and coefficients given by the inverse moments of j (l ):       ER (t  t0 ) 1 1 1 1 1 2 (t  t0 ) þ (t  t (t  t0 )3 þ    ¼1 )  0 l ER (0) 2! l2 3! l3   1 X (1)n 1 (t  t0 )n ¼ n! ln n¼0

Appendix C: Distribution of Linear Viscoelastic Relaxation Times

445

Employ the following strategy to calculate the inverse moments of j (l ) from the coefficients of the polynomial expansion for the stress relaxation modulus; Step 1: Obtain stress relaxation data in the linear regime, divide time-dependent stress by the magnitude of the initial jump strain to generate ER(t 2 t0), and normalize the stress relaxation modulus, as prescribed by the quantity on the left side of the previous equation. Step 2: Truncate the polynomial expansion after n terms. There are (n 2 1) coefficients that contain h1/lni, n ¼ 1, 2, 3, . . . , which can be evaluated by solving a system of (n 2 1) linear algebraic equations. Step 3: The required (n 2 1) linear algebraic equations might be obtained by evaluating the normalized stress relaxation modulus at (n 2 1) values of time t that are different from t0. For example, let’s include four terms (i.e., n ¼ 0, 1, 2, 3) in the polynomial expansion prior to truncation and explicitly evaluate ER at times t1, t2, and t3. One obtains a set of three coupled linear algebraic equations that must be solved for the first three inverse moments of the distribution of relaxation times, h1/lni, n ¼ 1, 2, 3:       ER (t1  t0 ) 1 1 1 1 1 2 (t1  t0 ) þ (t (t1  t0 )3 ¼1  t )  1 0 l ER (0) 2! l2 3! l3       ER (t2  t0 ) 1 1 1 1 1 2 (t2  t0 ) þ (t (t2  t0 )3 ¼1  t )  2 0 l ER (0) 2! l2 3! l3       ER (t3  t0 ) 1 1 1 1 1 2 (t3  t0 ) þ (t3  t0 )  (t3  t0 )3 ¼1 l ER (0) 2! l2 3! l3 The methodology described above calculates (n 2 1) inverse moments by sampling the stress relaxation modulus at (n 2 1) selected values of time t. The preferred strategy outlined below calculates these coefficients (i.e., h1/lni) such that they provide a more accurate reflection of stress relaxation by sampling the relaxation modulus continuously over the time interval of interest. This procedure is implemented by integrating the normalized stress relaxation modulus from t ¼ t0 to t ¼ tk, where k ¼ 1, 2, 3 to generate the appropriate number of linearly independent coupled algebraic equations for the coefficients that yield the inverse moments of the distribution. Hence, ðtk t¼t0

  ðtk   1 1 X X ER (t  t0 ) (1)n 1 (1)n 1 n (tk  t0 )nþ1 dt ¼ (t  t0 ) dt ¼ n! ln (n þ 1)! ln ER (0) n¼0 n¼0 t¼t0

If the summation in the previous equation is truncated after four terms (i.e., n ¼ 0, 1, 2, 3), then three equations must be generated and solved simultaneously to estimate the first three inverse moments. It is necessary to integrate the relaxation modulus over three different time intervals but, unlike the example below, the lower integration

446

Chapter 10 Mechanical Properties of Viscoelastic Materials

limit is not necessarily restricted to be the same in each case: ðt1 t¼t0

ðt2 t¼t0

ðt3 t¼t0

      ER (t  t0 ) 1 1 1 1 1 1 2 3 dt ¼ (t1  t0 )  (t1  t0 ) þ (t1  t0 )  (t1  t0 )4 ER (0) 2! l 3! l2 4! l3       ER (t  t0 ) 1 1 1 1 1 1 2 3 (t2  t0 ) þ (t2  t0 )  (t2  t0 )4 dt ¼ (t2  t0 )  ER (0) 2! l 3! l2 4! l3       ER (t  t0 ) 1 1 1 1 1 1 2 3 (t3  t0 ) þ (t3  t0 )  (t3  t0 )4 dt ¼ (t3  t0 )  ER (0) 2! l 3! l2 4! l3

Numerical Results When the Relaxation Modulus ER Follows a Single Exponential Decay The previous example is investigated in more detail by (i) analytically integrating the stress relaxation modulus if it conforms to a single exponential decay with time constant t ¼ 2 minutes, (ii) testing the sensitivity of the preferred algorithm to the number of terms that are included in the summation prior to truncation, and (iii) comparing the linear algebraic solution for the inverse moments with known values of h1/lni when the normalized distribution j (l ) is a delta function, spiked at l ¼ t. The relevant functions for this specific example are

j ( l ) ¼ d ( l  t) ER (t  t0 ) ¼ ER (0)

1 ð



   (t  t0 ) (t  t0 ) j (l) exp  d l ¼ exp  l t

l¼0

1 1   ð ð 1 1 1 1 j (l) dl ¼ d(l  t) dl ¼ n ¼ ln ln ln t

l¼0

l¼0

Analytical integration of the relaxation modulus from t0 to tk, k ¼ 1, 2, 3, allows one to evaluate the left side of the coupled linear algebraic equations that must be solved for the inverse moments. Once again, if truncation occurs after four terms in the summation, then the relevant equations are 

    (t1  t0 ) 1 1 ¼ (t1  t0 )  (t1  t0 )2 t 1  exp  t 2! l     1 1 1 1 3 (t1  t0 )  (t1  t0 )4 þ 3! l2 4! l3

Appendix C: Distribution of Linear Viscoelastic Relaxation Times

447

     (t2  t0 ) 1 1 ¼ (t2  t0 )  (t2  t0 )2 t 1  exp  t 2! l     1 1 1 1 3 (t2  t0 )  (t2  t0 )4 þ 3! l2 4! l3      (t3  t0 ) 1 1 t 1  exp  ¼ (t3  t0 )  (t3  t0 )2 t 2! l     1 1 1 1 3 þ  t )  (t (t3  t0 )4 3 0 3! l2 4! l3 When t ¼ 2 minutes, the exact solution for the first four inverse moments is       1 1 1 1 1 1 1 1 1 ; ; ; ¼ ¼ ¼ 2¼ ¼ 3¼ 2 2 3 l t 2 min t t l l 4 min 8 min3   1 1 1 ¼ 4¼ t l4 16 min4 These answers are compared with the linear algebraic results summarized in Table B.1 when the Taylor series expansion of exp(2t/l ) is truncated after four (i.e., n ¼ 0, 1, 2, 3), six (i.e., n ¼ 0, 1, 2, 3, 4, 5), or eight (i.e., n ¼ 0, 1, 2, 3, 4, 5, 6, 7) terms. Assuming that the zeroth inverse moment is unity, by definition, and 2% error is tolerable, the procedure outlined above provides accurate results for (i) the first inverse moment when four terms are included in the summation prior to truncation, (ii) the first three inverse moments when six terms are included, and (iii) the first five inverse moments when eight terms are included. For this simple illustrative example, the calculated nth-inverse moment of the distribution of relaxation times is predicted with less relative error when more terms are included in the polynomial expansion of exp(2t/l ) prior to truncation. The most important inverse moment is h1/li, when n ¼ 1, which represents the average rate of stress relaxation, or the average relaxation rate. It is increasingly difficult to provide a useful interpretation of h1/lni for larger values of n, but the results in Table B.1 illustrate the importance of including more terms in the polynomial expansion to obtain more accurate predictions of h1/li. Numerical Results When ER Follows an Exponential Decay with Two Relaxation Times The next example is relevant when relaxation behavior can be described by a sum of two exponential decay functions with normalized weighting factors and two relaxation times. In many realistic situations, the decay of carbon-13 magnetization at a particular site in the chemical structure of the repeat unit follows biexponential decay during solid state NMR relaxation experiments in the rotating reference frame, such that the average relaxation rate can be obtained from the first inverse moment. Now, the distribution function is given by the sum of two delta functions, spiked at t1 and t2,

448

Chapter 10 Mechanical Properties of Viscoelastic Materials

Table B.1 Comparison of Exact Inverse Moments of the Distribution of Viscoelastic Relaxation Times and Those Calculated via Linear Algebra When exp(2t/l ) is Expanded as a Polynomial and Truncated After Four (i.e., 0 n 3), Six (i.e., 0 n 5), or Eight (i.e., 0 n 7) Terms a Summation index n Four terms 1 2 3 Six terms 1 2 3 4 5 Eight terms 1 2 3 4 5 6 7

h1/lni via linear algebraic solution

h1/lni¼1/t n exact solution

Percentage error

4.99  1021 2.43  1021 9.33  1022

5.00  1021 2.50  1021 1.25  1021

1.22  1021 2.76 2.54  101

5.00  1021 2.50  1021 1.25  1021 5.98  1022 2.20  1022

5.00  1021 2.50  1021 1.25  1021 6.25  1022 3.13  1022

4.50  1024 1.87  1022 3.74  1021 4.38 2.96  101

5.00  1021 2.50  1021 1.25  1021 6.25  1022 3.11  1022 1.48  1022 5.27  1023

5.00  1021 2.50  1021 1.25  1021 6.25  1022 3.13  1022 1.56  1022 7.81  1023

1.17  1026 6.99  1025 2.19  1023 4.41  1022 6.06  1021 5.60 3.25  101

a

Stress relaxation follows a single exponential decay with one time constant (i.e., t ¼ 2 minutes). Integration intervals were chosen within the following range: 0.2 min , (tk 2 t0) , 2 min.

with normalized weighting factors w1 and w2 that sum to unity. The appropriate functions and parameters for this specific example are

j (l) ¼ w1 d(l  t1 ) þ w2 d(l  t2 ) ER (t  t0 ) ¼ ER (0)

ð1 l¼0

      (t  t0 ) (t  t0 ) (t  t0 ) d l ¼ w1 exp  þ w2 exp  j (l) exp  l t1 t2

  ð1 1 1 w1 w2 n ¼ n j (l ) d l ¼ n þ n l l t1 t2 l¼0

w1 ¼ 0:75; t1 ¼ 2 minutes w2 ¼ 0:25; t2 ¼ 3 minutes Analytical integration of the relaxation modulus over a sufficient number of different time intervals (i.e., tk 2 t0; k ¼ 1, 2, 3, . . . , 10) generates the required number of coupled linear algebraic equations that must be solved for the inverse moments of

Appendix C: Distribution of Linear Viscoelastic Relaxation Times

449

Table B.2 Comparison of Exact Inverse Moments of the Distribution of Viscoelastic Relaxation Times and Those Calculated via 10 Linear Algebraic Equations When exp(2t/l ) is Expanded as a Polynomial and Truncated After Eleven Terms (i.e., 0 n 10) a Summation index n 1 2 3 4 5 6 7 8 9 10

h1/lni via linear algebraic solution 21

4.58  10 2.15  1021 1.03  1021 5.00  1022 2.45  1022 1.20  1022 5.83  1023 2.67  1023 1.01  1023 2.26  1024

h1/lni ¼ w1/ft1gn þ w2/ft2gn exact solution 21

4.58  10 2.15  1021 1.03  1021 5.00  1022 2.45  1022 1.21  1022 5.97  1023 2.97  1023 1.48  1023 7.37  1024

Percentage error 3.05  1027 1.44  1025 3.36  1024 5.02  1023 5.27  1022 4.09  1021 2.37 1.02  101 3.19  101 6.93  101

a

Stress relaxation follows a sum of two exponential decay functions with two time constants (i.e., t1 ¼ 2 minutes, t2 ¼ 3 minutes) and normalized weighting factors (i.e., w1 ¼ 0.75, w2 ¼ 0.25). Integration intervals were chosen within the following range: 0.5 min , (tk 2 t0) , 5.5 min.

the distribution. The general form of these equations is provided below when stress relaxation is described by the sum of two exponential decay functions:       (tk  t0 ) (tk  t0 ) þ w2 t2 1  exp  w1 t1 1  exp  t1 t2   1 X (1)n 1 nþ1 ¼ n (tk  t0 ) (n þ 1)! l n¼0 Comparison between exact solutions for h1/lni and results from ten coupled linear algebraic equations is provided in Table B.2 when the Taylor series expansion of exp(2t/l ) is truncated after eleven terms (i.e., 0 n 10). Once again, the zeroth inverse moment is unity, by definition, because the distribution function j (l ) is normalized via two weighting factors that sum to unity. If 2% error is tolerable, then the procedure outlined above provides accurate results for the first six inverse moments when eleven terms are included in the summation prior to truncation.

C.4 Quantitative Evaluation of the Moments of a Triangular Distribution of Viscoelastic Relaxation Times Consider the following linear functions for j (l ) when there are no relaxation times less than t1 or greater than t3, and the largest fraction of a viscoelastic material relaxes with time constant t2. The generalized asymmetric triangular distribution function is

450

Chapter 10 Mechanical Properties of Viscoelastic Materials

described as follows:

8
when t1 l t2 when t2 l t3 when l t1 and l  t3

The overall objective is to postulate a triangular distribution of viscoelastic relaxation times and use actual stress relaxation measurements to calculate accurate first, second, and third inverse moments of the distribution function h1/lni, where n ¼ 1, 2, 3, based on the methodology described above by including a sufficient number of terms in the Taylor series expansion prior to truncation. Then, one predicts t1, t2, t3, and the average relaxation time defined by hli. Positive constants C1 and C2 are evaluated in terms of t1, t2, and t3 by invoking continuity of j at l ¼ t2, as well as normalization of the complete distribution function. Hence,

j ( t2 ) ¼ C 1 ( t2  t1 ) ¼ C 2 ( t 3  t2 ) tð3

j( l ) d l ¼ 1 ) C 1 ¼ C1 &C2

t1

2 2 ; C2 ¼ (t2  t1 )(t3  t1 ) (t3  t2 )(t3  t1 )

As mentioned above for the motivation to pursue these detailed calculations based on triangular distributions with terminal relaxation time t3, one uses realistic stress relaxation data on viscoelastic materials to estimate h1/lni that can be compared with predictions based on the proposed functional form of j (l ). The generalized nth-inverse moment is (i.e., n = 1, 2) tð3 tð2   tð3 1 1 1 1 ¼ j(l) d l ¼ C1 ( l  t1 ) d l þ C 2 (t3  l) d l ln ln ln ln t1

¼

t1

t2

C1 C1 t1 1n {t 22n  t 12n}  {t  t 11n} 2n 1n 2 C2 t3 1n C2 þ {t {t 2n  t 22n}  t 21n}  1n 3 2n 3

It is relatively straightforward to evaluate the previous expression when n ¼ 3 for the third inverse moment h1/l3i, and n ¼ 21 for the average relaxation time kll. Modified integrations are required when n ¼ 1 for the first inverse moment and n ¼ 2 for the second inverse moment of j (l ):       1 t2 t3 ¼ C1 {t2  t1}  C1 t1 ln þ C2 t3 ln  C2 {t3  t2} l t1 t2           1 t2 1 1 1 1 t3 ¼ C1 ln  C1 t1 þ C2 t3  C2 ln   t1 t1 t2 t2 t3 t2 l2 The following expressions are appropriate when the triangular distribution is symmetric, such that C1 ¼ C2, t2 is midway between t1 and t3, and the average relaxation

Appendix C: Distribution of Linear Viscoelastic Relaxation Times

451

time kll via tedious algebra is t2: C1 ¼ C2 ; t2 ¼ 12{t1 þ t3} hli ¼

tð3 t1

¼

lj(l) dl ¼ C1

tð2

l(l  t1 ) d l þ C2

t1

tð3

l(t3  l) dl

t2

C1 3 C1 t1 2 C 2 t3 2 C2 {t  t 31}  {t 2  t 21} þ {t 3  t 22}  {t 33  t 32} ¼ t2 3 2 2 2 3

C.5 Frequency Response via Fourier Transformation of the Step Response or Stieltjes Transformation of the Distribution Function Begin with the Maxwell – Wiechert model in Section 10.31 for the stress relaxation modulus ER(t) when the number of Maxwell elements in parallel approaches the continuous limit, and let t0 ¼ 0 without loss of generality when the step response is initiated. In terms of the distribution of viscoelastic relaxation times ED(l ), one writes ER (t) ¼

1 ð

n to ED (l) exp  dl l

l¼0

Results in Section 10.40 employ complex variable analysis to construct the complex dynamic modulus E (v) from the product of iv and one-sided Fourier transformation (i.e., from t to v) of the stress relaxation modulus (i.e., the step response function). In other words,

E (v) ¼ iv

1 ð

ER (t) exp{ivt} dt t¼0

By comparison, direct one-sided Fourier transformation of the pulse response function, which can be obtained from the step response via time differentiation, yields the frequency dependence of the complex dielectric constant for molecular dynamic analysis of viscoelastic materials using dielectric spectroscopy. As a consequence of sample inertia, the practical aspects of performing these transient viscoelastic experiments suggest that stress relaxation via step response measurements is tractable, whereas the mechanical pulse response experiment has limitations. Hence, the frequency response of viscoelastic materials can be obtained from Fourier transformation of (i) the pulse response via dielectric spectroscopy or (ii) the step response from stress relaxation. Upon substituting the integral expression for the relaxation modulus that contains the distribution function ED(l ) into the previous equation, reversing the order of integration (i.e., integrate first with respect to t, then with respect to l ), and performing integration with respect to t, one obtains well-known relations for the storage (i.e., E0 (v)) and loss (i.e., E00 (v)) moduli in terms of the distribution of

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Chapter 10 Mechanical Properties of Viscoelastic Materials

viscoelastic relaxation times: 2 1 3 1 ð ð n to E (v ) ¼ i v 4 ED (l) exp  d l5exp{ivt} dt l l¼0

t¼0 1 ð

¼ iv

¼

2

3     ð1  1 ivl 5 dt d l ¼ E D (l ) d l exp t þ iv l 1 þ ivl

1 ð

ED (l)4

l¼0

t¼0





1 ð

l¼0

l¼0

(vl)2 ED (l) dl þ i 1 þ (vl)2

1 ð

l¼0





vl ED (l) d l ¼ E 0 (v) þ iE00 (v) 1 þ (vl)2

If one employs the following substitutions for the complex integral expression on the far right side of the second line of the previous set of equations—p ¼ iv, q ¼ l21, and ED(l ) ¼ q 2h(q)—then the complex dynamic modulus can be expressed as E (v) ¼



1 l¼q1   ð  p¼iv ivl h(q) ED (l) dl ) p dq 1 þ ivl pþq ED (l)¼q2 h(q)

1 ð

l¼0

q¼0

Hence, these substitutions for v, l, and ED(l ) in terms of p, q, and h(q), respectively, allow one to write the complex dynamic modulus as the product of iv and the double Laplace transform, or the Stieltjes transform, of h(q) ¼ l2ED(l ). For example, the Laplace transform of h(q), from q to s, is given by Lq)s {h(q)} ¼

1 ð

h(q) exp{sq} dq ¼ D(s)

q¼0

The Laplace transform of D(s), from s to p, yields a double integral in which the order of integration can be reversed, and integration can be performed explicitly with respect to s: Ls)p {D(s)} ¼

1 ð

D(s) exp{ps} ds s¼0

¼

1 ð s¼0

¼

1 ð

q¼0

2 6 4

3

1 ð

7 h(q) exp{sq} dq5 exp{ps} ds

q¼0

2 h(q)4

1 ð

s¼0

3 exp{s(q þ p)} ds5dq ¼

1 ð



h(q) pþq

 dq ¼ M( p)

q¼0

In summary, the complex dynamic modulus is given by E (v) ¼ p{M(p)}, where p ¼ iv and M( p) is the Stieltjes transform of l2ED(l ).

References

453

FURTHER READING BIRD RB, ARMSTRONG RC, HASSAGER O. Dynamics of Polymeric Liquids, Volume 1: Fluid Mechanics. Wiley, Hoboken, NJ, 1977, Chap. 6. LUSKIN M. On the classification of some model equations for viscoelasticity. Journal of Non-Newtonian Fluid Mechanics 16:3– 11 (1984). CHURCHILL SW, CHURCHILL RU. A general model for the effective viscosity of pseudoplastic and dilatant fluids. Rheologica Acta 14:404–409 (1975). COLLYER AA. Department of Applied Physics, Sheffield Polytechnic. Time-independent fluids. Physics Education 8:333 (1973). COLLYER AA. Department of Applied Physics, Sheffield Polytechnic. Time-dependent fluids. Physics Education 9:38 (1974). COLLYER AA. Department of Applied Physics, Sheffield Polytechnic. Viscoelastic fluids. Physics Education 9:313 (1974). COLLYER AA. Department of Applied Physics, Sheffield Polytechnic. Demonstrations with viscoelastic liquids. Physics Education 8:111 (1973).

REFERENCES ABRAMOWITZ M, STEGUN IA. Handbook of Mathematical Functions. Dover Publications, Mineola, NY, 1965, pp. 228, 238–243. ADAM G, GIBBS JH. On the temperature dependence of cooperative relaxation properties in glass-forming liquids. Journal of Chemical Physics 43(1):139 (1965). ALDERSON KL, ALDERSON A, SMART G, SIMKINS VR, DAVIES PJ. Auxetic polypropylene fibres; Part 1; Manufacture and characterization. Plastics Rubber and Composites 31(8):344– 349 (2002). ALMGREN RF. An isotropic three-dimensional structure with Poisson’s ratio ¼ 21. Journal of Elasticity 15:427– 430 (1985). BEUCHE F. Derivation of the WLF equation for the mobility of molecules in molten glasses. Journal of Chemical Physics 24(2):418 (1956). BUECHE F. Mobility of molecules in liquids near the glass transition. Journal of Chemical Physics 30(3):748 (1959). BRANDEL B, LAKES RS. Negative Poisson’s ratio in polyethylene foams. Journal of Materials Science 36(24):5885– 5893 (2001). DAS PK, BELFIORE LA. Nonlinear stress relaxation in palladium(II) complexes with triblock copolymers of styrene and butadiene. Journal of Applied Polymer Science 93(3):1329–1336 (2004). DOOLITTLE AK. Studies in Newtonian flow; dependence of the viscosity of liquids on free-space. Journal of Applied Physics 22(12):1471 (1951). EVANS KE, DONOGHUE JP, ALDERSON KL. The design, matching and manufacture of auxetic carbon fibre laminates. Journal of Composite Materials 38(2):95– 106 (2004). FERRY JD. Viscoelastic Properties of Polymers, 3rd edition. Wiley, Hoboken, NJ, 1980. FUJITA H, KISHIMOTO A. Interpretation of viscosity data for concentrated polymer solutions. Journal of Chemical Physics 34(2):393–398 (1961). LAKES RS. Foam structures with negative Poisson’s ratio. Science 235:1038– 1040 (1987). MANSFIELD ML. One-dimensional models of polymer dynamics; spring-dashpot models. Journal of Polymer Science; Polymer Physics Edition 21:787–806 (1983). MAXWELL JC. On the dynamical theory of gases. Philosophical Transactions of the Royal Society A157:49–88 (1867). MCCRUM NG, READ BE, WILLIAMS G. Anelastic and Dielectric Effects in Polymeric Solids. Wiley, Hoboken, NJ, 1967, p. 192. ROVATI M. On the negative Poisson’s ratio of an orthorhombic alloy. Scripta Materialia 48(3):235– 240 (2003).

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Chapter 10 Mechanical Properties of Viscoelastic Materials

RUZZENE M, MAZZARELLA L, TSOPELAS P, SCARPA F. Wave propagation in sandwich plates with periodic auxetic core. Journal of Intelligent Material Systems and Structures 13(9):587– 597 (2002). RUZZENE M, SCARPA F, SORANNA F. Wave beaming effects in two-dimensional cellular structures. Smart Materials & Structures. 12(3):363–372 (2003). SCARPA F, YATES JR, CIFFO LG, PATSIAS S. Dynamic crushing of auxetic open-cell polyurethane foam. Proceedings of the Institution of Mechanical Engineers; Part C: Journal of Mechanical Engineering Science 216(12):1153– 1156 (2002). SCARPA F, CIFFO LG, YATES JR. Dynamic properties of high structural integrity auxetic open cell foam. Smart Materials & Structures 13(1):49– 56 (2004). TUMINELLO WH. Structure-Property Relations in Thermoplastic Rubbers, PhD thesis. Loughborough University of Technology, September 1973, pp. 61– 68. WANG YC, LAKES R, BUTENHOFF A. Influence of cell size on re-entrant transformation of negative Poisson’s ratio reticulated polyurethane foams. Cellular Polymers 20(6):373– 385 (2001). WEBBER RS, ALDERSON KL, EVANS KE. Novel variations in the microstructure of auxetic microporous ultrahigh molecular weight polyethylene, Part 1: Processing and microstructure. Polymer Engineering and Science 40(8):1894–1905 (2000). YANG X, YANG MB, LI ZM, FENG JM. The negative Poisson’s ratio effect of polyolefin blends. Acta Polymerica Sinica 221– 224 (2003).

PROBLEMS 10.1. Develop the viscoelastic constitutive relation between stress s and strain g for the Voigt model, which contains an elastic element (i.e., spring with static compliance J ¼ 1/E) in parallel with a viscous element (i.e., dashpot with Newtonian viscosity h). The entire configuration is subjected to a time-dependent stress s (t), applied initially at time t0. Hint: Each element experiences the same strain, and total stress is obtained by adding the stress in each element. Answer Follow the hint and consult the discussion in Section 10.19. The general solution for g (t) can be obtained from the Boltzmann superposition integral for any stress history, discussed in Appendix A.1, which is consistent with the final expression in Section 10.19.1 for g (t) via the integrating-factor method of solving first-order inhomogeneous ODEs. Hence,   ðt integrating 1 (t  Q) g (t) ) s (Q) exp  dQ factor h l(T) Q¼t0 Boltzmann

)

ðt

superposition

JC (t  Q)

@s dQ @Q

Q¼t0

where JC is the time-dependent creep compliance. If the one-time-constant Voigt model (i.e., l ¼ h/E ¼ Jh) is adequate to describe viscoelastic material response, then the analysis of creep experiments in Section 10.20 reveals that    (t  Q) JC (t  Q) ¼ J 1  exp  l(T) and the solution for g (t) via the Boltzmann superposition principle can be integrated by parts to yield the answer that was obtained by applying the integrating factor method to the equation of motion of the Voigt model (i.e., see in Section 10.19.1). Integration-by-parts analysis of the

Problems

455

generalized Boltzmann superposition integral for g (t) is discussed in significant depth in Appendix A (see Section A.2). For this particular problem, when time-dependent stress s (t) is applied initially to the Voigt model at time t0, the stress history vanishes for t , t0 and the Boltzmann integral reduces to Boltzmann

g (t) ) J superposition

  ðt  (t  Q) @s 1  exp  dQ l(T) @Q Q¼t0

Now, introduce the following substitutions and integrate by parts: u ¼ 12expf2(t 2 Q)/l(T )g and dv ¼ ds, which yields for constant observation time t, (du)t ¼ 2(dQ/l )expf2(t 2 Q)/ l(T )g and v ¼ s (Q). One obtains the following result:      ðt Boltzmann (t  Q) Q¼t J (t  Q) g (t) ) J s (Q) 1  exp  þ s (Q) exp  dQ superposition l(T) l l(T) Q¼t0 Q¼t0

     ðt (t  t0 ) 1 (t  Q) ¼ J s (t0 ) 1  exp  s (Q) exp  þ dQ l(T) h l(T) Q¼t0

The second term on the right side of the previous equation for g (t) is identical to the complete solution that employs the integrating factor method. The first term is discarded because applied stress is continuous such that s (t0) ¼ 0, which matches the boundary condition on stress as t0 ) 21 via the change of reference state in the original development of the Boltzmann superposition principle and its alternate forms. There is no inconsistency between invoking the condition s (t0) ¼ 0 and modeling creep response via a step function for s (t) that begins at t ¼ t0, because physically realistic mathematical models for the implementation of jump stress employ an exceedingly steep constant rate-of-stress (i.e., s0/1) for a very short time interval 1, beginning at t0. Hence, the stress history is given by t t0 ; s (t) ¼ 0 t0 t t0 þ 1; s (t) ¼

s0 (t  t0 ) 1

t  t0 þ 1; s (t) ¼ s 0 which corresponds to s ¼ 0 at t ¼ t0 to s ¼ s0 at t ¼ t0 þ 1. Inertial restrictions of the measurement device (i.e., servohydraulic mechanical testing systems) impose a lower limit on 1 in the millisecond regime. 10.2. How does the Voigt model respond immediately after the application of a “jump” stress s0? Answer There is no immediate response because the short-time deformation of the spring is impeded by the sluggishness of the dashpot in parallel. This characteristic of the Voigt model is captured by its creep compliance, discussed in Problem 10.1. Hence, JC(0) ¼ 0 and g (0) ¼ s0JC(0) ¼ 0. 10.3. (a) Sketch the creep response and the creep recovery of the Voigt model. A jump stress s0 is applied at time t0 and it is removed at time t1, where t1 . t0. Sketch g versus time.

456

Chapter 10 Mechanical Properties of Viscoelastic Materials (b) What is the time-dependent creep compliance JC(t) of the Voigt model? (c) What is the stress relaxation modulus ER(t) of the Voigt model?

Answer to Part (c) Analyze the equation of motion of the Voigt model specifically for constant strain g0 and divide the stress response by this constant level of strain to obtain an expression for the relaxation modulus. {s (t)}Total ¼ s (t) ¼ E g (t) þ h ER (t) ¼

@g ) E g0 @t g(t)¼g0

s (t) ¼E g0

The entire configuration does not achieve final deformation g0 instantaneously because the dashpot responds sluggishly. However, the elastic and viscous elements in the Voigt model asymptotically achieve g0 after approximately five relaxation time constants, give by 5h/E, with a rate of strain in the viscous element that is negligible when g0 is achieved. Under these conditions, the equation of motion reduces to Hooke’s law, providing a simple mathematical explanation for the fact that the Voigt model does not exhibit stress relaxation, because ER is time independent. 10.4. The Voigt –Kelvin viscoelastic model contains N Voigt elements in series, where each element consists of a spring and dashpot in parallel. Qualitatively describe why this model does not exhibit stress relaxation. Predict the stress relaxation modulus for the Voigt –Kelvin model. Answer This is a conceptually difficult problem because the dashpot in each Voigt element prohibits its partner spring from responding initially. Finally, when each spring deforms by an amount that is inversely proportional to its modulus, such that the product of the spring’s modulus and its deformation is the same for each Voigt element, the dashpot is prohibited from relieving the stress. In light of the fact that the stress relaxation modulus for one Voigt element is equivalent to the static modulus of its spring, based on the answer to Problem 10.3c, it is reasonable to conclude that the relaxation modulus of the Voigt–Kelvin model is equivalent to the effective modulus of N Voigt elements in series. Hence, the spring with modulus Ei in the ith Voigt element exhibits final deformation gi, retarded by the sluggishness of its partner dashpot, such that the product Eigi is the same for each of the N Voigt elements in series. The total strain g0, imposed on the entire configuration, is given by the sum of all gi, 1 i N. Hence, application of Hooke’s law for the deformation of each spring yields

g0 ¼

N X i¼1

gi ¼

N X si i¼1

Ei

Now, let si ¼ s0 for each Voigt element in series, and define the effective modulus of the Voigt –Kelvin model as the ratio of s0 to g0. The final result reveals that the inverse of the effective modulus, or the effective static compliance Jeffective, is obtained by summing the static

Problems

457

compliance Ji ¼1/Ei of the spring in each Voigt element: 

 N X 1 1 g 1 ¼ ¼ Jeffective ¼ 0 ¼ ER (t) Voigt – Kelvin Eeffective E s static 0 i¼1 i static

It is important to emphasize that inverse relations between modulus and compliance are applicable to static, time-independent, material properties that characterize this problem because the Voigt –Kelvin model does not exhibit stress relaxation. Inverse relations between relaxation modulus and creep compliance are invalid when these material properties are functions of time. 10.5. Draw the electrical circuit that is analogous to a spring and dashpot in series (i.e., the Maxwell model with one viscoelastic relaxation time). The list below provides analogies between mechanical and electrical properties. Each element in the Maxwell model experiences the same stress. Mechanical

Electrical

Strain Strain rate Stress Viscosity Compliance (i.e., 1/E)

Charge Current Voltage drop Resistance Capacitance

Answer The total rate of strain for the Maxwell model is obtained by adding the rate of strain in each element. Hence, the electrical analog of Maxwell’s model should reveal that the currents passing through the resistor and capacitor are additive, whereas both electrical elements experience the same voltage drop. A parallel combination of resistance and capacitance represents the electrical analog of a spring and dashpot in series. 10.6. What is the static “compliance” of the electrical analog of the Maxwell model? Answer Recall that static compliance is the inverse of static modulus. The table and answer to Problem 10.5 suggest that capacitance C represents the static compliance of the electrical analog of Maxwell’s model, given by a parallel combination of resistance R and capacitance C. Justification for this claim is obtained by comparing the equation of motion of the Maxwell model, in which strains and rates of strain are additive:       @g @g @g 1 @s s ¼ þ ¼ þ @t Total @t Elastic @t Viscous E @t h with the corresponding circuit law when currents i passing through the resistor and capacitor in parallel are additive: {i}Total ¼ {i}Capacitor þ {i}Resistor ¼ C

@V V þ @t R

The table in Problem 10.5 contains analogies that demonstrate equivalence between these mechanical and electrical problems.

458

Chapter 10 Mechanical Properties of Viscoelastic Materials

10.7. Zero-shear-rate viscosities of a viscoelastic material have been measured at every 2 8C increment in temperature, from Tg þ 15 8C to Tg þ 35 8C. No data are available at Tg. Describe the linear least squares procedure that should be implemented to analyze these 11 viscosity– temperature data pairs. In other words, two columns of numbers for h versus T in the appropriate units are available, and the polynomial model is y ¼ a0 þ a1 x (a) (b) (c) (d)

Identify the independent variable x. Identify the dependent variable y. Relate some important viscoelastic parameters to the intercept a0. Relate some important viscoelastic parameters to the slope a1.

Answer Combine (i) Doolittle’s equation for the dependence of zero-shear-rate polymer viscosity on fractional free volume and (ii) expansion of fractional free volume in a temperature polynomial that is truncated after the linear term. Use the glass transition temperature as the reference state for expansion of fractional free volume, and the first viscosity data point at Tg þ 15 8C as the reference state for construction of the WLF temperature shift factor aT. It is not possible to use the glass transition as the reference state for aT because viscosity data are not available at that temperature. When Tg is chosen as the reference state for fractional free volume, developments in Section 1.11 reveal that the coefficient of the linear term is essentially the discontinuous increment in thermal expansion at Tg, Da, which is convenient: f (T) ¼ f (Tg ) þ {aLiquid  [1  f (Tg )]aGlass}(T  Tg ) þ     f (Tg ) þ Da@Tg (T  Tg ) However, the use of different reference states for fractional free volume and the WLF shift factor introduces slight modification in the linear least squares procedure relative to the one described in Section 10.24 and in Problem 10.8. Begin by constructing the WLF shift factor that incorporates the Doolittle equation. Algebraic manipulation yields aT (T; Tref ) ¼

h(T) h(Tref )

  B B  Aþ ln{aT (T; Tref )} ¼ ln{h(T)}  ln{h(Tref )} ¼ A þ f (T) f (Tref )   B f (T)  f (Tref ) B  ln{aT (T; Tref )} ¼  f (Tref ) f (T) f (Tref ) ( ) f (Tg ) þ Da@Tg (T  Tg )  [ f (Tg ) þ Da@Tg (Tref  Tg )]  f (Tg ) þ Da@Tg (T  Tg ) ( ) Da@Tg (T  Tref ) B ¼ f (Tref ) f (Tg ) þ Da@Tg (T  Tg ) The appropriate linear relation is obtained via (i) dividing the previous expression for ln aT by (T – Tref ) and (ii) inverting the resulting equation: ( ) T  Tref f (Tref ) f (Tg ) þ T  Tg ¼ B Da@Tg  ln{aT (T; Tref )}

Problems

459

The following correspondence is made between this physical model for the temperature dependence of zero-shear-rate polymer viscosities and the first-order polynomial for linear least squares analysis: Independent variable:

x ¼ T  Tg , where the glass transition is known or measured separately via differential scanning calorimetry

Dependent variable: y ¼ Slope or first-order coefficient: Intercept or zeroth-order coefficient:

T  Tref Tref  T ¼  ln{aT (T; Tref )} ln {h(T)=h(Tref )}

f (Tref ) 1 ¼ { f (Tg ) þ Da@Tg (Tref  Tg )} B B f (Tg ) f (Tref ) f (Tg ) ¼ a0 ¼ B Da@Tg BDa@Tg a1 ¼

 { f (Tg ) þ Da@Tg (Tref  Tg )} The choice of B  1, which agrees with experimental data, is consistent with the fact that the temperature coefficient of fractional free volume is given by Da at the glass transition instead of the thermal expansion coefficient in the liquid state aLiquid. Since the glass transition temperature is known or measured via thermal analysis, and the reference state for construction of the WLF shift factor is coincident with the temperature of the initial viscosity data point, Tref ¼ Tg þ 15 8C, the slope and intercept from linear least squares analysis can be used to calculate fractional free volume at Tg and the discontinuous increment in thermal expansion coefficients at Tg. These calculations should be compared with the following universally accepted values: f(Tg)  0.025 and Da(@Tg)  4.8  1024 K21. The use of different reference states for fractional free volume and the WLF shift factor does not introduce any complexities that cannot be solved. The glass transition temperature is a very convenient choice of reference state for fractional free volume because it eliminates the need to determine three parameters (i.e., f (Tref ), aLiquid(@Tref ), and B) instead of only two (i.e., f(Tref ) and Da(@Tg), because B  1) via the slope and intercept from linear least squares analysis. 10.8. Williams, Landel, and Ferry invoked the principle of time –temperature superposition, calculated the temperature-dependent shift factor aT, and implemented linear least squares analysis of (T  Treference )={log aT (T; Treference )} versus T  Treference in terms of a first-order polynomial: y ¼ a0 þ a1 x where y ¼ (T  Treference )={log aT (T; Treference )} x ¼ T  Treference How are the slope a1 and intercept a0 of their polynomial model related to the classic WLF parameters C1 and C2 of the viscoelastic model? See the correspondences with Dr. R. F. Landel.

460

Chapter 10 Mechanical Properties of Viscoelastic Materials

Subject: WLF Shift Factor Hi Bob—While you were a graduate student in Prof. John Ferry’s lab at Wisconsin, I’m assuming that you measured stress relaxation moduli at several different temperatures and then shifted the data empirically to obtain a continuous curve for transient relaxation moduli that spanned several decades in time at the reference temperature. Did you then do the following to quantify the shift factor at each temperature and calculate C1 and C2? (i) Apply linear least squares analysis of aT(T; Treference) versus T 2 Treference based on a first-order polynomial in the form y ¼ a0 þ a1 x x ¼ 1=(T  Treference ) y ¼ 1= log[aT (T; Treference )] (ii) Calculate the WLF constants C1 and C2 at the reference temperature Treference via the slope and intercept of the linear model: Intercept ¼ a0 ¼ 1=C1 Slope ¼ a1 ¼ C2 =C1 This idea surfaced during a discussion of the principle of time–temperature superposition, followed by linearization of the functional form of aT(T; Treference) and subsequent linear least squares analysis, as mentioned above. Am I correct in assuming that you adopted this approach to calculate C1 and C2 at the reference temperature from your actual data? Larry, it was nice to hear from you. We only worked with dynamic data. Remember, John Ferry is the father of dynamic mechanical properties. With dynamic data, one has two sets of data to superpose. These shift factors should agree, especially upon recognizing that shifted real and imaginary compliances J0 and J00 at low temperatures are not the measured, but the corrected, values; and that shifting real and imaginary moduli G0 and G 00 does not work, in principle, at low temperatures. In other words, one should not attempt to superpose G0 and G 00 from low-temperature data. In reference to your question about the method to obtain C1 and C2, we essentially implemented the linear least squares analysis that you described above, by plotting; (T  Treference )= log{aT (T; Treference )} versus T  Treference and analyzing the slope and intercept. It was difficult to perform linear least squares regression on hand-cranked Marchand calculating machines. These are similar to the elementary calculators that are given away for free these days. However, one could obtain 13 figure results at each handle pull! 10.9. The glass transition temperature of atactic poly(methyl methacrylate), PMMA, is 115 8C when its number-average molecular weight is 1  105 daltons. Stress relaxation data are obtained isothermally at 125 8C. The critical molecular weight of PMMA at the entanglement threshold (i.e., Mcritical ) is 1.8  104 daltons. (a) By how much should these data at 125 8C be shifted on the log(time) axis to generate relaxation modulus versus time at 115 8C? (b) In what direction should these data at 125 8C be shifted on the log(time) axis to generate relaxation modulus versus time at 115 8C?

Problems

461

Answer: see Section 10.24 The Deborah number De ¼ 1024 when a 3-inch long sample (i.e., initial length) of 100-kilodalton PMMA at 125 8C is subjected to a constant strain rate of 0.1 inch per minute until the final engineering strain is 33% (i.e., 0.33). (c) Estimate the material response time l for 100-kilodalton PMMA at 125 8C. (d) Estimate the Deborah number De for 100-kilodalton PMMA at 125 8C if the 3-inch long sample (i.e., initial length) is subjected to a constant strain rate of 10 inches per minute until the final engineering strain is 33%. (e) Estimate the Deborah number De at 125 8C for atactic PMMA with a number-average molecular weight of 1106 daltons if the 3-inch long sample (i.e., initial length) is subjected to a constant strain rate of 0.1 inch per minute until the final engineering strain is 33%. 10.10. Use the Deborah number, De ¼ l(T )/(t 2 t0), to describe qualitatively the equivalence between time and temperature that represents the basis of the principle of time– temperature superposition as applied to the WLF shift factor. 10.11. Use the Deborah number, De ¼ l(T )/(t 2 t0), to describe qualitatively the equivalence between temperature and rate of strain for an InstronTM tensile testing experiment, where the time dependence of engineering strain is g (t) ¼ K(t 2 t0). Each experiment should be evaluated at the same level of strain. 10.12. Consider a three-time-constant Maxwell –Wiechert model for polyisobutylene that consists of three Maxwell elements in parallel, where each element contains a spring and dashpot in series. The parameters are E1 ¼ 1011 dynes=cm2

l1 ¼ 1010 second

E2 ¼ 109 dynes=cm2

l2 ¼ 107 second

E3 ¼ 106 dynes=cm2

l3 ¼ 103 seconds

(a) Sketch the distribution of viscoelastic relaxation times, ED(l ), on log–log coordinates. Be as quantitative as possible on both axes. (b) Sketch the distribution of viscoelastic relaxation times, HD(l ), on log–log coordinates. Be as quantitative as possible on both axes. (c) Sketch the relaxation modulus ER as a function of log(observation time). (d) Which dashpot exhibits the highest viscosity? (e) Calculate the zero-shear-rate polymer viscosity. A numerical answer is required here, with units of dynes-seconds per square centimeter (i.e., dyn-s/cm2). 10.13. A dog-bone shaped specimen with an initial length Linitial ¼ 2 inches is placed between the grips of an InstronTM tensile testing apparatus. At time t0, this viscoelastic material is subjected to a constant rate of strain, K ¼ 0.5 inch per minute, such that the time dependence of engineering strain is

g (t) ¼

K (t  t0 ) Linitial

462

Chapter 10 Mechanical Properties of Viscoelastic Materials The viscoelastic material is described by three Maxwell elements in parallel. The parameters are E1 ¼ 104 dyn=cm2

l1 ¼ 5 seconds

E2 ¼ 106 dyn=cm2

l2 ¼ 5 minutes

E3 ¼ 108 dyn=cm2

l3 ¼ 5 hours

(a) Relative to the shortest relaxation time l1, does this material behave like an elastic solid or a viscous liquid when the engineering strain g is 50% (i.e., g ¼ 0.5)? Hint: Consider the Deborah number. (b) Relative to the terminal relaxation time l3, does this material behave like an elastic solid or a viscous liquid when the engineering strain g is 50%? (c) Calculate a numerical value for Young’s modulus of elasticity with units of dyn/ cm2. Young’s modulus is the initial slope of the stress–strain curve. (d) How would your answer to part (c) differ if the viscoelastic material is described by three Maxwell elements in series with the same parameters given above? Calculate Young’s modulus of elasticity with units of dyn/cm2. 10.14. (a) Combine the Maxwell model for the stress relaxation modulus,   (t  Q) ER (t  Q) ¼ E exp  l(T) and the Boltzmann superposition integral for linear viscoelastic response,

s (t) ¼

ðt

ER (t  Q)

@g dQ @Q

1

to calculate the time dependence of induced stress s(t) during an InstronTM tensile testing experiment, where a constant rate of engineering strain is imposed on a viscoelastic material at time t0, such that for t ¼ t0

g (t) ¼

K (t  t0 ) Linitial

(b) Based on your answer to part (a), what is the asymptotic value of stress when the time scale for the relevant deformation process (i.e., t 2 t0) is much longer than the response time of the material l(T )? How does this asymptotic value of stress change if the tensile test is performed at higher temperature? (c) Based on your answer to part (a), calculate the slope of the stress –strain curve as follows: ds {ds=dt} Linitial ds ¼ ¼ dg {d g=dt} K dt (d) What is the slope of the stress –strain curve at time t0? (e) For time t . t0, does the slope of the stress –strain curve increase, decrease, or remain constant relative to the slope of the stress –strain curve at time t0? Provide a very brief explanation for your answer.

Problems

463

10.15. The time dependence of engineering strain g (t) during an InstronTM tensile testing experiment is described by

g (t) ¼

K (t  t0 ) Linitial

where the initial sample length in the stretch direction is Linitial, and the constant strain rate K has dimensions of length per time. Lower temperature response is analogous to testing the sample at higher rates of strain K, via equivalence of the dimensionless Deborah number. Consider the following calculations based on the Boltzmann superposition integral for linear viscoelastic response that employs a continuous distribution of relaxation time constants ED(l ) for the stress relaxation modulus ER(t 2 Q). The time scale for the relevant deformation process is (t 2 Q): ER (t  Q) ¼

ð1

  (t  Q) dl ED (l) exp  l

l¼0

(i) One must evaluate a double integral via the Boltzmann superposition theorem, where integration is performed initially with respect to relaxation time constant l and secondly with respect to observation time Q. The lower limit of integration for the time-dependent stress s (t) is Q ¼ t0 when the sample is subjected initially to a constant rate of strain. The upper integration limit is Q ¼ t. Hence, 8 9   =   ðt < 1 ð ðt @g K (t  Q) dl dQ s (t) ¼ ER (t  Q) ED (l)exp  dQ ¼ : ; Linitial @Q l Q¼t0

Q¼t0

l¼0

(ii) Reverse the order of integration and integrate first with respect to observation times Q. Now, the expression for s(t) can be presented in terms of an integral with respect to viscoelastic time constants l. 8 9 1 >   > ðt ð < = n o K t Q dQ dl s (t) ¼ ED (l) exp  exp > Linitial l > l : ; l¼0

¼

K Linitial

ð1

Q¼t0

   (t  t0 ) dl lED (l) 1  exp  l

l¼0

(iii) Differentiate the expression for s (t) with respect to present time t, and divide @ s/@t by the constant rate of strain (i.e., @ g/@t) to evaluate the slope of the stress –strain relation. The only dependence of stress on present time t appears in the exponential decay term within the previous integral with respect to time constant l. Partial time derivatives are employed in the following expression because both stress and strain are functions of position and time. One obtains the following result: Linitial @ s @ s=@t @ s ¼ ¼ ¼ @ g=@t @ g K @t

1 ð

  (t  t0 ) dl ¼ ER (t  t0 ) ED (l) exp  l

l¼0

(iv) The final result states that, within the linear viscoelastic regime, the instantaneous slope of the stress–strain curve is the stress relaxation modulus, ER(t 2 t0).

464

Chapter 10 Mechanical Properties of Viscoelastic Materials (a) Evaluate the initial slope of the stress –strain curve (i.e., @ s/@ g @ t ¼ t0) for a viscoelastic material that is described adequately by three Maxwell elements (i.e., E1, l1; E2, l2; E3, l3) in parallel. (b) Evaluate the initial slope of the stress–strain curve (i.e., @ s/@ g @ t ¼ t0) for a viscoelastic material that is described adequately by three Maxwell elements (i.e., E1, l1; E2, l2; E3, l3) in series. (c) Evaluate the initial slope of the stress– strain curve (i.e., @ s/@ g @ t ¼ t0) for a viscoelastic material that is described adequately by three Voigt elements (i.e., E1, l1; E2, l2; E3, l3) in series.

10.16. If an Instron tensile test is performed on a 5-inch sample of amorphous polycarbonate (i.e., Tg ¼ 150 8C) at ambient temperature (i.e., 25 8C), then the ultimate strain is approximately 10% or 20% when fracture occurs. Hence, it is reasonable to calculate the time dependence of stress during the constant rate-of-strain experiment via the Boltzmann superposition principle, as described in the previous two problem statements. As illustrated in Problem 10.15, the instantaneous slope of the stress –strain curve for polycarbonate in the linear viscoelastic regime is given by the stress relaxation modulus. (a) Draw a reasonable stress–strain curve for polycarbonate at 25 8C when the engineering strain rate K is 5 inches per minute. (b) Explain why the instantaneous slope of the stress –strain curve continuously decreases at higher strain. (c) Is the instantaneous slope of the stress –strain curve at g ¼ 0.10 during the experiment at K ¼ 5 inches/minute larger, smaller, or the same as the instantaneous slope of the stress –strain curve at g ¼ 0.20 during the experiment at K ¼ 10 inches/minute? (d) Explain your response to part (c) in terms of the temperature dependence of the relaxation modulus. Do not answer this question by comparing the rate of strain in each Instron stress–strain experiment. 10.17. Identify two real-life examples where viscoelastic damping, or the irreversible degradation of mechanical energy to thermal energy, is extremely important. 10.18. Indicate whether each of the following statements is True or False. (a) Maximum irreversible degradation of mechanical energy to thermal energy occurs in a forced-vibration dynamic viscoelastic experiment when the material response time l matches the excitation frequency v of the harmonic strain. (b) Irreversible degradation of mechanical energy to thermal energy is most efficient in a viscoelastic material when the temperature and excitation frequency of a forcedvibration dynamic experiment correspond to a maximum in the storage modulus E0 . (c) The frequency dependence of the storage modulus E0 (v), obtained from a forcedvibration dynamic experiment, is exactly the same as the time dependence of the relaxation modulus ER(t), obtained from a constant “jump” strain experiment. (d) The dynamic compliance J ¼ g(t; v)/s (t; v) is the inverse of the dynamic modulus E in a dynamic mechanical test, but complex variable theory is required to relate storage (J0 ) and loss (J00 ) compliances to storage (E0 ) and loss (E00 ) moduli.

Problems

465

For example, if E ¼ E 0 þ iE 00 then J ¼

1 1 ¼ E E0 þ iE 00

and one must multiply numerator and denominator of the previous equation by the complex conjugate of the denominator (i.e., E0 2 iE00 , where i equals the square root of –1): J ¼

 0  1 E  iE 00 E0 E00 ¼ 02 i 0 2 ¼ J 0  iJ 00 2 00 E0 þ iE 00 E 0  iE 00 (E ) þ (E ) (E ) þ (E 00 )2

Then, it is possible to identify the storage compliance as J0 ¼

(E 0 )2

E0 þ (E 00 )2

(E 0 )2

E 00 þ (E 00 )2

and the loss compliance as J 00 ¼

(e) The dynamic storage compliance J0 is very large at high excitation frequencies in a forced-vibration dynamic experiment. (f) The creep compliance J(t) for the Maxwell model is not the inverse of the relaxation modulus ER(t) for the Maxwell model. (g) If a polymeric material is chemically crosslinked above the percolation threshold, then its relaxation modulus will never decay to zero and its zero-shear-rate viscosity is finite. (h) Stress relaxation data obtained at 50 8C for an amorphous polymer should be shifted forward (i.e., to the right) to longer times to obtain the equivalent relaxation moduli versus observation time at the reference temperature, which is 25 8C. 10.19. Complex phasor analysis produces the following expressions for stress, strain, and rate of strain in a dynamic viscoelastic experiment, where the phase angle difference d(v, T ) between complex stress and complex strain is characteristic of the viscous nature of the material: Excitation function (i.e., strain):

g (v, t) ¼ gdc þ g0 exp(ivt)

Rate of Strain:

@ g (v, t) ¼ ivg0 exp(ivt) @t

Induced stress:

s (v, t) ¼ s0 exp{i(vt þ d)}

Perfectly elastic materials exhibit no phase angle difference between stress and strain (i.e., d ¼ 0). The complex dynamic viscosity h complex is defined as the complex stress

466

Chapter 10 Mechanical Properties of Viscoelastic Materials divided by the complex rate of strain. Hence,

s (v, t) s s  ¼ 0 exp(id) ¼ 0 (cos d þ i sin d) h complex ¼  @ g (v, t) ivg0 ivg0 @t s0 (sin d  i cos d) ¼ vg0 If hstorage measures mechanical energy storage and hLoss measures irreversible degradation of mechanical energy to thermal energy, then which of the two expressions below is predicted by complex variable analysis of the dynamic viscoelastic experiment?

h complex (v, T) ¼ hStorage  ihLoss

or

h complex (v, T) ¼ hLoss  ihStorage

10.20. Your observations of the creep, creep recovery, and stress relaxation response for a viscoelastic material are summarized below: Creep Response. The material deforms instantaneously to strain g1. The equilibrium level of strain g2 during the creep experiment is achieved after the stress is applied for 10 minutes, where g2 . g1. “Equilibrium” implies that the strain is no longer a function of time. Creep Recovery. The material exhibits no permanent deformation (i.e., permanent set). Stress Relaxation. The residual stress in the material never decays to a negligible fraction of its initial value. (a) Based on these observations, construct one plausible viscoelastic model (i.e., one combination of springs and dashpots) that will predict all of these responses accurately. (b) Estimate the viscoelastic time constant during the creep experiment. A numerical answer is required. 10.21. (a) Discuss the advantages and disadvantages of replacing Newton’s law of viscosity with the power-law model (i.e., s ¼ m(dg/dt)n) to describe the viscous characteristics of viscoelastic materials using various combinations of springs and dashpots. (b) Derive the equation of motion for a nonlinear Maxwell model that contains a series combination of elastic and viscous elements, in which the constitutive equation for the viscous element is described by the power-law model, with power-law exponent n and consistency index m. The elastic modulus of the spring is E. Answer For two elements in series, the strains and rates of strain are additive in each element. Focus on the rates of strain and add (dg/dt)Elastic þ (dg/dt)Viscous to obtain the total rate of strain, dg/dt. The result is a first-order nonlinear differential equation for the time rate of change of stress s for power-law fluids that exhibit elastic memory: "  1=n # ds dg jsj ¼E  m dt dt

Problems

467

(c) Use a numerical ordinary differential equation (ODE) solver and demonstrate that predictions from the equation of motion for the nonlinear Maxwell model reveal that sinusoidal steady state stress during dynamic testing is achieved at shorter times after initiation of the excitation strain function at t ¼ 0: g (t) ¼ g0 sin(vt), when the consistency index m is smaller for pseudo-plastic fluids (i.e., 0 , n , 1), because smaller values of m yield shorter material response times, l ¼ fmjdg/dtjn21g/E. Use the following initial conditions at t ¼ 0: g (t ¼ 0) ¼ 0, and {s (t ¼ 0)/m}1/n ¼ wg0v, where w represents the fraction of the initial rate of strain experienced by the viscous element (i.e., w  25%). Hint: It is not appropriate to choose the following initial conditions for stress and strain: s (t ¼ 0) ¼ g (t ¼ 0) ¼ 0, in anticipation of a phase angle difference between these harmonic functions that oscillate at the same frequency v ¼ 2p radians/s, even though the viscoelastic model is slightly nonlinear. 10.22. Intrinsic damping properties of viscoelastic samples via the torsion pendulum Parameters “a” and “b” in Section 10.42.1 contain material properties, G and h, of a viscoelastic sample. These material properties can be calculated via measurements of the oscillation frequency “b” of the torsion pendulum, and the “exponential rate of damping” characterized by “a”. Extremum conditions for angle of twist, cmax, on the same side of the equilibrium position map out the following time-dependent function; d ln cmax ¼ a dt Hence, measurements of “a” and “b” yield G and h when viscoelastic samples are analyzed via the 1-time-constant Voigt model that includes inertial effects of an attached disk. Torsion pendulum data yield free-vibration dynamic information that is sometimes compared with data from forced-vibration dynamic experiments, where the latter yield storage (in-phase, E0 ) and loss (out-of-phase, E00 ) moduli whose ratio is E00 /E0 ¼ tan d. Equations on Section 10.42.1 relate tan d from forced-vibration dynamic experiments to the logarithmic decrement D in free-vibration dynamic experiments (i.e., D ¼ p tan d). The disk’s moment of inertia IDisk affects the logarithmic decrement and the corresponding loss tangent under actual experimental conditions. How does one remove effects of IDisk on torsion pendulum data to calculate loss tangents that reflect intrinsic damping properties of viscoelastic samples which can be compared with forced-vibration data? Answer 1. Use experimental measurements for the pendulum’s “exponential rate of damping” with attached disk and the equation for parameter “a” in Section 10.42.1 to predict the sample’s viscosity h. Area moment of inertia J of the sample, the sample’s original length LSample, and the disk’s moment of inertia IDisk represent geometric parameters that can be calculated rather easily. Hence;   2IDisk LSample d ln cmax h¼  J dt

2. Use experimental measurements for the oscillation frequency of the pendulum with attached disk and the equation for parameter “b” in Section 10.42.1, to predict the sample’s shear modulus G, according to the one-time-constant Voigt model that

468

Chapter 10 Mechanical Properties of Viscoelastic Materials includes inertial effects of the attached disk. All other geometric parameters and the sample’s viscosity h are known or they have been calculated, previously. Hence; "   # {2bIDisk LSample }2 þ h2 J 2 IDisk LSample 2 d ln cmax 2 ¼ b þ  G¼ 4JIDisk LSample J dt

3. Use these predictions for the viscoelastic sample’s shear modulus G and viscosity h to predict the intrinsic damping characteristics of the sample via the penultimate equation in Section 10.42.1, upon replacement of the disk’s moment of inertia with the sample’s moment of inertia; n o 2p  d lndtcmax 2p D ¼ p tan d ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n o2 n offi 4GISample LSample ISample 2 ISample d ln cmax 1 h2 J b   1  IDisk dt IDisk In the presence of the disk, one measures experimental data and completely neglects inertial effects within the sample. This assumption for data analysis is valid when the oscillation frequency of the pendulum is small relative to the frequency of shear-wave propagation in the sample, where the latter travels at the speed of sound. In the absence of the disk, rapidly decaying high-frequency angle-of-twist data are simulated via an angular momentum balance on the sample which includes inertial effects within the sample and yields the same equations as those given in Section 10.42.1, using the moment of inertia ISample of the viscoelastic rod. Free-vibration damping properties of viscoelastic samples, measured using a torsion pendulum, are modeldependent according to the one-time-constant (i.e., l ¼ h/G) calculations outlined above, whereas “model-independent” forced-vibration viscoelastic properties (i.e., E0 , E00 , and tan d) are measured directly via phase angle differences between stress and strain that require sensitive electronic detection without instrumental artifacts (consult WH Tuminello’s PhD thesis from Loughborough University of Technology for a model-independent approach to extract intrinsic damping properties of viscoelastic materials using torsion-pendulum data: Structure-Property Relationships in Thermoplastic Rubbers, September 1973, pp. 61 –68).

Chapter

11

Nonlinear Stress Relaxation in Macromolecule – Metal Complexes Winter snow crimson lips a moment of pure forgetfulness. —Michael Berardi

Stress relaxation at large jump strains is analyzed for palladium complexes with polybutadiene that contain transition-metal catalyzed chemical crosslinks. Relaxation moduli are discussed in terms of a “correlated-states” model that employs nonlinear exponential functions. The effect of strain on viscoelastic relaxation times is discussed in terms of time – strain separability of the relaxation functions and conformational rearrangements of several neighboring segments that respond in cooperative fashion. Relaxation time constants increase when (i) experiments are performed at higher strain and (ii) higher concentrations of palladium chloride produce larger crosslink densities.

11.1 NONLINEAR VISCOELASTICITY When a spring is stretched beyond its elastic limit, its mechanical properties become deformation dependent (i.e., the spring “constant” depends on strain) and it may not recover equilibrium undeformed dimensions upon removal of the external force. The spring no longer behaves as an elastic material that exhibits linear response. Similar behavior is observed for viscoelastic materials when they are deformed beyond a few percent strain. In the large-deformation regime (i.e., beyond 50– 100% strain), considerable plastic flow occurs, which is completely irrecoverable. The theory of linear viscoelasticity is rather well developed. However, several practical applications subject materials to deformation in the nonlinear regime. Relations between stress and Physical Properties of Macromolecules. By Laurence A. Belfiore Copyright # 2010 John Wiley & Sons, Inc.

469

470

Chapter 11 Nonlinear Stress Relaxation in Macromolecule–Metal Complexes

strain have been proposed to characterize large-deformation response, but agreement with experiment is less satisfactory. The phenomenological analysis of stress relaxation in this chapter assumes that viscoelastic relaxation times depend on strain in the nonlinear regime.

11.2 OVERVIEW Above 200% strain, the mechanical response of triblock copolymers that contain styrene and butadiene (i.e., KratonTM D) is modified significantly by complexation with dichlorobis (acetonitrile)– palladium(II). This pseudo-square-planar transition-metal salt forms p-complexes with, and catalyzes the dimerization of, alkene groups in the main chain and the side group of Kraton’s butadiene midblock. Between 10% and 100% strain, the plastic flow regime is similar for undiluted KratonTM and its Pd2þ complexes, as illustrated in Figure 11.1, but the level of engineering stress is approximately two-fold larger for the complex that contains 4 mol % Pd(II). Nonlinear stress relaxation measurements in the plastic flow regime (i.e., beyond the yield point but before the large upturn in stress) are analyzed at several different levels of strain. Transient relaxation moduli have been modeled by a three-parameter biexponential decay with two viscoelastic time constants. The longer relaxation time 1.5e+7 4 mol % PdCl2 3 mol % PdCl2 2 mol % PdCl2 1 mol % PdCl2 0.5 mol % PdCl2 0 mol % PdCl2

Stress (N/m2)

1.0e+7

5.0e+6

0

25

50

75

100 200 300 400 500 600 700 Strain (%)

Figure 11.1 Ambient temperature engineering stress–strain response of KratonTM D-1101 complexes with palladium chloride at a strain rate of 50 mm/min (i.e., 2 inches/min). The molar concentration of PdCl2(CH3CN)2 is indicated in the legend. In each case, the polymeric palladium complex exhibits reinforced ductile mechanical response with an ultimate strain of 400% or greater. (KratonTM D-1101 (courtesy of Shell Development), SBS tri-block copolymer, 31 wt % polystyrene, polystyrene end-block MW  17.5 kDa, polybutadiene mid-block MW  78 kDa (granules).)

11.4 Effect of Palladium Chloride on the Stress –Strain Behavior

471

for KratonTM increases at higher strain, and in the presence of 4 mol % palladium chloride. A phenomenological model is proposed to describe the effect of strain on relaxation times. This model is consistent with the fact that greater length scales are required for cooperative segmental reorganization at larger strain. The resistance V to conformational reorganization during stress relaxation is estimated via integration of the normalized relaxation modulus versus time data. This resistance increases at higher initial jump strain because conformational rearrangements are influenced strongly by knots and entanglements at larger strain. The effect of strain on V is analyzed in terms of time – strain separability of the relaxation modulus. Linear behavior is observed for V versus inverse strain, and the magnitude of the slope (i.e., 2dV/ d(1/1)) is threefold larger in the absence of PdCl2(CH3CN)2.

11.3 RELEVANT BACKGROUND INFORMATION ABOUT PALLADIUM COMPLEXES WITH MACROMOLECULES THAT CONTAIN ALKENE FUNCTIONAL GROUPS Palladium(II) coordination in macromolecules that contain alkene functionality in the main chain or side group produces a variety of network topologies with crosslink densities that depend on polymer microstructure and metal cation concentration [Belfiore et al., 1995, 1996]. Characterization of these networks in terms of defect structures (i.e., branching, ineffective crosslinks, loops, etc.) is critically important to understand fracture properties and toughness. Recent studies of Pd2þ complexation in polybutadienes and polyisoprenes [Belfiore et al., 1995] have generated reactive blends with significant enhancement in fracture stress and resistance to deformation at large strain (i.e., above 200%). This response is most likely due to the presence of crosslinks and entanglements in the polymeric matrix, resulting from Pd-catalyzed dimerization of alkenes that couples different chains and increases their effective molecular weight [Bosse´ et al., 1995]. For example, polybutadiene – PdCl2 mixtures in THF or toluene below the gelation threshold exhibit light-scattering-detected average aggregation numbers (i.e., AN ¼ Mw,Complex/Mw,Pure Polymer via Zimm-plot intercepts) of 2 for low-viscosity solutions, whereas AN  9 for viscous THF solutions [Bosse´ et al., 1997]. An organometallic mechanism for Pd-catalyzed coupling of alkene functional groups in the main chain and/or side chain of polybutadienes that agrees with hightemperature infrared kinetic studies of the decreasing CvC stretching vibration in solid films is provided in Appendix C of Chapter 12.

11.4 EFFECT OF PALLADIUM CHLORIDE ON THE STRESS– STRAIN BEHAVIOR OF TRIBLOCK COPOLYMERS CONTAINING STYRENE AND BUTADIENE Previous investigations of Pd2þ complexes with cis-polybutadiene, 1,2-polybutadiene, and 3,4-polyisprene suggest that polymers containing alkene functionality either in the main chain or the side group can be modified chemically via

472

Chapter 11 Nonlinear Stress Relaxation in Macromolecule–Metal Complexes

dichlorobis(acetonitrile) – palladium(II) [Bosse´ et al., 1995]. This methodology has been employed to induce a significant upturn in the stress – strain response of KratonTM D-1101, above 200% strain, as illustrated in Figure 11.1. Relative to the undiluted triblock copolymer, the engineering fracture stress is more than five-fold larger in complexes that contain either 3 mol % or 4 mol % palladium chloride. Hence, significant modifications in the mechanical properties of an important engineering thermoplastic have been achieved at ambient temperature via Pd2þ-catalyzed dimerization of alkenes. These materials are lightly crosslinked, but the crosslink density increases when the Pd2þ concentration is larger.

11.5 CROSSLINKED POLYMERS AND LIMITED CHAIN EXTENSIBILITY Polymeric networks can be modeled as an assembly of macromolecules interconnected by chemical or physical crosslinks. Network characterization is achieved by monitoring these crosslinks, entanglements, and clusters in terms of the molecular weight between junction points, especially when viscoelastic properties are being measured. The classic approach to describe the stress – strain behavior of rubber-like materials in the linear viscoelastic regime is based on an analysis of polymer network strands that exhibit a Gaussian distribution of end-to-end chain dimensions within the framework of equilibrium thermodynamics (see Chapters 13 and 14). However, theory does not predict an increase in stress at moderate-to-large deformations. It has been postulated that this significant increase in stress is due to strain-induced crystallization of network strands [Queslel and Mark, 1989]. Surprisingly, this increase in stress is observed in experiments with model poly(dimethylsiloxane) networks where chain segments cannot crystallize. These experiments suggest that chains exhibit limited extensibility in which their non-Gaussian nature and nonaffine deformation are not predicted by the classical theory of rubber elasticity [Andrady et al., 1980]. This theory has been improved to capture the high-strain regime much better via the treatments of Erman and Flory [1978]. However, the number of model parameters increases and the parameters become more difficult to quantify experimentally because nonlinear viscoelastic constitutive relationships are complex.

11.6 NONLINEAR STRESS RELAXATION Coordination between reactive functional groups in the main chain or side group of selected polymers and transition-metal catalysts or nanoclusters has the potential to produce a variety of structures that range from loosely crosslinked rubber-like materials to densely crosslinked glasses. Characterization of these materials in terms of defect structures, such as branching, ineffective crosslinks, and loops, allows one to correlate fracture properties and toughness via chain microstructure. Industrially important macromolecule – metal complexes described in this chapter exhibit remarkable highstrain mechanical response, as illustrated in Figure 11.1. Loops, entanglements, and, most importantly, chemical crosslinks are responsible for the significant increase in high-strain mechanical response when Kraton thermoplastic elastomers form

11.6 Nonlinear Stress Relaxation

473

coordination complexes with palladium chloride. When materials exhibit large strains at failure, obvious questions arise that address potential energy dissipation mechanisms associated with the deformation process. Stress relaxation measurements that focus on viscous flow at constant strain provide a well-controlled macroscopic probe of the viscoelastic time constants that are associated with energy dissipation processes, particularly at high levels of strain. The Boltzmann superposition principle is not applicable when the response is nonlinear, but stress relaxation experiments with straindependent time constants qualitatively simulate energy dissipation at large strains [Ward and Hadley, 1995]. Brueller [2000] has extended the Boltzmann superposition principle to nonlinear viscoelasticity using creep experiments, not stress relaxation.

11.6.1 Time-Dependent Relaxation Functions via the Correlated States Model In general, stress relaxation can be viewed as the reorientation of chain segments via rotation about single bonds in the backbone that increases the conformational entropy of network strands. If entanglements are present, then they will function as topological knots and enhance mechanical properties at large strain. The coupling model of Ngai et al. [1986] describes relaxation phenomena rather successfully for a wide range of materials, including glass-forming viscous liquids, polymer melts, and vitreous ionic conductors. The experimental methodologies of Yee et al. [1988] address nonlinear viscoelasticity in glass-forming materials via (i) imposing a level of strain that induces nonlinear response and (ii) measuring stress relaxation after application of a small 0.5% perturbation in strain. Then, relaxation spectra are analyzed using a coupling model that includes one fractional exponential function, as suggested by Williams and Watts [1970] (i.e., the KWW model). One of the most important parameters in the KWW coupling model is the fractional exponent in Eq. (11.1):   1n  t ER (t) ¼ ER (t ¼ 0) exp  l

(11:1)

where ER(t) is the relaxation modulus, l is the effective viscoelastic time constant, and 12n is the fractional exponent (i.e., 0  n , 1). A smaller exponent, with n closer to unity, is consistent with a broader spectrum of relaxation times for real materials, whereas exponents that approach unity as n tends toward zero describe systems that exhibit a much narrower distribution of viscoelastic time constants. The effect of fractional exponent 12n on normalized stress relaxation moduli ER(t)/ER(t ¼ 0) is illustrated in Figure 11.2. Equation (11.1) is based on a “correlated states” model in which the normalized relaxation function w(t) is defined by

w(t) ¼

ER (t) ER (t ¼ 0)

(11:2)

Classical and quantum mechanical analyses of spatial perturbations in morphological structure, due to randomness of the environment, are described by time-dependent

474

Chapter 11 Nonlinear Stress Relaxation in Macromolecule–Metal Complexes

Normalized Relaxation Function

1.0 n = 0.0 n = 0.3 n = 0.5 n = 0.7 n = 0.8 n = 0.9

0.8 0.6 0.4 0.2 0.0 0.0

0.5

1.0 1.5 2.0 Dimensionless Time (t/l)

2.5

3.0

Figure 11.2 Fractional exponential relaxation functions, illustrating time-dependence of the normalized relaxation modulus, according to Eq. (11.1). When n approaches unity and the fractional exponent, 1 2 n, approaches zero, the initial decrease in ER(t) is steeper, but a larger fraction of ER(t ¼ 0) is retained for longer observation times.

modulations of cooperatively relaxing neighboring segments. This yields the following rate equation for the relaxation function [Plonka, 1991] that agrees with reaction rate theory when the conformational entropic contribution to the Gibbs free energy, which controls the behavior of the relaxation rate, decreases at longer time (i.e., the first-order “rate constant” exhibits time dependence): dw (eg v cutoff t)n ¼ w(t) dt tCorr

(11:3)

where vcutoff is an “upper-cutoff ” frequency associated with energy-level spacings for cooperative conformational adjustments of neighboring segments required for stress relaxation, g  0.577 is Euler’s constant, and tCorr represents the correlation time for thermally activated segmental motion that exhibits Arrhenius temperature dependence. Solution of the previous rate equation for w(t) subject to the initial condition, w ¼ 1 at t ¼ 0, yields the following “stretched” exponential relaxation function:  w(t) ¼ exp 

l

1n

t 1n (1  n)eng tCorr vncutoff

¼ (1  n)e

ng



  1n  t ¼ exp  l

(11:4)

tCorr vncutoff

When n ¼ 0, this relaxation function reduces to a single exponential decay with a viscoelastic time constant l that matches the correlation time tCorr for thermally activated segmental motion. If Eactivation represents the Arrhenius activation energy that describes the temperature dependence of correlation time tCorr, then the apparent

11.6 Nonlinear Stress Relaxation

475

activation energy for viscoelastic motion, in general, via Rgasfd ln l/d(1/T)g is Eactivation/(12n). When stress relaxation of viscoelastic materials is described by a “stretched” exponential decay, their zero-shear-rate viscosity h(T ) is calculated below via (i) integration of the relaxation modulus ER(t) or (ii) the first moment of the non-normalized distribution of relaxation times ED(l ) (see Sections 10.25 and 10.34). Hence, let z ¼ t/l and 12n ¼ 1/b:

h(T) ¼

1 ð

1 ð

ER (t) dt ¼ ER (t ¼ 0) w(t) dt

0

0

1   1n  ð t ¼ ER (t ¼ 0) exp  dt ¼ lER (t ¼ 0) exp(z1=b ) dz l 1 ð 0

0

¼ lER (t ¼ 0)bG(b) ¼

  1 ð lER (t ¼ 0) 1 G ¼ lED (l) dl  kll 1n 1n

(11:5)

0

where G represents the gamma function, and l  kll for a single exponential decay when n ¼ 0 (i.e., l and kll have different dimensions because ED(l )dl is a nonnormalized probability with dimensions of modulus, see Section 10.31). As n approaches unity and the fractional exponent 12n decreases toward zero, the average viscoelastic relaxation time kll increases, as illustrated in Figure 11.3. Conversely, as n approaches zero and the fractional exponent approaches unity, the average rate of viscoelastic relaxation, defined by 1/kll, increases.

107 Average Relaxation Time

106 105 104 103 102 101 100 10–1 0.00 Figure 11.3

0.20

0.40

n

0.60

0.80

1.00

Fractional exponential relaxation functions, illustrating the effect of n, or the fractional exponent, 12n, on the average relaxation time according to Eq. (11.5) when the stress relaxation modulus is described by Eq. (11.1). The vertical axis is dimensionless, kll/flER(t ¼ 0)g.

476

Chapter 11 Nonlinear Stress Relaxation in Macromolecule–Metal Complexes

11.6.2 Previous Investigations of Stress Relaxation in the Nonlinear Regime McKenna and Zapas [1986] imposed a large deformation initially, and then applied incremental strains at selected time intervals. Askadskii and Valetskii [1991] used a nonlinear integral expansion representation of time-dependent stress to model nonlinear stress relaxation based on the rate of entropy generation. Yoshioka [2002] employed a nonlinear model with one relaxation time to predict yielding in glassy polymers at large deformation. In many cases, the relaxation modulus or the viscoelastic strain energy function exhibits time – strain separability at large deformations. Osaki et al. [1996] concluded that time-dependent reptation based on the tube model and time- and strain-dependent retraction along the tube axis are uncoupled below 700% strain, but the factorability of time and strain is not possible at higher strains. Watanabe et al. [1996] concluded that time – strain separability can be applied to styrene – isoprene diblock copolymers in the terminal relaxation regime, where terminal relaxation times are strain independent below 200% strain. Ionic interactions in molten ionomers do not invalidate the separabilty of time and strain. The straindependent damping function reveals “strain softening” of ethylene – methacrylic acid copolymers at high degrees of Naþ and Zn2þ neutralization in the nonlinear viscoelastic regime [Takahashi et al., 1995]. Randomly dispersed rubber particles have a considerable effect on strain-dependent damping functions in molten acrylonitrile – butadiene – styrene terpolymers [Aoki et al., 2001], but the validity of time – strain separability of the nonlinear relaxation modulus is not challenged. However, Tanaka and Edwards [1992] demonstrated that the underlying physical basis of time – strain separability is questionable in their transient study of physically crosslinked gels. Kwon and Cho [2001] concluded that several well-known viscoelastic constitutive equations exhibit dissipative instabilities due to the hypothesis of time – strain separability. This hypothesis is violated during the short-time response immediately after imposing a jump strain [Chen et al., 2003].

11.7 RESULTS FROM STRESS RELAXATION EXPERIMENTS ON TRIBLOCK COPOLYMERS Stress relaxation moduli ER(t, 1) in the plastic flow regime, between 5% and 55% jump strains, are illustrated in Figure 11.4 for KratonTM D-1101, and Figure 11.5 for complexes of KratonTM D-1101 with 4 mol % PdCl2(CH3CN)2. At the same level of strain, relaxation moduli are larger for the KratonTM /Pd2þ complex relative to the undiluted block copolymer. Since engineering stress does not increase significantly at larger strain in the plastic flow regime, the corresponding relaxation moduli are smaller when experiments are performed at larger jump strain, because the relaxation modulus was calculated via division of the time-dependent engineering stress by the magnitude of the constant jump strain. Relaxation moduli in Figures 11.4 and 11.5 are qualiltatively consistent with an increase in creep compliance for nylons and cellulosic fibres at higher jump stress [Ward and Hadley, 1995]. Watanabe et al. [1996], Osaki et al. [1996], and Merriman and Caruthers [1981] report decreasing

11.7 Results from Stress Relaxation Experiments on Triblock Copolymers 1.2e+7

Relaxation Modulus (N/m2)

9.8% 13.9% 15.1% 19.5% 22.8% 25.9%

34.3% 35.7% 44.9% 54.5%

1.0e+7

477

8.0e+6

6.0e+6

4.0e+6

2.0e+6

0.0 0

3

6 9 Time (min)

12

15

Non-normalized stress relaxation modulus versus time for KratonTM D-1101 at ambient temperature, corresponding to jump strains between 10% and 55%, achieved at a rate of 254 mm/min, as indicated in the legend. The ratio of engineering stress to the initial jump strain is plotted on the vertical axis.

Figure 11.4

2.2e+7

1.8e+7 Relaxation Modulus (N/m2)

29.1% 35.5% 42.9% 45.2% 46.3%

5.9% 9.7% 14.5% 19.7%

2.0e+7

1.6e+7 1.4e+7 1.2e+7 1.0e+7 8.0e+6 6.0e+6 4.0e+6 2.0e+6 0

3

6 9 Time (min)

12

15

Non-normalized stress relaxation modulus versus time for complexes of KratonTM D-1101 with 4 mol % palladium chloride at ambient temperature, corresponding to jump strains between 6% and 46%, achieved at a rate of 254 mm/min, as indicated in the legend. The ratio of engineering stress to the initial jump strain is plotted on the vertical axis.

Figure 11.5

478

Chapter 11 Nonlinear Stress Relaxation in Macromolecule–Metal Complexes

stress relaxation moduli at larger strain in the nonlinear regime above 10% strain. For comparison with the stress relaxation moduli of KratonTM D-1101 in Figure 11.4, Merriman and Caruthers [1981] measured nonlinear stress relaxation in lightly crosslinked styrene–butadiene random copolymers in the vicinity of the glass transition temperature. The volume decrease during stress relaxation, that follows the instantaneous dilation upon deformation, was measured experimentally, and nonlinearity in the transient viscoelastic response was correlated with volume relaxation via fractional free volume. Matsuoka et al. [1978] reported that polycarbonate dilates under tensile deformation in the glassy state, and Poisson’s ratio decreases from 0.45 to 0.375 at higher strain.

11.8 EFFECT OF STRAIN ON STRESS RELAXATION Analysis of stress relaxation moduli in Figures 11.4 and 11.5 considers the area under the normalized relaxation modulus versus time curve to provide estimates of the resistance V to conformational reorganization in the polymer: V(1) ¼

tfinal ð tstart

 ER (t, 1) dt ER (t ¼ 0, 1)

(11:6)

where the lower integration limit (i.e., tstart) is 0.3 min and the upper integration limit (i.e., tfinal ) is 15 min. Sweeney et al. [1999] discuss the effects of microstructural rearrangements as semicrystalline polymers are stretched, to predict yielding and necking of isotactic polypropylene at 150 8C via an inhomogeneously strained network model. Both the stress relaxation modulus ER and the resistance V depend on strain 1 in the nonlinear regime. In the linear viscoelastic regime, V is independent

Table 11.1 Effect of Strain on the Resistance to Conformational Reorganization V and Viscoelastic Time Constants t During Stress Relaxation KratonTM Strain (%) 9.8 13.9 15.1 19.1 22.8 25.9 34.3 35.7 44.9 54.5

KratonTM with 4 mol % PdCl2

V (min)

t (min)

Strain (%)

V (min)

t (min)

12.8 13.3 13.4 13.5 13.7 13.7 13.8 13.8 13.9 14.0

63 84 91 105 119 121 130 133 149 164

5.9 9.7 14.5 19.7 29.1 35.5 42.9 45.2 46.3

13.0 13.2 13.4 13.5 13.5 13.5 13.7 13.7 13.6

101 110 119 133 149 154 179 170 167

11.9 Time–Strain Separability of the Relaxation Function

479

Area Under Relaxation Modulus vs. Time (minutes)

14.2 Undiluted Kraton™ Kraton™ w/ 4 mol % Pd(II)

14.0 13.8 13.6 13.4 13.2 13.0 12.8 12.6

0

2

4

6

8 10 12 Inverse Strain

14

16

18

Figure 11.6

Effect of jump strain during nonlinear stress relaxation on the “area under the curve” of the normalized relaxation modulus versus time (i.e., V via Eq. (11.6)) for KratonTM D-1101 and its complex with 4 mol % palladium chloride. Approximate linear behavior is observed for V versus inverse strain, with a three-fold larger slope when PdCl2(CH3CN)2 is absent.

of strain. Normalization of the relaxation modulus prior to performing the integration via Eq. (11.6) precludes any inconsistencies in using engineering stress instead of true stress in the nonlinear regime. The effect of strain on V is summarized in Table 11.1 for KratonTM and its complex with 4 mol % palladium chloride. These results from Table 11.1 are displayed in Figure 11.6, where V is correlated versus inverse strain. One concludes that (i) V, which is related to zero-shear-rate viscosity, increases when stress relaxation experiments are performed at larger jump strain, (ii) V versus inverse strain exhibits linear behavior, and (iii) the magnitude of the slope, 2dV/d(1/1), is three-fold larger (i.e., 0.13 vs. 0.04) for KratonTM without PdCl2(CH3CN)2.

11.9 TIME –STRAIN SEPARABILITY OF THE RELAXATION FUNCTION Strain dependence of the resistance to conformational reorganization in the nonlinear viscoelastic regime, based on the definition of V via Eq. (11.6) and the separability of time and strain when reptation and retraction can be decoupled [Ward, 1983;

480

Chapter 11 Nonlinear Stress Relaxation in Macromolecule–Metal Complexes

Aoki et al., 2001; Le Meins et al., 2002], is given by Eq. (11.7): tfinal ð h(1) V(1) ¼ g(t) dt 1

(11:7)

tstart

where g(t) is the time-dependent part of the stress response that decays during the relaxation experiment (i.e., analogous to the normalized relaxation modulus for linear viscoelasticity), h(1) resembles a damping function, and h(1)/1 approaches unity in the linear viscoelastic regime. When g(t) is analyzed in terms of a biexponential two-time-constant model for the transient relaxation moduli in Figures 11.4 and 11.5, the longer relaxation time is reported in Table 11.1 as a function of strain. In general, one observes that the longer relaxation time constant increases at higher strain. This effect is described in the following section. The presence of palladium chloride also increases these longer relaxation times, relative to undiluted KratonTM . If the conformational rearrangements that give rise to stress relaxation occur over length scales that are comparable to the distance between network junction points at low crosslink densities, then it seems reasonable that the concentration of the low-molecular-weight transition-metal complex should affect viscoelastic time constants.

11.10 CHARACTERISTIC LENGTH SCALES FOR COOPERATIVE REORGANIZATION AND THE EFFECT OF STRAIN ON VISCOELASTIC RELAXATION TIMES In the condensed state, polymer chain segments typically do not undergo conformational rearrangements without cooperativity from neighboring segments. Stress relaxation requires that nearest neighbor segments within the same chain or from different chains must move in synchronous fashion. Hence, barriers to segmental relaxation contain both intramolecular and intermolecular contributions. In general, chain segments between crosslink junctions respond to an applied deformation by reorganizing and dissipating stress. Free energy dissipation occurs when segments rotate about single bonds in the chain backbone to increase their conformational entropy, with a corresponding decrease in free energy. Characteristic dimensions, or length scales for cooperative segmental relaxing units, have been discussed by Matsuoka [1992], yielding an expression for the stress relaxation modulus that is very similar to Eq. (11.1), with much better physical significance of the model parameters. Viscoelastic time constants t for cooperative rearrangements are directly related to the characteristic dimensions over which segmental relaxation occurs, such that ln t scales linearly with the number z of cooperatively relaxing segmental units. As these length scales increase, the associated inertia reduces the relaxation frequencies and increases viscoelastic time constants. There is a critical strain 1critical, most likely in the linear regime, below which segmental relaxation processes are independent of conformational rearrangements from neighboring segments on different chains. In this regime, barriers to segmental relaxation are dominated by

11.10 Characteristic Length Scales for Cooperative Reorganization

481

intramolecular cooperativity and the corresponding relaxation times are rather short. At higher jump strain, there is a stronger coupling among all of the segmental relaxing units, the intermolecular contribution to the overall relaxation process increases at the expense of the intramolecular contribution, and viscoelastic time constants increase considerably. According to Matsuoka [1992], the probability that z cooperative segments will relax simultaneously is given by the product of z identical factors, where each factor represents the probability that one segment will relax. Hence, the apparent activation energy for this cooperative process is z-fold larger than the barrier that must be overcome for one segment to relax. If the temperature dependence of viscoelastic time constants t is based on a thermally activated rate process in Arrhenius form, then the following scaling law is appropriate: ln t 

z DG kB T

(11:8)

where DG is the free energy barrier for one segment to relax at temperature T, and kB is Boltzmann’s constant. Based on results from the previous section, the effect of strain on viscoelastic time constants is proposed in Eq. (11.9) for a rate process that is thermally and mechanically activated, requiring cooperativity when jump strain 1 is greater than 1critical:   z DG (11:9) t  f (1) exp kB T where f(1) increases at larger strain and approaches a nonzero constant in the limit of zero strain. The time constants in Table 11.1 for KratonTM and its complex with 4 mol % palladium(II) are presented graphically in Figure 11.7 to illustrate that f (1) in Eq. (11.9) depends logarithmically on strain. For example, nonlinear least squares analysis yields the following phenomenological result:

t (1)  a0 f1 þ a1 ln(1 þ a2 1)g

(11:10)

where a0, a1, and a2 are positive constants, and a0 represents the relaxation time at zero strain. For undiluted KratonTM , the inverse Hessian method reveals that several combinations of a0 and a1 in Eq. (11.10) satisfy the minimization procedure such that a0a1  68 minutes, and a2  0.18 when strain is reported in percent. For the complex between KratonTM and 4 mol % PdCl2(CH3CN)2, which produces a lightly crosslinked network, the method of steepest descent yields a0 ¼ 83.3 minutes, a1 ¼ 1.17, and a2 ¼ 0.033 when strain is reported in percent. Effects of strain on viscoelastic relaxation times for polymeric materials have not been discussed previously in much depth, either phenomenologically or theoretically. The dimensionless free energy barrier (i.e., z DG/kBT ) for stress relaxation in Eqs. (11.8) and (11.9) is higher when the length scale over which segmental relaxation occurs is larger, due to intermolecular coupling among more cooperatively relaxing units.

482

Chapter 11 Nonlinear Stress Relaxation in Macromolecule–Metal Complexes

Relaxation Time (minutes)

200

150

100

50

0

Kraton Kraton, Inverse Hessian Method Kraton/4 mol % Pd(II) Kraton/4 mol % Pd(II), Steepest Descent 0

10

20 30 40 Engineering Strain (%)

50

60

Figure 11.7 Effect of jump strain during nonlinear stress relaxation on the longer viscoelastic time constant for undiluted KratonTM and its complex with 4 mol % palladium chloride. Each set of transient stress relaxation data in Figures 11.4 and 11.5 was modeled as a three-parameter biexponential decay with two time constants. The longer relaxation time is plotted on the vertical axis. The strain dependence of t is given phenomenologically by Eq. (11.10). For undiluted KratonTM , the product of a0 and a1 is  68 minutes, and a2  0.18 via the inverse Hessian method of nonlinear least squares analysis, when strain is reported in percent. For the complex between KratonTM and 4 mol % PdCl2(CH3CN)2; a0 ¼ 83.3 minutes, a1 ¼ 1.17, and a2 ¼ 0.033 via the method of steepest descent.

11.11

SUMMARY

Nonlinear stress relaxation in styrene – butadiene – styrene triblock copolymers and their complexes with palladium chloride has been analyzed in the plastic flow regime via time – strain separability, the resistance to conformational reorganization V in the polymer, and the effect of strain on viscoelastic time constants. Pseudosquare-planar dichlorobis(acetonitrile) –palladium(II) forms p-olefin complexes with, and catalyzes the dimerization of, alkene groups in the main chain and the side group of Kraton’s butadiene midblock. Consequently, the high-strain mechanical properties of KratonTM are modified significantly by PdCl2(CH3CN)2. Transient relaxation moduli can be modeled by a three-parameter biexponential decay with two viscoelastic time constants. The longer relaxation time for KratonTM increases at higher strain, and in the presence of 4 mol % palladium chloride. A phenomenological model is proposed to describe the effect of strain on relaxation times. This model is consistent with the fact that greater length scales are required for cooperative segmental reorganization at larger strain. The resistance V to conformational reorganization during stress relaxation is estimated via integration of the normalized relaxation modulus versus time data. This resistance, denoted by V, increases at

References

483

higher jump strain, which is consistent with the fact that knots, entanglements, and low crosslink densities have a stronger influence on mechanical properties at larger strain.

REFERENCES ANDRADY AL, LLORENTE MA, MARK JE. Journal of Chemical Physics 72(4):2282 (1980). AOKI Y, HATANO A, TANAKA T, WATANABE H. Macromolecules 34(9):3100–3107 (2001). ASKADSKII AA, VALETSKII MP. Mechanics of Composite Materials 26(3):324 (1991). BELFIORE LA, BOSSE´ F, DAS PK. Polymer International 26(2):165 (1995). BELFIORE LA, DAS PK, BOSSE´ F. Journal of Polymer Science; Polymer Physics Edition 34(16):2675 (1996). BOSSE´ F, DAS PK, BELFIORE LA. Macromolecules 28:6993 (1995). BOSSE´ F, DAS PK, BELFIORE LA. Polymer Gels and Networks 5(5):387–413 (1997). BRUELLER OS. Proceedings of the 13th International Congress on Rheology Binding DM, editor. British Society of Rheology, Glasgow, UK, 2000, pp. 82– 84. CHEN CY, WU SM, CHEN ZR, HUANG TJ, HUA CC. Journal of Polymer Science, Polymer Physics Edition 41:1281–1293 (2003). ERMAN B, FLORY PJ. Journal of Chemical Physics 68:5363 (1978). KWON Y, CHO K. Journal of Rheology 45(6):1441– 1452 (2001). LE MEINS JF, MOLDENAERS P, MEWIS J. Industrial & Engineering Chemistry Research 41(25):6297– 6304 (2002). MATSUOKA S. Relaxation Phenomena in Polymers. Hanser, Munich, and Oxford University Press, New York, 1992, pp. 42–79. MATSUOKA S, BAIR HE, BEARDER SS, KERN HE, RYAN JT. Polymer Engineering and Science 18(14): 1073– 1080 (1978). MCKENNA GB, ZAPAS LJ. Polymer Engineering and Science 26:725 (1986). MERRIMAN HG, CARUTHERS JM. Journal of Polymer Science; Polymer Physics Edition 19(7):1055–1071 (1981). NGAI KL, RENDELL RW, RAJAGOPAL AK, TEITLER S. Dynamic Aspects of Structural Change in Liquids and Glasses, Annals of the New York Academy of Sciences, Vol. 484, Angell CA and Goldstein M, editors. New York Academy of Sciences, New York, 1986, pp. 150– 184. OSAKI K, WATANABE H, INOUE T. Macromolecules 29(10):3611–3614 (1996). PLONKA A. Polymer matrix relaxation and kinetics of trapped species. Radiation Physical Chemistry 37(4):555–557 (1991). QUESLEL JP, MARK JE. Comprehensive Polymer Science: The Synthesis, Characterization, Reactions & Applications of Polymers, Vol. 2, Booth C and Price C, editors. Pergamon Press, Elmsford, NY, 1989. SWEENEY J, COLLINS TLD, COATES PD, DUCKETT RA. Journal of Applied Polymer Science 72(4):563– 575 (1999). TAKAHASHI T, WATANABE J, MINAGAWA K, TAKIMOTO JI, IWAKURA K, KOYAMA K. Rheologica Acta 34(2):163–171 (1995). TANAKA F, EDWARDS SF. Journal of Non-Newtonian Fluid Mechanics 43(2-3):289– 309 (1992). WARD IM. Mechanical Properties of Solid Polymers, 2nd edition. Wiley-Interscience, Hoboken, NY, 1983, pp. 201– 204. WARD IM, HADLEY DW. An Introduction to the Mechanical Properties of Solid Polymers, Wiley, Hoboken, NJ, 1995, Chap. 10, pp. 194– 195, 198– 202. WATANABE H, SATO T, OSAKI K, YAO ML. Macromolecules 29(11):3890–3897 (1996). WILLIAMS G, WATTS D. Transactions of the Faraday Society 66:80 (1970). YEE AF, BANKERT RJ, NGAI KL, RENDELL RW. Journal of Polymer Science; Polymer Physics Edition 26:2463 (1988). YOSHIOKA S. Journal of the Society of Rheology of Japan 30(5):253–257 (2002).

Chapter

12

Kinetic Analysis of Molecular Weight Distribution Functions in Linear Polymers Angels descend violent and without quarter bestowing savage gifts. —Michael Berardi

Continuous and discrete molecular weight distribution functions are analyzed from a chemical kinetics viewpoint for condensation, free radical, and anionic polymerization mechanisms. Comparisons between these distributions and the output curves from gel permeation chromatography are mentioned. Moments-generating functions are introduced to evaluate number-average and weight-average degrees of polymerization. Examples of kinetic mechanisms for transition-metal-catalyzed hydrogenation and chemical-crosslinking reactions in industrially important polymers are discussed in Appendix B and Appendix C, respectively.

12.1 ALL CHAINS DO NOT CONTAIN THE SAME NUMBER OF REPEAT UNITS It is relatively accurate to claim that the molecular weight of every molecule in a pure small-molecule system is essentially the same, if one disregards the low natural abundance of common isotopes, such as carbon-13, in the chemical structure of low-molar-mass species. This claim is definitely inaccurate when monomers combine by any polymerization scheme to generate high-molecular-weight chains whose physical properties represent the focal point for discussion throughout this book. As indicated in the title of this chapter, polymers exhibit a distribution of chain lengths that depend on the kinetic mechanism by which these molecules are generated via chemical reactions. Statistical analysis is useful to describe four different average molecular weights (i.e., number-average, weight-average, viscosity-average, Physical Properties of Macromolecules. By Laurence A. Belfiore Copyright # 2010 John Wiley & Sons, Inc.

485

486

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

z-average), and experimental measurements are sensitive to specific averages. It is important to realize that macroscopic physical properties of polymers are sensitive not only to the average molar mass of the chains, but the actual distribution of chain lengths, particularly the high-molecular-weight tail, has a significant effect that should not be overlooked.

12.2 THE “MOST PROBABLE DISTRIBUTION” FOR POLYCONDENSATION REACTIONS: STATISTICAL CONSIDERATIONS When monomers such as a,v-hydroxycarboxylic acids react to yield ester linkages and small-molecule by-products like water, 2 HOZRZCOOH ) HOZRZCOOZRZCOOH þ H2 O the chain molecules (i.e., polyesters) that are produced after sufficient polymerization time reveal a distribution of molecular weights via chromatographic analysis. In other words, all chains do not contain the same number of repeat units. Kinetic analysis suggests that the mole (or number) fraction of molecules that contain x repeat units, hereafter referred to as x-mers, is described by the “most probable distribution.” This distribution function can be obtained by the following Lagrange multiplier optimization subjected to two constraints, as illustrated previously for the effect of free volume on diffusion coefficients in amorphous polymers (see Section 2.3) and the Langevin distribution for the orientation of freely jointed chain segments with respect to the end-to-end chain vector (see Section 7.7.1). Let NMonomer represent the total number of monomers that are present initially at time t ¼ 0, and NTotal be the total number of molecules of all sizes that exist in the reactive mixture after polymerization occurs for time t. If Nx-mer represents the total number of molecules that contain x repeat units at time t and fNx-merg describes the complete set of x-mers for all values of x, then the two constraints at any time within the batch polymerization reactor are given by 1 X x¼1

Nx-mer ¼ NTotal ;

1 X

xNx-mer ¼ NMonomer

x¼1

The total number of monomers present initially is constant and, after reaction occurs for time t, these monomers are distributed among all of the molecules in the mixture to maximize the multiplicity of states for a fixed instantaneous total number of molecules. Hence, at any time t during the polycondensation reaction, one seeks expressions for the complete set of x-mers, fNx-merg to maximize the number of distinguishable ways V that NMonomer can be distributed among all of the molecules for a fixed value of NTotal. Of course, the total number of esters and polyesters present in the reactive mixture decreases as the polymerization proceeds and chains of higher molecular weight are produced but, at any given time t, the optimization problem seeks the maximum number of distinguishable ways of distributing NMonomer among the set fNx-merg subject to a constant value of NTotal. It is important to emphasize that small-molecule

12.2 The “Most Probable Distribution” for Polycondensation Reactions

487

by-products of the polycondensation reaction are overlooked when the total number of molecules in the reactive mixture is addressed from statistical considerations. The multiplicity of states is expressed as V¼

NTotal ! 1 Q Nx-mer !

x¼1

One actually forms the Lagrangian L (in honor of the great French mathematician Joseph Louis Lagrange, 1736 – 1813, who was born in Torino, Italy under the name Giuseppe Luigi Lagrangia) in terms of the natural logarithm of V and maximizes ln V instead of V. Both constraints are included in L with Lagrange multipliers g and d as follows: ( ) 1 Y ln V ¼ ln(NTotal !)  ln Nx-mer ! x¼1

L ¼ ln(NTotal !)  ( þd

1 X

(

ln{Nx-mer !} þ g

x¼1 1 X

1 X

) xNx-mer  NMonomer

x¼1

)

Nx-mer  NTotal

x¼1

As illustrated on more than one occasion in this treatise, Sterling’s approximation for the factorial of a large argument is employed prior to implementing the optimization algorithm, where the gamma function is represented by G(n): pffiffiffiffiffiffiffiffiffi G(n þ 1) ¼ nG(n) ¼ n! ¼ nn 2p n exp(n)   1 1 139 571 þ      1þ 12n 288n2 51840n3 2488320n4   ln n!  12 ln(2p) þ n þ 12 ln(n)  n Now, Sterling’s approximation is used to simplify the Lagrangian: ln(NTotal !) 

1 X

  ln{Nx-mer !} ¼ 12 ln(2p) þ NTotal þ 12 ln(NTotal )  NTotal

x¼1



1  X

1 2 ln(2p) þ



  Nx-mer þ 12 ln(Nx-mer )  Nx-mer

x¼1



1  X

 1

L ¼ 12 ln(2p) þ NTotal þ 2 ln(NTotal )  ( þg

1 X x¼1

)

   þ Nx-mer þ 12 ln(Nx-mer )

x¼1

(

xNx-mer  NMonomer þ d

1 2 ln(2p)

1 X x¼1

) Nx-mer  NTotal

488

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

The complete set of x-mers fNx-merg, which summarizes the molecular weight distribution that corresponds to maximum entropy via Boltzmann’s equation (i.e., S ¼ k ln V), is obtained by maximizing L via the “method of Lagrange multipliers,” as follows:

@L @Nx-mer





¼ 1þ NTotal ,g,d, Nr-mer [r=x]

1 2Nx-mer

 þ ln(Nx-mer )

þ g x þ d ¼ 0; x ¼ 1, 2, 3, ...

1 X @L ¼ xNx-mer  NMonomer ¼ 0 @ g NTotal ,d, Nx mer [all x] x¼1

1 X @L ¼ Nx mer  NTotal ¼ 0 @ d NTotal ,g, Nx mer [all x] x¼1 In the first extremum condition, it is not rigorously possible to vary the number of molecules of one type of x-mer while the number of all other types of molecules of different chain length and the total number of molecules NTotal remain constant, because the constraint associated with Lagrange multiplier d is not satisfied. However, for very large numbers of molecules, NTotal is approximately constant when the number of molecules of one type of x-mer varies slightly. Furthermore, since the number of molecules of each type of x-mer, Nx-mer, is quite large also, the first extremum condition given above simplifies to the following result for discrete values of x at any given polymerization time t in the batch reactor: Nx-mer  exp(g x þ d  1) Lagrange multipliers g and d are evaluated by using this expression for Nx-mer in the two constraints. For example, the total number of x-mers for all values of x must add to NTotal: 1 X

Nx ¼ exp(d  1)

x¼1

1 X

exp(g x) ¼ exp(d  1){eg þ e2g þ e3g þ }

x¼1

g

¼

e exp(d  1) ¼ NTotal 1  eg

Evaluation of the power series for exp(g x) ¼ p x was obtained by (i) multiplying the series by exp(g), (ii) subtracting the modified series from the original series, both of which are very similar except for the leading term in the original series, and (iii) solving the resulting algebraic equation, which is valid when exp(g) is greater than zero but less than unity. Analysis of the other constraint reveals that exp(g) is, indeed, within this range (i.e., 0 , exp(g) , 1) because it relates NTotal to NMonomer via the fractional conversion p of monomer to polymer. Differentiate the previous power series for exp(g x) with respect to g, which allows one to evaluate the power series for fx exp(g x)g. Then, consider the constraint associated with

12.2 The “Most Probable Distribution” for Polycondensation Reactions

489

Lagrange multiplier g : 1 X

xNx-mer ¼ exp(d  1)

x¼1

1 X

x exp(gx) ¼ exp(d 1){1eg þ 2e2g þ 3e3g þ4e4g þ }

x¼1

( ) 1 d X eg NTotal ¼ exp(d  1) exp(g x) ¼ exp(d  1) ¼ ¼ NMonomer 2 g d g x¼1 1  eg (1 e ) NTotal ¼ NMonomer (1  eg ) ¼ NMonomer (1  p); p ¼ eg

Now, consider all of these results and construct the mole fraction of x-mers, defined as the ratio of Nx-mer to NTotal. One obtains a normalized distribution function Px that is valid for discrete values of x, where x  1: Nx-mer ¼ eg x exp(d  1) ¼ eg x NTotal Px ¼

1  eg ¼ NTotal eg (x1) {1 eg } eg

Nx-mer Nx mer ¼ 1 ¼ px1 (1 p) NTotal X Nx-mer x¼1

P It is rather straightforward to demonstrate that the infinite power series, 1 x¼1 Px , sums to unity via (i) multiplication of the original series by p, (ii) subtraction of the modified series from the original series, both of which only differ by the leading term of the original series, and (iii) solution of the algebraic equation that is valid when 0 , p , 1. The most probable distribution invariably reveals that the mole fraction of xmers decreases exponentially at larger values of x, with an extended “high-molecular-weight tail” that is more significant at larger fractional conversion p. For example, Px corresponding to p ¼ 0.99 is essentially “flat” from x ¼ 1 to x ¼ 60 when this distribution is compared with Px at p ¼ 0.90 on the same set of axes, even though the former distribution actually varies from 1% at x ¼ 1 to 0.5% at x ¼ 60. Now, let’s analyze the mass fraction of chains that contain x repeat units when the fractional conversion of monomers is p. This is also known as the weight fraction of x-mers, Wx, and it is constructed via (i) multiplication of the number of molecules of x-mers, Nxmer, by the molar mass of these molecules (i.e., x MWrepeat, where MWrepeat represents the repeat unit molecular weight) and (ii) division by the total mass of the mixture, which remains constant throughout the polymerization reaction, given by MWrepeatNMonomer. Small-molecule by-products are excluded from the calculations below. One obtains the normalized mass fraction distribution of x-mers that corresponds to fractional conversion p as follows: Wx ¼

x MWrepeat Nx-mer xNx-mer xNx mer xNTotal px1 (1p) ¼ ¼ 1 ¼ ¼ xpx1 (1p)2 MWrepeat NMonomer NMonomer X NMonomer xNx-mer x¼1

This mass fraction distribution function that corresponds to maximum entropy is analogous to the output curve from gel permeation chromatography (GPC), which yields

490

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

the “molecular weight distribution” for polycondensation reactions. Unlike the mole fraction distribution function Px, the mass fraction of x-mers at high conversion exhibits a maximum at one repeat unit below the number-average degree of polymerization kxnl, which is synonymous with the first moment of Px, (i.e., kxnl ¼ 1/(12p)) as discussed in Section 12.5. If differentiation of discrete distribution functions is allowed, then quantitative analysis of the maximum in Wx is summarized below: dWx ¼ (1p)2 px1 {1þx ln p} ¼ 0 dx 1 1 1 p 1 ¼  ¼ ¼ 1 ¼ kxn l1 xMaximum ¼ ln( p1 ) ln[1þ( p1 1)] p1 1 1p 1p

12.3 DISCRETE VERSUS CONTINUOUS DISTRIBUTIONS FOR CONDENSATION POLYMERIZATION Discrete distribution functions for Px and Wx, with x  1 can be expressed in the continuous limit via Taylor series expansion of exp(2b), where b represents the fraction of unreacted monomers (i.e., b ¼ 12p). Hence, at high fractional conversion (i.e., 1 2 b  1), exp(b ) ¼ 1  b þ

b2 b3 b4  þ    1  b ¼ p 2! 3! 4!

An approximate expression for the continuous mole fraction distribution of x-mers, P(x), is constructed by analogy with Px:

b exp{bx0 } Px ¼ (1  p)px1 ) P(x)  b exp{b(x  1)} ) 0 1x1

x ¼x1 0x0 1

Normalization of P(x) is demonstrated rather easily by (i) integrating b expf2b(x 2 1)g over a range of repeat units from 1 to 1, or (ii) replacing x21 with x0 and integrating b exp(2bx0 ) from 0 to 1. At 90% conversion of monomer (i.e., p ¼ 0.90), there are small differences between Px and P(x) when x ¼ 15 and x21 is replaced by x (not x0 ) in the continuous distribution but not in the discrete distribution. Two equivalent approaches to obtain the continuous mass fraction distribution function are illustrated below. First, (i) begin with the discrete distribution Wx, (ii) let b ¼ 1 2 p, (iii) replace p by exp(2b) at high conversion, and (iv) replace x21 by x for large chains. One obtains Wx ¼ xpx1 (1  p) 2 ) W(x)  b 2 x exp{b (x  1)} ) b 2 x exp{bx} 1x1 x-1x 0x1

The second approach (i) begins with the discrete definition of Wx in terms of Nx-mer, (ii) divides Nx-mer in the numerator and denominator of this definition for Wx by NTotal to

12.4 The Degree of Polymerization for Polycondensation Reactions

491

rewrite Wx in terms of Px, (iii) replaces Px by P(x), and (iv) transforms the summation to an integral, as illustrated below: Nx mer x xNx-mer xPx N Wx ¼ 1 ¼ 1 Total ¼ 1 X X Nx-mer X xNx-mer x xPx N x¼1

x¼1

Total

x¼1

bx exp(bx) ) W(x) ¼ ð 1 ¼ ð1 ¼ b2 x exp(bx) 0x1 xP(x) dx b x exp(bx) dx xP(x)

0

0

The integral that appears in the denominator of W(x) represents the first moment of the most probable distribution P(x), given by 1/b. This corresponds to the numberaverage degree of polymerization kxnl for polycondensation reactions. Analogous to the discrete distributions, it should be obvious that the continuous mole fraction distribution P(x) decreases exponentially when the number x of repeat units per chain increases. Furthermore, as p approaches unity at higher conversion of monomer, b decreases and the exponential decay is retarded at larger values of x due to the presence of a significant “high-molecular-weight tail.” The continuous mass fraction distribution function W(x) also mimics the output curve from gel permeation chromatography, exhibiting a maximum at dW ¼ b2 exp(bx){1  bx} ¼ 0 dx 1 xMaximum ¼ ¼ kxn l b This result is essentially equivalent to the calculation of xMaximum at the end of the previous section for the discrete distribution function Wx. Differences between W(x) and Wx, including the number of repeat units per chain where they exhibit maxima, vanish at high conversion of monomer to polymer, where p approaches unity and b approaches zero.

12.4 THE DEGREE OF POLYMERIZATION FOR POLYCONDENSATION REACTIONS Consider the following question that generated 25 responses internationally from members of the American Chemical Society’s Division of Polymer Chemistry discussion list. The responses have been edited for clarity and to insure anonymity. Question: Two polymer chemistry textbooks do not agree on the relationship between degree of polymerization (DP) and the number of repeat units in polymers produced by polycondensation reactions. One states that the reaction of monomer AA with monomer BB yields a degree of polymerization that is two-fold larger than the number of repeat units. The other textbook indicates that DP and the number of repeat units are equal. Which one is correct?

492

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

RESPONSE #1: DP is synonymous with the number of repeat units. Sometimes, the number of repeat units is more complex than analyzing the structure of simple polymers and identifying the repeat unit. For most synthetic polymers, it is sufficient to determine the number of times that the repeat unit appears. In block copolymers and other more complex polymers, the number of repeat units might not be trivial, so the author should illustrate the chemical structure of the repeat unit. For vinyl polymers such as polystyrene, DP corresponds to the average number of styrene-derived units in the polymer. For a condensation polymer such as poly(ethylene terephthalate), DP is the average number of ethylene(E) – terephthalate(T) units in the chain. For ethylene –terephthalate oligomeric materials, one might describe ETETE as a low-molecular-weight oligomer that contains 2.5 repeat units. The term average degree of polymerization is used for actual polymeric materials that consist of a mixture of chains of varying length. RESPONSE #2: The number-average DP is defined as the average number of monomer units in one polymer chain. This definition works, in general, for any kind of polymerization, including branched polymers where the chemical structure of the repeat unit is difficult to quantify. When difunctional AA is polymerized with difunctional BB, DP corresponds to twice the number of repeat units because the repeat unit contains two monomers. Flory [1953] defines DP as the number of structural units per polymer molecule, and there are two structural units per repeat unit in the AA þ BB system. RESPONSE#3: Many textbooks are quite obscure on this point. The definition of DP is the same for step-growth (i.e., condensation) and chain-growth (i.e., free radical) polymers. The DP of a given macromolecule is defined as the number of monomer units in the chain. The number-average DP is given by the ratio of the total number of monomer units to the total number of molecules. In step-growth polymerization, the total number of monomer units corresponds to the total number of monomer molecules (of any type) initially at time t ¼ 0. Unreacted monomers have a DP ¼ 1. The number of repeat units per macromolecule is 50% of DP for a stoichiometric AA þ BB polymerization only. If reactive functional groups A and B produce C, one additional B group is present, and the fractional conversion of A is 100%. Then, xAZRZA þ yBZR0 ZB ) BZ(R0 ZCZRZCZ)n ZR0 ZB From stoichiometric considerations, x ¼ n and y ¼ n þ 1. The number of repeat units per chain ¼ n, but the number-average DP ¼ x þ y ¼ 2n þ 1. When polymers are produced by free radical mechanisms, the total number of monomer units is identified by the number of reacted monomer molecules, because chain-growth polymers are generally obtained by precipitation, such that they are phase-separated from their monomer. RESPONSE #4: The confusion lies in the definition of the repeat unit. Neither AA nor BB is the monomer, so it might seem reasonable to use the average molar mass of AA and BB. The proper repeat unit is (AAZBB), which corresponds to the structural element that, when connected together repeatedly, generates the correct chemical chain structure. RESPONSE #5: In all three cases, (A)n, (AB)n, and (ABC)n, DP ¼ n, the number of repeat units. RESPONSE #6: The answer depends on the definition of the repeat unit. If AZB is one unit, then the degree of polymerization for (AB)n is twice the number of repeat units. RESPONSE #7: The most reasonable definition is number of repeat units ¼ DP/2. The reaction product of AA and BB is a dimer with DP ¼ 2, even though there is only one

12.4 The Degree of Polymerization for Polycondensation Reactions

493

repeat unit. In principle, the polymerization of AA þ BB yields an alternating copolymer. The repeat unit contains two monomers, but the degree of polymerization is based on all of the monomer units. The definition number of repeat units ¼ DP/2 is consistent with DP ¼ 1/(12p). RESPONSE #8: When AA monomer reacts with BB, each repeat unit corresponds to an AABB sequence, but the degree of polymerization is given by the average number of monomers in a single chain. Therefore, the number of repeat units is 50% of the degree of polymerization. For an AB monomer, the number of repeat units and the degree of polymerization are identical. For a random terpolymer of AA, A0 A0 , and BB, the repeat unit is not well defined. In this case, there are segments of average composition. The degree of polymerization is defined by the average number of monomers in a single polymer chain. In olefin polymerization, a more confusing situation arises. Consider the free radical polymerization of styrene, which yields an atactic polymer. When the stereochemistry is not regular, the concept of the repeat unit is extremely vague, analogous to the terpolymer example described above. For syndiotactic polystyrene, the repeat unit and degree of polymerization are the same. For polystyrene with considerably high isotacticity, one must consider the stereochemistry of each chiral center, even though one typically equates the number of repeat units and the degree of polymerization. This later description that overlooks stereochemistry is satisfactory if the gross chemical composition is the only factor of interest. This example becomes more complex if one accounts for head-to-tail and head-to-head attachment of monomer to the growing free radical chain. In the AA þ BB system, if one or both of the monomers possesses an asymmetric center, then the polymer will be atactic unless some external control is influenced by a catalyst. If the monomers are chiral, then one should consider the enantiomeric AA or BB to be different monomers. RESPONSE #9: You point to an interesting problem encountered when teaching polymerization kinetics. The problem arises because of the difference between polymers produced from AABB and AB monomers. The repeat unit for AABB involves both AA and BB monomers, whereas only one unit is required for polymers produced from AB monomers. The repeat unit and its molecular weight are related to the average DP (i.e., or kxnl), and these quantities are used to calculate Mn, Mw, and DP of the synthesized polymer as a function of the conversion p of monomer to polymer. For AB polymers, the repeat unit molecular weight is based on the molar mass of the AB monomer minus the small-molecule by-product. For AABB polymers, two chemical units are required for each repeat unit. Furthermore, the molecular weight of monomer AA is not the same as that of monomer BB. If there is a stoichiometric imbalance of A and B functional groups, then one should calculate the molar mass of one repeat unit from an average of the molecular weights of AA and BB. In other words, the average repeat unit molecular weight is given by (MWAA þ MWBB 2 MWby-products)/2. Hence, one AABB repeat unit corresponds to two kinetic repeat units, where each unit is written as 12AABB. The written repeat unit must include the complete chemical structures of monomers AA and BB as one unit, but this written repeat unit contains two equisized kinetic repeat units for condensation polymers produced from AA and BB monomers. RESPONSE #10: The degree of polymerization corresponds to the value of n in the condensed chemical structure of a single polymer chain, [ZAAZBBZ]n, based on the second approach as stated in the question. Confusion might be due to the definition of the repeat unit. For poly(ethylene terephthalate), if one considers that the terephthalate and ethylene segments are each individual structural units such that the polymer unit contains two repeat

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Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

units, then DP corresponds to 50% of the number of repeat units, but this is the minority viewpoint. RESPONSE #11: Refer to the following website for information on the Nomenclature for Macromolecules and specifically on your question in which DP is an abbreviation for degree of polymerization, that corresponds to the number of monomeric units in a polymer chain. Furthermore, there is distinction between monomer and structural repeat unit. http://www. iupac.org/publications/books. RESPONSE #12: Flory [1953] clarified this concept. The degree of polymerization corresponds to the number of repeat units, where a repeat unit is defined as the segment which generates the chain when repeated. For polymers produced from AA and BB monomers, the repeat unit is (AAZBB), which contains two structural units, or two residues. Discussions of the degree of polymerization for hompolymers produced by free radical mechanisms use repeat unit, structural unit, and residue interchangeably. Confusion occurs when this methodology is applied to condensation polymers that contain two residues per repeat unit. RESPONSE #13: When polymers are produced by free radical reactions, the same molecular unit (i.e., mer unit) repeats continuously, so it is rather straightforward that DP corresponds to the number of repeat units. When polymers are produced by condensation reactions that involve two different molecules, one could define the number of repeat units ¼ 50% of DP. However, DP ¼ number of repeat units provides a better representation because the mer unit is (AABB). RESPONSE #14: The IUPAC (International Union of Pure and Applied Chemistry) definition of DP is the number of monomeric units in a macromolecule (see definition 1.13 in the document, Glossary of Basic Terms at the following website): http://www.iupac.org/publications/pac/ 1996/pdf/68122287.pdf. In nylon-66, one monomeric unit is hexamethylene diamine, and the second monomeric unit is adipic acid. Hence, if one represents nylon-66 by the following chemical or structural repeat unit, [NHZCOZ(CH2)4ZCOZNHZ(CH2)6], which is the correct representation despite the apparent splitting of an amino linkage in hexamethylene diamine, then two monomeric units are required, so one should identify the number of repeat units as 50% of DP. RESPONSE #15: DP corresponds to the number of repeating units. The reason for the apparent discrepancy occurs when polymers are produced by condensation reactions that require two difunctional monomers (i.e., AA þ BB). For polymers produced from one bifunctional monomer (i.e., AB), DP is calculated in the same way as that for the polymerization of vinyl monomers because condensation reactions that involve one AB monomer yield a repeat unit with one connecting linkage, (AB)n. When polymers are produced from two difunctional monomers, the repeat unit contains two AB linkages, (AABB)n, rather than the one linkage that occurs with AB monomers. This difference must be considered when a stoichiometric imbalance or a monofunctional reactant is employed to control the degree of polymerization. For a complete explanation, Flory [1953] discusses this topic in his classic book, Principles of Polymer Chemistry. RESPONSE #16: The difficulty in answering this question might be due to a common misunderstanding about the definition of a repeat unit. A repeat unit contains a group of interconnected atoms in a molecular sequence that repeats itself completely to form a polymer. For example, poly(ethylene terephthalate), PET, and nylon-66 each contain two different units (i.e., AA and BB). Hence, the true repeat unit contains one AA unit and one BB unit (e.g., one molecule of terephthalic acid and one molecule of ethylene glycol in the case of PET). Based on this

12.4 The Degree of Polymerization for Polycondensation Reactions

495

definition, the relation between the number of true repeat units and DP is quite logical. Terephthalic acid is not the monomer for PET, it is only half of the monomer. RESPONSE #17: The answer depends on the definition of the repeat unit. Textbooks with a rigorous statistical treatment of polymerization kinetics define DP for macromolecules generated from AA and BB monomers as the number of (AB) linkages in the chain. Thus, AAZBBZAAZBBZAA has a DP of 4. This definition originated with the work of Flory [1953], in Principles of Polymer Chemistry. However more general textbooks that focus on the organic chemistry rather than the kinetics define the repeat unit as (AAZBB). Thus, AAZBBZAAZBB has a DP of 2. It is important to consult both DP discussions and determine if there is a difference in the definition of the repeat unit. RESPONSE #18: DP is defined as the number of monomer units, which is twice the number of repeat units for condensation polymers produced from two different monomers, AA þ BB. This convention is based on the classic textbook, Principles of Polymerization, by Odian [2004]. RESPONSE #19: For this interesting example, it is important to remember that two different monomers are required to produce the desired polymer, but these monomers combine to form one repeat unit. For PET, a repeat unit consists of one ethylene –terephthalate segment, derived from ethylene glycol and terephthalic acid. Hence, it is correct to equate DP to the number of repeat units, realizing that one repeat unit is formed from two different monomers. RESPONSE #20: The former statement is correct. When condensation polymers are produced from two difunctional monomers, such as AA þ BB, each repeat unit contains two structural units, one derived from monomer AA and the other derived from monomer BB. Hence, DP ¼ number of structural units in the average polymer chain, which translates to DP ¼ twice the number of repeat units. RESPONSE #21: DP, or kxnl, corresponds to the number of monomers or monomer residues in the chain. If chains are produced from condensation polymerization of AA þ BB (e.g., reactions of diols and diacids), then the repeat unit contains residues from two monomers minus a smallmolecule by-product. Hence, calculation of the polymer’s molecular weight requires use of the following formula: number-average molecular weight ¼ M0kxnl, where M0 represents one-half of the repeat unit molecular weight and kxnl is the number-average degree of polymerization or the average number of monomer residues in a single chain. The factor of 12 is necessary because the repeat unit is formed from two different monomers, so one must divide the repeat unit molecular weight by 2 to obtain an average molecular weight based on two residues minus the small-molecule by-product from the condensation reaction. Hence, the polymer’s molecular weight is obtained from the product of kxnl and the average molecular weight of a monomer residue. The procedure is slightly different if chains are produced from the condensation polymerization of AB monomers, such as a,v-hydroxycarboxylic acids (i.e., HOZRZCOOH), or free radical mechanisms. When the repeat unit contains a single monomer, one simply multiplies kxnl by the molecular weight of the repeat unit. RESPONSE #22: Both answers are correct, if one considers the following two definitions of DP. The first definition is DP ¼ number of repeat units in a single chain, which can be calculated via division of the polymer’s molecular weight by the repeat unit molecular weight. This definition is irrespective of the type of polymer. However, if one is interested in calculating DP for the reaction mixture with respect to monomer consumption, such that DP is proportional to the number of monomers consumed, then one must use one-half of the repeat unit molecular weight when condensation polymers are produced from AA þ BB because two monomers are

496

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

consumed per repeat unit. For AB monomers, one employs the entire molecular weight of the repeat unit because two monomers yield a dimer. RESPONSE #23: This question confuses many people. When condensation polymers are generated from AA þ BB, the total number of molecules present initially, Ninitial, the total number of molecules present after reaction occurs for time t, N(t), and DP or kxnl are defined by the Carouthers equation: kxnl ¼ 1/(1 2 p), where p is the extent of reaction, which is equal to the fraction of functional groups that have reacted, p ¼ [Ninitial – N(t)]/Ninitial. When diols react with diacids, for example, Ninitial includes all of the molecules at time t ¼ 0. The number-average molecular weight is given by Mn ¼ M0kxnl, where M0 is the average molar mass of the diol and the diacid. The concept is less ambiguous for a monomer like p-hydroxybenzoic acid in which the chemical structure of the repeat unit is almost identical to that of the monomer. For the AA þ BB system, the repeat unit contains two structural units. For the example of p-hydroxybenzoic acid, kxnl ¼ 2 implies that a dimer forms from two p-hydroxybenzoic units, minus water. For PET or nylon-66, kxnl ¼ 2 implies that the product contains one AA unit and one BB unit (minus water). It is really a question of definitions. If one does not distinguish between monomers and repeat units, then the answer could be ambiguous. RESPONSE #24: Step-growth, or condensation, polymers are produced from bifunctional reagents (AB) or two difunctional reagents (AA þ BB). The mer unit, building block, or repeat unit in either case is (AB). The difference between the statements in these two textbooks is a consequence of the definition of a repeat unit. If the repeat unit is (AB), then both answers are consistent. RESPONSE #25: Both definitions are correct, depending on the definition of DP. Consider the following polymer (AABB)n that contains n repeat units. Based on the number of molecules required to produce this polymer, the answer is 2n (i.e., n molecules of AA and n molecules of BB). This is important when one considers the relation between DP and the extent of reaction p. However, for most practical homogeneous polymers, DP represents the number of repeat units. Some ambiguity arises when the discussion focuses on copolymers that contain AA, BB, and AB monomers. Since copolymer structure, in general, does not have a regular repeat unit, it is more reasonable to consider the initial number of molecules that were consumed to generate the chains. It is important to use caution when defining DP for each case, based either on the number of repeat units or the number of reacting molecules.

12.5 MOMENTS-GENERATING FUNCTIONS FOR DISCRETE DISTRIBUTIONS VIA z-TRANSFORMS As mentioned in Section 12.3, the first moment of the most probable distribution represents the number-average degree of polymerization, defined by 1 X

kxn l ¼

xNx-mer

x¼1 1 X x¼1

Nx-mer

X 1 1 X Nx-mer x xPx NTotal Q1 x¼1 x¼1 ¼ 1 ¼ ¼ 1 X X Nx-mer

Q0 Px NTotal x¼1 x¼1

12.5 Moments-Generating Functions for Discrete Distributions via z-Transforms

497

where Qn represents the nth moment of the discrete distribution function Px, as defined below. Statistical analysis requires that the average of a distribution function must be constructed in terms of the ratio of the first moment to the zeroth moment. The zeroth moment is unity, by definition, for a normalized distribution function. When mass fractions Wx represent weighting factors in construction of the average number x of monomer units per chain, instead of mole fractions Px, one obtains the weight-average degree of polymerization, defined by (mass fraction distribution Wx is introduced in Sections 12.2 and 12.3) 1 X

kxw l ¼

1 X

xWx ¼

x¼1

x2 Px

x¼1 1 X

¼ xPx

Q2 Q1

x¼1

where higher-molecular-weight chains contribute more to the average value than smaller molecules. If the number-average degree of polymerization is considered to be the democratic average in which all chains contribute equally, regardless of their size, then the weight-average degree of polymerization is analogous to the electoral vote in U.S. presidential elections where winning votes in more populous states, like California and Texas, contribute more than winning votes in sparsely populated states, like Nevada and Wyoming. The “z-transform” of Px, denoted by Fz, represents a new function from which all moments of the distribution can be calculated. The moments-generating function Fz is defined by

Fz ¼

1 X

z x Px ¼ (1  p)

x¼1

1 X

z x px1 ¼ z(1  p)

x¼1

1 X

( pz)x1

x¼1

¼ z(1  p){1 þ pz þ ( pz)2 þ ( pz)3 þ ( pz)4 þ   } ¼

z(1  p) 1  pz

where the methodology to evaluate the power series for ( pz)x21 was described twice in Section 12.2. Now, the nth moment Qn of the discrete distribution function Px is written in terms of Px and Fz:

 d n Fz dz z¼1 x¼1 ( )   1 1 X X z(1  p) x Q0 ¼ Px ¼ Fz (z ¼ 1) ¼ z Px ¼ ¼1 1  pz z¼1 x¼1 x¼1 z¼1 ( )     1 1 X X dFz z(1  p) 1 x Q1 ¼ xPx ¼ z ¼ xz Px ¼ ¼ 2 dz z¼1 (1  pz) z¼1 1  p x¼1 x¼1 Qn ¼

1 X

x n Px ¼



z

z¼1

498

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

( ) 

 1 X d dFz 2 x z Q2 ¼ x Px ¼ z ¼ x z Px dz dz z¼1 x¼1 x¼1 z¼1   z(1 þ pz)(1  p) 1þp ¼ ¼ (1  pz)3 (1  p)2 z¼1 1 X

2

Explicit evaluation of the zeroth, first, and second moments of Px, illustrated above, is unique to polycondensation reactions. In terms of the average degrees of polymerization, or weighted averages of the number x of repeat units per chain, the number-average degree of polymerization is kxnl ¼ Q1/Q0 ¼ 1/(1 2 p), and the weight-average degree of polymerization is kxwl ¼ Q2/Q1 ¼ (1 þ p)/(1 2 p). One defines the polydispersity of the polymerization reaction as the ratio of the weightaverage to the number-average degrees of polymerization, or kxwl/kxnl ¼ Q0Q2/ (Q1)2. For polycondensation reactions, the polydispersity is given by 1 þ p, which asymptotically approaches 2 at high fractional conversions of monomer to polymer. Since polydispersity is proportional to the second moment of the distribution function Q2, and the latter is analogous to standard deviation or breadth of the distribution, larger polydispersity corresponds to a broader distribution of molecular weights. Polydispersity is an inherent feature of the mechanism and kinetics of the polymerization reaction. For example, if it were possible to produce all chains invariably with the same number of repeat units, in which the distribution function is spiked at one value of x, then the number-average and weight-average degrees of polymerization are exactly the same (i.e., kxnl ¼ kxwl ¼ x) and the polydispersity is unity. Anionic “living” polymerizations yield macromolecules with narrow molecular weight distributions. For these types of reactions described below, polydispersities asymptotically approach unity when the chain lengths are large. These polymers are useful when a distribution of molecular weights, or chain lengths, is undesirable. One particular example is the requirement for narrow molecular weight standards used to calibrate gel permeation chromatographs.

12.6 KINETICS, MOLECULAR WEIGHT DISTRIBUTIONS, AND MOMENTS-GENERATING FUNCTIONS FOR FREE RADICAL POLYMERIZATIONS 12.6.1 Free Radical Mechanism that Includes Termination by Recombination One must consider initiation, propagation, chain transfer, termination, and the corresponding kinetic rate expressions for each of these elementary steps in a free radical mechanism to obtain the mole fraction distribution function Xx. Furthermore, this distribution function for the fraction of chains that contain x repeat units per chain depends on the nature of the termination step. Free radical initiators, denoted by I, such as

12.6 Kinetics, Molecular Weight Distributions, and Moments-Generating Functions

499

benzoyl peroxide, dicumyl peroxide, azobis(isobutyronitrile) or diphenylpicrylhydrazyl, thermally decompose with time, according to initiation kinetic rate constant kinitiation, and produce two nearly identical radical fragments R via the following elementary reversible reaction:





kinitiation

I , 2R

The rate of initiation vinitiation is defined as the rate of production of radical fragments via the forward reaction. Hence, vinitiation ¼



dR ¼ 2f kinitiation [I] dt

where [I] represents the molar density of initiator molecules and f is the probability that radical fragments R do not recombine to form initiators. In other words, initiator decomposition is reversible, but the rate of initiation is written as the product of an irreversible first-order forward rate law and the probability that the backward reaction does not occur. Other mechanisms for the generation of free radicals include (i) ultraviolet radiation (i.e., photopolymerizations), in which temperature-insensitive initiation rates depend on the square of the magnitude of the electric field, (ii) thermal activation of the monomer, where no initiator molecules are required, and (iii) redox chemistry of water-soluble salts. Propagation occurs when radical fragments, R or RMx , attack monomer M to produce growing chains, RM or RM xþ1, with one additional repeat unit. Free radical molar density is conserved during propagation. These irreversible elementary steps are summarized as follows:





 RM þ M

kpropagation







 RM

R þ M ) RM kpropagation

)

2





kpropagation



RMx þ M ) RMxþ1 where the kinetic rate constant kpropagation for propagation or monomer disappearance does not depend on kinetic chain length (i.e., the value of x in radical molar density RMx ). When termination by recombination occurs, two growing radicals produce one polymer chain P as illustrated below:







ktermination

RMx þ RMy ) RPxþy R where the polymer chain on the right side of the coupling reaction contains, by convention, x þ y þ 2 monomer units (including initiator fragments at both ends). Now, constant-volume batch reactor mass balances are written for each RMx ,



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Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

0  x  1, based on the elementary steps introduced above for initiation, propagation, and termination by recombination. The objective is to relate initiation and termination rates via the pseudo-steady-state approximation. Transfer of the growing free radical chain RMx to solvent, initiator, monomer, and polymer is neglected. If chain transfer reactions occur, then they can have a significant effect on molecular weight distributions and the average degree of polymerization. In general, chain transfer reactions reduce the average degree of polymerization. Since each growing radical can be terminated by coupling to all other radicals, the following unsteady state mass balances are applicable in a constant-volume batch reactor:





1 X d[R ] ¼ vinitiation  kpropagation M[R ]  ktermination [R ] [RMy ] dt y¼0









1 X d[RM ] ¼ kpropagation M{[R ]  [RM ]}  ktermination [RM ] [RMy ] dt y¼0

















1 X d[RM2 ] ¼ kpropagation M{[RM ]  [RM2 ]}  ktermination [RM2 ] [RMy ] dt y¼0 .. . 1 X d[RMx ] ¼ kpropagation M{[RM x1 ]  [RMx ]}  ktermination [RMx ] [RMy ] dt y¼0













where M represents monomer molar density, [ ] denotes molar density of a radical species, and the probability f that initiator decomposition is successful in vinitiation is not required because the termination step in the balance for [R ] includes the recombination of two R fragments when y ¼ 0 in the summation of the first equation. The pseudo-steady-state approximation for each radical species RMx , 0  x  1, claims that the molar density of these growing chains is relatively low and does not change much. Hence, the left-hand side of each constant-volume batch reactor mass balance vanishes, allowing one to obtain an approximate expression for the molar density of each of these reactive species. Before completing this task, it is instructive to add all of these mass balances within the context of the pseudo-steady-state approximation. The rate of production of a radical species in one propagation step is offset by the rate of depletion of its precursor in the previous propagation step. Hence, all propagation rates cancel and the initiation rate is balanced by the sum of all termination rates. Addition of the previous set of balances yields

 



vinitiation ¼ ktermination

1 X x¼0



[RMx ]

1 X y¼0



[RMy ] ¼ ktermination

(

1 X



)2

[RMx ]

x¼0

where, once again, the probability f that initiator decomposition is successful is not required in the initiation rate because the recombination of two initiator fragments

12.6 Kinetics, Molecular Weight Distributions, and Moments-Generating Functions

501



R is included in the termination rate when x ¼ 0 and y ¼ 0. Now, let’s apply the pseudo-steady-state approximation to each individual constant-volume batch reactor mass balance for RMx , 0  x  1, and obtain approximate expressions for the molar density of each reactive free radical species:



( ktermination



[R ] ¼

vinitiation kpropagation M þ ktermination

1 X



[RMy ]



¼

kpropagation M þ ktermination

1 X

1 X [RMy ]



y¼0

kpropagation M kpropagation M þ ktermination



)2

y¼0

y¼0

[RM ] ¼

1 X [RMy ]







[R ] ¼ a[R ]

[RMy ]

y¼0



[RM2 ] ¼

kpropagation M kpropagation M þ ktermination

1 X

[RM]





[RM ] ¼ a2 [R ]

y

y¼0

.. .



[RMx ] ¼

kpropagation M kpropagation M þ ktermination

1 X







[RM x1 ] ¼ a x [R ]

[RMy ]

y¼0

where a represents the probability that a reactive radical species propagates. Since these active species can either propagate or terminate, high-molecular-weight chains are produced when a approaches unity. Multiplication of numerator and denominator of a by the sum of all active species, RMk , 0  k  1, reveals that the numerator corresponds to the rate of monomer disappearance, or the propagation rate, whereas the denominator yields the rate of all possible reactions, propagation and termination, that involve these active species since chain transfer reactions have not been considered in this kinetic mechanism of free radical polymerization. The previous result for the molar density of RMx is employed to evaluate the rate of production of x-mers, d[RPx22R]/dt, which is proportional to the mole fraction of x-mers Xx. If termination occurs exclusively by recombination, then the smallest chain molecule is RZR when two initiator radical fragments recombine. Each initiator fragment R is analyzed on the same scale as monomer M when the number of repeat units is determined in a given chain. Hence, an x-mer is identified by the following structure RPx22R, 2  x  1, where RZR represents a dimer (i.e., x ¼ 2). The rate of production of an x-mer occurs when two free radicals combine via several possible termination steps such that there are x 2 2 monomers in the chain and two initiator fragments cap the chain ends. The appropriate constant-volume batch reactor mass







502

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

balance for the rate of production of RPx22R is

     þ    þ [RM ][RM] þ [RM ][RM] þ [RM ][R]}

d[RPx2 R] 1 ¼ ktermination {[R ][RM x2 ] þ [RM ][RM x3 ] þ [RM2 ][RM x4 ] dt 2



x4

x3

2

x2

x2 X 1 [RMj ][RM x2j ] ¼ ktermination 2 j¼0









where the factor of 12 is required because coupling reactions between RMi and RMk in which i þ k ¼ x2 2 (2  x  1) have been counted twice. Assistance from the pseudo-steady-state approximation for each individual reactive intermediate, [RMx ] ¼ a x [R ], allows one to develop an expression for the discrete mole fraction distribution of x-mers Xx:



Xx 



x2 x2 X X d[RPx2 R] 1 1 ¼ ktermination [RMj ][RM x2j ] ¼ ktermination [R ]2 a x2 dt 2 2 j¼0 j¼0







1 ¼ ktermination [R ]2 (x  1)a x2 2



Xx ¼ C1 (x  1)a x2 where constant C1 is required for normalization. Since all Xx (2  x  1) must sum to unity, one evaluates C1 as follows: 1 X x¼2

Xx ¼ C1

1 X

(x  1)a x2 ¼ C1 {1 þ 2a þ 3a2 þ 4a3 þ   }

x¼2

( ) 1 d X d n a o C1 x a ¼ C1 ¼1 ¼ C1 ¼ d a x¼1 da 1  a (1  a)2 Xx ¼ (1  a)2 (x  1)a x2

In comparison, normalization of the most probable distribution function was guaranteed in statistical analysis of polycondensation reactions as a consequence of one of the constraints in Lagrange multiplier optimization, whereas normalization was invoked to determine the proportionality constant between the rate of production of x-mers and their mole fraction for free radical polymerization.

12.6.2

Discrete Moments-Generating Function Fz

The logical sequence of calculations described below performs the following tasks: (i) use z-transforms (i.e., as illustrated in Section 12.8, z-transformation is the discrete

12.6 Kinetics, Molecular Weight Distributions, and Moments-Generating Functions

503

analog of Laplace transforms) to obtain the moments-generating function for free radical polymerization when termination occurs exclusively by coupling of two growing chains, (ii) demonstrate that the zeroth moment Q0 of Xx is normalized, as required, (iii) evaluate the first moment Q1, which yields the number-average degree of polymerization kxnl and provides assistance in calculating the mass fraction distribution function Wx, (iv) evaluate the second moment Q2 of Xx to determine the weight-average degree of polymerization kxwl and the polydispersity for free radical reactions, and (v) compare the peak of the GPC output curve, obtained from the maximum value of Wx at constant a, with kxnl and kxwl. z-Transformation of the mole fraction distribution Xx yields Fz: Fz ¼

1 X

z x Xx ¼ (1  a )2

1 X

x¼2

z x (x  1)a x2 ¼ z2 (1  a)2

x¼2

1 X

(x  1)(a z) x2

x¼2

¼ z2 (1  a)2 {1 þ 2a z þ 3(a z)2 þ 4(a z)3 þ 5(a z)4 þ   } ( )   1 X d d az z2 (1  a)2 2 2 x 2 2 ¼ (a z) ¼ z (1  a) ¼ z (1  a) d(a z) x¼1 d(a z) 1  a z (1  az)2 Consistency is guaranteed because normalization was invoked to obtain the final form for the mole fraction distribution of x-mers. The zeroth moment of Xx, defined by Q0, is related to Fz via Q0 ¼ Fz(z ¼ 1) ¼ 1. The number-average degree of polymerization kxnl is given by Q1: Q1 ¼

1 X

 xXx ¼

x¼2

z

dFz dz

 ¼ z¼1

 2  2z (1  a)2 2 ¼ ¼ kxn l 3 (1  a z) z¼1 1  a

If the probability a that growing free radicals propagate, instead of terminating, and the fractional conversion p of monomer to polymer assume similar values for their respective polymerizations, then the number-average degree of polymerization is two-fold larger for free radical polymers that terminate by recombination relative to kxnl for condensation polymers. The mole fraction distribution function Xx and Q1 are useful to calculate the mass fraction distribution Wx, which simulates the output curve from gel permeation chromatography. One obtains the following result: Wx ¼

xXx xXx xXx 1 ¼ (1  a)3 x(x  1)a x2 ¼ ¼ 1 X Q1 kxn l 2 xXx x¼2

There are two routes to calculate the weight-average degree of polymerization. The first approach is based on the moments-generating function Fz, whereas the second approach employs a weight-fraction-weighted sum of the number of repeat units x

504

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

per chain. Both calculations summarized below are equivalent, but the latter approach yields an infinite series that cannot be evaluated by simple techniques:  2 

 d dFz 2z (2 þ a z)(1  a)2 2(2 þ a) z x Xx ¼ z ¼ ¼ Q2 ¼ 4 dz dz (1  a z) (1  a)2 z¼1 z¼1 x¼2   2(2 þ a) Q2 2þa (1  a)2  ¼ kxw l ¼ ¼  2 Q1 1a 1a 1 X

kxw l ¼

1 X x¼2

2



xWx ¼ 12(1  a)3

1 X

x2 (x  1)a x2

x¼2

¼ 12(1  a)3 (4 þ 18a þ 48a2 þ 100a3 þ 180a4 þ   ) The power of discrete moments-generating functions Fz should be obvious from the previous calculation of kxwl because the first method of evaluating the weightaverage degree of polymerization from Fz is tedious but straightforward, whereas the weight-fraction-weighted sum of the number of repeat units per chain yields an infinite sum that requires advanced techniques for converging series. The polydispersity of free radical polymerizations in which termination occurs exclusively by recombination of two growing active sites is given by the ratio of kxwl to kxnl, which yields (2 þ a)/2. The production of high-molecular-weight polymers, which is typical of free radical mechanisms, requires a high probability that reactive radical species propagate instead of terminate. This is consistent with a ) 1 and a polydispersity of 1.5, that should be compared with a polydispersity of 2 for polycondensation reactions. Hence, the breadth of the molecular weight distribution for vinyl polymers produced by free radical mechanisms with termination by coupling is slightly narrower than the distribution of molecular weights for condensation polymers.

12.6.3 Peak of the GPC Output Curve for Vinyl Polymers Produced by Free Radical Mechanisms When vinyl polymers are produced by free radical reactions that undergo termination by recombination, both the mole fraction and mass fraction distribution functions exhibit maxima at intermediate values of x. Straightforward analysis of Xx reveals that the mole fraction distribution peaks at (i.e., dXx/dx ¼ 0) xMaximum(Xx ) ¼ 1 þ

1 ln(a1 )

505

12.6 Kinetics, Molecular Weight Distributions, and Moments-Generating Functions

Use of the quadratic formula and selection of the negative square root (i.e., because ln a , 0) identify the degree of polymerization where Wx is largest (i.e., dWx/dx ¼ 0):

xMaximum(Wx )

x2 ln a þ x{2  ln a}  1 ¼ 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 1 1 þ þ ¼ þ 1 2 ln(a ) 4 {ln(a1 )}2

A comparison of these maxima in Xx and Wx with the number-average and weightaverage degrees of polymerization is summarized in Table 12.1 for several values of a. These results are consistent with those from polycondensation reactions, which indicate that the mass fraction distribution function exhibits a maximum slightly below the number-average degree of polymerization. However, differences between kxnl and xMaximum where dWx/dx ¼ 0 are indistinguishable when a is greater than or equal to 90%.

12.6.4 Continuous Distribution Functions for Free Radical Polymerizations that Terminate by Recombination The continuous mole fraction distribution of x-mers, X(x), is obtained from the discrete distribution function Xx via the definition of k ¼ 1 2 a, which represents the probability that growing free radicals terminate by coupling instead of propagating,

Table 12.1 Comparison of Average Degrees of Polymerization with Maxima in Mole and Mass Fraction Distribution Functions for Free Radical Reactions Terminated by Recombination

a 0.70 0.75 0.80 0.85 0.90 0.95 0.98 0.99 0.999 0.9999 0.99999

dXx/dx ¼ 0 at xMaximum

dWx/dx ¼ 0 at xMaximum

kxnl

kxwl

3.8 4.5 5.5 7.2 10.5 20.5 50.5 1.0  102 1.0  103 1.0  104 1.0  105

6.2 7.5 9.5 12.8 19.5 39.5 1.0  102 2.0  102 2.0  103 2.0  104 2.0  105

6.7 8.0 10.0 13.3 20.0 40.0 1.0  102 2.0  102 2.0  103 2.0  104 2.0  105

9.0 11.0 14.0 19.0 29.0 59.0 1.5  102 3.0  102 3.0  103 3.0  104 3.0  105

506

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

and approximating a(¼12 k) for high-molecular-weight chains as exp(2k) when k ) 0 (i.e., a ) 1). One obtains the following result for X(x): Xx ¼ (1  a)2 (x  1)a x2 ) X(x)  k 2 (x  1) exp{k(x  2)} 2x1

) k 2 x exp(kx) x1, x2x 0x1

Integration by parts is required to demonstrate that the continuous representation of the mole fraction distribution function X(x) is normalized. In other words, 1 ð

X(x) dx ¼ k2

x¼0

1 ð

8 9 1 x¼1 ð < x = 1 x exp(k x) dx ¼ k2  exp(k x) þ exp(k x) dx : k ; k x¼0

x¼0

¼k

x¼0

1 ð

exp(k x) dx ¼ 1

x¼0

Two “methods of attack” yield the same expression for the continuous representation of the mass fraction distribution function for x-mers, W(x). The first method employs the same substitutions and approximations presented above to generate X(x) from Xx. Beginning with Wx, one obtains Wx ¼ 12(1  a)3 x(x  1)a x2 ) W(x)  12 k 3 x (x  1) exp{k (x  2)} 2x1

) x1,x2x 0x1

1 3 2 2k x

exp{k x}

The second approach (i) is based on the relation between Wx and Xx at the discrete level, (ii) replaces Xx by X(x), and (iii) transforms the summation to an integral, as illustrated below: Wx ¼

xXx xX(x) k 2 x 2 exp(kx) 1 2 2 ð ð1 ) W(x) ¼ k x exp(kx) ¼ ¼ 1 1 X 0x1 kxn l 2 2 xX(x) dx k x exp( k x) dx xXx x¼2

0

k

2

0

1 ð

x 2 exp(kx) dx ¼

2 k

0

The integral that appears in the denominator of W(x) represents the first moment Q1 of the mole fraction distribution of x-mers, X(x), given by 2/k. Continuous representations of the mole fraction and mass fraction distributions for x-mers exhibit

12.6 Kinetics, Molecular Weight Distributions, and Moments-Generating Functions

507

peaks at intermediate values of x. For example, dX/dx ¼ 0 at x ¼ 1/k (i.e., maximum), and dW/dx ¼ 0 when x ¼ 0 (i.e., minimum) and x ¼ 2/k (i.e., maximum). Once again, these calculations provide sufficient evidence that the maximum in W(x) occurs when x ¼ kxnl.

12.6.5 Vinyl Polymers Produced via Termination by Disproportionation When termination occurs by disproportionation, two growing active species produce two polymer chains with unsaturation (i.e., CvC) at the terminus of one macromolecule. A schematic representation of this termination step is illustrated below:





ktermination

RMx þ RMy ) RPx þ RPy where each polymer chain on the right side of this reaction contains either x þ 1 or y þ 1 monomer units (including an initiator fragment at one end). Consider the rate of production of x-mers, which occurs according to the previous reaction when growing chain RM x1 is in the immediate vicinity of and collides with RMy (0  y  1). The kinetic expression of interest is analyzed via pseudo-steady-state approximations for radical molar densities [RM x1 ] and [R ], as well as the definition of a, from Section 12.6.1 that are not unique to termination by recombination. One obtains









1 X d[RPx1 ] ¼ ktermination [RM x1 ] [RMy ] ¼ dt y¼0





ktermination ¼ vinitiation a

x1

(

1 X

ktermination

1 X



)



[RMy ] a x1 [R ]

y¼0



[RMy ]

y¼0

kpropagation M þ ktermination

1 X



¼ vinitiation a x1 (1  a)

[RMy ]

y¼0

The rate of initiation vinitiation is balanced by the sum of the rates for all possible termination steps, as illustrated in Section 12.6.1. Hence, one sets vinitiation ¼ vtermination in the previous expression and defines the discrete mole fraction of xmers Xx as the ratio of the termination rate for the production of chains that contain x repeat units, including one initiator fragment end cap, to the sum of all termination rates that produce chains with any number of repeat units: d[RPx1 ] 1 d[RPx1 ] ¼ (1  a)a x1 Xx ¼ 1  dt ¼ X d[RPy1 ] vtermination dt dt y¼1 The range of x in this normalized discrete distribution function is from 1 (i.e., monomer or initiator fragement) to 1 for infinitely high-molecular-weight chains.

508

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

One concludes that the statistics of polycondensation reactions are identical to those for free radical polymerizations when termination occurs exclusively by disproportionation, yielding the most probable distribution in both cases for the mole fraction of x-mers. If one replaces fractional conversion p of monomer to polymer in all of the polycondensation results by the probability a that a growing free radical propagates, instead of terminating, then all statistical results for free radical polymerization via termination by disproportionation are obtained rather quickly. One concludes that kxnl is two-fold larger when termination occurs by coupling.

12.7 ANIONIC “LIVING” POLYMERIZATIONS AND THE POISSON DISTRIBUTION 12.7.1 Kinetic Mechanism, Rate Expressions, and the Mole Fraction Distribution of x-mers There is no spontaneous termination step in these polymerizations unless foreign material such as water, alcohol, or carbon dioxide is added deliberately to the reaction mixture. Hence, very high-molecular-weight polymers are produced by anionic techniques, with unusually narrow molecular weight distributions because all chains initiate upon injection of the “catalyst” and propagate at the same rate in the absence of chain transfer or termination until all of the monomer is consumed. If Rinitial and Minitial represent the initial molar densities of initiator and monomer at t ¼ 0, respectively, and each growing chain contains one initiator fragment as illustrated P below,then Rinitial is equivalent to the total molar density of all growing chains, 1 x¼0 [RMx ], at any time during the polymerization reaction and the number-average degree of polymerization, or equilibrium chain length, is calculated as follows after all of the monomer is consumed (i.e., when t ) 1): Minitial kxn l ¼ 1 þ Rinitial Hence, the previous equation is useful to design polymers with specified molecular weight upon (i) consumption of all of the monomer and (ii) addition of a termination reagent. For example, if n-butyl-lithium (i.e., RLi, where R is n-C4H9) is employed to initiate the polymerization of vinyl monomers (i.e., H2CvCHX) such as styrene (i.e., X is ZC6H5), methyl acrylate (i.e., X is ZCOOCH3), or acrylonitrile (i.e., X is ZCuN) in organic solvents like ether or tetrahydrofuran (THF), then the following anionic mechanism is appropriate when termination does not occur: Kequilibrium

þ C4 H9 Li , C4 H 9 þ Li THF

kinitiation

 C 4 H 9 þ H2 CvCHX ) C4 H9 CH2 CHX

12.7 Anionic “Living” Polymerizations and the Poisson Distribution

509

kpropagation

RM þ M ) RM2  kpropagation

RM2  þ M ) RM3  .. . kpropagation

RMx1  þ M ) RMx  where M represents vinyl monomer (i.e., H2CvCHX), the propagation kinetic rate constant kpropagation is independent of chain length, RMx  is an active anionic propagating species (i.e., carbanion) that contains x þ 1 monomer units (including initiator fragment R), and lithium cations reside in the vicinity of the active anions for charge neutrality. Unlike free radical mechanisms that exhibit spontaneous termination when two radicals combine, two active carbanions will not react due to charge repulsion. If one does not distinguish between kinetic rate constants for initiation and propagation, such that kinitiation  kpropagation ¼ kreaction, then the analysis of monomer consumption in an unsteady state constant-volume batch reactor proceeds as follows: 1 1 X X dM ¼ kinitiation M[R ]  kpropagation M [RMx  ]  kreaction M [RMx  ] dt x¼1 x¼0

¼ kreaction MRinitial M(t)  Minitial exp{kreaction Rinitial t} This result is not completely accurate because the requirement for (i) instantaneous initiation of all chains upon injection of the “catalyst” into the monomer solution and (ii) the production of high-molecular-weight polymers with a monodisperse distribution of chain lengths is kinitiation kpropagation. This inequality is satisfied when the basicity of substituent X in the vinyl monomer is strong enough to stabilize negative charge on the a-carbon of the growing carbanion in the initiation step. However, when ethylene is polymerized anionically via n-butyl-lithium, the 2 initiator anion C4H2 9 and the growing carbanions C4H9Mx are structurally similar. Under these conditions, carbanions are not stabilized preferentially in the initiation step relative to the propagation steps, and kinitiation  kpropagation. The expression for M(t) is employed to define the kinetic chain length y and evaluate its timerate-of-change:

y¼

Minitial  M(t) Minitial  M(t) Minitial ¼  {1  exp(kreaction Rinitial t)} 1 X Rinitial Rinitial  [RMx (t)] x¼0

dy  kreaction Minitial exp(kreaction Rinitial t)  kreaction M dt

510

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

Kinetic chain length y and time-dependent degree of polymerization kxnl are essentially equivalent. In fact, when polymerization is initiated at t ¼ 0, the initial conditions are R ¼ Rinitial, M ¼ Minitial, kxnl ¼ 1, and y ¼ 0. As polymerization proceeds, y and kxnl asymptotically approach each other. Now, let’s use this result for the time dependence of kinetic chain length to analyze the molar density of initiator anion R2: d[R ] d[R ] dy d[R ]  kreaction M ¼ ¼ kinitiation M[R ] dt d y dt dy [R (t)]  Rinitial exp(y) Proceeding to the next active anion, RM2(t), which is not present initially in the reaction mixture, unsteady state analysis in a constant-volume batch reactor yields d[RM ] d[RM ] dy d[RM ]  kreaction M ¼ dt d y dt dy ¼ kinitiation M[R ]  kpropagation M[RM ]  kreaction M{[R ]  [RM ]} d[RM ] þ [RM ]  [R ]  Rinitial exp(y) dy   d[RM ] d exp(y) þ [RM ] ¼ {[RM ] exp(y)}  Rinitial dy dy [RM (t)]  Rinitial y exp(y) Analysis of active anions RM2  (t) with degree of polymerization of 3 allows one to generalize these results for the time-dependent molar density of any active carbanion and the mole fraction distribution of x-mers when termination is absent in anionic polymerization: d[RM2  ] d[RM2  ] d y d[RM2  ] ¼  kreaction M dt dy dy dt ¼ kpropagation M[RM ]  kpropagation M[RM2  ] ¼ kpropagation M{[RM ]  [RM2  ]} d[RM2  ] þ [RM2  ]  [RM ]  Rinitial y exp(y) dy   d[RM2  ] d exp(y) þ [RM2  ] ¼ {[RM2  ] exp(y)}  Rinitial y dy dy [RM2  (t)]  12 Rinitial y 2 exp(y) The molar density of active anions RMx1  (t) with degree of polymerization x (including the initiator fragment R) is obtained by repeating this procedure several

12.7 Anionic “Living” Polymerizations and the Poisson Distribution

511

times. Generalization yields [RMx1  (t)] ¼

1 Rinitial y x1 exp(y) (x  1)!

In a constant-volume system, the mole fraction Xx of chains that contain x repeat units is given by the ratio of its molar density [RMx1  ] to the molar density of all growing chains in the reaction mixture, the latter of which is equivalent to the initial molar density of initiator fragments Rinitial. Hence, the discrete mole fraction distribution function is 1 Rinitial y x1 exp(y) [RMx1 ] y x1 (x  1)! ¼ exp(y) Xx ¼ 1 ¼ X Rinitial (x  1)!  [RMy ] 

y¼0

This is a discrete representation of the Poisson distribution function, which is valid for the mechanism and kinetics described above when x  1. The kinetic and statistical analysis of anionic polymerizations is based on the assumption that the initiation and propagation rate constants are of comparable magnitudes, which will not yield a monodisperse distribution of chain lengths. Regardless of the fact that kinitiation kpropagation is not entirely consistent with the normalized mole fraction distribution function derived in this section, when spontaneous termination is absent and the first two initiation steps equilibrate on the time scale of all of the propagation steps outlined at the beginning of this section, the resulting Poisson distribution is much narrower than the distribution of chain lengths for either condensation or free radical polymerizations. When the kinetic chain length y is large, the polydispersity index for anionic polymerizations approaches unity.

12.7.2 Discrete Moments-Generating Function, Average Degrees of Polymerization, and Polydispersity for Anionic Polymerization The discrete z-transform of Xx yields the required moments-generating function Fz. Based on the normalized mole fraction distribution from the previous section, one obtains Fz ¼

1 X x¼1

z x Xx ¼ exp(y)

1 1 X X z x y x1 (y z) x1 ¼ z exp(y) ¼ z exp{y (z  1)} (x  1)! (x  1)! x¼1 x¼1

which yields, as expected, a normalized zeroth moment of Xx because Fz(z ¼ 1) ¼ 1. Evaluation of the summation in Fz was based on the Taylor series representation of exp( y z), expanded about y z ¼ 0. The first and second moments of the normalized

512

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

mole fraction distribution function can be calculated rather easily from Fz:   dFz Q1 ¼ xXx ¼ z ¼ [z(1 þ y z) exp{y (z  1)}]z¼1 ¼ 1 þ y ¼ kxn l dz z¼1 x¼1 

 1 X d dFz z x2 Xx ¼ z Q2 ¼ dz z¼1 dz x¼1 1 X

¼ [z{1 þ 3y z þ (y z)2 } exp{y (z  1)}]z¼1 ¼ 1 þ 3y þ y 2 kxw l ¼

Q2 1 þ 3y þ y 2 ¼ ) yþ3 y 1 Q1 1þy

kxw l 1 þ 3y þ y 2 yþ3 ¼ ) ) 1 y 1 y þ 2 y !1 kxn l (1 þ y) 2 Hence, polymers produced by anionic techniques in the absence of terminating reagents yield the narrowest molecular weight distribution, in comparison with condensation and free radical polymerizations. The polydispersities are (i) 2 for polycondensation reactions when the fractional conversion p of monomer to polymer approaches unity, (ii) 2 for free radical reactions that terminate by disproportionation, and 1.5 for termination by recombination, when the probability a approaches unity that growing free radicals propagate instead of terminating, and (iii) 1 for anionically produced polymers when the kinetic chain length y is extremely large. The shape of the GPC output curve for living polymers can be simulated from the discrete mass fraction distribution function Wx: Wx ¼

xXx 1 X

xXx

¼

xy x1 exp(y) (x  1)!(1 þ y)

x¼1

The following analysis reveals that both the mole fraction and mass fraction distribution functions for anionic polymerizations exhibit maxima when x is in the vicinity of y for high-molecular-weight chains. The strategy is based on identifying maxima in ln Xx and ln Wx due to the presence of (x 2 1)! in the denominator of each distribution function. The mole fraction distribution achieves a maximum, denoted by xMaximum(X), when the following equation is satisfied: xMaximum(X)  12 þ ln(xMaximum(X)  1) ¼ 1 þ ln y xMaximum(X)  1 The mass fraction distribution function achieves a maximum, denoted by xMaximum(W ), when the following equation is satisfied: xMaximum(W)  12 1  þ ln(xMaximum(W)  1) ¼ 1 þ ln y xMaximum(W)  1 xMaximum(W)

12.7 Anionic “Living” Polymerizations and the Poisson Distribution

513

For example, when kinetic chain length y  10, numerical solution of the previous two equations yields xMaximum(X )  y þ 0.5, and xMaximum(W )  y þ 1.5, which reveals that both distribution functions exhibit maxima in the vicinity of kxnl ¼ y þ 1, even for rather short kinetic chain lengths.

12.7.3 Continuous Distribution Functions for Anionic Polymerization Begin with the discrete mole fraction distribution function Xx and multiply it by ( y /x), which is approximately equal to unity for values of x (i.e., in the vicinity of the kinetic chain length y ) where the mole fraction of x-mers is significantly different from zero. One obtains the following result upon invoking Sterling’s approximation for x!: y x1 y x exp(y) exp(x  y) nyoxþ1=2 Xx ¼ exp(y)   pffiffiffiffiffiffiffiffiffi (x  1)! x x! 2p y Once again, since ( y /x) is approximately equal to unity for values of x where Xx is significant, further analysis of the last factor on the far right side of the previous equation yields  ny oxþ1=2 ny ox

y x=y  n y x o  ¼ exp ln ¼ exp y ln x x x x  

 x  y x=y ¼ exp y ln 1  x Now, the argument of the logarithm is expanded in a power series about x ¼ y with first-order coefficient 21/y and truncated after the first-order term, followed by expansion of the logarithm in a Taylor series that is truncated after the secondorder term:

x  y x=y xy 1 þ  y x

x  y x=y x  y x  y 1 x  y 2 1 x  y 3 ln 1   ln 1      x y y 2! y 3! y

x  y x=y (x  y)2 y ln 1   (x  y)  x 2y 1

The continuous Poisson distribution for the mole fraction of x-mers X(x) in anionic polymers with kinetic chain length y is given by   exp(x  y) nyoxþ1=2 exp(x  y) (x  y)2 Xx  pffiffiffiffiffiffiffiffiffi ) X(x)  pffiffiffiffiffiffiffiffi exp (x  y)  x 2y 2py 2py  2 1 (x  y) ¼ pffiffiffiffiffiffiffiffiffi exp  2y 2p y

514

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

which exhibits a maximum at x ¼ y . The discrete and continuous mole fraction distribution functions are indistinguishable when y is large. Further justification for the approximations and truncated expansions that were employed to generate X(x) from Xx is obtained by evaluating the zeroth moment and first moment of X(x) via numerical integration at several values of y . The results are summarized below:

Q0 ¼

1 ð

0

1   ð 1 (x  y)2 p ffiffiffiffiffiffiffiffi dx ¼ 1 X(x) dx ¼ exp  2y 2py

Q1 ¼ kxn l ¼

0

1 ð 0

1   ð 1 (x  y)2 dx  y xX(x) dx ¼ pffiffiffiffiffiffiffiffi x exp  2y 2py 0

Analytical evaluation of the zeroth moment, Q0, yields an error function solution that verifies the normalization condition for large kinetic chain lengths y . Let y ¼ x 2 y in the argument of the exponential, integrate with respect to y from 2y to 1, rewrite the result in terms of the sum of two separate integrals, and invoke the fact that integration of an even function yields an odd function. One obtains the following integration of the Poisson distribution function: 1 ð

1 1   2

2

2

ð ð0 ð (x  y)2 y y y dx ¼ exp  dy ¼ exp  dy þ exp  dy exp  2y 2y 2y 2y y

0

¼

1 ð

y

0

¼

1 ð

0

 2

ðy 2

y y dy  exp  dy exp  2y 2y 0

2

2

y y dy þ exp  dy exp  2y 2y ðy

0

0

On the extreme right side of the previous equation, the first term yields sqrt(p y /2) via (i) evaluation of the square of the first integral and (ii) transformation from rectangular Cartesian to cylindrical coordinates:

I¼

ð1

1 2

2

ð y x dy ¼ exp  dx exp  2y 2y

0

2

I ¼

1 ð ð1

0 0

0

pð=2 1 2

2

ð x þ y2 r 1 dx dy ¼ dr ¼ py exp  dQ r exp  2y 2y 2 0

0

12.8 Connection Between Laplace Transforms and the Moments-Generating Function

515

Definition of the error function allows one to evaluate ðy

pffiffiffiffiffi y=2 pffiffiffiffi  2

pffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffi ð pffiffiffiffiffi z¼y= 2y y p 2 dy ) erf 2y exp  exp(z ) dz ¼ 2y y=2 pffiffiffiffi 2y 2 dy¼dz 2y

0

0

rffiffiffiffiffiffin pffiffiffiffiffiffiffio py ¼ erf y=2 2

A combination of all of this information reveals that the zeroth moment of the continuous Poisson distribution function for the mole fraction of x-mers with kinetic chain length y is, indeed, normalized: Q0 ¼

1 ð

0

1   ð 1 (x  y)2 dx X(x) dx ¼ pffiffiffiffiffiffiffiffiffi exp  2y 2p y 0

rffiffiffiffiffiffi rffiffiffiffiffiffin pffiffiffiffiffiffiffio 1 py py þ erf y=2 ¼ pffiffiffiffiffiffiffiffiffi 2 2 2p y

pffiffiffiffiffiffiffio 1n 1 þ erf y=2 ) 1:00 ¼ y8 2 because the error function approaches unity (i.e., to two decimal places, 1.00) for arguments (i.e., sqrt( y /2)) that are equal to or greater than 2. Previous results in this section are employed to calculate the continuous approximation for the mass fraction distribution function W(x) that is unique to anionic polymerization. Based on the continuous first moment that yields kxnl ¼ y , instead of y þ 1 via the discrete first moment, one obtains W(x) as follows: xX(x)

W(x) ¼ ð 1 xX(x) dx

  x (x  y)2 ¼ pffiffiffiffiffiffiffiffiffi exp  2y y 2p y

0

which exhibits a maximum at x  y þ 1. Numerical integration reveals that W(x) is normalized and the weight-average degree of polymerization kxwl ¼ y þ 1 for several values of the kinetic chain length y .

12.8 CONNECTION BETWEEN LAPLACE TRANSFORMS AND THE MOMENTS-GENERATING FUNCTION FOR ANY DISTRIBUTION IN THE CONTINUOUS LIMIT There is a continuous analog of z-transformations that were employed earlier in this chapter to develop the moments-generating function for discrete molecular weight

516

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

distributions. Beginning with the discrete moments-generating function, which converts the discrete mole fraction distribution function Px to Fz, replace z by exp(2s) and Px by P(x). Upon extrapolating the summation over index x to an integral with respect to x, with limits ranging from 0 to 1, one obtains the Laplace transform of P(x): 1 X

Fz ¼

x

z¼exp(s)

1 ð

Px !P(x) Fz !F(s)

0

z Px ) F(s) ¼

x¼1(or2)

P(x)esx dx

The next task is to develop a relation between the nth moment Qn of the continuous mole fraction distribution function P(x) and F(s). By definition, P Ð

1 X

Qn ¼

!

n

dx

x Px )

1 ð

Px !P(x)

x¼1(or2)

xn P(x) dx

0

In the discrete cases, nth derivatives of the moments-generating function Fz were evaluated at z ¼ 1 to yield Qn. Similar operations on F(s) yield continuous analogs of Qn, but the expressions are evaluated at s ¼ 0 because z ¼ exp(2s). Begin with F(s) and expand exp(2sx) in an alternating Taylor series about x ¼ 0 and use the definition of Qn to rewrite F(s) as follows: 1 ð

sx

F(s) ¼ P(x)e

dx ¼

0

1 ð

1  sx þ

 (sx)2 (sx)3 (sx)4  þ     P(x) dx 2! 3! 4!

0

1 1 1 ð ð ð s2 2 s3 3 s4 4 ¼ P(x) dx  s xP(x) dx þ x P(x) dx  x P(x) dx þ x P(x) dx     2! 3! 4! 1 ð

1 ð

0

0 2

¼ Q0  sQ1 þ

0 3

0

0

4

s s s Q2  Q3 þ Q4     2! 3! 4!

The desired relations between Qn and F(s) are 2

3

d F d F ; Q2 ¼ ; Q ¼  3 2 ds ds3 s¼0 s¼0 n s¼0 d F Qn ¼ (1)n dsn s¼0

Q0 ¼ {F(s)}s¼0 ; Q1 ¼ 

dF ds



In the following subsections, Laplace transforms of the continuous mole fraction distribution functions are evaluated for condensation, free radical, and anionic

12.8 Connection Between Laplace Transforms and the Moments-Generating Function

517

polymerization. Once F(s) is determined, the previous equations are employed to evaluate Q1 and Q2 so that continuous representations of kxnl, kxwl, and the polydispersity index can be compared with results from the analysis of discrete distribution functions.

12.8.1

Polycondensation Reactions

Begin with the continuous analog of the most probable distribution for condensation polymerizations and calculate its moments-generating function via Laplace transformation, as illustrated below: F(s) ¼

1 ð

{b exp(bx)}esx dx ¼ 

b b [exp{(s þ b)x}]x)1 x¼0 ¼ sþb sþb

0

This result is consistent with the normalized mole fraction distribution function, because F(s) ¼ 1 when s ¼ 0. The first moment of P(x) yields the number-average degree of polymerization: Q1 ¼ kxn l ¼ 

dF ds



¼ s¼0

b (s þ b)2

 ¼ s¼0

1 1 ¼ b 1p

which agrees exactly with previous calculations of kxnl via discrete expressions for Px and Fz in Section 12.5. The second moment of P(x) allows one to evaluate kxwl and the polydispersity index: 2

     d F d b 2b 2  ¼ ¼ ¼ 2 Q2 ¼ 2 3 2 ds s¼0 ds (s þ b) (s þ b) s¼0 b s¼0 kxw l ¼

Q2 2 kxw l ¼2 ¼ ; Q1 b kxn l

These results for Q2, kxwl, and the polydispersity index only agree with the discrete results in Section 12.5 when the fractional conversion p of monomer to polymer approaches 100% (i.e., p ) 1). Recall that, at high conversion, p(¼12 b) was replaced by exp(2b) to generate P(x) from Px in Section 12.3, and this approximation manifests itself in F(s) and the continuous expressions for Q2 and kxwl.

12.8.2 Free Radical Polymerizations that Terminate by Recombination The continuous mole fraction distribution function X(x) is exact when the probability a that active free radicals propagate, instead of terminating, approaches unity.

518

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

Integration by parts yields the Laplace transform of X(x) for vinyl polymers that terminate by coupling: 1 ð

F(s) ¼ X(x)e 0

sx

dx ¼ k

2

1 ð

{x exp(kx)}esx dx

0

8 <

x exp{(s þ k)x} ¼ k2  : sþk ¼

k2 sþk

1 ð

exp{(s þ k)x} dx ¼

x¼1 x¼0

9 = exp{(s þ k)x} dx ;

1 ð

1 þ sþk

x¼0

k2 (s þ k)2

x¼0

which is consistent with a normalized zeroth moment, because F(s ¼ 0) ¼ 1. Straightforward evaluation of the first and second moments of X(x) is obtained from the first and second derivatives, respectively, of F(s), as illustrated below:

dF Q1 ¼ kxn l ¼  ds





    d k2 2k2 2 ¼ ¼ ¼ 2 3 ds (s þ k) (s þ k) s¼0 k s¼0 s¼0

Continuous and discrete formulations of the first moment produce the same result for free radical polymerizations that terminate by recombination, because k ¼ 12 a. Once again, Q2, kxwl, and the polydispersity index based on the continuous distribution X(x) only agree with results from the discrete distribution Xx when molecular weights are high enough, such that a approaches unity. For example, 2

     d F d 2k2 6k2 6 Q2 ¼  ¼ ¼ ¼ 2 3 4 2 ds s¼0 ds (s þ k) (s þ k) s¼0 k s¼0 kxw l ¼

Q2 3 kxw l ¼ 1:5 ¼ ; Q1 k kxn l

Hence, the approximations invoked to generate continuous mole fraction distribution functions from the discrete distributions do not affect the first moment, or the numberaverage degree of polymerization, but agreement between the continuous and discrete results for Q2 and kxwl is exact only in the limit of extremely high-molecular-weight polymers. This claim is verified above for both condensation and free radical polymerizations.

12.8.3 Anionic Polymers Described by the Poisson Distribution Evaluation of the Laplace transform of the Poisson distribution involves integration techniques that were employed in Section 12.7.3 to demonstrate that X(x) is normalized and Q0 ¼ 1. Begin with the defining equation for F(s) and complete the square

12.8 Connection Between Laplace Transforms and the Moments-Generating Function

519

for the argument of the exponential function in the integrand:

F(s) ¼

1 ð

sx

X(x)e 0

2



1   ð 1 (x  y)2 sx e dx dx ¼ pffiffiffiffiffiffiffiffi exp  2y 2py 0

2

(x  y) x y  sx ¼  þ x   sx 2y 2y 2 y 1 2 y ¼   {x  2y (1  s)x þ y 2 (1  s)2} þ (1  s)2 2 2y 2

y s2 1  {x  y (1  s)}2 2 2y   1 ð 1 y s2 1 F(s) ¼ pffiffiffiffiffiffiffiffiffi exp y s þ exp  {x  y (1  s)}2 dx 2 2y 2p y ¼ y s þ

0

Since the moments-generating function F(s) and its derivatives with respect to s are evaluated at s ¼ 0, it is preferable to write terms such as 2(1 2 s), instead of (s 2 1), that appear with the correct sign when s ¼ 0. The integral of interest in the previous expression for F(s) yields an error function solution with assistance from the following substitution: 1 1 z ¼ pffiffiffiffiffi {x  y (1  s)}; dz ¼ pffiffiffiffiffi dx 2y 2y 1 1  ð ð pffiffiffiffiffi 1 exp(z2 ) dz exp  {x  y (1  s)}2 dx ¼ 2y 2y pffiffiffiffiffi 0 (1s)

8 > > pffiffiffiffiffi < ¼ 2y > > :

y=2

9 > > = 2 2 exp(z ) dz þ exp(z ) dz > > pffiffiffiffiffi ; 0

ð0

(1s)

1 ð

y=2

9 8 pffiffiffiffiffi (1s) y=2 > > 1 > > ð ð = pffiffiffiffiffi < 2 2 ¼ 2y  exp(z ) dz þ exp(z ) dz > > > > ; : 0 0 9 8 pffiffiffiffiffi (1s) y=2 > > 1 > > ð ð = pffiffiffiffiffi < 2 2 ¼ 2y exp(z ) dz þ exp(z ) dz > > > > ; :0 0 ¼

pffiffiffiffiffi 2y

pffiffiffiffi pffiffiffiffi h  pffiffiffiffiffiffiffii p p þ erf (1  s) y=2 2 2

520

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

The final result for the Laplace transform of the mole fraction distribution function that characterizes anionic polymerization is F(s) ¼

1 ð

X(x)e 0

sx

  ð1  1 y s2 1 2 dx ¼ pffiffiffiffiffiffiffiffiffi exp y s þ exp  {x  y (1  s)} dx 2 2y 2py 0

  pffiffiffiffi pffiffiffiffi h  pffiffiffiffiffiffiffii 1 y s2 pffiffiffiffiffi p p þ erf (1  s) y=2 2y ¼ pffiffiffiffiffiffiffiffi exp y s þ 2 2 2 2py     n h i o pffiffiffiffiffiffiffi 1 y s2 y s2 1 þ erf (1  s) y=2 exp y s þ ) exp y s þ ¼ y 1 2 2 2 s!0

This moments-generating function yields a normalized zeroth moment pffiffiffiffiffiffiffiof X(x) for large kinetic chain lengths, because Q0 ¼ F(s ¼ 0) ¼ 1 when erff y=2g ¼ 1. As mentioned in Section 12.7.3, y  8 guarantees that Q0 is normalized. Rigorous evaluation of the first moment of X(x) yields   

dF 1 y s2 exp y s þ ¼ Q1 ¼ kxn l ¼  2 ds s¼0 2 " #) rffiffiffiffiffi n n y o pffiffiffiffiffiffiffi o 2y exp  (1  s)2  y (1  s) 1 þ erf[(1  s) y=2] þ 2 p s¼0 " # rffiffiffiffiffi n o n o pffiffiffiffiffiffiffi 1 2y y exp  ¼ y 1 þ erf[ y=2] þ ) 12[y{1 þ 1} þ 0]  y y 1 2 2 p which agrees with numerical analysis of the continuous first moment of X(x) that does not employ the moments-generating function, as summarized in Section 12.7.3. Recall that the number-average degree of polymerization is given by y þ 1 via z-transformation of the discrete mole fraction distribution function Xx in Section 12.7.2, which agrees with the continuous first moment of X(x) when the kinetic chain length is large. Evaluations of the continuous second moment of X(x), kxwl, and the polydispersity index are obtained using the simplified version of the moments-generating function: F(s) ¼

1 ð

0

  y s2 X(x)esx dx ) exp y s þ y 1 2 s!0

   2

d F d y s2 ¼ y (s  1) exp y s þ Q2 ¼ 2 ds2 s¼0 ds s¼0    y s2 2 ¼ y {y (s  1) þ 1} exp y s þ ¼ y (y þ 1) 2 s¼0 kxw l ¼

Q2 kxw l y þ 1 ¼ ¼ y þ 1; ) 1 y 1 Q1 y kxn l

12.9 Expansion of Continuous Distribution Functions

521

For comparison, the discrete second moment is Q2 ¼ y 2 þ 3y þ 1, the weightaverage degree of polymerization based on Xx is kxwl ¼ ( y 2 þ 3y þ 1)/( y þ 1) ) y þ 3, and the discrete polydispersity index is ( y þ 3)/( y þ 2). Hence, approximate agreement is obtained between all of the important average quantities of interest when one analyzes the discrete and continuous molecular weight distribution functions for anionic polymerization. Recall that exact agreement was obtained for Q1 and the number-average degree of polymerization for polymers produced by condensation and free radical mechanisms.

12.9 EXPANSION OF CONTINUOUS DISTRIBUTION FUNCTIONS VIA ORTHOGONAL LAGUERRE POLYNOMIALS The objective of this section is to demonstrate that continuous probability distribution functions, such as the mole fraction distribution P(x) for polymers produced by condensation reactions, as discussed in Section 12.3, can be expanded using a set of orthogonal functions with coefficients that depend on the continuous moments of P(x). Laguerre polynomials Lk(x) represent one set of orthogonal functions that provide a basis for these expansions. Other possible orthogonal polynomials include Legendre, Hermite, and Chebyshev. The following second-order homogeneous ordinary differential equation with variable coefficients has a finite series solution defined by Lk(x) when the parameter k is a non-negative integer: d2 y dy þ (1  x) þ ky ¼ 0 dx2 dx y(@x ¼ 0) ¼ a0 ¼ 1 1 k X X y(x) ¼ Lk (x) ¼ an x n ) (1)n x

Solution:

n¼0

akþ1 ¼0 akþ2 ¼0

n¼0

k! xn (n!) (k  n)! 2

.. .

One initially postulates an infinite power series solution for y(x) ¼ Lk(x), 0  n  1, but all coefficients beyond ak (i.e., akþ1, akþ2, akþ3, etc.) vanish via the recurrence formula. For example, let y(x) ¼

1 X

an x n ;

n¼0

1 1 dy X d2 y X ¼ nan x n1 ; ¼ n(n  1)an x n2 dx n¼1 dx2 n¼2

Substitute these infinite series expressions for y, y0 , and y00 into Laguerre’s differential equation, redefine the summation index n such that each summation begins with n ¼ 0 and contains x n, and combine terms. The differential equation reduces to 1 X n¼0

{(n þ 1)2 anþ1 þ (k  n)an}x n ¼ 0

522

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

This equation is satisfied when the coefficient of x n vanishes for each term in the infinite series, yielding the following recurrence formula: anþ1 ¼ 

(k  n) an (n þ 1)2

a0 ¼ 1; a1 ¼  a4 ¼

k k(k  1) k(k  1)(k  2) ; ; a2 ¼ 2 2 ; a3 ¼  2 1 (2 )(1 ) (32 )(22 )(12 )

k(k  1)(k  2)(k  3) k! ; . . . an ¼ (1)n 2 2 2 2 2 (4 )(3 )(2 )(1 ) (n!) (k  n)!

The series solution for y(x) ends when n ¼ k because anþ1 and all subsequent coefficients vanish. Laguerre polynomials can be generated via Rodrigues’ formula: Lk (x) ¼

ex d k x k {e x } k! dx k

L0 (x) ¼ 1; L1 (x) ¼ 1  x; L2 (x) ¼ 1  2x þ 12x2 L3 (x) ¼ 1  3x þ 32x2  16x3 An important relation that allows one to prove orthogonality of the Laguerre polynomials is (i.e., t , 1) 1 X

t k Lk (x) ¼

k¼0

n xt o 1 exp  1t 1t

The Laguerre polynomials are orthogonal with respect to the weighting factor e 2x. For example, 1 ð

ex Lk (x)Lm (x) dx ¼ dkm

0

where dkm is the Kronecker delta, which equals 1 when k ¼ m, and 0 otherwise (i.e., k = m). In light of the weighting factor required for orthogonality of the Laguerre polynomials, one expands the mole fraction distribution function P(x) as follows: P(x) ¼ exp(x)

1 X

Ck Lk (x)

k¼0

It is possible to obtain a closed-form expression for the coefficients Ck (or Cm) via (i) multiplication of the previous equation by Lm(x), (ii) use of the finite series developed above for Lm(x), (iii) integration of P(x)Lm(x) with respect to x from 0 to 1, which yields the moments of the mole fraction distribution, and (iv) use of the orthogonality relation among Laguerre polynomials to isolate Ck (or Cm) on the right side of the

12.9 Expansion of Continuous Distribution Functions

523

previous equation. The result is 1 1 ð ð 1 X Lm (x)P(x) dx ¼ exp(x) Ck Lk (x)Lm (x) dx 0

k¼0

0

1 1 ð ð 1 1 X X m! r x (1) x P(x) dx ¼ C e L (x)L (x) dx ¼ Ck dkm ¼ Cm k k m (r!)2 (m  r)! r¼0 k¼0 k¼0

m X

r

0

0

Cm ¼

m X

(1)r

r¼0

m! Qr (r!)2 (m  r)!

where Qr corresponds to the continuous rth moment of the mole fraction distribution P(x).

12.9.1 Use of the Zeroth, First, Second, and Third Moments to Reconstruct the Most Probable Distribution for Polycondensation Reactions If one has limited information about Q0, Q1, Q2, and Q3 in terms of the fractional conversion p ¼ 1 2 b of monomer to polymer, then results from the previous section are employed for k ¼ 0, 1, 2, 3 to approximate P(x) using the Lk’s. For example, P(x)  ex {C0 L0 (x) þ C1 L1 (x) þ C2 L2 (x) þ C3 L3 (x) þ   } L0 ¼ 1; L1 ¼ 1  x; L2 ¼ 1  2 x þ 12 x2 ; L3 ¼ 1  3x þ 32 x2  16 x3 Expansion of P(x) via a limited number of Laguerre polynomials is valid if expressions for C0, C1, C2, and C3 in terms of the parameter b yield a continuous mole fraction distribution function that compares well with b exp(2bx). The quantities of interest are (i.e., the moments of P(x) have been calculated via Laplace transform analysis in Section 12.8.1) 3

  1 2 d F 6b 6 ¼ ¼ Q0 ¼ 1; Q1 ¼ ; Q2 ¼ 2 ; Q3 ¼  b ds3 s¼0 b (s þ b)4 s¼0 b3 C 0 ¼ Q0 ¼ 1 C1 ¼ Q0  Q1 ¼ 1  C2 ¼ Q0  2Q1 þ

1 b

1 2 1 Q2 ¼ 1  þ 2 b b 2!

3 1 3 3 1 C3 ¼ Q0  3Q1 þ Q2  Q3 ¼ 1  þ 2  3 b b 2 3! b where continuous moments Qn of P(x) in terms of b are employed instead of discrete moments of Px expressed in terms of p. Reconstruction of the most probable

524

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

distribution P(x) via a limited set of orthogonal Laguerre polynomials yields the following result: 





1 2 1 1 P(x)  ex 1(1) þ 1  (1  x) þ 1  þ 2 1  2x þ x2 b b b 2



 3 3 1 3 1 þ 1 þ 2 3 1  3x þ x2  x3 þ    b b 2 6 b This approximation for P(x) is indistinguishable from the continuous expression b exp(2bx), for all values of x, when b ¼ 0.7, which corresponds to 30% (or less) conversion of monomer to polymer with an extremely low number-average degree of polymerization of 1.4 (or less). Additional Laguerre polynomials are required in the truncated series expansion of P(x) to obtain agreement with the continuous mole fraction distribution function when the fractional conversion p (¼ 1 2 b) of monomer to polymer increases and higher-molecular-weight polymers are produced.

APPENDIX A: UNSTEADY STATE BATCH REACTOR ANALYSIS OF THE MOST PROBABLE DISTRIBUTION FUNCTION We now derive the most probable distribution for condensation polymerization of a, v -hydroxycarboxylic acids via kinetic analysis of the molar density of linear chains that contain x repeat units, Cx( p), where p(t) represents the time-dependent extent of reaction, or the fractional conversion of monomer to polymer. Let Cx be the molar density of x-mers at time t. The instantaneous total molar density of all types of molecules with reactive functional end groups is given by CTotal(t), obtained by summing Cx(t) over all values of x. One constructs unsteady state constant-volume batch reactor mass balances separately for one type of molecules (i.e., that contain x repeat units), sums all of these balances, and calculates the time dependence of CTotal(t) in terms of the extent of reaction p. The initial condition stipulates that the total molar density of all types of molecules at t ¼ 0 is given by the initial monomer molar P density Cinitial. Furthermore, at any time during the polymerization, CTotal(t) ¼ 1x1 Cx(t) ¼ Cinitialf1 2 p(t)g. Let’s analyze the rate of disappearance of x-mers via chemical reaction of x-mers with molecules of all sizes. In general, reactants disappear at a rate that is two-fold faster than the rate of product formation because functional end groups on two smaller molecules combine to produce one higher-molecular-weight polymer via bimolecular reactions. Trimolecular reactions are excluded from the analyses below. Hence, 

1 x1 X X dCx ¼ 2kRx Cx Cy  kRx Cy Cxy dt y¼1 y¼1

For example, when x ¼ 5, the disappearance terms (i.e., first summation) on the right side of the previous rate equation account for reactions that occur when pentamers attack monomers (i.e., y ¼ 1). The factor of 2 is required because pentamers are

Appendix A

525

also depleted when monomers attack pentamers. The same factor of 2 is not required in the generation terms (i.e., second summation) on the right side of the previous equation because pentamers are produced when monomers attack tetramers (i.e., y ¼ 1; kRxC1C4) and when tetramers attack monomers (i.e., y ¼ 4; kRxC4C1). The previous equation is summed from x ¼ 1 to x ) 1, but the lower limit in the summation with respect to x is x ¼ 2 for the generation term, to yield an expression for the timedependent total molar density of all types of molecules. One obtains 

1 X dCx x¼1



dt

¼ 2kRx

1 X x¼1

Cx

1 X

Cy  kRx

1 X x1 X

y¼1

C y C x- y

x¼2 y¼1

1 X x1 X dCTotal 2 2 ¼ 2kRx CTotal  kRx Cy Cx-y ¼ kRx CTotal dt x¼2 y¼1

It is necessary to expand the double summation on the second line of the previous set of equations to reveal that it corresponds to kRxfCTotalg2. Hence, if the factor of 2 were excluded from the disappearance term in the unsteady state mass balance on chains with x repeat units, then the time dependence of CTotal would uncharacteristically vanish, yielding erroneous results because, undoubtedly, the total molar density of all types of chain molecules with reactive functional end groups decreases as polymerization proceeds and low-molecular-weight molecules are converted to high-molecular-weight polymers. If one combines the previous simple second-order kinetic rate equation for CTotal(t) and the relation between CTotal, Cinitial, and p, it is possible to calculate the time dependence of the extent of reaction, dp/dt, so that individual constant-volume unsteady state batch reactor mass balances for chains with x repeat units can be re-expressed using independent variable p, instead of time t. The desired result is 

dCTotal 2 ¼ kRx CTotal dt CTotal ¼ Cinitial (1  p) dp ¼ kRx Cinitial (1  p)2 dt

Subject to an initial condition where the extent of reaction vanishes at t ¼ 0, the previous unsteady state mass balance yields the following time dependence of p: 1 ¼ 1 þ kRx Cinitial t 1  p(t) p(t) ¼ 1 

1 1 þ kRx Cinitial t

Systematically, one uses p(t) to analyze the unsteady state mass balance for monomers, dimers, trimers, and so on, to obtain a generalized expression for the molar density of x-mers, Cx( p). There is no generation term for monomers, so one accounts for monomer depletion via reactions of monomers with molecules of all sizes. The initial

526

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

condition states that monomer molar density C1 ¼ Cinitial when the extent of reaction p ¼ 0: 

1 X dC1 ¼ 2 kRx C1 Cy ¼ 2kRx C1 CTotal dt y¼1

¼

dC1 dp dC1 ¼ 2 kRx C1 Cinitial (1  p) ¼ kRx Cinitial (1  p)2 dp dt dp

dC1 dp ¼ 2 C1 1p C1 ( p) ¼ Cinitial (1  p)2 The unsteady state batch reactor mass balance for dimers includes depletion terms when dimers react with molecules of all sizes, and a generation term when two monomers combine. Remember that two monomers disappear when they react, but only one dimer is produced, hence the factor of 2 for monomer depletion above and a factor of 1 for dimer generation below. It is helpful to replace independent variable t by extent of reaction p, and introduce the following substitution u ¼ 1 2 p, with dp ¼ 2du. No dimers are present at t ¼ 0, so C2 ¼ 0 when p ¼ 0 and u ¼ 1: 

1 X dC2 ¼ 2 kRx C2 Cy kRx C1 C1 ¼ kRx {2C2 CTotal  [Cinitial (1  p)2 ]2} dt y¼1

¼

dC2 dp dC2 ¼ kRx Cinitial (1  p)2 dp dt dp

2 (1  p)4} ¼ kRx {2C2 Cinitial (1  p)  Cinitial

dC2 2C2 þ ¼ Cinitial u2 dp u   1 dC2 2C2 d C2 þ 3 ¼  2 ¼ Cinitial u u du du u2 C2 ¼ Cinitial (1  u) u2 C2 ( p) ¼ Cinitial p(1  p)2 Trimers are depleted when they react with molecules of all sizes. Trimer generation occurs when monomers attack dimers, and vice versa. The appropriate unsteady state constant-volume batch reactor mass balance for trimers is 

1 X dC3 ¼ 2kRx C3 Cy  kRx {C1 C2 þ C2 C1} dt y¼1

If one employs the same independent variable transformation from t to p and the same substitution from p to u described above for monomers and dimers, together with an

Appendix B

527

initial condition that stipulates no trimers at t ¼ 0, then the following trimer molar density expression is obtained: C3 ( p) ¼ Cinitial p2 (1  p)2 Whereas the depletion terms are similar for any x-mer, the generation terms become more complex as x increases. For example, three events are responsible for the production of tetramers: monomers attacking trimers, dimers attacking themselves, and trimers attacking monomers. Four events lead to the production of pentamers: monomers attacking tetramers and vice versa, and dimers attacking trimers and vice versa. Needless to say, the algebra involved in simplifying the unsteady state constant-volume batch reactor mass balance becomes more tedious as x increases. Extrapolation of the previous results for monomers, dimers, and trimers suggests that the following general expression is valid for the molar density of x-mers: Cx ( p) ¼ Cinitial px1 (1  p)2 For constant-volume no-flow systems, one defines the mole fraction P(x) of x-mers as follows: Cx ( p) Cinitial px1 (1  p)2 ¼ ¼ px1 (1  p) P(x; p) ¼ X1 C (1  p) initial C ( p) y¼1 y Hence, kinetic analysis of the mole fraction distribution function for polycondensation reactions yields the most probable distribution that was developed at the beginning of this chapter in Section 12.2 via statistical considerations and Lagrange-multiplier maximization of the multiplicity of states for a constant number of molecules in the reaction mixture at any time.

APPENDIX B: MECHANISM AND KINETICS OF ALKENE HYDROGENATION REACTIONS VIA TRANSITION-METAL CATALYSTS B.1 Introduction and Overview In Chapter 11, the nonlinear viscoelastic properties of triblock copolymers that contain styrene and butadiene (i.e., SBS KratonTM D series) were analyzed in the presence and absence of palladium chloride. It is a well-established fact that palladium(II) coordinates to alkene functional groups in the main chain and/or side group of Kraton’s butadiene midblock. The significantly modified mechanical response of Kraton– palladium complexes is a consequence of the dimerization of alkene functional groups in two different chains, which couples these macromolecules and increases their effective molecular weight in a lightly crosslinked network

528

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

structure. The primary objective of Appendix B is to introduce simple concepts from organometallic chemistry and propose a mechanism for the hydrogenation of alkenes to alkanes via transition metal catalysis. Subsequent kinetic analysis of the proposed mechanism provides a model for the interpretation of experimental rate data. Hydrogenation catalysts are useful to convert Kraton D polymers to the G series (i.e., S-E/B-S triblock copolymers) in which the butadiene midblock is transformed into a random copolymer of ethylene and 1-butene with greater high-temperature stability. The random copolymeric nature of the saturated midblock of Kraton G series arises because the butadiene midblock of Kraton D contains random sequences of 1,4-butadiene (cis and trans) and 1,2-butadiene. Hydrogenation of polymerized 1,4-butadiene yields polyethylene, whereas hydrogenation of polymerized 1,2-butadiene yields poly(1-butene). Following the discussion of hydrogenation reactions via transition-metal hydrides, palladium coordination to alkene functional groups and subsequent catalytic dimerization reactions are analyzed in Appendix C to explain the coupling of dissimilar chains via classic organometallic mechanistic chemistry.

B.2 Sequence of Independent Elementary Steps A four-step mechanism is proposed to convert alkenes to alkanes via homogeneous transition-metal catalysis. The catalysts are based on nanoclusters of nickel, cobalt, iron, iridium, rhodium, palladium, or platinum. Reversibility is considered in three of the four elementary steps, as illustrated below. Forward and backward kinetic rate constants for the ith elementary step are given by ki and k2i, respectively. Mass transfer across the gas – liquid interface is required to introduce molecular hydrogen into the solution. The following nomenclature is employed to identify short-lived and stable species that participate in the reaction sequence: A represents a linear or cyclic alkene. BH represents a linear or cyclic alkane. M represents a transition-metal nanocluster. X, or MH2, is a transition-metal hydride that forms via oxidative addition in Step #1. Y, or AMH2, is an h2-complex that forms when alkenes coordinate to transition metals. Z, or BMH, is a s-alkyl hydride that forms when Y undergoes migratory insertion. Oxidative addition of solubilized H2 to a transition-metal complex produces a transition-metal hydride MH2 with two metal – hydrogen bonds as the oxidation state of the metal center increases by two: k1

H2 þ M , MH2 k1

Rate #1 ¼ k1 [H2 ]M  k1 X

Appendix B

529

It is a well-known fact that alkenes undergo perpendicular coordination to transition metal hydrides [Hegedus, 1999]: k2

MH2 þ A , AMH2 k2

Rate #2 ¼ k2 [A]X  k2 Y Migratory insertion of hydrogen ligands at the least-substituted terminus of coordinated alkenes produces s-alkyl hydrides, BMH: k3

AMH2 , BMH k3

Rate #3 ¼ k3 Y  k3 Z The final product BH (i.e., an alkane) is obtained via irreversible reductive elimination from BMH. Catalytic regeneration of the transition-metal nanocluster M allows for repetition of the four-step mechanism with continuous reduction in the gas-phase partial pressure of hydrogen in a constant-volume batch reactor: k4

BMH ) BH þ M Rate #4 ¼ k4 Z

B.3 Kinetic Analysis of the Rate of Alkene Consumption The primary objective of the hydrogenation of alkenes is prediction of the rate of consumption of A and comparison with the reduction of H2 gas-phase pressure via experiments. The previous mechanism suggests that 

d[A] ¼ Rate #2 ¼ k2 [A]X  k2 Y dt

Pseudo-steady-state approximations (i.e., PSSA) are required to estimate constantvolume molar densities for short-lived reactive intermediates X, Y, and Z. For example, d[MH2 ] dX ¼ ¼ Rate #1  Rate #2  0 dt dt d[AMH2 ] dY ¼ ¼ Rate #2  Rate #3  0 dt dt d[BMH] dZ ¼ ¼ Rate #3  Rate #4  0 dt dt The strategy to predict the rate of alkene consumption during transition-metalcatalyzed hydrogenation is as follows: Step 1: Use the PSSA for BMH to express the molar density of BMH (i.e., Z) in terms of the molar density of AMH2 (i.e., Y ).

530

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

Step 2:

Put the expression for [BMH] into the PSSA for AMH2 and express the molar density of AMH2 (i.e., Y ) in terms of the molar density of MH2 (i.e., X ).

Put the expression for [AMH2] into the PSSA for MH2 and express the molar density of MH2 (i.e., X ) in terms of the molar densities of measurable species, such as the concentration of dissolved hydrogen and the transition metal catalyst (i.e., nanocluster). Step 4: Use results for the molar densities of AMH2 (i.e., Y ) and MH2 (i.e., X ) to predict the rate of consumption of alkenes during hydrogenation reactions. Step 3:

The simultaneous algebraic equations that must be satisfied are d[MH2 ]  0; Rate #1  Rate #2 dt d[AMH2 ]  0; Rate #2  Rate #3 dt d[BMH]  0; Rate #3  Rate #4 dt Begin by equating reaction rates for the third and fourth elementary steps: Rate #3  Rate #4 k3 Y  k3 Z  k4 Z Z ¼ [BMH] 

k3 Y k3 þ k4

Now, employ the previous result for Z and equate reaction rates for the second and third steps: Rate #2  Rate #3 k2 [A]X  k2 Y  k3 Y  k3 Z  k3 Y  Y ¼ [AMH2 ] 

k3 k3 Y k3 þ k4

k2 [A]X f (ki )

 f (ki ) ¼ k2 þ k3 1 

k3 k3 þ k4



This pseudo-steady-state prediction for the molar density of short-lived perpendicular alkene coordination complexes is useful to evaluate the molar density of transitionmetal hydrides, X or MH2, upon equating reaction rates for the first and second elementary steps. Hence, Rate #1  Rate #2 k1 [H2 ]M  k1 X  k2 [A]X  k2 Y  k2 [A]X 

k2 k2 [A]X f (ki )

Appendix B

531

k1 [H2 ]M g([A]; ki )   k2 g([A]; ki ) ¼ k1 þ k2 [A] 1  f (ki ) X ¼ [MH2 ] 

Pseudo-steady-state molar densities for X (i.e., MH2) and Y (i.e., AMH2) allow one to predict the rate of alkene consumption via the reversible reaction rate for the second elementary step: 

d[A] k1 k2 [H2 ]M[A] k1 k2 k2 [H2 ]M[A]  ¼ k2 [A]X  k2 Y  dt g([A]; ki ) f (ki )g([A]; ki )     k1 k2 [H2 ]M[A] k2 k1 k2 [H2 ]M[A] 1  1  f (ki ) w(ki ) g([A]; ki ) g([A]; ki )   k2 k3 w(ki ) ¼ 1 þ 1þ k3 k4

Hence, the alkene consumption rate is first-order with respect to dissolved hydrogen, first-order with respect to the metal nanocluster, first-order with respect to the alkene at low alkene concentrations where g([A]; ki)  k21, and zeroth-order with respect to the alkene at high alkene concentrations. If all four elementary steps in the proposed mechanism are irreversible such that the “backward” reactions can be neglected, then the previous kinetic analysis simplifies considerably: k1  k2  k3  0 f (ki ) ) k3 ; w(ki ) ) 1; g([A]; ki ) ) k2 [A]   d[A] k1 k2 [H2 ]M[A] 1  ) k1 [H2 ]M  w(ki ) dt g([A]; ki ) This suggests that the alkene consumption rate is first-order with respect to dissolved hydrogen and the transition-metal nanocluster, but independent of the concentration of the alkene. If poisoning of the catalyst does not occur and the metal nanocluster is regenerated completely during each cycle of the four-step mechanism, then M is essentially constant in the previous rate expression and it can be combined with k1 as an “apparent” first-order rate constant kapparent ¼ k1[M]. Hence, in the absence of any backward reactions, the irreversible four-step mechanism predicts the following superficially simple pseudo-first-order rate of alkene consumption during catalytic hydrogenation: 

d[A]  kapparent [H2 ] dt

Interphase mass transfer and interfacial equilibrium at the gas – liquid interface are required to relate the gas-phase partial pressure of hydrogen to the molar density of dissolved hydrogen. The method of initial rates suggests that the rate of alkene consumption should scale linearly with the initial concentration of the transition-metal

532

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

nanocluster in solution, and this claim does not depend on the reversible or irreversible nature of the first three steps in the hydrogenation mechanism. The temperature dependence of the simplified pseudo-first-order kinetic rate process is governed by the forward activation energy for the production of transition-metal hydrides MH2 via oxidative addition. A larger rate of alkene consumption should occur at higher temperature. If oxidative addition is reversible (i.e., k21 = 0) in the first elementary step that produces transition-metal hydrides, but the other three steps are irreversible (i.e., k22  k23  0), then the rate of alkene consumption reduces to the following complex expression: k2  k3  0 f (ki ) ) k3 ; w(ki ) ) 1; g([A]; ki ) ) k1 þ k2 [A]   d[A] k1 k2 [H2 ]M[A] 1 k1 [H2 ]M[A]  )  k1 w(ki ) dt g([A]; ki ) þ [A] k2 During the initial stages of hydrogenation, this is equivalent to the previous superficially simple first-order rate process at high alkene concentrations, where [A] k21/k2. During the later stages of hydrogenation when the alkene molar density is much smaller than k21/k2, one predicts pseudo-second-order kinetics where the rate of alkene consumption scales linearly with the molar densities of dissolved hydrogen and the alkene, as well as the initial concentration of the transition-metal nanocluster: 

d[A] k2 k3 0 k1 k2 ) M[H2 ][A] ¼ kapparent [H2 ][A] k1 dt [A] k1 =k2

The temperature dependence of the previous rate expression is governed by the forward and backward activation energies for oxidative addition and the forward activation energy for alkene coordination to the transition-metal hydride. The temperature derivative of the natural logarithm of the previous rate of alkene consumption yields d dT



 

 d[A] d k1 k2 d ¼ ln   ln {ln k1  ln k1 þ ln k2 } k1 dt dT dT

d Eactivation,i ln ki ¼ RT 2 dT 8 9 > > > > 

 = d d[A] 1 < ln   E  E þ E 2 activation forward; activation backward; activation forward; > dT dt RT > > > : oxidative addition oxidative addition alkene coordination; The difference between forward and backward Arrhenius activation energies for oxidative addition of H2 to a transition-metal nanocluster is equivalent to the enthalpy

Appendix B

533

change for this reversible reaction, DHoxidative addition. Hence, 9 8 > > 

 = < d d[A] 1 ln   DH þ E oxidative addition activation forward;> dT dt RT 2 > : alkene coordination; Oxidative addition requires thermal energy to dissociate an HZH bond, but the formation of two metal –hydride (i.e., MZH) bonds is exothermic. The energetics of the first elementary step can be formulated in terms of the difference between bond energies for one HZH bond and two MZH bonds. The temperature dependence of the overall hydrogenation reaction is 8 9 > > 

 < = h i d d[A] 1 ln   E  2E þ E HH MH activation forward; > dT dt RT 2 > bond : bond alkene coordination; One predicts larger rates of alkene consumption at higher temperature if the sum of the HZH bond energy and the forward activation energy for alkene coordination is greater than the strength of two metal – hydride bonds.

B.4 Kinetic Analysis of the Rate of Hydrogen Consumption This development might be preferred, relative to the rate of alkene consumption that was analyzed in the previous section, because the results below can be applied directly to the decrease in hydrogen gas pressure above the reactive liquid mixture. All kinetic rate expressions in this appendix are based on the molar density of dissolved hydrogen, whereas experimental data monitor exponential decreases in hydrogen gas pressure. Based on the first elementary step in the four-step mechanism and the pseudosteady-state approximation for the molar density of transition-metal hydrides MH2 that form via oxidative addition, one obtains the following prediction for the consumption of dissolved hydrogen: d[H2 ] ¼ Rate #1 ¼ k1 [H2 ]M  k1 [MH2 ] dt k1 [H2 ]M [MH2 ]  g([A]; ki )   d[H2 ] k1  ¼ k1 [H2 ]M 1  g([A]; ki ) dt 

which is first-order with respect to dissolved hydrogen and first-order with respect to the transition-metal nanocluster. At very low alkene concentrations during the later stages of the hydrogenation reaction, where g([A]; ki)  k21 is given by the kinetic rate constant for reductive elimination (i.e., reverse reaction) in the first elementary step, the rate of hydrogen consumption should vanish.

534

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

APPENDIX C: ALKENE DIMERIZATION AND TRANSITION-METAL COMPATIBILIZATION OF 1,2-POLYBUTADIENE AND cis-POLYBUTADIENE VIA PALLADIUM(II) CATALYSIS: ORGANOMETALLIC MECHANISM AND KINETICS The following sequence of elementary steps is postulated to model the kinetics of ternary solid state mixtures containing palladium chloride and both diene polymers (i.e., cis- and 1,2-polybutadienes). This mechanism accounts for the fact that kinetic mismatches exist between the coupling of alkene functional groups in binary 1,2-polybutadiene – Pd(II) complexes versus binary cis-polybutadiene – Pd(II) complexes in tetrahydrofuran (THF) and in the solid state. These catalytic reactions proceed at a rate that is approximately five-fold faster in solid binary complexes when alkene functionality exists in the sidechain (i.e., 1,2-polybutadiene) relative to the main chain (i.e., cis-polybutadiene). To counterbalance these kinetic mismatches that have been measured in binary systems, cis-polybutadiene and dichlorobis(acetonitrile)– palladium(II) are mixed separately in tetrahydrofuran prior to introducing a THF solution of 1,2-polybutadiene. Furthermore, the molar density of the cis-polymer is three-fold larger than the vinyl polymer. This mixing strategy should allow alkene functionality in the main chain of cis-polybutadiene to occupy vacant sites in the first-shell coordination sphere of Pd(II). If THF solutions of both polymers and Pd(II) were introduced simultaneously into a closed reaction vessel, then alkene coordination and subsequent palladium-catalyzed chemical crosslinking would occur primarily between 1,2-polybutadiene and Pd(II), because these reactions exhibit time constants that are five-fold shorter than similar reactions between cispolybutadiene and Pd(II). No steps are included in the mechanism that are inconsistent with established kinetic data from infrared spectroscopy. In the first step, alkene groups in the main chain of the cis-polymer displace both acetonitrile ligands and coordinate to palladium(II) in THF solution. 2cis-PBD þ PdCl2 (CH3 CN)2 ) PdCl2 (cis-PBD)2 þ 2CH3 CN

(C:1)

The polymeric palladium complex PdCl2(cis-PBD)2 undergoes high-temperature dimerization addition in the solid state, as summarized qualitatively below via an eight-step mechanism, to increase the effective molecular weight of the polymer chains and regenerate the palladium catalyst (i.e., HPdCl). PdCl2 (cis-PBD)2 ) {cis-PBD} : {cis-PBD} þ Pd(II)

(C:2)

The second step (i.e., reaction (C.2)) occurs only at high temperature in the solid state. The palladium catalyst that is regenerated in reaction (C.2) could form coordination complexes with both diene polymers in the solid state, as illustrated by reactions (C.3) and (C.4). 2(1, 2-PBD) þ HPdCl ) HPdCl(1, 2-PBD)2

(C:3)

cis-PBD þ 1, 2-PBD þ HPdCl ) HPdCl(cis-PBD)(1, 2-PBD)

(C:4)

Appendix C

535

Whereas reaction (C.3) occurs rather quickly in THF solutions of 1,2-polybutadiene and PdCl2(CH3CN)2 without cis-polybutadiene, it must be excluded from the mechanism when both diene polymers are present and a 2-hour mixing delay is employed because it leads to erroneous predictions about the rate at which the 1640 cm21 CvC stretch in 1,2-polybutadiene decreases in binary versus ternary solid state mixtures. Reaction (C.4) is suggested because the vinyl polymer coordinates to the transitionmetal ion much faster than cis-polybutadiene coordinates to PdCl2, based on kinetic data for monosubstituted versus disubstituted small-molecule alkenes [Hartley, 1973; Hegedus, 1999]. However, high-temperature infrared data for binary complexes of cis-polybutadiene and PdCl2 suggest that kinetic rate constant k4  0 because the “free” CvC stretch at 1653 cm21 does not decrease [Belfiore et al., 1999]. Hence, reaction (C.4) is neglected. In other words, when chemical crosslinks form at high temperature between two repeat units of cis-polybutadiene that are coordinated to palladium(II) in solid films, the regenerated catalyst does not form complexes with the remaining main-chain alkene groups in the solid state during the time scale of the kinetic measurements (i.e., 30– 40 minutes). The polymeric palladium complex with cis-polybutadiene that is generated in reaction (C.1), PdCl2(cis-PBD)2, undergoes the following sequential ligand displacement reactions in the presence of atactic 1,2-polybutadiene. PdCl2 (cis-PBD)2 þ 1, 2-PBD ) PdCl2 (cis-PBD)(1, 2-PBD) þ cis-PBD

(C:5)

PdCl2 (cis-PBD)(1, 2-PBD) þ 1, 2-PBD ) PdCl2 (1, 2-PBD)2 þ cis-PBD (C:6) Reactions (C.5) and (C.6) occur in THF as well as in the solid state at elevated temperatures. The polymeric palladium complexes on the right sides of reactions (C.5) and (C.6) undergo palladium-catalyzed dimerization addition in the solid state to increase the effective molecular weight of the chains and regenerate the catalyst (i.e., HPdCl). PdCl2 (cis-PBD)(1, 2-PBD) ) {cis-PBD} : {1, 2-PBD} þ Pd(II)

(C:7)

PdCl2 (1, 2-PBD)2 ) {1, 2-PBD} : {1, 2-PBD} þ Pd(II)

(C:8)

Reaction (C.7) represents a classic example of transition-metal compatibilization via reactive blending. Palladium chloride catalyzes the high-temperature dimerization of two diene polymers that are incompatible in the absence of the inorganic component. Reaction (C.7) is partially responsible for decreasing the palladium – p-complexed CvC absorbance at 1543 cm21 in cis-polybutadiene during transient infrared experiments [Belfiore et al., 1999], but its effect on 1,2-polybutadiene’s CvC stretch at 1640 cm21 is questionable because a residual carbon – carbon double bond survives the dimerization reaction. Reactions (C.2), (C.7), and (C.8) are not truly elementary steps. Each one summarizes the following sequence of eight steps via detailed organometallic considerations [Heck, 1985; Hegedus, 1999]: Step 1:

An alkene functional group in polybutadiene displaces acetonitrile (i.e., s-bonded to the metal via the lone pair on nitrogen; there are no p-bound

536

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

nitriles) in the first-shell coordination sphere of Pd(II) and coordinates to PdCl2(CH3CN),

Cl H3C

C

Pd

N

Cl

Step 2:

Nucleophilic attack by one of the anionic chloride ligands occurs at the more-substituted terminus of the coordinated alkene (i.e., chloropalladation of the alkene),

Cl H3C

C

N

Cl

Pd

H3C

Cl

Step 3:

C

N

Pd Cl

b-Hydrogen elimination generates pseudo-square-planar HPdCl, which participates in the following catalytic cycle:

THF Cl H3C

C

N

Pd

H

Cl

Step 4: Another alkene functional group in polybutadiene displaces acetonitrile or solvent (i.e., THF) in the first shell of Pd(II) and coordinates to HPdCl, forming an h2 p-olefin complex,

THF Cl

Pd H

Appendix C

537

Step 5: Migratory insertion of coordinated H at the less-substituted terminus of the coordinated alkene produces a palladium(II) s-alkyl intermediate,

THF Cl

Pd

Pd

Cl

THF CH3 H

Step 6:

Another alkene functional group in polybutadiene displaces solvent in the first shell and coordinates to Pd(II), generating an h1 s-alkyl, h2 p-olefin complex,

Cl

Pd CH3

Step 7: Migratory insertion of the s-alkyl occurs at the less-substituted terminus of the coordinated alkene,

Cl Cl

Pd Pd CH3

Step 8:

CH3

b-Hydrogen elimination generates HPdCl(THF)2 to continue the catalytic cycle at Step 4, and a four-carbon chemical crosslink between two different

538

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

chains or a four-carbon bridge within the same chain:

Cl Pd CH3

CH3

The four-carbon crosslink or bridge is ZCH(CH3)CHvCHZ. However, if migratory insertion of coordinated H in Step (5) occurs at the more-substituted terminus of the coordinated alkene, then the four-carbon crosslink or bridge between chains is ZCH2CH2CHvCHZ. This is illustrated below via modification of Steps (5), (6), (7), and (8). For example; Step 5:

Migratory insertion of coordinated H at the more-substituted terminus of the coordinated alkene produces a palladium(II) s-alkyl intermediate,

THF Cl

THF

Pd Cl H

Pd

CH2

THF

Step 6:

Another alkene functional group in polybutadiene displaces solvent in the first shell and coordinates to Pd(II), generating an h1 s-alkyl, h2 p-olefin complex,

THF Cl

Pd

CH2

Appendix C

539

Step 7: Migratory insertion of the s-alkyl occurs at the less-substituted terminus of the coordinated alkene,

THF Cl

Step 8:

Cl CH2

Pd

THF Pd

b-Hydrogen elimination generates HPdCl(THF)2 to continue the catalytic cycle at Step 4, and a four-carbon chemical crosslink between two different chains or a four-carbon bridge within the same chain:

Cl

THF Pd

The kinetic analysis described below considers reactions (C.2), (C.7), (C.8) as elementary steps. If reaction (C.8) is solely responsible for decreasing 1,2-polybutadiene’s CvC stretching vibration at 1640 cm21 during transient infrared experiments for ternary solid state mixtures, then   d dA1640 cm1 [1, 2-PBD] : [1, 2-PBD] ¼   k8 [PdCl2 (1, 2-PBD)2 ] dt dt ternary

(C:9)

where ki represents the kinetic rate constant for the ith elementary step in the six-step sequence proposed above via reactions (C.1) through (C.8), excluding reactions (C.3) and (C.4). The following nomenclature is employed to represent molar densities of the various species that participate in the proposed mechanism. X ¼ PdCl2(cis-PBD)(1,2-PBD) Y ¼ PdCl2(cis-PBD)2 Z ¼ PdCl2(1,2-PBD)2

540

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

V ¼ 1,2-PBD (atactic 1,2-polybutadiene) C ¼ cis-PBD (cis-polybutadiene) M ¼ PdCl2 or HPdCl The rate of decrease of 1,2-polybutadiene’s CvC infrared absorbance at 1640 cm21 in ternary mixtures, given by Eq. (C.9), is rewritten as   dA1640 cm1  k8 Z (C:10)  dt ternary The pseudo-steady-state approximation is invoked for the polymeric palladium complexes (i.e., reactive intermediates) denoted by X, Y, and Z when reactions (C.3) and (C.4) are neglected: dX ¼ k5 VY  k6 VX  k7 X  0 dt dY ¼ k1 C 2 M  k2 Y  k5 VY  0 dt dZ ¼ k6 VX  k8 Z  0 dt

(C:11) (C:12) (C:13)

The solution to Eqs. (C.11), (C.12), and (C.13) for the approximate molar densities of the three intermediate complexes, in the order in which they were solved, is Y ¼ [PdCl2 (cis-PBD)2 ] 

k1 C2 M k2 þ k5 V

X ¼ [PdCl2 (cis-PBD)(1, 2-PBD)] 

(C:14) k1 k5 C 2 VM (k2 þ k5 V)(k7 þ k6 V)

  k6 k1 k5 C 2 V 2 M Z ¼ [PdCl2 (1, 2-PBD)2 ]  k8 (k2 þ k5 V)(k7 þ k6 V)

(C:15) (C:16)

The rate of decrease of the CvC infrared absorbance at 1640 cm21 for 1,2-polybutadiene in ternary solid state mixtures, given by Eqs. (C.9) and (C.10), can be rewritten as   dA1640 cm1 k1 k5 k6 C 2 V 2 M  k8 Z   (C:17) dt (k2 þ k5 V)(k7 þ k6 V) ternary If (i) the vinyl polymer and the cis-polymer are present in excess relative to palladium chloride, (ii) k5V k2, and (iii) k6V k7, then the disappearance of the CvC stretch at 1640 cm21 obeys the following kinetic rate law:   dA1640 cm1  k1 C 2 M  dt ternary

(C:18)

which is first-order with respect to the transition-metal catalyst, palladium chloride. Equations (C.17) and (C.18) describe the rate of disappearance of the CvC

Appendix C

541

absorbance at 1640 cm21 for 1,2-polybutadiene in ternary solid state mixtures with cis-polybutadiene and PdCl2, subject to a mixing strategy where the palladium catalyst is exposed to the cis-polymer prior to introducing 1,2-polybutadiene into the reaction vessel. For binary mixtures of 1,2-polybutadiene and palladium chloride, the reaction scheme is 2(1, 2-PBD) þ PdCl2 (CH3 CN)2 ) PdCl2 (1, 2-PBD)2 þ 2CH3 CN

(C:3a)

PdCl2 (1, 2-PBD)2 ) {1, 2-PBD} : {1, 2-PBD} þ Pd(II)

(C:8a)

Reaction (C.8a) is solely responsible for the disappearance of 1,2-polybutadiene’s CvC stretch at 1640 cm21. Coordination of the vinyl side group in 1,2-PBD to PdCl2 does not affect the absolute absorbance of this infrared signal. Hence,   dA1640 cm1   k8a Zbinary (C:19) dt binary Now, the pseudo-steady-state approximation for PdCl2(1,2-PBD)2, based on reactions (C.3a) and (C.8a) in binary mixtures, is     d dZ [PdCl2 (1, 2-PBD)2 ] ¼ ¼ k3a V 2 M  k8a Zbinary  0 (C:20) dt dt binary binary Based on the pseudo-steady-state approximation in Eq. (C.20), transient decay of the 1640 cm21 signal for binary mixtures follows:   dA1640 cm1   k3a V 2 M (C:21) dt binary which should be compared with Eq. (C.18) for ternary mixtures. Transient decay of 1,2-polybutadiene’s CvC stretch at 1640 cm21 in binaries and ternaries follows first-order kinetics with respect to the transition-metal complex, palladium chloride, which could be the dominant effect when the other components are present in excess. Pseudo-first-order kinetic rate constants in Eq. (C.18) for the ternary (i.e., k1C 2) and Eq. (C.21) for the binary (i.e., k3aV 2) differ significantly, based on transient analysis of the real-time infrared absorbance for atactic 1,2-polybutadiene’s CvC stretch at 1640 cm21. This infrared signal decreases at a faster rate in binary complexes that do not contain cis-polybutadiene. The rate of disappearance of the palladium-p-complexed CvC stretch in cispolybutadiene is predicted for (i) binary mixtures of the cis-polymer with PdCl2 and (ii) ternary mixtures of both diene polymers with PdCl2. Reactions (C.1) through (C.8), excluding (C.3) and (C.4), apply to ternary mixtures according to the mixing strategy. The rate of disappearance of the CvC absorbance at 1543 cm21 is calculated by considering each elementary step that dissociates a di-hapto coordination bond between palladium(II) and the main-chain alkene. In light of the fact that reaction

542

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

(C.1) occurs in THF solution prior to performing the solid state infrared experiments, reaction (C.1) is neglected relative to reactions (C.2), (C.5), (C.6), and (C.7): 

  dA1543 cm1  2k2 Y þ k5 VY þ k6 VX þ k7 X dt ternary

(C:22)

where 2k2Y accounts for the fact that dimerization addition reaction (C.2) dissociates two palladium – alkene coordination bonds. Equations (C.14), (C.15), and (C.16) provide approximations for the molar densities of short-lived polymeric palladium complexes in ternary mixtures via the pseudo-steady-state approximation. These are employed to simplify the prediction given by Eq. (C.22). For example, k2 Y þ k5 VY ¼ (k2 þ k5 V)Y  k1 C 2 M

(C:23)

k2 Y þ (k7 þ k6 V)X  k2 Y þ k5 VY  k1 C 2 M

(C:24)

Also,

Hence, transient decay of the palladium – p-complexed CvC stretch in cis-polybutadiene at 1543 cm21 should obey the following rate law in ternary mixtures:  

dA1543 cm1 dt



 2k1 C 2 M

(C:25)

ternary

which is first-order with respect to the transition-metal catalyst, palladium chloride. It is not necessary to exclude reaction (C.4) to arrive at Eq. (C.25) because k4CVM will enhance the infrared absorbance of the palladium – p-complexed CvC stretch in cis-polybutadiene via Eq. (C.22), but the same term (i.e., k4CVM) also appears in Eq. (C.11) for the pseudo-steady-state approximation of PdCl2(cis-PBD)(1,2-PBD) and a modified version of Eq. (C.24). Transient analysis of the real-time infrared absorbance at 1543 cm21 reveals that the palladium – p-complexed CvC stretch in cis-polybutadiene decreases at a rate that is approximately two-fold faster than the rate at which the 1640 cm21 CvC stretch decreases in 1,2-polybutadiene for ternary solid state mixtures that contain a 3 : 1 molar ratio of the cis-polymer to the vinyl polymer, according to the mixing strategy outlined above. Kinetic predictions given by Eqs. (C.18) and (C.25) are consistent with these experimental data. When 1,2-polybutadiene is absent in THF solutions of cis-polybutadiene and palladium chloride, elementary steps (C.1) and (C.2) are the only ones that must be considered. In solid binary mixtures, reaction (C.2) is solely responsible for decreasing the CvC absorbance of cis-polybutadiene at 1543 cm21. Hence,   dA1543 cm1   2k2 Ybinary dt binary

(C:26)

543

References Table 12.2 Summary of Predicted Kinetic Rate Laws for 1,2-Polybutadiene and cis-Polybutadiene with Low Concentrations of Dichlorobis(acetonitrile) –Palladium(II)

Mixture Binaries with PdCl2 Ternaries with 3 : 1 molar ratio of cis-PBD/1,2PBD and strategic mixing

Pd – p-complexed CvC stretch in cis-PBD at 1543 cm21 f2dA1543/dtg

CvC stretch in 1,2-PBD at 1640 cm21 f2dA1640/dtg

Reactions (C.1) and (C.2) 2k1C 2M (Eq. (C. 28)) Reactions (C.1) ) (C.8) excluding (C.3) 2k1C 2M (Eq. (C.25))

Reactions (C.3) and (C.8) k3V 2M (Eq. (C.21)) Reactions (C.1) ) (C.8) excluding (C.3) and (C.4) k1C 2M (Eq. (C.18))

A1543 ¼ infrared absorbance of the Pd– p-complexed CvC stretch in cis-polybutadiene. A1640 ¼ infrared absorbance of the CvC stretch in 1,2-polybutadiene. ki ¼ kinetic rate constant for the ith elementary step in the physical mechanism. C ¼ molar density of cis-polybutadiene. V ¼ molar density of 1,2-polybutadiene (i.e., vinyl polymer). M ¼ molar density of dichlorobis(acetonitrile)palladium(II) or HPdCl.

Now, the pseudo-steady-state approximation for PdCl2(cis-PBD)2 based on reactions (C.1) and (C.2) in binary mixtures is     d dY [PdCl2 (cis-PBD)2 ] ¼ dt dt binary binary ¼ k1 C 2 M  k2 Ybinary  0

(C:27)

Hence, transient decay of the CvC signal at 1543 cm21 for binary mixtures obeys the following rate law:   dA1543 cm1  2k1 C 2 M  (C:28) dt binary which has the exact same functional form as Eq. (C.25) for ternary mixtures. In practice, the molar density of cis-polybutadiene, C, is slightly less in the 3 : 1 ternary solid state mixure relative to binary mixtures with PdCl2. Experiments reveal that chemical reaction time constants for f2dA1543/dtgbinary and f2dA1543/dtgternary are very similar between 100 8C and 140 8C via transient decay of the 1543 cm21 absorbance for palladium – p-complexes with cis-polybutadiene. This is consistent with the predictions given by Eqs. (C.28) and (C.25), respectively, as summarized in Table 12.2.

REFERENCES BELFIORE LA, SUN X, DAS PK, LEE JY. High-temperature infrared kinetics of transition-metal-catalyzed chemical reactions in solid state complexes of polybutadienes with palladium chloride. Polymer 40(20):5583– 5599 (1999). FLORY PJ. Principles of Polymer Chemistry. Cornell University Press, Ithaca, NY, 1953, p. 81.

544

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

HARTLEY FR. The Chemistry of Platinum and Palladium. Wiley, Hoboken, NJ, 1973, Chap. 11, p. 308; Chap. 13, pp. 362– 377. HECK R. Palladium Reagents in Organic Synthesis. Academic Press, New York, 1985. HEGEDUS LS. Transition Metals in the Synthesis of Complex Organic Molecules, 2nd edition. University Science Books, Sausalito, CA, 1999. ODIAN G. Principles of Polymerization, 4th edition. Wiley, Hoboken, NJ, 2004.

PROBLEMS 12.1. Several monodisperse fractions of an anionically polymerized material are dissolved in a common solvent. After evaporation of the solvent, the homogeneous solid residue, which can be considered polydisperse, contains gi grams of the fraction with molecular weight MWi. This is a discrete example of the non-normalized continuous distribution function g (MW), where g represents the mass of material with molecular weight MW. Answer the following questions in terms of the discrete gi – MWi pairs. (a) Obtain an expression for the mass fraction of species i, vi, with molecular weight MWi in the polydisperse solid mixture. Answer By definition, vi corresponds to the ratio of the mass of the solid mixture with molecular weight MWi relative to the total mass of the mixture. The mass of species i, gi, can be expressed as the product of MWi and the number of moles of species i. Division of numerator and denominator of the following expression by total moles of all species in the polydisperse mixture introduces mole fractions yi for species i with molecular weight MWi. Hence,

g vi ¼ Xi j

gj

yi MWi ¼X yj MWj j

(b) Obtain an expression for the mole fraction of species i, yi, with molecular weight MWi in the polydisperse solid mixture. Answer By definition, yi corresponds to the ratio of the number of moles of species i in the solid mixture relative to the total number of moles of all species in the mixture. Furthermore, it is possible to divide numerator and denominator of the following expression by total mass to introduce mass fractions. Hence, gi vi MWi MWi yi ¼ X g ¼ X vj j MWj MW j j j (c) Obtain an expression for the number-average molecular weight of the polydisperse solid mixture, Mn, by invoking additivity of the mass of each component. Answer Define Mn such that the product of Mn and the total number of moles of all species in the mixture yields the total mass of the mixture. It is relatively straightforward to add the mass of each

Problems

545

species to obtain total mass. After multiplication and division of gi by MWi, and division of both sides of the following equation by total moles, which introduces mole fractions, one obtains X X  gi  X gj Mn MWi ¼ gi ¼ MWj MWi j i i 8 9 > gi > > > > = X < MW > X i Mn ¼ yi MWi X gj MWi ¼ > > > i > i > > : ; MWj j (d) Obtain an expression for the number-average molecular weight of the polydisperse solid mixture, Mn, by invoking additivity of the moles of each component. Answer Mn is also defined such that division of total system mass by Mn yields the total number of moles of all species in the mixture. It is relatively straightforward to add the number of moles of each species to obtain total moles. Then, division of both sides of the following equation by total mass introduces mass fractions. Hence, ( ) P gj X gi j ¼ MWi Mn i X vi 1 ¼ Mn MWi i (e) Obtain an expression for the weight-average molecular weight of the mixture, Mw, in terms of mass fractions vi. Answer Begin with the defining equation in Section 12.5 for the weight-average degree of polymerization kxwl in terms of mass fraction Wx ¼ vx, and multiply both sides of the equation by the repeat-unit molecular weight. One obtains Mw ¼ kxw lMWrepeat ¼

1 X

{x MWrepeat }Wx ¼

X

vi MWi

i

x¼1

(f) Obtain an expression for the weight-average molecular weight of the mixture, Mw, in terms of mole fractions yi. Answer Combine the answers to parts (a) and (e) above: 8 9 P yi MWi2 < y MW > = X X> i i i P vi MWi ¼ Mw ¼ MWi ¼ P > yj MWj > yj MWj ; i i : j

j

546

Chapter 12 Kinetic Analysis of Molecular Weight Distribution Functions

12.2. (a) A polymer is analyzed for molecular weight determination by dissolving small amounts of it in tetrahrofuran. The dilute solution is injected into a gel permeation chromatographic column and the output curve is recorded at 25 8C and 65 8C. Draw the output curve at both temperatures on one set of axes. Label the axes and describe briefly in words why the output curve changes as a function of temperature even though the molecular weight of the polymer is the same at both temperatures. (b) The output curve from a gel permeation chromatographic experiment reveals that the molecular weight distribution is bimodal. The material that exits the column with shorter residence times has a narrow distribution of chain lengths. The material that exits the column with longer residence times has a broad distribution of chain lengths. Draw the output curve with molecular weight as the independent variable on the horizontal axis. 12.3. Consider the discussion in Section 1.2 and the information in Problem 1.6 to describe how the “order of a thermodynamic phase transition” affects kinetic rate processes that occur on each side (i.e., sol vs. gel) of the phase boundary. Courtesy of communications with Prof. Jan Feijen, Twente University, The Netherlands.

Chapter

13

Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers Voluptuous indolence drips from lifeless stone. —Michael Berardi

Statistical analysis of chain dimensions is presented for ideal macromolecules in noninteracting solvents and real chains in good solvents with excluded volume. Scaling concepts are introduced for the dimensions of real chains, intrinsic viscosity, and concentrated polymer solutions. Capillary viscometry is discussed in the Appendix. The Gaussian distribution is combined with Boltzmann’s entropy law to calculate retractive forces for ideal elastomers via entropy elasticity. This simplified statistical thermodynamic analysis of ideal elastomers in the presence of external fields yields relations between stress and strain that are discussed further in the following chapter.

13.1 GAUSSIAN CHAINS AND ENTROPY ELASTICITY The end-to-end vector of a high-molecular-weight polymer chain depends on its conformational characteristics that are described by the preferred rotational state (i.e., trans or gauche) of each backbone bond. Planar-zigzag (i.e., all trans) chains have much larger end-to-end vectors relative to random coils whose backbone bonds exhibit a higher fraction of gauche rotational states. Both planar-zigzag and random-coil conformations exhibit end-to-end vectors that are shorter than the contour length of the chain, which is obtained by traversing each backbone bond. There is a severe “entropy penalty” for molecular conformations whose end-to-end vectors approach the chain contour length. Probability considerations favor chains with shorter end-to-end vectors provided that space, other than the excluded volume, is available for these Gaussian chains to collapse upon themselves. The thermodynamic consequence of Physical Properties of Macromolecules. By Laurence A. Belfiore Copyright # 2010 John Wiley & Sons, Inc.

547

548

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers

the entropy penalty for chains with longer end-to-end vectors is the development of retractive forces, similar to spring forces, as the system strives to maximize its entropy.

13.2 SUMMARY OF THREE-DIMENSIONAL GAUSSIAN CHAIN STATISTICS In the absence of polymer – solvent interactions that typically induce chain expansion, the conformational characteristics of a single polymer chain can be described by threedimensional random walk statistics. The following assumptions are appropriate. (a) Long-chain molecules act independently of each other when the polymer solution is very dilute. Except for the analysis of chain conformations in the solid state, it is necessary to dissolve polymers in an appropriate solvent to study their conformational characteristics in dilute solution. Under “Q-solvent conditions,” which correspond to a particular temperature that depends on the polymer – solvent combination, the solvent does not induce chain expansion and the polymer, which is on the verge of precipitating from solution, exhibits unperturbed chain dimensions. Hence, in the presence of a solvent, properties of dilute polymer solutions can be measured by conventional techniques, but the conformation of long-chain molecules in Q-solvents is the same as their conformation in vacuo. “Good solvents” interact significantly with segments of the polymer chain, yielding dimensions from one end of the chain to the other end that are much larger than the “end-to-end” distance in a Q-solvent. (b) A single chain is described by n random steps, or chain segments, each of length l, where l represents the length of a carbon – carbon single bond or the effective length of a monomeric repeat unit, sometimes referred to as the “Kuhn statistical segment length.” (c) Freely jointed chains exhibit no bond angle or valence cone restrictions. Each conformation exhibits the same energy. Furthermore, it is perfectly acceptable for a chain segment to “fall back” on the previous segment such that both segments occupy the same region of space. This freedom is not enjoyed by real chains, as described by the principle of “excluded volume,” which increases the end-to-end distance of a real chain relative to a freely jointed chain. One-dimensional random walk statistics of a freely jointed chain, via the formalism of Bernoulli trials and the binomial distribution, suggest that the conformational characteristics in three-dimensional space should follow a Gaussian distribution. The probability density distribution function, with dimensions of inverse volume, for finding an end-to-end chain vector r after n steps, each of length l, is given by P(r; n) ¼ A exp{b2 r 2 } where r 2 is the square of the magnitude of the end-to-end chain vector r. This result implies that if one end of the polymer chain is anchored at the origin of a

13.2 Summary of Three-Dimensional Gaussian Chain Statistics

549

three-dimensional coordinate system, then it is most probable to find the other end at the origin also, but one must realize that there is no volume at r ¼ 0 (i.e., 4pr 2dr) to accommodate the end of the chain even if excluded volume is not considered. Analogously, if one flips a coin and take one step to the right each time the coin lands on its head, and one step to the left when the coin lands on its tail, then after a sufficient number of trials that justifies statistical analysis, the probabilities of obtaining heads and tails are 50 : 50 and the final position coincides with the initial position on a one-dimensional lattice. The Gaussian distribution parameters A and b2, that characterize unperturbed chain dimensions in a Q-solvent, are calculated from (i) normalization of the distribution function and (ii) evaluation of the mean-square end-toend chain length, kr 2l. If end-to-end chain vector r is measured from the origin of a rectangular Cartesian coordinate system and P(r; n) dr represents the probability that the end-to-end vector lies within the range from r to r þ dr, where orientation is an important consideration, then the normalization condition is written as follows: þ1 ð P(r; n) dr ¼ 1 1

where dr represents a differential volume element. In terms of the probability distribution function P(r; n) that is insensitive to orientation of the end-to-end chain vector, one expresses the differential volume element dr in spherical coordinates, such that dr ¼ r2 sin Q dr dQ dw and integrates over all possible angles Q and w (i.e., 0  Q  p, 0  w  2p). Now, since the Gaussian distribution function depends only on the square of the magnitude of the end-to-end chain vector, one factors the normalization expression and obtains the probability density distribution function P(r; n) that does not depend on orientation, with dimensions of inverse length, such that P(r; n)dr represents the normalized probability of finding the end-to-end chain vector r on the surface of a spherical shell of radius r about the origin with thickness dr. Hence, þ1 ð

2ðp

1

0

P(r; n) dr ¼

ðp

1 ð

0

0

1 ð

2

dw sin Q dQ r P(r; n) dr ¼ 4pA r 2 exp(b2 r2 ) dr ¼ 1 0 2

The distribution function P(r; n) is given by 4pAr exp(2b2r 2), and the previous statement of normalization yields the following result for the pre-exponential factor A: A¼

b3 p 3=2

The mean-square end-to-end chain length is defined as the second moment of P(r; n): 2

kr l ¼

1 ð

0

2

1 ð

r P(r; n) dr ¼ 4pA r4 exp(b2 r 2 ) dr 0

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Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers

The second moment calculation, together with normalization, yields the following results:  3=2 3 3 3 2 kr l ¼ , A¼ , b ¼ 2kr2 l 2p kr2 l 2b 2  3=2   3 3r2 2 P(r; n) ¼ 4p r exp  2 2kr l 2p kr2 l 2

13.3 VECTOR ANALYSIS OF THE MEAN-SQUARE END-TO-END CHAIN DISTANCE Initially, one constructs an expression for the end-to-end chain vector r by summing all n segment vectors hi as follows: r¼

n X

hi

i¼1

where hi represents the segment vector from the (i – 1)st mass point to the ith mass point. Now, it is necessary to evaluate the scalar “dot” product of r with itself by accounting for all dot products among the individual segment vectors. Hence,



r r¼

n X i¼1

hi

n X



hk

k¼1

There are n terms in the previous expression that contain the dot product of an individual segment vector with itself. Since each segment vector has the same length l, these n terms contribute nl2 to the square of the end-to-end chain distance. The previous expression also contains n(n 2 1) (i.e., an even number of) dot products of two different segment vectors. Since h i . h k is the same as h k . h i and the n terms in which i ¼ k have already been considered, the previous expression for the square of the end-to-end chain distance of a particular conformation reduces to



r r¼

n X i¼1

hi



n X k¼1

hk ¼ nl2 þ

n X

hi i,k¼1;(i=k)

h

k

The previous expression for the square of the end-to-end chain distance r 2 of a particular conformation must be averaged with respect to the distribution of end-to-end chain vectors r, where orientation is a necessary consideration due to the presence of the dot product of dissimilar segment vectors on the right side of the previous equation. In other words, a weighted average is required to calculate the mean-square displacement, where the weighting factor for a particular conformation with end-to-end chain vector r is given by P(r; n) dr, which represents the normalized probability that this conformation exists. The mean-square end-to-end chain distance is

13.3 Vector Analysis of the Mean-Square End-to-End Chain Distance

551

constructed as follows: 2

kr l ¼

ð1



1 ð

2

{r r}P(r; n) dr ¼ nl

1

P(r; n) dr þ

n X i,k¼1;(i=k)

1

1 ð



{hi hk }P(r; n) dr 1

The first term on the right side of the previous equation is simply nl2, due to the fact that P(r; n) is normalized. If aik represents the orientation angle between segment vectors h i and hk, then the dot product of these two vectors in the previous expression yields hi hk ¼ l2 cos aik



and one must average the cosine of this orientation angle with respect to the angular part of the distribution of chain vectors. One obtains the following result: 2

2

kr l ¼ nl þ l

2



b3 p 3=2

 X n i,k¼1;(i=k)

2ðp ð p

1 ð

{cos aik }sin Q dQ d w r2 exp(b2 r2 ) dr 0

0 0

Integration of the radial part of the distribution function over all possible chain lengths r yields ð1

pffiffiffiffi p r exp( b r ) dr ¼ 3 4b 2

2 2

0

This is obtained rather quickly via differentiation of the following result with respect to the parameter a, and then letting a ¼ b2: 1 ð

1 exp(ax ) dx ¼ 2 2

rffiffiffiffi p a

0

The final result for the mean-square end-to-end chain distance via this analysis is 2 3 2ðp ð p n X 1 {cos aik }sinQ dQ d w5 kr 2 l ¼ nl2 41 þ 4np i,k¼1;(i=k) 0 0

where the second term in large brackets [ ] on the right side of the previous equation represents the expansion of real chains relative to unperturbed “ideal” chains due to bond angle restrictions, valence cone restrictions, and polymer – solvent interactions in systems that follow Gaussian statistics at temperatures above the Q-point. For freely jointed chains with no bond angle or valence cone restrictions, h i and h k are uncorrelated, unless i ¼ k, and there is no preferred orientation for cos aik. Hence, the average value of cos aik with respect to the orientational part of the distribution of chain vectors vanishes and one finds that the mean-square end-to-end chain distance reduces to nl2.

552

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers

13.4 ONE-DIMENSIONAL RANDOM WALK STATISTICS VIA BERNOULLI TRIALS AND THE BINOMIAL DISTRIBUTION If one flips an “ideal coin” that exhibits an equal probability of landing on its head or tail, then the sequential probability that the coin will land on its head x times after n trials is given by P(x; n). For example, when both elemental probabilities are equal (i.e., 50 : 50),  n n! 1 P(x; n) ¼ x!(n  x)! 2 where x ranges from 0 to n. This result based on Bernoulli trials with equal elemental probabilities is a subset of the binomial distribution with unequal elemental probabilities p for the coin landing on its head and q when it lands on its tail (i.e., p þ q ¼ 1). Now, the sequential probability that a real coin will land on its head x times after n trials is given by P(x; n) ¼

n! px (1  p)nx x!(n  x)!

where, once again, x ranges from 0 to n. In both cases, the coefficient of the elemental probabilities corresponds to the xth coefficient in the binomial expansion of ( p þ q)n. The moments-generating function for P(x; n) is useful, in general, to calculate average chain dimensions (i.e., end-to-end chain distance and mean-square displacement). For a discrete distribution function, like P(x; n), the moments-generating function F(z) is defined by the “z-transform” of P(x; n): F(z) ¼

n X x¼0

z x P(x; n) ¼

n X

n! ( pz)x qnx ¼ ( pz þ q)n x!(n  x)! x¼0

The first moment, or the average value, of this normalized distribution function P(x; n) is given by   n X dF ¼ xP(x; n) ¼ {np( pz þ q)n1 }z¼1 ¼ np kxl ¼ dz z¼1 x¼0 which can be interpreted as the average number of times that a real coin lands on its head. For completeness, in terms of evaluating the second moment of a distribution function,    n X d dF 2 z ¼ x2 P(x; n) ¼ np þ n(n  1)p2 kx l ¼ z dz dz z¼1 x¼0 The kth moment of the distribution function P(x; n) is obtained by subjecting the moments-generating function F(z) to the operator fz d/dzg k times and then evaluating the result at z ¼ 1. By definition of the z-transform of P(x; n), the zeroth moment of P(x; n) is obtained by evaluating the moments-generating function F(z) at z ¼ 1, which yields unity for a normalized distribution. Random walk statistics

13.4 One-Dimensional Random Walk Statistics via Bernoulli Trials

553

on a one-dimensional lattice can be analyzed via Bernoulli trials when there are equal elemental probabilities that each step is taken toward the right (i.e., the coin lands on its head) or toward the left (i.e., the coin lands on its tail). The path followed by drunken sailors is analogous to random flight statistics. Furthermore, the concentration profile that satisfies the unsteady state one-dimensional diffusion equation (i.e., Fick’s second law of diffusion), subjected to a “spike input” initially at the edge of a semi-infinite rectangular slab, is analogous to the probability density distribution function for random flight statistics of freely jointed chains in the continuous limit. Begin at the origin of a one-dimensional lattice. After n random steps of unit length, nR steps occur to the right and nL steps are taken to the left. Hence, nR þ nL ¼ n and the net displacement to the right is nR 2 nL ¼ m, where nR ¼ (n þ m)/2 is analogous to x in the previous examples. The sequential probability of obtaining a net displacement of m units to the right of the origin after n independent random steps of unit length is given by rffiffiffiffiffiffi    n n! 1 2 m2 exp  ) P(m; n) ¼ n þ m n  m n1 2n np ! ! 2 2 2 The previous expression for the probability density distribution function P(m; n) is discrete in terms of the number n of Bernoulli trials. The range of m is from 2n (i.e., when nL ¼ n and nR ¼ 0) to þn (i.e., when nL ¼ 0 and nR ¼ n). Furthermore, the minimum change in m is 2 units because nL and nR change by a minimum of 1 unit each, nL and nR must sum to n, which represents a constant number of discrete random steps, and m ¼ nR 2 nL ¼ 2nR 2 n. One obtains a Gaussian distribution for P(m; n) in the continuous limit for an exceedingly large number of random steps.

13.4.1 Asymptotic Expression for P (m; n) via Bernoulli Trials The following steps are required to obtain a Gaussian distribution for P(m; n) after a very large number of random steps with equal elemental probabilities. Step 1:

Take the natural logarithm of the discrete Bernoulli trials result for P(m; n): ln{P(m; n)} ¼ ln{n!}  ln

Step 2:

nn þ m o nn  m o !  ln !  n ln{2} 2 2

Use Sterling’s approximation for the factorial, given by   pffiffiffiffiffiffiffiffiffi 1 1 n! ¼ nn 2p n exp(n) 1 þ þ     12n 288n2

ln n!  12 ln(2p) þ n þ 12 ln n  n

554

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers

Step 3:

Obtain the following result after combining terms without invoking any other approximations: ln P(m; n)   12 ln(2p) þ n ln n þ 12 ln n  n ln 2 nn mo  12(n þ m þ 1)ln 1þ 2 n nn o m 1  12(n  m þ 1)ln 2 n

Step 4:

Use a Taylor series expansion for ln(1 + x)  +x2x 2/2 + . . . to evaluate the last two terms on the right side of the previous equation, truncating each series after the quadratic term since m , n:  mo m m m2 ¼ ln n  ln 2 þ ln 1 þ  ln n  ln 2 þ  2 þ    2 n n n 2n nn  o   m m m m2 1 ¼ ln n  ln 2 þ ln 1   ln n  ln 2   2     ln 2 n n n 2n ln

Step 5:

nn

1þ

Further analysis of the last two terms on the right side of Step 3, with assistance from the Taylor series expansions in Step 4, yields (a)



nn n þ 1 h nn  mo moi ln 1þ þ ln 1 2 2 n 2 n   2 nþ1 m 2 ln n  2 ln 2  2  n 2

m2 m2  ln n þ ln 2 þ 2 2n 2n   h n   o n   o i m n m n m m 2m m2 ln 1þ  ln 1    n 2 2 n 2 n 2 n  n ln n þ n ln 2 þ

(b)

Step 6:

Combining terms in Steps 3 and 5 for an exceedingly large number of random steps implies that n  1 and it is reasonable to neglect m 2/2n 2 relative to m 2/2n in Step 5: ln P(m; n)  12 ln(2p) þ n ln n þ 12 ln n  n ln 2  n ln n þ n ln 2 þ

Step 7:

m2 m2 m2 m2  ln n þ ln 2 þ 2   12 ln(2p)  12 ln n þ ln 2  2n 2n n 2n

In the continuous limit where n  1, one obtains a Gaussian distribution function for the probability of obtaining a net displacement of m units to the right of the origin on a one-dimensional lattice after n independent random steps of unit length with equal elemental probabilities forward

13.5 Extrapolation of One-Dimensional Gaussian Statistics to Three Dimensions

555

and backward: rffiffiffiffiffiffi   2 m2 P(m; n)  exp  pn 2n This distribution function must be divided by 2 to achieve normalization because the minimum change in m is 2 units via the discrete formulation, and m ranges from 21 to þ1 in the continuous limit.

13.5 EXTRAPOLATION OF ONE-DIMENSIONAL GAUSSIAN STATISTICS TO THREE DIMENSIONS On a one-dimensional lattice, the statistical problem focuses on anchoring one end of a long-chain molecule to the origin and evaluating the probability of finding the other end at position rx to the right (i.e., in the positive x-direction) after n random steps, each of length lx, where rx ¼ mlx. Hence, one sets m ¼ rx/lx in the previous expression for P(m; n), divides P(m; n) by 2 to ensure normalization in the continuous limit, as described above, and equates probabilities in terms of both distribution functions to yield the final expression for P(rx; n), with dimensions of inverse length: P(rx ; n) drx ¼ P(rx ; n)lx dm ¼ 12P(m; n) dm sffiffiffiffiffiffiffiffiffiffiffiffiffi   1 1 rx2 exp  P(m; n) ¼ P(rx ; n) ¼ 2lx 2nl2x 2pnl2x It is rather straightforward to verify that P(rx; n) is, indeed, normalized. The average displacement on the one-dimensional lattice, given by krxl, or the first moment of the distribution P(rx; n) vanishes because the integrand of the following expression is an odd function: sffiffiffiffiffiffiffiffiffiffiffiffiffi þ1 þ1   ð ð 1 rx2 krx l ¼ rx P(rx ; n) drx ¼ rx exp  drx ¼ 0 2pnl2x 2nl2x 1

1

Hence, the average position of the end of a freely jointed long-chain molecule that exhibits Gaussian statistics on a one-dimensional lattice is at the origin, exactly where the other end is anchored. The second moment of the distribution P(rx; n), or the mean-square displacement on a one-dimensional lattice is calculated as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffi þ1 þ1   ð ð 1 rx2 2 2 2 drx ¼ nl2x rx P(rx ; n) drx ¼ 2 rx exp  krx l ¼ 2nl2x 2pnl2x 1

0

This is the one-dimensional analog of the three-dimensional result for mean-square displacement in which contributions from chain conformations to the left of the origin do not cancel those to the right. The final result for P(rx; n) is completely

556

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers

analogous to the solution of the following unsteady state one-dimensional diffusion equation in which a homogeneous semi-infinite rectangular medium is subjected to a delta-function tracer input at x ¼ 0 and t ¼ 0, where C is the tracer concentration [i.e., per unit length in the direction (i.e., x) in which diffusion occurs] with respect to the initial amount that has been distributed per unit area along the line at x ¼ 0. Due to the nature of the initial condition at t ¼ 0, this one-dimensional unsteady state diffusion equation is transformed with respect to time into the Laplace domain (i.e., t ) s) and solved as a second-order ordinary differential equation, with spatial coordinate x as the independent variable and Laplace variable s as a parameter. A Laplace inversion (i.e., s ) t) back into the time domain yields @C @2C ¼D 2 @t @x rffiffiffiffiffiffiffiffiffiffiffiffi   1 x2 exp  C( x, t)  4 Dt 4p Dt The mean-square displacement krx2 l ¼ nl2x is given by 2Dt, where t represents time and D is the tracer diffusion coefficient in the medium. The final solution for P(rx; n) can be extended to analyze random flight statistics of isotropic freely jointed chains in three dimensions. The following conditions apply: 1. The polymer chain consists of n random flight steps in three-dimensional space, where each step has length l. The mean-square end-to-end chain distance is kr 2l ¼ nl2 when there are no bond angle or valence cone restrictions of a freely jointed chain with no polymer – solvent interactions in a Q-solvent. 2. This three-dimensional problem is equivalent to three simultaneous onedimensional problems, where (i) n random steps, each of length lx, occur with equal probability forward and backward in the x-direction such that the net displacement in the positive x-direction is rx and the mean-square displacement is kr2x l ¼ n(lx)2; (ii) n random steps, each of length ly, occur with equal probability forward and backward in the y-direction such that the net displacement in the positive y-direction is ry and the mean-square displacement is kr2y l ¼ n(ly)2; and (iii) n random steps, each of length lz, occur with equal probability forward and backward in the z-direction such that the net displacement in the positive z-direction is rz and the mean-square displacement is kr2z l ¼ n(lz)2. The sequential probability of finding an end-to-end chain vector r after n random steps is P(r; n) dr ¼ {P(rx ; n) drx }{P(ry ; n) dry }{P(rz ; n) drz } where the differential volume element dr is equivalent to the product of drx, dry, and drz. Furthermore, the product of three Gaussian distribution functions on the

13.6 Properties of Three-Dimensional Gaussian Distributions

557

right side of the previous equation yields a Gaussian distribution for P(r; n). For each random flight conformation, the sum of the square of each net displacement along the positive x-, y-, and z-directions yields the square of the end-to-end chain length r 2. In other words, r 2 ¼ rx2 þ ry2 þ rz2 When the previous equation for r 2 is averaged with respect to the distribution of chain lengths, one obtains the following relation between all of the mean-square displacements mentioned above: kr2 l ¼ nl2 ¼ krx2 l þ kry2 l þ krz2 l ¼ n(l2x þ l2y þ l2z ) Since the three-dimensional random flight problem is applicable for isotropic media, chain segments are not oriented preferentially along any coordinate direction. In fact, krx2 l ¼ kry2 l ¼ krz2 l ¼ 13kr 2 l The final expression for the Gaussian distribution of chain lengths r in threedimensional space is sffiffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffi )    (  ry2 rz2 1 1 1 rx2 exp  2 exp  2 exp  2 P(r; n) ¼ 2krx l 2kry l 2krz l 2p krx2 l 2p kry2 l 2p krz2 l  ¼

3 2p kr2 l

3=2

   3 3=2  3r2  3  2 2 2 exp  2 rx þ ry þ rz ¼ exp  2 2kr l 2kr l 2p kr2 l

which is very similar to the Maxwell – Boltzmann distribution of molecular velocities for an ideal gas that exhibits three-dimensional motion. The three-dimensional Gaussian distribution function for end-to-end chain length r is asymptotically exact for a large number n of segments when r is significantly less than the contour length nl. It is not physically possible for P(r; n) to yield a nonzero probability for an end-to-end chain length that is greater than the contour length, but the Gaussian distribution does not vanish unless r tends toward infinity. This discrepancy, however small, between realistic predictions and the Gaussian expression for P(r; n) has been corrected by invoking the “Langevin” distribution via restricted Lagrange multiplier optimization, as discussed in Section 7.7 when the molecular optical anisotropy of random coils and rigid rod-like polymers was presented.

13.6 PROPERTIES OF THREE-DIMENSIONAL GAUSSIAN DISTRIBUTIONS AND THEIR MOMENTS-GENERATING FUNCTION One of the primary objectives of this section is to demonstrate that the Laplace transform of the Gaussian distribution function P(r; n) contains information about

558

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers

the moments kr nl, where the nth moment of P(r; n) is defined as follows þ1 ð

n

kr l ¼

1 ð

n

r P(r; n) dr ¼ 4p r n P(r; n) r2 dr

1

0

Now, in terms of the Laplace vector s, which is oriented along the z-direction of a rectangular Cartesian coordinate system such that end-to-end chain vector r makes an angle Q (i.e., polar angle Q in spherical coordinates) with respect to s, the Laplace transform of P(r; n) is L(s; n) ¼

þ1 ð



P(r; n) exp (r s) dr 1

¼

2ðp

1 ð

2

ðp

d w P(r; n)r dr exp (rs cos Q) sin Q dQ 0

0

0

In the previous expression, integration with respect to polar angle Q yields a hyperbolic sine function that can be expanded in a Taylor series in which only the odd terms survive. For example, if one makes the following substitution, u ¼ cos Q and du ¼ 2sin Q dQ, then ðp

exp(rs cos Q) sin Q dQ ¼ 

1 ð

exp(rsu) du ¼

1 {exp(rs)  exp(rs)} rs

1

0

  2 2 1 1 1 (rs) þ (rs)3 þ (rs)5 þ (rs)7 þ    ¼ sinh(rs) ¼ rs rs 3! 5! 7!

Substitution of the previous result into the Laplace transform of P(r; n) and identification of various moments of the Gaussian distribution yields L(s; n) ¼

þ1 ð



P(r; n) exp(r s) dr 1

¼ 4p

1 ð

1þ

 1 1 1 (rs)2 þ (rs)4 þ (rs)6 þ    P(r; n)r2 dr 3! 5! 7!

0

¼1þ

1 2 2 1 4 4 1 6 6 kr ls þ kr ls þ kr ls þ    3! 5! 7!

Hence, the Laplace transform L(s; n) of the Gaussian distribution function contains information about the even moments of P(r; n), which are identified on the far right side of the previous equation by kr 2nl, with n ¼ 0, 1, 2, . . . . It is reasonable to describe

13.6 Properties of Three-Dimensional Gaussian Distributions

559

L(s; n) as the moments-generating function of P(r; n). Since P(r; n) is an even function of the magnitude of the end-to-end chain vector because the Gaussian distribution was constructed to be orientation insensitive, all odd moments of P(r; n) are zero. Furthermore, the moments-generating function of the Gaussian distribution is an even function of the magnitude of the Laplace vector s, as illustrated by the final result below: L(s; n) ¼

þ1 ð

 P(r; n) exp (r s) dr ¼ exp 16kr2 ls2



1

Justification for the previous functional form of L(s; n) is obtained by reexpressing the integrand of the moments-generating function as a product of three integrals in rectangular Cartesian coordinates (i.e., dr ¼ drx dry drz) that are almost identical:

L(s; n) ¼

þ1 ð



P(r; n) exp(r s) dr 1



3 ¼ 2pkr2 l

3=2 þ1 ð



 3r2 exp(r s) dr exp 2kr2 l



1



3 ¼ 2pkr2 l

3=2 þ1 ð



 3rx2 exp exp(rx sx ) drx 2kr2 l

1



þ1 ð

( ) þ1   ð 3ry2 3rz2 exp(r exp(rz sz ) drz exp s ) dr exp y y y 2kr2 l 2kr2 l

1

1

Focus on integration with respect to the x-component of the end-to-end chain vector (i.e., drx) and expand the exponential function that contains the x-component of the Laplace vector (i.e., sx) in a Taylor series about rx sx ¼ 0. All odd functions in the integrand do not contribute to the final result. Furthermore, integration limits for the even functions can be changed from 21 ) þ1 to 0 ) þ1, if all results are multiplied by 2: 

þ1 ð

exp

 3rx2 exp(rx sx ) drx 2kr2 l

1

¼

þ1 ð 1

1  rx sx þ

   1 1 (1) j 3rx2 (rx sx ) j þ    exp drx (rx sx )2  (rx sx )3 þ    þ j! 2kr 2 l 2! 3!

560

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers

  ð 1 1 m X X s2m 3rx2 s2m 2m m d x x dr (1) 2 rx exp ¼ x (2m)! 2kr 2 l (2m)! dam m¼0 m¼0 1

rffiffiffiffi rffiffiffiffi  2 p p s ¼ exp x a a 4a

0

a¼

3 2kr 2 l

Assistance was obtained from the helpful hints in Problem 13.1 at the end of this chapter to evaluate the integrals in the summation above. It is necessary to evaluate several terms in the summation on the far right side of the previous equation to identify the Taylor series of an exponential function (i.e., exp(s2x =4a)). Finally, one evaluates three similar products to calculate the moments-generating function for Gaussian distributions:  L(s; n) ¼

3 2p kr2 l

3=2 þ1 ð

  3rx2 exp(rx sx ) drx exp 2kr 2 l

1



þ1 ð

( ) þ1   ð 3ry2 3rz2 exp(r exp s ) dr exp exp(rz sz ) drz y y y 2kr2 l 2kr2 l

1

1

3=2 (rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  )(rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ) 3 2 1 2 2 2 1 2 2 2 2 ¼ p kr l exp kr lsx p kr l exp kr lsy 2p kr2 l 3 6 3 6 (rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  )    2 1 1 ¼ exp kr2 ls2 p kr2 l exp kr2 ls2z  3 6 6 

If one expands the exponential function in the previous equation for L(s; n) in a Taylor series and compares both infinite series expressions for the moments-generating function for Gaussian distributions, L(s; n) ¼

þ1 ð



P(r; n) exp(r s) dr ¼ 1 þ

1 2 2 1 4 4 1 6 6 kr ls þ kr ls þ kr ls þ    3! 5! 7!

1

    1 2 2 1 2 2 1 1 2 2 2 kr ls ¼ exp kr ls ¼ 1 þ kr ls þ 6 6 2! 6     1 1 2 2 3 1 1 2 2 4 kr ls þ kr ls þ    þ 3! 6 4! 6 then any even moment of P(r; n), kr 2nl, can be expressed in terms of the nth power of the mean-square end-to-end chain length, kr 2l. The result is   (2n þ 1)! 1 2 n kr l kr l ¼ n! 6 2n

13.7 Mean-Square Radius of Gyration of Freely Jointed Chains

561

13.7 MEAN-SQUARE RADIUS OF GYRATION OF FREELY JOINTED CHAINS The mean-square end-to-end chain distance is difficult to visualize when large molecules contain loops, rings, branches, or multiarm star-like segments. For these types of structures, the radius of gyration provides a better description of molecular size. The mean-square end-to-end distance and the mean-square radius of gyration enjoy a very simple relation for ideal straight-chain molecules that obey the freely jointed assumptions, where the former is sixfold larger than the latter. If all chain segments or repeat units of a macromolecule have the same mass (i.e., mi ¼ m), then the center of mass is defined by the following equation for large molecules that contain n þ 1 point masses (i.e., 0  i  n) connected by n segments: n X

mi ri,CM ¼ m

i¼0

n X

ri,CM ¼ 0

i¼0

where mi is the mass of the ith repeat unit, assumed to be concentrated at point i, and ri,CM represents a vector from the center of mass of the chain to the ith point mass. The zeroth point identifies the beginning of the chain and the nth point corresponds to the chain end with end-to-end chain vector r. The mean-square radius of gyration is defined by summing ri,CM ri,CM over all n þ 1 mass points, dividing by the total number of mass points, and averaging (i.e., k l) the result with respect to the distribution of chain segments. Hence,



ks2 l ¼

n 1 X kri,CM ri,CM l n þ 1 i¼0



Now, rewrite ri,CM in terms of the vector from the center of mass to the beginning of the chain, r0,CM, and the vector from the beginning of the chain to the ith mass point, ri0 via vector addition. The definition of the center of mass yields n X i¼0

ri,CM ¼

n X

{r0,CM þ ri0} ¼ (n þ 1)r0,CM þ

i¼0

r0,CM ¼ 

n X

ri0 ¼ 0

i¼0

n n 1 X 1 X ri0 ¼  rj0 n þ 1 i¼0 n þ 1 j¼0

Next, one calculates the square of the magnitude of the vector from the center of mass to the ith mass point:







2 2 þ 2{r0,CM ri0 } þ ri0 ri,CM ri,CM ¼ {r0,CM þ ri0 } {r0,CM þ ri0 } ¼ r0,CM

562

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers

The previous two results are employed to sum the squares of the magnitudes of the vectors from the center of mass to each of the n þ 1 mass points: n X i¼0



2 (ri,CM ri,CM ) ¼ (n þ 1)r0,CM þ 2r0,CM



n X

ri0 þ

i¼0

n X

2 ri0

i¼0

( ) ( ) n n X X 1 2 ¼ ri0 þ ri0 rj0 n þ 1 i¼0 i¼0 j¼0 ( ) ( ) n n X X 2  r j0 ri0 n þ 1 j¼0 i¼0 ( ) ( ) n n n X X X 1 2 ri0  ri0 rj0 ¼ n þ 1 i¼0 i¼0 j¼0 n X







Define segment vector rji from the ith mass point to the jth mass point in the chain in terms of the difference between vectors from the beginning of the chain to the jth point relative to the ith point: rji ¼ rj0  ri0

 

 

2 2 rji rji ¼ rj0 þ ri0  2ri0 rj0 n o 2 2 ri0 rj0 ¼ 12 rj0 þ ri0  rji rji

This relation between various segment vectors allows one to evaluate the double summation on the far right side of the expression for the sum of the squares of the magnitudes of the vectors from the center of mass to each of the n þ 1 mass points. For example, ( ) ( ) n n n n X X X X 1 2 (ri,CM ri,CM ) ¼ ri0  ri0 rj0 n þ 1 i¼0 i¼0 i¼0 j¼0





n X n n o 1 X 1 2 2 rj0 þ ri0  rji rji 2 n þ 1 i¼0 j¼0 i¼0 " # n n X X 1 2 2 (n þ 1) ri0  rj0 ¼ 2(n þ 1) i¼0 j¼0 " # n n X n X X 1 1 2  (n þ 1) ri0 þ {rji rji } 2(n þ 1) 2(n þ 1) i¼0 j¼0 i¼0

¼

n X



2 ri0 



¼

n X n X 1 {rji rji } 2(n þ 1) i¼0 j¼0



13.7 Mean-Square Radius of Gyration of Freely Jointed Chains

563

Dividing the previous result by the total number of mass points (i.e., n þ 1) and averaging (i.e., k l) with respect to the distribution of chain segments yields the mean-square radius of gyration for chains of arbitrary shape and conformational model: ks2 l ¼

n n X n X 1 X 1 kri,CM ri,CM l ¼ krji rji l n þ 1 i¼0 2(n þ 1)2 i¼0 j¼0





This result is not restricted to freely jointed chains. However, the remainder of this analysis focuses on segment vectors from the ith mass point to the jth mass point in freely jointed chains. Similar to the random coil results in Section 13.3 for the mean-square end-to-end chain distance, in which individual segment vectors h i from the (i21)st mass point to the ith mass point are uncorrelated for different values of i, one obtains the following ensemble average with respect to any orientation-insensitive distribution of chain vectors:



krji rji l ¼ l2 j j  ij when the length of each segment vector between adjacent mass points in the chain is l. Obviously, when i ¼ 0 and j ¼ n, the previous equation yields the mean-square end-to-end distance of the entire chain, as developed earlier in this chapter. It is not necessary to invoke Gaussian statistics for rji when the ith and jth mass points in the chain are rather close such that the number of random steps between them is not too large. Simple vector analysis yields the previous equation for any value of the magnitude of the difference between i and j if the distribution function for freely jointed segments is independent of the orientation of each random step (see Section 13.3). Now, the mean-square radius of gyration for n random steps in a freely-jointed chain reduces to ks2 l ¼ ¼

n 1 X kri,CM ri,CM l n þ 1 i¼0



n X n n X n X X 1 l2 kr r l ¼ j j  ij ji ji 2(n þ 1)2 i¼0 j¼0 2(n þ 1)2 i¼0 j¼0



Consider an n þ 1 by n þ 1 square matrix in which the value of i in each cell increases from top to bottom and j increases from left to right. Each of the (n þ 1)2 cells in this square matrix is characterized by a unique set of values for i and j. Differences between i and j vanish in each of the n þ 1 cells along the main diagonal from upper left to lower right. Hence, it is possible to evaluate the absolute magnitude of the difference between i and j in each cell of the square matrix by focusing only on the diagonals that are above the main diagonal and multiplying the sum by 2. For example, there are n cells on the diagonal that is directly above the main diagonal from upper left to lower right, and the difference between i and j in each of these n cells is 1. Then, there are n 2 1 cells on the second diagonal above the main one from upper left to lower right, and the difference between i and j in each of these n 2 1 cells is 2. Finally, the diagonal

564

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers

in the upper right corner of the square matrix contains 1 cell in which i ¼ 0 and j ¼ n; hence, the difference between i and j is n. The double summation on the far right side of the previous equation is evaluated in detail when n ¼ 6. The numbers below represent ij pairs in all 49 cells of the 7  7 matrix: 2

00 6 10 6 6 20 6 6 30 6 6 40 6 4 50 60

01 11 21 31 41 51 61

02 12 22 32 42 52 62

03 13 23 33 43 53 63

04 14 24 34 44 54 64

05 15 25 35 45 55 65

3 06 16 7 7 26 7 7 36 7 7 46 7 7 56 5 66

1. There are 7 cells on the main diagonal, upper left to lower right; j 2 i ¼ 0 in each case. 2. There are 6 cells on the diagonal from 01 to 56; j 2 i ¼ 1, in each case. 3. There are 5 cells on the diagonal from 02 to 46; j 2 i ¼ 2, in each case. 4. There are 4 cells on the diagonal from 03 to 36; j 2 i ¼ 3, in each case. 5. There are 3 cells on the diagonal from 04 to 26; j 2 i ¼ 4, in each case. 6. There are 2 cells on the diagonal from 05 to 16; j 2 i ¼ 5, in each case. 7. There is 1 cell on the uppermost diagonal, where i ¼ 0 and j ¼ 6; j 2 i ¼ 6. Since n ¼ 6 in this example, there are n þ 12k cells on a given diagonal in which j 2 i ¼ k, and the index k ranges from 0 to n. After multiplication by 2 to account for all cells on diagonals below the main diagonal from upper left to lower right, it is possible to evaluate the required summation as follows: n X n X

j j  ij ¼ 2

i¼0 j¼0

n X

k(n þ 1  k) ¼ 2(n þ 1)

k¼0

n X k¼1

k2

n X

k2

k¼1

n X

n k ¼ (n þ 1) 2 k¼1

n X

n k 2 ¼ (n þ 1)(2n þ 1) 6 k¼1

Finite sums for k and k 2, 1  k  n, can be found in Gradshteyn and Ryzhik [1980]. Whereas the former sum of k, 1  k  n, is rather trivial, the sum of k 2, 1  k  n, requires the use of factorial polynomials, calculation of the anti-difference of the factorial polynomial representation of k 2, and evaluation of this anti-difference at argument n þ 1 relative to an argument of 1. The mean-square radius of gyration of a freely jointed chain consisting of n random steps, each of length l, is given

13.8 Mean-Square End-to-End Distance of Freely Rotating Chains

565

by the following expression when the number of steps is large: n X n n o X l2 l2 n n(n þ 1)2  (n þ 1)(2n þ 1) j j  ij ¼ 2 2 3 2(n þ 1) i¼0 j¼0 2(n þ 1)   l2 1 (2n þ 1) n(n þ 1) n þ 1  ¼ 3 2(n þ 1)2   n1 1 nþ2 1 ) nl2 ¼ nl2 6 nþ1 6

ks2 l ¼

13.8 MEAN-SQUARE END-TO-END DISTANCE OF FREELY ROTATING CHAINS As a general rule, chain dimensions are larger when the backbone bond angles (i.e., CZCZC, CZOZC, CZNZC, etc.) are restricted, relative to the size of freely jointed chains that can “collapse upon themselves” with no excluded volume. The following vector analysis for semirealistic chains with fixed bond angle p 2 Q and free rotation about the valence cone yields the Eyring – Sadron equation for the mean-square end-to-end chain distance. Essentially a continuum of rotational isomeric states is allowed, corresponding to no preference for the Ramachandran backbone bond rotation angles w and c that correspond to trans (w ¼ 1808), gauche þ (w ¼ 608), and gauche 2 (w ¼ 3008). There is no energy barrier between any of these chain conformations in this analysis. Even the cis or “eclipsed” conformation at w ¼ 08, which is highly unfavorable for real chains, is allowed (hence, the term “semirealistic” is used to describe these chains). The Stockmayer – Kurata ratio, introduced in the next section, accounts for rotational isomeric state preferences, thereby eliminating the cis conformation from consideration in the end-to-end distance calculation. As discussed in Section 13.3, the following expression must be evaluated and averaged with respect to the normalized distribution of end-to-end chain vectors P(r; n) to calculate the mean-square end-to-end chain distance:



r r¼

2

kr l ¼

n X

hi

i¼1 1 ð





n X k¼1

n X

hk ¼ nl2 þ

2

{r r}P(r; n) dr ¼ nl þ

1



hi hk

i,k¼1;(i=k) n X

i,k¼1;(i=k)

1 ð



{hi hk }P(r; n) dr 1

where h i represents a segment vector of length l from the (i 2 1)st mass point to the ith mass point in the chain. Without invoking the Gaussian distribution of end-to-end chain vectors that is only appropriate for freely jointed chains, one evaluates the following scalar “dot” products of segment vectors, where cos Q accounts for the

566

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers

projection of one segment vector onto its adjacent segment vector:

 h h h h

hi hiþ1 ¼ l2 cos Q i

iþ2

i

k

¼ l2 {cos Q}2 ¼ l2 {cos Q}jkij

Correlations between “next-nearest-neighbor” segments require a factor of cos2 Q in the scalar product of hi with hiþ2, because l cos Q represents the projection of hiþ2 onto the axis defined by hiþ1, and then another factor of cos Q projects l cos Q onto the axis defined by hi. Q is the supplement of the actual backbone bond angle, given by p 2 Q. For example, Q ¼ 70.58 and p 2 Q ¼ 109.58 when the chain backbone consists exclusively of carbon – carbon single bonds. Since the normalized distribution function of end-to-end vectors for semirealistic chains exhibits no preference for any Ramachandran backbone bond rotation angle on the valence cone, the ensemble average of cos Q is, simply, cos Q ¼ 13 for tetrahedral bond angles. The mean-square end-to-end distance of an n-segment chain, where the length of each segment is l, can be calculated from the following expression: ( ) ( ) n n X X 2 2 2 jkij jkij ¼l nþ kr l ¼ kr rl ¼ l n þ (cos Q) x



i,k¼1;(i=k)

i,k¼1;(i=k)

where x ¼ cos Q , 1 in the analysis below. The magnitude of the difference between k and i, or j and i, in an n by n, or n þ 1 by n þ 1, square matrix was considered in the previous section for the mean-square radius of gyration. In summary, there are (n 2 p) terms in which the magnitude of the differences between k and i equals p, where p ranges from 0 to n 2 1, and the entire sum must be multiplied by 2. Furthermore, all terms in which i and k are the same have already been removed from the double summation (i.e., nl2), so one must neglect all contributions when p ¼ 0. In the previous example, the summations ranged from 0 to n and the magnitude of the differences between j and i ranged from 0 to n. Now, the summations range from 1 to n, or 0 to n 2 1, so the magnitude of the differences between k and i ranges from 0 to n 2 1. In the previous example, failure to remove all contributions where i and j are the same did not affect the summation because the magnitude of the differences between i and j was simply added. Now, contributions to the power series are not negligible when the exponent is zero. Consider the double summation on the far right side of the previous equation for the mean-square end-to-end chain distance: n n1 n1 n1 X X X X xjkij ¼ 2 (n  p)x p ¼ 2n xp  2 px p i,k¼1;(i=k)

p¼1

p¼1

p¼1

The following techniques are employed to evaluate each finite sum on the far right side of the previous equation. Step 1:

The infinite power series of x p, where p ranges from 1 to 1 and x , 1, can be evaluated by subtracting the power series of x pþ1 from the original power series. Since there are an infinite number of terms in each series, all terms

13.8 Mean-Square End-to-End Distance of Freely Rotating Chains

567

cancel except the initial term in the series of x p when p ¼ 1. For example, 1 X x p ¼ x þ x2 þ x3 þ x4 þ x5 þ    p¼1

x

1 X

xp ¼

p¼1

(1  x)

1 X

1 X

x pþ1 ¼ x2 þ x3 þ x4 þ x5 þ   

p¼1

xp ¼ x

p¼1

Step 2:

An infinite series can be split into a finite series and another infinite series via continuity of the index p. Then, the finite power series of x p can be written as the difference between two infinite series, both of which have been evaluated in Step 1: 1 X

xp ¼

n1 1 n1 1 n1 X X X X X x xn ¼ xp þ xp ¼ x p þ x n1 xp ¼ xp þ 1x 1  x p¼1 p¼n p¼1 p¼1 p¼1

xp ¼

x  xn 1x

p¼1 n1 X p¼1

Step 3: The finite power series of px p, where p ranges from 1 to n 2 1, can be evaluated by differentiating the final result from Step 2 with respect to x, and then multiplying by x. Hence, ( )   n1 n1 X d X d x  xn p p ¼x px ¼ x x dx p¼1 dx 1  x p¼1   (1  x)(1  nx n1 )  (x  x n )(1) x(1  x n ) nx n x  ¼ 2 2 (1  x) (1  x) (1  x) The Eyring – Sadron equation for the mean-square end-to-end chain distance of freely rotating chains is exact for any number n of segments, where the length of each segment is l: ( ) n1 n1 X X 2 2 p p x 2 px kr l ¼ l n þ 2n p¼1

p¼1

  

 x  xn x(1  x n ) nx n 2 2  ¼ l n þ 2n 1x (1  x) 2 1  x   1 þ x 2x (1  x n ) ¼ nl 2 ; x ¼ cos Q  1  x n (1  x)2 One obtains the following relation for freely rotating chains that contain a large number of segments (i.e., n  1) with tetrahedral backbone bond angles

568

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers

(i.e., p 2 Q ¼ 109.58, Q ¼ 70.58, x ¼ cos Q ¼ 13): 2

kr l ) nl

2



n1

1þx 1x



) 2nl2 x¼1=3

13.9 CHARACTERISTIC RATIOS AND STATISTICAL SEGMENT LENGTH 13.9.1 Flory’s Characteristic Ratio and the Stockmayer –Kurata Rotational Isomeric Preference Ratio Several parameters have been defined to measure the deviation of realistic chain dimensions from either freely jointed or freely rotating chains in a Q-solvent where polymer – solvent interactions are indistinguishable from segment – segment interactions in the undiluted polymer and there is no excluded volume. Flory’s characteristic ratio Cn accounts for deviations of real chains in Q-solvents from mean-square freely jointed chain dimensions. Based on results from the previous section for freely rotating chains, Cn depends on the number of segments n in the chain, as illustrated below: kr2 l

Q-solvent

unperturbed 2

Cn ¼

nl

Cn !C1

) n1

fixed bond angles

) freely rotating chains

1 þ cos Q 2 cos Q (1  cosn Q)  1  cos Q n (1  cos Q)2

  1 þ cos Q ) 2 1  cos Q Q¼70:58

However, the characteristic ratio C1 asymptotically approaches a constant that depends on bond angles and valence cone preferences, but not n as illustrated by the previous equation, when the number of chain segments is infinitely large (i.e., actually n . 100– 200 bonds). The Stockmayer – Kurata ratio sSK  1 measures deviations of root-mean-square real-chain dimensions in Q-solvents from ideal chains with fixed bond angles and free rotation around the valence cone. Hence, sSK reflects preferences for particular rotational isomeric states. Since Flory’s characteristic ratio, introduced above, does not include rotational isomeric state preference in the determination of real-chain dimensions under Q-solvent conditions, the previous calculations for Cn and C1 are modified slightly: kr 2 l Cn ¼

Q-solvent unperturbed 2

nl

Cn !C1

2 ) sSK n1

fixed bond angles

) valence cone preference



2 sSK

1 þ cos Q 1  cos Q



  1 þ cos Q 2 cos Q (1  cosn Q)  1  cos Q n (1  cos Q)2 2 ) 2sSK Q¼70:58

13.9 Characteristic Ratios and Statistical Segment Length

13.9.2

569

Kuhn Statistical Segment Length

Consider the mean-square end-to-end distance of real chains under Q-solvent conditions, which includes fixed bond angles and preference for a particular rotational isomeric state on the valence cone, as described by C1 and sSK in the previous section. Now, the product of NK and the Kuhn statistical segment length AK yields the magnitude of the end-to-end vector for fully extended chains. There are n bonds and l represents the length of each carbon – carbon backbone bond. Rigorously, these chain-dimension formulas apply when all backbone bonds are identical, but approximations can be used to determine the effective length leffective of a repeat unit that contains dissimilar bonds (now, n should be interpreted as the number of repeat units). The projection of each bond length l along the end-to-end vector of fully extended chains is l sin{( p 2 Q)/2}, where the fixed bond angle is p 2 Q and Q is the supplement of the actual bond angle (employed in the previous section). Hence, the product of the number of segments NK and the statistical segment length AK is given by NK AK ¼ nl sin

  pQ 2

The mean-square end-to-end distance of real chains in Q-solvents provides another relation between NK and AK: 2 kr2 l Q-solvent ¼ NK A2K ¼ C1 nl2 ¼ sSK real chains

  1 þ cos Q nl2 1  cos Q

The previous two equations yield the following relation between the Kuhn statistical segment length AK, the Stockmayer – Kurata ratio sSK, the actual bond length l, and fixed bond angle QBond ¼ p 2 Q:     1 þ cos Q 1 þ cos(p  QBond ) 2 lsSK Q¼pQBond 1  cos Q 1  cos(p  QBond )     ) AK ¼ pQ QBond pQ¼QBond sin sin 2 2   1  cos(QBond ) 2 lsSK 1 þ cos(QBond )   ¼ QBond sin 2 2 lsSK

As indicated above, it is instructive to replace p 2 Q by the fixed bond angle QBond, and the supplement Q of the actual bond angle by p 2 QBond prior to trigonometric manipulation of the previous equation for AK. Then, replace 1 2 cos{QBond} by 2 sin2{QBond/2} via one of the half-angle formulas. The final expressions for AK

570

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers

and NK that apply to real chains with fixed bond angles and rotational isomeric state preferences in Q-solvents are provided below:   QBond 2 2lsSK sin freely jointed 2 ) l AK ¼ chains 1 þ cos (QBond ) NK ¼

n{1 þ cos (QBond )} freelyjointed ) n 2 chains 2sSK

13.10 EXCLUDED VOLUME AND THE EXPANSION FACTOR a FOR REAL CHAINS IN “GOOD” SOLVENTS: ATHERMAL SOLUTIONS Previous discussions in this chapter focus on ideal chains, with or without bond angle restrictions. By definition, there is no excluded volume associated with ideal chains. When excluded volume is an important consideration, polymers prevent segments within the same chain, as well as neighboring molecules, from occupying overlapping regions of space. Consequently, the root-mean-square end-to-end distance of ideal chains must be multiplied by the expansion factor a, which is greater than unity, to obtain the actual dimensions of real chains. There are a total of y chains, and y i represents the number of chains P that have the same end-to-end chain distance ri in the expanded state. Hence, i y i ¼ y . It is important to emphasize that all y i chains with the same end-to-end distance do not necessarily have the same rotational state for each backbone bond. Conformations that exhibit different end-to-end chain distances are not equally probable, as described by the Gaussian distribution function Pi (ri) in the unperturbed state. For example, the exponentially decreasing probability density distribution function, A exp(2b2r r), for freely jointed chains predicts that conformations with a larger end-to-end vector are less favorable. The answer to Problem 13.1a at the end of this chapter reveals that the most probable length of ideal chains with no restrictions on bond angle or orientation is obtained by maximizing 4pAr 2 exp(2b2r r), which includes the volume element of a spherical shell of radius r about the origin where one chain end is anchored, such that rMost Probable ¼ 1/b. Unperturbed chain conformations follow a Gaussian distribution in the ideal state, but polymer – solvent interactions cause chain expansion and produce nonGaussian conformations. Analogous to elongation ratios for stretched networks, discussed later in this chapter and in the next chapter, the expansion factor a provides a connection between freely jointed Gaussian chains in Q-solvents and expanded real chains in good solvents. If vi represents the total number of conformations that are available to a single chain with end-to-end distance ri, then the multiplicity of states V considers groups of conformations with different ri, instead of counting each conformation separately. Hence, the same normalized Gaussian probability density distribution Pi appears as a factor y i times in the multiplicity function because there are y i chains with the same end-to-end distance in the expanded state. One





13.10 Excluded Volume and the Expansion Factor a for Real Chains

571

constructs the following expression for V: Y

V ¼ y!

Groups of conformations w=different ri

vyi i ¼ y! yi !

Y Groups of conformations w=different ri

{v0 Pi wi}yi yi !

where v0 is the total number of conformations that are available to a single chain, regardless of the end-to-end vector, Pi identifies a subset of v0 with end-to-end chain distance ri in the absence of excluded volume (i.e., Gaussian distribution function), and wi reduces the total number of conformations with end-to-end chain distance ri as a consequence of excluded volume. These excluded volume factors (i.e., wi) are assumed to be the same for all ri, which simplifies the analysis. If all chain conformations are equally probable and excluded volume is not an important consideration, then one sets each factor of v0Piwi to unity in the multiplicity of states V. Upon taking the logarithm of the previous equation and employing a simplified version of Sterling’s approximation (i.e., ln x!  x ln x 2 x), one obtains X

lnV ¼ ln y! þ

yi ln{v0 Pi wi }

Groups of conformations w=different ri

ln y !  y ln y  y ¼ ln y

X

X

ln yi !

Groups of conformations w=different ri

yi  y

i



 V  ln V0

X Groups of conformations w=different ri

yi ln wi þ

ln V0 ¼ y ln v0   X yP i yi ln ) y ln w þ yi wi ! w Groups of conformations w=different ri

X Groups of conformations w=different ri



yi ln

yPi yi



V0 represents the total number of conformations available to an ensemble of y chains with unconstrained end-to-end vectors. The corresponding entropy is S0 ¼ k ln V0. The strategy is to employ Gaussian statistics for (i) freely jointed chains and (ii) the radius of gyration distribution for segments about the center of mass, together with a lattice model description of excluded volume, to evaluate both terms on the far right side of the previous equation for ln{V/V0}. Boltzmann’s entropy equation (i.e., S ¼ k ln V) provides a route to calculate equilibrium thermodynamic properties from this conformational study, and the expansion factor is chosen to minimize the Helmholtz free energy of the system of y chains at constant temperature T and total system volume V. Chain expansion does not increase the volume of dilute solutions, it only increases the probability for polymer – solvent interactions at the expense of segment – segment interactions within the same chain or between different chains. The discussion below is applicable to athermal solutions.

572

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers

13.10.1 Excluded Volume Parameter w via Probability Analysis on a Cubic Lattice and the Radius of Gyration Distribution Function Polymer chains contain N segments, each of length l. These segments are placed on a cubic lattice where each cell contains one chain segment. The volume of each cell is approximately l3. Consider a spherical shell of radius rk and thickness drk about the center of mass of a single chain with uniform segment density. The segment distribution about the center of mass, or the radius of gyration distribution, is assumed to follow Gaussian statistics. Comparisons between the exact expression and the Gaussian approximation for the radius of gyration distribution function can be found in Modern Theory of Polymer Solutions, by Yamakawa [1971]. Differences between the exact and approximate distributions are negligible for all radii of gyration. Within volume element dVk ¼ 4p(rk)2 drk about the center of mass, there are zk lattice cells and Nk chain segments. Hence, zk ¼ dVk/l3. Evaluate the probability that each of the Nk chain segments finds an empty lattice cell to occupy. Initially, this calculation is performed for all Nk segments within volume element dVk. Then, this analysis is extended to other regions of space that are described by spherical shells with all possible radii of gyration about the center of mass. For volume element dVk, there is unit probability that the first chain segment finds an empty lattice cell. All of the other Nk 2 1 segments experience decreasing probability of finding empty lattice cells in the same volume element dVk as a consequence of excluded volume. For example, the probability that the second chain segment finds an empty lattice cell after the first segment is placed in this region is given by 1 2 (1/zk), which is equivalent to the difference between the total volume of this spherical shell at radius rk and one lattice site (i.e., dVk 2 l3) relative to dVk. For 1  x  Nk, the probability that the xth chain segment finds an empty lattice cell in region k is 1 2 (x 2 1)/zk. The excluded volume parameter wk of interest in volume element dVk is evaluated by considering all Nk chain segments in this region of space and multiplying all sequential probabilities that correspond to segments finding empty lattice cells. The result is

wk ¼

Nk  Y x¼1

ln wk ¼

1

x1 zk



  Nk Nk N k 1 X X X x1 x1 x  ln 1  ¼ z z z k k x¼1 x¼1 x¼0 k

For sufficiently dilute solutions, the total number Nk of chain segments in this region will be small relative to the total number zk of lattice cells. Hence, one expands ln(1 2 y)  2y 2 y 2/2 2 y 3/3 2 . . . and truncates the Taylor series after the linear term (i.e., y ¼ (x 2 1)/zk). Next, one replaces the summation by an integral to obtain the following result for ln wk: ln wk  

N k 1 X x¼0

Nkð1

x )  zk

0

x (Nk  1)2 N2 dx ¼   k 2zk 2zk zk

13.10 Excluded Volume and the Expansion Factor a for Real Chains

573

This calculation of wk represents an estimate of the fractional decrease in the number of conformations that are available to polymer chains in volume element dVk due to excluded volume considerations. For freely jointed polymer chains that contain N segments, each of length l, one estimates Nk within dVk by invoking a Gaussian distribution of segments about the center of mass. The radius of gyration distribution P(s) for freely jointed chain segments within a spherical shell of radius rk about the center of mass is required, where s ¼ rk and the mean-square radius of gyration is ks 2l ¼ a2Nl2/6. Implicitly, one assumes that ks 2l is six-fold smaller than kr 2l in Q-solvents (i.e., see Section 13.7) and good solvents, which is accurate for the smoothed-density model, where one cannot distinguish any differences between the expansion factors for ks 2l and kr 2l [Yamakawa, 1971]. Hence, Nk ¼ NP(s ¼ rk) dVk, such that  3=2  3 3rk2 Nk ¼ 4pNrk2 drk exp  2ks2 l 2pks2 l ks2 l ¼ 16 kr2 l; kr 2 l ¼ a2 N l2 Now, one extends this single-chain excluded volume analysis to other regions of space that are characterized by spherical shells with radius of gyration 0  s  1 about the center of mass of the same chain, such that the mean-square end-to-end vector is kr 2l ¼ a2Nl2 and the mean-square radius of gyration is ks 2l ¼ kr 2l/6. Essentially, one (i) squares the Gaussian expression for Nk, (ii) evaluates wk in region k with volume element dVk where the spherical shell has radius of gyration rk about the center of mass of a single chain, (iii) calculates the probability that segments of the same chain find empty lattice cells in other regions of space, (iv) constructs products of wk’s in probabilistic fashion to obtain the complete excluded volume parameter for a single chain with prescribed end-to-end distance, and (v) employs this result for chains of all sizes, which is consistent with an assumption in the previous subsection. Since products of wk’s translate to sums of ln wk’s, and sums of ln wk’s over all regions of space can be replaced by integration with respect to rk, the excluded volume parameter of interest (i.e., w) is obtained with assistance from the third integration formula in Problem 13.1 at the end of this chapter: ln w ¼

1 X

ln wk ¼ 

k¼1

1 1X Nk2 2 k¼1 zk

 3  1 1X l3 9 18rk2 2 2 2 N (4 p r dr ) exp  k k 2 k¼1 4prk2 drk pa2 N l2 a2 N l2  3  1 X 9 18rk2 3 2 2 ¼ 2pl N rk drk exp  2 2 pa2 N l2 a Nl k¼1 ¼

3

)  2pl N

2



9 2 pa N l2

3 1 ð

rk2

 18rk2 exp  2 2 drk a Nl

0

¼

27 25=2 p 3=2

N 1=2 a3 ¼ 0:86N 1=2 a3

574

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers

Since the number of segments N in each polymer chain scales linearly with molecular weight MW (i.e., MW  N ), this analysis of the excluded volume parameter w yields the following molecular weight scaling law:

w  exp{MW 1=2 a3 }

13.10.2 Conformational Entropic Contribution to the Multiplicity of States for Real Chains in “Good” Solvents The previous expression for the excluded volume parameter w must be considered in the following expression for complete evaluation of the multiplicity function V and the associated conformational entropy of the system via kBoltzmannln V. Recall that Pi represents the fraction of chains with end-to-end distance ri in the expanded state when excluded volume effects are absent. The excluded volume parameter w reduces this fraction of chains described by Pi. The previous equation reveals that there is a larger reduction in the fraction of chains with given dimensions when the molecular weight is larger and the expansion factor a decreases toward unity. Since there are N segments per chain, and l represents the length of each segment, the following equations provide an appropriate representation of the actual situation:     X V yPi  y ln w þ ln yi ln yi V0 Groups of conformations w=different ri

Pi ¼ 4pri2



3 2pN l2

3=2

 3r 2 exp  i 2 2N l

There are y i chains in the expanded state with end-to-end distance ri and mean-square end-to-end vector kr 2l ¼ a2Nl2 when excluded volume is considered and Gaussian statistics are not applicable to describe the distribution of ri, even though a Gaussian distribution of segments about the center of mass was invoked in the previous subsection for the radius of gyration distribution function in the expanded state. It must be emphasized, however, that these real chains with end-to-end distance ri had unperturbed dimensions given by ri/a prior to expansion, as described by the Gaussian distribution function. Hence, within a spherical shell of radius ri/a about the origin where one end of the ideal chain is anchored, the orientation-independent distribution of freely jointed ideal chain ends with end-to-end distance ri/a is   yi r2 r2 ¼ 4pB i2 exp k2 i2 y a a It is important to emphasize that the fraction of unperturbed chains with end-to-end distance ri/a in a spherical shell of radius ri/a about the origin is the same as the fraction of real chains that experience expansion due to polymer – solvent interactions

13.10 Excluded Volume and the Expansion Factor a for Real Chains

575

with end-to-end distance ri in a spherical shell of radius ri about the same origin. Ideal chains follow a Gaussian distribution with mean-square end-to-end distance given by kr 2lUnperturbed ¼ Nl2, whereas the dimensions of the real chains are kr 2lExpanded ¼ a2Nl2. Constants B and k in the previous equation are determined from normalization and a freely jointed second moment given by kr 2lExpanded ¼ a2Nl2, with excluded volume in the expanded state. The second moment of y i/y is evaluated in the expanded state so that the results are representative of real chains. In summary, the fraction of real chains with end-to-end distance ri that have been expanded by a factor a relative to unperturbed ideal chains is obtained from a Gaussian distribution that was constructed from the dimensions of the corresponding unperturbed chains. The coefficient b2 in the exponential of Pi yields an ideal second moment given by kr 2lUnperturbed, whereas the coefficient k2 in the exponential of y i/y yields a real second moment of kr 2lExpanded. Both coefficients (i.e., b2 and k2) are equivalent because the expansion factor a is included in the exponential function for y i/y . With assistance from integration formula in Problem 13.1 at the end of this chapter, the appropriate equations to calculate B and k are ð1

1  

 ð yi 4pB r2 4pB p1=2 a3 dri ¼ 2 ri2 exp k2 i2 dri ¼ 2 ¼1 y a a a 4k3

0 1 ð

ri2

0

 

 yi 4pB r2 4pB 3p1=2 a5 ¼ a2 N l2 dri ¼ 2 ri4 exp k2 i2 dri ¼ 2 y a a a 8k5 1 ð

0

0 3

B¼

k 3 ; k2 ¼ p 3=2 a 2N l2

Finally, the fraction of chains with end-to-end distance ri in the expanded state, within a spherical shell of radius ri about the origin where one chain end is anchored, is given by

3=2   yi 3 3 ri2 ¼ 4pri2 exp  y 2pa2 N l2 2N l2 a2 Now, all of the information is available to evaluate the summation of terms on the right side of ln{V/V0} after replacing the summation by integration over all possible chain lengths. The calculation proceeds as follows:

yi ln

 

3=2   yP i 3 3 ri2 exp  ¼ y 4pri2 yi 2pa2 N l2 2N l2 a2 8  3=2  9 > > 3 3ri2 > > 2 > > exp  = < 4pri 2N l2 2pN l2  ln

   3=2 > 3 3 ri2 > > > > > ; :4pri2 exp  2 2 a2 2 2pa N l 2N l

576

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers

 

  3 ri2 3ri2 1 3 ln a exp  1  a2 2N l2 a2 2N l2

3=2       3 3ri2 1 3 ri2 2 3  1  2 þ ln a exp  ¼ y 4pri a 2N l2 2pa2 N l2 2N l2 a2 ¼ y 4pri2

3 2pa2 N l2

3=2

 exp 

Upon replacing the summation over all chain conformations with different ri in the expanded state by integration with respect to ri, integration formulas in Problem 13.1 at the end of this chapter yield the following result: 1     ð X yP i yP i dri yi ln ) yi ln yi yi Groups 0

of conformations w=different ri

3 ¼ 4py 2pa2 N l2

3=2

2

1  ð 4ln a3 r 2 exp  i

 3 ri2 dri 2N l2 a2

0





3 1 1 2 a 2N l2

1 ð

 ri4 exp 

3 2N l

ri2 2 a2



3 dri 5

0

3=2 "

  p1=2 2 2 2 3=2 a Nl ln a3 4 3     # 3 1 3p1=2 2 2 2 5=2  1 2 a Nl a 8 3 2N l2



 3 1 ¼ y ln a3  (a2  1) ¼ 3y ln a  (a2  1) 2 2

3 ¼ 4py 2pa2 N l2

13.10.3 Helmholtz Free Energy Minimization Yields the Equilibrium Chain Expansion Factor in Athermal Solutions The conformational entropy change is always negative when an ensemble of y chains with unperturbed dimensions (i.e., a ¼ 1) expands in good solvents such that a . 1 and excluded volume is prevalent. As prescribed by Boltzmann’s equation,   h i V S  S0 ¼ kBoltzmann ln ¼ kBoltzmann y 0:86N 1=2 a3 þ 3 ln a  32 (a2  1) V0 If the internal energy of the system does not depend on the chain expansion factor a, then maximization of the previous entropy expression with respect to a, corresponding to the minimum reduction in entropy associated with excluded volume, yields a

13.10 Excluded Volume and the Expansion Factor a for Real Chains

577

Dimensionless Entropy Change per Chain

0.0

–2.0

–4.0

–6.0 N = 5; a = 1.34 N = 20; a = 1.48 N = 50; a = 1.59

–8.0

–10.0

–12.0 1.0

N = 100; a = 1.68 N = 200; a = 1.78 1.3

1.7 2.0 2.3 Chain Expansion Factor (a)

2.7

3.0

Figure 13.1

Effect of the chain expansion factor a on the dimensionless conformational entropy change associated with expanding real polymers in good solvents when excluded volume is considered on a cubic lattice. The legend identifies the effect of the number of chain segments N, or molecular weight, on the optimum value of a that corresponds to maximum entropy and minimum Helmholtz free energy for athermal solutions.

minimum in the Helmholtz free energy at equilibrium. This methodology allows one to predict equilibrium chain dimensions in dilute solution that depend on the total number N of segments per chain, the length l of each segment, and the equilibrium expansion factor aequliibrium that is molecular-weight dependent but not segmentlength dependent. In the expanded state, the root-mean-square end-to-end chain length is given by kr 2l1/2 ¼ aequilibriumN 1/2l, which scales as molecular weight, or N, to a power greater than one-half. When the total number of polymer chains y under consideration and the molecular weight of each chain (i.e., MW  N ) are constant, Figure 13.1 illustrates how the dimensionless entropy change per chain (i.e., (S – S0)/y kBoltzmann) depends on molecular weight and the expansion factor a. The equilibrium expansion factor aequilibrium, which maximizes entropy and minimizes Helmholtz free energy, represents the solution to the following equation: 

@ (S  S0 ) @a

 N,T,V or N,T,y

¼ kBoltzmann y (3)0:86N 1=2 a4 equilibrium þ

3

aequilibrium

a5equilibrium  a3equilibrium ¼ 0:86N 1=2

  3aequilibrium ¼ 0

578

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers

revealing that aequilibrium increases for higher-molecular-weight chains. In fact, for large expansion of high-molecular-weight chains in very good solvents, where aequilibrium is much greater than unity, the cubic term can be neglected in the previous equation relative to the fifth power of aequilibrium, which predicts that the equilibrium chain expansion factor scales as the one-tenth power of molecular weight (i.e., aequilibrium  N 0.1). Under these conditions (i.e., high-molecular-weight chains in very good solvents), the root-mean-square end-to-end chain length scales as the six-tenths power of molecular weight, kr 2l1/2  aequilibrium N 0.5  N 0.6.

13.11 DEGENNES SCALING ANALYSIS OF FLORY’S LAW FOR REAL CHAINS IN “GOOD” SOLVENTS As mentioned at the end of the previous section, this development yields a universal scaling exponent, y ¼ 0.6, for the molecular-weight dependence of the root-meansquare end-to-end length of three-dimensional chains when polymer – solvent interactions cause considerable chain expansion with respect to unperturbed dimensions. There are at least two factors that contribute to the overall free energy of real chains: monomer – monomer repulsive interactions favor chain expansion and entropy elasticity favors recoil. Equilibrium chain conformations are determined by free energy minimization. If chains contain N monomer segments and kr 2l1/2  N y then the monomer molar density cmonomer scales as cmonomer  N/kr 2l3/2  N 1 – 3y . The unfavorable contribution of monomer – monomer repulsion to the overall free energy density (i.e., kBoltzmannTvexcludedfcmonomerg2, where vexcluded represents the excluded volume for each segment) is integrated over the volume of a single chain, yielding the following scaling law for repulsive interactions that cause chain expansion: Frepulsive  kBoltzmann Tvexcluded c2monomer kr2 l3=2  N 2(13y)þ3y  N 23y The competitive effect is due to entropy elasticity, which favors smaller end-to-end chain distances. The probability density distribution function P(r) may not be Gaussian for real chains with excluded volume, but the connection between entropy and multiplicity of states via the Boltzmann relation, and the entropic contribution to elastic free energy are constructed to mimic ideal chains where P(r)  exp(2r r). Hence,



Felastic  TSelastic ¼ kBoltzmann T ln P ¼ kBoltzmann T



kr rl  N 2y1 Na2

where Na 2 is the mean-square end-to-end distance of unperturbed chains that contain N segments, each of length a. Equilibrium chain dimensions, kr 2l1/2  N y , correspond to minimization of the overall free energy (i.e., Frepulsive þ Felastic) with respect to the mean-square end-to-end vector kr 2l. Evaluation of the universal scaling exponent

13.12 Intrinsic Viscosity of Dilute Polymer Solutions and Universal Calibration Curves

579

y for real chains proceeds as follows:      @ @ kr2 l 2 2 3=2 (Frepulsive þ Felastic )  kBoltzmann T vexcluded cmonomer kr l þ 2 @kr2 l @kr 2 l Na T T    2 @ kr l vexcluded N 2 kr2 l3=2 þ 2  kBoltzmann T ¼0 @kr 2 l Na T 3 1  vexcluded N 2 kr 2 l5=2 þ 2 ¼ 0 2 Na kr2 l5=2  vexcluded N 3 a2 kr2 l1=2  N 3=5 Notice that both contributions to the overall free energy (i.e., Frepulsive  N 223y ; Felastic  N 2y 21) scale as N 1/5 when y ¼ 0.6 for three-dimensional chains in good solvents. Biographical Sketch Pierre-Gilles de Gennes, 1991 Nobel laureate in physics and Professor of Physics at the College of France in Paris, passed away May 18, 2007 at the age of 74. De Gennes was a scientist with exceptional ability to tackle complex problems, including the properties of liquid crystals and polymers. His intellectual curiosity was prodigious. One exemplary characteristic was the magnificent way he lectured to nonspecialists, explaining clearly the problem, how it attracted his attention, his problem-solving approach, and the results he and his collaborators obtained. He never failed to point out what remained to be solved and how one could begin to work toward a better understanding of the phenomenon. His classic 1979 book, Scaling Concepts in Polymer Physics (Cornell University Press), had an enormous influence on the polymer science community. In short, Pierre-Gilles de Gennes was a giant in contemporary science.

13.12 INTRINSIC VISCOSITY OF DILUTE POLYMER SOLUTIONS AND UNIVERSAL CALIBRATION CURVES FOR GEL PERMEATION CHROMATOGRAPHY The objective of this section is to illustrate how chain dimensions, based on Gaussian statistics, and the hydrodynamic size of a single macromolecule allow one to predict the molecular weight dependence of the intrinsic viscosity of dilute polymer solutions. Furthermore, these predictions suggest how one should construct a universal calibration curve for molecular weight determination from gel permeation chromatography, when the separation device is employed to measure the molecular weight distribution of polymer B based on calibrated residence times using known monodisperse fractions of polymer A.

580

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers

13.12.1 Einstein’s Theory for the Viscosity of a Dilute Suspension of Spherical Particles in a Newtonian Solvent A crude model for a dilute suspension of spherical particles is based on a single solid sphere of radius R surrounded by a Newtonian solvent with viscosity hsolvent. The outer boundary of the solvent is identified by radius R0. This allows one to represent the volume fraction of solid particles f in the true suspension by (R/R0)3 in the model, where dilute suspensions correspond to f 1. The following assumptions are invoked to solve the hydrodynamic problem: (a) Fluid flow is ideal far from the particle. This corresponds to the potential flow regime where viscous forces are negligible. Hence, there is a balance between convective forces, pressure forces, and gravity forces far from the particle. (b) In the vicinity of the particle, viscous forces are important, and they are balanced by pressure and gravity forces in the creeping flow regime. This corresponds to very slow flow in the momentum boundary layer adjacent to the solid particle, where convective forces that scale as kvl2 are negligible (i.e., kvl is the average fluid velocity for flow past the particle). The first method of solution equates the volume-averaged viscous stress tensor in the true suspension and the crude hydrodynamic model. For the true suspension, Newton’s law of viscosity for an incompressible fluid yields ktlvolume-averaged ¼ hsuspension kg_ lvolume-averaged where the volume-averaged rate-of-strain tensor on the right side of the previous equation contains linear combinations of velocity gradients. The hydrodynamic model provides a crude estimate of the volume-averaged viscous stress tensor:    5 f kg_ lvolume-averaged ktlvolume-averaged ¼ hsolvent 1 þ 2 1f Comparison of the previous two equations yields an expression for the viscosity of a dilute suspension when the volume fraction of solid particles f is much less than one, which implies that second and higher order terms in the solid particle volume fraction can be neglected:    5 f hsuspension ¼ hsolvent 1 þ 2 1f     5 5 2 3 ¼ hsolvent 1 þ f[1 þ f þ f þ f þ   ]  hsolvent 1 þ f 2 2 The second method of predicting the viscosity of a dilute suspension of spherical particles in a Newtonian solvent equates the average rate of kinetic energy dissipation in the true suspension and the crude hydrodynamic model. The irreversible degradation of mechanical energy to thermal energy, given by the following integration over

13.12 Intrinsic Viscosity of Dilute Polymer Solutions and Universal Calibration Curves

system volume V,

581

ð  (t : rv) dV V

is always positive for Newtonian fluids (that cannot store elastic energy) and contributes to entropy generation in nonequilibrium systems [Belfiore, 2003]. The suspension viscosity based on this approach is  

1 þ 12f hsuspension ¼ hsolvent ¼ hsolvent 1 þ 12f 2 (1  f)

 [1 þ 2f þ 3f2 þ 4f3 þ   ]  hsolvent 1 þ 52f Both strategies yield the same expression for the suspension viscosity when truncation occurs after the terms that are linear in the solid particle volume fraction. The secondorder terms (i.e., 2.5f2 versus 4f2) are different in the two formulations summarized above. For concentrated cell suspensions, the first-order term is 2.5f and the secondorder term is 7.3f2 in a series expansion of the Vand equation: hsuspension/hsolvent ¼ f12 f 2qf2g2k, with k ¼ 2.5 and q ¼ 1.16. For a dilute suspension of spherical liquid droplets, with Newtonian viscosity hdroplet, in a Newtonian solvent with viscosity hsolvent, the effective viscosity of the suspension is " # ( ) hsolvent þ 52hdroplet hsuspension ¼ hsolvent 1 þ f þ  hsolvent þ hdroplet when the series is truncated after the term that is linear in the liquid droplet volume fraction f. This result is consistent with the two previous expressions for the viscosity of a dilute suspension of solid spherical particles when hdroplet  hsolvent.

13.12.2

Intrinsic Viscosity of Dilute Polymer Solutions

A hydrodynamic shell whose radius matches the root-mean-square end-to-end distance of a freely jointed chain surrounds a polymer molecule in a Newtonian solvent. Einstein’s theory for the viscosity of a dilute suspension of solid spherical particles is employed to evaluate the viscosity of the suspension. Then, one constructs the dimensionless relative viscosity hrelative as follows:

hrelative ¼

hsuspension  hsolvent 5  2 fPolymer hsolvent

The volume fraction of polymer in solution is directly related to its concentration, which can be reported in grams per cm3 or grams per 100 cm3 (i.e., g/dL). If CPolymer represents the polymer concentration with units of grams of polymer per dL of solution, then the polymer volume fraction is calculated as follows, where each chain is modeled as a spherical coil with an effective hydrodynamic radius given by its root-mean-square end-to-end distance, which includes the nonideal chain expansion factor a (see the answer to Problem 13.4 if one desires to use the

582

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers

mean-square radius of gyration instead of the mean-square end-to-end distance to estimate the volume of a random coil):   NAvogadro 4 1 CPolymer fPolymer ¼ p kr2 l3=2 MWPolymer 3 100 where Avogadro’s number (i.e., NAvogadro) and the molecular weight of the chain (i.e., MWPolymer) are required in the previous expression. The intrinsic viscosity [h], with units of dL/g, is the intercept on a graph of hrelative/CPolymer versus polymer concentration (i.e., CPolymer). It can also be evaluated by the following limit:     fPolymer hrelative 5 p NAvogadro 2 3=2  ¼ lim kr l [h] ¼ lim CPolymer !0 CPolymer 30 MWPolymer 2 CPolymer !0 CPolymer This analysis, based on Einstein’s theory for the viscosity of a dilute suspension of spherical particles, reveals that the product of intrinsic viscosity and polymer molecular weight scales as the 1.5 power of the mean-square end-to-end distance, where kr 2l0.5 is a measure of the hydrodynamic size of an isolated chain in dilute solution. Gel permeation chromatography is a separation technique that distinguishes different molecular weight fractions of a polymer by hydrodynamic size. It seems reasonable that hydrodynamic size scales as the square-root of molecular weight for freely jointed chains in a Q-solvent because the second moment of the Gaussian distribution reveals that kr 2l  nl2, where the number of random steps n is proportional to the degree of polymerization. The universal calibration curve for a specific column correlates residence time, retention time, or elution volume of a known molecular weight fraction of polystyrene, for example, with its hydrodynamic size. If the column is calibrated using known monodisperse fractions of polymer A, then residence time is correlated with molecular weight for polymer A. When the chromatographic column is used to identify a molecular weight fraction of polymer B that differs in chemical structure from polymer A, one measures the residence time of polymer B and locates the molecular weight of polymer A that corresponds to the same residence time. Since polymer A with known molecular weight and polymer B with unknown molecular weight exhibit the same residence time, they have identical hydrodynamic size. Hence, one equates the product of intrinsic viscosity and molecular weight for both polymers and calculates the corresponding molecular weight of polymer B. The intrinsic viscosity for a particular polymer depends on the solvent, temperature, and molecular weight, as described in the following section via the Mark – Houwink equation.

13.13 SCALING LAWS FOR INTRINSIC VISCOSITY AND THE MARK –HOUWINK EQUATION Intrinsic viscosity calculations based on Einstein’s theory for the viscosity of a dilute suspension of solid spheres yields the following result: [h] ¼ 6:3  1022

kr2 l3=2 MWPolymer

13.14 Intrinsic Viscosities of Polystyrene and Poly(Ethylene Oxide)

583

Since intrinsic viscosities can be calculated by extrapolating solution viscometry measurements to infinite dilution, and mean-square end-to-end chain dimensions are related to radii of gyration, which can be calculated from light scattering or neutron scattering data, the constant in the previous equation should be changed from the predicted value of 6.3  1022 to the experimental value of (2.0 + 0.6)  1021 for many polymers. The unperturbed dimensions of a freely jointed polymer chain under Q-solvent conditions are characterized by the mean-square end-to-end distance kr 2l, which scales linearly with molecular weight because the number of random steps n is proportional to the degree of polymerization. Hence, intrinsic viscosity scales as the 0.5 power of molecular weight in a Q-solvent. This result represents a subset of the following scaling law for intrinsic viscosity, which is known classically as the Mark – Houwink equation: [h] ¼ K{kMWPolymer lv }a ) K(MWPolymer )a monodisperse

The temperature-dependent Mark – Houwink parameters K and a (where the exponent a is different from the equilibrium chain expansion factor in good solvents) are unique to each polymer – solvent pair. For polydisperse polymers, the viscosity-average molecular weight should be used in the previous equation. Under Q-solvent conditions, the exponent a is 0.5, as discussed above. Classic examples of Q-solvent conditions that can be used to study unperturbed chain dimensions when polymer – solvent interactions are indistinguishable from segment – segment interactions in the undiluted polymer are summarized below: (i) Poly(isobutylene) in benzene at 24 8C. (ii) Natural rubber in n-propyl ketone at 14.5 8C. (iii) Poly(vinyl acetate) in ethyl n-butyl ketone at 29 8C. When a ¼ 1, the Mark – Houwink equation reveals proportionality between intrinsic viscosity and molecular weight, which is consistent with Staudinger’s rule developed in 1930 by the first Nobel laureate in polymer science (Hermann Staudinger). Tabulated Mark – Houwink exponents a can be used to determine the molecular weight dependence of mean-square end-to-end chain dimensions above the Q-temperature. For example, kr2 l  {[h]MWPolymer }2=3  {(MWPolymer )aþ1 }2=3  (MWPolymer )2(aþ1)=3

13.14 INTRINSIC VISCOSITIES OF POLYSTYRENE AND POLY(ETHYLENE OXIDE) Consider an atactic vinyl polymer, like polystyrene, with a number-average molecular weight of 105 daltons in tetrahydrofuran at the Q-temperature. Since the repeat unit (i.e., CH2CH(C6H5)) molecular weight is approximately 102 daltons, there are 103 monomer segments in the chain. Within the context of freely jointed chains in a Q-solvent that contain n random steps of length l, n corresponds to the degree of polymerization (i.e., 103) and l represents the linear dimension of a repeat unit

584

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers

that is modeled as a straight segment. The repeat unit consists of two carbon – carbon ˚ and a CZCZC bond angle of 109.58. single bonds with a CZC bond length of 1.54 A Hence, l is calculated as follows:   109:58 ¼ 2:51  108 cm l  2{1:54  108 cm} sin 2 ˚ , whereas the contour length The root-mean-square end-to-end chain distance is 80 A of the macromolecule, evaluated by traversing two carbon –carbon single bonds per ˚ . Based on freely jointed chain dimensions, the intrinsic viscosity repeat unit, is 3080 A is 0.32 dL/g using the Einstein constant of 6.3  1022 and 0.010 dL/g using the experimental constant of 2  1021 for a broad range of polymers. As a second example of intrinsic viscosity calculations, a 9  105 dalton molecular-weight sample of poly(ethylene oxide), with repeat unit chemical structure of CH2CH2O, is dissolved ˚, in water at the Q-temperature. The carbon –carbon single bond length is 1.54 A ˚ , and both the CZCZO and the carbon – oxygen single bond length is 1.43 A CZOZC bond angles are 109.58. The effective length of a random step l is obtained by modeling a repeat unit as a “straight segment” with one CZC bond and two CZO bonds. Hence,     109:58 109:58 þ 2{1:43  108 cm} sin l ¼ {1:54  108 cm} sin 2 2 ¼ 3:6  108 cm Furthermore, the PEO chains are freely rotating with restricted bond angles of 109.58, which increases the mean-square end-to-end chain distance by a factor of 2. The intrinsic viscosity at the Q-temperature is   MWPolymer 2 3=2 21 2:0  10 2 l MWrepeat ¼ 0:85 dL=g [h ] ¼ MWPolymer These examples provide typical ranges for the intrinsic viscosity of common polymers with reasonable molecular weights.

13.15 EFFECT OF pH DURING DILUTE-AQUEOUSSOLUTION PREPARATION OF SOLID FILMS ON THE GLASS TRANSITION A few examples are summarized below to illustrate this effect, where the pH of aqueous polyelectrolyte solutions, which can be manipulated via NaOH, affects the glass transition temperature of the solid residue that is recovered after evaporation of the solvent. When an aqueous solution that contains 1% poly(acrylic acid) is prepared at pH ¼ 2.7 (without NaOH), the solid residue exhibits a glass transition temperature of 135 8C. Addition of concentrated NaOH during solution preparation yields pH values of 5 and 7, with corresponding glass transition temperatures of 165 8C and 225 8C for the solid residues. Capillary viscometry measurements of these 1%

13.15 Effect of pH During Dilute-Aqueous-Solution Preparation of Solid Films

585

Table 13.1 Viscosities of Dilute a Aqueous Solutions of Poly(acrylic acid) and the Corresponding Glass Transition Temperatures of the Solid Residue After Solvent Evaporation Solution pH via NaOH 2.7 (no NaOH) 5 7 a b

Relative viscosity b

Tg of solid residue (8C)b

4 55 92

135 165 225

Dilute means 1 wt %. From Lee [2004].

aqueous solutions yield dimensionless relative viscosities (i.e., hrelative): h  hsolvent hrelative ¼ solution hsolvent that are summarized in Table 13.1. The actual viscosities of the solvent (i.e., H2O) and the solution are given by hsolvent and hsolution, respectively, in the previous equation. These viscosities are calculated from efflux times tefflux required to drain the bulb above a tilted capillary tube (see analysis in the Appendix of this chapter; m/r ¼ btefflux, where m is the Newtonian equivalent of h). The increase in relative viscosity can be explained by more Coulombic repulsion and chain expansion at higher NaOH concentrations and higher pH, because the degree of dissociation of acidic sidegroups is larger, and Hþ in the weakly dissociated COOH side group of poly(acrylic acid) is scavenged by OH2 counterions in the added base (i.e., NaOH). There is not much discussion in the research literature of the effect of pH during solution preparation on the Tg of solid films that are recovered after solvent evaporation. Another example of ion pairing is presented graphically in Figure 13.2 to illustrate how the pH of aqueous solutions decreases when mono-, di-, and trivalent metal cations liberate Hþ from the side group of poly(acrylic acid) to form acid – base complexes. Hence, pH decreases at higher metal cation concentrations, in agreement with the equilibrium scheme provided in Figure 13.2. There are at least two possible reasons why glass transition temperatures increase significantly when solid films are prepared at higher pH. As mentioned above, chain conformation in the solid state, which mimics Coulombic repulsion at higher pH in solution, should produce more extended chains that exhibit larger hydrodynamic size. The discussion of intrinsic viscosity reveals that chain expansion and hydrodynamic size directly affect solution viscosity, as illustrated in Table 13.1. The most important factors that affect the glass transition temperature of solid films are ion pairing and Naþ cation clustering (i.e., the ionomer effect). In solution, some counterions (i.e., Hþ, Naþ, etc.) will be loosely associated with the polyanion, whereas other counterions will be strongly associated (condensed counterions). There will also be specific binding effects. Ion-specific electrodes can be employed to measure the chemical potential of various counterions. The interpretation of free versus bound counterions that are separated from or attached to the polyanion (i.e., polymer chain with anionically charged side groups) depends on the physics of the measurement system employed to characterize polyelectrolytes.

586

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers 2.8

Solution pH

2.6 MClx

2.4 O

OH

2.2

O

O M Clx–1

Eu3+ Co2+ Ni2+ Zn2+ Cu2+ Na+

2.0

1.8

+ H+ + Cl–

0

1

2

3 4 Metal/PAA

5

6

Figure 13.2 Effect of metal cation concentration on pH in aqueous solutions of poly(acrylic acid), PAA [Lee, 2004].

13.16 DEGENNES SCALING ANALYSIS OF THE THRESHOLD OVERLAP MOLAR DENSITY c IN CONCENTRATED POLYMER SOLUTIONS AND THE CONCEPT OF “BLOBS” This section focuses on three-dimensional chains in good solvents (i.e., kr 2l1/2  aN y ) when coil overlap occurs in the intermediate regime between dilute solutions and solids. The threshold monomer concentration c is comparable to the local monomer concentration within a single chain. Hence, c  cmonomer 

N kr2 l3=2

 a3 N 13y ) a3 N 4=5 y ¼0:6

where a corresponds to the monomer segment length. If a 3 represents monomer volume on a cubic lattice and c describes the threshold number of monomers per volume of lattice required for coil overlap, then the critical polymer volume fraction w scales as

w  a3 c  N 13y ) N 4=5 y ¼0:6

In semidilute solutions, one concludes that critical polymer concentrations and critical polymer volume fractions decrease for higher-molecular-weight chains according to the previous two scaling laws. Polymer chains contact each other, and they are on the verge of interpenetration, in semidilute solutions when solute volume fractions w  w. One envisions a single chain in semidilute solution as a succession of interconnected (but independent) “blobs,” where Flory’s law of real chains describes the conformation of g monomer segments within each blob. In simple terms, a blob with

13.17 Entropically Elastic Retractive Forces via Statistical Thermodynamics

587

mesh size j describes a region of space in solution with volume 43 pj3 that contains g monomers which do not interact with other chains. The total number of interconnected blobs per chain is N/g. An important consideration here is that the root-mean-square end-to-end length of three-dimensional chains that contain N monomer segments with excluded volume is comparable to the blob radius j when w  w, because interpenetration has not begun and g  N. At higher solute volume fractions, w . w, chain dimensions are significantly larger than blob radius j and g , N, so j is a decreasing function of w and independent of the total number of segments N per chain. It is necessary to require that j must decrease at larger polymer volume fractions (i.e., w . w ) because segments within one blob do not interact with other chains, and this region of space in solution becomes smaller at higher solute concentrations. Application of Flory’s law of real chains within each blob, together with the restrictions mentioned immediately above, yields the following scaling law for the polymer volume fraction dependence of mesh size j: Flory0 s law

j(w) ) agy  kr 2 l1=2 y ¼ 0:6



  13y k w k N  aN y = f (N) w w

Hence, j  w2k and y þ k(1 2 3y ) ¼ 0 to eliminate any dependence of j on the total number of chain segments N when the polymer volume fraction is greater than its critical value, such that coil overlap and interpenetration occur, and chain dimensions are much larger than the mesh size of an individual blob. Dependence of mesh size j and the number of monomers g per blob on polymer volume fraction w is summarized below for three-dimensional chains with excluded volume: Flory0 s

y 3 law k¼ ) 3y  1 y¼0:6 4 j(w)  awk ) aw3=4 g(w)  wk=y ) w5=4

13.17 ENTROPICALLY ELASTIC RETRACTIVE FORCES VIA STATISTICAL THERMODYNAMICS OF GAUSSIAN CHAINS This is a classic example of the use of Gaussian chain statistics and macroscopic thermodynamics in the presence of an external force field to predict the stress – strain behavior of rubber-like solids that experience affine deformation. The principle of affine deformation states that individual network strands and crosslink junctions deform in the same way that macroscopic dimensions of the sample deform. One invokes the principle of affine deformation because Gaussian statistics are used to describe the conformation of individual network strands between crosslink junctions, whereas stress – strain analysis is based on deformation of macroscopic sample dimensions.

588

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers

13.17.1 Characteristic Mechanical Response of Elastomers High-molecular-weight polymers that exhibit rubber-like elasticity are known as elastomers. These materials respond almost instantaneously to an external force field with negligible viscous dissipation. Elastomers recover their original dimensions rather quickly when the external force field is removed. Hence, the creep recovery process occurs quickly. Chemically crosslinked elastomers exhibit complete recovery of original dimensions, whereas thermoplastic elastomers do not recover completely because there is significant stress – strain hysteresis due to chain – chain slippage when covalent chemical crosslinks are absent. The following structural properties of polymeric materials are required for elastomeric response: (i) high molecular weight; (ii) low glass transition temperatures, such that the material is used above its Tg; (iii) flexible chain structure with rapid interconversion among rotational isomers; (iv) weak intermolecular attractive forces; (v) minimum degree of order and high segmental mobility prior to stretching; and (vi) permanent crosslinks for rapid retraction with no irrecoverable deformation.

13.17.2 Connection Between Entropy and Probability Density Distribution Functions For ideal networks with random chemical crosslinks, Gaussian chain statistics describe the conformational characteristics of network strands between crosslink junctions. The statistical problem begins by invoking Boltzmann’s relation between entropy S and the multiplicity of states V. Since energy differences among all of the conformations for a particular network strand are not considered, the “counting problem” reduces to S ¼ k ln V, which appears on Boltzmann’s tombstone at the Zentralfriedhof (central cemetery) in the district of Simmering in Vienna, Austria (Fig. 13.3). This is the largest and most famous of Vienna’s 50 cemeteries, that opened in 1874, spanning 2.4 square kilometers, and containing 3.3 million interred. Analogous to the three-dimensional probability density distribution function P(r; n), V depends on the magnitude of the end-to-end chain vector r. If the total number of possible chain conformations available to the network strands is given by the acronym TNPC, and DV represents a volume element that contains a sufficient number of strands such that statistics are applicable, then the total number (i.e., multiplicity) of conformations with end-to-end chain vector r is given by  V(jrj) ¼ (TNPC)DV

3 2pkr2 l

3=2

  3r 2 exp  2 2kr l

13.17 Entropically Elastic Retractive Forces via Statistical Thermodynamics

Figure 13.3

589

Boltzmann’s tombstone at the Zentralfriedhof in Vienna, Austria.

Hence, on a “per-network-strand basis,” one calculates the entropy for freely jointed chains with end-to-end chain vector r, realizing that S, similar to P(r; n) and V, depends on the magnitude of r. Furthermore, S ¼ k ln V reveals that S scales as r 2. Hence, tensile deformation induces larger chain dimensions, a reduction in entropy, and a corresponding retractive force that is consistent with “entropy elasticity”:   3 3 3kr2  S(jrj) ¼ k ln(TNPC) þ k ln(DV) þ k ln 2kr 2 l 2 2p kr 2 l where the mean-square end-to-end chain distance of network strands kr 2l is given by nl2 in the absence of crosslink junctions. A distribution in the number of random steps n is required to account properly for a distribution in the molecular weight of individual network strands between crosslink junctions. The fact that all network strands are connected to each other via an intricate web of permanent junction points when the crosslink density surpasses the percolation threshold is considered in the development below via the effect of affine deformation on the square of the end-to-end chain vector, r 2. For example, r 2 is expressed in terms of the sum of squared displacements along three mutually orthogonal directions of a rectangular Cartesian coordinate system: r 2 ¼ x2 þ y2 þ z2 Next, one introduces relative elongations dx, dy, and dz in the x-, y-, and z-directions, respectively, such that di represents the instantaneous displacement in the ith coordinate direction divided by the initial dimension in the same direction prior to

590

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers

deformation. Hence,

dx ¼

x y z , dy ¼ , dz ¼ xinitial yinitial zinitial

The principle of affine deformation allows one to re-express the x-, y-, and z-projections of the square of the end-to-end chain vector for the deformed sample in terms of relative elongations, which correspond to normal components of the engineering strain tensor þ1. Let the initial dimensions of the undeformed crosslinked sample resemble a cube, such that xinitial ¼ yinitial ¼ zinitial at both the macroscopic and microscopic levels of description. Now, the square of the end-to-end chain vector in the undeformed crosslinked state is r2undeformed ¼ x2initial þ y2initial þ z2initial ¼ 3x2initial w=crosslinks

This is consistent with an isotropic crosslinked material in the undeformed state. The effect of deformation on the entropy S per network strand is expressed in terms of relative elongations:  2  3 x þ y2 þ z2 S ¼ constant  k 2 kr 2 lwithout crosslinks 9 8 r2undeformed > > = < 1 w=crosslinks (d2x þ d2y þ d2z ) ¼ constant  k 2 2 > ; :kr lwithout crosslinks > Two important concepts about rubber-like materials are invoked to re-express elongations dy and dz, transverse to the stretch direction, in terms of dx for uniaxial stretching in the x-direction. Since Poisson’s ratio is 0.5 for incompressible liquids and crosslinked solid materials that exhibit no volume change upon deformation in tension, compression, or shear, one equates initial volume to the instantaneous volume at any level of deformation, assuming that the constant-volume assumption is valid for all reasonable values of dx that are not too large: Volume initial w=crosslinks

¼ xinitial yinitial zinitial ¼ Volumeinstantaneous ¼ xyz w=crosslinks

dx dy dz ¼ 1 The mismatch between prediction and experiment for the stress – strain response of crosslinked rubber-like solids at large deformation can be traced, in part, to the fact that the constant-volume assumption is not valid beyond approximately 100% strain (i.e., dx  2). In other words, Poisson’s ratio decreases below the value of 0.5, indicating that transverse contraction of the material does not offset the volume increase due to uniaxial extension at large strain. The second important concept for isotropic materials subjected to uniaxial deformation is that lateral contraction is the same in both transverse directions, which implies that dy ¼ dz ¼ (dx)20.5. The effect of uniaxial deformation in the x-direction on the entropy per network strand of an

13.17 Entropically Elastic Retractive Forces via Statistical Thermodynamics

591

incompressible solid with isotropic lateral contraction is 9 8 r 2undeformed > >  = < 1 2 w=crosslinks 2 dx þ S ¼ constant  k 2 dx 2 > ; :kr lwithout crosslinks > Multiply the previous expression by Avogadro’s number NAvogadro, which represents the number of network strands per mole, where a mole is based on the molecular weight of n repeat units or chain segments between crosslink junctions, and evaluate the entropy for a mole of network strands before (i.e., dx ¼ 1) and after stretching (i.e., dx . 1). One obtains the following expression for this entropy change at elongation dx: 8 9 r2undeformed > >  < = 1 2 w=crosslinks 2 DS ¼ S(dx )  S(dx ¼ 1) ¼  R d þ  3 x 2 dx 2 > :kr lwithout crosslinks > ; It should be obvious that the entropy change is always negative, except in the undeformed state, because elongation dx  1. The quantity in brackets { } in the previous equation is known classically as the “front factor” g, and it provides molecular information about the dimensions of undeformed network strands in the presence of crosslink junctions relative to the unperturbed size of “equivalent” network strands in the “free” state when crosslinks are absent. Neutron scattering experiments on isotopically labeled (i.e., deuterated) networks shed light on the magnitude of the g factor, which is greater than unity.

13.17.3 Classical Thermodynamics in the Presence of External Force Fields Without focusing on molecular structure or chain conformations between crosslink junctions, the differential expression of the first law for an ideal rubber-like material that deforms at constant volume allows one to evaluate equilibrium retractive forces in terms of entropy gradients. If dq and dw represent inexact differentials for the heat absorbed and work done by the system, respectively, then dU ¼ dq  dw where dq ¼ T dS at temperature T and extensive entropy S. The system is the crosslinked elastomer, and in the presence of external vector force f that deforms the system by differential displacement dr, the work done by the system is



dw ¼ p dV  f dr at pressure p, and extensive system volume V. The negative sign in the previous expression for dw suggests that the external field performs work on the system as it is deformed. The first law in differential form reduces to



dU ¼ T dS  p dV þ f dr

592

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers

Differentiation of the first law with respect to displacement vector r at constant T yields an expression for the external vector force f:       @U @S @V ¼T p þf @r T @r T @r T For ideal rubber-like materials that deform at constant volume when Poisson’s ratio is 0.5, the volume derivative on the right side of the previous equation vanishes. Furthermore, there is no energy barrier to deformation for freely jointed and freely rotating Gaussian chains, in response to bending backbone bond angles or rotation about the valence cone. In other words, the internal energy derivative on the left side of the previous equation also vanishes because all conformations exhibit the same energy for ideal elastomers. In agreement with these assumptions, one calculates the external vector force in terms of the entropy gradient as follows     @S @S ¼ T(Ñr S)T ¼ T dr f ¼ T @r T @r T where dr is a unit vector in the direction of the macroscopic displacement. For materials that obey the principle of affine deformation, dr is also a unit vector in the direction of the end-to-end chain vector. At equilibrium, f also represents the retractive force that is established within the crosslinked system.

13.17.4 Hooke’s Law of Elasticity for Ideal Rubber-Like Materials The equilibrium retractive force in a crosslinked elastomer is a manifestation of “entropy elasticity,” because the presence of permanent crosslink junctions and the conformational description of network strands via Gaussian statistics correspond to entropy reduction upon deformation. Equilibrium states of a system are characterized by energy minimization and entropy maximization. The former criterion is appropriate for metal-like materials in which Hooke’s law is a consequence of “energy elasticity.” Energy increases, and it is stored reversibly, when metals are subjected to tensile deformation or compression. When entropy is the major contributor to macroscopic mechanical properties, and the multiplicity of conformational states is linked directly to the Gaussian probability density distribution function for individual network strands, the system strives for shorter end-to-end vectors between crosslink junctions and larger entropy in the presence of an external tensile force field. Information from the previous two subsections allows one to evaluate the system’s retractive force for 1 mole of network strands at constant temperature when the mean-square end-to-end distance of an ensemble of network strands without crosslinks is independent of the state of deformation:  @ k ln(TNPC) þ k ln(DV) f ¼ TNAvogadro dr @r    3 3 3kr 2 3RTr þ k ln  ¼ 2 2 2 2kr lwithout crosslinks 2 2pkr lwithout crosslinks kr lwithout crosslinks

13.17 Entropically Elastic Retractive Forces via Statistical Thermodynamics

593

Hence, the retractive force is collinear with the end-to-end chain vector r, and the previous macroscopic force – displacement constitutive equation is analogous to Hooke’s law of elasticity with spring force constant, or modulus of elasticity E, given by E¼

3RT kr 2 l

without crosslinks

This modulus has dimensions of force per length, instead of force per area, because force has not been converted to stress and displacement has not been re-expressed in terms of strain or elongation in the constitutive equation. Most importantly, elastic moduli scale linearly with temperature according to the previous equation when retractive forces are “entropy driven” for crosslinked elastomers, whereas the deformation of metals in which retractive forces are “energy driven” exhibits elastic moduli that decrease linearly at higher temperature. An interesting consequence of this result can be demonstrated by attaching weights separately to a metal wire and a rubber band in a well-controlled temperature chamber. It might be necessary to attach a much heavier weight to the wire such that observable elongations are obtained. However, weights should be chosen to induce small displacements that are within the regime where Hooke’s law is applicable (i.e., the linear elastic regime). After equilibrium is achieved and displacements are measured, one increases the temperature and re-evaluates the equilibrium displacements. The product of elastic modulus and displacement must be the same at both temperatures for the rubber band because the force has not changed, and the same statement is true for the wire. The increase in elastic modulus of the rubber band induces a decrease in displacement at higher temperature, whereas the reverse applies to the wire. Hence, the weight attached to the rubber band rises to higher elevations when the temperature increases, but the weight attached to the wire descends to lower elevations. This is an example of the macroscopic differences between entropy elasticity and energy elasticity.

13.17.5 Constitutive Relation Between Stress and Elongation for Ideal Rubber-Like Materials Let’s return to the effect of uniaxial deformation in the x-direction on the entropy per network strand of an incompressible rubber-like solid with isotropic lateral contraction, via the penultimate equation in Section 13.17.2: 9 8 r 2undeformed > >  = < 1 2 w=crosslinks 2 d þ S ¼ constant  k x 2 dx 2 > ; :kr lwithout crosslinks > There are NAvogadro network strands per mole and N moles of network strands, where MWC represents the average molecular weight of chain segments between crosslink junction points. Hence, the extensive entropy of the elastomeric network is obtained via multiplication of the previous equation by both NAvogadro and N. The rubber-like material is subjected to uniaxial extension in the x-direction with displacement x ¼ xinitialdx that is collinear with the retractive force. Consequently, the generalized “dot product” expression for the contribution from the external field in the differential

594

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers

form of the first law of thermodynamics (i.e., f . dr) is replaced by fx dx or, in terms of relative elongation dx in the x-direction, fx xinitial ddx. For an ideal rubber-like solid that exhibits no volume change upon extension, the equilibrium retractive force in the x-direction is obtained from the x-component of the final equation in Section 13.17.3:   T @S fx ¼  xinitial @ dx T 9 8 2 3 r 2undeformed > >   = < T @6 1 2 7 w=crosslinks ¼ NNAvogadro 4constant  k d2x þ 5 2l > kr d xinitial @ dx 2 > x : without crosslinks ; 9 8 ! r 2undeformed > > = < NRT 1 w=crosslinks dx  2 ¼ 2 xinitial > dx ; :kr lwithout crosslinks > The engineering stress txx (i.e., normal tensile stress) due to fx is obtained via division of the previous equation by the initial cross-sectional area (i.e., yinitialzinitial ) perpendicular to the x-direction. One obtains (txx )Engineering ¼

fx yinitial zinitial

8 > <

NRT ¼ Volume initial w=crosslinks

1 ¼ ngRT dx  2 dx

r 2undeformed

9 > =

1 dx  2 2l > > kr dx : without crosslinks ; w=crosslinks

!

!

where n is the molar crosslink density (i.e., moles of network strands per initial sample volume). The instantaneous “true” stress is larger than the engineering stress because one divides fx by the instantaneous cross-sectional area (i.e., yz) instead of the initial cross-sectional area, and yz , yinitialzinitial due to lateral contraction of the sample upon extension. A negative value of Poisson’s ratio is required for materials to expand laterally upon uniaxial deformation [Almgren, 1985; Lakes, 1987]. Hence, (txx )True

   (txx )Engineering fx fx yinitial zinitial ¼ ¼ ¼ yz yinitial zinitial y z dy dz   1 ¼ dx (txx )Engineering ¼ ngRT d2x  dx

If one calculates the equilibrium crosslink density rcrosslink (i.e., grams of network strands per initial sample volume) and MWC by swelling the rubber-like material in various solvents and measuring the volume of imbibed solvent, then n ¼ rcrosslink/MWC.

Appendix: Capillary Viscometry

595

APPENDIX: CAPILLARY VISCOMETRY A.1 Transient Analysis of Draining an Incompressible Newtonian Fluid from a Spherical Bulb with a Tilted Capillary Tube to Simulate the Performance of Capillary Viscometers for the Determination of Momentum Diffusivities and Fluid Viscosities This problem combines the unsteady state macroscopic mass balance and the Hagen – Poiseuille law for laminar tube flow, together with the volume of fluid in a partially filled sphere. The overall objectives are to (i) predict the capillary constant b, based solely on geometric parameters of the viscometer, and (ii) compare this prediction with experimental values obtained by calibrating a capillary viscometer using fluids with known viscosity and density. The system is defined as fluid within the bulb plus the capillary, and one seeks the time required to drain only the bulb above a capillary that is oriented at angle Q with respect to gravity. Hence, this is an example of the unsteady state macroscopic mass balance where the fictitious inlet plane “floats” on the upper surface of liquid in the bulb such that the average velocities of the fluid and the surface are equal. Consequently, there is no contribution from convective mass transfer across the inlet plane. Fluid flow across the stationary outlet plane at the exit from the capillary is described by the Hagen – Poiseuille law for incompressible Newtonian fluids. The macroscopic mass balance for an incompressible fluid with time-varying system volume, no inlet contribution, and one stationary outlet plane reduces to

r

dVsystem pR4 DP ¼ rQHP ¼ r tube dt 8mL

where the capillary has radius Rtube and length L, and P represents dynamic pressure. Laminar flow occurs through a cylindrical capillary tube of length L, regardless of whether the capillary is vertical or tilted at angle Q with respect to gravity. The angle of tilt is considered in the dynamic pressure difference DP from tube inlet to tube outlet. If h(t) describes the height of fluid within the spherical bulb above the capillary at any time t, and the “zero of potential energy” is placed arbitrarily at the exit from the capillary, then fluid pressure at the capillary entrance is pambient þ rgh(t), based on approximate hydrostatic conditions in the bulb, and dynamic pressure at the capillary entrance is given by the sum of fluid pressure and gravitational potential energy per unit volume of fluid. Since the capillary entrance is at higher elevation than the capillary exit, by a distance L cos Q, one evaluates dynamic pressure at the capillary inlet as follows: Pinlet ¼ pambient þ rgh(t) þ rgL cos Q There is no contribution from gravitational potential energy to dynamic pressure at the capillary exit because it coincides with the potential energy reference plane. Ambient

596

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers

pressure exists on the upper surface of liquid in the bulb and at the capillary exit. Hence, the dynamic pressure difference DP ¼ Pinlet 2 Poutlet is given by rgfh þ L cos Qg. One must solve the following time-dependent ODE to relate momentum diffusivity, m/r, to efflux time:  d dVsystem d ¼ Vpartially filled sphere þ pR2tube L ¼ Vpartially filled sphere dt dt dt ¼

pR4tube g {h(t) þ L cos Q} 8(m=r)L

Fluid volume within the capillary tube is constant during the analysis of efflux times because one measures the time required to drain the bulb, not the capillary. The next task is to evaluate the volume of fluid in a partially filled sphere of radius Rsphere when the fluid achieves height h(t). This calculation is performed in cylindrical coordinates by stacking an infinite number of cylinders with infinitesimal thickness dz and radius v(z), such that v(z) vanishes when z ¼ 0 and z ¼ 2Rsphere, but v(z) ¼ Rsphere when the sphere is 50% filled according to the calculation directly below. Let the spherical bulb sit on the origin of an xyz-coordinate system such that the center of the sphere is found at a distance z ¼ Rsphere upward from the origin in the z-direction. If the sphere is filled with fluid to height z that can be greater than or less than the sphere radius, then the liquid surface is circular and the following relation allows one to predict the radius v(z) of the circular surface of liquid: {v(z)}2 þ (z  Rsphere )2 ¼ R2sphere {v(z)}2 ¼ 2zRsphere  z2 Now, calculate the volume of an infinite number of cylinders with radius v(z) and thickness dz stacked upon each other using a differential volume element in cylindrical coordinates. When fluid achieves height h(t) in this partially filled sphere, one evaluates the following triple integral to obtain the liquid volume:

Vpartially filled sphere ¼

ððð

dV ¼

2ðp

dQ 0

¼p

h(t) ð

h(t) ð

vð(z)

dz 0

0

r dr ¼ p

h(t) ð

{v(z)}2 dz

0

 {2zRsphere  z2 } dz ¼p Rsphere h2 (t)  13h3 (t)

0

As expected, the liquid volume vanishes when h ¼ 0, it achieves the normal volume of a sphere (i.e., 43pR3sphere ) when h ¼ 2Rsphere, and it achieves 50% of the normal sphere volume when h ¼ Rsphere. The time-rate-of-change of system volume in the unsteady state mass balance is obtained via differentiation of the volume of this partially filled

Appendix: Capillary Viscometry

597

sphere with respect to time, because the fluid height h(t) is time dependent as the sphere drains. One obtains the following result via separation of variables: d dh pR4 g Vpartially filled sphere ¼ p{2Rsphere h(t)  h2 (t)} ¼  tube {h(t) þ L cos Q} dt dt 8(m=r)L ð tefflux gR4tube dt m 0 ¼ btefflux ¼ ð 2Rsphere r h(2Rsphere  h) dh 8L (h þ L cos Q) 0 with the following integration limits: h ¼ 2Rsphere initially at t ¼ 0, and h ¼ 0 at the efflux time required to drain the bulb. The results of this analysis yield the functional dependence of the capillary constant b: b¼

ð 2Rsphere 8L 0

gR4tube h(2Rsphere  h) dh (h þ L cos Q)

The capillary constant depends on the (i) dimensions of the capillary tube, (ii) orientation of the capillary with respect to gravity, (iii) volume (or radius) of the spherical bulb, and (iv) strength of the gravitational field. The capillary constant does not depend on temperature or the physical properties of the fluid, provided that the fluid is incompressible and Newtonian.

A.2 Detailed Evaluation of the Capillary Constant and Comparison with Experimental Results The next task is to evaluate the complex integral expression in the previous equation for the capillary constant. Begin with the following substitution so that the denominator of the integrand can be rewritten in terms of only one variable C. Let C ¼ h þ L cos Q. Integration proceeds as follows: 2Rð sphere 0

h(2Rsphere  h) dh ¼ (h þ L cos Q)

2Rsphereð þL cos Q

(C  L cos Q)(2Rsphere  C þ L cos Q) dC C

L cos Q

The integrand reduces to a simple function of C that can be integrated rather easily: (C  L cos Q)(2Rsphere  C þ L cos Q) ¼ 2Rsphere  C þ 2L cos Q C L cos Q(2Rsphere þ L cos Q)  C

598

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers

Integration from L cos Q to 2Rsphere þ L cos Q yields the following expression: 2Rsphereð þL cos Q

 L cos Q(2Rsphere þ L cos Q) 2(Rsphere þ L cos Q)  C  dC C

L cos Q

¼ 2{Rsphere þ L cos Q}{2Rsphere þ L cos Q  L cos Q}   12 (2Rsphere þ L cos Q)2  (L cos Q)2  L cos Q(2Rsphere þ L cos Q)ln

2Rsphere þ L cos Q L cos Q

¼ 2Rsphere (Rsphere þ L cos Q)   2Rsphere  L cos Q(2Rsphere þ L cos Q) ln 1 þ L cos Q Finally, the capillary constant can be written in terms of the gravitational acceleration constant and several geometric parameters that characterize the spherical bulb and the tilted capillary tube:

b¼

gR4tube 8L

  2Rsphere 2Rsphere (Rsphere þ L cos Q)  L cos Q(2Rsphere þ L cos Q) ln 1 þ L cos Q

Geometric parameters and capillary constants are summarized below for two different Cannon – Fenske capillary viscometers. If longer efflux times are desirable to minimize errors associated with end effects and experimental reproducibility, then one should use a viscometer with a smaller capillary constant. Geometric Characteristics Bulb volume, assumed to be spherical (mL) Bulb radius, Rsphere (cm) Capillary length, L (cm) Capillary radius, Rtube (cm) Capillary tilt angle with respect to gravity (degrees) Capillary constant, predicted (cm2/s2) Capillary constant, experimental (cm2/s2)

Size #100

Size #150

8 1.24 7.6 0.041 15 1.53  1024 1.5  1024

8 1.24 6.7 0.050 15 3.45  1024 3.5  1024

A.3 Draining Power-Law Fluids from a Right Circular Cylindrical Tank via a Tilted Capillary Tube: Comparison of Efflux and Half-Times The capillary viscometer in the previous section is reanalyzed when an incompressible power-law fluid is drained from a cylindrical tank instead of a spherical bulb. The

Appendix: Capillary Viscometry

599

unsteady state mass balance with no inlet stream and one outlet is analogous to the previous development, except that it is necessary to (i) modify the time-varying system volume and (ii) use a generalized expression for the volumetric flowrate of nonNewtonian fluids through straight tubes with radius Rtube and length L in the laminar regime. The dynamic pressure difference from capillary inlet to capillary outlet in the generalized Hagen – Poiseuille law for tilted tubes is identical to that in the previous section if h(t) represents the variable height of fluid in a cylindrical tank. Hence, the starting point for this analysis to drain the tank, with radius Rtank, but not the capillary tube, is  dVsystem d dh ¼ Vpartially filled tank þ pR2tube L ¼ pR2tank dt dt dt

1=n n 3þ(1=n) rg ¼ {h(t) þ L cos Q} pRtube 1 þ 3n 2 mL If the initial height of fluid in the tank is H (i.e., h ¼ H at t ¼ 0), then one defines the half-time t1/2 and the efflux time tefflux as h ¼ H/2 at t ¼ t1/2 and h ¼ 0 at t ¼ tefflux, respectively. For vertical capillary tubes, tefflux is evaluated explicitly in Problem 5C.2 on p. 267, in Dynamics of Polymeric Liquids, Volume 1 [Bird et al., 1977]. The remainder of this analysis compares half-times and efflux times for incompressible Newtonian fluids, when n ¼ 1 and m ¼ m. The overall objective is to prove, unequivocally, that the efflux time is greater than twice the half-time for any set of initial conditions and viscometer geometries, including all orientations (i.e., angle Q) of the exit capillary with respect to gravity. For fluids that obey Newton’s law of viscosity, the previous expression reduces to R2tank

dh gR4tube {h(t) þ L cos Q} ¼ 8(m=r)L dt

This unsteady state macroscopic mass balance for incompressible Newtonian fluids yields a much simpler result for the momentum diffusivity via the half-time or the efflux time, relative to the final expression for m/r from the previous section when a spherical bulb is drained. From a practical viewpoint, there are two geometric parameters (i.e., H and Rtank) that must be related to the volume of the bulb above the capillary tube (i.e., Volumebulb  pR 2tankH ). In contrast, when the bulb volume is modeled as a sphere instead of a right circular cylinder, one identifies the sphere radius via Volumebulb ¼ 43pR3sphere . Hence, even though two parameters (i.e., H and Rtank) are related by one equation (i.e., Volumebulb  pR 2tankH ), one predicts the momentum diffusivity for this “tank-draining” problem as follows: gR4tube m ¼ ðH r 2 8Rtank L 0

ð tefflux dt 0

dh h þ L cos Q

gR4tube ¼ ðH 2 8Rtank L

ð t1=2 dt 0

dh H=2 h þ L cos Q

600

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers

Obviously, one can predict momentum diffusivities for incompressible Newtonian fluids via laboratory measurements of efflux times or half-times. The capillary constant based on efflux times is smaller than the capillary constant based on half-times, because the product of the appropriate capillary constant and either the half-time or the efflux time yields the momentum diffusivity, which is the same for a given fluid regardless of whether one measures the time required to drain either one-half of the total volume of the tank (or bulb) or the total volume of fluid above the capillary tube. The rather simple relation between half-time and efflux time, based on the previous equation, is   ðH H þ L cos Q dh ln tefflux 0 þ L cos Q h þ L cos Q ).2 ¼ ð H0 ¼ ( t1=2 dh H þ L cos Q ln 1 H=2 h þ L cos Q 2H þ L cos Q Numerical substitutions for the (i) initial height H of fluid in the cylindrical tank, (ii) length L of the capillary tube, and (iii) angle of tilt Q with respect to gravity reveal that the ratio of tefflux to t1/2 is always greater than 2. In fact, this ratio (i.e., tefflux/t1/2) becomes significantly greater than 2 when H is larger, L is smaller, and Q approaches p/2. When the capillary tube is horizontal (i.e., Q ¼ p/2), it is important to emphasize that the half-time is finite: ( ) 8(m=r)LR2tank H þ L cos Q 8(m=r)LR2tank ln{2} t1=2 ¼ ln ) 1 gR4tube gR4tube Q¼p=2 2H þ L cos Q but an infinite amount of time is required to drain the total volume of fluid in the tank. These trends can be rationalized in terms of a dynamic pressure difference from capillary inlet to capillary outlet that decreases at longer times because the hydrostatic pressure at the capillary inlet is directly proportional to the instantaneous height of fluid in the reservoir.

REFERENCES ALMGREN RF. An isotropic three-dimensional structure with Poisson’s ratio ¼ 21. Journal of Elasticity 15:427 –430 (1985). BELFIORE LA. Transport Phenomena for Chemical Reactor Design. Wiley, Hoboken, NJ, 2003, Chap. 25. BIRD RB, ARMSTRONG RC, HASSAGER O. Dynamics of Polymeric Liquids, Volume 1. Wiley, Hoboken, NJ, 1977, p. 267. GRADSHTEYN IS, RYZHIK IM. Tables of Integrals, Series, and Products, Corrected and Enlarged Edition. Academic Press, New York, 1980, p. 1. LAKES R. Foam structures with a negative Poisson’s ratio. Science 235:1038–1040 (1987). LEE CKS. Interactions Between Water-Soluble Polymers and Inorganic Complexes from the Transition Metal and Lanthanide Series, MS thesis, Colorado State University (2004). YAMAKAWA H. Modern Theory of Polymer Solutions. Wiley, Hoboken, NJ, 1971, Chap. 2, pp. 26–35; Chap. 3, pp. 70– 75.

Problems

601

PROBLEMS 13.1. For a freely jointed polymer chain with no bond-angle or valence-cone restrictions, the probability distribution function for finding an end-to-end chain length r (i.e., scalar) after n random segmental steps, each of length l, is P(r; n) ¼ 4pAr 2 exp{b2 r 2 } 3 3 ¼ 2kr 2 l 2nl2  3=2 3 ¼ 2pnl2

b2 ¼ A¼

b3 p 3=2

(a) Obtain an expression for the most probable end-to-end chain length, regardless of orientation angles Q and w in spherical coordinates. It should be emphasized that there is a vanishingly small volume at the origin of the coordinate system, where r ¼ 0, such that it is not possible to accommodate any chains with end-to-end distance r ¼ 0. Answer rMost probable

rffiffiffiffiffi pffiffiffi 1 2n  0:82l n ¼ ¼l 3 b

(b) Obtain an expression for the average end-to-end chain length, krl, that is defined by the first moment of the distribution function P(r; n). Answer

rffiffiffiffiffiffi pffiffiffi 2 8n p ffiffiffi ffi ¼l  0:92l n krl ¼ 3p b p

(c) Obtain an expression for the root-mean-square end-to-end chain length, kr 2l1/2, which corresponds to the square root of the second moment of the distribution function P(r; n). Answer kr 2 l1=2 ¼

1 b

rffiffiffi pffiffiffi 3 ¼l n 2

The following integral expressions are helpful to solve this problem: 1 ð

pffiffiffiffiffiffiffiffi exp(ax2 ) dx ¼ 12 p=a

0 1 ð 0

x exp(ax2 ) dx ¼ 1=2a

602

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers 1 ð 0 1 ð 0 1 ð 0

81 9 ð = d < 2 x exp(ax ) dx ¼  exp(ax ) dx ; da : 2

2

0

81 9 <ð = d x3 exp(ax2 ) dx ¼  x exp(ax2 ) dx ; da : 0

81 9 = 2 <ð d x4 exp(ax2 ) dx ¼ 2 exp(ax2 ) dx ; da : 0

13.2. Consider uniform segment density, postulate a Gaussian distribution of segments about the center of mass, and obtain an expression for the radial segment density distribution function r(r) of a freely jointed polymer chain that contains n segments, each of length l, and n þ 1 mass points with no bond-angle or valence-cone restrictions. Multiplication of this radius of gyration distribution function r(r) by volume element 4pr 2 dr yields the number of mass points, or segments, of a single chain that exist within a spherical shell of radius r from the center of mass, and thickness dr. The mean-square radius of gyration ks 2l ¼ nl2/6 for freely jointed chains. Answer If r (r; n) is described by a Gaussian function, then it is necessary to determine the coefficients C and s2 such that

r (r; n) ¼ C exp(s 2 r 2 ) subject to the following conditions. (i) Integration of 4pr 2r (r; n) dr from r ¼ 0 to r ) 1 should yield the total number of mass points (i.e., n þ 1) in an isolated polymer chain. (ii) The mean-square radius of gyration ks 2l, given by the ratio of the second moment of r(r; n) to the zeroth moment of r(r; n) is nl2/6. It is necessary to include the zeroth moment of the radial segment density distribution function in the expression for ks 2l because r(r; n) is not normalized. These two conditions, together with the definite integrals provided in Problem 13.1, are sufficient to evaluate C and s 2. For example, 1 1  pffiffiffiffi  pffiffiffiffi3 ð ð n1 p p 4p r 2 r (r; n) dr ¼ 4pC r 2 exp(s 2 r 2 ) dr ¼ 4pC ¼ C ¼nþ1 ) n 4s 3 s 0

0

 pffiffiffiffi 3 p 3 nl2 8s 5 ð01 ¼ ¼ ð01 ¼  pffiffiffiffi  ¼ ks2 l ¼ 2s 2 6 p r 2 r (r; n) dr r 2 exp(s 2 r 2 ) dr 4p 3 4 s 0 0 ð1

4p

r 4 r (r; n) dr

ð1

r 4 exp(s 2 r 2 ) dr

One obtains the following expression for the radial segment density distribution function:  3=2 9 9 ; C ¼ n nl2 p nl2  3=2   9 9r 2 r (r; n) ¼ n exp p nl2 nl2

s2 ¼

Problems

603

˚ , and 13.3. Consider a freely jointed chain that contains 103 segments, each of length l ¼ 4 A evaluate the segment density, with dimensions of segments per volume, at the following radial positions with respect to the center of mass of the chain: (i) r ¼ 0 (i.e., at the center of mass). (ii) r ¼ root-mean-square radius of gyration ¼ ks 2l1/2 ¼ fnl2/6g1/2. (iii) r ¼ root-mean-square end-to-end chain distance ¼ fnl2g1/2. Answer Evaluate the last equation of Problem 13.2 at the center of mass of the chain (i.e., r ¼ 0):   9 3=2 r (r ¼ 0; n ¼ 103 , l ¼ 4 A8 ) ¼ n ¼ 2:4  1021 segments=cm3 pnl2 If one travels radially outward from the center of mass a distance given by the root-mean-square radius of gyration for freely jointed chains, then the segment density decreases by slightly less than one order of magnitude relative to the segment density at the center of mass:     9 3=2 9 ¼ 5:35  1020 segments=cm3 r (r ¼ ks2 l0:5 ; n ¼ 103 , l ¼ 4 A8 ) ¼ n exp 6 pnl2 If one travels radially outward from the center of mass a distance given by the root-mean-square end-to-end distance for freely jointed chains, then the segment density decreases by almost four orders of magnitude relative to the segment density at the center of mass:   pffiffiffi 9 3=2 3 8 r (r ¼ l n; n ¼ 10 , l ¼ 4 A ) ¼ n exp{9} ¼ 2:96  1017 segments=cm3 pnl2 (iv) For (ii) and (iii) above, calculate the fraction of the total number of segments (i.e., 103) that exist within the (ii) root-mean-square radius of gyration and (iii) root-mean-square end-to-end distance, with respect to the center of mass of the chain. Numerical integration is required. 13.4. Calculate the mean-square radius of gyration for uniform segment density distribution within a hollow spherical shell. The inner and outer radii of the shell are Rinner and R, respectively, which corresponds to a radius ratio of k ¼ Rinner/R. The radial segment density distribution function is given by: 8 < 0 for 0  r , Rinner r (r) ¼ Nonzero constant (r0 ) for Rinner  r  R : 0 for R , r ) 1 Analyze the limiting cases where k ) 0 for a solid sphere, and k ) 1 for a spherical shell with infinitesimal wall thickness. Answer Use the definition of the mean-square radius of gyration in the solution to Problem 13.2 and include the uniform radial profile summarized above: ð1 4p 2

ð01

ks l ¼ 4p

0

ðR

r 4 r(r) dr ¼ r 2 r(r) dr

R ð Rinner Rinner

r 4 r0 dr ¼ r 2 r0 dr

3 {R5  R5inner } 3 2 {1  k5 } ¼ R 5 {R3  R3inner } 5 {1  k3 }

604

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers

For solid spheres where k ) 0, ks2 l ¼ 35R2 . One application of l’Hoˆpital’s rule for spherical shells with infinitesimal wall thickness yields   3 {1  k5 } 3 2 5k4 ks2 l ¼ R2 lim ¼ R2 R lim ¼ 5 k)1 {1  k3 } 5 k)1 3k2 Hence, it might seem reasonable to approximate the volume of a random-coil that completely occupies a sphere of radius R (i.e., k ) 0) in terms of its mean-square radius of gyration: 3 4 3pR

¼

2 3=2 4 3p (R )

¼

4 3p

   2 3=2 5 2 3=2 nl 2 3=2 random ks l  9ks l ) 9 ¼ 0:61n1:5 l3 coils 3 6

This approximation for the volume of a random coil could have been used in Section 13.12.2 to construct universal calibration curves for molecular weight determination via gel permeation chromatography. 13.5. (a) Determine the scaling law relation between intrinsic viscosity and molecular weight for real chains when the root-mean-square end-to-end chain length is proportional 0.6 to the 3/5th power of molecular weight (i.e., kr 2l1/2  MWPolymer ) in good solvents when T . Q. Answer Use the result of Einstein’s theory for the viscosity of a dilute suspension of solid spheres and introduce Flory’s law of real chains in good solvents, as given in the problem statement: [h] 

b 3 b¼3=5 kr 2 l3=2 {kr 2 l1=2 }3 {MWPolymer } 4=5 3b1    MWPolymer ) MWPolymer MWPolymer MWPolymer MWPolymer

(b) Calculate the intrinsic viscosity of a 1.6-Mda molecular-weight sample of poly(ethylene oxide) in water at the Q-temperature. The repeat unit structural formula of PEO is (CH2CH2O or ZCZCZOZ). Account for the fact that the chain is not “freely jointed,” but that the bond angles are restricted to be 109.58. The carbon –carbon ˚ , and the carbon –oxygen backbone bond length is backbone bond length is 1.54 A ˚ 1.43 A. 13.6. (a) A polymer is analyzed for molecular weight determination by dissolving small amounts of it in tetrahydrofuran. The dilute solution is injected into a gel permeation chromatographic column and the output curve is recorded at 25 8C and 65 8C. Draw the output curve at both temperatures on one set of axes. Label the axes and describe briefly in words why the output curve changes as a function of temperature even though the molecular weight of the polymer is the same at both temperatures. (b) The output curve from a gel permeation chromatographic experiment reveals that the molecular weight distribution is bimodal. The material that exits the column with shorter residence times has a narrow distribution of chain lengths. The material that exits the column with longer residence times has a broad distribution of chain lengths. Draw the output curve with molecular weight as the independent variable on the horizontal axis.

Problems

605

13.7. A phenomenological treatment of rubber elasticity produces the following constitutive relation between engineering stress s and elongation d ¼ 1 þ 1, where 1 is the engineering strain. This phenomenological relation is known as the Mooney–Rivlin equation and provides agreement with experimental data up to elongations as large as d ¼ 4, which is equivalent to engineering strains as large as 300% (i.e., 1 ¼ 3). The Mooney –Rivlin equation is written in terms of two undetermined constants, C1 and C2, which are evaluated empirically to obtain the best match with the actual stress–strain curve for a rubberlike material.    1 C2 C1 þ t¼2 d 2 d d (a) You are given a set of stress –strain data in the form t versus 1 for a lightly crosslinked elastomer. Describe the data manipulation procedure that is required to calculate the Mooney–Rivlin constants for this material based on a linear relationship (i.e., slope-intercept method). (b) You know the crosslink density of this elastomer in units of grams per cubic centimeter because it is possible to swell the rubber-like material in various solvents and measure the volume uptake of each solvent. You also can estimate the front factor g from neutron scattering experiments on isotopically labeled networks. Remember that the front factor provides molecular information about the dimensions of network strands in the presence of junction points relative to the unperturbed size of “equivalent” network strands in the “free” state without crosslinks. Use this information about the crosslink density r and the front factor g, together with the Mooney–Rivlin constants C1 and C2 that have been calculated in part (a), to estimate the average molecular weight of network strands between crosslink junctions. This prediction of MWC should be compared with calculations from equilibrium swelling measurements. (c) Obtain a phenomenological constitutive equation for true stress via the Mooney – Rivlin relation given above. Remember that engineering stress t is based on the initial cross-sectional area of the sample ( y0z0), transverse to the direction of uniaxial extension (i.e., the x-direction). True stress is based on the instantaneous cross-sectional area transverse to the stretch direction. When one considers either engineering stress or true stress, it is reasonable to assume that there is no volume change due to extension for a rubber-like material with a Poisson ratio of 12 (i.e., 0.5). It is also reasonable to assume that the solid material is isotropic with respect to mechanical deformation. (d) Sketch the dimensionless entropy change (DS/R with respect to the undeformed sample) per mole of network strands for a rubber band as a function of elongation d, where d is a measure of the strained length of the sample in the direction of uniaxial extension. Remember that the crosslink junctions deform in the same way that the macroscopic dimensions of the sample deform for a rubber-like material that obeys the principle of “affine deformation.” 13.8. In an all-carbon-backbone chain, the probability density distribution function, with dimensions of inverse volume, for end-to-end chain vector r after n random steps, each of length l, is given by the following non-Gaussian expression that illustrates how

606

Chapter 13 Gaussian Statistics of Linear Chain Molecules and Crosslinked Elastomers P(r; n) depends only on the magnitude of the end-to-end vector r: ( ) r3 P(r; n) ¼ z exp  g kr 2 l3=2 where z depends on n and l, and g depends on n. The dimensions of z are inverse volume, and g is dimensionless. Since P(r; n) is normalized, one writes þ1 ð 1

P(r; n) dr ¼

2ðp

ðp ð1 dw sin Q dQ r 2 P(r; n) dr

0

¼ 4pz

0

ð1

(

0

r3

2

r exp 

0

g kr 2 l3=2

) dr ¼ 1

For moderate elongations that are not too large, the mean-square end-to-end chain length kr 2l depends on n and l, but not strain. The previous normalization condition indicates that kr 2 l ¼



3 4pzg

2=3

(a) Obtain an expression for the vector force generated by ideal non-Gaussian macromolecular chains upon extension that exhibit no energetic restrictions to conformational rearrangements. Then, identify the direction in which this vector force acts. Answer If N represents an exceedingly large number of conformations that are available to these polymer chains, then the multiplicity of states V for chains with end-to-end vector r is V(r; n) ¼ NP(r; n) Now, one employs the Helmholtz free energy (i.e., A ¼ U 2 TS) to calculate the isothermal retractive force when internal energy U is independent of strain, or end-to-end distance r. This condition, U = f (r) and S ¼ g(r), is described by entropy elasticity. The relevant thermodynamic expression for retractive force f, together with Boltzmann’s equation that connects entropy S and the multiplicity of states V is   @ ln V 3kTr 2 f ¼ frAgT ¼ TfrSgT ¼ kTfr ln VgT ¼ dr kT ¼ dr @r T g kr 2 l3=2 Since the probability density distribution function P(r; n), multiplicity V, entropy S, and Helmholtz free energy A depend on the magnitude of the end-to-end chain vector r, one concludes that the retractive force f acts exclusively in the r-direction in spherical coordinates, along a straight line that connects both ends of the chain, where one end is anchored at the origin. (b) Why does the retractive force calculated in part (a) exhibit a stronger dependence on end-to-end chain distance r relative to Gaussian chains that represent the focus of this chapter?

Problems

607

Answer For the non-Gaussian distribution function described in this problem, the conformational entropy decreases faster with chain extension relative to Gaussian chains. Hence, for any end-to-end chain distance r that is greater than the root-mean-square end-to-end distance, the non-Gaussian distribution predicts that there are fewer chains, smaller multiplicity, and lower entropy relative to Gaussian chains. These trends are all consistent with larger entropically elastic retractive forces for non-Gaussian chains that exhibit stronger entropy gradients upon extension. This particular non-Gaussian distribution creates a larger driving force for stretched chains to contract and increase their conformational freedom.

Chapter

14

Classical and Statistical Thermodynamics of Rubber-Like Materials I have seen a land where serpents walk with a wolf’s head. —Michael Berardi

C

lassical thermodynamics is employed to analyze the equation of state for ideal elastomers, using some results from the previous chapter. Flory’s approximations are discussed in detail, allowing one to obtain equation-of-state information from experimental data. Statistical analysis is based on three-dimensional Gaussian chains and Boltzmann’s entropy law. Another view of chain expansion in good solvents on the Flory – Huggins lattice is presented and compared with a similar analysis from the previous chapter. There is a significant amount of complementary discussion in Chapters 13 and 14.

14.1 AFFINE DEFORMATION Crosslinked elastomers behave similarly to a network of interconnected springs that store mechanical energy and exhibit essentially no irrecoverable deformation upon removal of external stress. Both classical and statistical thermodynamics of rubber elasticity are employed to develop stress –strain relations for these materials. Statistical analysis of linear chain dimensions from the preceding chapter is adopted to describe segment lengths between crosslink junctions. The important assumption focuses on the effect of deformation at the macroscopic and microscopic levels. In other words, if one stretches a rubber-like solid along a particular coordinate direction, then Gaussian chain segments between crosslinks experience similar extension along the same coordinate direction. Macroscopic stress – strain behavior is analyzed via probability density distribution functions for groups of network strands with the same end-to-end vector and the retractive forces that are generated when entropy Physical Properties of Macromolecules. By Laurence A. Belfiore Copyright # 2010 John Wiley & Sons, Inc.

609

610

Chapter 14 Classical and Statistical Thermodynamics of Rubber-Like Materials

decreases upon extension. The assumption of affine deformation provides a bridge between the statistical thermodynamic analysis of network strands and macroscopic mechanical properties.

14.2 OVERVIEW One of the simplest models that describe the mechanical response of crosslinked rubber-like solids with reasonable success is a spring with static modulus of elasticity E. The discussion in this chapter provides support for fact that this is an “entropy” spring, not an “energy” spring which is characteristic of metals. One focuses on classical and statistical concepts to evaluate the retractive force and the elastic modulus by considering network strands between crosslink junction points. Three-dimensional Gaussian distributions describe the number of conformational states that are available to network strands with end-to-end chain vector r. When these materials are stretched reversibly, chemical crosslinks are responsible for the establishment of a retractive force that opposes the applied force. Rather simple analysis of this problem in the previous chapter yielded Hooke’s law of elasticity with a temperature-dependent elastic modulus. More sophisticated analyses in the previous chapter and this chapter produce stress – strain relations that are applicable at much higher elongations, beyond the elastic limit of simple “energy” springs.

14.3 ANALOGIES The equation of state of an ideal rubber-like solid exhibits many similarities to the equation of state of an ideal gas. Under isothermal conditions, the internal energy of an ideal gas is independent of system volume. Likewise, the internal energy of ideal rubbers does not depend on sample length at constant temperature and volume. One employs the differential form of the first law of thermodynamics for reversible processes, together with the fact that internal energy is not a function of (i) sample volume for gases or (ii) deformed length for rubbers, to arrive at the equations of state. For an ideal gas, system pressure is expressed in terms of the volume dependence of entropy at constant temperature. For ideal rubbers, the retractive force is calculated from the dependence of entropy on sample length at constant temperature and volume. Statistical concepts are introduced to evaluate entropy via the Boltzmann equation. Pressure is a normal force, whereas retractive forces can act in tension or shear, but tension is most common.

14.4 CLASSICAL THERMODYNAMIC ANALYSIS OF THE IDEAL EQUATION OF STATE FOR RETRACTIVE FORCE FROM CHAPTER 13 Let’s analyze the force – elongation relation for ideal rubbers via the first law of thermodynamics in the presence of external fields. The following expression from

14.4 Classical Thermodynamic Analysis of the Ideal Equation

611

Section 13.17.5 for the retractive force fx in the x-direction provides reasonable predictions of experimental results, except at very large elongations (i.e., dx . 200– 300%): 9 8 2 ! ! rundeformed > > = < fx NRT 1 1 w=crosslinks ¼ dx  2 ¼ ngRT dx  2 2 > yinitial zinitial Volume initial dx dx ; :hr iwithout crosslinks > w=crosslinks

where N represents the number of moles of network strands from one crosslink junction to another, and n is the effective crosslink density (i.e., N per unit volume of the undeformed sample). The “g-factor” provides molecular information about the dimensions of undeformed network strands in the presence of crosslink junctions relative to the unperturbed size of “equivalent” network strands in the “free” state when crosslinks are absent.

14.4.1

Classical Thermodynamic Strategy

Begin with the differential form of the first law of thermodynamics in the presence of an external force field that includes a work term, in addition to pV – work, due to extension of the sample. If f is the equilibrium retractive force established in chemically crosslinked solids that opposes the applied force and L is the instantaneous length of the sample in the stretch direction, then the differential work done by the system on the surroundings is



dw ¼ p dV þ f dL ¼ p dV  fx dL Using vector notation, the retractive force f is collinear with sample displacement dL, but stretching induces a displacement (i.e., with respect to the sample’s original dimensions) in the opposite direction, relative to the retractive force. Using standard terminology for thermodynamic variables and state functions, the first law for reversible processes in an external force field yields the following differential expression for the extensive internal energy U(S, V, L): dU ¼ T dS  p dV þ fx dL Transform complete thermodynamic information from internal energy U(S, V, L) to the extensive Gibbs free energy G(T, p, L) via Legendre transformation [Belfiore, 2003]. G ¼ U þ pV  TS ¼ H  TS dG ¼ dU þ p dV þ V dp  T dS  S dT dG ¼ S dT þ V dp þ fx dL Calculate the equilibrium retractive force via the dependence of the Gibbs free energy on sample length at constant temperature and pressure. The previous total differential expression for G in terms of its natural variables (i.e., temperature T, pressure p, and

612

Chapter 14 Classical and Statistical Thermodynamics of Rubber-Like Materials

sample length L in the x-direction) yields the following result for the x-component of the retractive force fx:  fx ¼

@G @L



    @H @S ¼ T @L T, p @L T,p T, p

The thermodynamic equation of state for retractive forces in crosslinked elastomers has been expressed in terms of the dependence of enthalpy H and entropy S on sample length L at constant temperature and pressure. A Maxwell relation based on second mixed partial derivatives of the Gibbs free energy allows one to calculate the effect of sample length on entropy via the temperature coefficient of the retractive force. 

@S @L

  @fx ¼ @T p, L T,p     @H @fx fx ¼ þT @T p, L @L T, p



Using the equation of state for ideal rubber-like materials, one calculates the temperature dependence of retractive force at constant pressure and length. Since this equation of state for fx is expressed explicitly in terms of temperature and elongation in the x-direction dx, one begins by postulating that fx ¼ fx(T, p, dx). The total differential of fx is       @fx @fx @fx dT þ dp þ ddx dfx ¼ @T p,dx @p T,dx @ dx T,p The effect of temperature on fx at constant pressure and sample length is obtained from the previous expression, where the two quantities on the right side of the following equation that contain fx are calculated directly from the equation of state. For example,        @fx @fx @fx @ dx ¼ þ @T p,L @T p,dx @ dx T,p @T p,L !   @fx 1 fx ¼ ngR dx  2 yinitial zinitial ¼ @T p,dx T dx !   @fx 2 1 þ 2=d3x d3x þ 2 ¼ ngRT 1 þ 3 yinitial zinitial ¼ fx ¼ f x @ dx T,p dx dx  1=d2x dx4  dx 

Now, it is possible to evaluate the effect of sample length L on the enthalpy of an ideal crosslinked elastomer whose equation of state is given directly above Section 14.4.1. The result will be useful to analyze thermoelastic inversion, where the effects of thermal expansion and entropy elasticity cancel each other such that the retractive force is

14.4 Classical Thermodynamic Analysis of the Ideal Equation

613

independent of temperature at constant pressure and deformed sample length L. Classical thermodynamic analysis proceeds as follows: 

@H @L





@fx ¼ fx  T @T T,p

 p,L

By definition, the second term on the right side of the previous equation vanishes at the thermoelastic inversion point, so it is possible to relate the x-component of retractive force to the effect of sample length on enthalpy at constant temperature and pressure. Expansion of fx in terms of T, p, and dx, together with the equation of state yields 

@H @L

 T,p

(      )  @fx @fx @fx @ dx ¼ fx  T ¼ fx  T þ @T p,L @T p,dx @ dx T,p @T p,L (   )   fx d3x þ 2 @ dx d3x þ 2 @ dx þ fx 4 ¼ fx  T ¼ Tfx 4 T dx  dx @T p,L dx  dx @T p,L 

The volumetric coefficient of thermal expansion aV in the absence of stress is useful to evaluate the effect of temperature on elongation dx at constant pressure and deformed sample length L in the x-direction. Isotropic rubber-like solid materials expand and contract with no preference in any coordinate direction in response to temperature changes, due to the random nature of the crosslinking process. Furthermore, stretching occurs at constant volume because Poisson’s ratio is essentially 0.5. Hence, system volume V scales as the third power of the initial sample dimension in the x-direction, which coincides with the stretch direction. These properties of ideal crosslinked solids imply that aV is threefold larger than the linear coefficient of thermal expansion aL, based on the following definitions and calculations with proportionality constant k: V ¼ kL3initial     @ ln V @ ln Linitial aV ¼ ¼3 ¼ 3aL @T @T p p Since elongation in the x-direction is defined by dx ¼ L/Linitial, its temperature dependence at constant pressure and deformed length is related to these thermal expansion coefficients: (    1    ) @ dx @Linitial L 1 @Linitial ¼  13 dx aV ¼L ¼ @T p,L @T p,L @T p,L Linitial Linitial Expansion of the undeformed solid at higher temperature, while the instantaneous sample length remains constant, yields a negative temperature coefficient for dx, as illustrated by the previous equation. All of the results in this section allow one to

614

Chapter 14 Classical and Statistical Thermodynamics of Rubber-Like Materials

evaluate the effect of sample length L on system enthalpy at constant temperature and pressure for ideal rubber-like solids with random crosslinks:     @H d3 þ 2 @ dx 1 d3 þ 2 ¼ Tfx 4x ¼ Tfx aV x3 @L T, p dx  dx @T p, L 3 dx  1

14.5 ANALOGOUS DEVELOPMENT FOR THE EFFECT OF SAMPLE LENGTH ON INTERNAL ENERGY: THE CONCEPT OF IDEAL RUBBER-LIKE SOLIDS A sequence of logical calculations, similar to those described in the previous section in terms of system enthalpy, allows one to evaluate the effect of sample length on internal energy at constant temperature and volume. Relative to the previous equation, replace (i) H by U and (ii) p by V. Hence, the effect of temperature on elongation dx at constant volume V and instantaneous deformed sample length L is required. Since dx ¼ L/Linitial, and V ¼ k L 3initial, it should be obvious that elongation is not a function of temperature when V and L remain constant. These restrictions yield the definition of an ideal rubber-like solid, where instantaneous sample length has no effect on internal energy at constant temperature and system volume. Justification for these statements is summarized below. 1. Transform complete thermodynamic information about the system from the internal energy U(S, V, L) to the Helmholtz state function A(T, V, L). In the presence of external force fields, A ¼ U  TS dA ¼ S dT  p dV þ fx dL 2. Express retractive force fx in terms of the effect of instantaneous sample length L on the Helmholtz free energy at constant temperature and system volume. Then, introduce internal energy and entropy into the thermodynamic result:  fx ¼

@A @L



 ¼

T,V

@U @L



 T

T,V

 @S @L T,V

3. Use a Maxwell relation based on the total differential expression for Helmholtz free energy to rewrite the previous equation for retractive force as follows:    @S @fx ¼  @T L,V @L T,V     @U @fx fx ¼ þT @T L,V @L T,V 

14.5 Analogous Development for the Effect of Sample Length on Internal Energy

615

4. The equation of state for ideal rubber-like materials provides an expression for retractive force explicitly in terms of temperature and elongation. This is useful to evaluate the effect of temperature on fx at constant deformed length and system volume but, initially, one postulates that fx depends on temperature, volume, and elongation, fx(T, V, dx). The total differential of the retractive force is       @fx @fx @fx dfx ¼ dT þ dV þ d dx @T V,dx @V T,dx @ dx T,V 5. Now, one calculates the effect of sample length L on the internal energy of ideal crosslinked solids via the previous two equations:     @U @fx ¼ fx  T @T L,V @L T,V         @fx @fx @fx @ dx ¼ þ @T L,V @T V,dx @ dx T,V @T L,V 6. The equation of state for fx reveals that the first term on the right side of the previous equation is fx/T, because fx varies linearly with temperature at constant system volume and elongation. Furthermore, restrictions on the temperature dependence of elongation at constant deformed length and system volume cause the second term on the right side of the previous equation to vanish. Consequently, internal energy is independent of deformed sample length at constant temperature and system volume. This represents the definition of ideal rubber-like solids, and it is supported theoretically by the thermodynamic analysis presented in this section. Problem Use classical thermodynamics and the equation of state for an ideal rubber-like solid to calculate the effect of deformed sample length on system volume at constant temperature and pressure. Hint: Begin with H ¼ U þ pV, and differentiate this relation with respect to deformed sample length L at constant temperature and pressure. Postulate U(T, V, L), write the total differential of internal energy using these independent variables, and calculate the effect of deformed sample length on U at constant T and p. A Maxwell relation is required based on the Helmholtz free energy. Now, invoke the definition of an ideal rubber-like solid and adopt results from this section and the previous one in terms of U and H, and their dependence on L. The final result is   @V 1 d3 þ 2 ¼ bfx x3 @L T, p 3 dx  1 where b represents the isothermal coefficient of volume compressibility, that appears in the previous equation via application of the triple-product rule. Flory’s

616

Chapter 14 Classical and Statistical Thermodynamics of Rubber-Like Materials

approximation discussed below requires that the derivative on the left side of the previous equation must be negligible for rubber-like materials with compressibility coefficients that are five or six orders of magnitude smaller than b for ideal gases (i.e., bIdeal gas ¼ 1/p).

14.6 THERMOELASTIC INVERSION The following experimental results illustrate competition between thermal expansion and entropy elasticity, and the fact that they cancel each other at the thermoelastic inversion point. Retractive forces are measured as a function of temperature at constant pressure and sample length in the deformed state. Engineering strain, or elongation, is not rigorously constant during these measurements because the undeformed sample dimensions respond to temperature changes according to aV or aL. Consequently, at low strain g (i.e., g , 10%), the effect of thermal expansion causes retractive forces to decrease at higher temperatures. At higher strain, entropy elasticity is more important than thermal expansion, and fx increases at higher temperature. This effect is based on the conformational properties of network strands between crosslink junction points. For example, since entropy increases at higher temperature, network strands seek an end-to-end vector r that corresponds to a larger multiplicity of states V, via Boltzmann’s relation between entropy and multiplicity (i.e., S ¼ k ln V), where k is Boltzmann’s constant. Shorter distances between crosslink junctions are consistent with the fact that more conformational states are available to a single strand. The principle of affine deformation provides a simple connection between macroscopic dimensions of the sample and the conformational characteristics of individual chains. Shorter end-to-end chain distances between crosslink junctions at higher temperature translate into crosslinked rubber-like materials that strive to contract. However, when experiments are performed at constant deformed sample length in the stretch direction, thermodynamic contraction is thwarted and fx increases at higher temperature. At the thermoelastic inversion point, retractive forces are insensitive to temperature changes because the effects of thermal expansion and entropy elasticity fortuitously cancel. Under these conditions,   @fx ¼0 @T p,L and the effect of sample length on enthalpy at constant temperature and pressure yields the retractive force, as illustrated below:   @H 1 d3 þ 2 fx ¼ ¼ Tfx aV x3 @L T,p 3 dx  1 If TaV ¼ 0.298, for T ¼ 298 K and aV ¼ 1  1023 K21, then the previous equation is satisfied when elongation dx ¼ 1.1 (i.e., 10% strain). If the volumetric coefficient of thermal expansion is aV ¼ 5  1024 K21, then the previous equation is satisfied at ambient temperature when elongation dx  1.05 (i.e., 5% strain). Since reasonable

14.7 Temperature Dependence of Retractive Forces

617

expansion coefficients can be found within this range (i.e., 5  1024 K21  aV  1  1023 K21), one concludes that thermoelastic inversion occurs somewhere between 5% and 10% strain at ambient temperature. Based on the previous equation, one observes a linear increase in engineering strain, dx 21, at the thermoelastic inversion point when TaV is larger.

14.7 TEMPERATURE DEPENDENCE OF RETRACTIVE FORCES THAT ACCOUNTS FOR THERMAL EXPANSION When the temperature dependence of the undeformed sample length is considered, it is possible to analyze fx – T data at constant pressure and engineering strain such that equilibrium retractive forces always increase at higher temperature. These results, reconfigured in terms of constant strain instead of constant deformed sample length, provide a description of the deformation process based on molecular orientation. It is necessary to combine definitions of engineering strain g and the linear coefficient of thermal expansion aL to provide a guide for adjusting sample length in the deformed state such that engineering strain remains constant at all temperatures. For example,   @ ln Linitial L  Linitial aL ¼ ; g¼ @T Linitial p Hence, thermal expansion describes the effect of temperature on the initial, or undeformed, sample length and engineering strain accounts for the stretching process. If aL does not vary much over the temperature range Tinitial ) T, then one adjusts the deformed sample length L to maintain constant strain at any temperature T. The following expression for deformed sample length in the stretch direction is valid at constant pressure: 9 8 T = < ð aL dT L(g, T) ¼ (1 þ g)Linitial (T) ¼ (1 þ g)Linitial (Tinitial ) exp ; : Tinitial

¼ (1 þ g)Linitial (Tinitial ) exp{aL (T  Tinitial )} Now, all equilibrium retractive force versus temperature data obtained at constant pressure and engineering strain exhibit positive slopes. The actual procedure, together with important approximations, is summarized below. 1. Measure equilibrium retractive forces versus temperature at constant pressure and engineering strain. Use the previous equation to adjust the stretched length of the sample at each temperature to account for effects of thermal expansion. 2. Use results from the development in Section 14.5 based on the Helmholtz free energy, in which temperature T, system volume V, and instantaneous sample

618

Chapter 14 Classical and Statistical Thermodynamics of Rubber-Like Materials

length L are important independent variables in the presence of an external force field. Equilibrium retractive forces are related to internal energy and entropy via the following thermodynamic relations that were discussed previously in this chapter. The overall objective is to identify energetic and entropic contributions to the retractive force, based solely on experimental data. A Maxwell relation is required to isolate the effect of sample length on internal energy at constant temperature and system volume, such that (@U/@L)T,V can be calculated directly from fx versus temperature data. This quantity, (@U/@L)T,V, vanishes for ideal rubber-like materials via theoretical considerations described above, and one should compare this theoretical result with thermodynamic predictions based on actual data:       @A @U @S ¼ T fx ¼ @L T,V @L T,V @L T,V     @S @fx Maxwell:  ¼ @T L,V @L T,V     @U @fx ¼ fx  T @T L,V @L T,V 3. One encounters problems experimentally when the previous equation is employed to evaluate (@U/@L)T,V because it is difficult, if not impossible, to calculate the slope of fx versus temperature at constant system volume V and deformed sample length in the stretch direction. The difficulty is associated with the fact that thermal expansion hinders sample volume from remaining constant in response to temperature changes. Even though experimentally measured retractive forces can be used in the previous equation, (@ fx/@T )L,V is not easy to calculate because measurements are obtained at constant pressure and engineering strain. Flory’s approximation is invoked to address this dilemma:     @fx @fx  @T L,V @T p,g The assumptions that must be satisfied to justify this approximation are (i) rubber-like materials are isotropic in the presence of tensile deformation such that the volumetric coefficient of thermal expansion is three-fold larger than the linear coefficient of thermal expansion, (ii) mechanical isotropy is preserved under tension such that the linear coefficient of isothermal compressibility exhibits no directionality, (iii) thermal expansion coefficients are independent of temperature, and (iv) Poisson’s ratio is very close to 12 such that (@V/@L)T,p  0 when engineering strain is less than 250%. Thermodynamic analysis of (@V/@L)T,p is provided in Section 14.5 in conjunction with the equation of state for ideal rubber-like materials.

14.8 Derivation of Flory’s Approximation for Isotropic Rubber-Like Materials

619

4. If Flory’s approximation is justified, then experimental data for equilibrium retractive forces versus temperature are separated into energetic and entropic contributions as follows:     @U @fx  fx  T @T p,g @L T,V     @S @fx  @T p,g @L T,V Experimental data allow one to evaluate all terms on the right side of the previous two equations, such that thermal expansion effects are considered and fx versus temperature exhibits a positive slope at all engineering strains. At constant temperature and system volume, thermodynamic quantities on the left side of the previous two equations depend only on molecular orientation. Subsequent analyses in this chapter apply to ideal rubber-like materials, for which experimental data support the definition given by (@U/@L)T,V  0. The connection between classical and statistical thermodynamic results for these materials is obtained from the equation of state:   @S fx  T @L T,V where entropy is related to the multiplicity of states via Boltzmann’s equation.

14.8 DERIVATION OF FLORY’S APPROXIMATION FOR ISOTROPIC RUBBER-LIKE MATERIALS THAT EXHIBIT NO VOLUME CHANGE UPON DEFORMATION To reiterate some of the major experimental roadblocks from the previous section, it is nearly impossible to calculate the slope of equilibrium retractive force fx versus temperature at constant system volume V and deformed sample length L in the stretch direction because V responds to temperature changes. Flory’s approximation allows one to evaluate (@U/@L)T,V experimentally from fx versus temperature at constant pressure and engineering strain. Actual data reveal that (@U/@L)T,V  0. Hence, at constant temperature and system volume, the effect of sample length on entropy can be evaluated from a statistical thermodynamic analysis of chain conformations between network junction points in crosslinked materials. This analysis via the NVT-ensemble yields fx due solely to molecular orientation. Begin by expressing equilibrium retractive forces as a function of temperature T, pressure p, and deformed sample length L in the stretch direction. Hence, fx ¼ fx(T, p, L). The total differential of fx is       @fx @fx @fx dT þ dp þ dL dfx ¼ @T p,L @p T,L @L T,p

620

Chapter 14 Classical and Statistical Thermodynamics of Rubber-Like Materials

Now, one evaluates the temperature dependence of fx at constant pressure p and engineering strain g in the stretch direction. The result is         @fx @fx @fx @L ¼ þ @T p,g @T p, L @L T, p @T p,g Use the expression for L(T, g) in Section 14.7 to calculate (@L/@T )p,g when the linear coefficient of thermal expansion aL is independent of temperature. Since L depends explicitly on T and g at constant pressure, L(g, T) ¼ (1 þ g)Linitial (Tinitial ) exp{aL (T  Tinitial )}   @L ¼ LaL @T p,g The previous three equations yield the final result for this segment of the derivation of Flory’s approximation:       @fx @fx @fx ¼  LaL @T p, L @T p,g @L T,p The second route to the previous equation is based on the temperature dependence of equilibrium retractive forces at constant system volume V and deformed sample length L via the total differential of fx:         @fx @fx @fx @p ¼ þ @T V, L @T p, L @p T, L @T V, L Application of the triple-product rule for partial derivatives is required for both parts of the last term on the right side of the previous equation. A generic presentation of the triple-product rule is based on the following multivariable function w ¼ w(x, y, z). The total differential of w is       @w @w @w dx þ dy þ dz dw ¼ @x y,z @y x,z @z x,y If one differentiates this expression with respect to y at constant w and x, the result is       @w @w @z ¼ @y x,z @z x,y @y x,w The following correspondences yield an expression for (@ fx/@p)T,L: w ) fx, y ) p, x ) T, and z ) L. Hence,       @fx @fx @L ¼ @p T,L @L T,p @p T, fx

14.8 Derivation of Flory’s Approximation for Isotropic Rubber-Like Materials

621

For (@ p/@T )V, L, the correspondences are w ) p, y ) T, x ) L, and z ) V. One obtains       @p @p @V ¼ @T V, L @V T, L @T p, L All of these partial differential expressions are considered to obtain the temperature dependence of equilibrium retractive forces at constant pressure and deformed sample length in the stretch direction. In addition, system volume is introduced to re-express (@L/@p)T, fx as a product of two derivatives via the chain rule:         @fx @fx @fx @p ¼  @T p, L @T V, L @p T, L @T V, L           @fx @fx @L @p @V ¼  @T V, L @L T, p @p T, fx @V T, L @T p, L             @fx @fx @L @V @p @V ¼  @T V, L @L T, p @V T, fx @p T, fx @V T, L @T p, L When rubber-like materials with Poisson’s ratio of 0.5 are stretched at constant temperature and pressure, no volume change occurs such that (@V/@L)T,p  0. As a consequence of this restriction on system volume, the product of five partial derivatives in the last term on the far right side of the previous equation can be simplified considerably. For example, if rubber-like materials behave as isotropic solids, then     @V @V  ¼ 3V aL @T p, L @T p,g Linear coefficients of thermal expansion aL describe the effect of temperature on undeformed sample dimensions at constant pressure, as well as the effect of temperature on the dimensions of deformed materials at constant pressure and engineering strain. It is assumed that external force fields do not affect aL considerably. The previous equation is obtained by expressing total system volume V in terms of temperature T, pressure p, and deformed sample length L in the stretch direction. The total differential of V is       @V @V @V dT þ dp þ dL dV ¼ @T p, L @p T, L @L T, p The effect of temperature on total system volume at constant pressure and engineering strain g is         @V @V @V @L ¼ þ @T p,g @T p, L @L T, p @T p, g

622

Chapter 14 Classical and Statistical Thermodynamics of Rubber-Like Materials

where the second term on the right side of the previous equation vanishes if deformation does not induce any volume change. The total differential of V is also useful to evaluate the effect of pressure on total system volume at constant temperature T and equilibrium retractive force fx. For example,         @V @V @V @L ¼ þ @p T, fx @p T, L @L T, p @p T, fx Once again, the second term on the right side of the previous equation vanishes and rearrangement yields 

@V @p





T, fx

@p @V

 1 T, L

Isochoric deformation at constant temperature and pressure yields the following simplification for the temperature dependence of equilibrium retractive forces at constant pressure and deformed sample length in the stretch direction: 

@fx @T





@fx  @T p, L



  3V aL

V, L

@L @V

 T, fx



@fx @L

 T,p

The final task to complete the derivation of Flory’s approximation requires an analysis of (@V/@L) at constant temperature and retractive force. The primary assumption is that linear coefficients of isothermal compressibility are the same in all coordinate directions when rubber-like materials are subjected to tensile forces. In rectangular Cartesian coordinates, total system volume is obtained from a product of instantaneous sample dimensions in three mutually perpendicular directions. Hence, V ¼ Lx L y Lz dV ¼ L y Lz dLx þ Lx Lz dL y þ Lx L y dLz Since stretching occurs in the x-direction, sample length Lx in this direction is synonymous with length L in all equations presented in this chapter. Isothermally, one employs the definition of linear compressibility coefficients to relate changes in sample dimensions to pressure changes, so that one does not confuse the desired result with (@V/@L)T,p, which is vanishingly small for any material with Poisson’s ratio near 0.5. For example, in the ith coordinate direction,   1 @Li bi ¼  ; dLi ¼ bi Li dp Li @p T

14.9 Statistical Thermodynamic Analysis

623

The differential of total system volume is expressed in terms of all three bi’s, and pressure changes are written using Lx and bx:   Lx L y Lz bx þ by þ bz 3V dLx dLx ¼ dV ¼ Lx L y Lz (bx þ by þ bz ) dp ¼ Lx bx Lx The previous equation contains no explicit dependence on retractive force fx because the assumption of mechanically isotropic materials under tension implies that bx ¼ by ¼ bz. Since Lx ¼ L, one obtains the desired result that does not correspond to constant pressure conditions:   @V 3V ¼ @L T, fx L Substitution of the inverse of this result into a previous equation in this section,             @fx @fx @L @fx @fx @fx   3V aL   LaL @T p, L @T V, L @T V,L @L T,p @V T, fx @L T,p yields an expression for the temperature dependence of equilibrium retractive forces at constant pressure p and deformed sample length L in the stretch direction that agrees with the fifth equation in this section if     @fx @fx  @T V, L @T p,g

14.9 STATISTICAL THERMODYNAMIC ANALYSIS OF THE EQUATION OF STATE FOR IDEAL RUBBER-LIKE MATERIALS 14.9.1

Effect of Temperature on Chain Conformations

The actual shape of the intramolecular potential energy function for various backbone bond rotation (i.e., torsional) angles governs random-coil dimensions that are sensitive to temperature changes. The possible rotational isomeric states for each backbone bond are trans, gaucheþ, or gauche2. As a general rule, mean-square end-to-end chain dimensions decrease when there is a higher concentration of gauche rotational states in the chain backbone. Hence, if the “all-trans state” is most stable, then the planar zigzag conformation exists at lowest energy and an increase in temperature induces rotational mobility to surmount the eclipsed conformational energy barrier between different preferred isomeric states, thereby converting some trans states to gauche+ states. The overall effect at the single-chain-level is that the mean-square end-to-end distance decreases at higher temperature. The opposite effect occurs when gauche conformations are more stable and an increase in temperature induces transformations from gauche states to trans states. Now, single-chain dimensions

624

Chapter 14 Classical and Statistical Thermodynamics of Rubber-Like Materials

increase at higher temperature. Invariably, cis or eclipsed conformations are forbidden because they exist at highest energy and represent the rotational barrier when the intramolecular potential is portrayed as a function of the Ramachandran angles.

14.9.2

Affine Deformation

This principle has been mentioned earlier in this chapter and in the previous chapter. Essentially, it is possible to analyze changes in macroscopic sample dimensions by focusing on the conformational characteristics of groups of network strands between crosslink junction points. If chemical crosslinking occurs randomly via catalysis, UV radiation, vulcanization, and so on, then random walk Gaussian statistics that describe the conformational behavior of freely jointed chains are also useful for individual network strands. The three-dimensional Gaussian distribution function for the endto-end chain vector r was constructed from a product of three one-dimensional Gaussian distributions that describe projections of the end-to-end chain vector along three mutually orthogonal coordinate axes. In the undeformed state, the breadth of each one-dimensional Gaussian distribution is the same because crosslinking occurs in random fashion, the medium is isotropic, and network strands are not oriented preferentially along any coordinate direction. When uniaxial deformation occurs, each of these one-dimensional Gaussian distributions retains its Gaussian-like characteristics with a distorted breadth, or second moment. The breadth of the onedimensional Gaussian distribution in the stretch direction increases, whereas contraction of the distribution function occurs in the other two coordinate directions that are transverse to stretching. This is somewhat analogous to the effect of elongation on the macroscopic dimensions of any material that exhibits a Poisson’s ratio greater than zero. For reasonable levels of strain that are not too large (i.e., ,100%), deformation has a negligible effect on the breadth of the three-dimensional Gaussian distribution that describes the conformational characteristics of network strands between crosslink junctions, because distortions in each one-dimensional distribution function cancel. Consequently, the mean-square end-to-end distance of network strands between crosslink junctions is essentially independent of the state of deformation, provided that one-dimensional Gaussian distributions can describe the conformational characteristics of network strands via their projections along three mutually perpendicular coordinate axes. More detail is provided in Section 14.9.5 when uniaxial deformation favors network strand conformations with a larger component of the end-to-end vector along the stretch direction.

14.9.3

Gaussian Statistics

It is assumed that chemical crosslinking occurs in an isotropic unoriented molten polymer such that the conformational characteristics of network strands between crosslink junctions can be described by a three-dimensional Gaussian distribution function. The network contains a total of y strands, and y i represents the number of strands in the deformed state whose end-to-end vector frigDeformed has an x-component

14.9 Statistical Thermodynamic Analysis

625

in the range from xi to xi þ dx, a y-component in the range from yi to yi þ dy, and a z-component in the range from zi to zi þ dz, where dx dy dz is a differential volume element in the vicinity of the end-to-end vector. Even though the orientation-insensitive Gaussian distribution prior to deformation was analyzed in significant detail using spherical coordinates, a rectangular Cartesian volume element is employed in this discussion to facilitate the use of simple elongation ratios that describe the state of deformation. With respect to the initial sample dimensions, elongations in three mutually perpendicular coordinate directions (i.e., x, y, and z) are defined as follows:

dx ¼

x y z , dy ¼ , dz ¼ xinitial yinitial zinitial

As a consequence of homogeneous affine deformation and the “reciprocal strain ellipsoid,” discussed in Appendix B.3 of Chapter 10, there are y i network strands in the deformed state that had the following x-, y-, and z-components of their end-toend vector frigUndeformed prior to stretching: x-component in the range from xi =dx to {xi þ dx}=dx y-component in the range from yi =dy to {yi þ dy}=dy z-component in the range from zi =dz to {zi þ dz}=dz Hence, if P(frigUndeformed) dxinitial dyinitial dzinitial describes the normalized probability of finding network strands with end-to-end vector frigUndeformed prior to stretching, and the three-dimensional Gaussian distribution is applicable, then P({ri }Undeformed ) dxinitial dyinitial dzinitial ¼ "      #)  3=2 ( 3 3 xi 2 yi 2 zi 2 dx dy dz exp  þ þ 2 2 dy dz dx dy dz 2phr iUndeformed 2hr iUndeformed dx allows one to relate y i and y via

yi ¼ yP({ri }Undeformed ) dxinitial dyinitial dzinitial X yi ¼ y i

14.9.4 Multiplicity of Network Strand Conformations After Deformation When elongations are not too large, Gaussian distribution functions can be used to describe the conformational characteristics of network strands that are subjected to uniaxial deformation. The normalized probability of finding individual strands whose end-to-end vector frigDeformed has an x-component in the range from xi to xi þ dx, a y-component in the range from yi to yi þ dy, and a z-component in the

626

Chapter 14 Classical and Statistical Thermodynamics of Rubber-Like Materials

range from zi to zi þ dz is P({ri }Deformed ) dx dy dz ¼   3=2  3 3 2 2 2 exp  [x þ y þ z ] dx dy dz i i 2phr2 iDeformed 2hr2 iDeformed i One constructs the multiplicity of states by considering the total number of distinguishable conformations for y network strands in which there are groups of y i strands with end-to-end vector frigDeformed in the deformed state. If all conformations were equally likely, then the multiplicity function V that accounts for distinguishability would be given by y! Y V¼ y i! Conformations w=different {ri }Deformed

Similar to the statistical analysis of excluded volume in Section 13.10, it is necessary to modify the multiplicity of states by considering the normalized Gaussian probability distribution for finding groups of network strands within the differential volume element dx dy dz of the end-to-end vector frigDeformed. Since there are y i strands with the same end-to-end vector, but not necessarily the same conformation when the rotational state of each backbone bond is considered, the products in the following expression consider groups of chain conformations with different frigDeformed instead of counting each network strand separately. Hence, the same normalized probability appears as a factor y i times in the multiplicity function because there are y i strands with the same end-to-end vector in the deformed state. One writes Y {P({ri }Deformed ) dx dy dz}yi V ¼ y! y i! Conformations w=different {ri }Deformed

This expression for V, which is consistent with the partition function for the canonical ensemble that includes the degeneracy of network strands with the same end-to-end vector in the deformed state, provides the required statistics for Boltzmann’s entropy via S ¼ k ln V.

14.9.5 Manipulation of the Multiplicity Function and Extrapolation to the Continuous Limit One evaluates the discrete expression for ln V as follows: X yi ln{P({ri }Deformed ) dx dy dz}  ln V ¼ ln y ! þ Conformations w=different {ri }Deformed

X Conformations w=different {ri }Deformed

ln y i !

14.9 Statistical Thermodynamic Analysis

627

With assistance from Stirling’s approximation for the factorial of an ensemble of network strands, ln y !  y ln y  y ¼ ln y

X

yi  y

Conformations w=different {ri }Deformed

ln y i !  y i ln y i  y i the previous expression for ln V reduces to X



y y i ln P({ri }Deformed ) dx dy dz ln V ¼ y i Conformations



w=different {ri }Deformed

Using the concept of the reciprocal strain ellipsoid, the ratio of y i to y is given by the Gaussian probability distribution function for network strands with end-to-end vector frigUndeformed prior to stretching. If the number of chain segments and the length of each segment do not change before and after stretching, then the second moment of the Gaussian distribution function should be independent of the state of deformation, assuming that the Gaussian distribution is valid. Consequently, the mean-square end-to-end distance of network strands should be the same before and after stretching. In other words, kr 2lUndeformed  kr 2lDeformed, even though uniaxial deformation increases the scalar component of the end-to-end vector for individual network strands in the stretch direction. Alternatively, one can envision that the breadth of the Gaussian distribution about its mean value (i.e., k(r 2 rAverage)2l) should be independent of the state of deformation, but the effect of uniaxial tensile deformation causes the average end-to-end vector rAverage of network strands that connect two crosslink junctions to be skewed in the stretch direction. Hence, the Gaussian distribution of end-to-end distances might be constructed as follows: "

#3=2 3 P(ri ) ¼ 2ph(r  rAverage )2 i ( ) 3  exp  (ri  rAverage ) (ri  rAverage ) 2h(r  rAverage )2 i



where rAverage vanishes in the undeformed state, but deformation skews the distribution and its mean value such that rAverage has a significant component in the stretch direction that increases with deformation. Consideration of the Gaussian distribution prior to and after stretching with a deformation-insensitive second moment yields the following discrete result for the multiplicity of states, where the summation includes

628

Chapter 14 Classical and Statistical Thermodynamics of Rubber-Like Materials

groups of network strands y i with the same end-to-end distance:  3=2   X y y 3 y i ln P({ri }Deformed ) dx dy dz ¼ ln V ¼ yi dx dy dz 2phr 2 i Conformations w=different {ri }Deformed

X

(

Conformations w=different {ri }Deformed

þ z2i

" ! ! 3 2 1 2 1 x ln dx dy dz þ  1 þ yi 2  1 2hr 2 i i d2x dy

(      )# !#) " 1 3 xi 2 yi 2 zi 2 dx dy dz  1 exp  þ þ dx dy dz 2hr 2 i d2z

In the continuous limit when the number of network strand conformations and endto-end vectors for segments between crosslink junctions is exceedingly large, the summation in the previous equation can be replaced by three-dimensional integration where rectangular Cartesian coordinates x, y, and z range from 21 to þ1. One obtains the following result:  3=2 ( y 3 F(dx )F(dy )F(dz ) ln dx dy dz : ln V ¼ dx dy dz 2phr2 i " ! 3 1  1 G(dx )F(dy )F(dz ) þ 2hr 2 i d2x ! ! #) 1 1 þ 2  1 F(dx )G(dy )F(dz ) þ  1 F(dx )F(dy )G(dz ) dy d2z ( ) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð 3j2 2 F(di ) ¼ exp  2 2 d j ¼ di phr2 i 3 2hr idi 1

G(di ) ¼

1 ð 1

(

3j2 j exp  2 2 2hr idi 2

)

1 d j ¼ hr2 id3i 3

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 phr2 i 3

No additional assumptions are required to simplify the previous expression for the multiplicity of states: h i ln V ¼ y  12 (d2x þ d2y þ d2z  3) þ ln{dx dy dz } The mean-square end-to-end distance of network strands that connect two crosslink junction points does not appear in the final expression for ln V because homogeneous affine deformation via the “reciprocal strain ellipsoid” allows one to evaluate the Gaussian distribution using Cartesian coordinates x, y, and z for the end-to-end vector after deformation, together with kr 2lDeformed, and coordinates x/dx, y/dy,

14.9 Statistical Thermodynamic Analysis

629

and z/dz prior to deformation. The same distribution function for network strand conformations can be employed before and after deformation because the corresponding coordinates of the end-to-end vector (i.e., x/dx ) x, y/dy ) y, z/dz ) z) are related by simple elongation ratios in three coordinate directions.

14.9.6 Boltzmann’s Entropy Expression for Isochoric Deformation with Isotropic Lateral Contraction The classic value of Poisson’s ratio is 0.5 for crosslinked elastomers, which implies that stretching occurs at constant volume. The corresponding “deformation invariant” is xinitial yinitial zinitial ¼ xyz

dx dy dz ¼ 1 Uniaxial tensile deformation in the x-direction implies that dx  1, and one invokes the concept of isotropic lateral contraction, which is interpreted as dy  dz. Hence, isotropic networks subjected to homogeneous tensile deformation in the x-direction contract similarly in both transverse directions. Boltzmann’s entropy for a network of y strands in the deformed state becomes   1 2 2 S ¼ k ln V ¼  k y dx þ  3 2 dx This result provides a quantitative measure of entropy in any state of deformation relative to the undeformed state.

14.9.7

Stress – Strain Relations

The equation of state for ideal rubber-like solids in Sections 13.17.3 and 14.7 is employed, together with Boltzmann’s entropy equation for network strands in the previous subsection, to evaluate the equilibrium retractive force that opposes deformation in the stretch direction: 

@S fx  T @Lx

 ¼ T,V

T Lx,initial



@S @ dx

!   1 1 2 ¼ y kT 2dx  2 2 Lx,initial dx T,V



Engineering stress is obtained from the previous expression for fx via division by the initial cross-sectional area of the sample Ainitial, transverse to the stretch direction. Hence,

s Engineering ¼

fx Ainitial



y  kT Ainitial Lx,initial



1 dx  2 dx

!

1 ¼ y Effective kT dx  2 dx

!

630

Chapter 14 Classical and Statistical Thermodynamics of Rubber-Like Materials

where y Effective is the effective crosslink density of the network (i.e., total number y of strands per volume). This result for sEngineering is essentially the same as the one at the end of the previous chapter, but the more sophisticated development here does not include the “front factor” g, which provides molecular information about the dimensions of undeformed network strands in the presence of crosslink junctions relative to the unperturbed size of “equivalent” network strands in the “free” state when crosslinks are absent. As illustrated in Section 13.17.5, one obtains the instantaneous “true” stress that accounts for the decrease in transverse cross-sectional area at higher deformation via multiplication of the previous equation by elongation in the x-direction (i.e., dx). These predictions for true stress and engineering stress deviate from the actual mechanical response of crosslinked elastomers at very high strains (i.e., above 300%) because (i) Gaussian statistics might not describe the conformational characteristics of network strands that connect two junction points, (ii) the mean-square endto-end distance is not insensitive to the level of deformation, (iii) the constant-volume approximation is not realistic, and (iv) stress-induced orientation of network strands could be responsible for crystallization that provides additional reinforcement and causes an abrupt upturn in the stress – strain curve.

14.10 EFFECT OF BIAXIAL DEFORMATION AT CONSTANT VOLUME ON BOLTZMANN’S ENTROPY AND STRESS VERSUS STRAIN Begin with the multiplicity of states in Section 14.9.5 for y network strands that deform in three orthogonal coordinate directions at constant volume. Under these conditions, the product of all three elongation ratios (i.e., dx, dy, dz) must be unity. Biaxial stretching is described by two independent elongation ratios in the x- and y-directions, such that dx . 1 and dy . 1. Constant-volume deformation requires that rubber-like materials must contract in the z-direction, with dz ¼ fdxdyg21. The multiplicity function reduces to   1 2 2 2 ln V ¼ y  (dx þ dy þ dz  3) þ ln{dx dy dz } 2 ( ) dx dy dz ¼1 y 2 1 2 )  d þ dy þ 2 2  3 2 x dx dy dz ¼{dx dy }1 Boltzmann’s entropy relation, S ¼ k ln V, allows one to evaluate retractive forces via the equation of state for ideal rubber-like materials with no dependence of internal energy on deformed sample length at constant temperature T and system volume V. For biaxial stretching, it is possible to calculate retractive forces in both stretch directions, fx and fy, via the dependence of entropy S on deformed sample lengths Lx and Ly, respectively, at constant T and V. Evaluation of fx must be performed at constant T, V, and dy, such that dz offsets changes in dx to maintain constant volume. Similarly, fy is evaluated at constant T, V, and dx, with dz offsetting changes in dy. Since deformed sample dimensions are given by Lx ¼ dxLx,initial and Ly ¼ dyLy,initial, the equation of

14.11 Effect of Isotropic Chain Expansion in “Good” Solvents

631

state yields the following expressions for fx and fy: ( )     @S kT @ ln V y kT 1 ¼ ¼ dx  3 2 fx ¼ T @Lx T,V, Ly Lx,initial @ dx T,V,dy Lx,initial dx dy ( )     @S kT @ ln V y kT 1 fy ¼ T ¼ ¼ dy  2 3 @Ly T,V, Lx Ly, initial @ dy T,V,dx Ly, initial dx dy The corresponding “true” normal stresses, sxx and syy, in tension (i.e., dx . 1, dy . 1) or compression (i.e., dx , 1, dy , 1) are obtained via division of fx and fy, respectively, by the instantaneous cross-sectional areas, LyLz and LxLz, that are perpendicular to the respective stretch directions. For example, ( ) fx fx fx dx y 1 2 sxx ¼ ¼ ¼ ¼ kT dx  2 2 L y Lz dy L y,initial dz Lz,initial L y,initial Lz,initial V dx dy ( ) fy fy fy dy y 1 syy ¼ ¼ ¼ ¼ kT d2y  2 2 Lx Lz dx Lx,initial dz Lz,initial Lx,initial Lz,initial V dx dy These results for sxx and syy allow one to calculate the first normal stress difference N1 for biaxial deformation, defined by N1 ¼ sxx  syy ¼ y Effective kT{d2x  d2y } where y Effective is the effective crosslink density of the network, given by the number of network strands per unit volume. Compare the previous normal stress difference for biaxial strain in rubber-like solids with the equation for N1 in Appendix B.7 of Chapter 10 when perfectly elastic isotropic solids experience simple two-dimensional shear in the x – y plane, and it should be obvious that normal stress differences scale as the square of deformation. The first normal stress difference, derived above, vanishes when elongation ratios in the x- and y-directions are equivalent (i.e., dx ¼ dy ¼ d). This is described as balanced biaxial deformation, which generates the following normal stresses:

sxx ¼ syy ¼ y Effective kT{d2  1=d4 }

14.11 EFFECT OF ISOTROPIC CHAIN EXPANSION IN “GOOD” SOLVENTS ON THE CONFORMATIONAL ENTROPY OF LINEAR MACROMOLECULES DUE TO EXCLUDED VOLUME As illustrated in Section 14.9.5, the multiplicity function for an ensemble of y chains in which there are groups of y i chains with the same end-to-end length after deformation is given by h i ln V ¼ y  12 (d2x þ d2y þ d2z  3) þ ln{dx dy dz }

632

Chapter 14 Classical and Statistical Thermodynamics of Rubber-Like Materials

where di represents the elongation ratio in the ith-coordinate direction. In other words, at the single-chain level of description, di is the ratio of the projections of the root-mean-square end-to-end vector in the ith-coordinate direction (i) after deformation and (ii) prior to deformation. The primary objective of this section is to employ previous results from the statistical thermodynamic analysis of rubber elasticity and revisit the effects of excluded volume on isotropic chain expansion. The problems are similar because expansion of linear chains in good solvents is analogous, at the molecular level, to isotropic swelling of network strands, both of which generate entropically elastic internal retractive forces. The previous expression for the multiplicity of states does not include the assumption of isochoric deformation with isotropic lateral contraction for unidirectional extension because chain expansion in good solvents occurs isotropically in all three coordinate directions with a corresponding increase in system volume. The isotropic deformation processes of interest here are due to (i) swelling of network strands and (ii) expansion of linear chains by good solvents. These processes generate entropically elastic retractive forces internally because there is an entropy decrease upon expansion that both systems attempt to recover via contraction. The results developed below represent an elastic contribution to the Gibbs free energy that must be considered in parallel with polymer – solvent mixing via the Flory – Huggins lattice theory, which was discussed in detail in Section 3.4. Recall that the logarithmic term on the right side of the previous equation for V is identically zero in rubber elasticity analysis because rubber-like materials expand or contract at approximately constant volume in the presence of external-forces when Poisson’s ratio is very close to one-half. Now, it is necessary to re-express the multiplicity function V in terms of the chain expansion factor a when all three elongation ratios are greater than unity. For isotropic chain expansion, it is reasonable to develop relations between a and each di by assuming that the projection of the root-meansquare end-to-end chain vector is the same in each orthogonal direction of a rectangular Cartesian coordinate system such that the effect of expansion is the same for all di. In terms of the root-mean-square end-to-end vector and its projections along the x-, y-, and z-coordinate axes prior to (i.e., unperturbed state) and after chain expansion, one defines the following elongation ratios for an ensemble of chains: 1=2

dx ¼

hx2 iExpanded 1=2

hx2 iUnperturbed

1=2

; dy ¼

hy2 iExpanded 1=2

hy2 iUnperturbed

1=2

; dz ¼

hz2 iExpanded 1=2

hz2 iUnperturbed

These definitions for groups, or ensembles, of chains are equivalent to the single-chain definitions of elongation ratios in Section 14.9.3. For isotropic expansion due to polymer – solvent interactions, not the application of unidirectional tensile or compressive forces, one writes 1=2

1=2

1=2

hx2 iUnperturbed ¼ hy2 iUnperturbed ¼ hz2 iUnperturbed 1=2

1=2

1=2

hx2 iExpanded ¼ hy2 iExpanded ¼ hz2 iExpanded such that dx ¼ dy ¼ dz . 1. Now, in terms of the mean-square end-to-end distance prior to (i.e., unperturbed state) and after chain expansion in good solvents, one defines

14.12 Effect of Polymer–Solvent Energetics on Chain Expansion

633

the expansion factor for real chains as follows:

a2 ¼

hr2 iExpanded hx2 iExpanded þ hy2 iExpanded þ hz2 iExpanded ¼ ¼ d2x ¼ d2y ¼ d2z hr 2 iUnperturbed hx2 iUnperturbed þ hy2 iUnperturbed þ hz2 iUnperturbed

The elastic contribution to the multiplicity function and the conformational entropy of a system of y chains that have experienced isotropic expansion in good solvents, or swelling, is re-expressed in terms of the expansion factor via Boltzmann’s equation:

SElastic ¼ kBoltzmann ln V ¼ y kBoltzmann  12 (3a2  3) þ ln{a3 }

¼ 3ykBoltzmann  12 (a2  1) þ ln a Now, one constructs an expression for the extensive conformational entropy change for the following process: an ensemble of y chains in their unperturbed state is swollen or expanded a-fold isotropically to a final state where the mean-square end-to-end distance is larger by a factor of a2 relative to kr 2lUnperturbed:

DSElastic ¼ SElastic (a)  SElastic (a ¼ 1) ¼ 3y kBoltzmann  12 (a2  1) þ ln a If the internal energy and enthalpy of the ensemble are not affected by the fact that the system responds to swelling or chain expansion by developing internal retractive forces, then the entropically driven elastic contribution to the extensive Gibbs free energy change for this process at constant temperature T and pressure p is

DGElastic ¼ T DSElastic ¼ 3y kBoltzmann T 12 (a2  1)  ln a When macromolecules expand in good solvents, the probability that chain segments interact energetically with surrounding solvent molecules increases at the expense of segment – segment interactions within the same chain or between different chains. These effects are included in the complete thermodynamic development of excluded volume and equilibrium chain expansion via lattice models for the Gibbs free energy of mixing which contain the polymer – solvent interaction parameter. Hence, solvent quality affects the expansion factor a via the dimensionless interaction free energy of mixing, described by the x-parameter, on the Flory – Huggins lattice.

14.12 EFFECT OF POLYMER– SOLVENT ENERGETICS ON CHAIN EXPANSION VIA THE FLORY – HUGGINS LATTICE MODEL The complete expression for the extensive Gibbs free energy of mixing NSolvent molecules of solvent and y ¼ NPolymer molecules of polymer, which contains entropic and energetic effects, was developed using mean-field analysis in Section 3.4.5 as  DGmixing ¼ kT NSolvent ln wSolvent þ NPolymer ln wPolymer þ x NSolvent wPolymer

634

Chapter 14 Classical and Statistical Thermodynamics of Rubber-Like Materials

where k is Boltzmann’s constant, wi represents volume fraction of component i, and x is the composition-independent Flory – Huggins polymer – solvent interaction parameter or the dimensionless interaction free energy of mixing. Definitions of wSolvent and wPolymer, together with their dependence on NSolvent at constant NPolymer, are provided below: xNPolymer NSolvent wSolvent ¼ ; wPolymer ¼ NSolvent þ xNPolymer NSolvent þ xNPolymer     @ wPolymer @ wSolvent NSolvent ¼ NSolvent ¼ wSolvent wPolymer @NSolvent T, p, NPolymer @NSolvent T, p, NPolymer   @ wPolymer 1 NPolymer ¼  w2Polymer @NSolvent T, p, NPolymer x For composition-independent x, the dependence of DGmixing on NSolvent is (     @DGmixing NSolvent @ wSolvent ¼ kT ln wSolvent þ @NSolvent T, p, NPolymer wSolvent @NSolvent T, p, NPolymer   NPolymer @ wPolymer þ wPolymer @NSolvent T, p, NPolymer " #)   @ wPolymer þ x wPolymer þ NSolvent @NSolvent T, p, NPolymer     1 ¼ kT ln(1  wPolymer ) þ 1  wPolymer þ xw2Polymer x   wPolymer 1 1  x w2Polymer )  kT x 1 2 Each monodisperse polymer chain occupies x interconnected lattice sites. The previous expression for the dependence of the extensive Gibbs free energy of mixing on the number of solvent molecules at constant temperature, pressure, and NPolymer is, essentially, the chemical potential or partial molar Gibbs free energy of mixing of the solvent on a molecule basis, not a molar basis, for very dilute solutions (i.e., wPolymer 1) that contain extremely high-molecular-weight polymer chains (i.e., x 1). Taylor series expansion of ln(1 2 wPolymer)  2 wPolymer 2 w2Polymer/2 2 w3Polymer/32   and truncation after the quadratic term are required to obtain the final result for (@DGmixing =@NSolvent )T, p, NPolymer when the x-parameter depends only on temperature, not the composition of the solution. The primary objective here is to consider the dependence of the Flory – Huggins expression for the free energy of mixing, that accounts for an increase in polymer – solvent interactions as chains expand, on the number of solvent molecules and, subsequently, the dependence of the number of solvent molecules on the chain

14.12 Effect of Polymer–Solvent Energetics on Chain Expansion

635

expansion factor, a. Hence, one employs the Gibbs free energy, not the Helmholtz function, because the volume element under consideration increases in size to encompass a fixed number of polymer chains from the unperturbed state to the expanded state. Calculations are performed at constant temperature, pressure, and NPolymer, but not constant volume. As the volume element under consideration increases to account for chain expansion, the number of polymer molecules NPolymer will not change in dilute solutions, but their size increases in good solvents. Solvent molecules are modeled as structureless point particles on the lattice, so NSolvent increases to fill the additional lattice sites in larger volume elements and one seeks an expression for the change in NSolvent with respect to a. The effect of chain expansion on the extensive Gibbs free energy of mixing for the Flory – Huggins lattice is formulated as follows:       @DGmixing @DGmixing @NSolvent ¼ @a @NSolvent T, p, NPolymer @a T, p, NPolymer T, p, NPolymer     wPolymer 1 1 @NSolvent  x w2Polymer )  kT x 1 @a 2 T,p,NPolymer Since (@NSolvent =@ a)T, p, NPolymer is always positive because more solvent molecules are required to occupy additional lattice sites when chain expansion is larger, the previous equation correctly predicts that the Gibbs free energy of mixing on the Flory – Huggins lattice decreases favorably for dilute solutions of high-molecular-weight polymers when the interaction parameter x is less than its critical value of one-half (see Appendix A at end of this chapter). This is expected for homogeneous solutions that don’t exhibit phase separation. The remainder of this section focuses on explicit evaluation of the previous equation. The strategy at the single-chain level of description considers the increase in the number of solvent molecules within spherical shells of radius r about the center of mass of an isolated polymer chain, where the differential volume element expands from 4p r 2 dr (i.e., unperturbed state) to 4pa3r 2 dr in response to an increase in the shell radius from r to ar and shell thickness from dr to a dr. Ultimately, one multiplies the final result by y chains in the entire solution. In the expanded state, the differential volume element is dVExpanded ¼ 4pa3r 2 dr, the fraction of this volume element occupied by solvent is wSolvent dVExpanded, and the volume of one solvent molecule in solution is given by the ratio of the solvent’s molar volume vSolvent, or its partial molar volume, to Avogadro’s number NAvogadro. Hence, the differential increase in the number of solvent molecules with uniform density in dVExpanded is   wSolvent dVExpanded NAvogadro   dNSolvent ¼ wSolvent a3 r2 dr ¼ 4p vSolvent vSolvent NAvogadro Since the solvent density is uniform throughout the solution, the two previous equations yield the contribution to (@DGmixing =@ a)T,p,NPolymer from differential

636

Chapter 14 Classical and Statistical Thermodynamics of Rubber-Like Materials

volume element dVExpanded:     NAvogadro @NSolvent ¼ 12p wSolvent a2 r2 dr @a vSolvent T, p, NPolymer       @DGmixing 1 @NSolvent 2  x wPolymer  kT @a @a 2 T, p, NPolymer T, p, NPolymer    NAvogadro 1  x (1  wPolymer )w2Polymer r2 dr  12pa2 kT vSolvent 2 The complete expression for the effect of chain expansion on the Flory – Huggins free energy of mixing is obtained by summing the previous expression over all regions of space, as differential volume element dVExpanded encompasses the entire solution at constant chain expansion factor a. Mathematically, one replaces this sum by integration with respect to r and obtains the following result based on single-chain analysis: 

@DGmixing @a

 T,p,NPolymer

   NAvogadro 1 x  12pa kT vSolvent 2 2



1 ð

(1  wPolymer )w2Polymer r2 dr

r¼0

Now, it is necessary to invoke a Gaussian distribution of polymer chain segments about the center of mass of each macromolecule via the radius of gyration distribution function P(s) to evaluate the radial dependence of wPolymer in dilute solution (i.e., 12 wPolymer  1), where the radius of gyration s ¼ r prior to chain expansion and s ¼ ar after expansion. Returning to differential volume element dVExpanded in the expanded state, which contains NdVExpanded chain segments, the polymer volume fraction in this differential volume element is

wPolymer 

l3 NdVExpanded l3 NP(s ¼ r) dVUnperturbed l3 ¼ ¼ 3 NP(s ¼ r) 4pa3 r2 dr 4pa3 r 2 dr a

where N represents the total number of segments in one polymer chain, P(s ¼ r) dVUnperturbed is the fraction of segments within differential volume element 4p r 2 dr in the unperturbed state that had radius of gyration s ¼ r about the center of mass of each chain via Gaussian statistics prior to expansion, the segment length is l, and l3 is the approximate volume of each lattice cell that contains one segment. It is important to emphasize that the total number of polymer segments within unperturbed differential volume element 4p r 2 dr, prior to chain expansion, is the same as the number of segments in the expanded volume element dVExpanded, after isotropic chain expansion, because solvent molecules occupy the additional lattice cells that

14.12 Effect of Polymer–Solvent Energetics on Chain Expansion

637

are created when the volume element increases in size to encompass the expanded chain. The previous equation correctly reveals that the local volume fraction of polymer scales as a23 because the actual volume of all N chain segments remains the same as the volume element expands. The radius of gyration distribution function, which describes the segment density distribution about the center of mass of each isolated chain, is assumed to follow Gaussian statistics for unperturbed freely jointed chains [Yamakawa, 1971]. Hence, the fraction of segments that exists in a spherical shell of thickness ds and radius of gyration s about the center of mass is P(s) dVUnperturbed ¼ 4ps2 A exp(k2 s2 ) ds Parameters A and k are determined by invoking normalization and a second moment given by 16kr 2lUnperturbed for freely jointed chains when the number N of segments per chain is large. The relevant equations to calculate A and k are 1 ð

1 ð

P(s) dVUnperturbed ¼ 4pA s2 exp(k2 s2 ) ds ¼ 4pA

0

 1=2  p ¼1 4k3

0

2

hs i ¼

1 ð 0

2

1 ð

s P(s) dVUnperturbed ¼ 4pA s4 exp(k2 s2 ) ds 0

 1=2  3p 1 ¼ hr2 iUnperturbed ¼ 4pA 8k5 6 A¼

k3 9 ; k2 ¼ 2 ; hr2 iUnperturbed ¼ N l2 hr iUnperturbed p 3=2

These results for the unperturbed radius of gyration distribution function yield the following expression for the polymer volume fraction that must be integrated over all space:

wPolymer

" #3=2 ( ) l3 9 9r 2 ¼ 3N exp  2 a phr2 iUnperturbed hr iUnperturbed

One obtains the same dependence of wPolymer on the expansion factor a by calculating the number of chain segments in expanded differential volume element dVExpanded via a Gaussian approximation for the radius of gyration distribution function in the expanded state with shell radius as and shell thickness a ds, where s ¼ r. Now, the mean-square end-to-end distance after chain expansion replaces kr 2lUnperturbed in P(s), similar to the distribution function for ni/n in Section 13.10.2 for real chains in the expanded state. This is consistent with the smoothed-density model [Yamakawa, 1971], where it is not possible to distinguish between the expansion factors for ks 2l and kr 2l, and the mean-square radius of gyration is six-fold smaller

638

Chapter 14 Classical and Statistical Thermodynamics of Rubber-Like Materials

than the mean-square end-to-end distance for real chains and freely jointed ideal chains. Following this approach within dVExpanded, one obtains an expression for the polymer volume fraction of a single chain that contains N segments, with segment length l, approximate segment volume l3, and expanded dimensions such that kr 2l ¼ a2Nl2:

wPolymer 

wPolymer

l3 NdVExpanded l3 NP(s ¼ ar) dVExpanded ¼ ¼ l3 NP(s ¼ ar) 4pa3 r2 dr 4pa3 r 2 dr     9 3=2 9a2 r 2 exp  P(s ¼ ar)  phr 2 i hr 2 i " #3=2 ( ) 9 9a2 r 2 3 l N exp  2 2 pa2 hr 2 iUnperturbed a hr iUnperturbed

Proceeding with calculations of the effect of a on the Flory – Huggins free energy of mixing for dilute solutions (i.e., 12 wPolymer  1), one obtains      1 ð @DGmixing NAvogadro 1 2 x  12pa kT w2Polymer r2 dr @a v 2 Solvent T, p, NPolymer 







r¼0

 NAvogadro 1  x l6 N 2 ¼ 12pa2 kT vSolvent 2 " #3 ð1 ( ) 2 9 18r dr  r 2 exp  2 pa2 hr 2 iUnperturbed hr iUnperturbed r¼0

 NAvogadro 1  x l6 N 2 ¼ 12pa kT vSolvent 2 ( )3=2 " #3 9 p1=2 hr2 iUnperturbed  pa2 hr 2 iUnperturbed 4 18     6 NAvogadro 3(9)3 1 l N2  ¼ kT x 2 vSolvent a4 hr2 i3=2 (18)3=2 p 3=2 Unperturbed  3   1=2 l NAvogadro 1 81 N x ¼  3=2 3=2 kT vSolvent a4 2 2 p 2

This quantitative single-chain result suggests that chain expansion provides a favorable (i.e., negative) contribution to the free energy of mixing in good solvents because the Flory –Huggins interaction parameter is less than its critical value of 0.5 for dilute solutions of high-molecular-weight chains. Since polymer molecular weight is proportional to the number N of segments per chain, the following scaling

14.13 Gibbs Free Energy Minimization Yields the Equilibrium Chain Expansion Factor

law is applicable:

639

  @DGmixing  N 1=2 a4 @a T, p, NPolymer

This is consistent with the molecular-weight and chain-expansion-factor dependence of the derivative of the logarithm of the free volume parameter w with respect to a in Section 13.10.1 (i.e., @ ln w/@ a  þN 0.5a24), but with opposite sign because one must realize that w accounts for entropic effects whereas DGmixing includes both entropic and energetic effects.

14.13 GIBBS FREE ENERGY MINIMIZATION YIELDS THE EQUILIBRIUM CHAIN EXPANSION FACTOR Add results from the previous two sections for the elastic and mixing contributions to the extensive Gibbs free energy change of an ensemble of y ¼ NPolymer chains that expand and simultaneously experience greater interaction with the solvent. The optimum expansion factor aequilibrium minimizes the total Gibbs free energy change, such that at constant temperature T, pressure p, and number of polymer chains y ,     @DGmixing @DGElastic þ ¼0 @a @a T,p,y T,p,y     3   1=2  l NAvogadro 1 81 1 N  3=2 3=2 ¼0 x y kBoltzmann T 3 a  a vSolvent a4 2 2 p  3   l NAvogadro 27 1 5 3  x N 1=2 aequilibrium  aequilibrium ¼ 3=2 3=2 vSolvent 2 2 p This result for the equilibrium chain expansion factor illustrates explicit dependence on molecular weight (i.e., MW  N ) and implicit dependence on temperature and solvent quality via the Flory – Huggins interaction parameter (i.e., x). The x-parameter is unique for each polymer – solvent pair, and x decreases at higher temperature as the solvation power of the solvent increases. The critical value of x is 0.5 for dilute solutions of high-molecular-weight polymers at the Q-temperature (see Appendix A at end of chapter), suggesting that the balance between chain expansion due to polymer – solvent interactions and entropically elastic retractive forces occurs in the unperturbed state where aequilibrium ¼ 1. Phase separation occurs when x exceeds its molecular-weight-dependent critical value (see Table 3.1), but the previous equation requires severe modification that is beyond the scope of this chapter to evaluate the expansion factor in the concentrated polymer phase because the dilute-solution approximations are not valid. The repeat-unit, or segment, molar volume of the polymer is approximately given by l3NAvogadro, which is the same order of magnitude but slightly larger than the solvent’s molar volume vSolvent. Hence, the ratio of these molar volumes in the previous equation is slightly greater than unity but probably less than 5. Finally, the front factor in the previous equation (i.e., 27={2p}1:5 ¼ 1:72) is two-fold

640

Chapter 14 Classical and Statistical Thermodynamics of Rubber-Like Materials

larger than the one in the comparable expression for aequilibrium in Section 13.10.3, where excluded volume was analyzed as a probabilistic entropic effect instead of the focusing on the energetics of mixing. However, when the previous equation is evaluated for athermal solutions (i.e., x ¼ 0), where the interaction free energy of mixing vanishes and entropic effects dominate the mixing process, and the repeatunit molar volume of the polymer is the same as that for the solvent (i.e., l3NAvogadro  vSolvent), the equilibrium expansion factor is calculated from   27 1  0 N 1=2  0:86N 1=2 a5equilibrium  a3equilibrium  3=2 3=2 2 2 p This equation is exactly the same as the final equation in Section 13.10.3.

APPENDIX A: CHEMICAL OR DIFFUSIONAL STABILITY OF POLYMER –SOLVENT MIXTURES Statistical analysis of the placement of linear polymer chains and structureless solvent molecules on the three-dimensional Flory – Huggins lattice, as described in Section 3.4 and presented in Section 14.12 of this chapter, yields the following expression for the partial molar Gibbs free energy of the solvent in dilute polymer solutions:       @DGmixing 1 2 ¼ RT ln(1  wPolymer ) þ 1  wPolymer þ xwPolymer @nSolvent T,p,nPolymer x ¼ RT ln aSolvent where ni represents the mole numbers of component i, wPolymer is the polymer volume fraction, x is the molar volume ratio of polymer to solvent, or the number of segments per chain, and x is the concentration-independent dimensionless Flory – Huggins interaction free energy of mixing. Energetic effects in the previous equation, via the x-parameter, are analogous to the two-parameter van Laar model for the excess nonideal Gibbs free energy of mixing, where the van Laar quantities of interest represent a dimensionless interaction parameter and the ratio of molar volumes of both components. The temperature dependence of x governs the entropic versus enthalpic contributions to the interaction free energy of mixing, as discussed in Sections 3.4.4 and 5.3.2. The previous equation represents the difference between the solvent’s chemical potential in solution mSolvent and in a pure-component reference state m0Solvent at 1 atmosphere total pressure. Chemical stability of homogeneous binary solutions requires that [Belfiore, 2003]     @ mSolvent @ ln aSolvent ¼ RT @ySolvent T,p @ySolvent T,p ( )    d wPolymer @ ln aSolvent dwSolvent .0 ¼ RT @ wPolymer d wSolvent dySolvent T,p

Appendix B

641

where yi is the mole fraction of component i. Since the solvent volume fraction wSolvent increases when ySolvent is higher, but the polymer volume fraction decreases at higher concentrations of solvent (i.e., wPolymer ¼ 1 2 wSolvent), chemical stability is reformulated in terms of the polymer volume fraction dependence of the solvent activity [Belfiore, 2003]:   1 ln aSolvent ¼ ln(1  wPolymer ) þ 1  wPolymer þ xw2Polymer x ( ) @ ln aSolvent 1 1 ¼ þ 1  þ 2xwPolymer , 0 @ wPolymer 1  wPolymer x T,p

Even though the x-parameter is composition-independent, it has a critical upper limit imposed by chemical stability that varies with composition. The previous inequality must be satisfied to ensure that the solution is homogeneous and phase separation does not occur. The critical value of the Flory – Huggins interaction parameter is defined when the inequality becomes an equality at the stability limit on the spinodal curve where the compositional dependence of DGmixing changes from positive to negative curvature:

xCritical ¼

1  {1  1=x}(1  wPolymer ) 1 1 ) ) x!1 2(1  wPolymer ) wPolymer !0 2 2wPolymer (1  wPolymer )

For very high-molecular-weight polymers (i.e., x ) 1) at infinite dilution (i.e., wPolymer ) 0), xCritical ¼ 0.5 at the Q-temperature. The critical value of the interaction parameter was discussed in significant detail in Section 3.5.4. Phase separation occurs when x exceeds its critical value. As the solvent power increases at higher temperature, x decreases. Athermal solutions correspond to x ¼ 0, and exothermic heats of mixing are described by negative values of x.

APPENDIX B: GENERALIZED LINEAR LEAST SQUARES ANALYSIS FOR SECOND-ORDER POLYNOMIALS WITH ONE INDEPENDENT VARIABLE Experimental measurements yield N data pairs (i.e., xi and yi, 1  i  N ), and it is desired to model the data with the following quadratic function: y(x) ¼ a0 þ a1 x þ a2 x2 The objective of this exercise is to use all of the data pairs and determine the optimum values of the parameters a0, a1, and a2 in the second-order polynomial. Even though the polynomial is not linear, the three parameters can be calculated from simultaneous solution of three linear algebraic equations. The nature of the model function determines whether linear or nonlinear analysis is required to determine the parameters. Sometimes, simple algebraic manipulation together with “taking the log of the entire equation” reduces a nonlinear problem to one that requires linear analysis. The quadratic function y(x) that best matches the discrete data pairs is determined by comparing

642

Chapter 14 Classical and Statistical Thermodynamics of Rubber-Like Materials

y(xi) and yi, 1  i  N. Since differences between y(xi) and yi can be positive or negative, a very poor match between model and data might yield small overall differences when y(xi) 2 yi is summed over all data pairs if some of the differences fortuitously cancel. To avoid this problem in the evaluation of any polynomial model, the error is constructed as follows: N N X X Error(a0 , a1 , a2 ) ¼ {y(xi )  yi }2 ¼ {a0 þ a1 xi þ a2 x2i  yi }2 i¼1

i¼1

so that all differences between model and data contribute to larger error. The same final result could be achieved by summing the absolute value of the difference between model and data over all points. If Error, as defined above, is plotted hypothetically in four-dimensional space as a function of the three parameters, then the best combination of a0, a1, and a2 produces a global minimum on this multidimensional surface. This condition is expressed mathematically as   N X @Error ¼2 {a0 þ a1 xi þ a2 x2i  yi } ¼ 0 @a0 a1 ,a2 i¼1   N X @Error ¼2 xi {a0 þ a1 xi þ a2 x2i  yi } ¼ 0 @a1 a0 ,a2 i¼1   N X @Error ¼2 x2i {a0 þ a1 xi þ a2 x2i  yi } ¼ 0 @a2 a0 ,a1 i¼1 Minimization is assured because there is no upper bound to the Error. These three linear algebraic equations are rearranged to calculate a0, a1, and a2: N N N X X X xi þ a2 x2i ¼ yi Na0 þ a1 a0

N X

xi þ a1

i¼1

a0

N X

i¼1

i¼1

N X

N X

x2i þ a2

i¼1

x2i þ a1

i¼1

N X

i¼1

x3i ¼

i¼1

x3i þ a2

i¼1

N X

N X

xi yi

i¼1

x4i ¼

i¼1

N X

x2i yi

i¼1

The following summations over all of the xi – yi data pairs are defined to simplify the final solution for the three parameters: S1 ¼

N X

xi ; S2 ¼

i¼1

S5 ¼

N X

x2i ; S3 ¼

i¼1 N X i¼1

yi ; S6 ¼

N X

x3i ; S4 ¼

i¼1 N X i¼1

xi yi ; S7 ¼

N X

x4i

i¼1 N X

x2i yi

i¼1

If it is desired to exclude one or more data points from the analysis, then the seven summations defined above must be modified accordingly and N is reduced. Multivariable

Appendix C

643

minimization of the Error is accomplished by solving the following coupled linear equations: Na0 þ S1 a1 þ S2 a2 ¼ S5 S1 a0 þ S2 a1 þ S3 a2 ¼ S6 S2 a0 þ S3 a1 þ S4 a2 ¼ S7 There are many situations where a linear model is desired (i.e., y ¼ a0 þ a1x). The optimum values of the first-order coefficient a1 (i.e., slope) and the zerothorder coefficient a0 (i.e., intercept) can be calculated from a subset of the information provided above for a second-order polynomial model. It is not necessary to minimize the Error with respect to the second-order coefficient a2. Furthermore, a2 ¼ 0 in the other two linear equations. Hence, Na0 þ S1 a1 ¼ S5 S1 a0 þ S2 a1 ¼ S6 If the linear polynomial has a known value of the intercept a0, then the linear least squares procedure identifies the best “slope” via minimization of the Error with respect to a1. The value of a1 is obtained by solving the second of the two simultaneous linear equations for a0 and a1: S1 a0 þ S2 a1 ¼ S6 where the known value of a0 is used together with the appropriate summations to calculate a1. The previous expression for the first-order coefficient a1 is simplified further when the intercept a0 is forced to vanish: a1 ¼ S6/S2.

APPENDIX C: LINEAR VERSUS NONLINEAR LEAST SQUARES DILEMMA The data in Table 14.1 have been obtained for the concentration dependence of the modulus of elasticity for solid state binary mixtures of a rubber-like polymer and a crosslinking agent. Percolation theory provides a nonlinear model to predict the elastic modulus (E) as a function of the crosslinker concentration ( p): E ¼ E0 ( p  pcritical )b p  pcritical

(C:1)

where b is the critical exponent, pcritical is the crosslinker concentration at the percolation threshold, and E0 is the elastic modulus of the network at concentrations slightly above the percolation threshold. When hot water finds a continuous path through fresh coffee grinds and drips into a pot of fresh-brewed coffee, the system is above the percolation threshold with respect to fluid flow. Analogously, when all of the polymer chains are chemically linked together by the crosslinking agent, the network is above the percolation threshold and the elastic modulus increases “exponentially”

644

Chapter 14 Classical and Statistical Thermodynamics of Rubber-Like Materials Table 14.1 Effect of the Concentration of a Crosslinking Agent, Dichlorobis(acetonitrile) –Palladium(II), on the Elastic Modulus of Atactic 1,2-Polybutadiene a Concentration of the crosslinking agent (mol % wrt the polymer)

Young’s modulus of elasticity of the crosslinked solid (kilopounds per square inch)

1.0 2.0 3.0 3.5 4.0

0.332 10.8 32.7 72.1 200

a

The polymer was synthesized by A. F. Halasa at Goodyear Research and Development in Akron, Ohio.

with respect to the crosslinker concentration via Equation (C.1). The three parameters (E0, pcritical, and b) can be calculated via a nonlinear least squares algorithm using Eq. (C.1) or log E ¼ log E0 þ b log( p  pcritical ) for p . pcritical

(C:2)

It should be emphasized that nonlinear least squares regression is required to determine the three parameters via Eq. (C.1) or (C.2). If the percolation threshold concentration of crosslinking agent is known to be 1 mol % (i.e., pcritical  1), then Eq. (C.1) requires a nonlinear least squares algorithm to calculate E0 and b via E ¼ E0 ( p  1)b

p1

(C:3)

but Eq. (C.2) requires a linear least squares algorithm to calculate E0 and b via log E ¼ log E0 þ b log( p  1)

for p . 1

(C:4)

Standard application of linear least squares regression methodology via Eq. (C.4) is based on an analysis of the four data points where p . pcritical ¼ 1. Obviously, the first data point (i.e., 1 mol % and 0.332 kpsi) cannot be evaluated via Eq. (C.4) because log( p 2 1) is undefined. The last four data points are processed accordingly and matched to a first-order polynomial (i.e., y ¼ a0 þ a1x), where (i) the independent variable (i.e., x) is log( p 21); (ii) the dependent variable (i.e., y) is log E; (iii) the first-order coefficient in the model (i.e., a1 ¼ slope) is the critical exponent b; and (iv) the zeroth-order coefficient in the model (i.e., a0 ¼ intercept) is log E0.

Appendix C

645

The results are E0 ¼ 8:99 kpsi, b ¼ 2:46, Minimized Error #1 ¼

4 X {yi  a0  a1 xi }2 ¼ 0:394 i¼1

Next, all five data points are analyzed via Eq. (C.3) as follows: (i) Educated guesses are made for E0 and b. (ii) Equation (C.3) is used to calculate the elastic modulus of the network for each concentration of crosslinking agent, with pcritical ¼ 1 mol %. (iii) Then, one evaluates the square of the difference between the actual experimental elastic modulus and the model given by Eq. (C.3). 5 P {Ei  E0 ( pi  1)b }2 : (iv) The objective function is evaluated: Error ¼ i¼1

(v) Trial and error is employed to find the best combination of E0 and b that minimizes the objective function. This is an application of nonlinear least squares analysis. The results are E0 ¼ 0:717 kpsi, b ¼ 5:120, Minimized Error #2 ¼

5 X {Ei  E0 ( pi  1)b }2 ¼ 200 (kpsi)2 i¼1

One should not compare Minimized Error #1 with Minimized Error #2 because the first one uses four data points and the logarithm of modulus whereas the second one uses five data points and the square of modulus. If the “best” values of E0 and b from linear least squares analysis via the logarithmic model and the first-order polynomial (i.e., y ¼ a0 þ a1x) are used as inputs for the nonlinear algorithm described in (i) through (v) above, then the results are 5 X E0 ¼ 8:99 kpsi, b ¼ 2:46, Error #1 ¼ {Ei  E0 ( pi  1)b }2 ¼ 4826 (kpsi)2 i¼1

Obviously, this combination of E0 ¼ 8.99 kpsi and b ¼ 2.46 is not optimal from the viewpoint of the nonlinear model because Error #1 ¼ 4826 (kpsi)2 is much greater than Minimized Error #2 ¼ 200 (kpsi)2. Both of these error functions employ all five data points and the square of modulus. The same conclusion is drawn (i.e., Error #1 . Minimized Error #2) if the first data point is neglected when Eq. (C.3) predicts E ¼ 0. Now, the “best” values of E0 and b from trial and error nonlinear least squares analysis are used as inputs for the linear logarithmic model and the first-order polynomial. One obtains the following results: E0 ¼ 0:717 kpsi, b ¼ 5:120, Error #2 ¼

4 X {yi  a0  a1 xi }2 ¼ 7:42 i¼1

Error #2 is larger than Minimized Error #1, where both error functions employ four data points and the logarithm of elastic modulus.

646

Chapter 14 Classical and Statistical Thermodynamics of Rubber-Like Materials

Here’s the dilemma: Which combination of model parameters (i.e., E0 and b) provides the best match with the experimental data when pcritical  1?

REFERENCES BELFIORE LA. Transport Phenomena for Chemical Reactor Design. Wiley, Hoboken, NJ, 2003, pp. 787– 790, 815. YAMAKAWA H. Modern Theory of Polymer Solutions. Wiley, Hoboken, NJ, 1971, Chap. 2, pp. 26–35; Chap. 3, pp. 70– 75.

PROBLEMS 14.1. Consider a rubber-like solid material subjected to an external force field. The retractive force f is measured as a function of temperature. In these experiments, the external pressure and the length of the sample in the strained state are held constant. The following observations are made: (i) When the mechanical strain is less than 10% ( g , 0.10), the retractive force (required to maintain the specimen at a fixed length in the strained state) decreases at higher temperatures. (ii) When the mechanical strain is greater than 10%, the retractive force increases at higher temperatures. In both experimental observations described above, mechanical strain is defined as follows:

g ¼ [L  L0 (T)]=L0 (T) where L is the strained length of the sample, L0 is the undeformed length of the sample, and T is temperature. (a) Based on your knowledge of the theory of rubber elasticity, provide an in-depth explanation of the experimental observations described above. (b) How should retractive force f versus temperature be analyzed experimentally to reveal that the retractive force always increases at higher temperatures. 14.2. The mechanical response of an ideal rubber-like solid is modeled in the simplest way by a Hookean spring with a modulus of elasticity given by G. A force of 10,000 dynes is applied to the solid material and the experimental strain is measured to be on the order of 150%. The strain g is defined in the statement to Problem 14.1. When equilibrium conditions are attained in practice, the strained length of the rubber-like solid is 50 cm if the experiment is performed at ambient temperature (i.e., 25 8C). What equilibrium sample length (expressed in centimeters) will be measured if the experiment is performed at 60 8C? Do not neglect the effect of thermal expansion in the undeformed material. 14.3. (a) A composite material is modeled as two crosslinked elastic segments in series, with moduli E1 and E2. The initial length of each segment is L1,initial and L2,initial, and the system is stretched uniaxially until the extended length is Ldeformed. Identify the position of the interface that connects both elastic segments in the deformed state, in general, and obtain the limiting response when either the first segment or the

Problems

647

second segment is infinitely rigid. Courtesy of communications with Prof. Travis S. Bailey, Dept. of Chemical and Biological Engineering, Colorado State University. Answer The first segment is placed to the left of the second segment, and the origin of a one-dimensional coordinate system coincides with the left side of the first segment. Each segment of the composite material experiences the same force, or engineering stress, as a consequence of series alignment of two springs. Rigorously, strain is a “point” property, so the total deformation is obtained by adding the deformation in each segment. Application of Hooke’s law of elasticity, or the stress –elongation relation for crosslinked elastomers with negligible contribution from the {dx }2 term in the ideal equation of state, provides a route to identify the location of the interface, Linterface, with respect to the origin of the one-dimensional coordinate system: E1 {strain1} ¼ E2 {strain2} strain1 ¼ strain2 ¼

L interface ¼

L interface  L 1,initial L 1,initial L deformed  L interface  L 2,initial L 2,initial   L deformed E1 þ E2 1 L2,initial E1 E2 þ L1,initial L2,initial

If the first segment is infinitely rigid, then E1 E2 and the previous relation reduces to Linterface ) L1,initial, which is reasonable if all of the deformation is experienced by the second segment. When the second segment is essentially undeformable, due to the fact that E2 E1, the interfacial position is given by Linterface ) Ldeformed – L2,initial because only the first segment deforms. (b) How does an increase in temperature affect the position of the interface in the deformed state when neither segment is infinitely rigid and (i) segment #1 is above its thermoelastic inversion point, but segment #2 is below; (ii) segment #1 is below its thermoelastic inversion point, but segment #2 is above; and (iii) both segments are above their thermoelastic inversion points.

Part Four

Solid State Dynamics of Polymeric Materials

Chapter

15

Molecular Dynamics via Magnetic Resonance, Viscoelastic, and Dielectric Relaxation Phenomena We conjure half dreams, tell incandescent tales, and call them Reality. —Michael Berardi

Relaxation phenomena for NMR are developed from the Liouville equation and compared with mechanical relaxation via the fluctuation – dissipation theorem. Connections between correlation functions and spectral densities are discussed for NMR and stress relaxation. Correlation times for stress relaxation are introduced via Rouse dynamics with negligible hydrodynamic interaction. Relations between the real and imaginary parts of the complex dynamic viscosity, and the complex dynamic modulus, are discussed within the context of the Kramers – Kronig theorem using the Maxwell model as an illustrative example. Experimental results for bisphenol-A polycarbonate are used to compare measurements from NMR, mechanical, and dielectric relaxation on this industrially important polymer. Complex impedance analysis of dielectric relaxation measurements for Maxwell and Voigt models is compared with thermally stimulated discharge of polarized electrets. This chapter illustrates how motional-induced relaxation across a broad spectrum of physical chemistry and rheological experiments provides complementary information about the solid state dynamics of polymeric materials.

15.1 FLUCTUATION – DISSIPATION Thermally induced motion of polymer chain segments introduces time dependence into key physical variables, such as the end-to-end vector for a single chain. Next, a Physical Properties of Macromolecules. By Laurence A. Belfiore Copyright # 2010 John Wiley & Sons, Inc.

651

652

Chapter 15 Molecular Dynamics via Magnetic Resonance

time-averaging process is applied to the end-to-end vector, generating information about the decay of stress in viscoelastic materials as a consequence of chain motion. Stress relaxation in viscoelastic solids can be parameterized by characteristic time constants, known as relaxation times. Fourier transformation of the stress relaxation modulus yields a frequency-dependent dynamic modulus that contains information about material response frequencies, which are inversely related to viscoelastic relaxation times. Of critical importance, mechanical energy is dissipated into thermal energy (i.e., heat) during stress relaxation. This phenomenon arises because resonance is satisfied when significant overlap occurs between material response frequencies and the frequencies of external excitations that fluctuate periodically in time. In general, there is a connection between molecular mobility and relaxation.

15.2 OVERVIEW The underlying principles that govern mobility-induced relaxation across a broad spectrum of physical chemistry share a common thread whose origin is embedded in the fluctuation – dissipation theorem. Similarities between magnetic resonance and viscoelastic relaxation are discussed theoretically via Fourier transformation of the relevant time-correlation functions that are consistent with the Maxwell model. The quantum mechanical Liouville equation for simple two-state problems describes transition rates between these states in terms of magnetic dipolar fluctuations among coupled spins. Second-order-correct solutions to the Liouville equation for selected elements of the density matrix are formulated using correlation functions and their associated spectral densities that mirror viscoelastic and dielectric resonance processes due to molecular motion. The connection and limitations between theory and experiment are compared for three types of relaxation phenomena, using bisphenol-A polycarbonate as an illustrative example. Experimental strategies are outlined to analyze the frequency –temperature correspondence of activated rate processes in rheologically simple materials that exhibit a spectrum of relaxation time constants with one activation energy.

15.3 BRIEF INTRODUCTION TO QUANTUM STATISTICAL MECHANICS The objective of the following discussion is to calculate the partition function Z of a single molecule, or an ensemble of spins, that populate several states with a distribution that is described by the “canonical ensemble.” Partition function formalism and the concept of spin temperature are employed in Section 15.5 to describe NMR relaxation via spin temperature equilibration with the lattice, or the environment. In the language of quantum mechanics, consider the following postulates: (a) Each state of a time-varying system is described by a complex time-dependent wave function C(r, t), where r represents a set of generalized spatial coordinates and t is time.

15.3 Brief Introduction to Quantum Statistical Mechanics

653

(b) C satisfies the Schro¨dinger wave equation, @C 2pi ¼ HC @t h pffiffiffiffiffiffiffi where h is Planck’s constant, i ¼ 1, and H is the Hamiltonian operator for the system, which includes contributions from kinetic and potential energies of the molecules. Complex variable analysis of the wave equation ensures that the time evolution of the eigenvectors, or eigenstates, of Hermitian operators like H exhibits reversibility. This characteristic of quantum mechanics would be missing if complex numbers were not used to quantify the wavefunction. Without complex numbers, the Schro¨dinger equation is essentially a classical diffusion equation, modified by the interaction potential between molecules. (c) The time-dependent “expectation” value of a thermodynamic observable (i.e., property) is illustrated in terms of the Hamiltonian operator, for which the eigenvalue of any state within the ensemble is given by a real number: ð C HC dr ¼ f (t) hHi ¼ all coordinate space

C is the complex conjugate of the wavefunction. The previous expression provides an estimate of the total energy of the system, whose classical thermodynamic analog is the internal energy. (d) Equilibrium thermodynamic properties, such as internal energy, are calculated from the time-dependent expectation value of the total system energy as follows: 8 t0 ¼t 9 <1 ð = hHi dt0 hHiequilibrium ¼ lim t)1: t ; t 0 ¼0

(e) Analogous to the construction of molecular orbital wavefunctions based on linear combinations of atomic orbitals, the actual wavefunction C(r, t) for a single molecule or an ensemble of spins is expressed in terms of a set of time-independent orthonormal basis functions wi (r) and complex time-dependent weighting factors Ci (t): X C(r, t) ¼ Ci (t)wi (r) stationary states i

where the summation includes all possible stationary states that are occupied by the system at equilibrium. Each wi is an eigenfunction, eigenvector,

654

Chapter 15 Molecular Dynamics via Magnetic Resonance

or eigenstate of the time-independent Hamiltonian operator with stationary-state real eigenvalues given by Ei. Hence, Hwi ¼ Ei wi For example, hydrogen-like orbitals are described in Pauling [1970], and they represent the solution to the time-independent Schro¨dinger wave equation for hydrogen-like atoms, forming a complete set of orthonormal basis functions denoted by wi above.

15.3.1

The Density Matrix

As defined above, construct the expectation value of the total energy under dynamic conditions and expand the time-dependent wavefunction C in terms of an orthonormal basis set wi. The Hamiltonian operates on the basis wavefunctions wi, not the time-dependent weighting factors Ci (t): ð

hHi ¼



C HC dr ¼

all coordinate space

¼

X ij

ð all coordinate space

ð

Ci (t)Cj (t)

( X

) Cj (t)wj (r)

j

( H

X

) Ci (t)wi (r) dr

i

wj (r)Hwi (r) dr

all coordinate space

Products of the complex time-dependent weighting coefficients in the linear expansion of C correspond to elements of the density matrix [McWeeny, 1960]:

rij (t) ¼ Ci (t)Cj (t) Since each wi is an eigenfunction of the time-independent Hamiltonian operator with eigenvalue (i.e., stationary state energy) Ei, the dynamic expectation value of the total energy reduces to ð ð X X   Ci (t)Cj (t) wj (r)Hwi (r) dr ¼ rij (t)Ei wj (r)wi (r) dr hHi ¼ ij

all coordinate space

ij

all coordinate space

Integration of products of orthonormal basis functions wi and wj over all coordinate space yields a Kronecker delta dij:  ð 1 if i ¼ j  wj (r)wi (r) dr ¼ dij ¼ 0 if i = j all coordinate space

15.4 The Ergodic Problem of Statistical Thermodynamics

655

Hence, hHi ¼

X ij

ð

rij (t)Ei

wj (r)wi (r) dr ¼

X

rij (t) Ei dij ¼

ij

all coordinate space

X

rii (t)Ei

i

Diagonal elements of the density matrix, or the density of stationary states, play an important role in calculating dynamic expectation values when orthonormal basis functions are used to construct C. At equilibrium, the total energy of a single molecule or an ensemble of spins is

hHiequilibrium

8 t0 ¼t 9 8 t0 ¼t 9 <1 ð X = X <1 ð = ¼ lim rii (t0 )Ei dt 0 ¼ Ei lim rii (t0 ) dt 0 t)1: t t)1: t ; ; i i t0 ¼0

t0 ¼0

The summation includes all possible stationary states with energy Ei that are occupied by the system. The time-independent equilibrium probability Pi that the system populates each of these states, or the occupational probability, is given by a Boltzmann distribution for the canonical ensemble. In other words, 8 t0 ¼t 9 <1 ð = rii (t 0 ) dt 0 ¼ A exp(bEi ) Pi ¼ lim t)1: t ; t0 ¼0

where A is dimensionless and b has dimensions of reciprocal energy (actually, b ¼ (kT)21). The total energy of an ensemble of spins is X X Ei Pi ¼ A Ei exp(bEi ) hHiequilibrium ¼ i

i

The previous expression states that the total energy of a system at equilibrium is obtained by weighting the energy of each available stationary state by the occupational probability of that state, where the latter is given by the Boltzmann distribution.

15.4 THE ERGODIC PROBLEM OF STATISTICAL THERMODYNAMICS A canonical ensemble represents a large number of closed systems in thermal contact with each other. Each system in the ensemble occupies stationary states with energy Ei, and Pi describes the occupational probability of each state. One postulates that ln Pi is an additive constant of the trajectories of the molecules in each system, where energy, linear momentum, and angular momentum are constants along each trajectory. Since linear and angular momentum vanish at equilibrium, occupational probabilities for

656

Chapter 15 Molecular Dynamics via Magnetic Resonance

stationary states obey the following Boltzmann distribution: ln Pi  a  bEi Pi  A exp(bEi ) where a and A are dimensionless, and b ¼ (kT)21 with dimensions of reciprocal energy. Lagrange multiplier maximization of the multiplicity of states for a simple “counting problem”, subject to constraints on the total number of systems in the ensemble with fixed total energy, yields the previous Boltzmann distribution via calculations that are analogous to those in Sections (i) 2.3 for fractional free volume, (ii) 7.7.1 for segment orientation (i.e., inverse Langevin distribution), and (iii) 12.2 for the mole fraction of x-mers via polycondensation reactions (i.e., most probable distribution). Since occupational probabilities must sum to unity when all possible states are considered, X X Pi ¼ A exp(bEi ) ¼ 1 i

i

Z¼

X

exp(bEi ) ¼

i

Pi ¼

1 A

1 exp(bEi ) Z

Hence, the partition function Z is defined as a sum of Boltzmann factors for discrete stationary states that are available to the canonical ensemble. In the high-temperature classical limit, where Z ) Z1 and bEi  1, the occupational probability for any state Pi  1/Z1.

15.5 NMR RELAXATION VIA SPIN TEMPERATURE EQUILIBRATION WITH THE LATTICE The partition function formalism described above for a system of N spins provides the methodology to evaluate the occupational probability of state i with energy Ei. The relevant expression for Pi is given by the previous equation that includes the partition function, where spin temperature is represented by the parameter bSpin. Consider the situation where all states of the ensemble of spins are coupled, to some extent, such that molecular motion within the lattice triggers fluctuations that induce transitions among these coupled states at different energies. Let Vik represent the transition rate constant, or frequency, for energy exchange when the ensemble evolves from state i to state k during NMR relaxation. From the viewpoint of energy-conserving relaxation processes, Vik is larger when the two states i and k that exchange energy are characterized by energies that are closer together. The following equation describes the time rate-of-change of the occupational probability of state i, dPi/dt, such that the coupling between state i and all other states provides the mechanism for energy exchange between these states, which undoubtedly affects Pi. The appropriate rate equation is

15.5 NMR Relaxation via Spin Temperature Equilibration with the Lattice

657

analogous to a constant-volume batch reactor mass balance in the presence of multiple first-order chemical reactions, each of which is reversible. Hence, X dPi ¼ {Vki Pk  Vik Pi } dt all states k,(k=i)

where occupational probabilities Pi assume the role of molar densities in this ratebased “master equation.” The total energy of the ensemble of spins decreases in response to lattice-induced molecular motion that triggers energy exchange among the various spin states at different energy. The spin microsystem is sufficiently small that the concept of spin temperature applies at all times. If this concept of local equilibrium is justified, then the system traverses a sequence of equilibrium states defined by different spin temperatures bSpin that decrease at longer times and approach the spin temperature of the lattice bLattice as relaxation proceeds. It is important to emphasize that bSpin is rigorously the inverse spin temperature, based on its definition. NMR relaxation due to molecular motion within the lattice causes bSpin and the bulk magnetization to decrease via Curie’s law of magnetism as equilibration with the lattice occurs. Curie’s law is described in more detail via density matrix formalism in the next section. It reveals that the equilibrium expectation value of the z-component of the bulk magnetization vector kMzl for paramagnetic materials, parallel to the static magnetic field, due to aligned spins or overpopulation of the lower-energy state (i.e., mS ¼ þ 12) relative to the higher-energy state (i.e., mS ¼  12) for a spin-onehalf nucleus when its gyromagnetic ratio is greater than zero, is proportional to bSpin and magnetic field strength. Hence, at constant field strength, the decrease in bulk magnetization during NMR relaxation is consistent with a decrease in bSpin as disorder within the spin system occurs due to misalignment, or dephasing, of individual magnetic moments. Now, it is acceptable to construct the time rate-of-change of the equilibrium expectation value of the Hamiltonian for the ensemble of spins, because the system traverses a sequence of interconnected equilibrium states in response to spin – lattice relaxation. The following analysis of NMR relaxation represents a discrete microscopic analog of mechanical and dielectric relaxation phenomena. All three relaxation processes in solid polymers represent a subset of the fluctuation – dissipation theorem, as discussed in Section 15.7. The time rate-ofchange of the total system energy during spin – lattice relaxation is adopted from the statistical thermodynamic expression for kHlequilibrium: X dbSpin d d dPi hHiequilibrium ¼ hHiequilibrium ¼ Ei dt d bSpin dt dt stationary ¼

X stationary states i

Ei

X

states i

{Vki Pk  Vik Pi }

all states

k,(k=i)

This equation is manipulated to arrive at a rather simple first-order differential equation that describes the time dependence of spin temperature bSpin. In the high-temperature

658

Chapter 15 Molecular Dynamics via Magnetic Resonance

limit, where each stationary state energy Ei  kTSpin, or bSpinEi  1, one evaluates the dependence of total system energy on temperature as follows: 8 9 d d <1 X = hHiequilibrium ¼ Ei exp(bSpin Ei ) d bSpin d bSpin :Z stationary ; states i



1 Z1

X

Ei2 exp(bSpin Ei )

stationary states i

  1 X 2 1 1 2 3  E 1  bSpin Ei þ (bSpin Ei )  (bSpin Ei ) þ    Z1 stationary i 2! 3! states i

1 X 2  E Z1 stationary i states i

Z1 represents the high-temperature limit of the partition function, which was evaluated prior to differentiation of total system energy with respect to bSpin. Now, one invokes the Principle of Microscopic Reversibility for each pair of states, i and k, such that there is no further change in occupational probabilities Pi or Pk when thermal equilibrium between the lattice and the spins in these two states is achieved at lattice temperature bLattice. This allows one to evaluate the ratio of transition frequencies for spin exchange between states i and k, similar to evaluation of the ratio of forward and backward kinetic rate constants for elementary reversible gas-phase chemical reactions in terms of the equilibrium constant based on gas-phase partial pressures. In other words, for each pair of dissimilar states i and k (i.e., k = i), the Principle of Microscopic Reversibility states that Vki {Pk }@bLattice ¼ Vik {Pi }@bLattice Vik {Pk }@bLattice ¼ ¼ exp{bLattice (Ek  Ei )}  1  bLattice (Ek  Ei ) þ    Vki {Pi }@bLattice The time dependence of NMR spin temperature during spin – lattice relaxation due to thermally induced stochastic motion of the lattice is described by

d bSpin  dt

  Pk Vik Ei Pi Vki  Pi Vki stationary X

X

states i,k

stationary states i,k

1 X 2 E  Z1 stationary i states i



 Ei Pi Vki

Pk  [1  bLattice (Ek  Ei )] Pi





1 X 2 E Z1 stationary i states i

Analogous to the evaluation of the ratio of occupational probabilities Pk/Pi at lattice temperature bLattice via the Boltzmann distribution, the system of spins is sufficiently

15.5 NMR Relaxation via Spin Temperature Equilibration with the Lattice

659

small that the concept of local equilibrium is justified and temperature bSpin is defined for all spin states traversed by the system during spin – lattice relaxation. Consequently, the ratio of occupational probabilities Pk/Pi at temperature bSpin in the previous rate equation for dbSpin/dt is described by the Boltzmann distribution. One obtains the following result: Pk ¼ exp{bSpin (Ek  Ei )}  1  bSpin (Ek  Ei ) Pi X Ei Pi Vki (bLattice  bSpin )(Ek  Ei ) d bSpin  dt

stationary states i,k



1 X 2 E Z1 stationary i states i

bSpin Ei 1

X

) (bSpin  bLattice )

Ei Vki (Ei  Ek )

stationary states i,k

Pi 1=Z1

X

Ei2

stationary states i

To reiterate, bSpin decreases as molecular motion of the lattice induces periodic fluctuations that cause aligned magnetic moments to dephase with subsequent decrease in bulk magnetization. As described by the previous rate expression, spin – lattice relaxation of NMR magnetization implies that spin temperatures approach the lattice temperature. Hence, (i) bSpin decreases toward bLattice, or (ii) TSpin increases toward TLattice. The time constant for this relaxation process is known as the spin – lattice relaxation time T1, defined by the previous equation as X Ei Vki (Ei  Ek ) d bSpin (bSpin  bLattice ) 1  ; ¼ T1 dt T1

stationary states i,k

X

Ei2

stationary states i

Straightforward manipulation of the previous expression for the spin – lattice relaxation time reveals that T1 is always positive and bSpin decreases toward bLattice as relaxation occurs, because bSpin . bLattice. This relation between bSpin and bLattice is satisfied when spin – lattice relaxation occurs in the rotating frame of reference, where the reference frame rotates at the Larmour frequency of the nuclear spins, the effective magnetic field strength in the rotating frame is approximately three orders of magnitude smaller than the static field strength in the laboratoryfixed reference frame, and the time constant for spin – lattice relaxation is known as the rotating-frame relaxation time, T1r. When the transition rate constants that connect states i and k are approximately equal (i.e., Vik  Vki), the previous rate expression for dbSpin/dt identifies spin – lattice relaxation times in either frame of reference

660

Chapter 15 Molecular Dynamics via Magnetic Resonance

as follows: X

Ei Vki (Ei  Ek ) ¼

stationary states i,k

1 X {Vki Ei (Ei  Ek ) þ Vik Ek (Ek  Ei )} 2 stationary states i,k

¼

1 X Vki (Ei  Ek )2 2 stationary states i,k

X 1 ¼ T1

X

Ei Vki (Ei  Ek )

stationary states i,k

X

¼

Ei2

Vki (Ei  Ek )2

stationary states i,k

2

stationary states i

X

Ei2

.0

stationary states i

Thought-provoking exercise: Spin-temperature equilibration revisited Reformulate the time rate-of-change of total energy during spin-lattice relaxation as the system traverses a sequence of interconnected equilibrium states, with assistance from the expression for hHiequilibrium in Section 15.3.1. For example: 8 9 X d 1< X dPi dPk = þ Ei Ek hHiequilibrium ¼ dt ; dt 2 :stationary dt stationary states i

states k

X 1 X Ei ¼ fVki Pk  Vik Pi g 2 stationary all states i

þ

X 1 X Ek fVik Pi  Vki Pk g 2 stationary all states k

¼

states k(k=i)

states i(i=k)

1 X ðEi  Ek ÞfVki Pk  Vik Pi g 2 stationary states i,k

Demonstrate that the previous equation yields a simple first-order ODE for the time dependence of spin temperature bspin. Furthermore, one obtains the same expression for the spin-lattice relaxation time T1, derived previously in this section, without invoking the approximation that transition rate constants which connect states i and k must be equal (i.e., Vik  Vki). Spin –lattice relaxation occurs at a faster rate and T1 is shorter when the transition rate constants Vki that describe molecular-motion-induced coupling between pairs of spin states at different energies are larger. As suggested by the rate expression for dbSpin/dt and the definition of spin –lattice relaxation times T1 or T1r, the assumption of local equilibrium and the identification of a spin temperature for all states traversed

15.6 Analysis of Spin–Lattice Relaxation Rates

661

by the system during spin – lattice relaxation are consistent with single exponential decay of NMR magnetization. It should be emphasized that exponentials containing bSpin, bLattice, and energy levels of various states were expanded in Taylor series and truncated after the zeroth-order or first-order term. These approximations are primarily responsible for predictions in this section that NMR relaxation is a first-order rate process described by a single exponential decay. Actual data for solid polymers reveal that at least two time constants are required for adequate descriptions of spin –lattice relaxation of NMR magnetization in either the laboratory or rotating frame of reference [Belfiore, 1982], suggesting that NMR relaxation might not be described adequately by one first-order rate process and the Taylor series expansions should retain more than the first two terms prior to truncation.

15.6 ANALYSIS OF SPIN –LATTICE RELAXATION RATES VIA TIME-DEPENDENT PERTURBATION THEORY AND THE DENSITY MATRIX 15.6.1

Development of the Liouville Equation

This development begins by constructing the expectation value for one of the components of the bulk magnetization vector M, using density matrix formalism that was introduced earlier in this chapter for an ensemble of nuclear spins that has not achieved equilibrium. During motional-induced spin– lattice relaxation, the system does not traverse a sequence of equilibrium states and the concept of spin temperature is not valid at all times [Slichter, 1978]. The primary objective is to consider the time dependence of the z-component of the bulk magnetization vector Mz, which is aligned in the direction of the static magnetic field. The orthonormal basis set fwig, upon which the actual wavefunction C(r, t) for the ensemble of nuclear spins is expanded, is not an eigenfunction of any component of the transient bulk magnetization vector. Density matrix notation for the expectation value of the z-component of M has been developed, in general, in Section 15.3.1 via replacement of kHl by kMzl . The following equation allows one to predict bulk magnetization along the static field at equilibrium and during transient spin – lattice relaxation: hMz i ¼

X ik

Ci (t)Ck (t)

ð all coordinate space

wk (r)Mz wi (r) dr ¼

X

rik (t)Mz(ki) ¼ Trace{rMz }

ik

Mz(ki) is the ki element of the Mz operator with respect to the fwig basis set and TracefrMzg sums all diagonal elements of the product of the density matrix with Mz, as defined by matrix multiplication via the summation in the previous equation. Explicit evaluation of kMzl at equilibrium in the presence of a static magnetic field yields a first-order approximation to Curie’s law of magnetism for paramagnetic materials that was mentioned in the previous section. The approximation is first-order because the exponential operator for the density matrix (i.e., r ¼ (1/Z) expf2H/kTg) is expanded in terms of the time-independent

662

Chapter 15 Molecular Dynamics via Magnetic Resonance

Hamiltonian operator H and truncated after the linear term. The temperature-independent diamagnetic (i.e., negative) contribution to the susceptibility is negligible, except at very high temperature. For example, if Sz is the z-component of spin angular momentum (i.e., þ 12 or 2 12, in units of h/2p, for spin  12 nuclei; 2S  Sz  þS, in increments of 1 unit) for an isolated spin with gyromagnetic ratio gs, and B0 is the strength of the static magnetic field along the z-axis, then expressions for the Mz operator, the static Hamiltonian Hstatic for spins in the B0 field, and the expectation value for the z-component of the bulk magnetization are given by Mz ¼ gs h Sz ; Hstatic ¼ Mz B0    kT  Estatic 1 Hstatic þ  hMz i ) Trace Mz 1  kT Z1   1 B0 Trace{Mz } þ Trace{Mz2 }  Z1 kT These expressions are based on the strong Zeeman interaction of an isolated spin with the static magnetic field in the laboratory frame of reference. Since the matrix representation of Mz at equilibrium is diagonal and traceless (i.e., zero trace) in the basis set of the static Hamiltonian (i.e., 2S  Sz  þS ), the previous expression for kMzl yields Curie’s law in the high-temperature classical limit when the zerothorder approximation for the partition function Z1 is 2S þ 1 and Trace{Mz2 } is S(S þ 1)(2S þ 1)/3. The final results are summarized below:   SX z ¼þS Estatic,Sz  exp  {1    } ¼ 2S þ 1 Z) kT Sz ¼S Sz ¼S SX z ¼þS

kT  Estatic

kT  Estatic

hMz i )

2 Sz ¼þS B0 B0 g2s h 2 X B0 gs2 h S(S þ 1) Trace{Mz2 }  {S2z } ¼ 3kT Z1 kT (2S þ 1)kT Sz ¼S SX z ¼þS Sz ¼S

{S2z } ¼ 2

SX z ¼S

{S2z } ¼ 26 S(S þ 1)(2S þ 1)

Sz ¼0

where the evaluation of the sum of S2z requires an application of factorial polynomials, as illustrated in Section 13.7. Prior to the establishment of equilibrium, the time dependence of kMzl requires knowledge of the time dependence of the density matrix, which is accessible from perturbation theory analysis of the Schro¨dinger wave equation. For example, substitution of the basis-function-expansion for wavefunction C(r, t) into the wave equation in Section 15.3 yields X Ci (t)wi (r) C(r, t) ¼ stationary states i

X @C dCi 2pi 2pi X ¼ ¼ HC ¼  wi (r) Ci (t)Hwi (r) dt @t h h stationary stationary states i

states i

15.6 Analysis of Spin–Lattice Relaxation Rates

663

where the Hamiltonian operator H for an ensemble of spins in a strong magnetic field contains static and fluctuating contributions. The time rate-of-change of complex weighting factor Ck is obtained via (i) multiplication of the previous equation by the complex conjugate of basis function wk, and (ii) integration over all coordinate space, realizing that the basis functions fwig form an orthonormal set. One obtains ð X X dCi ð dCi 2pi X ¼ wk (r)wi (r) dr ¼ dki Ci (t) wk (r)Hwi(r) dr dt dt h stationary stationary stationary states i

all coordinate space

states i

states i

all coordinate space

dCk 2pi X ¼ Ci (t)Hki h stationary dt states i

where Hki is the ki-element of the Hamiltonian operator with respect to the fwig basis set. Complex conjugation of the differential equation for dCk/dt provides the second piece of information required to construct an expression for the time rate-of-change of any element of the density matrix. The result is dCk 2pi X ¼ C  (t)Hki dt h stationary i states i

dr jk d(Cj Ck ) dC  dCj 2pi X ¼ ¼ ¼ Cj k þ Ck {Cj Ci Hki  Ck Ci Hji } dt dt dt dt h stationary states i

2pi X ¼ {r Hik  Hji rik } h stationary ji states i

where Hki ¼ Hik :

ð

 ð wk (r)Hwi (r) dr ¼ wi (r)Hwk (r) dr

because the Hamiltonian operator is Hermitian and its eigenvalues are real. In other words, matrix representations of H are invariant to (i) constructing the complex conjugate of each element, followed by (ii) interchanging the rows and columns. These operations yield the Hermitian adjoint of H, which is indistinguishable from H because the operator is Hermitian. The previous differential equation for the time dependence of the jk-element of the density matrix represents the quantum mechanical equivalent, via Ehrenfest’s theorem, of the Liouville equation in statistical mechanics for the classical density of points. Iterative approximations are required to solve the Liouville equation because elements of the density matrix appear in the time derivative on the left side, and in matrix multiplication with time-dependent elements of the Hamiltonian operator on the right side. The following analysis is based on iterative approximations for the solution of the Liouville equation that are

664

Chapter 15 Molecular Dynamics via Magnetic Resonance

second-order-correct. A two-state problem is considered (i.e., states 1 and 2, with stationary-state energies E1 and E2, respectively, that yield solutions to the following eigenvalue equation for each eigenfunction wi; Hstaticwi ¼ Eiwi). Only state 1 is populated initially at time t ¼ 0, with r11(0) ¼ 1 and r12(0) ¼ r21(0) ¼ r22(0) ¼ 0 to simplify the mathematics required. The rate at which transitions occur from state 1 to state 2, due to magnetic dipolar couplings between these states via nonzero off-diagonal elements of the time-dependent fluctuating contribution to the Hamiltonian operator H12 and H21, is obtained by calculating the ensemble average of the time rate-ofchange of r22(t) because the entire system occupies state 1 initially. For more complex systems in which several states are populated at time t ¼ 0, the actual rate of spin – lattice relaxation represents an extrapolation of the functional form of the simple two-state problem for dkr22l/dt. Detailed mathematical analysis of the multistate problem with several states initially populated is exceedingly difficult. Begin by evaluating the time derivative for the jk-element of the density matrix when j ¼ k ¼ 2 and the summation on the right side of the Liouville equation includes two stationary states, i ¼ 1 and i ¼ 2. The appropriate expression is dr22 2pi 2pi ¼ {r21 H12  H21 r12 þ r22 H22  H22 r22 } ¼ {r21 H12  H21 r12 } dt h h where all quantities on the right side of the previous equation that involve matrix multiplication between elements of r and those of the Hamiltonian operator should be evaluated at time t. Hence, it is necessary to obtain expressions for r21(t) and r12(t) by writing the Liouville equation specifically for these elements of the density matrix: dr21 2pi ¼ {r21 H11  H21 r11 þ r22 H21  H22 r21 } dt h dr12 2pi ¼ {r11 H12  H11 r12 þ r12 H22  H12 r22 } j ¼ 1, k ¼ 2: dt h j ¼ 2, k ¼ 1:

First-order-correct approximations for r21(t) and r12(t) are generated by (i) setting r11 ¼ 1 and r22 ¼ 0 on the right sides of the previous two Liouville equations and (ii) integrating from t0 ¼ 0 to t0 ¼ t, realizing that each matrix element of the Hamiltonian operator contains static and time-dependent contributions when microBrownian molecular motion of the lattice induces periodic fluctuations in dipolar interaction energies between coupled nuclear magnetic moments. The integrating factor method of solving first-order ODEs is used to analyze the Liouville equations for r21 and r12 by rearranging the previous two expressions. For example, rearrangement of the first equation for r21 yields d r21 2pi 2pi þ {H22  H11 }r21 ¼  H21 dt h h When the coefficient of the first-derivative term is unity, the integrating factor is defined by the exponential function of the integral of the coefficient of the

15.6 Analysis of Spin–Lattice Relaxation Rates

665

zeroth-derivative term on the left side of the previous equation. Hence, the Liouville equation for r21 is multiplied by the following integrating factor:   ð 2pi (H22  H11 ) dt exp h Then, both terms on the left side of the Liouville equation for r21 are combined to yield the following first-order ODE with initial condition r21 ¼ 0 at t ¼ 0:      ð ð d 2pi 2pi 2pi (H22  H11 ) dt H21 exp (H22 H11 ) dt r21 exp ¼ dt h h h 9 8 = <2pi ðt r21 (t) exp [H22 (t 000 )  H11 (t 000 )] dt000 ; : h t 000 ¼0

ðt

2pi ¼ h

t0 ¼0

8 9 0 <2p i ðt = H21 (t 0 ) exp [H22 (t 00 )  H11 (t 00 )] dt00 dt0 : h ; t 00 ¼0

It is important to realize that each matrix element of the Hamiltonian operator is time dependent, denoted by t 0 , t 00 , or t 000 in the relevant integrals of the previous equation so that one can identify the appropriate dependence on the correct time variable. When the integrating factor, or the exponential function on the left side of the previous equation, is rewritten on the right side, it is acceptable to include this function of time t within the integrals with respect to both t 0 and t 00 because the integrating factor is not a function of t0 or t00 . Hence, one obtains the following first-order-correct solution for r21(t): 2pi r21 (t) ¼  h

ðt t0 ¼0

8 0 <2pi ðt H21 (t 0 ) exp [H22 (t 00 )  H11 (t 00 )] dt00 : h t 00 ¼0



2pi h

ðt t000 ¼0

9 = [H22 (t 000 )  H11 (t 000 )] dt 000 dt0 ;

Analogous treatment of the Liouville equation for r12 proceeds as follows: d r12 2pi 2pi þ {H11  H22 }r12 ¼ H12 dt h h It is possible to identify similarities between this Liouville equation for r12 and the previous Liouville equation for r21. In other words, replacement of (i) H22 – H11 by H11 – H22, and (ii) H21 by – H12 in the first-order-correct solution for

666

Chapter 15 Molecular Dynamics via Magnetic Resonance

r21 yields the first-order-correct solution to the Liouville equation for r12. The result is 8 0 ðt <2pi ðt 2pi 0 r12 (t) ¼ H12 (t ) exp [H11 (t 00 )  H22 (t 00 )] dt00 : h h t 0 ¼0

t00 ¼0



2pi h

ðt

[H11 (t 000 )  H22 (t 000 )] dt000

t 000 ¼0

9 = ;

dt0

These first-order-correct approximations for r21(t) and r12(t) are used to generate a second-order-correct approximation for the time rate-of-change of r22(t) via the Liouville equation. In an effort to use condensed notation for the previous two firstorder-correct solutions, r21(t) and r12(t) are expressed as follows: ðt

2pi r21 (t) ¼  h

H21 (t 0 ) exp{ f (t 0 )  f (t)} dt 0

t0 ¼0

ðt

2pi r12 (t) ¼ h

H12 (t 0 ) exp{f (t 0 ) þ f (t)} dt0

t 0 ¼0

ðt

2pi f (t) ¼ h

[H22 (t 00 )  H11 (t 00 )] dt00

t 00 ¼0

The temporal rate at which state 2 is populated from state 1, via fluctuations in magnetic dipolar couplings as described by time-dependent contributions to H12 and H21, is analyzed below: d r22 2pi ¼ {r21 (t)H12 (t)  H21 (t)r12 (t)} dt h 82 3  2 < ðt 2p 4 H21 (t 0 ) exp{ f (t 0 )  f (t)} dt05H12 (t) ¼ h : 2 þ H21 (t)4

t0 ¼0

ðt t 0 ¼0

39 = H12 (t 0 ) exp{f (t 0 ) þ f (t)} dt05 ;

Matrix elements of the off-diagonal coupling terms (i.e., H12 and H21) at time t in the previous equation can be included within the integrals with respect to t 0 , a variable transformation from t 0 to t via t 0 ¼ t – t is imposed to remove t 0 from the integration, and ensemble averaging, which is equivalent to time averaging with respect to time t for systems that obey the ergodic hypothesis [Kummerer and Maassen, 2003], yield the rate at which transition occur from state 1 to state 2 for this simplified

667

15.6 Analysis of Spin–Lattice Relaxation Rates

two-state problem. Based on the definition of ensemble averaging, one obtains the following result: dhr22 i ¼ dt

8  2 < ðt 2p hH21 (t  t)H12 (t) exp{ f (t  t)  f (t)}i d t h : t¼0

þ

ðt t¼0

9 = hH21 (t)H12 (t  t) exp{f (t  t) þ f (t)}i d t ;

15.6.2 Effect of Small Time-Dependent Fluctuations on the Liouville Equation for r22 Transition frequencies or relaxation rates for the two-state problem described above must consider a Hamiltonian operator for magnetic resonance that contains a dominant time-independent contribution Hstatic due to the Zeeman interaction of Mz with the z-component of the static magnetic field B0 in the laboratory frame of reference (i.e., Hstatic ¼ 2MzB0) and a much smaller transient contribution Hfluctuation due to stochastic motion of the lattice. As a consequence of the fact that the lattice is not rigid, homonuclear or heteronuclear dipolar interaction energies between two coupled nuclei (i.e., 2 mA mB/r 3AB) exhibit time-dependence because the distance between (i.e., rAB), and relative orientation of, two magnetic dipole moments (i.e., mA mB) are affected by molecular motion. Hence, the appropriate Hamiltonian operator is





H ¼ Hstatic þ Hfluctuation where the ensemble average or time average of all matrix elements of Hfluctuation vanishes because fluctuations in the dipolar coupling between two nuclear magnetic moments occur randomly. Each state within the ensemble experiences the same Hstatic, but Hfluctuation is different. Furthermore, each wavefunction wi in the basis set is an eigenfunction of Hstatic with energy Ei, but wi is not an eigenfunction of Hfluctuation. Consequently, important matrix elements of the Hamiltonian operator (i.e., H21 and H12) at times t and t – t in the previous equation for the ensemble average of the time rate-of-change of r22 exhibit no contribution from Hstatic because the basis set fwig is orthonormal. In other words, ð ð i = j: Hij(static) ¼ wi (r)Hstatic wj (r) dr ¼ wi (r)Ej wj (r) dr ð

¼ Ej wi (r)wj (r) dr ¼ Ej dij ¼ 0 However, Hij(fluctuation) is nonzero when i= j because state 1 is coupled to state 2 via dipolar interaction energies. If there is no coupling between these two states, then all off-diagonal matrix elements of the total Hamiltonian operator vanish and state 2 will never be populated by motional-induced transitions from state 1. It is important to

668

Chapter 15 Molecular Dynamics via Magnetic Resonance

realize that, if state 1 is coupled to state 2, then all off-diagonal matrix elements of Hfluctuation are nonzero, but the time average or ensemble average of these off-diagonal matrix elements vanishes when fluctuations occur randomly. Hence, kHij(fluctuation)l vanishes, but the time average or ensemble average of products of these fluctuating matrix elements is nonzero. The time rate-of-change of kr22l reduces to 8  2 < ðt dhr22 i 2p ¼ hH21(fluctuation) (t  t)H12(fluctuation) (t) exp{ f (t  t)  f (t)}i d t dt h : t¼0

þ

ðt

9 = hH21(fluctuation) (t)H12(fluctuation) (t  t) exp{f (t  t) þ f (t)}i dt ;

t¼0

which contains ensemble averages of products of two off-diagonal matrix elements of the time-dependent Hamiltonian Hfluctuation with respect to the fwig basis set. Ensemble averaging of the function f(t) can be expressed in terms of the transition frequency v from stationary state 1 to stationary state 2: 2pi h f (t)i ¼ h

ðt

hH22 (t 00 )  H11 (t 00 )i dt 00

t00 ¼0

2pi ¼ h

ðt

hH22(static) þ H22(fluctuation) (t 00 )  H11(static)  H11(fluctuation) (t 00 )i dt00

t00 ¼0

¼

2pi (E2  E1 )t ¼ ivt h

Fluctuating contributions to H11 and H22 vanish upon averaging them over the time interval 0  t00  t. Now, the exponential function in the Liouville equation for kr22l is independent of time t (i.e., it is only a function of the time difference t), so it can be removed from the ensemble average, yielding 8  2 < ðt

dhr22 i 2p ¼ H21(fluctuation) (t  t)H12(fluctuation) (t) exp{ivt} dt h : dt t¼0

þ

ðt t¼0





H21(fluctuation) (t)H12(fluctuation) (t  t) exp{ivt} d t

9 = ;

Products of fluctuating off-diagonal elements of the time-dependent Hamiltonian operator at times t – t and t in the previous equation are described by time correlation functions. These products are the consequence of obtaining iterative approximations for dr22/dt via Liouville’s equation and truncating the result after the second-ordercorrect terms. The relevant correlation function G21(t) based on Hfluctuation depends only on the time difference t for stationary perturbations (i.e., probability distribution

15.6 Analysis of Spin–Lattice Relaxation Rates

669

functions are invariant to a shift in the time axis), as well as states 1 and 2. Within the framework of the interaction representation [Slichter, 1978; Dayie et al., 1996; Kupriyanova, 2004], G21(t) with dimensions of the square of energy is defined by ð ð   w2 (r)Hfluctuation (t  t)w1 (r) dr w1 (r)Hfluctuation (t)w2 (r) dr G21 (t) ¼ ¼

ð

w2 (r)Hfluctuation (t)w1 (r) dr

ð

w1 (r)Hfluctuation (t

þ t)w2 (r) dr

In the previous equation, integration occurs over all coordinate space. Now, the transition rate from state 1 to state 2 is given by the following classic expression when time t is much longer than the correlation times tC that parameterize G21(t): 2 3  2 ðt ðt dhr22 i 2p 4 ¼ G21 (t) exp(ivt) d t þ G21 (t) exp(ivt) dt5 h dt t¼0

 ¼  ¼

8 2 < ðt

2p h : 2p h

t¼0

G21 (t) exp(ivt) dt 

t¼0

2 ðt

Q¼0



9 = G21 (Q) exp(ivQ) dQ ; 

G21 (t) exp(ivt) d t ) limt..tc

t¼t

¼

 ðt

2p h

2 1 ð G21 (t) exp (ivt) d t 1

 2p 2 J21 (v) h

where J21(v) is the spectral density of the relevant fluctuation characterized by G21(t). For more complex systems in which several states are populated initially, timeindependent relaxation rates at t  tC, summarized by the inverse of the spin – lattice relaxation time, are expressed as a sum of spectral density functions evaluated at frequencies that depend on whether the dominant fluctuation is due to homonuclear (i.e., 1H – 1H) or heteronuclear (i.e., 1H – 13C) dipolar interaction energies. In general, correlation functions and their associated spectral densities are related by two-sided Fourier transformation, as illustrated below: J21 (v) ¼

1 ð

G21 (t) exp{ivt} dt 1

G21 (t) ¼

1 2p

1 ð

J21 (v) exp{ivt} d v 1

Regardless of the characteristic time constants tC for the relevant fluctuation, known as correlation times that parameterize G21(t), the fact that J21(v) and G21(t) form a Fourier pair reveals that the strength of the spectral density (or the relaxation strength), defined by the following integral of J21(v) over all possible frequencies

670

Chapter 15 Molecular Dynamics via Magnetic Resonance

(i.e., – 1  v  1) is not a function of tC. Hence, when t ¼ 0; 1 ð J21 (v) d v ¼ 2pG21 (t ¼ 0) 1

¼ 2p

ð

w2 (r)Hfluctuation (t)w1 (r) dr

ð



w1 (r)Hfluctuation (t)w2 (r) dr

The time-dependent Hamiltonian that describes randomly fluctuating interaction energies among dipolar coupled nuclei due to thermally activated molecular motion is a Hermitian operator, which implies that matrix elements of Hinteraction with respect to the fwig basis set are indistinguishable from its Hermitian adjoint. In other words, ð  ð w1 (r)Hfluctuation (t)w2 (r) dr ¼ w2 (r)Hfluctuation (t)w1 (r) dr 1 ð 1

ð ð  J21 (v) d v ¼ 2p w2 (r)Hfluctuation (t)w1 (r) dr w2 (r)Hfluctuation (t)w1 (r) dr * ð

2 +



¼ 2p

w2 (r)Hfluctuation (t)w1 (r) dr

. 0, = f (tC )

Hence, G21(t ¼ 0) is real and greater than zero. For a given fluctuation that is thermally activated, correlation times tC decrease at higher temperature and the corresponding spectral density extends to higher frequency, but the area under J21(v) versus v remains constant. For example, if fluctuations described by G21(t) conform to a single exponential decay for positive and negative values of t with one correlation time tC and a real pre-exponential factor, then the dimensionless correlation function, normalized by h/2p, is   j tj G21 (t) ¼ exp  tC Two-sided Fourier transformation of G21(t) must consider the regime 21  t  0 separately from 0  t  1. Detailed calculations yield the following spectral density function: 1 1   ð ð t exp(ivt) d t G21 (t) exp{ivt} dt ¼ exp  J21 (v) ¼ tC 1

þ

0

ð0 1

¼

  t exp(ivt) dt exp þ tC

    tC 1  ivtC tC 1 þ ivtC 2tC þ ¼ 1 þ ivtC 1  ivtC 1  ivtC 1 þ ivtC 1 þ v2 tC2

which reduces to J21(v ¼ 0) ¼ 2tC. Hence, J21(v) is a symmetric absorption function, and it is essentially constant at its zero-frequency value (i.e., 2tC) for

15.6 Analysis of Spin–Lattice Relaxation Rates

671

vtC  1. The slope dJ21/dv is positive for v , 0 and negative for v . 0 such that the spectral density exhibits a Lorentzian lineshape with a maximum at v ¼ 0 and inflection points given by pffiffiffi 3 d2 J21 ¼ 0; @v ¼ + 2 dv 3 tC If one “approximates” J21(v) versus v as a rectangle with height J21(v ¼ 0) and width given by the distance between the inflection points at positive and negative v, then 1 ð 1

* ð

2 +



J21 (v) dv ¼ 2pG21 (t ¼ 0) ¼ 2p w2 (r)Hfluctuation (t)w1 (r) dr

1 ð

¼ 2tC

1

dv ¼ 4tC 1 þ v2 tC2

1 ð 0

dv ¼ 4[arctan (vtC )]vv)1 ¼0 ¼ 2p 1 þ v2 tC2

pffiffiffi pffiffiffi 2 3 4 3  gCorrection J21 (v ¼ 0) ¼ g 3tC 3 Correction which reveals that G21(t ¼ 0) ¼ 1, the correction factor for the rectangular approximation is gCorrection  2.72 (i.e., essentially the base of the natural logarithm) and the strength of the spectral density (i.e., the area under the curve of J21(v) versus v) is not affected by the correlation time. This quantum description of magnetic resonance (i.e., NMR) relaxation is echoed in the analysis of stress relaxation, as prescribed by the fluctuation – dissipation theorem in Section 15.7.

15.6.3 Analysis of the Liouville Equation for Simple Two-State Problems Let’s begin with the generalized form of the Liouville equation for any element of the density matrix and explicitly evaluate this equation for all elements (i.e., r11, r12, r21, and r22) in a simple two-state system. The entire system occupies state 1 initially, such that r11 ¼ 1 and the other three elements vanish at time t ¼ 0. d r jk 2pi X ¼ {r Hik  Hji rik } dt h stationary ji states i

d r11 dt d r12 j ¼ 1, k ¼ 2; dt d r21 j ¼ 2, k ¼ 1; dt d r22 j ¼ 2, k ¼ 2; dt j ¼ 1, k ¼ 1;

2pi {r12 H21  H12 r21} h 2pi ¼ {r11 H12 þ r12 H22  H11 r12  H12 r22} h 2pi ¼ {H21 r11 þ r21 H11  H22 r21 þ r22 H21} h 2pi ¼ {H21 r12 þ r21 H12} h ¼

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Chapter 15 Molecular Dynamics via Magnetic Resonance

Make the following substitutions: r11 ¼ x1, r12 ¼ x2, r21 ¼ x3, r22 ¼ x4, such that (i) all four elements of the density matrix and (ii) all four time derivatives can be expressed as separate 4 1 column vectors. The 4 4 coefficient matrix A contains various combinations of matrix elements of the total Hamiltonian Hik (excluding the phase factor 2pi/h): 2 3 H12 0 0 H21 6 H12 H22  H11 0 H12 7 6 7 4 H21 0 H11  H22 H21 5 0 H21 H12 0 The previous set of four coupled ODEs for all elements of the density matrix can be written in condensed notation as dx/dt ¼ Ax. Diagonalization of the coefficient matrix A yields four uncoupled first-order ODEs that can be integrated rather easily. If states 1 and 2 are completely uncoupled, such that H12 ¼ H21 ¼ 0, and it is not possible to populate state 2 via motional induced transitions from state 1, then the 4 4 coefficient matrix A simplifies considerably: 2 3 0 0 0 0 6 0 H22  H11 0 07 6 7 40 0 H11  H22 0 5 0 0 0 0 with eigenvalues given by 0, 0, E1 – E2, and E2 – E1. The completely uncoupled two-state problem is described by the following set of equations for all elements of the density matrix: d r11 ¼0 dt d r12 2pi ¼ {H22  H11}r12 dt h d r21 2pi {H11  H22 }r21 ¼ h dt d r22 ¼0 dt When state 1 is completely populated and state 2 is vacant at time t ¼ 0, the previous set of four uncoupled equations exhibits a trivial solution, as expected:

r11 (t) ¼ 1 r12 (t) ¼ r12 (t ¼ 0) exp

r21 (t) ¼ r21 (t ¼ 0) exp r22 (t) ¼ 0

8 <2pi ðt : h

t0 ¼0

8 <2pi ðt : h

[H22 (t 0 )  H11 (t 0 )] dt 0

t0 ¼0

[H11 (t 0 )  H22 (t 0 )] dt 0

9 = ; 9 = ;

¼0

¼0

15.7 Classical Description of Stress Relaxation

673

When states 1 and 2 are coupled, such that H12 and H21 do not vanish, perturbation theory analysis of the rate at which state 2 is populated via thermally induced transitions from state 1 reveals that matrix elements of the total Hamiltonian operator should exhibit the following characteristics: (i) Diagonal elements must contain a static contribution that reflects the energy of each stationary state, but fluctuations are not necessary because their time average vanishes, as illustrated by the evaluation of k f(t)l in Section 15.6.2. Furthermore, H11 and H22 only appear as a difference in the ODEs for r12 and r21, so fluctuations in each of these matrix elements will cancel if they exhibit the same amplitude and frequency. (ii) Off-diagonal elements must contain a fluctuating contribution that represents the origin of motional-induced transitions between the two states, but static contributions vanish because the basis set fwig contains eigenfunctions of the static part of the total Hamiltonian operator.

15.7 CLASSICAL DESCRIPTION OF STRESS RELAXATION VIA AUTOCORRELATION OF THE END-TO-END CHAIN VECTOR AND THE FLUCTUATION– DISSIPATION THEOREM According to Callen and Welton [1951], systems exhibit dissipative characteristics if they are capable of absorbing energy in response to perturbations that fluctuate periodically in time. In statistical physics, the fluctuation– dissipation theorem states that changes or fluctuations in system behavior are dissipated as the system returns to equilibrium [Deutch and Oppenheim, 1968; Pathria, 1986; de Groot and Mazur, 1984]. For small perturbations, system response is linear, such that molecular models can be employed to yield quantitative predictions of material properties based on linear response theory. These predictions must agree with the fluctuation – dissipation theorem, which suggests that explicit relationships exist between molecular dynamics and macroscopic response in dynamic experiments. The fluctuation – dissipation theorem, which relates equilibrium fluctuations to dissipative processes and nonequilibrium properties, provides accurate predictions when external fields are weak relative to the potential energy of intermolecular interactions. According to Pathria [1986]: The most striking feature of the fluctuation–dissipation theorem is that it relates fluctuations of a physical quantity pertaining to the equilibrium state of a system to dissipative processes that occur only when systems are subjected to external forces that drive them away from equilibrium. This theorem enables one to determine nonequilibrium properties of statistical systems based on knowledge of thermal fluctuations that occur when the system exists in one of its equilibrium states. Hence, the statistical mechanics of irreversible processes reduces to the statistical mechanics of equilibrium states, provided that one can describe the time-dependent fluctuations that occur. . . . Any system that exhibits dissipative mechanisms can be described by equations which must agree with the fluctuation –dissipation theorem.

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According to de Groot and and Mazur [1984]: For systems under the influence of time-dependent excitations, the fluctuation –dissipation theorem relates correlation functions of stochastic Brownian motion to relaxation functions that contain the complex susceptibility for dissipation processes which generate entropy.

Using the nomenclature in de Groot and Mazur [1984] to summarize the essence of the fluctuation– dissipation theorem, a(t) is a fluctuating variable for some pertinent stationary random process that is autocorrelated via time averaging or ensemble averaging over the period of random fluctuations tperiod to yield

r(t) ¼ ha(t)a(t þ t)i ¼

1

tperiod ð

tperiod

a(t)a(t þ t) dt

t¼0

A complex spectral density S(v) is constructed from one-sided Fourier transformation of r(t) via the Wiener – Khinchin – Einstein theorem. Hence,

S(v) ¼ X(v)  iY(v) ¼

t )1 ð

r(t) exp(ivt) dt t ¼0

The fluctuation– dissipation theorem relates the real (i.e., X(v)) and imaginary (i.e., Y(v)) parts of S(v) to the imaginary and real parts, respectively, of a complex susceptibility k(v). The ratio of the induced response to the harmonic excitation is defined as the susceptibility. The connection between S(v) and k(v) is

kimaginary (v) 2 kBoltzmann T p v 2 kreal (v) Y(v) ¼ kBoltzmann T p v

X(v) ¼

Aside from the factor of 2 kBoltzmannT/p, the real and imaginary parts of the susceptibility are obtained from the real and imaginary parts, respectively, of ivS(v) ¼ vY(v) þ ivX(v). From the viewpoint of stress relaxation and dynamic testing of linear viscoelastic materials in the following sections, (i) a(t) represents the endto-end chain vector rETE(t), which is directly related to the microscopic stress tensor, (ii) r(t) is the stress relaxation modulus, (iii) S(v) is the complex dynamic viscosity, (iv) X(v) is the loss component of dynamic viscosity, (v) Y(v) is the storage component of dynamic viscosity, (vi) ivS(v) is the complex dynamic modulus, (vii) kreal(v) is essentially the storage modulus E 0 (v), and (viii) kimaginary(v) is essentially the loss modulus E 00 (v). The connection between de Groot and Mazur’s [1984] dynamic variables and dynamic mechanical spectroscopy is reiterated in a formal discussion of stress relaxation and the dissipation of mechanical energy to thermal energy. As illustrated in Section 15.6.2 for fluctuating magnetic dipolar interactions among coupled nuclear

15.7 Classical Description of Stress Relaxation

675

moments, molecular motion in dynamical systems is characterized by time correlation functions for the relevant fluctuation, which are typically parameterized by characteristic constants known as correlation times tC. In general, time-dependent properties like the end-to-end chain vector rETE(t) for affine deformation, which is directly related to the microscopic stress tensor and nonequilibrium retractive forces via entropy elasticity (see Sections 13.17.3 and 13.17.4), can be autocorrelated at times t and t þ t during stress relaxation experiments [Resibois and de Leener, 1977; Masubuchi et al., 2001]. If a single correlation time is sufficient to describe the effect of molecular motion on the end-to-end chain vector at times t and t þ t [Masubuchi et al., 2001], then the dimensionless autocorrelation function for stress relaxation, j (t), for t . 0, reveals that the time-dependent mean-squared displacement of a single chain in the presence of an external force field decreases in magnitude according to correlation time tC:

j (t ) ¼

  hrETE (t) rETE (t þ t)i t ¼ exp  hrETE (t) rETE (t)i tC





Brackets k l in the previous equation represent an ensemble average, over all chains, of the scalar product of the end-to-end chain vector at two different observation times that are separated by time t (i.e., where time t is not necessarily fixed). For systems that obey the ergodic hypothesis, such that occupational probabilities for stationary states follow Boltzmann statistics, time averages and ensemble averages are identical [Kummerer and Maassen, 2003]. For viscoelastic materials that require either single or multiple correlation times to describe the dissipation of fluctuations in stress relaxation experiments, the autocorrelation function j (t) for the end-to-end chain vector in the previous equation exhibits behavior that is similar to G21(t) for fluctuating dipolar interaction energies between coupled magnetic moments during NMR relaxation. Softer materials exhibit shorter correlation times which, according to the previous equation, quickly reduce any similarities between rETE at times t and t þ t, and the system is essentially “uncorrelated.” On the other hand, rigid elastic materials with negligible viscous dissipation are described by extremely long correlation times, implying that rETE is essentially the same at times t and t þ t for a highly correlated system. According to the fluctuation– dissipation theorem [Deutch and Oppenheim, 1968], two-sided Fourier transformation (i.e., 21  t  1) of the autocorrelation function for stress relaxation j (t) yields the frequency-dependent spectral density, which describes how mechanical energy associated with a jump strain during stress relaxation is distributed among, or dissipated by, the natural frequencies of a viscoelastic material. One-sided Fourier transformation (i.e, 0  t  1) of the previous equation yields a complex susceptibility function x (v), in which the real and imaginary parts of x(v) are related by the Kramers – Kronig theorem [Bird et al., 1977; de Groot and Mazur, 1984] (see Section 15.7.3). The imaginary part of ivx (v) represents the frequency-dependent absorption or dissipation function for the loss process [Aklonis and MacKnight, 1983], which maps out the characteristic frequencies of periodic fluctuations in dynamical systems associated with thermally induced molecular motion that is responsible for stress relaxation.

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Chapter 15 Molecular Dynamics via Magnetic Resonance

These relations and their connection to the complex dynamic modulus are illustrated below:

x(v) ¼

1 ð

j(t) exp(ivt) d t ¼

t ¼0

 ¼

1 ð

t ¼0

exp{t(iv þ 1=tC )} iv þ 1=tC

  t exp(ivt) d t exp  tC

t ¼0 ¼ t)1

  tC 1  ivtC 1 þ ivtC 1  ivtC

tC vtC2 i ¼ 2 1 þ (vtC ) 1 þ (vtC )2 To be consistent with the linear viscoelastic analysis in Section 10.40, the dimensionless complex dynamic modulus E (v; T ) is given by E  (v; T) ¼ ivx(v) ¼ E0 (v; T) ¼

(vtC )2 vtC þi ¼ E0 (v; T) þ iE00 (v; T) 1 þ (vtC )2 1 þ (vtC )2

(vtC )2 vtC ; E 00 (v; T) ¼ 1 þ (vtC )2 1 þ (vtC )2

where E0 (v; T ) and E00 (v; T ) represent dimensionless storage and loss moduli, respectively, for a dynamical system whose molecular mobility causes stress relaxation that can be modeled by an exponential decay with one correlation time tC. According to the analysis in Section 10.41, the connection between (i) energy dissipation during each cycle of oscillation in forced-vibration dynamic mechanical spectroscopy and (ii) Fourier transformation of the stress relaxation autocorrelation function, or the stress relaxation modulus ER(t) [Williams, 1971; Aklonis and MacKnight, 1983] is given by, Energy dissipation ¼ Volume

ððð

s dg ¼

System Volume

¼

vg20

2pð=v

2pð=v

t¼0

  @g s dt @t v

{E 0 (v, T) sin(vt) þ E 00 (v, T) cos(vt)} cos(vt) dt

t¼0

8 1 9 < ð = ¼ pg02 E00 (v, T) ¼ pg02 Im iv ER (t) exp(ivt) dt : ; t¼0

where Im selects the imaginary part of the complex dynamic modulus. This represents the essence of the fluctuation– dissipation theorem in statistical physics. In other words, spontaneous micro-Brownian molecular motions for a stationary process,

15.7 Classical Description of Stress Relaxation

677

where the probability distribution functions are invariant to a shift in the time axis, produce fluctuations in the end-to-end chain vector that can be autocorrelated to yield the dimensionless stress relaxation modulus [Masubuchi et al., 2001], and subsequently Fourier transformed to reveal how these stochastic perturbations are distributed among the natural frequencies of viscoelastic materials. Dissipation of mechanical energy into thermal energy occurs during stress relaxation when resonance conditions are satisfied (i.e., at maxima in E00 where material response (i.e., natural) frequencies match the frequencies of external perturbations that fluctuate periodically with time). The responses described above by j (t), x(v), and E (v; T ) are consistent with the dynamic mechanical behavior of the Maxwell model for linear viscoelasticity [Williams, 1971; Aklonis and MacKnight, 1983]. The loss modulus E00 (v; T ) exhibits a maximum at the temperature-dependent characteristic frequency for thermal motion, identified by 1/tC.

15.7.1

Single-Chain Dynamics via the Rouse Model

As a straightforward but rather complex extension of elastic dumbbell models for flexible polymers [Bird et al., 1977], high-molecular-weight chains are described as N þ 1 beads connected by N massless springs. Each bead represents a subchain that contains several repeat units, and the springs follow Hooke’s law of elasticity, which implies that they are infinitely extendable and exhibit linear elastic response. It is possible to construct expressions for elastic forces within the springs that describe finitely extendable nonlinear elastic response [Bird et al., 1977]. Let ri be the position vector from the origin of a stationary reference frame to the center of the ith mass point, and ai represents the segment vector from the ith bead to the (i þ 1)st bead along the connecting spring. Hence, the connector vectors are ai ¼ riþ1 2 ri. A balance between elastic and hydrodynamic forces on each bead yields the concept of linearized relaxation modes, or eigenmodes, with terminal relaxation times that scale linearly with N 2, or the square of molecular weight. One constructs elastic retractive forces that arise from entropy elasticity, where the subchain between the ith bead and the (i þ 1)st bead exhibits Gaussian statistics and the internal energy U of these ideal subchains is conformation independent. In agreement with these approximations, the relation between entropy and the Gaussian probability density distribution function is adopted from Section 13.17.2:

 



3 (riþ1  ri ) (riþ1  ri ) 3 ai ai ¼ S0  k 2 Si)iþ1 ðjriþ1  ri jÞ  S0  k 2 h(riþ1  ri ) (riþ1  ri )i 2 hai i Excluding chain ends, beads i ¼ 2 to i ¼ N experience elastic forces from both neighboring beads, and these forces act along connecting springs toward the neighboring bead. The total elastic force on the ith bead is (2  i  N ) f i ¼ T{rS}i)iþ1 þ T{rS}i1)i 

3kT {ai  ai1 } ha2i i

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Chapter 15 Molecular Dynamics via Magnetic Resonance

A microhydrodynamic force balance on the ith bead neglects inertial, gravity, and pressure forces. Furthermore, the time average of the random Brownian force vanishes, which circumvents the analysis of stochastic partial differential equations. Rubinstein and Colby [2003] analyze the Rouse model with a delta-function correlated random Brownian force, which is essentially uncorrelated. The discussion below equates the total elastic force to the Stokes’ law drag force in a Newtonian medium, where the latter is given by the product of a friction coefficient zi, which decreases at higher temperature, and the creeping flow velocity, @ri/@t, of the bead with respect to a stationary reference frame. Actually, the sum of these two forces on each bead (i.e., hydrodynamic and elastic) vanishes, where the hydrodynamic drag force acts in the direction of the relative motion of the Newtonian fluid with respect to the solid bead. For very high-molecular-weight chains (i.e., N  1), essentially all beads are identical, and the force balance on the ith bead is extrapolated to the continuous limit, where h is a dimensionless spatial coordinate along the connecting springs, or the chain backbone, and the root-mean-square segment length of each Gaussian subchain, ha2i i1=2 , represents the characteristic length for spatial scaling:

zi

continuous @ri 3kT 3kT 3kT @ 2 ri ¼ f i  2 {ai  ai1 } ¼ 2 {riþ1  2ri þ ri1 } ) limit @t hai i hai i ha2i i @ h2 N..1

The previous eigenvalue equation describes the time dependence of the position vector to the ith bead, and the continuum analog of the discrete finite-difference expression for elastic retractive forces yields an unsteady state diffusion equation for each position vector ri with a Stokes – Einstein diffusion coefficient given by 3kT/zi. Analogous to reptation, diffusion is considered in only one coordinate direction, that is, along the connector springs in the chain backbone. The factor of 3kT in the diffusion coefficient and correlation times for stress relaxation originates from the elastic force in the connector springs via an application of Boltzmann’s entropy equation to Gaussian chains. Similarity between this unsteady state diffusion equation for ri and the tube model for reptation is obtained by invoking continuity of position and velocity between adjacent beads when h ¼ 0 and h ¼ 1 to achieve a smooth transition in ri and @ri/@t from one chain end (i.e., i ¼ 1) to the other (i.e., i ¼ N þ 1). The end-to-end vector for the entire chain is defined by rETE (t) ¼ rNþ1 (t)  r1 (t) ¼

N X

ai (t)

i¼1

The overall strategy involves (i) solving the continuum analog of the microhydrodynamic force balance, or unsteady state diffusion equation, for ri (t), (ii) constructing the time dependence of the end-to-end chain vector via the previous equation, (iii) autocorrelating the end-to-end chain vector via time averaging, which, according to the ergodic hypothesis, is equivalent to ensemble averaging, and (iv) identifying molecular parameters that yield correlation times for dimensionless stress relaxation moduli.

15.7 Classical Description of Stress Relaxation

679

15.7.2 Solution of the Unsteady State Diffusion Equation for the Position Vector to the ith Bead in High-Molecular-Weight Chains Let’s focus on the time dependence of the magnitude ri of the position vector ri via the continuum analog of the microhydrodynamic force balance on the ith bead in a single chain: @ri 3kT @ 2 ri ¼ @t zi ha2i i @ h2 Postulate a separation-of-variables solution, such that ri (t, h) ¼ Fi (t)Ki (h), substitute this expression for ri (t, h) into the unsteady state diffusion equation, and divide by ri (t, h). One obtains the following ordinary differential equations for Fi (t) and Ki (h) in terms of the separation constant li . 0: 1 dFi 3kT 1 d 2 Ki ¼ ¼ l i Fi (t) dt zi ha2i i Ki (h) dh2 dFi ¼ li Fi (t) dt d 2 Ki li z ha2 i ¼  i i K i (h ) 2 dh 3kT A negative separation constant yields solutions for Fi (t) and Ki (h) that reveal transient behavior of ri which decays exponentially at longer times, and periodic spatial behavior with respect to the dimensions of the ith subchain (i.e., 0  h  1). Dimensionless eigenvalues mi depend on specific details of the boundary conditions for ri at h ¼ 0, 1. For example, Fi (t)  exp{li t} Ki (h) ¼ Bi cos(mi h) þ Ci sin(mi h)

m2i ¼

li zi ha2i i 3kT

The smallest nonzero eigenvalue that forces K1(h ¼ 1) to vanish (i.e., m1 ¼ p) yields the longest (i.e., terminal) relaxation time 1/l1 for each subchain, with a Stokes – Einstein diffusion coefficient Dsegment given by the ratio of the mean-square segment length of the subchain to its relaxation time. Hence, terminal relaxation

Dith segment ¼

li ha2i i

3m2 kT 3p 2 kT time ¼ i ) m1 ¼p zi (T) zi (T)

Polymer chains exhibit rigid elastic response on time scales that are shorter than the relaxation time of the subchain, 1/li. Of significant interest here is that correlation times tC for the end-to-end vector of an ideal chain are comparable to (i.e., two-fold larger than) the longest Rouse relaxation time for the entire chain [Rubinstein and Colby, 2003], which scales as the product of N 2 and the terminal relaxation time for each subchain. Hence, the dynamic response of each bead in high-molecular-weight

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Chapter 15 Molecular Dynamics via Magnetic Resonance

linear chains with negligible hydrodynamic interaction, except for a creeping-flow drag force on the bead under investigation, identifies the functional form of characteristic time constants that describe stress relaxation. Finally, the dimensionless autocorrelation function for stress relaxation is   hrETE (t) rETE (t þ t)i t ¼ exp  j( t) ¼ hrETE (t) rETE (t)i tC





terminal relaxation

N2 N 2 ha2i izi (T) time tC  ) m1 ¼p li 3p 2 kT where the temperature dependence of individual friction coefficients for each bead in a generalized bead – spring chain is described by an Arrhenius function such that zi decreases at higher temperature. The temperature dependence of 1/tC has the same functional form as kinetic rate constants for gas-phase reactions via transition-state, or absolute-rate, theory. One of the primary deficiencies of the Rouse model, developed in 1953, is that it exhibits no dependence of viscosity on shear rate for steady state shear flows [Bird et al., 1977].

15.7.3 Kramers –Kronig Theorem via Fourier Transformation Let’s begin with the integral form of Maxwell’s viscoelastic constitutive equation in Section 10.11. The time-dependent stress s (t) experienced by uncrosslinked materials in the linear regime, which agrees with the Boltzmann superposition integral in Section 10.35, contains the stress relaxation modulus ER(t2Q) and the rate-ofstrain history @ g/@Q: 

Q¼t ð

s (t) ¼

ER (t  Q)

 @g dQ @Q

Q)1

Dynamic properties of linear viscoelastic materials can be predicted from the previous equation when the excitation (i.e., forcing) function is harmonic at frequency v. The strain history is expressed using complex variable notation (including the dc level of strain gdc required to ensure that solids are always under tension), g (Q) ¼ gdc þ g0 expfivQg, and the previous integral expression for complex time-dependent stress s (t; v) becomes (after multiplication by expfivtg and its complex conjugate): 

s (t; v) ¼ ivg0

Q¼t ð

ER (t  Q) exp{ivQ} dQ

Q)1

¼ ivg0 exp{ivt}

Q¼t ð

Q)1

ER (t  Q) exp{iv(t  Q)} dQ

15.7 Classical Description of Stress Relaxation

681

Identify the factor outside of the previous integral as the complex rate of strain at time t, change variables in the integrand from t 2 Q to s (i.e., s ¼ t 2 Q), and construct the complex dynamic viscosity h (v) as the ratio of complex stress to the complex rate of strain. s)1 ð s (t; v) s (t; v) ¼ ER (s) exp{ivs} ds h (v) ¼    ¼ @ g (t) ivg0 exp{ivt} s¼0 @t v 

¼ hLoss (v)  ihStorage (v) Fourier sine relation: hStorage (v) ¼

s)1 ð

ER (s) sin(vs) ds s¼0

Fourier cosine relation: hLoss (v) ¼

s)1 ð

ER (s) cos(vs) ds s¼0

One should recognize that the stress relaxation modulus ER(s) and the complex dynamic viscosity h (v) are a Fourier pair via one-sided Fourier transformation of ER(s) to yield h (v). If one inverts the Fourier cosine relation between hloss(v) and ER(s), then for s 0, 2 Inverted Fourier cosine relation: ER (s) ¼ p

v)1 ð

hLoss (v) cos(vs) dv v¼0

Analogously, inversion of the Fourier sine relation between hStorage(v) and ER(s) yields, for s . 0, 2 Inverted Fourier sine relation: ER (s) ¼ p

v)1 ð

hStorage (v) sin(vs) dv v¼0

Now, (i) substitute the inverted Fourier cosine relation between ER(s) and hLoss(v) into the Fourier sine relation between hStorage(v) and ER(s), (ii) use the method of generalized functions, as described by Lighthill [1964], and (iii) exercise caution in treating the behavior of ER(s) at s ¼ 0, to obtain the following relation between the storage and loss components of the complex dynamic viscosity (better known as one of the famous Kramers – Kronig equations):

hStorage (v) ¼

s)1 ð s¼0

ER (s) sin(vs) ds )

2v p

x)1 ð x¼0

hLoss (x) dx v2  x2

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Chapter 15 Molecular Dynamics via Magnetic Resonance

Let’s illustrate the sequence of logical steps required to compute hStorage(v), or v21E0 (v), when the previous Kramers – Kronig equation is applied to the one-timeconstant (i.e., t) Maxwell model for linear viscoelastic response. The nonnormalized distribution of relaxation times is given by the product of the static modulus E of the spring and a Dirac delta function spiked at l ¼ t. Application of the Boltzmann superposition principle to dynamic properties in Section 10.38 reveals that this relaxation time distribution function is consistent with the following loss components of the complex dynamic viscosity and the complex dynamic modulus: 1 E hLoss (v) ¼ E 00 (v) ¼ v v

1 ð

d(l  t)

l¼0

vl Et dl ¼ 2 1 þ (vl) 1 þ (vt)2

The Kramers – Kronig theorem uses the previous expression for hLoss to calculate the frequency dependence of hStorage, because the real and imaginary components of the ratio of induced stress to the harmonic forcing function (i.e., strain or rate of strain) cannot be postulated independently. There exists an equivalent relation between the real and imaginary components of the susceptibility in magnetic resonance. It is necessary to evaluate the following integral for hStorage: 1 2v hStorage (v) ¼ E 0 (v) ¼ v p

x )1 ð

hLoss (x) 2 dx ¼ E vt v2  x2 p

x¼0

x )1 ð

1 dx (v2  x2 )(1 þ x2 t2 )

x¼0

When the single relaxation time t is real and positive, the MapleTM library evaluates the definite integral and provides expressions for the storage components of the complex dynamic viscosity and the complex dynamic modulus that agree with the Maxwell model:

2 E (v) ¼ vhStorage (v) ¼ E v2 t p 0

x )1 ð

1 (vt)2 dx ¼ E (v2  x2 )(1 þ x2 t2 ) 1 þ (vt)2

x¼0

15.7.4 Kramers –Kronig Theorem via Complex Variables and Cauchy’s Integral One-sided Fourier transformation of ER(s) to yield h (v) represents the starting point for complex variable analysis of the dynamic viscosity h (z), which is constructed to be a function of the complex variable z ¼ v þ iz, where v identifies the real axis and z represents the imaginary axis of the complex plane. For example,

15.7 Classical Description of Stress Relaxation

683

in terms of the real (i.e., F) and imaginary (i.e., C) parts of the complex dynamic viscosity, 

h (z) ¼ F(v, z ) þ iC(v, z) ¼

s)1 ð

ER (s) exp{i(v þ iz)s} ds

s¼0

F(v, z ) ¼

s)1 ð

ER (s) exp(zs) cos(vs) ds s¼0

C(v, z ) ¼ 

s)1 ð

ER (s) exp(zs) sin(vs) ds s¼0

The previous integral expressions for F(v, z ) and C(v, z ) are well behaved and do not diverge when z  0. Hence, the Cauchy – Riemann equations are satisfied on the entire real axis and the lower half of the imaginary axis (i.e., z  0 or p  w  2p, where w is the polar angle in the complex plane such that w ¼ 0 coincides with the positive real axis), as required for analytic functions of the complex variable z. In other words, the following derivative tests (i.e., Cauchy – Riemann equations) are satisfied by the real and imaginary parts of the complex dynamic viscosity to ensure that the first derivative of h with respect to the complex variable z yields an expression that does not depend on the multitude of paths by which one approaches an arbitrary point z in the complex plane. Hence, h (z) is analytic because @F @C ¼ ¼ @v @z

s)1 ð

sER (s) exp(zs) sin(vs) ds s¼0

@F @C ¼ ¼ @z @v

s)1 ð

sER (s) exp(zs) cos(vs) ds s¼0

The Kramers – Kronig theorem considers the following contour integral around closed path C in the lower half of the complex plane (i.e., where h (z) is analytic) that excludes z ¼ v. Hence, this contour integral vanishes via the classic Cauchy integral theorem, based on an application of Stokes’ theorem to line integrals of exact differentials, because the integrand exhibits no poles, or singularities, within the region bounded by path C which excludes z ¼ v: þ  h (z) dz ¼ 0 Czv The result of this analysis of the Kramers – Kronig theorem, or the principle of causality as described by de Groot and Mazur [1984], yields relations between the absorption F(v) ¼ hLoss(v) ¼ E 00 (v)/v and the dispersion 2C(v) ¼ hStorage(v) ¼ E 0 (v)/v, where the storage and loss moduli are given by E 0 (v) and E 00 (v),

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Chapter 15 Molecular Dynamics via Magnetic Resonance

respectively. Notice that the absorption function for the irreversible dissipation process is given by the real part of the complex dynamic viscosity and the imaginary part of the complex dynamic modulus E (v), because E (v) ¼ E 0 (v) þ iE 00 (v), whereas n (v) ¼ hLoss(v) – ihStorage(v) (see Section 10.39).

15.8 COMPARISONS AMONG NMR, MECHANICAL, AND DIELECTRIC RELAXATION VIA MOLECULAR MOTION IN POLYMERIC MATERIALS: ACTIVATED RATE PROCESSES Three examples of dynamic processes have been discussed in this chapter: (i) NMR relaxation via spin-temperature equilibration with the lattice when systems traverse an interconnected sequence of equilibrium states, (ii) second-order perturbation theory analysis of the Liouville equation for specific elements of the density matrix in rather simple nonequilibrium systems when thermally induced stochastic motion of the lattice causes magnetic dipolar interaction energies to fluctuate periodically with time, and (iii) autocorrelation and Fourier transformation of the end-to-end chain vector, yielding stress relaxation moduli and complex dynamic viscosities that satisfy the Kramers – Kronig theorem and agree with the fluctuation – dissipation theorem. Relaxation is a consequence of chain mobility in high-molecular-weight solids, but the phenomenon occurs in nonpolymeric materials as well. Viscoelastic time constants that characterize the decay of stress are analogous to correlation times tC that parameterize NMR correlation functions like G21(t), discussed in Section 15.6.2. The fluctuation– dissipation theorem is applicable to magnetic resonance and viscoelastic relaxation. This was demonstrated by the identification of correlation functions for (i) dipolar interaction energies between coupled nuclear magnetic moments and (ii) the end-to-end chain vector. Fourier transformation of the relevant correlation functions yields spectral densities, susceptibilities, and complex dynamic viscosities that describe absorption and energy dissipation when perturbations fluctuate periodically in time. Arrhenius-like apparent activation energies can be evaluated for these dynamic processes if thermally induced molecular motion produces periodic fluctuations in material response.

15.8.1

NMR Relaxation

The slope of ln tC versus reciprocal absolute temperature yields an apparent activation energy for localized cooperative chain mobility that is responsible for NMR relaxation. Experimental test frequencies reside in the kilohertz or megahertz regimes for spin– lattice relaxation in the rotating-frame or laboratory-frame of reference, respectively. However, this methodology to determine activation energies suffers from the requirement that models for the appropriate correlation functions are necessary to calculate correlation times tC from spin – lattice relaxation times T1, where the latter are obtained from direct measurement of the first-order decay of NMR signal intensities. To circumvent this difficulty, model-independent strategies are based on

15.8 Comparisons Among NMR, Mechanical, and Dielectric Relaxation

685

the measurement of spin – lattice relaxation times for nuclear spins of interest (i.e., typically 1H or 13C) when the Larmour frequency vNMR of the spin system in the laboratory frame (i.e., MHz regime for T1) or rotating frame (i.e., kHz regime for T1r) of reference remains constant. If temperature-dependent measurements of spin –lattice relaxation can be obtained at constant frequency vNMR, then one correlates frequency – temperature combinations that produce the most efficient relaxation characterized by minima in either T1 or T1r. Hence, the slope of ln vNMR versus reciprocal absolute temperature yields activation energies for high-frequency cooperative motion, probed by magnetic resonance relaxation, that is localized in the chain backbone or the side groups. Carbon-13 NMR relaxation experiments focus on the decay of magnetization that exhibits chemical specificity, because chemically or morphologically inequivalent 13C nuclei are characterized by isotropic chemical shift interactions, which depend strongly on their electronic environment or the molecule’s crystallographic symmetry. In other words, when relaxation experiments are performed via high-resolution carbon-13 NMR spectra of polymeric solids, requiring sophisticated techniques such as (i) magic-angle sample spinning, (ii) 1H– 13C cross-polarization, and (iii) heteronuclear 1H – 13C dipolar decoupling, one focuses on measurements of T1 and T1r via the pulse sequences in Figure 15.1, for specific 13C signals that are unique to particular carbon sites in the chemical structure of the repeat unit. Hence, NMR activation energies provide a glimpse of chain dynamics in the vicinity of the carbon-13 nucleus whose signal intensities at known chemical shift are analyzed to extract spin – lattice relaxation times, from which ln vNMR versus reciprocal absolute temperature is correlated when relaxation is most efficient. In some cases, cooperative chain dynamics smear the high-resolution nature of NMR relaxation experiments, because motion of a particular functional moiety in the chain backbone or side group cannot occur without cooperation from its neighbors. The 13C NMR experiment is simplified considerably when synthetic design can be employed for isotopic enrichment of one carbon-13 site in the polymer’s repeat unit, but this strategy does not filter interference from cooperative chain dynamics. For example, the polycarbonate of bisphenol-A (i.e., BPAPC, tradename LexanTM ) can be synthesized with 13C-enriched phosgene (i.e., Cl213CvO) via condensation polymerization: CH3 C

O

13

O

C

CH3 O

yielding a glassy amorphous polymer in which the carbonyl carbon dominates the solid state carbon-13 NMR spectrum. Isotopic labeling at the carbonyl carbon site is advantageous because this 13C resonance at 152 ppm is not resolved in highresolution natural abundance spectra (i.e., without 13C enrichment), due to overlap with stronger signals from both substituted aromatic carbons at 148 – 149 ppm (i.e., relative to tetramethylsilane). Hence, macromolecular synthesis using monomers that are 13C-enriched at specific sites in the repeat unit allows one to probe chain dynamics in the vicinity of the isotopic label. When magic-angle sample spinning

686

Chapter 15 Molecular Dynamics via Magnetic Resonance (a) Carbon-13 T1 pulse sequence 90° 1H

Highpower decoupling

200 MHz

Magnetic field Magnetization

Cross polarization

90°

Spin–lattice relaxation t

13C

90°

Data acquisition

Free induction decay

50 MHz

Time

(b) Carbon-13 T1p pulse sequence

Magnetic field Magnetization

1H 200 MHz

High-power decoupling Cross polarization

Spin–lattice relaxation t

Data acquisition Free induction decay

13C

50 MHz

Time

Figure 15.1 Radiofrequency pulse sequences for 1H and 13C to measure carbon-13 spin–lattice relaxation in (a) the laboratory frame of reference T1 and (b) the rotating reference frame T1r via highresolution carbon-13 NMR spectra of solids, obtained using 1H – 13C cross-polarization and heteronuclear dipolar decoupling. Relaxation of 13C magnetization occurs during the time interval denoted by t, followed by data acquisition in the presence of high-power dipolar decoupling.

is employed, the high-resolution carbon-13 NMR spectrum of polycarbonate, 13 C-enriched at the carbonyl carbon, reveals a relatively narrow signal with an isotropic chemical shift of 152 ppm and a full-width at half-height of 5 ppm (Fig. 15.2). In the absence of magic-angle spinning, each orientation of the carbonyl group with respect to the static magnetic field exhibits a unique absorption due to the anisotropic nature of the chemical shielding tensor, where one-third of the Trace of the second-rank chemical shielding tensor, or the average of the three principal components on the main diagonal from upper left to lower right, yields an isotropic chemical shift of 152 ppm for the carbonyl carbon in BPAPC. The carbonyl powder pattern for BPAPC reveals that the principal components of the chemical shielding tensor are approximately 100 ppm, 127 ppm, and 230 ppm, as illustrated in Figure 15.3.

15.8 Comparisons Among NMR, Mechanical, and Dielectric Relaxation

687

5 ppm

200

160

120 80 Chemical shift (ppm)

40

0

High-resolution magic-angle spinning 13C solid state NMR spectrum of bisphenol-A polycarbonate, carbon-13 enriched at the carbonyl position via isotopically labeled phosgene. The signal of interest exhibits a chemical shift of 152 ppm, relative to tetramethylsilane. The poly(oxymethylene) sample container for magic-angle spinning (i.e., tradename DelrinTM ) is responsible for the strong signal near 90 ppm.

Figure 15.2

Analogous synthetic design that introduces deuterium enrichment at specific positions in a polymer’s repeat unit allows one to probe molecular dynamics in the vicinity of the label via solid state 2H NMR, where lineshapes (i.e., Pake doublets) are dominated by electric quadrupole interactions. By invoking two-site models for ring flips that allow one to simulate deuterium NMR lineshapes at various temperatures, rotational correlation times and flip angles for aromatic ring motion in polycarbonate have been investigated by deuterium NMR spectroscopy below the glass transition on samples that contain deuterated ring carbons.

300

250

200 150 100 Chemical shift (ppm)

50

0

Figure 15.3 Non-spinning 13C solid state NMR spectrum of bisphenol-A polycarbonate, carbon-13 enriched at the carbonyl position via isotopically labeled phosgene. This powder pattern reveals orientationdependent interactions of the anisotropic chemical shielding tensor of the carbonyl carbon nucleus with the static magnetic field in the laboratory reference frame. 1H– 13C cross-polarization is employed to generate carbon-13 signal intensity, and heteronuclear dipolar decoupling eliminates interactions between 1 H and 13C during data acquisition.

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Chapter 15 Molecular Dynamics via Magnetic Resonance

15.8.2

Mechanical Relaxation

Similar strategies have been employed in dynamic mechanical testing of viscoelastic materials, where one correlates frequency – temperature combinations that correspond to maxima in the loss modulus E00 . Since this methodology does not require detailed models for the distribution of viscoelastic relaxation times, as described in Appendix C of Chapter 10, one measures the temperature dependence of E00 at constant frequency vmechanical when a thin sample is subjected to tensile strain given by

g(t) ¼ gdc þ g0 sin(vmechanical t) These oscillatory perturbations exhibit a low-frequency limit of approximately 0.1 Hz that depends on instrumental stability and a high-frequency limit between 5 and 10 Hz due to sample inertia. Single-time-constant viscoelastic models in Chapter 10 reveal that loss moduli E00 exhibit maxima at higher temperatures when experiments are performed at higher perturbation frequencies vmechanical, because viscoelastic relaxation times decrease at higher temperature. Hence, one focuses on a particular viscoelastic loss process in which the slope of ln vmechanical versus inverse absolute temperature at maximum E00 yields an apparent activation energy for the dynamic process. When viscoelastic relaxation requires cooperative motion from a larger number of intramolecular and/or intermolecular segments that must undergo conformational rearrangements in harmony, relaxation times and the corresponding activation energies increase. However, unlike high-resolution NMR relaxation measurements that are unique to particular carbon-13 sites in the polymeric repeat unit, due to the chemically or morphologically specific nature of the isotropic chemical shift interaction under magic-angle spinning conditions, mechanical relaxation lacks specificity. It seems reasonable to attribute viscoelastic loss and energy dissipation to bulky chain segments that contribute significantly to the principal moments of inertia about a set of coordinate axes whose origin coincides with the segment’s center of mass. This empirical hypothesis can be verified by analyzing dynamic mechanical loss spectra for a related series of polymers whose repeat units are modified slightly via chemical substitution. In other words, if one calculates significantly different activation energies from ln vmechanical versus reciprocal absolute temperature corresponding to viscoelastic loss maxima when the aromatic ring in glassy polycarbonate is functionalized with a bulky tert-butyl substituent that hinders its ability to rotate or oscillate about an axis in the plane of the ring, CH3 C

O

CH3

C

O

O

functionalized CH3 C H3C H3C

CH3 C

CH3

O

C O

O

15.8 Comparisons Among NMR, Mechanical, and Dielectric Relaxation

689

then mechanical relaxation can be attributed to ring mobility and its hindrance due to functionalization.

15.8.3

Dielectric Relaxation

These measurements of the phase-angle difference between oscillatory current and voltage are extremely sensitive to polar functional groups in viscoelastic materials. Polymers are modeled via the electrical analog of the Maxwell or Voigt element and null impedance matching in two separate channels of a lock-in amplifier yields dielectric parameters and the viscoelastic loss tangent. Hence, dielectric spectroscopy of polycarbonate (i.e., BPAPC) (Fig. 15.4) provides a site-specific probe of carbonate mobility that is complementary to carbon-13 solid state NMR relaxation measurements when the carbonyl carbon in the repeat unit is isotopically enriched. Experimentally, thin disks with a thickness of approximately 1 – 2 mm are sandwiched between electrodes in a parallel-plate-capacitor configuration and the tangent of the phase-angle difference (see Section 10.9) between current and voltage under ac steady state conditions is monitored as a function of temperature and frequency (i.e., vdielectric). Now, it is possible to extend the transition map (i.e., temperature – frequency combinations that correspond to maximum dielectric loss) into the ultraslow frequency regime (i.e., 1024 Hz), as well as providing a bridge between the upper limit of mechanical relaxations (i.e., 10 Hz) and the lower limit of magnetic resonance relaxation in the rotating frame of reference (i.e., 20– 50 kHz). The Arrhenius graph in Figure 15.5 focuses on the broad dielectric relaxation (i.e., b)

Tg 10–2

tan de

b

10

A

–3

B

C

A: 102 Hz B: 103 Hz C: 104 Hz 10–4 –150

Figure 15.4

–100

–50 0 50 Temperature (°C)

100

150

Dynamic dielectric spectroscopy of bisphenol-A polycarbonate in the mid-kilohertz regime, illustrating viscoelastic loss at the glass transition and at the sub-Tg low-temperature b-process. Frequency–temperature combinations that yield loss maxima for the b-process are correlated for activated rate processes via the Arrhenius graph in Figure 15.5.

690

Chapter 15 Molecular Dynamics via Magnetic Resonance Temperature (°C) –51

–56

–60

–65

–69

–73

–77

–81

Frequency (Hz)

104

103

102 4.5

4.6

4.7

4.9 5.0 4.8 1 3 –1 – × 10 (K ) T

5.1

5.2

Figure 15.5 Frequency–temperature transition map for the low-temperature b-loss process in bisphenol-A polycarbonate via dynamic dielectric spectroscopy, yielding an Arrhenius activation energy of 13 kcal/mol, as summarized in the data analysis in Section 15.8.4.

process in BPAPC that occurs more than 200 8C below its glass transition temperature for probe frequencies between 102 Hz and 104 Hz. This localized motion that involves cooperative reorganization of the carbonate group is characterized by an activation energy of 13 kcal/mol (i.e., via the product of 2Rgas and the slope of ln vdielectric versus 1/T, where Rgas is the gas constant) for rheologically simple materials that exhibit a continuous spectrum of viscoelastic time constants because the Arrhenius pre-exponential factor t1 is different for each sub-relaxation process, but each activation energy is the same.

15.8.4 Temperature – Frequency Combinations that Correspond to Maximum Dielectric Loss for the b Process in BPAPC Below the Glass Transition Temperature, with an Activation Energy of 13 kcal/mol vdielectric (Hz) 102 103 104

T (88 C) 282 267 253

15.9 Activation Energies for the Aging Process in Bisphenol-A Polycarbonate

691

 3   4  10 Hz 10 Hz 1:987(cal=mol-K) ln 102 Hz 102 Hz   13 kcal=mol 1 1 1 1   67 8C þ 273 82 8C þ 273 53 8C þ 273 82 8C þ 273

1:987(cal=mol-K) ln

Extrapolation of the previous transition map for the b-process in BPAPC to ambient temperature (i.e., 298 K) reveals that this localized sub-Tg motion is described by correlation frequencies vcorrelation (i.e., inverse correlation times) in the megahertz regime (i.e., 30 MHz);   vcorrelation (25 8C) 1:987(cal=mol-K) ln 102 Hz  13 kcal=mol 1 1  25 8C þ 273 82 8C þ 273 Hence, solid state carbon-13 NMR spin – lattice relaxation (i.e., 13C T1) experiments in the laboratory frame of reference at ambient temperature are sensitive to localized chain dynamics of BPAPC below the glass transition temperature. It is reasonable to correlate polycarbonate’s sub-Tg dielectric relaxation process, for probe frequencies between 102 Hz and 104 Hz, with ambient-temperature carbon-13 NMR spin – lattice relaxation measurements. Further justification for this empirical correlation is obtained by introducing a 13C isotopic label at the carbonyl carbon in BPAPC and comparing (i) spin – lattice relaxation data for specific orientations of the carbonyl group relative to the static magnetic field, via the carbonyl powder pattern without magic-angle sample spinning (see Fig. 15.3), with (ii) dielectric relaxation of the sub-Tg loss process for the same isotopically labeled polymer in the vicinity of 103 Hz. With assistance from all three types of relaxation experiments, one can construct a “snapshot” of the multitude of molecular dynamic processes in polymeric materials that encompass approximately 10 –12 decades in frequency.

15.9 ACTIVATION ENERGIES FOR THE AGING PROCESS IN BISPHENOL-A POLYCARBONATE (COURTESY OF THE ACS DIVISION OF POLYMER CHEMISTRY DISCUSSION LIST) Question: In the design of aging tests for nonfilled standard bisphenol-A polycarbonate, it is desirable to use an Arrhenius-type formula to simulate the temperature dependence of the rate of degradation. What is the magnitude of the activation energy that describes this degradation process? RESPONSE #1: When materials are aged in humid environments at various temperatures, one must be sure that the aging processes are essentially identical, except for an acceleration in the rate of humid aging at higher temperature. To determine if this statement is obeyed, one could perform fatigue experiments at several temperatures and measure the number of cycles required for failure versus temperature at constant frequency. If there is no discontinuity in a graph of logfcycles-to-failureg versus logf1/T(absolute)g, then Arrhenius analysis is valid, with an activation energy for humid aging given by the product of the gas constant R and the slope of the log –log graph. It is implicitly assumed here that the combined rate of moisture absorption

692

Chapter 15 Molecular Dynamics via Magnetic Resonance

and degradation, measured by cyclic fatigue, will increase at higher temperature. Hence, fewer cycles-to-failure are required when humid aging occurs at higher temperatures, below Tg. RESPONSE #2: Mercier et al. [1965] have tabulated WLF shift factors for BPAPC, from which it might be possible to extract some equivalent information for mechanical activation energies (see Section 10.27). However, that might not necessarily help in the determination of activation energies for chemical rate processes. RESPONSE #3: Activation energies depend on the environment (air versus nitrogen, especially). There does not seem to be much useful information on this topic for polycarbonate in the Polymer Handbook. Previous experiments were performed on laboratory-grade polycarbonate that was unstabilized. Activation energies for degradation should be affected by the “degree of stabilization.” Thermogravimetric analysis might be useful to obtain the desired information. RESPONSE #4: It is necessary to specify the process for which an activation energy is required. Activation energies are specific to a reaction, or transformation. For example, thermal degradation of polycarbonate would be different from chemical degradation in strong base. The former is a thermal activation, the latter is a chemical reaction between PC and a base, each having different activation energies. Experiments have been performed on adhesive bonds in humid environments using accelerated aging at elevated temperatures to obtain an activation energy or time required for failure. RESPONSE #5: The Arrhenius equation is often embraced as the universal temperaturedependent formula for most rate processes. Whereas the equation quantifies the dependence of reaction rate constants on temperature, its application as a predictive model is questionable. It is important to remember that rate constants represent proportionality constants between reaction rates and reactant molar densities. Unfortunately, reactions taking place in many applications are not well defined, because reactant molar densities might be difficult to measure, and the presence of catalysts (metals) most likely induce significant reductions in known activation energies. Useful information on this topic might be available in the research literature; specifically, publications by Gillen and Clough at Sandia Laboratories in Albuquerque, New Mexico. RESPONSE #6: Consult a 1979 thesis from the Department of Materials Science at the University of Cambridge on aging studies of polycarbonate, specifically quenched versus annealed PC. The use of an apparent activation energy of 150 kcal/mol for PC is problematic, because it is greater than carbon –carbon bond energies. This occurs because the glass transition in PC requires a large volume for backbone rearrangement via significant cooperative chain mobility. RESPONSE #7: There are several processes that can be related to polycarbonate degradation. Physical aging that results in loss of ductility is due to densification of the polymer or annealing. This depends on the nature of the end groups and might not be universal for all PCs. The activation energy for physical aging appears to be 35 –37 kcal/mol [Bair et al., 1981]. The rate of hydrolysis is highly dependent on the concentration of phenolic end groups and the presence of acidic or basic impurities. Most commercial PCs are completely endcapped, so the initial phenolic content should be low. However, most commercial PCs also contain some phosphite processing stabilizers, which produce acidic species that catalyze hydrolysis. Therefore, the rate and activation energy for hydrolysis vary considerably. Data from Zinbo and Golovoy [1992] reveal an activation energy of 18.5 kcal/mol for molecular weight degradation due to “humidity aging.” Bair et al. [1981] report an activation energy of 19 kcal/mol for BPA formation upon hydrolysis. A third process is generation of yellow color upon thermal aging. An activation

15.10 Complex Impedance Analysis of Dielectric Relaxation Measurements

693

energy of 18 kcal/mol for yellowing during the thermal aging process in a controlled temperature chamber can be found in Polymer Durability [Clough et al., 1996]. Finally, the activation energy for photodegradation is reported to be 4–5 kcal/mol [Pickett and Gardner 2001; Pickett et al., 2008].

15.10 COMPLEX IMPEDANCE ANALYSIS OF DIELECTRIC RELAXATION MEASUREMENTS VIA ELECTRICAL ANALOGS OF THE MAXWELL AND VOIGT MODELS OF LINEAR VISCOELASTIC RESPONSE 15.10.1

Maxwell Model

Consider a parallel arrangement of resistor R and capacitor C that is connected to a ) ffiffiffiffiffiffi ¼V voltage source oscillating at frequency v. Hence, V(t; vp ffi 0 exp( jvt), where V0 is the peak-to-peak amplitude of the oscillations and j ¼ 1. Both circuit elements experience the same voltage drop and the currents are additive. This is completely analogous to the fact that the series configuration of a spring and dashpot experiences the same stress in the Maxwell model and strain rates are additive. The impedances of interest in the RC circuit are (i) R for the resistor (i.e., real) and (ii) ( jvC)21 for the capacitor (i.e., imaginary). The equivalent complex impedance Z for this parallel assembly of R and C is obtained via the following inverse sums: 1 1 1 ¼ þ ¼ G þ jvC ¼ Z R ( jvC)1 tan Q ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G2 þ (vC)2 exp( jQ)

vC ¼ vRC G

where G is the conductance (i.e., real) of the material. The overall objectives are to (i) measure R and C via null impedance matching as a function of temperature and frequency when viscoelastic materials are placed between two parallel-plate electrodes and (ii) calculate the complex dielectric constant (i.e., 10 2 j100 ). There are at least two approaches to calculate the induced current. Application of Ohm’s law (i.e., V ¼ ZI) yields I¼

V ¼ V0 {G þ jvC} exp( jvt) ¼ V0 Z

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G2 þ (vC)2 exp{ j(vt þ Q)}

which identifies Q as the phase angle difference between current and voltage. Since dynamic dielectric experiments provide direct measurements of Q, the viscoelastic loss tangent tan d is calculated from the loss angle d, where d ¼ p/2 – Q. If the circuit is purely capacitive, dissipating no power, then resistance R ) 1 because the parallel branch that contains the resistor is actually an open circuit, Q ¼ p/2, and current leads voltage by 908. It is possible to define a complex capacitance C for the parallel

694

Chapter 15 Molecular Dynamics via Magnetic Resonance

combination of R and C as follows: C Cvacuum

¼ 10  j100

where Cvacuum is the free-space capacitance of a parallel-plate assembly of two electrodes separated by thickness d, and the real and imaginary parts of the complex dielectric constant are given by 10 and 100 , respectively. One solves Laplace’s equation in one dimension to evaluate the linear electric potential V and the constant electric field E between the capacitor plates. Then, in the absence of a dielectric material q ¼ 10EA between the electrodes, Cvacuum is given by the magnitude of the charge Ð stored on either plate, via simple application of Gauss’ law (i.e., E . dS ¼ EA ¼ q/10), divided by the electric potential difference across the plates, DV ¼ Ed. Hence, Cvacuum ¼ 10 A/d, where A is the area of each plate and 10 is the permittivity of free space. When a dielectric material with static dielectric constant 1 is placed between the parallel plates of the capacitor, the capacitance C is augmented by the dielectric constant of the material (i.e., C ¼ 1Cvacuum ¼ 10 A1/d ). One reexpresses the charge – voltage relation for a capacitor (i.e., q ¼ CV) using complex capacitance C for the RC circuit, and differentiates q ¼ CV with respect to time t to evaluate the induced current. For example, I¼

dq dV ¼ C ¼ Cvacuum {10  j100 }jvV0 exp( jvt) dt dt

¼ vCvacuum V0 {100 þ j10 } exp( jvt) Current passing through resistor R, which dissipates electrical energy, is in-phase with oscillatory voltage V and contains the imaginary part of the complex dielectric constant 100 . In contrast, current passing through capacitor C, which stores electrical energy without power dissipation, is completely out-of-phase with voltage V and scales linearly with the real part of the complex dielectric constant 10 . Comparison of the previous two expressions for complex current yields relations for linear response of the electrical analog of the Maxwell model during dynamic experiments, better known as dielectric relaxation or dynamic dielectric spectroscopy of viscoelastic materials: 10 ¼

C Cvacuum

; 100 ¼

1 100 1 ; tan d ¼ 0 ¼ vRCvacuum 1 vRC

Since the real part of the complex dielectric constant 10 scales linearly with capacitance C, and C is analogous to the static compliance J ¼ 1/E of an ideal spring, one predicts that the temperature and frequency dependencies of (i) 10 via dielectric spectroscopy and (ii) the storage compliance J0 via dynamic mechanical testing are analogous. Both 10 and J0 increase at higher temperature or lower frequency, characteristic of liquid-like response. Purely capacitive (i.e., elastic) materials that have open circuits for the resistance branch (i.e., R ) 1) exhibit no dielectric loss because both the imaginary part of the complex dielectric constant 100 and the loss tangent tan d vanish. The dielectric spectra of bisphenol-A polycarbonate in Figure 15.4 were generated via temperature-dependent measurements of R and C at probe frequencies between 102

15.10 Complex Impedance Analysis of Dielectric Relaxation Measurements

695

and 104 Hz. When polymers are modeled via the electrical analog of the Maxwell element and null impedance matching in two separate channels of a lock-in amplifier yields the corresponding values of R and C at frequency v and temperature T, dielectric parameters and the loss tangent are calculated via the previous set of equations.

15.10.2

Voigt Model

The dielectric properties of viscoelastic materials that are sandwiched between two parallel-plate electrodes can be obtained from null impedance matching of resistance R0 and capacitance C0 when these circuit elements are assembled in series, such that they experience the same current and the voltage drops are additive. A different set of equations is required to calculate the dielectric properties based on bridge measurements of R0 and C0 that can be related to those (i.e., R and C) for the Maxwell model. Now, the equivalent complex impedance Z is given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j 1 0 ¼ R02 þ 2 02 exp ( jQ0 ) Z¼R  vC 0 vC 1 tan Q0 ¼  0 0 vR C Ohm’s law and the concept of complex capacitance C via the complex dielectric constant 10 – j100 yield the following expressions for induced current in response to a harmonic voltage forcing function V(t; v) ¼ V0 exp( jvt): I¼

V V0 exp( jvt) vC 0 V0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp{ j(vt  Q0 )} ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 þ v2 R02 C 02 1 R02 þ 2 02 exp( jQ0 ) vC

¼ vCvacuum V0 {100 þ j10 } exp( jvt) Comparison of the real and imaginary parts of the previous expressions for induced current provides appropriate relations to calculate the dielectric properties of viscoelastic materials that conform to the Voigt model. One obtains 10 ¼

C0 C0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (sin Q0 ) ¼ Cvacuum (1 þ v2 R02 C 02 ) Cvacuum 1 þ v2 R02 C 02

100 ¼

C0 vR0 C02 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos Q0 ¼ Cvacuum (1 þ v2 R02 C 02 ) Cvacuum 1 þ v2 R02 C 02

tan d ¼

100 ¼ vR0 C 0 10

Now, perfectly elastic materials that do not dissipate mechanical or electrical energy are modeled by an RC circuit with R0 ¼ 0 via the Voigt element. These materials exhibit the following dielectric properties: Q0 ¼ 2p/2, 10 ¼ C0 /Cvacuum, 100 ¼ 0, tan d ¼ 0, d ¼ 0. Once again, current leads voltage by 908 for nondisspative purely capacitive circuits (i.e., as expected based on ELI the ICE man). In principle, one

696

Chapter 15 Molecular Dynamics via Magnetic Resonance

should obtain the same model-independent dielectric properties of a viscoelastic material via (i) measurements of R and C together with the appropriate relations that are unique to the Maxwell model in the previous subsection, or (ii) measurements of R0 and C0 together with the previous set of equations in this section for the Voigt model. In other words, null impedance balancing based on (i) R and C in parallel versus (ii) R0 and C0 in series should not yield different results for the temperature and frequency dependence of the complex dielectric constant 10 – j100 when the same viscoelastic material, subjected to a harmonic voltage source, is analyzed by two different impedance bridges. Hence, it is reasonable to equate the corresponding dielectric properties from both models and relate R, R0 , C, and C0 at the same frequencyv. The results are 10 ¼ 100 ¼

C Cvacuum

¼

C0 Cvacuum (1 þ v2 R02 C 02 )

1 vR0 C 02 ¼ vRCvacuum Cvacuum (1 þ v2 R02 C 02 )

1 ¼ vR0 C 0 vRC C0 1 C¼ ; RC ¼ 2 0 0 2 1 þ tan d v RC

tan d ¼

15.11 THERMALLY STIMULATED DISCHARGE CURRENTS IN POLARIZED DIELECTRIC MATERIALS These dc experiments, which contain charging, storage, and discharging intervals, closely parallel the ac steady state dielectric loss experiments described in the previous two sections. The primary objective is to correlate depolarization currents with thermally stimulated molecular motion of electric dipoles. Polar polymers can be charged in the molten state by subjecting them to dc electric fields on the order of 30 kV/cm for at least 1 hour. Polarization due to electronic and ionic displacements within atoms and molecules occurs almost instantaneously when external electric fields are applied. However, the polarization of primary interest, due to permanent electric dipoles, is rather sluggish because this polarization is hindered by internal frictional forces. Response times during the charging interval decrease significantly when polar materials are heated above the highest thermal transition temperature (i.e., Tg or Tm) prior to the establishment of an external dc electric field. It is possible to create electrets (i.e., dielectric materials that are permanently charged in the absence of externally applied electric fields) by (i) cooling well below the glass transition temperature, (ii) removing the external field, and (iii) creating an open circuit. The field – temperature treatment is summarized below: †

Heat amorphous materials from ambient to well above their glass transition temperature in the absence of any external field.

15.11 Thermally Stimulated Discharge Currents in Polarized Dielectric Materials

697

†

Apply a dc electric field on the order of 30 kV/cm in step-function format for at least 1 hour (i.e., tformation – tinitial ) at temperature Tformation and do not remove the field until the temperature is well below Tg. Vacuum deposition of silver or gold on both surfaces of disk-shaped samples (i.e., metallization) will minimize imperfect sample – electrode contact that could inject ions into the sample from the breakdown of air gaps at the sample – electrode interface.

†

Cool the material, at approximately the same rate as the heating segment, back to ambient or subambient temperature, depending on Tg, and remove the dc field at time tremoval when the temperature is Tstorage. After a sub-Tg storage interval, tstorage – tremoval, at temperature Tstorage , Tg in the absence of the field where the electret is open-circuited, increase the temperature at a specified heating rate that is not related to the heating or cooling rate that was employed during the charging segment.

†

†

Short-circuit the electret and measure the discharge current as a function of increasing temperature, beginning at Tstorage.

Randomized reorientation of permanent electric dipoles in electrets represents the most important and dominant component of the discharge current. However, charge concentration gradients provide a diffusional contribution to the current density that produces nonuniform polarization.

15.11.1

Charging Process

A simple model invokes the Debye equation for a single relaxation process [van Turnhout, 1971] when dielectric materials contain no excess charge carriers that might exist as a consequence of inefficient metallization. Dipole polarization density P(t), with dimensions of charge-cm/cm3, is spatially averaged over a macroscopic control volume that encompasses all nonuniform contributions to the current density. In the presence of dc electric field E, P(t) satisfies [de Groot and Mazur, 1984] dP 1 1 P ¼ 10 {1S  11} E þ tdipole (T) dt tdipole (T) P(@t ¼ tinitial ) ¼ 0 where tdipole is a temperature-dependent dipole relaxation time of the viscoelastic medium, 10 is the permittivity of free space (i.e., vacuum), and 1S and 11 are the static and high-frequency dielectric constants of the material, respectively. The dipolar relaxation strength is identified as 1S – 11, which also appears as a parameter in ac steady state dielectric relaxation experiments. If these dielectric constants are not strong functions of temperature and can be treated as constants, then the solution of the previous equation during the charging interval must account for isothermal formation of the electret at temperature Tformation during time interval tformation – tinitial, as well as any charging that occurs during the cooling process in the presence of the field when the temperature decreases from Tformation to T at cooling rate rcooling

698

Chapter 15 Molecular Dynamics via Magnetic Resonance

(i.e., dT/dt ¼2rcooling). One obtains the following result via application of the integrating factor method for first-order ODE’s: 8 t 93 < ð = 1 0 5 dt P(t; T) ¼ 10 {1S  11}E 41  exp  : tdipole (T 0 ) ; 2

tinitial

8 t ð < formation 1 4 dt 0 ¼ 10 {1S  11}E 1  exp  : tdipole (Tformation ) 2

tinitial

ðt



tformation

"

93 = 0 5 dt tdipole (T 0 ) ; 1

(

¼ 10 {1S  11}E 1  exp 



1

(tformation  tinitial ) tdipole (Tformation )

Tformation ð

1

tdipole (T 0 )

rcooling

)# dT

0

T



  (tformation  tinitial )  10 {1S  11}E 1  exp  tdipole (Tformation ) Since dipole relaxation times increase dramatically as one cools electrets from the molten state at temperature Tformation to temperatures below Tg, the slight increase in polarization that occurs during cooling has been neglected relative to poling in the molten state. The Debye equation predicts linear increases in polarization during the charging process at higher dc electric fields and dipolar relaxation strength, which is consistent with the fact that polar materials with larger static dielectric constants 1S can store more charge. If one expresses the temperature dependence of dipole relaxation times in Arrhenius fashion with pre-exponential factor t1, then 

 Eactivation tdipole (Tformation ) ¼ t1 exp RTformation   1 P(tformation ; Tformation ) ¼ 10 {1S  11 }E 1  exp  (tformation  tinitial ) t1   Eactivation exp  RTformation It is possible to approach the ultimate charge density in dielectric materials, 10f1S – 11gE, by poling them in the presence of the dc electric field for longer times, tformation 2tinitial, and at higher temperatures, Tformation, in the molten state, such that tformation 2tinitial  tdipole.

15.11 Thermally Stimulated Discharge Currents in Polarized Dielectric Materials

15.11.2

699

Discharge Process

The Debye equation for dielectric materials that exhibit a single relaxation process reduces to the following homogeneous first-order ordinary differential equation for charge density after the externally applied electric field is removed and the electret is short-circuited to measure the depolarization current, beginning at time tstorage when the temperature is Tstorage. The nonisothermal discharge process is described by dP 1 P¼0 þ dt tdipole (T)

8 9 > > < ðt = 1 0 P(t; T) ¼ P(tstorage ; Tstorage ) exp  dt > > tdipole (T 0 ) ; : tstorage

9 8 > > ðT = < 1 1 0 dT ¼ P(tstorage ; Tstorage ) exp  > > tdipole (T 0 ) ; : rheating Tstorage

where time and temperature during measurement of the depolarization current are related by the heating rate, dT/dt ¼ rheating. It is necessary to consider three separate time intervals that account for †

isothermal poling in the molten state from time tinitial to tformation at temperature Tformation,

†

nonisothermal formation of the electret via cooling from time tformation at temperature Tformation in the molten state to time tremoval at temperature Tstorage below the glass transition temperature, and

†

isothermal decay of polarization at temperature Tstorage below Tg from time tremoval to tstorage in the absence of the external electric field when the electret is open-circuited,

to evaluate dipole polarization P(tstorage; Tstorage) at the beginning of the discharge process. The first two time intervals in the presence of the external field have been analyzed in the previous subsection to predict the accumulated charge density after time interval tremoval – tinitial. Hence, " ( (tformation  tinitial ) P(tremoval ; Tstorage ) ¼ 10 {1S  11 }E 1  exp  tdipole (Tformation ) 

1

Tformation ð

1

tdipole (T)

rcooling

)# dT

Tstorage

Now that the electret has been formed and this polarization is essentially “frozen-in” at temperature Tstorage below the glass transition, upon removal of the external dc electric

700

Chapter 15 Molecular Dynamics via Magnetic Resonance

field at time tremoval, the Debye equation predicts single exponential decay of P(tremoval; Tstorage) isothermally at temperature Tstorage from time tremoval to tstorage. The result is   (tstorage  tremoval ) P(tstorage ; Tstorage ) ¼ P(tremoval ; Tstorage ) exp  tdipole (Tstorage ) Finally, charge density during nonisothermal depolarization is described by the following equation as temperature increases from Tstorage to T at heating rate rheating: 9 8 > > ðT = < 1 1 0 dT P(t; T) ¼ P(tstorage ; Tstorage ) exp  0 > > tdipole (T ) ; : rheating Tstorage

9 8 > > ðT = < 1 1 0 ¼ kE10 {1S  11 } exp  dT > > tdipole (T 0 ) ; : rheating Tstorage

The filling state of the electret is characterized by the parameter k, which represents a ratio of the cumulative charge density just prior to nonisothermal discharge, P(tstorage; Tstorage), to the maximum attainable (i.e., ultimate) polarization of dielectric materials in the presence of dc electric fields, 10f1S – 11gE. Results from this subsection yield the following expression for k: 8 93 2 Tformation > > ð < (t = 1 1 6 7 formation  tinitial )  dT 5 k ¼ 41  exp  > tdipole (T) > rcooling : tdipole (Tformation ) ; Tstorage

  (tstorage  tremoval ) exp  tdipole (Tstorage ) which approaches unity when dielectric materials are (i) poled for long times in the molten state at temperatures that are sufficiently high enough to allow reorientation of permanent electric dipoles with negligible resistance, and (ii) stored at temperatures well below Tg where the external field is removed and the electret is open-circuited.

15.11.3 Depolarization Currents and Activation Energies for Rheologically Simple Materials Upon short-circuiting the electret, one heats the sample at a constant rate rheating and measures the current density jdischarge that is released as a function of time or temperature. The relevant equation for jdischarge, based on the Debye model with one dipole reorganizational time constant, is   dP 1 {P(t)}discharge ¼ jdischarge ¼ tdipole (T) dt discharge

15.11 Thermally Stimulated Discharge Currents in Polarized Dielectric Materials

701

Interestingly enough, the cumulative charge density recovered from the electret during thermally stimulated depolarization is nearly independent of heating and cooling rates, because any additional polarization that develops in the presence of the external field during cooling from temperature Tformation to Tstorage is typically negligible. This claim is verified by the miniscule effect of the factor that contains rcooling on the “filling state” parameter k. Activation energies for dipole reorientational motion during thermally stimulated depolarization can be predicted via identification of the temperature Tmaximum at which discharge current thermograms exhibit maxima. Each thermally activated motional process that exhibits a unique activation energy should yield a separate maximum in jdischarge versus temperature. The analysis proceeds as follows at Tmaximum: " #     {P(t)}discharge d tdipole djdischarge 1 dP ¼  þ ¼0 2 dT tdipole dT discharge dT tdipole Tmaximum Tmaximum

Temperature dependence of dipole polarization during discharge requires knowledge of the heating rate and the Debye equation:     dP dP dt 1 ¼ ¼ {P(t)}discharge dT discharge dt discharge dT rheating tdipole The effect of temperature on dipole reorganizational time constants introduces Arrhenius activation energies that characterize thermally activated motion of dielectric materials: dtdipole Eactivation ¼ tdipole dT RT 2 A combination of the previous three equations, together with experimental identification of the temperature Tmaximum at which discharge current thermograms exhibit maxima, yields a nonlinear expression to predict activation energies for dipole reorganization during thermally stimulated depolarization: " # {P(t)}discharge Eactivation 1 {P(t)}discharge  ¼0 2 tdipole RT 2 rheating tdipole Tmaximum   Eactivation Eactivation ¼1 t1 exp rheating 2 RTmaximum RTmaximum This equation is valid for each relaxation time in the continuous spectrum if a distribution of reorganizational time constants is required for adequate description of the dynamic process, because a linear superposition of many subrelaxation processes is invoked to simulate the time dependence of dipole polarization during discharge via a modified Debye equation. However, dielectric materials must be rheologically simple before one can employ the principle of time – temperature superposition to correlate depolarization charge densities, fP(t; T )gdischarge, at different

702

Chapter 15 Molecular Dynamics via Magnetic Resonance

temperatures. Rheologically simple materials exhibit a continuous spectrum of viscoelastic time constants because the Arrhenius pre-exponential factor t1 is different for each subrelaxation process, but each activation energy is the same. Hence, each dipole relaxation time tdipole,i in the spectrum has the same temperature dependence (i.e., d ln tdipole,i/dT ). If each subrelaxation process in the spectrum has a different activation energy, or a different pre-exponential factor and activation energy, then the viscoelastic material is not rheologically simple and the principle of time – temperature superposition is not applicable. One obvious difficulty with the use of thermally stimulated depolarization experiments to predict activation energies for dipole reorganizational motion stems from the appearance of the Arrhenius pre-exponential factor t1 in the nonlinear equation required to predict Eactivation. If dipole reorganizational time constants tdipole were known at Tmaximum, then Eactivation ¼

2 RTmaximum rheating tdipole (@Tmaximum )

but these relaxation times are not readily available and experimental evaluation of tdipole is difficult because depolarization occurs nonisothermally. Hence, it is possible to probe the chain dynamics of polar polymers via depolarization current thermograms, but activation energies might be too difficult to evaluate using the methodology described above. Empirically, it should be feasible to correlate (i) dielectric relaxation measurements for specific viscoelastic loss processes under ac steady state conditions as a function of temperature and oscillation frequency with (ii) depolarization current thermograms at various heating rates.

15.12

SUMMARY

The unifying theme of motional-induced relaxation across a broad spectrum of physical chemistry is discussed quantitatively within the context of the Liouville equation for elements of the density matrix, the fluctuation – dissipation theorem, and experimental results from dynamic dielectric spectroscopy for the glassy polymer, bisphenol-A polycarbonate. Correlation functions for the (i) end-to-end chain vector during stress relaxation and (ii) magnetic dipolar fluctuations among coupled nuclei yield Lorentzian spectral densities that agree with the Maxwell model when a single correlation time is sufficient to describe molecular motion. The analogous theoretical treatment of dielectric relaxation that parallels the discussion in this chapter has been developed by Bo¨ttcher and Bordewijk [1952], which is consistent with the fluctuation – dissipation theorem. For example, direct one-sided Fourier transformation of the pulse response function, which can be obtained from the step response via time differentiation, yields the frequency dependence of the complex dielectric constant for molecular dynamic analysis of viscoelastic materials using dielectric spectroscopy. As a consequence of sample inertia, the practical aspects of performing these transient viscoelastic experiments suggest that stress relaxation via step response measurements

References

703

is tractable, whereas the mechanical pulse response experiment has limitations at very short times. Hence, the frequency response of viscoelastic materials can be obtained from Fourier transformation of (i) the pulse response via dielectric spectroscopy, or (ii) the step response from stress relaxation. Localized motion in the main chain or side group of amorphous polymers can be influenced by chemical functionalization of the repeat unit and detected by the analytical technique that is most sensitive to the dynamic behavior of the material.

REFERENCES AKLONIS JJ, MACKNIGHT WJ. Introduction to Polymer Viscoelasticity. Wiley-Interscience, Hoboken, NJ, 1983, pp. 27–29, 143–146. BAIR HE, FALCONE DR, HELLMAN MY, JOHNSON GE, KELLEHER PG. Hydrolysis of polycarbonate to yield bisphenol A. Journal of Applied Polymer Science 26(6):1777–1786 (1981). BELFIORE LA. Molecular Dynamics of Polycarbonate-Diluent Systems, PhD thesis. University of WisconsinMadison, 1982. BIRD RB, ARMSTRONG RC, HASSAGER O, CURTISS CF. Dynamics of Polymeric Liquids, Volume 1: Fluid Mechanics, p. 303; Volume 2: Kinetic Theory, Chaps 10–12. Wiley, Hoboken, NJ, 1977. BO¨TTCHER CFJ, BORDEWIJK P. Theory of Electric Polarization. Elsevier, New York, 1952, pp. 523– 524. CALLEN HB, WELTON TA. Irreversibility and generalized noise. Physical Review 83(1):34–40 (1951). CLOUGH RL, BILLINGHAM NC, GILLEN KT. Polymer Durability, ACS Advances in Chemistry Series Vol. 249. American Chemical Society, Washington DC, 1996, pp. 59–76. DAYIE KT, WAGNER G, LEFEVRE JF. Theory and practice of nuclear spin relaxation in proteins. Annual Reviews of Physical Chemistry 47:243–282 (1996). DE GROOT SR, MAZUR P. Non-Equilibrium Thermodynamics. Dover, New York, 1984, Chap. 8 and pp. 143– 148, 153, 155, 400. DEUTCH JM, OPPENHEIM I. Time correlation functions in nuclear magnetic resonance, in Advances in Magnetic Resonance, Volume 3, Waugh JS, editor. Academic Press, New York, 1968, pp. 58–63. KUMMERER B, MAASSEN H. An ergodic theorem for quantum-counting processes. Journal of Physics A, Mathematical and General 36(8):2155–2161 (2003). KUPRIYANOVA GS. Nuclear magnetic relaxation of spin-12 scalars coupled to quadrupolar nuclei in the presence of cross-correlation effects. Applied Magnetic Resonance 26(3):283–305 (2004). LIGHTHILL MJ. Introduction to Fourier Analysis and Generalised Functions. Cambridge University Press, Cambridge, UK, 1964. MASUBUCHI Y, TAKIMOTO JI, KOYAMA K, IANNIRUBERTO G, MARRUCCI G. Brownian simulations of a network of reptating primitive chains. Journal of Chemical Physics 115(9):4387– 4394 (2001), Eqs (16)– (18). MCWEENY R. Some recent advances in density matrix theory. Reviews of Modern Physics 32(2):335–369 (1960). MERCIER JP, AKLONIS JJ, LITT M, TOBOLSKY AV. Viscoelastic behaviour of the polycarbonate of bisphenol A. Journal of Applied Polymer Science 9(2):447–459 (1965). PATHRIA RK. Statistical Mechanics. Pergamon, Oxford, UK, 1986, pp. 474–477. PAULING L. General Chemistry. Freeman, San Francisco, 1970, Appendix V. PICKETT JE, GARDNER MM. Effect of environmental variables on the weathering of engineering thermoplastics. Polymer Preprints 42(1):423– 426 (2001). PICKETT JE, GIBSON DA, GARDNER MM, RICE ST. Effects of temperature on the weathering of engineering thermoplastics. Polymer Degradation and Stability 93(3):684– 691 (2008); Effects of irradiation conditions on the weathering of engineering thermoplastics. Polymer Degradation and Stability 93(8):1597–1606 (2008). RESIBOIS M, DE LEENER M. Classical Kinetic Theory of Fluids. Wiley, Hoboken, NJ, 1977, pp. 295–305. RUBINSTEIN M, COLBY RH. Polymer Physics. Oxford University Press, New York, 2003, pp. 312, 358 –360.

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Chapter 15 Molecular Dynamics via Magnetic Resonance

SLICHTER CP. Principles of Magnetic Resonance, 2nd edition. Springer-Verlag, New York, 1978, Chaps 2 and 5. VAN TURNHOUT J. Thermally stimulated discharge currents in polymeric electrets. Polymer Journal 2:173– 191(1971). WILLIAMS DJ. Polymer Science and Engineering. Prentice-Hall, Englewood Cliffs, NJ, 1971, pp. 303–311. ZINBO M, GOLOVOY A. Determination of the long-term hydrolytic stability of polycarbonate engineering resins. Polymer Engineering and Science 32(12):786–791 (1992).

Chapter

16

Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers and Molecular Complexes From darkness we come, and to darkness we return, until the lesson of fire we learn. —Michael Berardi

Solid state NMR spectroscopy is discussed extensively throughout this chapter as a diagnostic probe of nanoscale dimensions in phase-separated copolymers and hydrogen-bonded molecular complexes. Magnetic spin diffusion measurements are modeled phenomenologically via Fick’s second law, and an optimization algorithm is presented to determine the best parameters in the diffusion equation. An example of two-dimensional 1H spin-diffusion spectroscopy identifies dipolar communication between 1H nuclei in different species on a time scale of 100 ms, which is three orders of magnitude shorter than the spin-diffusion mixing time employed by R. R. Ernst and co-workers to illustrate miscibility in a blend of polystyrene and poly(vinyl methyl ether) that was prepared in toluene [Caravatti et al., 1985, 1986]. The chapter concludes with a summary of NMR experiments on suggested polymers and metal complexes.

16.1 MAGNETIC RESONANCE Carbon-13 and 1H magnetic moments experience torques and precess at radiofrequencies (i.e., tens to hundreds of megahertz) about the axis of a strong static magnetic field in superconducting NMR spectrometers. Radiofrequency power sources introduce secondary time-dependent magnetic fields such that 1H and 13C magnets absorb energy when their precession frequency matches the excitation frequency of the power source, causing transitions between two different spin quantum states. The specific Physical Properties of Macromolecules. By Laurence A. Belfiore Copyright # 2010 John Wiley & Sons, Inc.

705

706

Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers

absorption frequency and the corresponding energy difference between these spin states depend on the strength of the static magnetic field, as well as the chemical and electronic environment about the magnetic moment of interest. Hence, solid state NMR spectra of polymers contain structural information that is chemically and morphologically specific. Functional groups absorb energy at characteristic frequencies, crystalline and amorphous regions are distinguishable in many cases, and molecular mobility has a significant effect on carbon-13 absorption lineshapes. All of these effects on NMR spectra of solid polymers are illustrated in this chapter. NMR pulse programs can be devised to (i) generate magnetization in one domain or chemically specific region of a molecule and (ii) detect the diffusion or transport of this magnetization into other domains or chemically specific regions. Measurements of magnetic spin diffusion can be analyzed via Fick’s second law (i.e., the unsteady state diffusion equation), yielding nanoscopic information about domain size and molecular proximity.

16.2 OVERVIEW Nanodomain structure in multiphase polymers plays a fundamental role in determining various macroscopic physical properties of these materials in the solid state. Applications of solid state NMR spectroscopy discussed in this chapter provide a qualitative diagnostic probe of domain structure in industrially important copolymers and multieutectic molecular complexes at length scales where continuum hypotheses are no longer valid. Evidence for interdomain communication in triblock copolymers of styrene and butadiene, or styrene and hydrogenated polybutadiene, is obtained indirectly via carbon-13 nuclear spins in a modified version of the Goldman-Shen [1966] experiment. NMR spin-diffusion experiments rely on spatially dependent dipolar communication, or spin exchange, due to magnetic dipolar interaction energies between coupled nuclei to provide qualitative information about morphology at nanoscale dimensions. This technique monitors dipolar couplings between 1H nuclei in domains of different mobility and addresses molecular proximity at the nanoscale via spin exchange, because rates of spin diffusion scale inversely with the sixth power of internuclear distances. Goldman – Shen experiments are useful to measure 1 H spin diffusion in phase-separated blends and copolymers when direct detection of solid state 1H chemical shift information cannot discriminate between different phases. In this chapter, transient spin diffusion within the 1H spin manifold is measured on the time scale of 100 ms to 10 ms for (i) commercial block and random copolymers that contain both rigid and mobile domains and (ii) stoichiometric hydrogen-bonded molecular complexes of poly(ethylene oxide) and resorcinol. Morphological characteristics (i.e., average domain size) of styrene– butadiene triblock copolymers are analyzed phenomenologically via solution of the unsteady state diffusion equation (i.e., Fick’s second law of diffusion) for the spin-diffusion process.

16.3 THE SPIN-DIFFUSION PROBLEM Consider a detectable tracer that is implanted in a mobile matrix phase at time t ¼ 0. This is accomplished experimentally by exploiting the molecular mobility of the

16.4 Interdomain Communication via Magnetic Spin Diffusion

707

matrix relative to the dispersed rigid phase in two-phase block or random copolymers to establish a gradient in magnetization. The system equilibrates as 1H magnetization diffuses across the interface and into the dispersed phase that has the symmetry of spheres or disorganized flat plates. The interphase is relatively narrow and diffusional resistance is larger in the matrix because dipolar interaction energies among coupled spin-12 nuclei are weakened by molecular motion. The primary objectives are to (i) establish magnetization gradients in two-phase systems that exhibit differences in mobility, (ii) measure 1H magnetic spin diffusion into the rigid dispersed phase via the carbon-13 spin system when it is difficult to discriminate between 1H signals in both domains, and (iii) model transient spin diffusion via Fick’s second law with no generation term to characterize domain size.

16.4 INTERDOMAIN COMMUNICATION VIA MAGNETIC SPIN DIFFUSION: DESCRIPTION OF THE MODIFIED GOLDMAN –SHEN EXPERIMENT This NMR technique employs chain dynamics to probe domain size in two-phase materials that contain rigid and mobile domains, where successful discrimination of 1 H chemical shifts is not feasible. In 1H spin-diffusion studies, from which domain sizes and molecular proximity may in principle be determined, it is attractive to observe proton resonances directly via 1H NMR. This methodology is employed later in this chapter to address hydrogen bonding in stoichiometric molecular complexes of poly(ethylene oxide) and resorcinol that separate two eutectic transformations in the temperature – composition binary phase diagram [Belfiore et al., 1990]. However, solid state 1H spectra of polymers are often insufficiently resolved to fully separate signals from different domains. When this unfortunate situation occurs, the Goldman – Shen [1966] experiment is useful to measure 1H spin diffusion indirectly via carbon-13 sites directly bound to the 1H nuclei of interest [Belfiore et al., 1992]. The success of this modified Goldman – Shen experiment relies on the ability to resolve 13C resonances in each phase. Of critical importance is the establishment of magnetization, or spin-temperature, gradients that provide the driving force for magnetic spin diffusion. This is facilitated by a substantial difference in mobility between the two domains of interest, and the consequent difference between 1H spin –spin relaxation rates. It is important to realize that the Goldman – Shen experiment is not useful to study spin diffusion and molecular proximity in phase-separated systems when transverse 1H spin –spin relaxation rates are not sufficiently different in the two domains of interest. The modified Goldman – Shen experiment is illustrated schematically in Figure 16.1. This experiment is a “second-generation” analog of the technique proposed by Goldman and Shen [1966] more than forty years ago. Initially, an 1H 908 pulse (i.e., 90x) produces 1H polarization in the transverse (x-y) plane. After the time delay designated by t1 in Figure 16.1, 1H magnetization in the rigid domains is severely attenuated by 1H homonuclear spin– spin relaxation processes due to static dipolar interactions within the 1H spin manifold. 1H polarization that survives the t1 time delay is returned to the +z-axis by the second 908 pulse (i.e., 902x). At the beginning of the t2 time interval, a gradient in 1H magnetization

708

Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers Magnetization Gradient

Proton Spin Diffusion 90x

90–x t1

1H

90x t2

Spin Lock

High Power Decoupling

Dipolor Dephasing in Rigid Domains Cross Polarize

13C

13C

Observe Magnetization

Analog of the Golden–Shen Experiment

Figure 16.1 Schematic pulse sequence for the slightly modified Goldman –Shen experiment to measure proton (i.e., 1H) spin diffusion between two domains of dissimilar mobility via the carbon-13 spin system. The t1 dipolar dephasing delay is fixed at 15 ms, and the t2 mixing period parameter is systematically varied during an experiment. Specific details are provided in the text.

between the rigid and mobile domains has been prepared. However, the modified Goldman – Shen experiment does not sample 1H magnetization directly and, hence, cannot detect spatial characteristics of spin-temperature gradients in the 1H spin manifold. If 1H dipolar interactions are operative across domain boundaries, which depends strongly on the strength of dipolar couplings in the interfacial regions, then the initial magnetization gradient will decrease with time, analogous to phenomenological transient diffusion-like processes that are described by Fick’s second law for solids [Carslaw and Jaegar, 1984], and 1H polarization is redistributed between the rigid and mobile domains until spin-temperature equilibration is achieved. By incorporating 1 H– 13C cross-polarization in the pulse sequence together with magic-angle sample spinning and high-power 1H– 13C dipolar decoupling (during data acquisition when 13 C signals are detected), one can map the effect of 1H spin diffusion between two domains of differing mobility onto the carbon-13 nuclear spin manifold. If the thermal mixing time during cross-polarization (CP) is relatively short (on the order of 50 –100 ms), then one can effectively thwart long-range 1H– 13C dipolar communication and obtain carbon intensities that are proportional to the magnetization of directly attached protons. Results for random ionic copolymers of ethylene and methacrylic acid (SurlynTM ), and triblock copolymers of styrene with butadiene (or completely hydrogenated butadiene) (KratonTM ), indicate that a dipolar dephasing delay (t1) of approximately 10 – 20 ms is sufficient to nullify 1H polarization in the rigid domains [Belfiore et al., 1992]. 1H polarization in the mobile domains is reduced to a much lesser degree during the dephasing interval because 1H homonuclear dipolar couplings in the mobile regions are partially averaged by molecular motion. Thus, a magnetization or spin-temperature gradient exists between the rigid and mobile regions of the sample at the beginning of the spin-diffusion mixing period, t2. 1H magnetization

16.6 Magnetic Spin-Diffusion Experiments on Random Copolymers

709

is subsequently redistributed between the two domains by spin diffusion during the mixing period. Several experiments are performed in which the t2 parameter is varied systematically. Spin exchange occurs between two coupled 1H nuclei that undergo energy-conserving spin flips between mS ¼ þ12 and mS ¼ 12 spin states (i.e., 1H flip-flops, one transition occurs to higher energy and the other to lower energy) via S þS 2 terms in the homonuclear dipolar Hamiltonian, where S þ ¼ Sx þ pffiffiffiffiffiffiffiffiffi iSy and S 2 ¼ Sx 2 iSy, with i ¼ (1), are the raising and lowering operators, respectively, based on the x- and y-components of 1H spin angular momentum. This doublequantum transition redistributes 1H polarization between the rigid and mobile 1H reservoirs during the spin-diffusion mixing period, denoted by t2. Both reservoirs also approach spin-temperature equilibration with the lattice at a rate that should be much slower than the spin-diffusion rate. Experimental artifacts due to 1H spin – lattice relaxation are effectively nulled by storing 1H magnetization along the +z-axis at the beginning of the t2 mixing period and subtracting alternate signals in the time domain before Fourier transformation. It is important to emphasize that the Goldman – Shen experiment fails to monitor undistorted 1H spin diffusion if the rate of spin diffusion is comparable to the rate of 1H spin –lattice relaxation in the laboratory frame of reference.

16.5 MATERIALS Random Copolymer of Ethylene and Methacrylic Acid. NucrelTM (DuPont), 85 wt % ethylene, 15 wt % methacrylic acid, unneutralized, melt viscosity ¼ 60 decigram/min at 190 8C (pellets). Zinc Ionomer. SurlynTM 1706 (Dupont), 85 wt % ethylene, 15 wt % methacrylic acid, 60% of the COOH groups are neutralized with zinc, melt viscosity ¼ 0.7 decigram/min at 190 8C (pellets) SBS Triblock Copolymer. KratonTM D-1101 (Shell Development), 31 wt % polystyrene, polystyrene endblock MW  17.5 kDa, polybutadiene mid-block MW  78 kDa (granules). Hydrogenated SBS Triblock Copolymer (S-EB-S). KratonTM G-1651 (Shell Development), 32 wt % polystyrene, polystyrene endblock MW  29 kDa ethylene/butylene midblock MW  122 kDa (granules).

16.6 MAGNETIC SPIN-DIFFUSION EXPERIMENTS ON RANDOM COPOLYMERS THAT CONTAIN DISORGANIZED LAMELLAE 1

H magnetic spin-diffusion data from the modified Goldman – Shen experiment are illustrated in Figure 16.2 for SurlynTM 1706. Carbon-13 magnetization unique to the ethylenic CH2 segments in the crystalline domains is generated via 1H homonuclear dipolar coupling to mobile amorphous CH2 segments across interfacial boundaries, and subsequent 1H – 13C cross-polarization within the crystallites.

710

Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers

(a)

(b)

Two-Phase Ethylenic Morphology Disoriented Lamallae

Carbon-13 Analog of the Goldman–Shen Experiment Zn-Ionomer; 15% MAA

{ Rigid Micro-crystallites { Mobile Amorphous Mobile Tracer Diffusion

Rigid

20 ms 7.5 ms 5 ms 500 μs 60

50

40

30

20

Chemical Shift

10

ppm

50

20 40 30 Chemical Shift

0 10 ppm

Figure 16.2 (a) Schematic representation of disorganized lamellae and the corresponding 13C NMR signals that are unique to the rigid microcrystalline and mobile amorphous domains. Magnetic spin diffusion (i.e., tracer diffusion) occurs in the direction indicated by the arrows. (b) High-resolution solid state 13C NMR spectra in the CH2 chemical shift region during the 1H spin-diffusion experiment. The material is a random copolymer of ethylene and methacrylic acid in which the carboxylic acid side groups are partially neutralized with Zn2þ (SurlynTM 1706). Spin-diffusion mixing times, denoted by t2 in Figure 16.1, are indicated at the right of each spectrum.

The proton dipolar dephasing delay (t1) was 15 ms, the spin-diffusion mixing time (t2) spanned the range from 0.1 ms to 100 ms with emphasis on the short millisecond time scale, the 1H – 13C contact (i.e., thermal mixing during cross-polarization) time was 75 ms, and 1H magnetization was stored alternately along the +z-axis at the start of the spin-diffusion mixing period to suppress spin –lattice relaxation processes. Redistribution of proton polarization between the two domains occurs on a time scale of 10 ms. This claim is supported by the transient behavior of the 1H spindiffusion process illustrated in Figure 16.3, based on carbon-13 intensities of the rigid crystalline ethylenic CH2 signal at  34 ppm. Data in Figure 16.3 reveal that the 1H spin-diffusion process is governed primarily by one time constant of 3.3 ms. For comparison, 1H spin diffusion in a cocrystallized stoichiometric molecular complex of poly(ethylene oxide) and resorcinol that separates two eutectic transformations in the temperature – composition binary phase diagram occurs on a time scale of 100 ms, as discussed in Section 16.10.3. Hence, spin-temperature equilibration between the crystalline and amorphous domains of SurlynTM 1706 is one to two orders of magnitude slower and occurs over much larger length scales relative to the previously mentioned hydrogen-bonded PEO molecular complex, because the spherulitic superstructure of this polyethylene-like twophase material is disrupted by the presence of the atactic comonomer (methacrylic

Magnetization

16.7 Magnetic Spin-Diffusion Experiments on Triblock Copolymers

Random Copolymer CH3 CH2CH2

CH2C –

COO Zn

0

Figure 16.3

711

4 8 12 16 Spin-Diffusion Time (ms)

++

20

1

H spin-diffusion data from the slightly modified Goldman–Shen experiment. The material is a random copolymer of ethylene and methacrylic acid in which the carboxylic acid side groups are partially neutralized with Zn2þ (SurlynTM 1706). The data points represent carbon-13 NMR signal intensities at 34 ppm for the crystalline CH2 segments, generated via 1H dipolar communication (across interfacial boundaries) with the amorphous ethylene segments. The horizontal time axis corresponds to the spin-diffusion mixing period (denoted by t2) in Figure 16.1.

acid), resulting in a fringed micellar pattern of thin disorganized ethylenic crystallites [Longworth, 1975]. In another example of 1H magnetic spin diffusion where 1H NMR signals are measured directly via high-resolution solid state spectra, Caravatti et al. [1985] report the absence of 1H dipolar communication on the 100 ms time scale for an immiscible solid state blend of polystyrene and poly(vinyl methyl ether), cast from chloroform. These three examples of magnetic spin diffusion provide a glimpse of 1H dipolar communication on time scales that range from 100 ms in cocrystallized PEO– resorcinol molecular complexes, to 3 –10 ms in two-phase random copolymers that contain thin disorganized lamellae, to .100 ms in macroscopically phaseseparated polymer – polymer blends.

16.7 MAGNETIC SPIN-DIFFUSION EXPERIMENTS ON TRIBLOCK COPOLYMERS THAT CONTAIN SPHERICALLY DISPERSED HARD SEGMENTS 16.7.1

Cross-Polarization Dynamics

There is a significant difference between local chain mobility in the mobile polybutadiene matrix and the rigid polystyrene dispersed phase of KratonTM D series SBS triblock copolymers. This claim is based on a study of 1H– 13C cross-polarization dynamics, where carbon-13 magnetization unique to each block is measured as a function of the thermal mixing (or cross-polarization contact) time when heteronuclear spin diffusion occurs. Hence, magnetization is transferred from the cold (i.e., polarized)

712

Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers

1

H spin manifold to the hot (i.e., disoriented) 13C spin system in the rotating reference frame where the energy difference between both spin-12 states for each nuclide (i.e., 1H and 13C) is matched at 50 kHz via the Hartmann – Hahn [1962] condition to ensure efficient magnetization transfer. Hence, mutual spin – spin flips occur in energyconserving fashion as carbon-13 polarization develops. The CP contact time data in Figure 16.4a,b reveal that 1H – 13C dipolar communication is much stronger within the polystyrene endblocks of KratonTM D relative to the polybutadiene midblock, based on steeper initial slopes in the graph (Fig. 16.4a) at very short contact times. The inverse of the initial slope of 13C magnetization versus cross-polarization (CP) contact time for all of the data in Figure 16.4 yields the characteristic spindiffusion time constant TCH during cross-polarization. TCH is shorter, the rate of magnetization transfer from 1H to 13C is faster, and the initial slope of 13C magnetization versus contact time is steeper when 1H – 13C dipolar interactions are stronger in more rigid domains. From the viewpoint of spatial discrimination, heteronuclear dipolar interaction energies scale inversely with the third power of 1H – 13C internuclear distances, and the rate constant f1/TCHg for magnetization transfer scales inversely with the sixth power of 1H – 13C internuclear distances. In well-defined binary mixtures or segmented block copolymers where one component or block is completely deuterated, intermolecular 1H– 13C cross-polarization transfer, or spin diffusion, from the protonated species to the deuterated species represents convincing evidence that both components are intimately mixed, or both blocks are not completely phase separated, at the molecular level due to the strong dependence of the rate of magnetization transfer on

(a)

(b)

(c) Styrene–Butadiene RANDOM Copolymer

SBS {Tri block} Styrene End blocks

Butadiene Mid block

45%

Butadiene C H

Downfield

Backbone

CH and CH2

4 8 12 16 Contact Time (ms)

Styrene

–CH2– –CH2–

4 8 12 16 20 Contact Time (ms)

Butadiene

4 8 12 16 20 Contact Time (ms)

H– 13C cross-polarization contact time data for triblock copolymers of (a) styrene and (b) butadiene, and (c) styrene –butadiene random copolymers. For the KratonTM D series SBS triblock copolymers, carbon-13 magnetization in the vinyl backbone and aromatic side group of the polystyrene endblocks is illustrated in part (a) as a function of the 1H – 13C cross-polarization contact time. Analogous data for aliphatic carbon magnetization (both ZCH2 signals) in the polybutadiene midblock are provided in part (b). Contact time data for 13C magnetization in the styrene– butadiene random copolymer are presented in part (c).

Figure 16.4

1

Log {peak intensity}

Log {peak height}

Upfield

16.7 Magnetic Spin-Diffusion Experiments on Triblock Copolymers

713

heteronuclear dipolar distances. This phenomenon is discussed in Sections 16.11.8 and 16.11.9 of this chapter, where intermolecular 1H – 13C magnetization transfer via heteronuclear spin diffusion occurs (i) between protonated additives and completely deuterated polymers in plasticized polymer-diluent blends and (ii) between protonated polymers and transition-metal carbonyl complexes that contain no hydrogen nuclei. Based on the initial rate of magnetization transfer via 1H– 13C spin diffusion in Figure 16.4, local rigidity within the lattice decreases in the following order: (1) polystyrene endblocks of KratonTM D, (2) random copolymers of styrene and butadiene, and (3) polybutadiene midblock of KratonTM D. Thermodynamic incompatibility and microphase separation between polystyrene endblocks and the polybutadiene midblock are responsible for the domain structure of SBS triblock copolymers. Chemical dissimilarity between the segments, coupled with microphase separation, yields (i) differences in local chain mobility and (ii) distinguishable carbon-13 NMR chemical shifts in the matrix and dispersed phases. It must be emphasized that chemical dissimilarity between segments of styrene – butadiene random copolymers is responsible for the same 13C chemical shift discrimination that exists in KratonTM SBS triblock copolymers, but the nature of the random copolymer microstructure and the absence of microphase separation do not produce differences in chain dynamics that are required to establish magnetization gradients via the Goldman – Shen experiment. This claim is verified by the fact that the initial slope of 13C magnetization versus

(a) Schematic representation of a two-phase morphology in KratonTM triblock copolymers that contain dispersed rigid spheres of polystyrene in a rubbery matrix. Magnetic spin-diffusion occurs in the direction indicated by the arrows. (b) 1H magnetic spin-diffusion data from the slightly modified Goldman– Shen experiment. The materials are commercial triblock copolymers of styrene and butadiene (KratonTM D-1101, upper data points), and styrene with completely hydrogenated butadiene (KratonTM G-1651, lower data points). These data represent protonated aromatic carbon-13 NMR signal intensities at 130 ppm for the styrenic ring, generated via 1H dipolar communication (across interfacial boundaries) with the mobile butadiene (SBS) or ethylene/butylene (S-EB-S) matrix. The horizontal time axis corresponds to the spin-diffusion mixing period (denoted by t2) in Figure 16.1.

Figure 16.5

714

Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers

cross-polarization contact time is essentially the same for all three chemically distinguishable 13C resonances in styrene – butadiene random copolymers (graph in Fig. 16.4c), where two 13C resonances are unique to the butadiene comonomer and the aromatic 13C signal is styrene specific.

16.7.2 Interdomain Communication via Magnetic Spin Diffusion Modified Goldman – Shen data are illustrated in Figure 16.5 for phase-separated commercial triblock copolymers of styrene and butadiene (SBS, KratonTM D series), and styrene with completely hydrogenated butadiene (S-EB-S, KratonTM G series). Redistribution of 1H polarization between the mobile midblock and the rigid endblocks occurs on the 10– 20 ms time scale in Figure 16.5. The timedependent interdomain communication process in KratonTM has been modeled phenomenologically via coupled unsteady state radial (Fickian) diffusion equations appropriate to a concentrically spherical core-shell morphology. Details of the mathematical simulation and parameter estimation are described in the following sections. Appendix B in Chapter 12 presents a mechanism for the hydrogenation of alkenes to alkanes via transition-metal catalysis and analyzes the kinetics of these industrially important reactions. In both types of triblock copolymers discussed in this section, (a)

(b) Styrene–Ethylene/Butylene–Styrene Triblock 32% Styrene

Styrene–Butadiene–Styrene–Triblock 31% Styrene B B

S

S

S

S S

160

B

S

S

140

120

100

80

60

40

20 ppm

B

S

160 140 120 100 80

60

40

20

0 ppm

40

20

0 ppm

Ethylene/ Butylene

B S S 160

140

B

S 120

100

80

60

40

20 ppm

160 140 120 100 80

60

13

Figure 16.6 High-resolution solid state C NMR spectra of (a) KratonTM D and (b) KratonTM G triblock copolymers at short (upper) and long (lower) 1H – 13C cross-polarization contact times. Chemically inequivalent 13C signals from the mobile matrix phases (i.e., polybutadiene (a), random copolymer of ethylene and butene (b)) are favored in the lower NMR spectra. Chemical shift distinction between polystyrene endblocks and the polybutadiene midblock is indicated by S and B for SBS triblock copolymers in the spectra in part (a) and S-EB-S triblock copolymers in the upper right spectrum in part (b). Dots identify aliphatic carbon-13 resonances for the ethylene/butylene midblock of S-EB-S triblock copolymers in the lower right spectrum of part (b).

16.8 Phenomenological Transient Diffusion Models for Two-Phase Systems

715

protonated aromatic carbon-13 magnetization near 130 ppm, unique to the rigid styrene endblocks, is generated via 1H homonuclear dipolar communication (i.e., via dipolar interaction energies) with the mobile butadiene or ethylene/butylene midblocks across interfacial boundaries, and subsequent 1H– 13C cross-polarization (CP) within the rigid polystyrene dispersed spherical domains. Discrimination between rigid and mobile segments in these triblock copolymers is illustrated in Figure 16.6. In each case, high-resolution carbon-13 signals in the rigid polystyrene domains are observed in the upper NMR spectra at relatively short 1H – 13C crosspolarization contact (i.e., thermal mixing) times, whereas the mobile matrix phase is favored in the lower NMR spectra at longer CP contact times.

16.8 PHENOMENOLOGICAL TRANSIENT DIFFUSION MODELS FOR TWO-PHASE SYSTEMS WITH SPHERICAL POLYSTYRENE DOMAINS IN A POLYBUTADIENE MATRIX 16.8.1 Analytical Predictions for Unsteady State Radial Diffusion into an Isolated Sphere Magnetic spin-diffusion data in Figure 16.5 are simulated via the unsteady state diffusion equation for species A within an isolated sphere of radius RS. It is important to emphasize that two continuum models are employed in Sections 16.8.1 and 16.8.2 – 16.8.4 to describe statistical phenomena, where homonuclear 1H– 1H dipolar couplings across the interface and within the rigid polystyrene domains provide the mechanism for magnetization transport via spin-temperature gradients between the mobile matrix and the rigid spheres. Qualitative characteristics of the isolated-sphere model are summarized in sequential order: Step 1:

Consider unsteady state radial diffusion in spherical coordinates for the magnetization density of species A, CA(r, t), within an isolated sphere.

Step 2:

Employ a canonical transformation, CA(r, t) ¼ f(r, t)/r to simplify the radial contribution to the Laplacian in spherical coordinates so that the partial differential equation (PDE) for f (r, t) contains constant coefficients.

Step 3:

Re-express f (r, t) ¼ V(r, t) þ b(r) and identify a linear function for b(r) so that the PDE for V(r, t) with constant coefficients has homogeneous spatial boundary conditions (i.e., V ¼ 0) at r ¼ 0 and r ¼ RS.

Step 4:

Use separation of variables to solve the PDE for V(r, t), identifying the eigenvalues and eigenfunctions in an infinite-series solution. Integrate the magnetization density profile for species A throughout the isolated sphere to obtain a time-dependent function mA(t) that mimics the spin-diffusion data in Figure 16.5:

Step 5:

mA (t)  4p

r¼R ðS r¼0

CA (r, t)r2 dr

716

Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers

Step 6:

Construct an analog of the diffusion lag-time tLag for transient penetrant flux across a permeable membrane (i.e., see Section 2.6.2) in terms of mA(t) relative to its steady state value: tLag ¼ lim

t0ð )t 

t)1 t0 ¼0

Step 7:

 mA (t0 ) R2 dt 0 ¼ S 1 mA (t ) 1) 15D

Calculate tLag for 1H spin-diffusion data in triblock copolymers that contain spherically dispersed hard segments (see Fig. 16.5) via the previous equation and correlate 1H spin-diffusion coefficients D with the average size of the spherical domains. Furthermore, average domain sizes can be estimated from calculations of tLag for realistic spin-diffusion data, together with an 1H spin-diffusion coefficient of 10212 cm2/s. For comparison, diffusion coefficients for 1H magnetization transport are 6.2  10212 cm2/s in polyethylene [Douglass and Jones, 1966], and 5.0  10212 cm2/s in poly(ethylene terephthalate) [Havens and VanderHart, 1985]. An excellent reference on 1H spin diffusion in polymeric materials is provided by Demco et al. [1995].

When concentration diffusion occurs exclusively in the radial direction, into an isolated polystyrene spherical domain with radius RS and a narrow interface, Fick’s second law of diffusion for the magnetization density of species A, CA(r, t), satisfies a second-order partial differential equation with variable coefficients because the surface area of a spherical shell at radius r, normal to radial diffusional flux, scales as the square of radial position. The appropriate equation is   @CA 1 @ @CA ¼D 2 r2 @t @r r @r Three boundary conditions are required for a unique solution: CA ¼ 0; CA ¼ CA,butadiene  constant;

r , RS ; t ¼ 0 r  RS ; all t

@CA ¼ 0 due to symmetry; @r

r ¼ 0; all t

where CA,butadiene is the relatively constant magnetization density of species A in the mobile polybutadiene matrix. The radial contribution to the Laplacian (i.e., r2CA) in spherical coordinates, on the right side of the diffusion equation, can be simplified considerably by re-expressing CA(r, t) in terms of f(r, t) as follows: 1 CA (r, t) ¼ f (r, t) r

16.8 Phenomenological Transient Diffusion Models for Two-Phase Systems

717

Straightforward partial differentiation of CA with respect to either r or t, as prescribed by the diffusion equation, reveals that f (r, t) satisfies a simpler diffusion equation with constant coefficients [Belfiore, 2003]: @f @2f ¼D 2 @t @r Since CA is finite with vanishing slope at r ¼ 0 due to symmetry, the modified boundary conditions for f(r, t) are r , RS ; t ¼ 0

f ¼ 0;

f ¼ RS CA,butadiene  constant; r ¼ RS ; all t f ¼ 0 and CA is finite;

r ¼ 0; all t

In an effort to make the spatial boundary condition homogeneous at the outer edge of the isolated sphere, one re-expresses f(r, t) as follows: f (r, t) ¼ rCA (r, t) ¼ V(r, t) þ b(r) The objective is to identify b(r) such that V(r, t) satisfies the unsteady state diffusion equation. Upon substituting f(r, t) into the diffusion equation, one obtains  2  @V @ V d2 b ¼D þ 2 dr @t @r 2 If one constructs b(r) such that the spatial boundary conditions on V(r, t) are homogeneous, then r ¼ 0; all t V ¼ 0; b ¼ 0; V ¼ 0; b ¼ RS CA,butadiene ; r ¼ RS ; all t The objective is accomplished if b(r) ¼ rCA,butadiene (i.e., d 2b/dr 2 ¼ 0), and one seeks a separation-of-variables solution for V(r, t) ¼ x(r)c(t) that satisfies @V @2V ¼D 2 @t @r V ¼ 0; r ¼ 0; all t V ¼ 0; r ¼ RS ; all t V(r, t ¼ 0) ¼ rCA,butadiene ; r , RS ; t ¼ 0 Recall that 1 V(r, t) þ b(r) 1 ¼ CA,butadiene þ x(r)c(t) CA (r, t) ¼ f (r, t) ¼ r r r Substitution of the postulated functional form for V(r, t) into the unsteady state diffusion equation yields two ordinary differential expressions that are equated to a negative separation constant, 2k2 where k is real, so that c(t) is not unbounded as t )1.

718

Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers

Divide both sides of the diffusion equation by V to obtain 1 dc D d2 x ¼ ¼ k2 c dt x dr 2

c(t)  exp{k2 t} x(r) ¼ A sin(gr) þ B cos(gr) rffiffiffiffiffi k2 g¼ D The homogeneous boundary condition on V(r ¼ 0, t), in addition to the fact that CA(r ¼ 0, t) must be finite, is satisfied when B ¼ 0. The second homogeneous boundary condition on V(r ¼ RS, t) is satisfied when gRS is an integer multiple of p (i.e., np, n ¼ 0, 1, 2, . . . ). Hence, the nth eigenvalues are rffiffiffiffiffi k2n np ¼ gn ¼ D RS

k2n ¼

n2 p2 D R2S

The nth eigenfunction and the complete solution for V(r, t) are Vn (r, t) ¼ xn (r)cn (t) ¼ An cn (0) sin{gn r} exp{k2n t}  2 2    1 1 X X npr n pD Vn (r, t) ¼ Cn sin t V(r, t) ¼ exp  R R2S S n¼0 n¼1 The initial condition on V(r, t ¼ 0) reveals that Cn ¼ Ancn(0), n ¼ 1, 2, 3, . . . , is given by the Fourier sine coefficients of – rCA,butadiene, as illustrated below:   1 X npr Cn sin V(r, t ¼ 0) ¼ rCA,butadiene ¼ RS n¼1 RðS



r¼0

RðS       1 X k pr npr k pr dr ¼ sin dr rCA,butadiene sin Cn sin RS RS RS n¼1 r¼0

1 ¼ RS 2 2CA,butadiene Ck ¼  RS

RðS

1 X

Cn dkn

n¼1



 k pr 2(1)k dr ¼ r sin RS CA,butadiene kp RS

r¼0

To predict the magnetization that has diffused into an isolated sphere of radius RS versus time and compare predictions of the lag time defined in Step 6 above with

16.8 Phenomenological Transient Diffusion Models for Two-Phase Systems

719

experimental data, as illustrated in Figure 16.5, it is necessary to integrate CA(r, t), via solution of the unsteady state diffusion equation, throughout the volume of the sphere. The transient function mA(t), defined in Step 5 above, is modified slightly via division by CA,butadiene. The desired quantity is RðS    ððð  CA (r, t) CA (r, t) mA (t) ¼ dV ¼ 4p r 2 dr CA,butadiene CA,butadiene Vsphere

r¼0

Substituting results from this section into the previous integral expression yields explicit evaluation of mA(t), with dimensions of volume, zero initial value, and infinite initial slope at t ¼ 0: RðS

mA (t) ¼ 4p

(

r¼0

 2 2 )   1 2RS X (1)n npr n pD sin 1þ t r2 dr exp  pr n¼1 n RS R2S

 2 2  1 4 3 8 3X 1 n pD ¼ pRS  RS exp  t 2 p n¼1 n 3 R2S Since mA(t) is “normalized” by CA,butadiene, which corresponds to the constant magnetization density at the periphery of an isolated sphere, it is reasonable that the steady state value of mA is given by the sphere volume (i.e., the leading term on the right side of the previous equation). The lag time analog for this unsteady state diffusion model is shown schematically by the area in the upper left-hand region of Figure 16.7, and tLag is predicted as follows:

tLag ¼ lim

t0ð )t 

t)1 t0 ¼0

 mA (t 0 ) dt0 1 mA (t ) 1)

1 6 X 1 ¼ 2 lim p n¼1 n2 t)1

t 0ð )t t0 ¼0

 2 2  1 n pD 0 6R2S X 1 R2S 0 dt exp  t ¼ ¼ p4 D n¼1 n4 15D R2S

where infinite series evaluations of 1/n 2 and 1/n 4 are given by p2/6 and p4/90, respectively [Abramowitz and Stegun, 1972]. This transient model suggests that the average radius of spherical polystyrene domains in KratonTM triblock copolymers is given by the square-root of 15DtLag, where the lag time is calculated experimentally from 1H magnetic spin-diffusion data in Figure 16.5 when magnetization on the vertical axis is normalized with respect to its steady state value.

16.8.2

Diffusion in a Core-Shell Morphology

Consider an isolated concentric core-shell morphology as illustrated in Figure 16.8, where the interphase is extremely narrow. The radius ratio of the rigid polystyrene

720

Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers 1.0

Transient Mass of Penetrant

0.9

Lag-Time Area

0.8 0.7 0.6 0.5 0.4 0.3 0.2

Dimensionless Lag-Time = 0.067 via Integration

0.1 0.0 0.0

0.1

0.2 0.3 0.4 Dimensionless Diffusion Time (t)

0.5

Figure 16.7 Quantitative evaluation of the accumulation of mass within an isolated sphere via the solution of Fick’s second law of diffusion. Dimensionless time t on the horizontal axis is defined by Dt/fRSg2. The dimensionless lag time tLag ¼ DtLag/fRSg2 ¼ 1/15 via integration area on the upper left side of this graph. Numerical calculations were performed by including 50 terms in the infinite series expression for mA(t), revealing zero initial value at t ¼ 0 and infinite initial slope that agrees with analytical evaluation of fdmA/dtgt¼0.

core RS to the polybutadiene shell RB matches the cube root of the polystyrene volume fraction (i.e., w1/3) in these triblock copolymers. Initially, 1H magnetization exists exclusively in the outer shell and the time evolution of volumetrically averaged signals within the core is predicted via the phenomenological equations discussed in the following section.

Figure 16.8 Schematic representation of a polystyrene spherical core surrounded by a butadiene shell that provides a simple model for magnetic spin diffusion from the mobile shell to the rigid core. Geometric parameters are defined that can be related to the styrene volume fraction w in SBS triblock copolymers (i.e., w  30%).

16.8 Phenomenological Transient Diffusion Models for Two-Phase Systems

721

16.8.3 Coupled Partial Differential Equations and Their Supporting Boundary and Initial Conditions 1

H magnetization density CS and CB in each domain obeys the unsteady state radial diffusion equation in spherical coordinates. Subscripts S and B describe properties of the polystyrene core and polybutadiene shell, respectively. The appropriate partial differential equations (i.e., Fick’s second law of diffusion) for CS and CB, with no generation terms due to spin – lattice relaxation, are   @CS 1 @ @CS ¼ DS 2 r2 0  r  RS ; @t @r r @r   @CB 1 @ @CB RS  r  RB ; ¼ DB 2 r2 @t @r r @r Continuity of diffusional flux and magnetization density are invoked at the interface between the two domains, where r ¼ RS in the concentric sphere model. Hence, the two initial conditions and four boundary conditions that are required to calculate CS(r, t) and CB(r, t) in this isolated system, with interfacial coupling and zero flux at the outer boundary, reveal similarities to the phenomenological description of unsteady state radial diffusion at the continuum level, as described by Fick’s second law. Even though continuity of magnetization density at the interface seems reasonable, the continuum analog (i.e., unsteady state mass transfer with interfacial equilibrium) of this boundary condition equates chemical potentials of the two phases, not molar densities, unless the two-phase system behaves ideally. From a heat-transfer perspective, continuity of magnetization density at the interface is consistent with the equivalence of magnetic spin temperature in both domains at r ¼ RS, where temperature and magnetization are related via Curie’s law. The appropriate conditions are Magnetization gradient at t 5 0 CS ¼ 0;

0  r , RS

CB ¼ CB,initial ; RS , r  RB Coupling across the interface at r 5 RS, t > 0 CS ¼ CB DS Symmetry

Zero flux

@CS @CB ¼ DB @r @r

  @CS ¼0 @r r¼0   @CB ¼0 @r r¼RB

722

Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers

From a numerical analysis (i.e., finite difference) viewpoint, it is necessary to address the discontinuity in the initial condition at the interface (i.e., r ¼ RS), where both flux and magnetization density should be continuous. This is achieved by invoking a simple approximation for flux continuity at r ¼ RS when t ¼ 0 and the rigid core exhibits no signal: DS

CS (r ¼ RS , t ¼ 0)  0 CB (r ¼ RB , t ¼ 0)  CB (r ¼ RS , t ¼ 0)  DB RS  0 RB  RS CS (h ¼ w1=3 , t ¼ 0) ¼ ¼

w1=3 ¼

CS (r ¼ RS , t ¼ 0) ¼ CB (h ¼ w1=3 , t ¼ 0) CB (r ¼ RB , t ¼ 0) CB (r ¼ RS , t ¼ 0) CB (r ¼ RB , t ¼ 0) RS DS ; g¼ RB DB

w1=3 þ g(1  w1=3 ) where the ratio of magnetic spin-diffusion coefficients g . 1 as a consequence of stronger dipolar interactions in the rigid polystyrene core, and the polystyrene core volume fraction w , 1. Fick’s second law of diffusion for magnetization density in each domain is written using dimensionless variables, some of which are defined in the previous set of equations. Time is dimensionalized using a ¼ (RS)2/DS, which represents a characteristic time constant for diffusion within the polystyrene core. Hence, dimensionless time is defined as t ¼ t/a, dimensionless radial coordinate h is given by r/RB in either domain, both magnetization densities are dimensionalized via CB(r ¼ RB, t ¼ 0) ¼ CB,initial, and the appropriate partial differential equations for dimensionless magnetization densities CS and CB in each domain (i.e., Ci (h, t ) ¼ Ci (r, t)/CB,initial; i ¼ S, B) are   @CS 1=3 2=3 1 @ 2 @CS h ¼w 0hw ; @t  h2 @ h @h   @CB 1 2=3 1 @ 2 @CB w h w1=3  h  1; ¼ @t  g h2 @ h @h CS (h ¼ w1=3 , t ¼ 0) ¼ CB (h ¼ w1=3 , t ¼ 0) 

w1=3

subject to two initial conditions and four boundary conditions, given by Magnetization gradient at t 5 0 CS ¼ 0; 0  h , w1=3 CB ¼ 1; w1=3 , h  1 Coupling across the interface at h 5 w 1/3, t > 0 CS ¼ CB

g

@CS @CB ¼ @h @h

16.8 Phenomenological Transient Diffusion Models for Two-Phase Systems

Symmetry

723

  @CS ¼0 @ h h¼0

Zero flux

  @CB ¼0 @ h h¼1

16.8.4 Bulk Magnetization that Diffuses Across the Mobile– Rigid Interface into the Central Core: FiniteDifference Simulations for KratonTM Triblock Copolymers After integrating the coupled partial differential equations in the previous section and obtaining bulk magnetization density profiles as a function of time and radial position from the central point in the rigid core (i.e., h ¼ 0) to the outer edge of the spherical shell (i.e., r ¼ RB or h ¼ 1), one should verify that the propagation or accumulation of error in the numerical integration scheme, due to (i) truncation or (ii) step-size adjustments, does not violate mass conservation. The following global check of the finite-difference results that is specific to an isolated core – shell model of spherically dispersed rigid domains in a mobile matrix phase will identify any inconsistencies in the numerical algorithm at each time step: RðB

4p

4 CB,initial r dr ¼ pCB,initial {R3B  R3S } ¼ 4p 3 2

r¼RS

RðS

2

CS (r, t)r dr þ 4p

r¼0 1=3

1 {1  w} ¼ 3

wð



2

CS ( h , t ) h d h þ

h¼0

ð1

RðB

CB (r, t)r 2 dr

r¼RS

CB (h, t  )h2 d h

h¼w1=3

Initially, all magnetization resides in the outer polybutadiene shell. Simulations should be discarded if the previous equation is not satisfied at all times. The first term on the right side of this dimensionless mass balance corresponds to the fraction of the total bulk magnetization that diffuses across the interface from the mobile polybutadiene shell to the rigid polystyrene core at time t . Hence, the time dependence of the following quantity, 3 1w

1=3 wð

CS (h, t  )h2 d h

h¼0

is presented in Figure 16.9, and should be compared with experimental 1H spindiffusion data for KratonTM triblock copolymers in Figure 16.5. A nonlinear

724

Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers

Fraction of Bulk Magnetization in Core

0.35 30% Polystyrene Core Volume Fraction 0.30 0.25 0.20 Diffusivity Ratio = 0.1

0.15

Diffusivity Ratio = 1 Diffusivity Ratio = 3

0.10

Diffusivity Ratio = 5 0.05 0.00 0.0

Diffusivity Ratio = 10 0.1

0.2

0.3 0.4 0.5 0.6 Dimensionless Time, tDs/R2s

0.7

0.8

Figure 16.9 Transient simulations of dimensionless volume-averaged bulk magnetization that diffuses into the central core, across the sharp core– shell interface, representing a simple model of the Goldmann– Shen experiment for KratonTM triblock copolymers that contain spherically dispersed polystyrene hard domains with a volume fraction of 30%. The diffusivity ratio in the legend corresponds to DPolystyrene/ DPolybutadiene ¼ DS/DB, revealing that there is greater resistance to magnetization transport across the sharp interface when the 1H spin-diffusion coefficient within the rigid core is greater than that in the mobile matrix, which is realistic. Simulation parameters: 101 points in the radial direction, with Dh ¼ 1022; 156 steps in dimensionless time, Dt ¼ 5  1023. All simulations conserve initial mass in the shell to within 0.15% (i.e., .0.9985, where 1.0 is exact).

optimization procedure developed by Gauss and Legendre [Beightler et al., 1979] is described later in this chapter to identify best-fit parameters of the magnetic spindiffusion model that agree with experimental data in Figures 16.3 and 16.5. When spin-temperature equilibration between the rigid and mobile domains is achieved at long times (i.e., actually t . 5a is probably sufficient), the following calculations based on an overall mass balance identify the asymptotic limit of the dimensionless magnetization density profile in either domain, because CS(0  h  w1/3, t . 5)  CB(w1/3  h  1, t . 5)  constant (i.e., 1 2 w) for all radial positions h: 1=3   wð 1 {1  w} ¼ CS 0  h  w1=3 , t  . 5 h2 dh 3 h¼0

  ð1 þ CB w1=3  h  1, t  . 5

h2 d h

h¼w1=3

CS



   0  h  w1=3 , t  . 5  CB w1=3  h  1, t  . 5  1  w

Hence, the asymptotic limit of the fraction of the total bulk magnetization that diffuses across the interface from the mobile polybutadiene shell to the rigid polystyrene core is

Local Molar Density at Fixed Radial Position

16.8 Phenomenological Transient Diffusion Models for Two-Phase Systems

725

1.0 30% Polystyrene Core Volume Fraction 0.9

Rigid Core Diffusivity/Mobile Shell Diffusivity = 2

0.8

0.7

0.6

0.5 0.0

0.1

0.2

0.3 0.4 0.5 0.6 Dimensionless Time, tDs/R2s

0.7

0.8

Figure 16.10

Transient simulations of magnetic spin diffusion at fixed radial positions in the polybutadiene shell, r ¼ 0.68 RB (lowest) to r ¼ 0.92 RB (highest), 0.68  h  0.92, for KratonTM triblock copolymers that contain spherically dispersed polystyrene hard domains. The discontinuous “step” initial condition and the requirement of coupling across the sharp interface are responsible for the “undershoot” illustrated in profiles between 68% and 73% of the outer shell radius when the sharp interface exists at 67% of RB, corresponding to 30% polystyrene by volume. Simulation parameters: 101 points in the radial direction, with Dh ¼ 1022; 156 steps in dimensionless time, Dt ¼ 5  1023.

equivalent to the polystyrene volume fraction if one assumes that the density mismatch between the two domains is negligible: 3 1w

1=3 wð

h¼0

1=3 wð

3 CS (h, t  . 5)h2 d h  {CS (0  h  w1=3 , t  . 5)} 1w

h2 dh  w

h¼0

These asymptotic predictions agree with numerical simulations at long times. Illustrated are the transient behavior of dimensionless volume-averaged bulk magnetization that diffuses into the central core (Fig. 16.9), as well as several local magnetization density profiles at fixed radial positions near the sharp interface (Figs. 16.10 and 16.11).

16.8.5 Multivariable Parameter Estimation via Nonlinear Least Squares Analysis Four parametric values are required for a unique description of this concentric core – shell model of spherically dispersed polystyrene rigid domains in a polybutadiene or ethylene/butene random copolymer matrix: (i) 1H magnetic spin-diffusion coefficients in each domain (i.e., DS and DB), (ii) radius of the rigid polystyrene core, RS, and (iii) outer radius of the mobile matrix shell, RB. These four parameters

Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers Local Molar Density at Fixed Radial Position

726

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

30% Polystyrene Core Volume Fraction Rigid Core Diffusivity/Mobile Shell Diffusivity = 2

0.1

0.2

0.3 0.4 0.5 0.6 Dimensionless Time, tDs/R2s

0.7

0.8

Figure 16.11 Transient simulations of magnetic spin diffusion at fixed radial positions in the polystyrene core, r ¼ 0.55 Rs (lowest) to r ¼ 0.99 Rs (highest), 0.55w1/3  h  0.99w1/3, Rs ¼ w1/3 RB, for KratonTM triblock copolymers that contain spherically dispersed polystyrene hard domains with a volume fraction w ¼ 30%. No overshoot within the core is predicted in the vicinity of the sharp interface (i.e., h ¼ w1/3) when DPolystyrene/DPolybutadiene . 1. Simulation parameters: 101 points in the radial direction, with Dh ¼ 1022; 156 steps in dimensionless time, Dt ¼ 5  1023.

are contained in the polystyrene volume fraction w ¼ (RS/RB)3, ratio of 1H magnetic spin-diffusion coefficients g ¼ DS/DB, and the characteristic time constant for diffusion in the rigid core a ¼ (RS)2/DS. As mentioned earlier in this section on phenomenological transient diffusion models, the radius ratio of the rigid polystyrene core RS to the polybutadiene shell RB matches the cube root of the polystyrene volume fraction (i.e., w1/3) in these triblock copolymers, and transmission electron micrographs provide a snapshot of the average size of the spherically dispersed rigid domains (i.e., RS). Hence, RS and RB can be estimated from experimental data (i.e., TEM) on triblock copolymers with known composition w. The adjustable parameters are embedded in a and g. The least-squares procedure developed by Gauss and Legendre [Beightler, et al., 1979] minimizes the sum of squares of the difference Q between model predictions based on a set of adjustable parameters fVig, 1  i  N, and M actual experimental data points, where M . N because there should be more data points than adjustable parameters. One constructs the Error and then proceeds with minimization: Error ¼

M X

[Qk {Vi }]2 ¼ QT Q

k¼1

Qk represents the difference between a model prediction using the complete set fVig of adjustable parameters and the kth data point. If Q corresponds to the column vector of M functions, where each one is denoted by Qk (1  k  M ), and QT is the transpose of this column vector, then optimization requires that the derivative of the Error with

16.8 Phenomenological Transient Diffusion Models for Two-Phase Systems

727

respect to the ith adjustable parameter Vi should vanish, and one invokes the following equation for each Vi, where 1  i  N (i.e., there are N equations of the form): M M M X X X d{Error} @Qk ¼2 [Qk {Vi }] ¼2 [Qk {Vi }]Jki ¼ 2 JikT [Qk {Vi }] ¼ 0 dVi @Vi k¼1 k¼1 k¼1

Jki ¼ JikT ¼

@Qk @Vi

Jki is the Jacobian of the difference Qk with respect to adjustable parameter Vi. The M  N Jacobian matrix exhibits a rank of N (i.e., N , M ) because all of the N columns of J are linearly independent. If one (i) approximates the derivative of the Error with respect to Vi at the new set of N adjustable parameters fVi þ DVig as follows,   M X dError 2 [JikT {Vi }][Qk {Vi þ DVi }]  0 dVi {Vi þDVi } k¼1 (ii) expands the Taylor series for QkfVi þ DVig about QkfVig, and (iii) truncates this series after the linear terms, yielding a set of coupled linear algebraic equations as discussed below, then  N  X @Qk (DVj ) þ Qk {Vi þ DVi } ¼ Qk {Vi } þ @Vj {Vi } j¼1  Qk {Vi } þ

N X

[Jkj {Vi }](DVj )

j¼1

A combination of the previous two equations must be invoked to minimize the Error with respect to each adjustable parameter, yielding an independent set of fDVjg, 1  j  N. Hence, one arrives at a multivariable linear algebraic equation, given below, that must be written once for each Vi, 1  i  N, for complete minimization of the Error: " #   M N X X dError T 2 [Jik {Vi }] Qk {Vi } þ [Jkj {Vi }](DVj )  0 dVi {Vi þDVi } j¼1 k¼1 ( ) N M M X X X T [Jik {Vi }][Jkj {Vi }] (DVj )   [JikT {Vi }][Qk {Vi }] j¼1

k¼1

k¼1

Matrix multiplication of the N  M transpose of the Jacobian matrix with the M  N Jacobian matrix (i.e., J TJ ), M X

[JikT {Vi }][Jkj {Vi }]

k¼1

yields the ij element of the nonsingular N  N matrix J TJ that can be inverted to facilitate solving the system of linear algebraic equations.

728

Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers

16.8.6

The Algorithm

Step 1: Make initial guesses for the pair of adjustable parameters that are embedded in the characteristic diffusion time constant a and the magnetic spin diffusion coefficient ratio g. Step 2: Use the most up-to-date values of the set of adjustable parameters fVi,Newg and calculate M difference functions Qk, 1  k  M, by (i) employing all of the experimental data points, (ii) solving the coupled partial differential equations, and (iii) predicting the volume-averaged bulk magnetization that diffuses into the rigid core at M different time steps, t ¼ at , corresponding to the experimental times at which magnetic spin-diffusion data are available. Step 3: Now, it is necessary to solve the pair of partial differential equations 2N times and use finite difference methodology to evaluate all elements of the Jacobian matrix, Jki ¼ @Qk/@Vi, by choosing each adjustable parameter Vi to be slightly larger and slightly smaller than its current value. Step 4: Obtain the transpose of the Jacobian matrix. Step 5: Evaluate the gradient of the Error with respect to the complete set of adjustable parameters fVig, rError ¼ 2J TQ. If all N elements of this column vector 2J TQ are within a preset tolerance of zero, then the current set of adjustable parameters represents the optimum solution. If any element of the column vector 2J TQ is outside the tolerance limit, then solve the following system of linear algebraic equations for 1  i  N, ( ) N M M X X X T [Jik {Vi }][Jkj {Vi }] (DVj )   [JikT {Vi }][Qk {Vi }] j¼1

k¼1

k¼1

to determine how each adjustable parameter should change (i.e., DVj, 1  j  N ). Step 6: Update all of the adjustable parameters, such that Vj,New ¼ Vj,Old þ DVj, 1  j  N, and return to Step 2. Repeat this loop until all N elements of rError ¼ 2J TQ are within the tolerance limit.

16.9 SOLID STATE NMR ANALYSIS OF MOLECULAR COMPLEXES 16.9.1

Spectroscopic Detection of Phase Coexistence

Spectroscopic methods are sensitive to phase behavior when a signal from the “key component” is influenced strongly by neighboring components in a blend or complex. In some cases, a spectroscopic probe can detect phase coexistence when one of the phases is transparent to more conventional probes, like thermal analysis. For example, when the melting point depression phase boundary converges with the glass transition

16.9 Solid State NMR Analysis of Molecular Complexes

729

phase boundary as the crystallizable component is diluted by the noncrystalline component, thermal analysis might only reveal the dominant glass transition that overlaps a weak melting endotherm at the same temperature (see Figs. 16.22 and 16.26). If the key component is present in the disordered crystalline and dominant amorphous phases, and if the interaction between dissimilar species in the amorphous phase is strong enough, then infrared or solid state NMR spectroscopies will detect phase coexistence because the signal for the key component in each environment is distinct. Hence, the overlap between Tg and Tm as described above is circumvented because the crystalline and amorphous regions are characterized by key component signals at (i) different NMR chemical shifts or (ii) different vibrational frequencies. When eutectic phase behavior occurs, thermal analysis reveals one melting endotherm for a two-phase mixture at the eutectic composition. This is misleading because both phases that comprise the eutectic mixture melt incongruently at the same temperature with no excess of either phase. In off-eutectic mixtures, the excess phase melts at higher temperature to produce a thermogram that reveals two endotherms. In both eutectic and off-eutectic mixtures of poly(ethylene oxide) with either resorcinol or 2-methylresorcinol, 13C solid state NMR spectroscopic results in this chapter identify phase-sensitive signals of the small-molecule aromatic in each phase. Hence, the number of NMR absorptions for chemically identical carbon sites in resorcinol correlates with the number of coexisting phases. However, the number of phases does not correlate with the number of melting endotherms in the DSC trace at the eutectic composition. There are other examples of crystalline – amorphous polymer – polymer blends where calorimetry and 13C solid state NMR have been employed in harmony to probe polyester crystallinity and hydrogen bonding [Belfiore et al., 1993].

16.9.2 Solid State NMR Detection of Molecular Complexes Mixing phenomena in polymer blends is an area of practical importance because the solid state phase behavior of a blend governs its physical properties. At the sitespecific level of probing interactions between dissimilar species in a blend, one can identify functional groups that potentially give rise to exothermic energetics and favorable mixing. High-resolution solid state NMR spectroscopy is the analytical technique of choice to fingerprint the components of a strongly interacting blend and monitor changes in crystal structure (if appropriate), conformation, molecular packing, hydrogen bonding, metal –ligand coordination, or other electron transfer reactions that accompany the formation of a molecular complex. Thermal analysis via differential scanning calorimetry is extremely useful to generate phase diagrams and provide phenomenological interpretations of solid state NMR spectra for blends in a two-phase region. When phase separation occurs, it is important that one of the components (the critical component) must be present in both phases and that the carbon-13 NMR spectrum of this critical component must contain at least one signal that is sensitive to its nearest neighbors. These conditions are satisfied for complexes of poly(ethylene oxide) and resorcinol that contain more than 33 mol %

730

Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers

resorcinol. The phenolic carbon resonance of the critical component (resorcinol) identifies the lattice structure of the molecular complex, designated as phase b, via a 3-ppm difference in isotropic 13C chemical shift relative to the phenolic carbon resonance of resorcinol in its undiluted crystalline state. The PEO – resorcinol system is unusual because it exhibits bieutectic phase behavior with eutectic compositions in the vicinity of 10 and 50 mol % resorcinol. The phase behavior of several binary polymeric systems that exhibit single eutectic response has been investigated. More recently, poly(ethylene oxide) complexes with a variety of isomers and derivatives of dihydroxybenzene have been identified that contain either two or three solid– solid – liquid (eutectic) transition lines in the temperature – composition projection of the phase diagram. Molecular proximity between dissimilar components on the order of a few angstroms is synonymous with micromixing and the formation of molecular complexes. Spin diffusion between dipolar-coupled 1H nuclei in dissimilar blend components represents a sophisticated diagnostic probe of spatial proximity. Two well-known NMR techniques are useful to monitor communication between dipolar-coupled 1H nuclei and address the question of molecular proximity. The first technique exploits the high-resolution nature of 1H solid state NMR spectra via combined rotation and multiple pulse spectroscopy (CRAMPS) for a homogeneous solid solution of poly(ethylene oxide) and resorcinol in phase b, whose 1H NMR spectrum is well resolved. The second example, discussed earlier in this chapter, is reserved for the measurement of 1H spin diffusion in phase-separated blends and copolymers when the 1H CRAMPS experiment does not provide useful information about molecular proximity. In this case, commercial block and random copolymers that contain both rigid and mobile domains are investigated. Both types of spin-diffusion experiments rely on spatially dependent spin – spin communication between 1H nuclei to provide qualitative information about dipolar distances. These distances are on the order of ˚ in the PEO – resorcinol molecular complex (phase b), and time-dependent 2 –5 A 1 H spin diffusion between dissimilar nearest-neighbor molecules occurs, and most likely equilibrates, on a time scale of 1024 s. In contrast, transient interdomain communication between dipolar-coupled 1H nuclei is at least one to two orders of magnitude slower (1 – 10 ms) for industrially important phase-separated copolymers of styrene and butadiene (KratonTM ), and ethylene with methacrylic acid (SurlynTM ). The critical parameter in these 1H spin-diffusion experiments is the mixing time during which spin exchange or magnetization transfer takes place in the presence of homonuclear dipolar couplings. Due to the complexity of the blends and copolymers investigated, a detailed analysis of dipolar distances between dissimilar molecules or copolymer microdomains is not attempted.

16.10 HIGH-RESOLUTION SOLID STATE NMR SPECTROSCOPY OF PEO MOLECULAR COMPLEXES: CORRELATIONS WITH PHASE BEHAVIOR The overall objective here is to bridge the gap between macroscopic and molecular probes of strong interaction in hydrogen-bonded blends. The bieutectic phase behavior of poly(ethylene oxide) and resorcinol is rather unique, and solid state NMR signals of

16.10 High-Resolution Solid State NMR Spectroscopy of PEO

731

resorcinol’s phenolic carbons identify different crystallographic symmetry in phases b and g, as described below.

16.10.1 Bieutectic Blends of Poly(ethylene oxide) and Resorcinol: Comparison with Trieutectic Phase Behavior in Binary Mixtures of PEO with 2-Methylresorcinol Carbon-13 NMR spectra of resorcinol in the undiluted state and in blends with poly(ethylene oxide) are illustrated in Figure 16.12a. The aromatic carbon chemical shift region between 100 and 160 ppm is unique to resorcinol, which serves as the critical component in these mixtures. Spectrum A of Figure 16.12a represents the crystal form of undiluted resorcinol (phase g) at a cross-polarization thermal mixing time of 1 ms and a pulse sequence repetition delay of 60 s. Spectra B, C, and D of various PEO– resorcinol blends in Figure 16.12a were obtained using a cross-polarization thermal mixing time of 1 ms and a pulse repetition delay of 2 s. These blend spectra reveal that the phenolic carbon resonance between 155 ppm and 160 ppm is sensitive to crystallographic symmetry. The morphological characteristics of the molecular complex (phase b) are described best by cocrystallization, and support for this (a)

13C

Resorcinol Resonances

D

9%

C

36%

B

67%

A

HO

Undiluted Crystalline 2-Methylresorcinol

Precipitate from Aqueous Solution

OH

HO C H3

Interaction with PEO

OH

170 160 150 140 130 120 110 100 90 Carbon-13 Chemical Shift (ppm)

Figure 16.12

(b)

20 13C

15 10 5 0 –5 ppm Solid State Chemical Shift

(a) High-resolution carbon-13 solid state NMR spectra for methanol-cast blends of poly(ethylene oxide) and resorcinol in the aromatic chemical shift region via cross-polarization with a contact time of 1 ms: A—undiluted crystalline resorcinol, pulse repetition delay ¼ 60 s; B—67 mol % resorcinol, pulse repetition delay ¼ 2 s; C—36 mol % resorcinol, pulse repetition delay ¼ 2 s; D—9 mol % resorcinol, pulse repetition delay ¼ 2 s. The dashed line on the left side of part (a) represents the chemical shift of resorcinol’s phenolic carbon in the molecular complex with PEO (phase b). Arrows on the right side of part (a) identify the aromatic resonance of resorcinol that is ortho to both hydroxyl groups. (b) For comparison, high-resolution 13C solid state NMR signals of 2-methylresorcinol’s methyl group in the undiluted crystalline state (upper spectrum, 60 s pulse repetition delay) and in an aqueous precipitate with poly(ethylene oxide) (lower spectrum, 2 s pulse repetition delay). In each case, the cross-polarization contact time was 1 ms.

732

Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers

claim is provided by 1H spin-diffusion results which suggest that dissimilar molecules ˚ . Hence, the phenolic carbon resonance of resorcinol functions are separated by 2 – 5 A as a probe of crystal structure, conformation, molecular packing, and near-neighbor environment, due to hydrogen bonds that are operative in this system. Temperature – composition projections of the phase diagram illustrated in Figure 16.13a for PEO and resorcinol were generated primarily from melting endotherms measured via differential scanning calorimetry (DSC). Thermograms are superimposed on the PEO– resorcinol phase diagram in Figure 16.14 to reveal the connection between first-order endothermic phase transitions and actual phase boundaries. Two molecular-weight-dependent solid – solid –liquid (eutectic) phase transitions at 40 8C and 80 8C are characteristic of this system. The phase diagram offers a phenomenological interpretation of the 13C resonances for the phenolic carbon of resorcinol illustrated in Figure 16.12a. Begin with the 67 mol % resorcinol blend at 15 8C, which exhibits phenolic carbon chemical shifts at 155 ppm and 158 ppm. If this mixture overcomes diffusional limitations and equilibrates from a chemical viewpoint, then the two coexisting phases (b þ g) at 15 8C are found from the intersection of a horizontal (isothermal) tie-line with the boundaries of the two-phase region under investigation (see Fig. 16.13a). Hence, for an overall mixture composition of 67 mol % resorcinol, the two coexisting solid state phases are (i) a resorcinol-rich phase g that is crystallographically similar and “NMR-indistinguishable” from undiluted resorcinol, and (ii) the homogeneous molecular complex (phase b) whose composition corresponds to 2 : 1 stoichiometry. The phenolic carbon resonance of resorcinol in phase b is indicated by the dashed line on the left side of Figure 16.12a at 158 ppm. The crystallographic symmetry of phase g produces a phenolic carbon signal at 155 ppm, which is identical to the chemical shift of the phenolic carbon in undiluted resorcinol. Spectrum C in

Figure 16.13 (a) Temperature – composition projection of the binary phase diagram for methanol-cast blends of poly(ethylene oxide) and resorcinol at constant pressure, illustrating two eutectic transitions at 40 8C and 80 8C. The molecular complex is labeled phase b. Arrows on the compositional axis identify blends whose carbon-13 solid state NMR spectra are presented in Figure 16.12a. (b) For comparison, trieutectic phase behavior is illustrated for binary mixtures of poly(ethylene oxide) with 2-methylresorcinol, where the three-phase (i.e., b, g, liquid) transition slightly above 70 8C between 20 and 30 mol % 2-methylresorcinol represents the crossover between eutectic and peritectic phase transitions.

16.10 High-Resolution Solid State NMR Spectroscopy of PEO

733

Temperature (°C)

110 90 70 50 30 10 0

20

40 60 Mole % of Resorcinol

80

100

Figure 16.14

Temperature – composition projection of the binary phase diagram for poly(ethylene oxide), MW  9  105 Da, and resorcinol, illustrating bieutectic phase behavior at constant pressure. Actual thermograms are presented at 10 different mixture compositions, superimposed on the phase diagram. These DSC thermograms reveal either one or two melting events, and the connection between these first-order melting transitions and the eutectic or liquidus phase boundaries is illustrated by the dashed lines. All thermograms were recorded at a DSC heating rate of 5 8C/min.

Figure 16.12a at 36 mol % resorcinol is characteristic of a two-phase (b þ g) mixture dominated by the molecular complex (phase b) whose phenolic carbon signal appears at 158 ppm. The weak phenolic carbon resonance at 155 ppm suggests that the 36 mol % resorcinol mixture resides within the same two-phase region as the 67 mol % mixture at ambient temperature, albeit at different proportions of the two coexisting phases. In this respect, differential scanning calorimetry and 13C NMR spectroscopy have been used constructively to extend the solid state boundary between solid solution b and the b þ g two-phase region to lower temperatures where the DSC thermograms are featureless. When the overall mixture composition lies to the left of the molecular complex (phase b) in Figure 16.13a, only the phenolic-carbon resonance at 158 ppm is observed, corresponding to resorcinol in the molecular complex. This is illustrated in spectrum D of Figure 16.12a for an overall mixture composition of 9 mol % resorcinol. The data in Figures 16.12 and 16.13 provide convincing evidence that carbon-13 solid state NMR detects bieutectic phase behavior in blends of poly(ethylene oxide) with resorcinol.

16.10.2 Rigid versus Mobile Domains in PEO– Resorcinol Molecular Complexes via 1 H– 13C Cross-Polarization Dynamics A snapshot of the transfer of magnetization from 1H to 13C via heteronuclear spin diffusion in a binary mixture of poly(ethylene oxide) and resorcinol is illustrated in Figure 16.15 at short (i.e., 50 ms) and long (i.e., 20 ms) mixing times during thermal

734

Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers OH PEO (900 kDa) Resorcinol Complex

OH

Molded @ 150 °C

50 ms Rigid

OCH2 of PEO 20 ms Mobile

180

160

140

120 100 80 Carbon-13 Chemical Shift

60

40 ppm

Figure 16.15 High-resolution carbon-13 solid state NMR spectra of poly(ethylene oxide) and resorcinol. Upper spectrum: heteronuclear spin diffusion occurs for 50 ms during 1H– 13C crosspolarization, selecting the rigid b-phase stoichiometric molecular complex. Lower spectrum: the mobile amorphous phase that contains predominantly PEO is selected when heteronuclear spin diffusion occurs for 20 ms during 1H– 13C cross-polarization.

contact of the two spin manifolds. The upper carbon-13 spectrum is characteristic of phase b that separates two eutectics. Strong heteronuclear dipolar couplings between 1H and 13C within the molecular complex produce carbon-13 magnetization after 50 ms of spin diffusion, which is characteristic of a rigid b-phase that exhibits cocrystallization. Notice that resorcinol’s phenolic carbon chemical shift at 160 ppm is indicative of phase b, not phase g that exhibits a phenolic carbon chemical shift at 155 ppm. Morphological discrimination is evident in both 13C spectra of Figure 16.15, because the cocrystallized molecular complex is favored at shorter spin-diffusion times whereas the mobile amorphous phase is favored at much longer spin-diffusion times. There is a considerable amorphous fraction of PEO, as evidenced by the narrow resonance at 70 ppm in the lower spectrum. Hence, PEO is partitioned between the rigid molecular complex and the mobile amorphous phase, but resorcinol resides predominantly in the molecular complex (i.e., phase b).

16.10.3 Molecular Proximity and Spin Diffusion in PEO– Resorcinol Complexes via 1H Combined Rotation and Multiple Pulse Spectroscopy (CRAMPS) Two applications of high-resolution 1H solid state NMR are presented here to support the concept of cocrystallization in the poly(ethylene oxide) –resorcinol molecular

16.10 High-Resolution Solid State NMR Spectroscopy of PEO

735

complex (i.e., phase b in Fig. 16.13a). Both applications exploit the high-resolution nature of 1H CRAMPS, and the simplicity of the solid state 1H spectrum for the homogeneous solid solution of PEO and resorcinol. The first investigation provides a 1H analog of 13C NMR indirect detection of micromixing in strongly interacting systems. However, 1H NMR spectroscopy of resorcinol’s hydroxyl group probes hydrogen bonding directly. Partial charge density surfaces for the minimum-energy conformation of resorcinol are illustrated in Figure 16.16, revealing that resorcinol’s OH groups function as proton donors and proton acceptors for intramolecular and intermolecular hydrogen bonding interactions. The one-dimensional 1H CRAMPS spectrum of the PEO– resorcinol solid solution is presented in Figure 16.17. The data reveal that the aromatic 1H signal of resorcinol resonates at a higher chemical shift in phase b (i.e., stoichiometric molecular complex in the lower spectrum of Fig. 16.17) relative to phase g (undiluted state of pure resorcinol in the upper spectrum of Fig. 16.17). The chemical shift of the hydroxyl 1H resonance is not affected to the same degree, but the NMR splitting of the OH signal in phase g is absent in phase b. Infrared data are consistent with the NMR splitting of the hydroxyl 1H resonance in phase g because the FTIR spectrum exhibits two peaks for the broad hydroxyl vibrational absorption at 3180 cm21 and 3280 cm21. In phase b where the 1H NMR splitting is absent, the OH stretching vibration exhibits one peak at 3346 cm21, suggesting that hydrogen bonds are weaker, on the average, in the stoichiometric molecular complex relative to the selfassociation of resorcinol in phase g. Of primary importance here is a discussion of intermolecular 1H spin diffusion (via 1H CRAMPS) between dissimilar species in a mixture. These experiments provide substantial evidence that dissimilar molecules (or chain segments) reside in a near-neighbor environment when micromixing is thermodynamically or kinetically favored. Resolution of isotropic 1H chemical shifts is required in the 1H CRAMPS version of the spin-diffusion experiment for (i) the generation of a “magnetization

Figure 16.16

(a) Wire mesh and (b) solid partial charge density surfaces for the minimum-energy conformation of resorcinol are illustrated via semiempirical extended-Hu¨ckel computational chemistry algorithms using MOPAC 97 (i.e., Molecular Orbital PACkage) within Chem3D. Partial negative charge is exhibited by the hydroxyl oxygens, partial positive charge is carried by the OH hydrogens, and the four aromatic CH hydrogens are neutral.

736

Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers Aromatics High-Resoultion 1H CRAMPS

Resorcinol HO

OH

15

10

Phase γ

5

0

–5

CH2 of PEO Poly(ethylene oxide)/ Resorcinol Molecular Complex

15 1H

Phase β

10 5 0 –5 Solid State Chemical Shift (ppm)

Figure 16.17 High-resolution solid state 1H spectra of undiluted crystalline resorcinol (upper spectrum, undiluted phase g) and the molecular complex of PEO with resorcinol (lower spectrum, phase b) via the CRAMPS technique. The concentration of resorcinol in the stoichiometric molecular complex is 33 mol %.

gradient” that drives the spin-exchange process via 1H dipolar couplings, and (ii) the detection of spin diffusion in a two-dimensional contour representation of the data. Hence, the molecular complex of PEO and resorcinol (i.e., phase b) is an attractive candidate for analysis via the two-dimensional 1H spin-exchange experiment because (i) the 1H solid state spectrum of each component is rather simple, (ii) spectral overlap between dissimilar blend components is essentially nonexistent or, at most, minimal, and (iii) micromixing and cocrystallization of dissimilar molecules or chain segments is favorable based on chemical structure, thermodynamic driving forces, the presence of interacting functional groups, and the nature of phase b in the temperature – composition projection of the phase diagram. In reference to requirement (iii) mentioned above, the 1H CRAMPS experiment is capable of monitoring 1H spin exchange in intimately mixed as well as phase-separated blends. The duration of the spin-diffusion mixing period necessary to detect “off-diagonal spin-exchange contours” generated by 1H dipolar couplings depends on the solid state morphology of the blend under investigation. The molecular complex of PEO and resorcinol in phase b is analyzed via two-dimensional 1H spin-diffusion spectroscopy in Figures 16.18 and 16.19. The high-resolution 1H spectrum of the solid complex is quite simple as illustrated in

16.10 High-Resolution Solid State NMR Spectroscopy of PEO

737

Poly(ethylene oxide)/Resorcinol Molecular Complex

w2 Resorcinol Hydroxyl OH

Resorcinol Aromatic CH

PEO CH

Spin Diffusion Mixing Time = 100 ms 1H CRAMPS (BR-24) during • Chemical Shift Evolution • Data Acquisition

w1

Two-dimensional 1H spin-diffusion spectroscopy on the molecular complex of PEO and resorcinol (phase b). The mixing period persists for 100 ms in the presence of 1H dipolar interactions. The horizontal and vertical arrows identify off-diagonal resonance contours generated via 1H dipolar communication (aromatic-hydroxyl) within resorcinol. The slanted arrows identify off-diagonal resonance contours generated via intermolecular dipolar couplings between PEO and resorcinol.

Figure 16.18

Resorcinol Aromatic CH

w1

Resorcinol Hydroxyl OH

w2

PEO CH

Poly(ethylene oxide)/Resorcinal Molecular Complex 1 H Spin Diffusion Mixing Time = 40 ms

Two-dimensional 1H spin-diffusion spectroscopy on the molecular complex of PEO and resorcinol (phase b). In this case, dipolar couplings are operative for 40 ms during the mixing period. Only the aromatic-hydroxyl off-diagonal 1H resonance contours, generated via dipolar communication within resorcinol, can be observed when the spin-diffusion mixing time is reduced to 40 ms.

Figure 16.19

738

Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers

Figure 16.17. Representative one-dimensional projection spectra in Figures 16.18 and 16.19 mimic the high-resolution nature of the blend spectrum in Figure 16.17. This allows one to distinguish contours of the methylene 1H nuclei of PEO from the aromatic and hydroxyl proton contours of resorcinol. In Figures 16.18 and 16.19, the resonance contours on the main diagonal from lower left to upper right are assigned, respectively, to the hydroxyl protons of resorcinol, the aromatic protons of resorcinol, and the methylene protons of PEO. After 100 ms of 1H dipolar interaction, spin diffusion between PEO and resorcinol can be observed in Figure 16.18. The horizontal and vertical arrows identify off-diagonal contours generated via 1H dipolar communication (aromatic-hydroxyl) within resorcinol. The slanted arrows identify off-diagonal contours generated via intermolecular dipolar couplings between PEO and resorcinol. Hence, 1H magnetization transport between PEO and resorcinol in phase b is operative on the 100-ms time scale, suggesting that intermolecular 1H ˚ . The bieutectic phase behavior of PEO and resordistances are in the range of 2 – 5 A cinol illustrated in Figure 16.13, coupled with the observation of extremely efficient 1 H spin diffusion in Figure 16.18, suggests that cocrystallization is operative in the stoichiometric molecular complex (i.e., phase b). The data in Figure 16.19 reveal that spin diffusion between PEO and resorcinol is absent when homonuclear 1H dipole – dipole interactions are operative for 40 ms. Four of the six off-diagonal contours, which are observed in Figure 16.18 after 100 ms of 1H dipolar communication, are absent in Figure 16.19. Dipole– dipole couplings between (i) the methylene 1H nuclei of PEO and the hydroxyl protons of resorcinol, and (ii) the methylene 1H nuclei of PEO and the aromatic CH protons of resorcinol do not produce off-diagonal contours in the two-dimensional 1H NMR experiment on a time scale of 40 ms. This is a consequence of dipolar couplings between 1H nuclei that are separated ˚ . Intramolecular (aromatic-hydroxyl) 1H dipolar couplings within by more than 2 A resorcinol are stronger than the intermolecular couplings discussed above, and they are observed when the spin-diffusion mixing time is reduced to 40 ms. This intramolecular interaction produces the pair of off-diagonal contours in Figure 16.19 that are identified by the slanted arrows. In summary, the time scale of the 1H spin-diffusion process is of paramount importance in an analysis of molecular proximity. The ability to differentiate between micromixing at the segment level and phase separation with domain sizes that can be detected via transmission electron microscopy depends critically on the spin-diffusion mixing time and the morphological features of the system. The observation of off-diagonal spin-exchange contours after 100 ms of dipolar interaction in phase b of PEO and resorcinol (see Fig. 16.18) supports the concept of cocrystallization and ˚ level. micromixing at the 2 – 5-A

16.11 CARBON-13 SOLID STATE NMR SPECTROSCOPY: LABORATORY EXPERIMENTS AND DATA ANALYSIS This site-specific experimental technique is useful to identify chemical structures of polymeric repeat units in the solid state, particularly when it is not possible to dissolve

16.11 Carbon-13 Solid State NMR Spectroscopy: Laboratory Experiments

739

the material prior to investigation. One can probe molecular mobility in phaseseparated copolymers and semicrystalline homopolymers, to distinguish between rigid and mobile domains. Detection of miscibility in strongly interacting polymer blends and the formation of transition-metal coordination complexes is possible if one of the components (i) does not contain 1H nuclei, (ii) is completely deuterated, or (iii) contains 13C nuclei whose chemical shift interaction is very sensitive to local environment.

16.11.1

Chemical Shift Interactions

Initially, high-resolution carbon-13 solid-state NMR spectra of the following polymers should be obtained to illustrate how chemical functionality affects the isotropic 13 C chemical shift interaction in the presence of cross-polarization, high-power dipolar decoupling, and magic-angle sample spinning: (i) Polyethylene (ii) Isotactic polypropylene (iii) Poly(ethylene oxide) (iv) Polystyrene (v) Poly(vinyl phenol) (vi) Poly(methyl methacrylate) When spectra are compared with the chemical structure of the polymer’s repeat unit, it is possible to correlate 13C chemical shifts for the following functional groups: methyl, methylene, methine, quaternary, methoxy, aromatic, phenolic, and carbonyl.

16.11.2 Powder Spectrum Without Magic-Angle Sample Spinning The chemical shift anisotropy powder pattern of poly(ethylene oxide) should be measured in a 13C static magnetic field of 50 MHz (i.e., 200 MHz spectrometer). The linewidth of the OCH2 powder spectrum should be compared with the corresponding 13C linewidth in the presence of magic-angle sample spinning at 3500 – 4000 Hz. In both cases, cross-polarization and high-power dipolar decoupling are employed. The isotropic 13C chemical shift of 70 ppm for poly(ethylene oxide) can be obtained by averaging the three principal components of the chemical shielding tensor (i.e., 33, 83, and 91 ppm), which are obtained from the powder spectrum without magic-angle sample spinning. A similar power-pattern spectrum, obtained without magic-angle sample spinning, for bisphenol-A polycarbonate, 13C-enriched at the carbonyl position, is illustrated in Figure 15.3, which should be compared with the high-resolution spectrum in Figure 15.2 that employs magic-angle spinning at 5 kHz (see Section 15.8.1).

740

Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers

16.11.3 Chemical Shift Discrimination Between Crystalline and Amorphous Regions, as Well as the Cation Environment, in Ethylene – Methacrylic Acid Copolymers and Ionomers Carbon-13 solid state NMR spectroscopy identifies at least four segments in these copolymers that differ either chemically or morphologically. Let’s begin with an analysis of the predominant ethylene segments that comprise at least 85 wt %, based on copolymer composition. Earlier in this chapter, magnetic spin diffusion between crystalline and amorphous regions was studied via ethylene segments in these copolymers and ionomers. It should be emphasized that ethylenic CH2 segments are chemically similar in all regions of these materials. However, the presence of some gauche rotational states for carbon – carbon backbone bonds in the amorphous regions provides the basis for morphological discrimination relative to the crystalline CH2 segments that exhibit an all-trans conformation. Spectra in Figure 16.20a identify (a)

Carbon-13 Solid State NMR

(b)

Cation Environment {13C NMR} CH3

CH3 CH2 C

CH2 CH2

Tl+ + COO– Na++ Zn + H

Crystalline

CH2 CH2

CH2 C

Tl+ + COO– Na Zn++ + H

Amorphous

Tl+ Tl+

Na+

Na+

Zn++ 15% MAA

Zn++

4% MAA

15% MAA 4% MAA

LDPE 50

40

30

Chemical Shift

20 ppm

200 195 190 185 180 175 ppm Chemical Shift

Figure 16.20 High-resolution carbon-13 NMR spectra of ethylene– methacrylic-acid random copolymers and ionomers. (a) Crystalline and amorphous CH2 resonances of the ethylene segments near 30 ppm. The all-trans conformational state of backbone bonds in the crystalline regions yields a CH2 chemical shift of 33 ppm, whereas the presence of some gauche rotational states for carbon–carbon backbone bonds in the amorphous regions gives rise to a 2-ppm shift toward lower chemical shielding (relative to TMS) via the g-gauche effect. Spectra of all copolymers and ionomers are compared to that of low-density polyethylene (LDPE), which has a significantly higher crystalline content and a much more intense CH2 signal at 33 ppm relative to the amorphous CH2 resonance at 31 ppm. (b) Effect of neutralizing cations (i.e., Zn2þ, Naþ, and Tlþ ) on the carboxylate carbon resonance near 185 ppm. Relative to the lower two spectra for DuPont NucrelTM copolymers that contain 4 wt % and 15 wt % (i.e., 5.4 mol %) methacrylic acid, it is possible to identify unneutralized (185 ppm) and zinc-neutralized (187– 188 ppm) carboxylic acid carbon resonances in the partially neutralized zinc ionomer (i.e., SurlynTM ).

16.11 Carbon-13 Solid State NMR Spectroscopy: Laboratory Experiments

741

a 2-ppm difference between crystalline and amorphous CH2 resonances in the ethylene segments. The intense resonance envelope between 30 and 35 ppm in Figure 16.20a contains a negligible contribution from CH2 signals in the methacrylic acid segments. For example, the weak resonance near 15 ppm is assigned to the a-methyl carbon in methacrylic acid. It is relatively straightforward, though not necessarily quantitative, to develop empirical correlations between percent crystallinity and the ratio of the CH2 resonance intensities at 33 ppm versus 31 ppm. The unneutralized methacrylic acid carbon chemical shift at 185 ppm in Figure 16.20b is chemically distinct from the zinc-neutralized carboxylate carbon signal at 187– 188 ppm. Additional information about the carboxylate carbon resonance is provided in the next section on transition-metal coordination complexes with poly(4-vinylpyridine).

16.11.4 Effect of Transition-Metal Coordination to Poly(4-vinylpyridine) on the Carboxylate-Carbon Chemical Shift in Zinc Acetate and Zinc Laurate Zinc Acetate Dihydrate and Poly(4-vinylpyridine) 13

C NMR spectroscopy of the carboxylate-carbon is employed as a diagnostic probe of coordination to d-block metal cations in the solid state, via the lone pair on nitrogen in the side group of poly(4-vinylpyridine). The pseudo-octahedral ligand arrangement in zinc acetate dihydrate and the proposed displacement of one lattice water in the coordination sphere by a pyridine side group are illustrated in Figure 16.21. Synergistic glass-transition-temperature response has been measured in the vicinity of 10– 30 mol % zinc acetate, where Tg of the blends is greater than that of

Figure 16.21

Molecular models of (a) zinc acetate dihydrate and (b) zinc acetate monohydrate coordinated to one pyridine side group in P4VP illustrating the concept of a “coordination pendant group.” These pseudo-octahedral models are adopted from the ligand arrangement in zinc acetate dihydrate, based on its crystal structure [VanNiekerk et al., 1953]. It is proposed that one pyridine side group in the polymer displaces one weak-base water of hydration in the coordination sphere of the divalent zinc cation.

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Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers

the undiluted polymer. The maximum enhancement in Tg is approximately 20 8C at 16 mol % zinc. The melting transition of undiluted zinc acetate near 245 8C is not observed in blends that contain less than 90 mol % zinc. NMR spectra in the carboxylate-carbon chemical shift region, which is unique to zinc acetate, reveal that the relatively sharp resonance in the vicinity of 185 ppm is broadened and shifted gradually to 179 ppm as zinc acetate is diluted. There is no 13C NMR evidence for the coexistence of both crystalline and amorphous phases that contain spectroscopically detectable fractions of zinc acetate. Synergistic glass-transition-temperature response has been measured for coordination complexes whose 13C NMR data are illustrated in the lower three spectra of Figure 16.22a. When Tg of the blends exceeds that of the undiluted polymer, the peak of the broad carboxylate-carbon resonance is measured

(a)

(b) 200

200 Zinc Acetate/Poly(4-vinylpyridine)

Zinc Laurate/Poly(4-vinylpyridine)

Temperature (°C)

Tg (°C)

150

100

50

150

Tm Tg

100

Amorphous Blends

50

Amorphous Blends 0

0

10

20

30

40

50

60

70

80

90

100

0

0

10

20

Zinc Acetate (% mole)

30

2-phase Blends 40

50

60

70

80

90

100

Zinc Laurate (% mole)

NMR

Carboxyl Resonance

(CH3COO)2Zn

Carboxyl Resonance Zn(CH3(CH2)10COO)2

100%

Mole% Zn Acetate 100%

85% 72%

65%

65%

Mole% Zn Laurate

50% 40%

34% 28%

32%

16%

18%

10% 190

185

180

175

170 ppm

Solid State 13C NMR Chemical Shift

190

185

180

175

170 ppm

Solid State 13C NMR Chemical Shift

Figure 16.22 Correlations between DSC-generated temperature –composition phase diagrams (upper) and solid state NMR spectroscopy of the carboxylate carbon (lower) for transition-metal coordination complexes between (a) zinc acetate with poly(4-vinylpyridine) and (b) zinc laurate with poly(4-vinylpyridine).

16.11 Carbon-13 Solid State NMR Spectroscopy: Laboratory Experiments

743

at 179 ppm. At higher concentrations of zinc acetate in which Tg of the blends no longer surpasses that of undiluted poly(4-vinylpyridine), the peak position of the carboxylate-carbon resonance is measured at progressively higher chemical shifts, which approach 185 ppm, characteristic of completely crystalline zinc acetate. To emphasize the importance of structural variations in the pyridine ring, blends of zinc acetate with poly(2-vinylpyridine) exhibit a melting transition when the concentration of zinc ranges from 40 to 100 mol %, and the polymer’s glass transition is marginally affected in these two-phase mixtures. Zinc Laurate and Poly(4-vinylpyridine) Now, the zinc salt contains 10 CH2 spacer groups between CH3 at the end of the alkyl tail and the carboxylate group. The interaction-sensitive carboxylate-carbon resonance of zinc laurate is useful to correlate site-specific results from solid state NMR with the temperature – composition projection of the phase diagram for blends with poly(4-vinylpyridine). 13C NMR and differential scanning calorimetry identify twophase behavior over a wide concentration range. Both Tg of the polymer and Tm of the zinc salt are depressed with no evidence of synergism. There are two contributors to the overall resonance envelope of zinc laurate’s carboxylate carbon, separated by approximately 4 ppm. The relatively sharp signal near 185 ppm is characteristic of completely crystalline zinc laurate. The formation of amorphous coordination complexes between these two components produces a new mixing-induced signal at 181 ppm when P4VP is the dominant component. The larger full-width-at-halfheight of the 181-ppm resonance relative to the sharp signal at 185 ppm suggests that complexation occurs in the amorphous phase, and it is detected below the glass transition temperature. In both transition-metal coordination complexes with poly(4-vinylpyridine), illustrated in Figure 16.22, the rather broad carboxylatecarbon resonance in the vicinity of 180 ppm is an indicator of metal – ligand interactions between the zinc cation and the structurally accessible nitrogen lone pair in the side group of the polymer. Two-phase behavior is observed from the viewpoint of solid state NMR at 32, 40, and 65 mol % zinc laurate, where the crystalline and rigid amorphous phases are detected simultaneously on the lower right of Figure 16.22b. The appearance of the unresolved shoulder at 185 ppm in the NMR spectrum at 32 mol % zinc laurate suggests that solid state NMR can detect the crystalline phase, which is in low abundance and slightly disordered, whereas the DSC thermogram for this mixture is dominated by the depressed glass transition process of poly(4-vinylpyridine) in the 60– 70 8C temperature range. The liquidus line representing Tm depression converges on the concentration-insensitive glass-transition phase boundary in the vicinity of 30– 35 mol % zinc laurate. This poses a mobility restriction for zinc laurate crystallization in the mixture that contains 32 mol % zinc, thereby retarding the kinetics that govern crystallization of the small-molecule-rich phase from the molten state. At 18 mol % zinc laurate, all metal cations are coordinated to pyridine side groups in amorphous complexes, as detected by a single resonance envelope for the carboxylate carbon in the lowest spectrum of Figure 16.22b.

744

Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers

16.11.5 Molecular Mobility via Dipolar Dephasing Experiments Heteronuclear dipolar interactions between 1H and 13C can be exploited to distinguish between crystalline and amorphous CH2 signals in polyethylene via the dipolar dephasing experiment (see Fig. 16.2). Conformational differences between polyethylene chains in the crystalline and amorphous domains give rise to two 13CH2 signals at different chemical shifts. The signal at higher chemical shift, between 33 and 34 ppm, is assigned to ethylenic carbons in the crystalline regions where chains adopt the alltrans planar zigzag conformation. Amorphous polyethylene segments, which undoubtedly contain some gauche backbone-bond rotational states, give rise to a 13 C signal at 31 ppm (see Fig. 16.20a). The empirical g-gauche effect predicts this difference in isotropic 13C chemical shifts based on conformational analysis. Carbon – carbon backbone bonds in a gauche rotational state, characterized by rotation angles of 608 (i.e., gaucheþ) or 3008 (i.e., gauche2) force carbon atoms that are separated by three bonds in the chain backbone to reside in closer spatial proximity relative to the trans conformation, characterized by rotation angles of 1808. Newman projections allow one to visualize spatial proximity when a particular bond adopts different rotational isomeric states. The gauche conformation has been found empirically to decrease 13C chemical shifts in amorphous polyethylene relative to the crystalline regions. Introduction of a 50 ms delay without high-power 1H decoupling between cross-polarization and acquisition segments of the pulse sequence for 13C free induction decays allows one to detect polyethylene segments in mobile domains because 1H– 13C dipolar interactions are partially averaged by molecular motion. Severe attenuation of 13CH2 signals occurs in the crystalline regions because there is insufficient mobility to reduce the strength of these dipolar interactions. Mobile carbons, nonprotonated carbons, and methyl carbons will survive the dipolar dephasing delay. Hence, this technique is useful for chemical and morphological discrimination between various types of carbon-13 resonances (i.e., spectral editing).

16.11.6 Molecular Mobility in Block Copolymers via Selective Cross-Polarization and Modified Goldman –Shen Experiments Selective Cross-Polarization Heteronuclear dipolar interactions between 1H and 13C can also be employed to observe rigid or mobile domains in styrene – butadiene– styrene triblock copolymers via cross-polarization experiments at short (i.e., 100 ms) and long (i.e., 20 ms) thermal mixing (i.e., spin-diffusion) times. When spin diffusion occurs for 100 ms during cross-polarization, one selectively observes 13C resonances in the rigid polystyrene domains because 1H– 13C dipolar interactions are stronger and polarization transfer from 1H to 13C is more efficient, relative to the mobile phase. At longer spin-diffusion times, both phases can be observed. The nature of cross-polarization

16.11 Carbon-13 Solid State NMR Spectroscopy: Laboratory Experiments

745

dynamics, discussed in Section 16.7.1 and illustrated in Figure 16.4, allows one to selectively observe mobile polybutadiene domains when the contact time is 20 ms. Carbon-13 NMR spectra in Figure 16.6 illustrate the use of selective cross-polarization experiments to probe domains with significantly different mobility in these triblock copolymers. Modified Goldman – Shen Mobility discrimination between dissimilar segments of block copolymers is also accessible from the modified Goldman – Shen experiment, described earlier in this chapter and illustrated below for a segmented polyurethane elastomer that contains rigid diphenylmethane di-isocyanate (MDI) hard segments and poly(tetramethylene ether glycol) (PTMEG) soft segments. The 1H dipolar dephasing filter was 15 ms, the mixing (spin-diffusion) time was 10 ms, the 1H– 13C cross-polarization contact time was 100 ms, and 1H magnetization was stored alternately along the +z-axis (i.e., collinear with the static magnetic field) at the start of the mixing period to suppress spin – lattice relaxation processes. It is evident from the lower spectrum in Figure 16.23 that 15 ms of 1H dipolar dephasing is sufficient to null the rigid OCH2 signal (dashed line) characteristic of the butanediol hard-segment chain extender before 1H spin diffusion is initiated. Furthermore, 100 ms of cross-polarization thermal contact between 1H and 13C is short enough to suppress long-range 1H – 13C dipolar communication and maintain the null in the hard-segment OCH2 signal intensity in the lower spectrum. Redistribution of 1H polarization between the hard and soft segments occurs (and most likely equilibrates) on the order of 10 – 100 ms, based on the appearance of the composite OCH2 signal in the upper spectrum of Figure 16.23.

16.11.7 Effect of Hydrogen Bonding on Isotropic 13C Chemical Shifts in Amorphous and Semicrystalline Polymer –Polymer Complexes Poly(dimethylacrylamide) and Poly(vinylphenol) As illustrated earlier in this chapter, hydrogen-bonded molecular complexes of poly(ethylene oxide) and resorcinol reveal sensitivity of resorcinol’s phenolic 13C isotropic chemical shift to the phases that are present in this bieutectic mixture. The connection between phenolic carbon chemical shifts and the PEO – resorcinol phase diagram (see Figs. 16.12 and 16.13) is rather unique because the presence of a stoichiometric molecular complex (i.e., phase b) that separates two eutectic (i.e., solid– solid – liquid) phase transformations is more common in metal alloys than polymer science. However, polymer – solvent molecular complexes are responsible for the formation of thermoreversible gels [Guenet, 1992, 2008]. Analogous to resorcinol, the phenolic 13C chemical shift of poly(vinylphenol) is sensitive to hydrogen bonding interactions in the amorphous phase of strategically selected polymer – polymer blends. In this example, poly(vinylphenol) and poly(dimethyl acrylamide) are mixed in a common solvent, and the 1 : 1 stoichiometric complex based on

746

Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers

Rigid OC H2

10 ms Mixing

No Mixing

100

80

60

40

20

0

ppm

Carbon-13 Solid State Chemical Shift

Figure 16.23 High-resolution carbon-13 solid state NMR spectra of an MDI-based polyurethane elastomer from the slightly modified Goldman– Shen experiment. This linear polyether– polyurethane was supplied by the UpJohn Company in North Haven, Connecticut. The polydisperse urethane hard segment is based on 4,40 -diphenylmethane di-isocyanate (MDI), chain extended with butanediol. Hydroxyl-equivalent data indicate that the mean hard-segment length consists of three to four urethane repeat units, which is sufficient for hard-segment crystallization to occur. The soft segment is a 1000-molecular-weight poly(tetramethylene ether glycol) (PTMEG) and the overall soft-segment weight fraction is 42%. Lower spectrum: the initial condition before 1H spin diffusion is initiated. Two strong signals at 27 ppm and 72 ppm that survive the 15-ms dipolar-dephasing filter are assigned to internal CH2 and OCH2 carbon-13 resonances in the polyether (PTMEG) soft segment. Upper spectrum: after 10 ms of 1H spin diffusion, a “near-equilibrium” spectrum is observed based on 100 ms of cross-polarization thermal contact between 1 H and 13C, and a 2-s pulse repetition delay. The dashed line and the arrow identify the hard-segment OCH2 resonance of butanediol at 66 ppm, generated via 1H dipolar couplings across interfacial boundaries with the mobile polyether-rich microdomains. The internal methylene CH2 signal of butanediol (i.e., hard-segment chain extender) at 25 ppm resonates at a slightly different chemical shift relative to the CH2 signal of the polyol soft segment at 27 ppm. Hence, the overall CH2 resonance envelope between 20– 30 ppm in the upper spectrum is slightly broader than that in the lower spectrum because the hardsegment butanediol signals have been regenerated after 10 ms of 1H spin diffusion. The CH2 resonance near 40 ppm is characteristic of MDI in the upper spectrum, after these hard-segment signals are regenerated by 1 H spin diffusion. Other hard-segment assignments for MDI-based polyurethane elastomers are protonated aromatic carbons at 120 ppm and 130 ppm, and substituted (i.e., nonprotonated) carbons in MDI’s aromatic rings at 137 ppm. The urethane carbonyl carbon resonates at 154 ppm.

repeat-unit molecular weights for each polymer is recovered after solvent evaporation in a fume hood and further drying at elevated temperature under vacuum. As illustrated in the vicinity of the dashed line in Figure 16.24, the phenolic 13C chemical shift of poly(vinylphenol) is found between 155 and 160 ppm, depending on the nature of the hydrogen bonds. In the undiluted state, hydrogen bonding (i.e., self-association) between aromatic hydroxyl groups in similar repeat units on either the same chain or different chains yields a phenolic 13C chemical shift at 155 ppm in spectrum (b) of Figure 16.24, similar to resorcinol. When intermolecular hydrogen bonds (i.e., NZCvO. . . .HOZC6H4) between dissimilar species are favored in the stoichiometric complex of these two

16.11 Carbon-13 Solid State NMR Spectroscopy: Laboratory Experiments

747

1:1 Complex

197 Hz (c) Aromatic CH (b)

140 Hz

C O N(CH3)2

OH

(a) 200

180

160

140

120

100 ppm

Carbon-13 Solid State Chemical Shift

Figure 16.24

High-resolution carbon-13 solid state NMR spectra of (a) poly(dimethyl acrylamide), (b) poly(vinyl phenol), and (c) a stoichiometric 1 : 1 complex of both amorphous polymers. Experiments were performed on a NT-150 spectrometer with a 13C Larmor frequency of 37.735 MHz. The 908 pulse width for 1H was 5 ms, which corresponds to dipolar decoupling at 50 kHz. The 1H– 13C cross-polarization contact time was 2 ms, also at 50 kHz, and magic-angle sample spinning was performed at 3600 Hz. (Source: Unpublished research collaboration with Prof. TK Kwei, Polytechnic University, Brooklyn, NewYork.)

amorphous polymers, the phenolic 13C chemical shift of poly(vinylphenol) approaches 160 ppm and the carbonyl 13C resonance of poly(dimethyl acrylamide) near 175 ppm is broadened by approximately 40% (i.e., from 140 to 197 Hz in a static magnetic field where the Larmor frequency for 13C nuclei is 37.735 MHz). Other experimental parameters used to generate the three 13C spectra in Figure 16.24 are included in the caption. Poly(vinylphenol) with either Poly(ethylene oxide) or Poly(vinyl methyl ketone) Two additional examples of the effect of hydrogen bonding between dissimilar species on the phenolic carbon resonance of poly(vinylphenol) (PVPh) are illustrated in Figure 16.25. The hydroxyl substituent on the aromatic side group interacts energetically with the ether oxygen in poly(ethylene oxide) (Fig. 16.25a) and the carbonyl oxygen in poly(vinyl methyl ketone) (Fig. 16.25b). In both cases, NMR sensitivity toward these interactions yields new phenolic carbon resonances near 156 ppm (i.e., dashed line in Fig. 16.25) that significantly overlap the undiluted 13C signal near 153– 154 ppm in poly(vinyl phenol). Hence, carbon-13 solid state NMR spectra of most polymer – polymer blends reveal a composite phenolic carbon resonance that is insufficiently resolved, due to self-association versus hydrogen bonding between dissimilar chain segments. The superficial appearance of the spectra in Figure 16.25 might suggest a smooth transition from one absorption at 153 – 154 ppm in undiluted

748

Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers

(a)

(b)

Semi-Crystalline Blends of Poly(vinyl phenol) and Poly(ethylene oxide)

(CH2 CH) x

Amorphous Blends of Poly(vinyl phenol) and Poly(vinyl methyl ketone)

OH

wt % PVPh H G F E D C B A

20 40

(CH2 CH) x OH

A

wt % PVPh 100

B 90 C 80

50 D 60 70 80 90

100 150 ppm 160 155 13 C Solid State Chemical Shift

70 E 60 F G H

40 30 20

I 10 160 155 150 ppm 13 C Solid State Chemical Shift

Figure 16.25 High-resolution carbon-13 solid state NMR spectra of the phenolic carbon resonance of poly(vinyl phenol) in hydrogen-bonded blends with (a) poly(ethylene oxide) and (b) poly(vinyl methyl ketone). Blends in part (a) are semicrystalline when the PEO concentration is 50 wt % or greater (i.e., spectra F, G, H), but all blends in part (b) are completely amorphous.

poly(vinyl phenol) to a predominant signal near 156 ppm when this amorphous polymer is diluted significantly by the other component, but the overall resonance envelope is described best by two distinct signals that are not well resolved. If one compares infrared and 13C NMR results for amorphous blends of poly(vinylphenol) (PVPh) and poly(vinyl methyl ketone) (PVMK) with those for composition-dependent semicrystalline blends of poly(vinylphenol) and poly(ethylene oxide) (PEO) the following similarities and differences are observed. In both cases illustrated in Figure 16.25, the broad phenolic carbon resonance of PVPh is sensitive to the presence of the other polymeric component in the amorphous phase. Consequently, two poorly resolved phenolic carbon signals, separated by 2 ppm, can be detected in the overall resonance envelope when competing equilibria produce hydrogen bonds of different strength that are NMR sensitive, with the signal at lower chemical shift representative of hydroxyl self-association. When PVPh is increasingly diluted with either PVMK or PEO, the single glass transition process characteristic of compatibility in the amorphous phase decreases to temperatures (i) slightly above ambient for mixtures of PVPh and PVMK or (ii) considerably below ambient for mixtures of PVPh and PEO. The Tglass/TCrystallization/TMelt phase diagram for composition-dependent semicrystalline blends of PVPh and PEO is compared with the Tg – composition phase diagram for amorphous blends of PVPh and PVMK in Figure 16.26. In both cases, the increase in chain mobility that accompanies the decrease in Tg produces considerable narrowing in the linewidth of the phenolic carbon resonance.

16.11 Carbon-13 Solid State NMR Spectroscopy: Laboratory Experiments (b)

(a) PEO PVPh Semi-Crystalline 900 K 1500–7000 Polymer–Polymer Blends (CH2 CH2O) x Temp. – Composition Phase Diagram (CH2 CH)x

Amorphous Polymer–Polymer Blends 100 Temp. – Composition Phase Diagram 90 80

90

OH

70

Tm Tg, °C

70 Temperature, °C

749

50 30

Tg

60 50

PVMK (CH2 CH)x —O C—

CH3

PVPh (CH2 CH)x

40

OH

–10

30

η = 3.5 Ψ = –96 K

–30

20

10

–50 100

Tc

80

40 60 PEO (wt %)

20

0

10 100 80 60 40 20 0 PVMK (wt %)

Figure 16.26

Temperature – composition phase diagrams for polymer– polymer blends of poly(vinylphenol) with either (a) poly(ethylene oxide), Mn  900 kilodaltons, or (b) poly(vinyl methyl ketone), Mn  500 kilodaltons. Blends in part (a) are semicrystalline when the PEO concentration is 50 wt % or greater, but all blends in part (b) are completely amorphous. Parameters in the Kwei equation (i.e., Tg vs. composition, see Section 6.9 and the Appendix of Chapter 6) for amorphous blends of PVPh and PVMK are h ¼ 3.5 and c ¼296 K. Question for discussion: Explain why PEO/PVPh blends exhibit melting endotherms at 50 wt % and 60 wt % PEO, but crystallization exotherms are not observed for blends that contain less than 65 wt % PEO as a consequence of the decrease in mobility, due to the fact that TC would be below the glass transition temperature.

Infrared spectroscopy reveals that hydrogen bonds between PVPh and PVMK (i.e., y OH  3400 cm21) are weaker than the self-association of hydroxyl groups in PVPh (i.e., y OH  3350 cm21), whereas hydrogen bonds between PVPh and PEO (i.e., y OH  3220 cm21) are stronger than the self-association of hydroxyl groups in undiluted PVPh. For comparison, the non-hydrogen-bonded, or free, OH stretching vibration in poly(vinyl phenol) occurs at 3525 cm21, and the free OH stretch of phenol is observed at 3610 cm21 at infinite dilution in tetrachloroethylene. Explanation: PVPh/PEO Both the hydroxyl oxygen of PVPh and the ether oxygen of PEO compete for the hydroxyl proton as a hydrogen-bonding partner. Stronger hydrogen bonds and weaker OH stretching vibrations result from interaction between the hydroxyl proton and the oxygenic functional group that is more electronegative. Functional group electronegativities are required (see footnote to Table 16.1), which can be calculated from atomic electronegativities proposed by Pauling [1960]. The electronegativity of the hydroxyl-oxygen substituent on the aromatic ring in PVPh (i.e., 2.90) is slightly less than that of the aliphatic ether oxygen (i.e., 2.95) because methylene groups in PEO donate electron density into the ether oxygen, whereas the aromatic ring of PVPh extracts electron density from

750

Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers

Table 16.1 Functional Group Electronegativities and Brønsted Ionization Equilibrium Constants for Model Compounds that Represent Poly(ethylene oxide), Poly(vinyl phenol), and Poly(vinyl methyl ketone)

Polymer

Maximum number of Model structure (central bonds removed from atom is located to the left the central atom that of the asterisk)a have been considered

PEO PVPh PVMK

OCH2CH2O CH2CH2O Aromatic-O H (C3H5)(CH3)C vO

3 3 3

Functional group electronegativity

pKa of the superacid

2.95 2.90 2.74

23.5 26.4 27.0

a

Asterisk identifies the “central atom” for group electronegativity calculations, based on Pauling’s electronegativity scale and the superatom approximation: P VCentral Atom ECentral Atom þ Ni Ei Functional group electronegativity EGroup ¼

Atoms& groups

VCentral Atom þ NTotal

where VCentral Atom and ECentral Atom are the valence of the central atom and its atomic electronegativity, respectively. Ni and Ei are the number of bonds of atomic or group i connected to the central atom and the atomic or group electronegativity of i (atom or group), respectively. NTotal is the total number of atoms and groups connected to the central atom.

the hydroxyl oxygen’s free electron pairs. This claim is supported by superacid dissociation constants (i.e., pKa) for low-molecular-weight model compounds in aqueous solution at 298 K [March, 1985]. Since C6H5OHþ 2 (i.e., pKa  26.4) is more acidic than R2OHþ (i.e., pKa  23.5), then in terms of conjugate bases, C6H5OH is a weaker base than ROR, where C6H5OH is a model for poly(vinylphenol) and ROR simulates poly(ethylene oxide). This is consistent with the fact that stronger hydrogen bonds form between the hydroxyl group in PVPh and the ether oxygen in PEO, relative to the self-association of hydroxyl groups in PVPh. Hence, the chemical shift of PVPh’s phenolic carbon is sensitive to the presence of the other blend component, but hydrogen-bond strength is not the only factor that affects carbon-13 resonance positions. Explanation: PVPh/PVMK A simple thermodynamic analysis of the infrared results for these blends favors chemical stability of both components in a homogeneous amorphous phase. Single-Tg behavior illustrated in Figure 16.26 represents macroscopic evidence for compatibility, a concept that is supported spectroscopically for the blends of interest, but it is not proved unambiguously by the infrared and 13C NMR analyses alone. Conjugate acids of low-molecular-weight aliphatic ketones, R2CvOHþ, pKa  27, analogous to PVMK, are slightly more acidic than conjugate acids of aromatic hydroxyl groups, C6H5OHþ 2 , pKa  26.4, analogous to PVPh. Furthermore, functional group electronegativity calculations suggest that the hydroxyl-oxygen substituent on the aromatic ring in PVPh (i.e., 2.90) has a greater ability to attract electron density, maintain partial negative charge, and form stronger hydrogen bonds than aliphatic ketones (i.e., 2.74). Hence, the basicity and

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electronegativity of the hydroxyl-oxygen substituent on the aromatic ring in PVPh (i.e., C6H5OH) are greater than those of the carbonyl oxygen in PVMK (i.e., R2CvO). This interpretation of hydrogen-bond strength is consistent with infrared results that reveal stronger hydrogen bonds due to self-association of hydroxyl groups in undiluted poly(vinylphenol). Functional group electronegativities, tabulated in Table 16.1, might be a much better indicator of hydrogen bond strength than Brønsted ionization equilibrium constants (i.e., pKa), but Table 16.1 reveals that the trends are similar. Thought-Provoking Exercise: Consider the energetics of mixing on the Flory –Huggins lattice and explain why blends of poly(vinylphenol) and poly(vinyl methyl ketone) are completely miscible, even though hydrogen bonds between similar segments in undiluted PVPh (i.e., self-association) are stronger than hydrogen bonds between dissimilar segments in PVPh/ PVMK blends. Answer: The thermodynamics of mixing in binary systems that contain high-molecularweight species stipulates that the dimensionless interaction free energy of mixing, denoted by xAB, must be negative (i.e., exothermic) to avoid phase separation, because the favorable entropy of mixing is negligible for polymer –polymer blends. If xAB is negative, then the free energy of mixing will also satisfy the curvature condition for chemical stability. The energetics of hydrogen-bond formation are typically exothermic, and spectroscopic data for blends of PVPh and PVMK reveal that hydrogen bonds between dissimilar chain segments are weaker than the self-association of hydroxyl groups in PVPh. The appropriate scheme is A– A þ B – B , 2 A –B, where A represents a segment of PVPh, B is a segment of PVMK, and each phenolic side group in undiluted PVPh participates in only one hydrogen bond with another phenolic side group on the same chain or different chain. Section 3.4.4 provides an analysis of the interaction free energy of mixing on the Flory–Huggins lattice and defines xAB for binary mixtures: n 1AA þ 1BB o kBoltzmann TxAB ¼ z 1AB  2 where z is the lattice coordination number, T represents absolute temperature, and 1ij is the segment interaction energy between species i and species j. The segment interaction energies that account for hydrogen bonding are 1AA and 1AB, whereas 1BB describes weak dipole– dipole interactions within undiluted PVMK. The following assumptions are employed, with partial assistance from infrared spectroscopy: 1BB j1ABj , j1AAj. Furthermore, 1AA and 1AB must be negative segment interaction energies to simulate “potential well” conditions, with a deeper well necessary to describe stronger self-association of hydroxyl groups in undiluted PVPh relative to hydrogen bonding in the blends. Hence, j1AA j . j1AB j 1BB  0 If the assumptions and infrared analysis described above are correct, then the energetics of mixing will be exothermic and phase separation will not occur when j1ABj . j1AAj/2. Predictions from quantum mechanics that account for charge transfer between the proton donor and proton acceptor suggest that hydrogen bond energies (i.e., the magnitudes of 1AA and 1AB) are on the order of 3– 10 kcal/mol [Pimentel and McClellan, 1960; Oliveira et al.,

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Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers

2006]. For the particular hydrogen bonds (i.e., OZH. . .O) that are operative in undiluted PVPh and in blends of PVPh with PVMK, these bond energies are between 3 and 7 kcal/mol [Coulson, 1961]. Furthermore, OH stretching frequencies between 3200 and 3400 cm21 suggest that oxygen(O1) –oxygen(O2) distances (i.e., O1ZH. . .O2) in these amorphous solids are ˚ [Coulson, 1961], and the hydrogen atom position is skewed toward approximately 2.7–2.8 A ˚ in O1ZH. . .O2) to which it was the hydroxyl oxygen (i.e., O1ZH bond length 0.97 A bound originally. Infrared data for OH stretching vibrations in undiluted PVPh and blends of PVPh with PVMK reveal that j1AAj 2 j1ABj  50 cm21  0.14 kcal/mol. Hence, the difference between segment interaction energies is approximately one to two orders of magnitude smaller than typical hydrogen bond energies, providing strong support for the fact that j1ABj . j1AAj/2, xAB , 0, the energetics of mixing are exothermic, and phase separation should not occur in blends of PVPh and PVMK.

Poly(acrylic acid) and Poly(ethylene oxide) PEO-rich blends are semicrystalline, and PAA-rich blends are completely amorphous. This claim is supported by the thermograms from differential scanning calorimetry in Figure 16.27, which reveal an absence of PEO melting when PAA is the dominant component (i.e., 73 wt %) in these blends. The NMR spectra illustrated in Figure 16.28 provide evidence that the carboxylicacid carbon resonance near 180 ppm in the side group of PAA is sensitive to selfassociation of COOH groups versus hydrogen bonding between dissimilar chain segments (i.e., with poly(ethylene oxide)). The COOH resonance envelope in Figure 16.28 is skewed toward lower chemical shifts (i.e., slightly below 180 ppm)

PAA + PEO Blends

Pure PAA

Endotherm

27% PEO 72% PEO

Pure PEO

40

50

60

70

80 90 100 110 120 130 140 150 Temperature (°C)

Figure 16.27 Thermograms from differential scanning calorimetry for hydrogen-bonded blends of poly(acrylic acid) with poly(ethylene oxide). Blend composition is identified at the right of each thermogram. PEO melting is observed near 60 8C in the lower two heating traces. Blends are completely amorphous at 27 wt % PEO, but semicrystalline at 72 wt % PEO.

16.11 Carbon-13 Solid State NMR Spectroscopy: Laboratory Experiments Model Blends

{13C NMR}

753

Polymer–Polymer Blend

〈Mn〉 PEO

CH2CH2–O

3400

PAA

CH2CH

5000

PEO

CH2CH2 O

27%

PAA

CH2CH

73%

COO–H+

COO–H+ Annealed @140 °C 12 Hours in Vacuum Physical Mix 28% Water Evaporation @ Ambient Temp. in Vacuum

73% PAA 200

100 °C in Vacuum as Received 190 180 170 160 ppm Chemical Shift

200

160 ppm 180 Chemical Shift

Figure 16.28

High-resolution carbon-13 solid state NMR spectra of the carboxylic-acid carbon resonance of poly(acrylic acid), PAA, in hydrogen-bonded blends with poly(ethylene oxide). Blends are completely amorphous at 73 wt % PAA, but semicrystalline at 28 wt % PAA. Powdered samples of both polymers were combined in the absence of solvent or thermal treatment to generate the physical mixture whose 13C spectrum is illustrated in the upper left.

when PEO is present, and this signal slightly below 180 ppm increases at the expense of the signal above 180 ppm at higher PEO concentrations. There are several examples of the carboxylic-acid and carboxylate-carbon resonances in this chapter that exhibit sensitivity to near-neighbor proximity in solid state polymer blends, ionomers, and molecular complexes. Isotropic NMR chemical shifts, obtained via magic-angle sample spinning and high-power 1H – 13C dipolar decoupling, respond to strong interactions that perturb electronic wavefunctions associated with the nucleus under observation. 13C NMR spectra of ZCOOH in Figure 16.28 can identify (i) self-association, or hydrogen bonding between ZCOOH side groups on the same chain or different chains in undiluted poly(acrylic acid), (ii) hydrogen bonding between dissimilar chain segments in polymer – polymer blends, (iii) water absorption, and (iv) the effect of thermal treatment in vacuum.

16.11.8 Detection of Molecular Proximity in Polymer –Polymer and Polymer– Small-Molecule Blends via 1H – 13C Spin Diffusion When One of the Components Is Completely Deuterated Obtain a high-resolution 13C solid state NMR spectrum of a binary mixture that is completely or partially miscible. It is also necessary to record 13C spectra of each

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Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers

pure component. The appropriate blend of interest must be chosen such that 13C signals for the pure components do not overlap significantly. For example, poly(ethylene oxide) with resorcinol and poly(methyl methacrylate) with 2,20 dinitrobiphenyl meet the requirement specified above. Let’s consider one of these polymer – small-molecule blends. If the polymer is protonated, then polarization of the various 13C nuclei in the repeat unit will be obtained intramolecularly via single Hartmann – Hahn contacts with the abundant 1H nuclear reservoir. The highresolution 13C NMR spectrum of atactic poly(methyl methacrylate) is illustrated in Figure 16.29a with the appropriate peak assignments. However, if the polymer is completely deuterated, then the mechanism by which 13C magnetization is generated in the undiluted homopolymer (i.e., cross-polarization via intramolecular 1H – 13C dipolar interactions) is effectively thwarted. Hence, the perdeuterated polymer should not exhibit a high-resolution 13C solid state NMR spectrum via cross-polarization, and this claim should be verified (see the lower spectrum in Fig. 16.29b). Now, use solution-blending techniques and introduce a protonated small molecule in a binary mixture with the perdeuterated polymer. If the two components are compatible at the molecular level, then the possibility of generating 13C magnetization in the perdeuterated polymer via 1H – 13C dipolar couplings is revived. In this case, however, the cross-polarization pathway is intermolecular in origin. Since the rate of intermolecular 1H – 13C spin diffusion is inversely proportional to the sixth power of the internuclear distance between coupled spins (i.e., 1H and 13C), one has

(b) d8-PMMA/Dinitrobiphenyl Thermal mixing time during magnetic spin diffusion (i.e., cross polarization) = 10 msec 1H–13C

(a) CH2

CH3 C

x

C O O CH3

O2N NO2

30% DNB

α-CH3

200

150

100

50

0 ppm

250 200 150 100 50 Chemical Shift

d8-PMMA 0 ppm

Figure 16.29 (a) Poly(methyl methacrylate)—high-resolution carbon-13 NMR spectrum via 1H– 13C cross-polarization, magic-angle sample spinning, and dipolar decoupling during data acquisition. (b) Intermolecular polarization transfer via 1H– 13C magnetic spin diffusion during cross-polarization in a miscible binary mixture, where the polymer, poly(methyl methacrylate), is essentially completely deuterated. Magnetic dipolar interactions between 1H nuclei in the aromatic ring of 2,20 -dinitrobiphenyl and 13 C nuclei in the polymer chain provide the mechanism that generates the upper spectrum of PMMA via intermolecular cross-polarization.

16.11 Carbon-13 Solid State NMR Spectroscopy: Laboratory Experiments

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constructed a very sensitive, but qualitative, probe of molecular proximity and miscibility. This concept is illustrated in Figure 16.29b for a compatible mixture of 2,20 -dinitrobiphenyl and perdeuterated poly(methyl methacrylate) when the spin diffusion mixing time is 10 ms.

16.11.9 Detection of Molecular Proximity in Poly(4-vinylpyridine) –Ru21 Complexes When One of the Components Does Not Contain Hydrogen Nuclei d-Block metals must have a vanishing electron magnetic moment to obtain highresolution 13C NMR signals of directly bound ligands. In this respect, the NMR experiments described herein focus on diamagnetic d6 octahedral complexes. The dimer of dichlorotricarbonyl– ruthenium(II) is attractive because this d6 complex is diamagnetic in an octahedral environment (see Sections 6.5.2 and 6.6.2), and it forms coordination complexes with pyridine as illustrated in Figure 16.30. The heavy-metal center with carbonyl ligands in the first-shell coordination sphere contributes to strong-field zero-electron-spin behavior and eliminates spectral broadening effects due to potential paramagnetic metal centers. Two magnetically active quadrupolar isotopes of ruthenium exist, 99Ru and 101Ru, with natural isotopic

Coordination complexes between Ru2þ and pyridine (i.e., model ligand for the polymer, P4VP) in which the dimer’s dichloride bridge is cleaved, one pyridine ligand occupies the vacant site, and the second pyridine ligand displaces CO in the first shell. (a) Several cis and trans isomers of RuCl2(CO2)(C5H5N)2 are illustrated with the predicted number of CO stretches based on molecular symmetry and chemical applications of group theory. (b) Schematic representation of one of the coordination complexes with C2v symmetry and the spatial pathway (indicated by black arrows) by which 1H– 13C magnetic spin diffusion occurs from 1H nuclei in the pyridine ring to carbonyl 13C nuclei in the same coordination sphere. Spectrum A represents carbonyl 13C resonances of the ruthenium dimer with no 1H nuclei. Spectra B and C represent carbonyl 13C resonances in a coordination complex between Ru2þ and poly(4-vinylpyridine). Spectrum B was generated via the cross-polarization pulse sequence that relies on dipolar communication between 1H and 13C. Expanded versions of these three spectra are presented in Figure 16.31.

Figure 16.30

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Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers

abundances of approximately 13% and 17%, respectively. Hence, it is feasible to generate high-resolution 13C solid state NMR spectra of the carbonyl ligands in the coordination sphere of ruthenium. Carbonyl 13C signals in the undiluted d-block dimer, [Ru(CO)3Cl2]2, are accessible via the “Bloch decay” pulse sequence with magic-angle sample spinning. In this case, 1H – 13C cross-polarization is not feasible because the molecule contains no protons (i.e., 1H nuclei). Standard cross-polarization/magic-angle-spinning/dipolar-decoupled 13C NMR experiments should generate signals for the ruthenium carbonyl salt when it forms a coordination complex with poly(4-vinylpyridine). Now, the well-established cross-polarization mechanism of heteronuclear spin diffusion between the 1H spin manifold of the polymer and the carbonyl 13C nuclei of the d-block salt provides a tool to evaluate mixing characteristics at the molecular level. This application of NMR spectroscopy is rather unusual due to (i) the choice of the d-block metal salt, [Ru(CO)3Cl2]2, and (ii) details associated with the 1 H– 13C cross-polarization process for solids. The overall objective is to identify poly(4-vinylpyridine) and the ruthenium salt as nearest neighbors in the blend that contains stoichiometric proportions of ruthenium to pyridine. Molecular-level information of this nature should support the hypothesis that the glass transition temperature of this complex is above 300 8C (i.e., difficult to detect as a consequence of thermal decomposition), even though Tg for undiluted poly(4-vinylpyridine) is approximately 150 8C. The experiment is designed for rigid solids with strong 1 H– 13C dipolar couplings. Proton-enhanced 13C NMR spectra of solids are generated via heteronuclear spin diffusion between dipolar-coupled nuclei that typically reside within the same molecule. 1H nuclei (i.e., protons) are polarized and spin-locked in the rotating reference frame such that the 1H manifold is highly ordered and characterized by a spin temperature approximately three orders of magnitude smaller than the equilibrium spin temperature of the abundant protons in the static magnetic field. The description of a cold 1H spin system is based on an application of Curie’s law of magnetism for equilibrium magnetization when the concept of spin temperature is valid, which states that magnetization is proportional to field strength and inversely proportional to spin temperature (i.e., see Section 15.6.1). Hence, when magnetization that equilibrates in the static field is rotated by 908 and spin-locked (i.e., phase shifted by 908) in the rotating frame of reference, one rationalizes that the spin temperature decreases by three orders of magnitude. This dramatic decrease in spin temperature is required to counterbalance the decrease in field strength by three orders of magnitude (i.e., from static megahertz fields in the laboratory frame to kilohertz fields in the rotating frame), because Curie’s law of magnetism in the rotating frame must yield the same ratio of field strength to spin temperature if the bulk magnetization has not changed. Now, consider the 13C spin system that is extremely warm via application of Curie’s law and the fact that 13C polarization is nonexistent in the rotating reference frame before spin diffusion occurs. The large spin-temperature difference between 1 H and 13C magnetic moments provides the driving force, and heteronuclear dipolar couplings provide the mechanism for energy-efficient polarization transfer from 1H to 13C in the rotating reference frame. Mutual spin – spin flips due to S þI 2 and S 2I þ terms in the heteronuclear dipolar Hamiltonian (where S þ ¼ Sx þ jSy, and

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pffiffiffiffiffiffiffiffiffiffi I 2 ¼ Ix 2 jIy, with j ¼ (1), are the raising and lowering operators, respectively, based on the x- and y-components of 1H and 13C spin angular momenta) are designed to be energy conserving. Hence, the overall process occurs typically on the short millisecond time scale if (i) the two nuclei are in close proximity and (ii) the static contribution to the spectral density function that characterizes micro-Brownian motion is large enough, as it is in rigid solids. When one considers the undiluted ruthenium dimer, it is obvious that the cross-polarization process is not feasible because the molecule has no protons (i.e., H nuclei). In this respect, the one-pulse Bloch-decay sequence using a repetition delay of 60 seconds has been employed to obtain the 13 C spectrum of the carbonyl ligands. There are at least two crystallographically inequivalent CO groups that give rise to 13C chemical shifts at 181 ppm and 182.5 ppm in Figure 16.31. The crystal structure of [Ru(CO)3Cl2]2 should be used to explain the presence of two 13CO signals, because there are four in-plane (equatorial) and two

(a) A (b)

B C ppm 190 185 180 175 170 Carbon-13 Solid State Chemical Shift (ppm)

Figure 16.31

ppm 270 260 250 240 230 220 Carbon-13 Solid State Chemical Shift (ppm)

(a) High-resolution carbon-13 solid state NMR spectra of the carbonyl ligands in the undiluted ruthenium dimer (spectrum A) and the P4VP–ruthenium complex (spectra B and C). Spectrum A was obtained via the Bloch-decay pulse sequence with a repetition delay of 60 s and 50 Hz of line broadening. Spectrum B was obtained for the complex that contains 1 mole of ruthenium per 1 mole of pyridine side groups in the polymer. This spectrum was generated via 2 ms of 1H– 13C cross-polarization thermal contact (i.e., heteronuclear spin diffusion) and 50 Hz of line broadening, suggesting that the two dissimilar molecules are in close proximity. Spectrum C represents the same polymeric ruthenium complex, but the Bloch-decay pulse sequence was used with a repetition delay of 30 s and 50 Hz of line broadening. The dashed lines highlight the strongest carbonyl 13C resonances in the undiluted dimer at 181 ppm and 182.5 ppm, and in the polymeric complex at 186 ppm. The arrow in spectrum A identifies a weaker resonance of the crystalline dimer at 185.5 ppm that exhibits a distinct spinning sideband at 251.5 ppm. (b) High-resolution carbon-13 solid state NMR spectrum of the carbonyl ligands in the undiluted ruthenium dimer via the Bloch-decay sequence with a repetition delay of 60 s and 50 Hz of line broadening. Magic-angle spinning was performed at a nominal rate of 5 kHz. This spectrum in the vicinity of 250 ppm focuses on the first spinning sideband pattern at higher chemical shift relative to the isotropic signals, illustrated in spectrum A in part (a), between 180 ppm and 190 ppm. The weaker signal at 251.5 ppm represents the first spinning sideband for the parent signal indicated by the arrow in spectrum A in part (a).

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Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers

Figure 16.32 Two isomers of the dimer of dichlorotricarbonylruthenium(II) with (a) C2h symmetry and (b) C2v symmetry. The isomer on the left contains a center of inversion and exhibits three infrared-active CuO stretches that belong to the following irreducible representations in C2h: Au þ 2Bu. Hence, there are fewer infrared-active CuO stretches relative to the number (i.e., 6) of CuO ligands. The isomer on the right does not contain a center of inversion or a three fold or higher proper rotation axis in its molecular point group (i.e., C2v ). There are five infrared-active CuO stretches that belong to the following irreducible representations in C2v : 2A1 þ 2B1 þ B2. Hence, the symmetry rule for infrared-active carbonyl stretches does not apply to complexes with more than one metal center. In both cases, crystal structure studies indicate that a dichloride bridge connects both ML4 fragments in a pseudo-octahedral configuration.

out-of-plane (apical) carbonyl ligands relative to the orientation of the dichloride bridge, as illustrated in Figure 16.32a, whereas the isomer in Figure 16.32b contains two equatorial and four apical carbonyl ligands. When one considers the ruthenium coordination complex with poly(4-vinylpyridine), the proximity of 1H nuclei in the polymer to 13C nulcei in the d-block metal salt can be addressed qualitatively via magnetization transport. This task is achieved by obtaining the proton-enhanced 13C NMR spectrum of the carbonyl ligands in the polymeric ruthenium complex at a cross-polarization contact (i.e., spin diffusion) time of 2 ms. It should be obvious that the ruthenium salt and poly(4-vinylpyridine) are nearest neighbors, because heteronuclear spin diffusion is operative and one observes carbonyl 13C signals via 2 ms of cross-polarization thermal contact in Spectrum B of Figure 16.31a. This is a typical mixing time for heteronuclear spin diffusion when 1H and 13C nuclei reside within the same molecule, suggesting that dipolar distances between poly(4-vinylpyridine)’s 1H nuclei and carbonyl 13C sites in the ruthenium salt are comparable to intramolecular distances. Heteronuclear Dipolar Interaction Energy and Flip-Flop Terms in the Hamiltonian Operator Consider two different nuclear spins, identified by I and S, in which each one interacts with the magnetic field generated by the other. The classical expression for the interaction energy EDipole is based on the scalar (i.e., dot) product of two magnetic moment

16.11 Carbon-13 Solid State NMR Spectroscopy: Laboratory Experiments

759

vectors, mI and mS, as well as a contribution that considers the orientation of each spin with respect to the internuclear vector r: EDipole ¼







mI mS (m r)(mS r) 3 I r3 r5

The first term in the previous expression is most common, whereas the second term in EDipole vanishes for spins at the same vertical position that are either up or down. Next, one introduces quantum mechanical operator formalism into the dipolar interaction energy via the connection between magnetic moments and spin angular momenta I and S (mI ¼ gIh I, mS ¼ gsh S ), where h is Planck’s constant divided by 2p. The final expression for the Hamiltonian operator contains the raising (i.e., I þ and S þ) and lowering (i.e., I 2 and S 2) operators for each spin, based on the x- and y-components of angular momentum. The primary objective of this exercise is to identify selected terms in the dipolar Hamiltonian that contain products of I þS 2 and I 2S þ, which correspond to one spin flipping up and the other flipping down (i.e., flipflops). This contribution to the Hamiltonian allows transitions to occur between the appropriate spin states. It provides the mechanism for energy-conserving heteronuclear spin diffusion among coupled magnetic moments during 1H– 13C cross-polarization when the Hartmann – Hahn [1962] condition is established to match energy-level splittings for both spin-12 nuclei in the rotating frame of reference. It should be mentioned that the dipolar Hamiltonian is typically treated as a small perturbation relative to the strong Zeeman interaction that spins experience in a static magnetic field. In terms of the x-, y-, and z-components of spin angular momenta and the internuclear vector between coupled dissimilar spins, one expresses the dipolar Hamiltonian operator HDipole as follows: HDipole

  gI gS h 2 3 ¼ Ix Sx þ Iy Sy þ Iz Sz  2 {xIx þ yIy þ zIz }{xSx þ ySy þ zSz} r3 r

The flip-flop contributions to HDipole are identified by re-expressing the x- and y-components of spin angular momenta in terms of the raising and lowering operators. For example, I þ ¼ Ix þ jIy ; I  ¼ Ix  jIy 1 1 Ix ¼ {I þ þ I }; Iy ¼ {I þ  I  } 2 2j pffiffiffiffiffiffiffiffiffiffi where j ¼ (1). Similarly, Sx and Sy are written in terms of S þ and S 2. Polar angle Q and azimuthal angle w in spherical coordinates are used to describe the relative orientation of both spins that are separated by distance r. Tedious operator algebra yields the final expression for the dipolar Hamiltonian, which is Hermitian: HDipole ¼ gI gS h 2

q¼þ2 X q¼2

Fq (r, Q, w)Aq

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Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers

where Fq(r, Q, w) describes the relative orientation and separation of both spins that exhibit random fluctuations due to thermal motion within the lattice, and Aq contains angular momentum operators that describe transitions among the various spin states. All of these orientation functions and angular momentum operators that comprise the dipolar Hamiltonian for two coupled dissimilar spins are summarized below: 1 (1  3 cos2 Q); A0 ¼ Iz Sz  14 {I þ S þ I  Sþ} r3 1 F1 ¼ 3 sin Q cos Q exp(jw); A1 ¼  32 {I þ Sz þ Iz Sþ} r 1 F1 ¼ 3 sin Q cos Q exp( jw); A1 ¼  32 {I  Sz þ Iz S} r 1 F2 ¼ 3 sin2 Q exp(2jw); A2 ¼  34 I þ Sþ r 1 F2 ¼ 3 sin2 Q exp(2jw); A2 ¼  34 I  S r F0 ¼

Operator A0 contains the flip-flop mechanism that connects the appropriate states for energy-conserving heteronuclear spin diffusion, as discussed earlier in this chapter from the viewpoint of morphology and molecular mobility in block copolymers and macromolecule – metal complexes.

16.11.10

Subjects for Discussion

1. Do the carbonyl 13C signals of the undiluted ruthenium dimer appear at the same chemical shifts as those in the coordination complex with poly(4-vinylpyridine)? 2. Construct a model of the polymer/Ru2þ coordination complex as follows: (i) Adopt the crystal structure of the undiluted ruthenium dimer. (ii) Cleave the dichloride bridge. (iii) Let the pyridine nitrogen in the polymeric side group occupy the vacant site in the coordination sphere of ruthenium. (iv) Let another pyridine nitrogen in the polymeric side group displace one of the CO ligands. 3. Employ molecular symmetry and chemical applications of group theory to devise a strategy for determining the number of pyridine side groups in the coordination sphere of ruthenium based on infrared spectral comparison of the undiluted dimer and the polymeric ruthenium complex. Hint: Focus on the terminal carbonyl (i.e., CuO) absorptions between 1950 and 2150 cm21, and assume that electroneutrality requires two anionic chloride ligands in the coordination sphere of each divalent ruthenium complex after the

16.11 Carbon-13 Solid State NMR Spectroscopy: Laboratory Experiments

761

dichloride bridge is cleaved by pyridine ligands in the side group of the polymer [McCurdie and Belfiore, 1999].

16.11.11 Analysis of Infrared-Active Carbonyl Ligands in the Coordination Sphere of Ruthenium Complexes via Molecular Symmetry and Chemical Applications of Group Theory Symmetry analysis reveals that monometallic complexes with a single metal center exhibit unique stretching vibrations for each CO ligand if the molecular point group does not contain either the center of inversion or a threefold or higher proper rotation axis. In other words, symmetry reduces the number of infrared-active CuO stretches relative to the number of CO ligands in a metal complex. If two isomers of the ruthenium dimer exist simultaneously, as illustrated in Figure 16.32, then five different ligand arrangements about a single ruthenium center are possible for RuCl2(CO)2(Py)2 after the dimer is cleaved by pyridine side groups in the amorphous polymer. These five ligand arrangements are illustrated in Figure 16.30a, and symmetry analysis via chemical applications of group theory is provided in Table 16.2. “Py” is used as an acronym for pyridine side groups coordinated to ruthenium via the nitrogen lone pair. The first four ligand arrangements (i.e., a – d) in Table 16.2 do not possess a center of inversion. Carbonyl ligands exhibit a cis configuration for the first three entries (i.e., a– c) in Table 16.2 and, hence, two CO signals should appear in the infrared spectrum for each of these complexes. When both CO ligands are in a trans arrangement, the complex given by configuration (d) in Table 16.2 exhibits C2v symmetry, and it is possible that collinearity of the CO ligands will generate one CO infrared signal, even though a center of inversion or proper rotation axis  1208 (i.e., threefold or higher) does not exist. When all identical ligands are trans, the molecule exhibits D2h symmetry (i.e., entry e in Table 16.2) and possesses a center of inversion. Hence, only one carbonyl absorption that belongs to the B2u irreducible representation should appear in the infrared spectrum of this complex.

Table 16.2 Geometrical Arrangements, Symmetry Considerations, and Infrared-Active Carbonyl Stretches for Monometallic Ruthenium Complexes, RuCl2(CO)2Py2 Isomer of RuCl2(CO)2Py2 (see Fig. 16.30) a b c d e a

Molecular Anionic Pyridine Carbonyl point group Cl2 ligands (Py) ligands ligands C1 C2v C2v C2v D2h

cis trans cis cis trans

cis cis trans cis trans

cis cis cis trans trans

Number of infrared CO stretches and symmetry types 2, 2, 2, 2 a, 1,

[A þ A] [A1 þ B1] [A1 þ B1] [A1 þ B1] [B2u]

Only one CuO stretch is observed due to collinearity of the carbonyl ligands in a trans configuration.

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Chapter 16 Magnetic Spin Diffusion at the Nanoscale in Multiphase Polymers

16.11.12 Group Theory Analysis of Carbonyl Ligands in Monometallic Complexes of Ruthenium The objective of the final exercise in this chapter provides detailed insight about the structure of macromolecule – metal complexes and infrared-active absorptions of carbonyl ligands in the first-shell coordination sphere of ruthenium(II). Since there are two carbonyl ligands in each monometallic complex in Table 16.2, one generates a two-dimensional reducible representation GCO that describes how the internal displacement vectors for all of the CuO stretching modes in the excited vibrational state are affected by each symmetry operation in the appropriate point group of the molecule. Then, it is necessary to determine how GCO can be reduced in terms of the irreducible representations of the molecular point group. Finally, one qualitatively identifies nonvanishing terms in the expectation value of the dipole moment operator, because the infrared transition moment connects the excited vibrational states and the totally symmetric ground state for each internal displacement vector that is infrared-active. The results of this analysis are summarized in Table 16.2 for all five isomers of RuCl2(CO)2Py2, where Py is a model ligand for pyridine groups in the side chain of poly(4-vinylpyridine).

16.12

SUMMARY

When polymer blend components are tailored such that strong intermolecular association between dissimilar chain segments is favorable, then the isotropic NMR chemical shifts of strategic 1H and 13C sites in the critical component detect microenvironmental changes due to mixing. If blend composition dictates that two solid state phases coexist below the eutectic solidification temperature(s), then the NMR spectra suggest that crystallographic symmetry influences 1H and 13C chemical shifts when multiple signals are observed for chemically equivalent nuclei. Hydrogen-bonded blends that exhibit multiple eutectic phase transformations and d-block coordination complexes that exhibit synergistic mechanical and thermal properties represent unique classes of strongly interacting systems that can be studied successfully via high-resolution solid state NMR spectroscopy. Thermodynamic phase diagrams, generated via differential scanning calorimetry and polarized optical microscopy (in some cases), are extremely useful to interpret carbon-13 NMR spectra of polymer blends in a two-phase region. Under favorable conditions, the direct observation of spin diffusion via 1H CRAMPS between dipolar-coupled 1H nuclei in dissimilar molecules or chain segments on a time scale of 1024 s or less provides convincing evidence for molecular mixing and cocrystallization in the b-phase of poly(ethylene oxide) and resorcinol. 1 H spin diffusion across domain boundaries is operative on a time scale that is one to two orders of magnitude slower (i.e., 1 – 10 ms) in ionic and triblock copolymers. A comparison of two NMR experiments that measure spin diffusion between dipolar-coupled 1H nuclei is presented: direct detection in a two-dimensional contour representation via 1H CRAMPS, and indirect detection in a one-dimensional mode via the modified Goldman – Shen pulse sequence. The spin-diffusion data suggest that the

References

763

time scale during which spin exchange occurs in the presence of homonuclear dipolar couplings is a critically important parameter in any analysis of molecular proximity.

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Index

absorbance, infrared, CvC stretch, 539 –543 absorption function, spectral density, 670, 675 –676 acetone, carbon disulfide, Gibbs free energy, E-field, 164 acid–base equilibrium constants, 200 –201 acidity constants, pKA, 200 –201 acids, hard and soft, 199 activation energy alkene hydrogenation, 532 –533 aging, BPAPC, 691 –693 BPAPC, dielectric relaxation, 689– 691 diffusion, 58 –60 discharge current, 701– 702 stretched exponential, 475 WLF shift factor, 392 activities entropy of mixing, 108 free energy of mixing, 106 –107 activity coefficient partial molar volume of mixing, 129 van Laar model, 161 activity vs. occupational probability, Guggenheim’s lattice, 110, 115 Adams– Gibbs theory, conformational entropy, 125 additive rule of mixtures, Tg, 242 affine deformation, rubbers, 609 –610, 624 affinities, thermodynamic order parameters, 175 aging, chemical, BPAPC, activation energy, 691 –693 air, dielectric breakdown, 141 algorithm, multivariable parameter estimation, 728

alignment, electric-field-induced, 137–138, 150 alkenes consumption, kinetics, 529–533 dimerization, kinetics, 534–543 hydrogenation, 527–533 amorphous phase, mass balance, 333– 334 analogies, rubber-like solid vs. ideal gas, 610 angle of shear, strain ellipsoid, 436–437 angle of twist, torsion pendulum, 423–425 angular momentum balance, torsion pendulum, 422–423 spin, raising/lowering operators, 759–760 anionic polymerization, 508–515 anionic, moments-generating function 511–512, 519–520 anisotropy, optical, 255– 279 annealing temperature, optimum, 304–307, 323 Arrhenius activation energies, BPAPC, 690–693 assumptions Flory– Huggins lattice, 79 –80 Gordon –Taylor equation, 186 Tg depression, order parameter, 178–181 asymptotic result, Bernoulli trials, 553–555 athermal mixtures excluded volume, 76 –79 Guggenheim’s lattice, 105–117 autocorrelation, end-to-end chain vector, 674–675, 680 auxetic materials, negative Poisson’s ratio, 356 average density, mixtures, 349–351

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

765

766

Index

average molar mass, 545 average molecular weights, 545 average MW anionic, 512, 520 free radical, 503 –504, 518 polycondensation, 496 –498, 517 average relaxation time, triangular distribution, 451 Avrami equation heterogeneous nucleation, 290 homogeneous nucleation, 292 –293 linear least squares analysis, 295 –296 rate of crystallization, 335 –336 Avrami exponent, isothermal DSC, 335 –337 azeotrope, minimum-boiling, methyl acetate, water, 162 backbone stabilization energy, Kwei equation, 243 balanced biaxial deformation, rubbers, 631 bases, hard and soft, 199 basis functions, orthonormal, wave function, 653 –654, 663, 667 batch reactor crystallization, 331 –335 polycondensation kinetics 524 –527 bead-spring model, Rouse dynamics, 677 –680 Belfiore, family history, 765 –767 benzaldehyde, volume vs. temp., 346 –349 Bernoulli trials, chain statistics, 552 –555 biaxial deformation, rubber elasticity, 630 –631 bieutectic mixtures, PEO-resorcinol, 731 –738 bimodal distribution, relaxation modulus, 399 –400 binary mixtures chemical stability, 89 –105, 640 –641 dependence of Flory x on MW at critical point, 99 Gibbs free energy, 91, 93 lever rule, 103 binary molecular diffusivity, 104 binodal points, 94–95 Flory –Huggins model, 100 –103 binomial distribution, random walks, 552 –553

biological macromolecules, 251 birefringence calculations, 275– 277 intrinsic, 278– 279 orientation, 279 rubbery polymers, 278–279 birefringent spherulite polarized light, 258– 259 unpolarized light, 284–285 blends, hydrogen bonding, 13C NMR, 745–753 blobs, deGennes, scaling laws, 586–587 block copolymers cross-polarization dynamics, 744–745 magnetic spin diffusion, 711– 715 PdCl2, stress –strain, 470–472 phase separation, 100 stress relaxation, 476–479 vs. random copolymers, cross polarization, 711–714 Boltzmann distribution, canonical ensemble, 655–656 Boltzmann integral alternate forms for strain, 426– 428 g(t), dynamic experiments, 432–433 s(t), dynamic experiments, 411–412 stress –strain testing, 462–463 Voigt model, 454–455 Boltzmann superposition integral alternate forms for s(t), 406–407 Kramers –Kronig, 680– 682 strain, 425–428 stress, 405–407 Boltzmann’s entropy concentrated solutions, 88– 89 excluded volume lattice model, 77 Flory –Huggins lattice, 82 Gaussian chains, 588–591, 605–606 isotropic chain expansion, 633 rubber elasticity, 629 trimers and hexamers, 132– 133 bond dissociation energies, carbonyl complexes, 238–239 bond distance, OH, infrared, 752 bond energies, alkene hydrogenation, 532–533 bone, trabecular, cancellous, 189 boundary conditions, stress relaxation, 384 Boyle temperature, 75

Index BPAPC (polycarbonate of bisphenol A) 13 C NMR, experimental, 685– 687 aging, activation energy, 691 –693 dielectric relaxation, 689 –691 functionalized, mechanical relaxation, 688 Bro¨nsted ionization equilibrium constants, 200 –201, 233 Brownian motion, 50 c , threshold molar density, 586 –587 C3v symmetry, 3-coordinate complexes above Tg, 237 C4v symmetry, 5-coordinate complexes above Tg, 220 –221, 229 calibration curve, GPC, 582, 604 Cannon–Fenske capillary viscometers, 598 canonical ensemble, Boltzmann distribution, 655 –656 cantina fratelli Pelligrini, Lavis, Trentino, Italia, 286 capacitance, complex, dielectric relaxation, 693 –695 capacitors and springs, 359 –362 capillary constant, viscometry, 597 –598 capillary tubes, viscometry, 595 –600 capillary viscometry, momentum diffusivity, 595 –600 carbon dioxide, supercritical, 189 carbon disulfide, acetone, Gibbs free energy, E-field, 164 carbon-13 NMR BPAPC, experimental, 685 –687 PEO-resorcinol, 731– 734 polyester blends, 308– 310 solid state experiments, 738 –758 carbonyl carbon chemical shift hydrogen bonding, 308 –310, 747 P4VP/Ru2þ, 757 spinning sideband, [RuCl2(CO)6]2, 757 carbonyl complexes bond dissociation energies, 238 –239 with molybdenum, 216 –223 carbonyl ligands, Jørgensen’s f-factor, 217 carbonyl stretches, Ru2þ complexes, group theory, 760 –762 carboxylate carbon, 13C NMR, 740 –743, 753 catalysis, via transition metals, 527 –543

767

cation environment, SurlynTM ionomers, 740–741 Cauchy’s integral, Kramers–Kronig theorem, 682– 683 Cauchy–Riemann eq., Kramers– Kronig theorem, 683 cell suspensions, Vand equation, 581 cellular foam, negative Poisson’s ratio, 356 center of chirality, 250 cessation, chain motion below Tg, 388 chain conformation protein folding, 251 temperature dependence, 623–624 chain dimensions, MW scaling law, 583, 604 chain dynamics, Rouse model, 677–680 chain expansion excluded volume, 570–578, 631–640 Flory– Huggins lattice, 633–635 free energy minimization, 576–578, 639–640 isotropic, Boltzmann’s entropy, 633 chain expansion factor vs. elongation ratio, 632–633 chain extensibility, crosslinked polymers, 472 chain folding, 253–254, 279–282, 307 energetics, multiple melting, 307– 317 chain length anionic, 508–509 mean-square ETE, freely jointed 549–551, 601 mean-square ETE, freely rotating, 565–568 chain motion, frozen below Tg, 388 characteristic ratio, Flory, 568 charge density, resorcinol, 735 charging process, electrets, 697–698 chemical aging, BPAPC, activation energy, 691–693 chemical crosslinks, percolation threshold, 643–644 chemical potential, 5 electric field, 144– 145, 148, 157– 158 osmotic pressure, 73–74 polymer, diluent, 183–184 polymer, Flory– Huggins model, 318 pressure dependence, 128–129

768

Index

chemical potential (Continued) solvent, Flory –Huggins model, 87, 634, 640 stability requirement, 96–97, 640 –641 zero-field, 148 –149 chemical requirement, phase equilibrium, 10, 95 chemical shielding tensor NMR, BPAPC, 685 –687 PEO, 739 chemical shifts, 13C NMR, hydrogen bonding, 745 –753 chemical stability binary mixtures, 89–105, 640 –641 electric fields, 157 Flory –Huggins lattice, 97 –99, 640– 641 requirements, 93, 640 –641 chi parameter chemical stability, 98, 641 dependence on MW at critical point, 99 effect on phase compositions, 102 Flory –Huggins, 180 –181 Flory–Huggins, PVPh/PVMK, 751–752 melting point depression, 319 –321 chiral center, 250 Chow TS, Tg depression, 185 chromatography, GPC calibration curve, 582, 604 output curve, 489 –491, 504 –505, 512 –513 circuit law, Maxwell model, 457 Clapeyron equation, 10 –11 electric-field analog, 150 l’Hopital’s rule, electric field, 150, 152 –153 classical thermodynamics enthalpy of mixing, 130– 131 entropy of mixing, 130 external force, 591 –592, 606, 611 Gibbs free energy of mixing, 128 –129 rubber elasticity, 610 –623 Clausius– Mossotti equation, 276 cobalt chloride, poly(vinylamine), volume vs. temp., 346 –349 cobalt complexes, 227 coefficient of linear compressibility, 622 –623 coexistence, phases, NMR detection, 728 –733

Cohen and Turnbull, free volume theory of diffusion, 51– 55 combinatorial entropy of mixing, concentrated solutions, 89 common tangent binodal points, 95 Flory –Huggins model, 100–103 compatibilization, via transition metals, 194–195, 534–543 complex capacitance, dielectric relaxation, 693–695 dielectric constant, relaxation, 694–696 dynamic modulus, 674, 676 dynamic viscosity, 674 impedance, dielectric relaxation, 693–696 susceptibility, Fourier transform, 674–676 complex variables dynamic compliance, 464–465 dynamic mechanical expts., 413–416 Stieltjes transform 452 complex viscosity forced vibration experiments, 415 phasor analysis, 465–466 complexes, Tg enhancement, 198 compliance creep, continuous distribution, 430–431 creep, Voigt model, 428 dynamic testing, 433–435 dynamic, complex variables, 464– 465 elastic, 361 Maxwell, electrical analog, 457 compositional dependence dielectric permeability, 159 fractional free volume, 23– 25 Gibbs free energy, 91–93 interaction parameter, 124 Tg, 39–45, 185–186, 242–243 Tg, entropy continuity, 15 –18 Tg, Gibbs–DiMarzio theory, 120–122 Tg, lattice model, 183–184 Tg, via fractional free volume, 25 –26 Tg, volume continuity, 18– 20 Tg, with order parameter, 177– 183 WLF shift factor, 394–397 compressibility coefficients, linear, 622–623

Index discontinuity at Tg, 7, 12, 30 –31, 37 isothermal, 332 –333 concentrated solutions entropy of mixing, 88 –89, 179 –180 Gibbs free energy of mixing, 183 concentric spheres, KratonTM , radial diffusion, 719 –725 condensation polymers, MW distribution, 486 –491, 524 –527 reaction, average MW, 496 –498, 517 reaction, degree of polymerization, 491 –496 configurational isomers, 250 conformation, folding funnel, 251 conformational entropy Adams –Gibbs theory, 125 excluded volume, 571 –577, 631 –633 Gibbs –DiMarzio theory, 117 –123 lattice cluster theory, 123 –126 conformational rearrangements, 10 reorganization, stress relaxation, 478 –479 conformations, temperature dependence, 623 –624 constitutive equation Maxwell model, 362 –367 stress– strain, rubbers, 594, 605 constraints, Lagrange multipliers, 488 –489 contact angle, surface tension effects on Tg, 35 continuous distribution functions anionic polymerization, 513 –515 free radical polymerization, 505 –507 most probable, 490 –491 continuous limit, multiplicity function, 628, 630 continuous moments-generating function, 515 –521 continuous MW distribution, Laguerre polynomials, 522 –524 controlled release, drug delivery near LCST, 96 convolution theorem, 380– 381 compliance, modulus, 372, 430 cooperative reorganization, stress relaxation, 480 –482

769

coordination crosslinks, 193–194 via nickel complexes, 211–215 coordination number, Guggenheim’s lattice, 109 coordination pendant group effect on Tg, 212 P4VP/zinc-acetate, 741 coordination, via transition metals, methodology, 193–205 coordinatively unsaturated complexes, 209, 238 copolymer block vs. random, cross polarization, 711–714 ethylene/methacrylic acid, 740–741 PdCl2, stress relaxation, 476–479 PdCl2, stress –strain, 470–472 poly(lactic acid), 189 random, magnetic spin diffusion, 709–711 SBS triblock, magnetic spin diffusion, 711–715 core-shell morphology, radial diffusion, 719–725 multivariable parameter estimation, 725–728 cork, Poisson’s ratio, 356 Corradini, Paolo, 282 correlated states model, stress relaxation, 473–475 correlation functions vs. spectral densities, 668–671, 674–676 correlation time, 669–671, 675, 684 temperature dependence, Rouse, 680 Couchman–Karasz, Tg depression, 17, 185–186 counterions, poly(acrylic acid), Tg, pH, 584–586 coupled PDE’s, radial diffusion, KratonTM , 721–726 coupling, free radical MW distribution, 505–507 reactions, 499–502 CRAMPS, proton NMR, PEO-resorcinol, 734–738 creep compliance continuous distribution, 430–431 Laplace transform, 371–372 Maxwell model, 369

770

Index

creep compliance (Continued) Voigt model, 376, 428, 454 vs. relaxation modulus, 428– 430 creep Boltzmann superposition integral, 426 –428 Maxwell þ Voigt elements, 377– 378 Voigt– Kelvin model, 430 –431 creep, creep recovery, 367 Maxwell model, 368 –370 mechanical models, 466 Voigt model, 376 –377 critical exponent, percolation theory, 643 –644 free energy, spherulite, 301 –302, 327 free volume for jumps, 52, 55, 57 –58, 60 MW, entanglements, 403 radius, spherulite, 301 –302, 327 critical point, MW dependence of chi parameter, 99 cross polarization flip-flop mechanism, 759 –760 intermolecular, P4VP/[RuCl2(CO)6]2, 755 –758 intermolecular, PMMA/dinitrobiphenyl, 753 –755 crosslink density, rubber-like solids, 594, 605, 611 crosslinked elastomers, retractive forces, 587 –594, 605 –607 crosslinked polymers, chain extensibility, 472 crosslinks dissociation at Tg, 213 –215, 222, 231, 236, 239 percolation threshold, 643 –644 via coordination, 193– 194 via coordination in Ni(II) complexes, 211 –215 cross-polarization dynamics block copolymers, 744 –745 KratonTM SBS triblocks, 711 –714 PEO-resorcinol, 733– 734 crystal density, 252 glasses, 286 growth rate, dependence on T, 303 crystalline mass fraction DSC heating, 337 –339

isothermal DSC, 336–337 via density, 339– 340 crystalline phase, mass balance, 333 crystalline volume fraction, density, 340 crystallite imperfections, 302 crystallization half-time, 296, 323 heterogeneous “reaction”, 352 internal energy change, 333–335 isothermal analysis, DSC, 295, 323, 331–335 mass balances, 333–334 maximum rate, 297–299 temperature, optimum, 304–307, 323 thermodynamics, hydrogen bonding, 310–317 crystallization kinetics, 287–299 isothermal, 335–336 optimum rate, 304– 307, 323 supercooling, 302 crystallization rate constant dependence on temperature, 302–304 heterogeneous, 291, 327 isothermal DSC, 335–337 crystallographic unit cell volume, 340 cubic lattice, excluded volume, 572–574 Curie restriction, 104 Curie’s law of magnetism, 657, 661–662, 756 curvature criterion, chemical stability, 96 –97 cylindrically symmetric bonds, polarizability tensor, 268–271 D3h symmetry, 5-coordinate complexes, .Tg, 218–220, 228–229 D5h symmetry, 5-coordinate complexes, .Tg, 221–222, 230 dashpots and resistors, 360–362 Deborah number, 373 dynamic mechanical experiments, 410–411 superposition principle, 386 time-temperature equivalence, 461– 462 Debye equation charging process, electrets, 697–698 discharge process, electrets, 699–701 deformation affine, rubbers, 609–610, 624

Index biaxial, rubber elasticity, 630 –631 homogeneous, 435 deformation-dependent Gaussian distribution, 627 normal stress difference, 442–443, 631 deGennes blobs, c , scaling laws, 586 –587 Pierre-Gilles, scaling law, biography, 578 –579 degradation, BPAPC, activation energy, 691 –693 degree of polymerization, polycondensation, 491 –496 supercooling, crystallization kinetics, 302 delta-function distribution, relaxation modulus, 399 –400, 413 tracer, diffusion equation, 556 densification, below Tg, 8–10 densified glasses, 9 density crystalline polymer, 252 fraction of crystallinity, 339 –340 functional calculations, bond energies, 238 –239 isotactic poly(1-butene) via pychnometry, 344– 345 isotactic polypropylene, 324 –325 matrix, quantum mechanics, 654 –655, 661 –672 multicomponent mixture, 349 –351 poly(vinylamine), CoCl2(H2O)6, 348 –350 via pychnometry, 341 via WAXD, 340 –341 depolarization current, electret, 699 –701 birefringence, 278 depression, melting point in blends, 317 –322 deuterated PMMA/dinitrobiphenyl, spin diffusion, 753 –755 diblock copolymers, phase separation, 100 dichlorotricarbonylruthenium(II), isomers, 758 dielectric breakdown, 141, 166 dielectric constant complex, relaxation, 694– 696

771

definition and magnitude, 143, 276 pressure dependence, examples, 155 temperature dependence, examples, 153 dielectric experiments, dynamic, 362 dielectric loss, Maxwell, Voigt, 694–696 dielectric permeability compositional dependence, 159 definition, 143 Euler’s theorem, 160 partial molar, 159 dielectric relaxation BPAPC, activation energy, 689–691 complex impedance, 693–696 dielectric spectroscopy, pulse response, 702–703 dielectric susceptibility definition, 143 discontinuity at Tm, 150 of mixing, 158–159, 165–166 partial molar, 146, 158 pressure dependence, discontinuity, 154–155 temperature dependence, discontinuity, 152–153 differentiating integrals, Leibnitz rule, 381–383 diffuse interface, spinodal decomposition, 104–105 diffusion activation energies in polymers, 58 –60 half-time, 61– 65 lag time, 65 –66 lag time, spheres, 716, 719– 720 multiplicity of permutations for, 52 temperature dependence, 56 –58 theory of Cohen and Turnbull, 51 –55 theory of Vrentas and Duda, 55– 60 through membranes, 50–51, 61– 66 via Lagrange multiplier optimization, 53–54 vs. temperature, linear least squares analysis, 56 –57 diffusion coefficient binary molecular, 104 effect of molecular weight, 66–69 gases in polymers, 61–62 Stokes-Einstein, 678–679 via free volume, 55

772

Index

diffusion equation Fick’s 2nd law, spherical domains, 715 –725 Rouse model, 677– 680 spinodal decomposition, 104 –105 tracer input, 556 unsteady state diffusion, 62–66 diffusional flux membranes, 64–65 supercritical CO2, 189 diffusional stability binary mixtures, 89–105, 640 –641 chemical potential, 96 –97, 640 –641 Flory –Huggins lattice, 97 –99, 640– 641 in electric fields, 157 requirements, 93, 640 –641 diffusivity number-average, 68 weight-average, 68–69 diluent, chemical potential, Flory chi, 183 –184 dimensional analysis, coupled PDE’s, radial diffusion, 722 dimensionality of crystal growth, 288 dipolar dephasing, polyethylene, 13C NMR, 744 Hamiltonian, mutual spin flip-flops, 709, 712, 756 –760 dipolar interaction energy, heteronuclear, 759 fluctuating Hamiltonian, 667, 669 1 H-13C spin diffusion, 754– 760 dipolar relaxation strength, Debye eq., electrets, 697– 698 dipole moment, 142 induced by electric field, 257 –258, 260, 262 dipole polarization Debye equation, 697– 700 vector, 142– 145 dipole relaxation time, Debye eq., electrets, 697 –702 discharge current, polarized dielectrics, 696 –702 discontinuity at Tg, examples, 177 at Tg, Prigogine-Defay ratio, electric field, 155 –156 at Tm, dielectric susceptibility, 150

compositional dependence of Tg 182–184 Flory chi, Tg depression, 183–184, 187–188 isothermal compressibility, 7, 12, 30– 31, 37 pressure dependence, dielectric susceptibility, 154– 155 specific heat, 6, 16, 41– 44 temp. dependence, dielectric susceptibility, 152– 153 thermal expansion coefficient, 6, 12, 29, 55, 59–60, 391 discontinuous observables at Tg, r-component mixture, 7 thermophysical properties, 4 discrete distribution, anionic, 511– 512 discrete moments-generating function, 496–498, 502–504, 511–512 disorganized lamellae, optical microscopy, 260–262 displacement mean-square, Gaussian chains, 555–556 vector, 358– 359 disproportionation, free radical kinetics, 507–508 dissipation mechanical-thermal energy, 580–581 vs. fluctuation, 651–652, 657, 671, 673–677 dissociation coordination crosslinks at Tg, 213–215, 222, 231, 236, 239 energy, bond, carbonyl complexes, 238–239 distributed-mass model, force balance, 421 distribution continuous, anionic, 513–515 continuous, free radical, 505–507 continuous, most probable, 490–491 mass fraction, anionic, 512, 515 mole fraction, anionic, 511, 513 distribution, viscoelastic relaxation times, 397–400, 443–451, 461 retardation times, 430–431 Doolittle equation monomeric friction coefficient, 395

Index viscosity vs. temp., 458 zero-shear viscosity, 390– 391, 395 d-orbital energy 3-coordinate complexes, 237 pentagonal planar complexes, 221 –222, 230 square pyramid complexes, 220 –221, 229 trigonal bipyramid complexes, 219, 228 double Laplace transform, distribution function, 452 driving force enthalpy relaxation, 9 volume relaxation, 9 drug delivery, controlled release near LCST, 96 DSC heating trace, 337 –339 isothermal crystallization, 331 –335 DSC thermograms, liquid crystals, 330 DSC vs. NMR PAA/PEO, 752 –753 PEO-resorcinol, 729– 733 PVPh/PVMK, PVPh/PEO, 747 –752 dynamic compliance complex variables, 464 –465 forced vibration experiments, 433 –434 shifting, RF Landel, 459–460 dynamic mechanical experiments, 368, 410 –425 bimodal distribution, 399 complex variables, 413 –416 energy dissipation, 417 –419 loss compliance, 433 –435 loss modulus, 412 –416 phase lag, 361 –362 phasor analysis, 413 –415, 465 –466 storage compliance, 433– 435 storage modulus, 412 –416 dynamic modulus complex variables, 451 –452 complex, 674, 676 forced vibration experiments, 412, 414, 416 Fourier transform, 415 –418 vs. distribution function, 452 vs. relaxation modulus, 451 –452 dynamic pressure, capillary viscometry, 595

773

dynamic viscosity complex, 674 forced vibration experiments, 415 Kramers– Kronig theorem, 681–684 dynamics, cross polarization block copolymers, 744– 745 KratonTM , SBS triblock, 711–714 PEO –resorcinol, 733–734 effect of deformation, normal stress difference, 442– 443, 631 efflux time vs. half-time, viscometry, 599–600 Ehrenfest equations entropy continuity, 13–14 entropy continuity, electric field, 152 integrated forms, 14, 36 via order parameters, 176– 177 volume continuity, 11 –12 volume continuity, electric field, 154–155 Ehrenfest inequalities, via order parameters, 177 Ehrenfest’s theorem, Liouville equation, 663 eigenvalue problem PDE, radial diffusion, 718 unsteady state diffusion, 63–64 eigenvalues, stress ellipsoid, 438, 441, 443 eigenvectors eigenstates, eigenfunctions, eigenvalues, 653–654, 664, 667 stress ellipsoid, 438 eighteen-electron rule, 209 Einstein model, intrinsic viscosity, 582, 604 Einstein’s viscosity, dilute suspensions, 580–581, 604 Einstein –Wiener –Khinchin theorem, spectral density, 674 elastic free recovery, Boltzmann superposition, 407– 410 elastic modulus, 360 nonlinear least squares, 643–645 rubber-like solids, 592–593, 606 elastic retractive forces, Rouse model, 677–678 elastic solids, stress-strain analysis, 435–443 elasticity, rubber-like, 587–594, 605–607

774

Index

electret, 137, 141, 150 electrets filling state, 700 temperature-field history, 696 –697 electric displacement vector, 143 electric field chemical potential, 144 –145, 148, 157 –158 chemical stability criteria, 157 effects in polymers, 139 –141 effects in small molecules, 138–139 effect on LCST, 149 effect on melting, 148– 150 effect on Tg, 150 –156 effect on volume, 153, 158 first law of themodynamics, 144 Gibbs free energy, 144, 157– 158, 160 Maxwell relations, 144 –145 partial molar properties, 144–147 Tg, entropy continuity, 151 –153 Tg, volume continuity, 153 –155 work term, 142– 143 electric permittivity, definition, 143 electric saturation, mixtures, 166 electrical analog Hooke’s law, 359 –362 Maxwell model, 367, 457 Newton’s law, 360– 362 Voigt model, 375 electric-field analog Clapeyron equation, 150 Prigogine –Defay ratio, 155 –156 electric-field-induced alignment, 137 –138, 150 phase separation, 137 –138, 166 electron delocalization, polarizability, 256 electronegativity functional group, 750 Pauling, 241 –242 electrostriction, 166 electro-thermodynamics, nomenclature, 167 –168 ellipsoid strain, 435 –438 stress, 438 –442 elongation ratio vs. chain expansion factor, 632 –633 enantiomeric effect, degree of polymerization, 493

end-to-end chain length freely jointed, 549–551, 601 freely rotating, 565–568 most probable, 601 end-to-end chain vector, autocorrelation, 674–675, 680 energetics of crystallization, hydrogen bonding, 310–317 mixing, Flory–Huggins, 83 –86 energy balance first-law of thermo, 331–332 isothermal crystallization, 334–335 energy dissipation, forced vibration experiments, 417–419, 676 energy elasticity vs. entropy elasticity, 592–593, 610 energy storage, forced vibration experiments, 419 enhancement in Tg, 192–194 Gibbs –DiMarzio theory, 122 enrichment, 13C isotope, NMR, BPAPC, 685–687 entanglements, critical MW, 403 enthalpy dependence on temperature, 16 partial molar, in electric field, 147 total differential, 16 enthalpy of fusion, isotactic polypropylene, 324–326 fusion, melting point depression, 319–321 hydration, hexa-aqua complexes, 196–197 interaction, Flory–Huggins, 86 enthalpy of mixing classical thermodynamics, 130–131 interaction, 180–181 interaction parameter, 134 vs. volume of mixing, 131 enthalpy relaxation, below Tg, 8–10 entropically elastic retractive force, Rouse model, 677–678 entropy conformational, excluded volume, 571–577, 631–633 dependence on temperature and pressure, 13, 16 linear additivity for mixtures, 15

Index partial molar, in electric field, 146 vs. elongation, elastomers, 589 –591, 605 with order parameters, 174, 181 –182 entropy, Boltzmann isotropic chain expansion, 633 rubber elasticity, 629 tombstone, 588 –589 entropy continuity compositional dependence of Tg, 15 –18 Ehrenfest equation, 13–14, 36 Ehrenfest equation, electric field, 152 Tg, electric field, 151 –153 entropy elasticity, 547 –548, 578, 587– 594, 606, 610 vs. energy elasticity, 592 –593, 610 entropy generation, viscous dissipation, 581 maximization, chain expansion, 576 –578 of interaction, Flory–Huggins, 86 entropy of mixing classical thermodynamics, 108, 130 concentrated solutions, 88– 89 concentrated, combinatorial, 179– 180 excluded volume lattice model, 77 Flory– Huggins lattice, 80 –83 interaction, 180 –181 interaction parameter, 134 monomers and dimers, 111 monomers and r-mers, 115 –117 trimers and hexamers, 133 environmental aging, BPAPC, activation energy, 691 –693 equation of state rubbers, retractive force, 611 –615, 619, 629 –631 van der Waals gas, 75 equilibrium chain expansion factor, 576 –578, 639 –640 chain length, anionic, 508 contact angle measurements, 35 glassy state, 8 melting point, Hoffman-Weeks analysis, 322 thermodynamics at Tg, 37– 39 vs. kinetics, glass transition, 8, 178 –179 equivalence, temperature & rate of testing, 461

775

ergodic problem, statistical thermodynamics, 655– 656 error function, 514–515, 519–520 ethylene/methacrylic-acid ionomers, 13C NMR, 740–741 random copolymers, 709–711 Euler’s integral theorem dielectric permeability, 160 Gibbs free energy, 5, 90 Gibbs free energy, electric field, 144 eutectic transitions, PEO-resorcinol, 731–738 excluded volume, 548 athermal mixture, 76 –79 chain expansion, 570–578, 631–640 lattice model, 76–79 osmotic pressure, 78–79 second-virial coefficient, 79 exothermic energetics of mixing, volume contraction, 128–131 PVPh/PVMK, 751–752 expanded chains, good solvents, 570–578, 631–640 expansion factor vs. elongation ratio, 632–633 expectation values, thermodynamic observables, 653–655, 661–662 exponential integral, relaxation modulus, 400–402 external stress, thermodynamics, 591– 594, 605–606 Eyring-Sadron eq., freely rotating chains, 567 facial trivacant complexes, above Tg, 237 factorial polynomials, finite summations, 564, 662 Farad’s law, capacitance, 361–362, 694–695 Faraday’s law, 142 Fick’s second law delta-function tracer, 556 membrane diffusion, 62–66 spherical domains, KratonTM , 715–725 film thickness, effect on Tg, 31 –34 finite strain, elastic solids, 435–443 finite summations, 566–567 factorial polynomials, 564, 662 First Avenue School, 1968 graduation, 767

776

Index

first law of thermodynamics electric field, 144 energy balance, 331 –332 first moment of distribution, effect on viscosity, 404 relaxation function, nonlinear, 475 first normal stress difference elastic solids, 442– 443 rubber-like solids, 631 first-order phase transition, 4 phase transition, in electric field, 148 –150 polynomial, linear least squares, 643 rate constant, time-dependent, 474 first-order-correct solution, Liouville equation, 664 –666 five-coordinate complexes above Tg, 218 –223, 228 –232 LFSE, 222 –223, 231 –232 flip-flop mechanism, spin diffusion, 759 –760 flip-flops, dipolar Hamiltonian, 709, 712, 756 –760 Flory approximation, retractive force, 619 –623 characteristic ratio, 568 chi parameter, melting point depression, 319 –321 chi parameter, discontinuity, Tg depression, 183 –184, 187 –188 law of real chains, PG deGennes, 578 –579, 586 –587 Flory–Huggins energetics, PVPh/PVMK, 751 –752 Flory–Huggins interaction parameter, 180 –181 at critical point, 99 chemical stability, 98, 641 Flory–Huggins lattice assumptions, 79–80 Boltzmann’s entropy, 82 chain expansion, 633 –635 chemical stability, 97 –99, 640– 641 effect of x on phase compositions, 102 enthalpy of interaction, 86 entropy of interaction, 86 entropy of mixing, 80 –83 free energy of interaction, 83 –86

Gibbs free energy of mixing, 86, 317, 634 model, 79–89 mole fractions, 83 multiplicity of states, 80 –82 nomenclature, 79, 83 osmotic pressure, 87 phase separation algorithm, 100–103 polymer chemical potential, 318 second virial coefficient, 87 solvent chemical potential, 87, 634, 640 volume fractions, 83, 634 fluctuating Hamiltonian, dipolar interaction, 667–673 fluctuation vs. dissipation, 651–652, 657, 671, 673–677 fluctuation-dissipation theorem, 651–652, 657, 671, 673–677, 684 energy dissipation, 418, 676–677 folding funnel, proteins, 251 folding, within lamellae, 253–254, 279–282, 307 force balance, torsion pendulum, 420–422 force vs. elongation, rubber-like solids, 592–594, 605–606, 611, 646 Fourier coefficients, diffusion equation, 63 –64 Fourier series solution, radial diffusion, 718–719 Fourier transform correlation function, spectral density, 669–670, 674–677 Kramers –Kronig theorem, 681– 683 pulse response, 702–703 relaxation modulus, 415–416, 418, 451–452 Fox equation, 18 assumptions, 40 binary mixtures, 41 fractional exponential, stress relaxation, 473–475 fractional free volume, 22 at Tg, 391 fracture testing, 368 fragile glasses, lattice cluster theory, 125 free energy minimization, chain expansion, 576–578, 639–640

Index free energy of interaction Flory– Huggins, 83 –85 van Laar model, 161 free energy effect on crystallization temperature, 305 –307 entropically elastic contribution, 631 –633 of mixing, classical thermodynamics, 106 –107, 115 lamellar thickness, 311 –317 spherulite, 299 –302, 327 vs. isotropic chain expansion, 633 –639 free radical average MW, 503 –504, 518 continuous MW distribution, 505 –507 disproportionation, 507 –508 moments-generating function, 502 –504, 518 polymerization, 498 –508 free recovery, elastic, Boltzmann superposition, 407– 410 free vibration expts., torsion pendulum, 419 –425, 467 –468 free volume, 21 –22 analysis of diffusion, 49–60 at Tg, 391 binary mixtures, 41, 395 –397 cooperativity, 51 critical size for diffusion, 52, 55, 57–58, 60 dependence on composition, 23–25 dependence on molecular weight, 26– 29 effect on growth rate, 303– 305 linear additivity, multicomponent mixtures, 25 –26 molecular weight dependence, 46 pressure dependence, 29–31 shift factor, linear least squares, 458 –459 temperature dependence, 22 –23, 55 theory of Cohen and Turnbull, diffusion, 51–55 theory of Vrentas and Duda, diffusion, 55–60 freely jointed chains, radius of gyration, 561 –565, 602 –604 freely rotating chains, ETE distance, 565 –568

777

friction coefficient, Doolittle equation, 395 Fujita & Kishimoto, Tg depression, 185 functional group electronegativity, 750 functionalized BPAPC, mechanical relaxation, 688 gamma-gauche effect, polyethylene, 740, 744 gas, van der Waals, 75 gases, diffusion coefficients in polymers, 61– 62 gauche rotational isomeric states, 251–252, 565 gauche-gamma effect, polyethylene, 740, 744 Gauss’ law, parallel-plate capacitor, 694 Gaussian chains entropy elasticity, 587–594 excluded volume, 573–576, 636–638 moments-generating function, 557–560 optical anisotropy, 277 radius of gyration, 561–565, 602–604, 637 Gaussian distribution 1-dimensional, 552–555 3-dimensional, 555–560, 601 effect of deformation on, 627 Gaussian statistics ideal chains, 548–560, 601 rubber elasticity, 624–628 gel permeation chromatography, 28, 489–491, 504–505, 512–513 gels, thermoreversible, polymer-solvent complexes, 745 g-factor, neutron scattering, elastomers, 591, 611 Gibbs free energy, 4 binary mixtures, 91 carbon disulfide, acetone, E-field, 164 dependence on composition, 91–93 Euler’s integral theorem, 5, 90 excluded volume lattice model, 78 in electric fields, 144, 157–158, 160 intercepts, tangents, 93 methanol, water, E-field, 163 methyl acetate, water, E-field, 162 minimization, chain expansion, 639–640 multicomponent mixtures, 92 osmotic pressure, 73

778

Index

Gibbs free energy (Continued) vs. isotropic chain expansion, 633–639 with order parameters, 172 Gibbs free energy of mixing classical thermodynamics, 128 –129 concentrated solutions, 183 Flory –Huggins lattice, 86, 317, 634 interaction, 180 Gibbs–DiMarzio theory conformational entropy, 117 –123 vs. lattice cluster theory, 124 –126 Gibbs–Duhem equation, 5, 10, 11, 91–92 binary mixtures, 96 glass transition analysis via l’Hopital’s rule, 37– 38 compositional dependence, 185 –186, 242 –243 compositional dependence, order parameter, 177 –183 dependence on molecular weight, 27– 29 discontinuous observables at, 7 effect of electric field, 150 –156 effect of particle size, film thickness, 31– 34 effect of pH, 584 –585 effect on activation energies for diffusion, 58– 60 entropy continuity, electric field, 151 –153 equilibrium vs. kinetics, 8, 178 –179 Gibbs –DiMarzio theory, 117 –123 graphical analysis via DSC, 28 iso-free-volume state, 24 –27 lattice cluster theory, 123 –126 modification via metal complexes, 208 –209 nanoscale effects, 34 pressure dependence, via order parameters, 176 poly(lactic acid), 189 volume continuity, electric field, 153 –155 vs. MW, linear least squares analysis, 46– 47 glass transition temperature definition, 4 depression, TS Chow, 185 enhancement, 192 –194, 240 enhancement, P4VP/Ni(II), 210

poly(vinylphenol) blends, 749 vs. composition, 185–186, 242–243 glasses densified, 9, 12 nonequilibrium, 9 glassy state, equilibrium, 8 glycerol, thermal expansion coefficient, 342 Goldman–Shen experiment magnetic spin diffusion, 707– 715 polyurethane, 13C NMR, 745–746 good solvents, chain expansion, 570–578, 631–640 Gordon–Taylor equation, 40 –45, 242 assumptions, 186 entropy continuity 17 –18 linear least-squares analysis, 20 –21, 44 volume continuity, 18–20 GPC (gel permeation chromatography) calibration curve, 582, 604 output curve, 489–491, 504–505, 512–513 ground state crossover, Tanabe –Sugano, 206, 234 group theory, molecular symmetry, complexes, 760–762 growth rate, crystals dependence on temperature, 303 free volume, 303–305 without impingement, 288 Guggenheim’s lattice theory, 105–117 entropy of mixing, 111, 115–117 monomers and dimers, 108–111 monomers and polymers, 111–117 gyration, radius of, Gaussian chains, 561–565, 602–604, 637 Hagen–Poiseuille law, capillary viscometry, 595, 599 half-time crystallization kinetics, 296, 323 diffusion in polymers, 61 –65 vs. efflux time, viscometry, 599–600 vs. maximum rate of crystallization, 297–299 Hamiltonian operator heteronuclear dipolar, 758– 760 quantum mechanics, 653– 655, 657–658, 660–673 hard-and-soft, acids, bases, 199

Index heat capacity, 332 –334 helical conformations, 250 –251 helical pitch, 251 –252 Helmholtz free energy minimization, chain expansion, 576 –578 heptyloxybenzoic acid, DSC thermogram, 330 Hermitian operator, Hamiltonian, 663, 670 heterogeneous nucleation, 288 –292 nucleation, snowflakes, 294 “reactions”, crystallization, 352 heteronuclear spin diffusion P4VP/[RuCl2(CO)6]2, 755 –758 PMMA/dinitrobiphenyl, 753– 755 hexyloxybenzoic acid, DSC thermogram, 330 high-spin, d-electron configuration, 205 –207 high-temperature limit, partition function, 656, 658, 662 Hoffman’s crystallization theory, 310 –317 Hoffman–Weeks analysis, equilibrium Tmelt, 322 homogeneous deformation, 435 homogeneous nucleation crystallization, 292 –293 dependence on temperature, 302 –304 thermodynamics, 299 –302 Hooke’s law of elasticity, 359 –360 nonlinear, elastomers, 606 –607 rubber-like solids, 592 –593, 646 –647 humid aging, BPAPC, activation energy, 691 –693 hydration enthalpy, hexa-aqua complexes, 196 –197 hydrogels, controlled release near LCST, 96 hydrogen bonding carbon-13 NMR, 745 –753 crystallization, thermodynamics, 310 –317 polymer blends, 308 –310 strength, PVPh/PVMK, 751 –752 hydrogen consumption, kinetics, 533 hydrogenation of alkenes, 527 –533 hydrogen-bonded molecular complexes, 728 –738 hydrostatic momentum balance, 11

779

ideal chains, Gaussian statistics, 548–560, 601 ideal gas vs. rubber-like solid, analogies, 610 ideal mixtures, 15 ideal rubber-like solids, internal energy, 614–615, 619 impedance bridge, dielectric relaxation, 693–696 complex, dielectric relaxation, 693–696 imperfect crystals, 302 impingement spherulites, 289–292, 327 truncation factor, 290 increase in Tg, Gibbs– DiMarzio theory, 122 inertial models, torsion pendulum, 420–425, 467–468 infrared absorbance, CvC stretch, 539–543 CO stretches, Ru2þ complexes, group theory, 760– 762 hydrogen bonds, PVPh/PVMK, 751–752 integral form, Maxwell model, 364–366 integral transforms, 443, 451–452 integrating factor Liouville equation, 664–665 Voigt model, 454–455 integro-differential equation, stress relaxation, 383– 384 interaction enthalpy of mixing, 134, 180–181 entropy of mixing, 134, 180–181 free energy, Flory –Huggins, 83 –86 free energy, van Laar model, 161 Gibbs free energy of mixing, 180 interaction parameter chemical stability, 98, 641 compositional dependence, 124 dependence on structure, 124 Flory– Huggins, 83 –86, 180–181 melting point depression, 319–321 MW dependence, 124 MW dependence at critical point, 99 pressure dependence, 124 temperature dependence, 133–135 theta solvent, 87 interaction representation, time-dependent perturbation theory, 668–669

780

Index

intercepts, Gibbs free energy vs. composition, 93 interchain coordination, 194 interelectronic repulsion, 204 –207, 233 –234 interfacial tension effects on Tg, 34 –35 intermolecular spin diffusion P4VP/[RuCl2(CO)6]2, 755 –758 PMMA/dinitrobiphenyl, 753– 755 internal energy ideal rubber-like solids, 614 –615, 619 in electric field, 144 total differential, 332 –333 internal energy change, crystallization, 333 –335 interspherulitic connectivity, 279 –282 intrachain coordination, 194 intrinsic birefringence, 278 –279 damping, torsion pendulum, 467 –468 intrinsic viscosity dilute solutions, 579 –584, 604 Einstein model, 582, 604 Mark –Houwink equation, 582 –583 MW scaling law, 583, 604 inverse Langevin function, Taylor series, 277 moments, relaxation time distribution, 444 –450 problems, dynamic mechanical expts., 434 ionomer effect, Tg, poly(acrylic acid), 584 –585 SurlynTM , carbon-13 NMR, 740 –741 irreducible representations, CO stretches, group theory, 761 –762 irreversible degradation, viscous dissipation, 580 –581 thermodynamics, 104 iso-free-volume state, glass transition, 24 –27 isomers, configurational, 250 isotactic definition, 250 poly(1-butene), 252 poly(1-butene), pychnometry, 344 –345 poly(methyl methacrylate), WAXD, 286

polypropylene, thermophysical data, 323–326 isotactic polystyrene optimum crystallization temperature, 306 thermal transitions, 322 WAXD, 286 isothermal calorimetry crystallization, 295, 323 theory, 331– 335 isothermal compressibility, 332–333 discontinuity at Tg, 7, 12, 30–31, 37 isothermal crystallization, energy balance, 334–335 isotopic enrichment, 13C NMR, polycarbonate, 685–687 isotropic chain expansion, Boltzmann’s entropy, 633 Jørgensen’s f-factor carbonyl ligands, 217 empirical correlation, 233 Jørgensen’s g-factor, molybdenum, 217 parameters, 204, 207, 233 jump strain, stress relaxation, 370 stress, Voigt model, 455 jumps critical free volume required for, 52, 55, 57– 58, 60 via Brownian motion, 50–51 Kelley–Bueche, Tg depression, 185 Khinchin–Wiener –Einstein theorem, spectral density, 674 kinetic chain length, anionic, 509, 513 kinetic processes, below Tg, 9 kinetic rate laws, Pd-catalysis, 543 kinetics of crystallization isothermal, 335–336 optimum rate, 304– 307, 323 kinetics alkene dimerization, 534–543 alkene hydrogenation, 529–533 anionic polymerization, 509–510 crystallization half-time, 296, 323 free radical reactions, 500–502 free radical, disproportionation, 507–508 homogeneous nucleation, 292–293

Index heterogeneous nucleation, 288 –291 isothermal crystallization, 295, 323 maximum rate of crystallization, 297 –299 most probable distribution, 524 –527 phase boundaries, 546 phase separation, 104 –105 vs. equilibrium, glass transition, 8, 178 –179 Kramers –Kronig theorem, 680 –684 Kratky–Porod persistence length Langevin distribution, 267 Tg increase, 122 –123 KratonTM , PdCl2 stress relaxation, 476 –479 stress– strain, 470 –472 KratonTM , SBS, magnetic spin diffusion, 711 –715 Kuhn statistical segment length, 569 –570 Kwei equation, Tg vs. composition, 242 –243 l’Hopital’s rule analysis of Tg, 37–38 applied to Clapeyron equation, 11, 12 Clapeyron equation, electric field, 150, 152 –153 lag time diffusion, spheres, 716, 719 –720 membrane diffusion, 65–66 Lagrange multiplier optimization Boltzmann distribution, 656 diffusion, 53 –54 Langevin distribution, 265 –266 polycondensation, 487 –489 strain ellipsoid, 437 –438 stress ellipsoid, 442 Laguerre polynomials, 521 –524 lamellae, 253 –254 disordered, optical microscopy, 260 –262 lamellar thickness, effect of hydrogen bonding, 310 –317 Landel RF, shifting dynamic compliance data, 459 –460 Langevin distribution function, 263 –267 inverse function, Taylor series, 277 optical anisotropy, 271– 274 Taylor series, 274

781

lanthanide complexes pH, poly(acrylic acid), 585– 586 Tg enhancement, 240 Laplace transforms compliance, modulus, 371– 372, 428–430 diffusion equation, 556 Gaussian distribution, 557–560 moments-generating function, 515–521 relaxation time distribution, 452 Voigt element, 379– 381 Laplace’s equation, parallel-plate capacitor, 694 Laplacian, spherical coordinates, 716–717 lattice cluster theory glass transition, 123– 126 vs. Gibbs– DiMarzio theory, 124–126 lattice coordination number, Guggenheim’s theory, 109 lattice enthalpy, hexa-aqua complexes with LFSE, 196–197 lattice models, 71 –72 excluded volume, 572–574 Guggenheim, 105–117 Sanchez –Lacombe, 126– 127 Tg depression, discontinuity in @ x/@T, 183–184 LCST, effect of electric field, 149 Leibnitz rule, differentiating integrals, 381–383 length scale, stress relaxation, 480–482 lever rule, binary mixtures, 103 LexanTM polycarbonate, experimental NMR, 685–687 LFSE (ligand field stabilization energy) 3-coordinate complexes, 237 5-coordinate complexes, 222– 223, 231–232 lattice enthalpies of hexa-aqua complexes, 196–197 Mo(CO)6, poly(vinylamine), 218, 223 poly(4-vinylpyridine)/nickel acetate, 211 ligand displacement at Tg, 213–215, 222, 231, 236, 239 ligand exchange, Tg enhancement, 202 ligand field model, Tg enhancement, 209–216

782

Index

ligand field splitting 5-coordinate complexes, 222, 232 Jørgensen’s parameters, 204, 207 Mo(CO)6, PVAm, 222 nickel complexes, 210 –211 quantum mechanics, 218 –219 rule of average environment, 207 –208, 210, 222, 231 –232 ligand field stabilization energy, 191 –192 Mo(CO)6, PVAm, 218, 223 Ni complexes, 211 ligand field stabilization model, Tg enhancement, 234– 235 light intensity, optical microscopy, 258 –259, 262, 285 limited chain extensibility, 472 limiting cases, Maxwell model, 363 –364 linear combinations of atomic orbitals, 653 –654, 662 linear compressibility coefficients, 622 –623 linear least squares analysis (LLSA) Avrami equation, 295 –296 diffusion coefficients vs. temperature, 56– 57 first-order coefficient, 346 Gordon –Taylor equation, 20 –21, 44 isothermal DSC, 336 –337 melting point depression, 319 –321 Mooney–Rivlin equation, 605 osmotic pressure, 75, 79 second order polynomial, 641– 643 Tg enhancement, 215–216 Tg vs. MW, 46 –47 WLF concentration shift factor, 395 –397 WLF equation, 387 –388 viscosity vs. temperature, 458 –459 linear viscoelasticity, Maxwell model, 362 –367 Liouville equation, time-dependent perturbation theory, 661 –673 liquid crystals DSC thermograms, 330 plasticizers, 188 lithium perchlorate, nucleating agent, 294 living polymerization, 508– 515 local equilibrium, spin temperature, 658 –659 logarithmic decrement intrinsic damping, 467 –468

torsion pendulum, 424–425, 468 vs. loss tangent, 419, 468 Lorentzian lineshape, spectral density, 670–671 Lorentz –Lorenz equation, 276–277 loss compliance complex variables, 464– 465 dynamic experiments, 433–435 loss modulus bimodal distribution, 399 dynamic experiments, 412–416 Fourier transform, 415–416 Kramers –Kronig theorem, 682– 684 loss tangent torsion pendulum, 424, 468 vs. logarithmic decrement, 419, 468 loss viscosity complex variables, 465– 466 forced vibration experiments, 415 Kramers –Kronig theorem, 681– 684 lower critical solution temperature (LCST), 96, 134 low-spin, d-electron configuration, 205–207 lumped-mass model, force balance, torsion pendulum, 421 magnetic resonance, 705– 706 magnetic spin diffusion copolymers, 706–715 vs. radial diffusion, 723–726 magnetism, Curie’s law, 657, 661– 662, 756 major axis, strain ellipsoid, 437 Maltese cross, polarized optical microscopy, 259 Margules model effect of x on phase compositions, 102 Tg enhancement, 214–216 vs. Flory-Huggins model, 85 Mark–Houwink equation, intrinsic viscosity, 582–583 mass balances crystallization, 333–334 lever rule, 103 unsteady state, viscometry, 595– 600 mass fraction distribution anionic, 512, 515 free radical, 503 most probable, 489–490

Index mass fraction of crystallinity density, 339 –340 DSC heating, 337 –339 isothermal DSC, 336 –337 mass fraction vs. mole fraction, 544 mass spectrometry, diffusion measurements, 61 mass-average degree of polymerization, 496 –498, 504, 512, 517 –520 material response time, power law model, 467 mathematical models, linear viscoelasticity, 355 –356 maximum rate of crystallization, 297 –299 Maxwell model complex impedance, 693 –694 creep, creep recovery, 368 –370 electrical analog, 457 integral forms, 364 –366 mechanical analog, 366 –367 nonlinear, power law, 466 –467 permanent set, 370 stress relaxation, 370 –371 viscoelasticity, 362 –367 vs. Voigt model, dielectric relaxation, 696 Maxwell relation based on Gibbs free energy, 332 in electric fields, 144–145 Maxwell –Boltzmann distribution, velocity, 557 Maxwell –Wiechert model, 373, 397 –398, 461 mean-square displacement Fourier transform, 416 –417 Gaussian chains, 555 –556 mean-square ETE chain length freely jointed 549 –551, 601 freely rotating, 565 –568 mean-square radius of gyration, 561– 565, 602 –604, 637 mechanical analog, Maxwell model, 366 –367 energy, irreversible degradation, 580 –581 models, creep, relaxation, 466 properties, KratonTM , PdCl2, 470 –472, 476 –479 properties, semicrystalline polymers, 279 –282

783

requirement for phase equilibrium, 10 vs. electrical, viscoelasticity, 457 mechanism alkene dimerization, 534–539 anionic polymerization, 508–509 free radical kinetics, 499 hydrogenation catalysts, 528–529 melting temperature depression, blends, 317–322 equilibrium, Hoffman –Weeks analysis, 322 thermodynamics, 252–253 melting transition, effect of electric field, 148–150 membrane diffusion, 50– 51, 61 –66 membrane osmometry, osmotic pressure, 72–75 memory fluids, 356 memory function, Maxwell model, 365 metal complexes KratonTM , stress relaxation, 476– 479 Pd(II), KratonTM , stress–strain, 470–472 Tg enhancement, 198 metastable states, 95 methanol, water, Gibbs free energy, E-field, 163 method of generalized functions, Kramers–Kronig theorem, 681 methyl acetate, water, Gibbs free energy, E-field, 162 methylresorcinol-PEO, NMR vs. DSC, 731–732 microhydrodynamic force balance, Rouse model, 678 microscopic reversibility, principle of, 658 stress tensor, autocorrelation, 674–675 minimization, free energy, chain expansion, 576–578, 639–640 minimum-boiling azeotrope, methyl acetate, water, 162 mobility, dipolar dephasing, polyethylene, 744 modulus dynamic, complex, 674, 676 dynamic testing, 412–416 of elasticity, 360 of elasticity, rubber-like solids, 592–593, 606

784

Index

modulus (Continued) vs. temperature, rubber-like solids, 592 –593, 606, 646 molar density, threshold, c , 586 –587 molar mass, averages, 545 mole fraction distribution anionic, 511, 513 disproportionation, 507 free radical, 502 Laguerre polynomials, 522 –524 most probable, 489 –490, 524 –527 mole fraction Flory –Huggins lattice, 83 vs. mass fraction, 544 molecular complexes, NMR vs. DSC, 728 –738 molecular weight dependence diffusivities, 66 –69 Flory x at critical point, 99 fractional free volume, 26– 29 free volume, 46 interaction parameter, 124 of Tg, 45–47 of Tg, via free volume, 27 –29 molecular weight distribution anionic, 511 –515 polycondensation, 486 –491, 524 –527 molybdenum carbonyl complexes, 216 –223 molybdenum, Jørgensen’s g-factor, 217 moments anionic polymerization, 512 –515, 520 –521 of inertia, torsion pendulum, 422, 467 –468 polycondensation reactions, 497 –498, 517, 523 moments-generating function average MW, 496 –498, 502 –504, 511 –512, 515 –521 continuous, 515– 521 Gaussian chains, 557 –560 momentum balance, hydrostatic conditions, 11 momentum diffusivity vs. efflux time, 597 –600 monodisperse distribution, anionic, 509, 511

monomeric friction coefficients, WLF shift factor, 394–395 monomers and dimers entropy of mixing, 111 nearest neighbors, 109 occupational probabilities, 108–110 monomers and polymers nearest neighbors, 112 occupational probabilities, 111–114 monomers and r-mers, entropy of mixing, 115–117 Mooney–Rivlin equation, rubber-like solids, 605 morphology, diblock copolymers below UCST, 100 most probable distribution Laguerre polynomials, 522–524 polycondensation, 486–491, 524–527 most probable ETE chain length, Gaussian, 601 multicomponent mixtures density, 349–351 Gibbs free energy, 92 multiple melting, chain folding, 307–309 multiplicity function continuous limit, 628, 630 network strands, 625–628 segment orientation, 264 multiplicity of permutations, for diffusion, 52 multiplicity of states excluded volume, 76, 571 Flory –Huggins lattice, 80 –82 polycondensation, 486–487 trimers and hexamers, 132 multivariable parameter estimation, core-shell, 725–728 mutual spin-spin flips, dipolar Hamiltonian, 709, 712, 756–760 MW dependence, terminal relaxation time, 402–403 MW distribution function free radical, 502–503 Laguerre polynomials, 522–524 most probable, 486–491, 524–527 MW scaling law chain expansion factor, 577–579, 638–639

Index excluded volume parameter, 573 –574, 639 free energy vs. chain expansion, 638 –639 intrinsic viscosity, 583, 604 MW, number-average, 545 MW, weight-average, 545 MWcritical, entanglements, 403 nanoclusters, transition metals, hydrogenation, 527 –533 nanorheology, particle-tracking, 416 –417 nanoscale effects on Tg, 34 nearest neighbors, Guggenheim’s lattice, 109, 112 negative deviations, nonideal mixing, E-fields, 165 nematic-isotropic phase transition, 330 network strands, multiplicity function, 625 –628 neutron scattering, g-factor, elastomers, 591 Newton’s law of viscosity, 360– 361 Newtonian fluids, capillary viscometry, 595 –600 nickel acetate, poly(4-vinylpyridine), 209 –216 nickel complexes, 227 ligand field splitting, 210–211 NMR carbon-13, polyester blends, 308 –310 phase coexistence, PEO-resorcinol, 728 –738 spin-lattice relaxation time, 659 –661 NMR experiments, 13C, solid state, 738 –758 NMR relaxation experimental, 684 –687 spin-temperature equilibration, 656 –661 time-dependent perturbation theory, 661 –673 NMR vs. DSC poly(acrylic acid)/poly(ethylene oxide), 752 –753 PEO-resorcinol, 729– 733 PVPh/PVMK, PVPh/PEO, 747 –752 nomenclature electro-thermodynamics, 167 –168 Flory– Huggins lattice, 79, 83 nonequilibrium glasses, 9, 37

785

non-Gaussian chains, entropy elasticity, 605–607 nonideal mixing electric fields, 165 thermodynamic properties, 128–131 nonlinear Maxwell model, power law, 466–467 stress relaxation, 476–482 viscoelasticity, 469–470 nonlinear least squares analysis elastic modulus, 643–645 radial diffusion, 725–728 volume vs. temp., 342–349 non-spinning 13C NMR, polycarbonate, 686–687 normal stress, 358 biaxial deformation, 631 difference, elastic solids, 442–443 nucleating agent, PEO, LiClO4, 294 nucleation heterogeneous, 288–292 homogeneous, crystallization, 292–293 rate constants, 293 snowflakes, 294 nucleation and growth, 287 phase separation, 105 without impingement, 288 NucrelTM , random copolymers, 13C NMR, 740–741 null impedance matching, dielectric relaxation, 693– 696 number-average degree of polymerization, 496–498, 503, 512, 517–520 diffusion coefficient, 68 molecular weight, 545 observables, quantum mechanical expectation value, 653–655, 661–662 occupational probability Boltzmann distribution, 655–656 monomers and dimers, 108– 110 monomers and polymers, 111–114 vs. activity, Guggenheim’s lattice, 110, 115 octahedral ligand field splitting, quantum mechanics, 218– 219 OH stretch, infrared, PVPh, 749–752 Ohm’s law, impedance, 361–362, 693

786

Index

Onsager coefficient, 104 optic axis, 257 optical anisotropy, 255 –279 Gaussian chains, 277 Langevin distribution, 271 –274 rigid-rod polymers, 274 optical microscopy birefringence, 275 –277 birefringent spherulite, 258 –259, 284 –285 disordered lamellae, 260 –262 nucleating agent, LiClO4, 294 quarter-waveplate, 289 thermal treatment, 283– 284 optimization, via Lagrange multipliers with constraints Boltzmann distribution, 656 diffusion, 53 –54 Langevin distribution, 265 –266 polycondensation, 487 –489 strain ellipsoid, 437 –438 stress ellipsoid, 442 optimum crystallization temperature, 304 –307, 323 lamellar thickness, hydrogen bonding, 314 –317 order parameters affinities, 175 Ehrenfest inequalities, 177 in thermodynamics, 171– 172 Tg depression, assumptions, 178–181 organometallic reactions, mechanisms, kinetics, 527– 543 orientation birefringence, 279 distribution, percent of rigid-rod limit, 267 Langevin distribution, 263 –267 orthonormal basis functions, wave function, 653 –654, 663, 667 oscillatory experiments, phase lag, 361 –362 mechanical perturbation, dissipation, 417 –419 response, angle of twist, 423 –424 osmotic pressure excluded volume lattice model, 78– 79 Flory –Huggins model, 87

linear least squares analysis, 75, 79 membrane osmometry, 72– 75 second virial coefficient, 75 overshoot, radial diffusion, core-shell, 725 P4VP, poly(4-vinylpyridine) Ni(II), ligand field stabilization energy, 211 [RuCl2(CO)6]2, 1H-13C spin diffusion, 755–758 Tg enhancement, Co(II), Ni(II), Ru(II), 225, 235 zinc-complexes, 13C NMR vs. DSC, 741–743 palladium catalysis, 1,2- and cis-PBD, 534–543 palladium chloride, polybutadiene, modulus, 644 parallel-plate capacitor, Laplace’s eq., Guass’ law, 694 parameter estimation, multivariable, 725–728 partial differential equation, radial diffusion, 715–725 partial molar dielectric permeability, 159 dielectric susceptibility, 146, 158 enthalpy, in electric field, 147 entropy, 130 entropy, in electric field, 146 properties, in electric field, 144–147 volume of mixing, activity coefficient 129 volume, 128 volume, in electric field, 146, 158, 160 volume, osmotic pressure, 73– 74 volume, zero-field, 161 partial specific internal energy, 333 particle size, effect on Tg, 31– 34 partition function, Boltzmann factors, 656, 662 Pauling electronegativities, 241–242 functional group electronegativity, 750 PdCl2, KratonTM (SBS triblock) stress relaxation, 476–479 stress –strain, 470–472 PDE, homogeneous boundary conditions, 717

Index PEEK, thermophysical properties, 352 pendant groups, via coordination, 212 pentagonal planar complexes, above Tg, 221 –222, 230 pentyloxycinnamic acid, DSC thermogram, 330 PEO, chemical shielding tensor, 739 PEO-resorcinol, NMR vs. DSC, 731 –738 percolation threshold, elastic modulus, 643 –644 perfectly elastic solids, 440 –443 permanent set, Maxwell model, 370 permittivity, definition, 143 perpetual motion machines, second kind, 417 persistence length Kratky–Porod, Langevin distribution, 267 Tg enhancement, 122 –123 perturbation theory, Liouville equation, 661 –673 pH, poly(acrylic acid), metal cations, 584 –586 phase coexistence, NMR detection, 728 –733 phase compositions, dependence on Flory x, 102 phase diagrams eutectic transitions, 732 –733 hydrogen-bonded blends, 749 Tg vs. composition, P4VP/Zn2þ, 742 phase equilibrium chemical requirement, 6, 95 first-order process, 10–11 second-order processes, 11 –14 phase lag, dynamic experiments, 361 –362 phase rule, 90, 128 phase separation algorithm, Flory–Huggins model, 100 –103 common tangent, 95 diblock copolymers, 100 electric-field-induced, 137– 138, 166 lever rule, 103 nucleation and growth, 105 spinodal decomposition, 104 –105 phase transitions, 3 first-order, 4 first-order, in electric field, 148– 150 kinetics, sol vs. gel, 546

787

nth-order, 5 second-order, in electric field, 150–156 third-order, 38–39 phasor analysis, dynamic mechanical expts., 413–415, 465–466 phenolic carbon chemical shift, PVPh, blends, 747–748 PHIS, Tg enhancement, Co(II), Ni(II), Ru(II), 226, 235 physical aging, 8–10 pizza dough, free recovery (Piedicastello: Alessandro, Monika, Petra), 286, 410 pKA, acid-base equilibrium constants, 200–201 plasticizer efficiency binary mixtures, 40–41, 395–397 fractional free volume, 23– 25 plasticizers effect on Tg of polycarbonate, 178 liquid crystalline, 188 PMMA/dinitrobiphenyl, 1H-13C spin diffusion, 753–755 point groups, molecular symmetry, 198, 760–762 Poisson distribution, anionic, 511, 513–515 Poisson’s ratio, 356–357, 590 polarizability tensor ensemble averaging, 271– 274 rectangular coordinates, 269–271 spherical coordinates, 256, 268 polarization coefficient, definition, 143 density, 142–143, 276 vector, 142– 145 polarization, dipole, Debye equation, 697–700 polarized dielectrics, discharge current, 696–702 polarized optical microscopy disordered lamellae, 260–261 Maltese cross, 259 nucleating agent, 294 thermal treatment, 283–284 poling, molten state, electrets, 697–698 poly(1,4-butylene adipate), 13C NMR, 309 poly(1-butene), isotactic, 252 density, 344–345

788

Index

poly(4-vinylpyridine) Co(II), Ni(II), Ru(II), 224– 238 nickel acetate, 209 –216 [RuCl2(CO)6]2, 13C NMR, 755 –758 zinc-complexes, 741 –743 poly(acrylic acid) pH, Tg, 584 –586 poly(ethylene oxide), 13C NMR, 752–753 poly(dimethylacrylamide)/ poly(vinylphenol), 13C NMR, 745 –747 poly(ether ether ketone), PEEK, 352 poly(ethylene adipate), 13C NMR, 309 poly(ethylene oxide), 252 aqueous, intrinsic viscosity, 584, 604 chemical shielding tensor, 739 poly(acrylic acid), 13C NMR, 752 –753 poly(vinylphenol), 13C NMR, 747– 752 resorcinol, NMR, 731 –738 spherulites, 256 spherulitic impingement, 289 poly(lactic acid), Tg, 189 poly(L-histidine), Co(II), Ni(II), Ru(II), 224 –238 poly(methyl methacrylate) dinitrobiphenyl, 13C NMR, 753 –755 isotactic, WAXD, 286 poly(N-isopropylacrylamide), smart hydrogels, 96 poly(oxymethylene), 252 poly(propylene), isotactic, 252 thermophysical data, 323 –326 poly(vinyl methyl ketone)/ poly(vinylphenol), 13C NMR, 747 –752 poly(vinylamine), CoCl2(H2O)6 density, 348 –350 volume vs. temperature, 346–349 poly(vinylamine), molybdenum complexes, 216 –223 poly(vinylphenol), 13C NMR poly(dimethylacrylamide), 745 –747 polyester blends, hydrogen bonding, 308 –310 poly(ethylene oxide), 747 –752 poly(vinyl methyl ketone), 747 –752 polybutadiene 1,2- and cis-, 534–543 PdCl2, elastic modulus, 644

polycaprolactone, 13C NMR, 309 polycarbonate aging, activation energy, 691–693 13 C NMR, experimental, 685–687 dielectric relaxation, 689–691 effect of plasticizers on Tg, 178 functionalized, ring motion, 688 Tg depression, 178, 187–188 polycondensation degree of polymerization, 491–496 Laguerre polynomials, 522–524 moments-generating function, 496–498, 517 MW distribution, 486–491, 524–527 polydispersity anionic, 512, 520–521 free radical, 504, 518 polycondensation, 498, 517 polyelectrolytes, poly(acrylic acid), 584–586 polyester blends, 13C NMR, 308– 310 polyethylene, 252 dielectric breakdown, 141 dipolar dephasing, 13C NMR, 740, 744 gamma-gauche effect, 740, 744 polymer blends hydrogen bonding, 13C NMR, 308– 310, 745–753 multiple melting, 307–309 polymer chemical potential, Flory –Huggins model, 183–184, 318 volume fraction vs. MW at UCST, 99 polymer-diluent blends, melting point depression, 319–320 polymerization anionic, 508–515 condensation, 486–498, 523–527 free radical, 498–508 polymer–polymer blends, melting point depression, 320–321 polymers, electric field effects, 139–141 polymer–solvent complexes, thermoreversible gels, 745 polymorphism, 251–252 polynomials linear least squares, 641– 643 Laguerre, 521–524

Index polypropylene, isotactic, thermophysical data, 323 –326 polysaccharide, hydrated, free recovery, 410 polystyrene intrinsic viscosity in THF, 583 –584 isotactic, thermal transitions, 322 isotactic, WAXD, 286 optimum crystallization temperature, 306 Tg depression, 188 thermal expansion coefficient, 342 thermophysical properties, 36 polyurethane, spin diffusion, 13C NMR, 745 –746 positive deviations, nonideal mixing, E-fields, 165 powder pattern, 13C NMR, polycarbonate, 686 –687 power-law fluid, capillary viscometer, 598 –599 nonlinear Maxwell model, 466 –467 pressure dependence chemical potential, 128 –129 dielectric constant, example, 155 fractional free volume, 29– 31 glass transition temperature, 36– 37 interaction parameter, 124 of Tg, via order parameters, 176 pressure difference, capillary viscometry, 595 pressure gradient, in external field, 11 Prigogine-Defay ratio, 14, 176 electric-field analog, 155– 156 principal axes strain ellipsoid, 437 –438 stress ellipsoid, 441 –443 principal values shielding tensor, PEO, 739 strain ellipsoid, 435 –436 stress ellipsoid, 438 –441, 443 principle of causality, Cauchy’s integral, Kramers–Kronig, 683 microscopic reversibility, 658 probability of occupation monomers and dimers, 108– 110 monomers and polymers, 111 –114 protein folding, 251

789

proton NMR, CRAMPS, PEO-resorcinol, 734–738 proton spin diffusion KratonTM SBS triblock copolymers, 711–715 PEO-resorcinol, 735– 738 polyurethane, 13C NMR, 745–746 SurlynTM random copolymers/ionomers, 709–711 pseudo-steady-state approximation, 500–501, 529–531, 540–543 pulse response dielectric spectroscopy, 702–703 vs. step response, 702– 703 pulse sequences Goldman-Shen experiment, 707–709 NMR relaxation, 686 purple heart, AV Belfiore, 766 pychnometry measurements, density, 340–341 poly(1-butene), 344–345 polypropylene, 324–325 thermal expansion coefficients, 341–349 quantum mechanics introduction, 652–655 ligand field splitting, 218–219 quarter-waveplate, optical microscopy, 289 Racah interelectronic repulsion energy, 204–207, 234 radial diffusion overshoot, core-shell, 725 spherical coordinates, 715–725 vs. spin diffusion, 723–726 radial segment density distribution, Gaussian, 602–604 radial wave function, 3-d orbital, 219 radiofrequency pulses, NMR relaxation, 686 radius of gyration distribution function, 573, 602–604, 636–638 freely jointed chains, 561–565, 602–604, 637 raising/lowering operators, angular momentum, 759–760 Ramachandran bond rotational angles, 251, 565–566

790

Index

random copolymers, magnetic spin diffusion, 709 –711 random walk statistics, 1-dimensional, 552 –555 random-coil polymers, optical anisotropy, 277 rate constants crystallization, isothermal DSC, 335 –337 heterogeneous nucleation, 291, 327 nucleation, 293 time-dependent, 474 rate laws, kinetics, Pd –catalysis, 543 rate of crystallization, Avrami equation, 335 –336 rate-of-strain elastic springs, 379 tensor, 359 viscous element, 383 Voigt element, 379 –383 reactive blending, 199 real chains Flory’s law of, PG deGennes, 578 –579, 586 –587 good solvents, 570 –578, 631 –640 real gases, virial expansion, 75 reciprocal strain ellipsoid, 625, 627 –629 recombination continuous distribution, 505 –507 free radical reactions, 499 –502 recoverable deformation, Boltzmann superposition, 407 –410 reduced symmetry, above Tg, 203 reference temperature, WLF shift factor, 393 refractive index, birefringence, 275 –277 relative electric permittivity, definition, 143 relative viscosity, 581, 585 relaxation modulus bimodal distribution, 399 –400 continuous distribution, 397 –400, 443 –451 exponential integral, 400 –402 Fourier transform, 415 –416, 418, 451 –452 fractional exponent, 473 –475 Kramers –Kronig theorem, 680– 683 Laplace transform, 371 –372 single time constant, 364, 371 –372 stress-strain analysis, 462 –463

stretched exponential, 473– 475 Taylor series, 444–446 time-strain separability, 479– 480 time-temperature equivalence, 372– 373 vs. creep compliance, 428–430 vs. viscosity, 389–390, 404, 406 relaxation strength, spectral density, 669–671 relaxation time distribution, 397–400, 430–431, 443–451, 461 effect of strain on, 478–482 first-moment, fractional exponent, 475 terminal relaxation time vs. MW, 403 relaxation, dielectric, polycarbonate, 689–691 relaxation, NMR experimental, 684–687 spin-temperature equilibration, 656–661 time-dependent perturbation theory, 661–673 reptation, 4, 9, 150, 316, 388, 476, 479, 678 repulsion, interelectronic, 204–207, 233–234 requirements, chemical stability, 93, 640–641 resistors and dashpots, 360–362 resorcinol, charge density, 735 resorcinol-PEO, NMR vs. DSC, 731–738 retractive force crosslinked elastomers, 587–594, 605–607, 611 entropy elasticity, 677–678 equation of state, rubbers, 611–615, 619, 629–631 stress –strain, 587–594, 605 temperature dependence, 617–623, 646 rheologically simple materials, 431–432, 701–702 activation energy, 690–691, 701–702 rigidity of diluents, effect on Tg, 120–122 rigid-rod polymers, optical anisotropy, 274 rotational isomeric preference, Stockmayer– Kurata, 565, 568 rotational states, trans and gauche, 251–252, 565 Rouse model, diffusion equation, 677–680

Index rubber elasticity classical thermodynamics, 610 –623 Gaussian statistics, 624 –628 stress vs. strain, 629 –631 thermodynamics, 587 –594, 605 –607 rubber-like solids affine deformation, 609– 610, 624 external force, entropy, 591 –594, 606, 611 thermoelastic inversion, 616– 617, 646 –647 vs. ideal gas, analogies, 610 rubbery polymers, birefringence, 278 –279 rule of average environment, ligand field splitting, 207– 208, 210, 222, 231 –232 mixtures, additive, Tg, 242 ruthenium complexes carbonyl chloride, [RuCl2(CO)6]2, isomers, 758 13 C NMR spinning sidebands, 757 CO stretches, group theory, 760 –762 P4VP, spin diffusion, 755 –758 Tg enhancement, 225 –227 Sanchez–Lacombe, lattice fluid theory, 126 –127 SBS triblock copolymers, diffusion equation, 715 –725 scaling law blobs, deGennes, 586– 587 chain expansion factor, 577 –579, 638 –639 diffusion coefficient vs. MW, 67 excluded volume parameter, 573 –574, 639 free energy vs. chain expansion, 638 –639 intrinsic viscosity, 583, 604 terminal relaxation time vs. MW, 403 viscosity vs. MW, 404 Schro¨dinger equation, 653 second virial coefficient excluded volume, 79 Flory– Huggins model, 87 osmotic pressure, 75 second-order phase transition, in electric field, 150 –156

791

polynomial, linear least squares, 641–643 processes, phase equilibrium, 11 –14 second-order-correct solution, Liouville equation, 664–673 second-rank tensors, symmetric 357–359 segment density distribution, Gaussian chains, 602–604 length, statistical, Kuhn, 569–570 orientation, Langevin distribution, 263–267 segmental relaxation, nonlinear, 480–482 self-association, OH groups, infrared, 749–752 semicrystalline polymers, mechanical properties, 279–282 semicrystallinity, definition, 329 separation-of-variables diffusion equation, Rouse model, 679 PDE, radial diffusion, 717–718 shear stress, 358 shear, 2-dimensional strain ellipsoid, 436–438 stress ellipsoid, 441–443 shielding tensor, 13C NMR polycarbonate, 685– 687 poly(ethylene oxide), 739 shift factor activation energy, 392 composition dependence, 394–397 reference temperature, 393 solutions, linear least squares, 395–397 theoretical justification, 389–391 time-temperature equivalence, 386–394 viscosity, linear least squares, 458–460 Vogel’s equation, 394 WLF equation, 387–388 shifting dynamic compliance data, RF Landel, 459–460 sidebands, 13C NMR, magic-angle spinning, [RuCl2(CO)6]2, 757 single crystals, 253 single relaxation time, inverse moments, 446–448 six-coordinate complexes, LFSE, 232 size of diluents, effect on Tg, 121– 122 slope, stress-strain analysis, 461–464

792

Index

small molecules, electric field effects, 138 –139 smart materials, controlled release near LCST, 96 smectic-nematic phase transition, 330 smoothed-density model, radius of gyration, 573, 602 –604, 636 –638 snowflakes, heterogeneous nucleation, 294 sol-gel phase boundary, kinetics, 546 solid state 13 C NMR experiments, 308– 310, 684 –687, 738 –758 phase transition, DSC, 330 solubility constants, for diffusion, 61–62 specific heat discontinuity, 6,16 at Tg, 41 –44 near Tmelt, 338 vs. temperature, third-order transition, 39 spectral density Lorentzian lineshape, 670 –671 vs. correlation functions, 668 –671, 674 –676 sphere diffusion eq., SBS triblock copolymers, 715 –725 volume, partially filled, 596 spherulite boundary strengthening, 281 –282 critical radius, 301 –302, 327 free energy, 299 –302, 327 polarized light, 258– 259 poly(ethylene oxide), 256 unpolarized light, 284 –285 spherulitic impingement, 289 –292, 327 superstructure, 249 spin angular momentum, raising/lowering operators, 759 –760 spin diffusion CRAMPS, PEO-resorcinol, 735 –738 Goldman-Shen experiment, 706 –715 Goldman-Shen expt., polyurethane, 745 –746 P4VP/[RuCl2(CO)6]2, 755 –758 PMMA/dinitrobiphenyl, 753– 755 vs. radial diffusion, 723 –726 spin-lattice relaxation, NMR pulse sequences, 686 time constant, 659 –661

spinning sidebands, carbonyl chemical shift, [RuCl2(CO)6]2, 757 spinodal decomposition, diffuse interfaces 104–105 spinodal points, 94 spin–spin flips, dipolar Hamiltonian, 709, 712, 756–760 spin-temperature equilibration, NMR relaxation, 656–661 springs and capacitors, 359–362 square pyramid complexes, above Tg, 220–221, 229 stability requirement, chemical potential, 96 –97, 640–641 stabilization energy, backbone, Kwei equation, 243 stable states, 94 –95 statistical segment length, Kuhn, 569–570 statistical thermodynamics ergodic problem, 655–656 rubber elasticity, 623–631 steady shear flow, Boltzmann superposition, 406 steady-state approximation, 500–501, 529–531, 540–543 step response vs. pulse response, 702–703 stereochemistry, degree of polymerization, 493 Stieltjes transform, relaxation time distribution, 452 Stirling’s approximation, 53, 81, 89, 264, 267, 487, 553, 571, 627 Stockmayer –Kurata ratio, rotational states, 565, 568 Stokes –Einstein diffusion coefficient, 678–679 storage compliance complex variables, 464– 465 dynamic experiments, 433–435 storage modulus bimodal distribution, 399 dynamic experiments, 412–416 Fourier transform, 415–416 Kramers –Kronig theorem, 682– 684 storage viscosity complex variables, 465– 466 forced vibration experiments, 415 Kramers –Kronig theorem, 681– 684

Index strain dependence, stress relaxation, 477 –482 strain Boltzmann integral, 425– 428 dynamic testing, 432 –433 ellipsoid, 435– 438 engineering and true, 357 principal values of, 435 –436 tensor, 358– 359 Voigt element, convolution theorem, 381 strength, relaxation, spectral density, 669 –671 stress Boltzmann integral, 405– 407, 429 dynamic testing, 411 –412 ellipsoid, 438– 442 engineering and true, 357, 594 principal values of, 438 –441, 443 quadric, 442 vs. elongation, rubber-like solids, 594, 605 vs. strain, rubber elasticity, 629 –631 stress relaxation, 367 autocorrelation function, 674 –675, 680 boundary conditions, 384 effect of strain, 477– 482 fluctuation-dissipation, 673 –677, 684 fractional exponent, 473 –475 integro-differential equation, 383 –384 inverse moments, 444 –450 KratonTM , PdCl2, 476 –479 Maxwell þ Voigt elements, 378– 385 Maxwell model, 370 –371 mechanical models, 466 nonlinear, 476 –482 temperature dependence of, 385 –387 Voigt model, 374, 456 Voigt– Kelvin model, 456 –457 stress relaxation modulus continuous distribution, 397 –400, 443 –451 exponential integral, 400 –402 Fourier transform, 415 –416, 418, 451 –452 Kramers– Kronig theorem, 680 –683 stress– strain curve, 462 –463 Taylor series, 444 –446 vs. viscosity, 389 –390, 404, 406

793

stress –strain crosslinked elastomers, 587–594, 605 KratonTM , PdCl2, 470–472 testing, 461–464 stress tensor, 357–358 volume-averaged, 580 stretched exponential, stress relaxation, 473–475 stretching frequency, OH, PVPh, 749–752 Sturm–Liouville problem, unsteady state diffusion, 62–64 sub-Tg motion, BPAPC, dielectric relaxation, 689– 691 summations, finite, 566– 567 factorial polynomials, 564, 662 super-atom approximation, electronegativity, 750 supercooling, crystallization kinetics, 302 supercritical carbon dioxide, 189 superposition principle, Boltzmann integral strain, g(t), 425–428 stress, s(t), 405– 407 surface free energy contact angle, 34–35 spherulite, 300 surface tension effects on Tg, 34 –35 SurlynTM , ionomers carbon-13 NMR, 740–741 magnetic spin diffusion, 709– 711 susceptibility complex, Fourier transform, 674–676 NMR, Kramers –Kronig theorem, 681–682 suspension viscosity, Einstein’s equation, 580–581, 604 symmetry above Tg, 203 group theory, ruthenium complexes, 760–762 molecular point groups, 198 stress & strain ellipsoids, 442 unit cell, 252 syndiotactic, definition, 250 tacticity, 250 Tanabe –Sugano diagram, crossover, 206, 234 tangents, Gibbs free energy vs. composition, 93

794

Index

Taylor series, stress relaxation modulus, 444 –446 temperature dependence alkene hydrogenation, 532 –533 chain conformations, 623 –624 correlation time, Rouse, 680 critical free energy, 302 critical radius, 302 crystal growth rate, 303 crystallization rate constant, 302 –304 dielectric constant, example, 153 diffusion coefficients, 56–58 dipole relaxation time, 698 discharge current, 701 –702 elastic modulus, rubbers, 592 –593, 606, 646 Flory x-parameter, 133 –135 fractional free volume, 22– 23, 55 homogeneous nucleation, 302 –304 retractive force, 617 –623, 646 specific heat, near Tmelt, 338 stress relaxation, 385 –387 zero-shear viscosity, 458– 459 temperature-field history, electrets, 696 –697 temperature-pressure relation at Tg, entropy continuity, 13–14 at Tg, volume continuity, 12 at Tmelt, 11 tensors, second-rank, symmetric 357 –359 terminal relaxation time, 400 –404 Rouse model, 677– 680 triangular distribution, 450 vs. MW, 403 ternary complexes, PdCl2, PBD, 539 –543 tetrahedral complexes, 206 –208 below Tg, 236 –238 tetrahedral lattice, Gibbs–DiMarzio theory, 117 –123 Tg (glass transition temperature) effect on diffusional activation energies, 58– 60 entropy continuity, electric field, 151 –153 equilibrium vs. kinetics, 8, 178 –179 graphical analysis via DSC, 28 molecular weight dependence, 45–47 nanoscale effects, 34

poly(lactic acid), 189 poly(vinylphenol) blends, phase diagrams, 749 pressure dependence, via order parameters, 176 surface tension effects, 34 –35 volume continuity, electric field, 153–155 Tg depression by plasticizers, 178 Couchman–Karasz model, 185–186 Fujita–Kishimoto model, 185 Gibbs –DiMarzio theory, 117–121 Kelley–Bueche model, 185 lattice model, Flory chi, 183–184, 187–188 polycarbonate, 178, 187– 188 polystyrene, 188 TS Chow model, 185 via metal complexes, 208–209 with order parameter, assumptions, 178–181 Tg enhancement, 192–194 Gibbs –DiMarzio theory, 122 ligand exchange, 202 ligand field model, 209–216 Margules model, 214–216 metal complexes, 198 P4VP, Co(II), Ni(II), Ru(II), 210, 225, 235 P4VP/zinc-acetate, 742 poly(histidine), Co(II), Ni(II), Ru(II), 226, 235 via metal complexes, 208, 210, 240 D(LFSE), correlation, 234–235 Tg vs. composition, 39 –45, 185–186 entropy continuity, 15–18 Gibbs –DiMarzio theory, 120–122 Gordon –Taylor equation, 40 –45 Kwei equation, 242– 243 linear additivity, 18 via fractional free volume additivity, 25– 26 volume continuity, 18–20, 39 with order parameter, 177–183 Tg vs. molecular weight linear least squares analysis, 46– 47 via fractional free volume, 27–29

Index Tg vs. pH during preparation, PAA, 584 –585 Tg vs. pressure entropy continuity, 13–14 pressure, volume continuity, 12 Tg, discontinuities at examples, 177 Prigogine-Defay ratio, electric field, 155 –156 Tg, effect of film thickness, 31 –34 particle size, 31 –34 TGA measurements, 345 thermal expansion coefficients, 332 –334 discontinuity at Tg, 6, 12, 29, 55, 59–60, 391 liquid, 8 nonlinear least squares analysis, 342 –349 PVAm, CoCl2(H2O)6, 346– 349 pychnometry, 341 –349 thermal expansion, retractive force, 617 –623 requirement for phase equilibrium, 10 treatment, optical microscopy, 283 –284 thermodynamic lattices, 71 –72 observables, expectation values, 653 –655, 661 –662 order parameters, 171– 172 properties, nonideal mixtures, 128 –131 work, due to electric field, 142 –143 thermodynamics crystallization, hydrogen bonding, 310 –317 entropy of mixing, 108 external force, rubbers, 591– 594, 605 –606, 611 first law, 331 –332 first law in electric field, 144 free energy of mixing, 106 –107, 115 homogeneous nucleation, 299 –302 of irreversible processes, 104 statistical, rubber elasticity, 623 –631 thermodynamics, classical enthalpy of mixing, 130 –131 entropy of mixing, 130 Gibbs free energy of mixing, 128 –129 rubber elasticity, 610 –623

795

thermoelastic inversion, rubber-like solids, 616–617, 646–647 thermograms discharge current, electret, 700–701 DSC, liquid crystals, 330 thermogravimetric analysis experiments, 345 thermophysical properties, discontinuities, 4 thermoreversible gels, polymer-solvent complexes, 745 theta solvent, 548 Flory– Huggins interaction parameter, 87 intrinsic viscosity, 583 theta temperature, 75–76,79 third-order phase transition, 38–39 three-coordinate complexes above Tg, 236–238 threshold molar density, c , 586–587 tilted capillary tubes, viscometry, 595–600 time-dependent perturbation theory, NMR relaxation, 661–673 rate constants, 474 time-strain separability, stress relaxation, 479–480 time-temperature equivalence, 460–461 equivalence, relaxation modulus, 372–373 shifting, 460 superposition, 385–397 torsion pendulum free vibration experiments, 419– 425, 467–468 intrinsic damping, 467–468 tracer input, delta-function, diffusion eq., 556 trans rotational isomeric states, 251–252, 565 transient profiles, radial diffusion, core-shell, 724–726 transition map, dielectric relaxation, BPAPC, 690 transition-metal catalysts, hydrogenation, 527–533 compatibilization, 194–195, 534–543 complexes, Tg enhancement, 240

796

Index

transition-metal (Continued) coordination, methodology, 193 –209 pH, poly(acrylic acid), 585– 586 transmitted light intensity, optical microscopy, 258 –259, 262, 285 triangular distribution, stress relaxation, 449 –451 triblock copolymers (KratonTM SBS) magnetic spin diffusion, 711– 715 PdCl2, stress –strain, 470–472 phase separation, 100 stress relaxation, 476 –479 unsteady state diffusion, 715 –725 trieutectic mixtures, PEO/ 2-methylresorcinol, 731 –732 trigonal bipyramid complexes, above Tg, 218 –220, 228 –229 planar complexes, above Tg, 237 trimers and hexamers entropy of mixing, 133 multiplicity of states, 132 volume fractions, 132 triple-product rule, 620 truncation factor, impingement, 290 two relaxation times, inverse moments, 447 –449 two-dimensional proton NMR, CRAMPS, 737 –738 shear, strain ellipsoid, 436 –438 shear, stress ellipsoid, 441 –443 two-state problem, Liouville equation, 664 –673 unit cell symmetry, 252 volume, WAXD, 340 universality, WLF eq. & parameters, 388 –389, 393 unpolarized light, birefringent spherulite, 284 –285 unstable states, 95 unsteady state diffusion membranes, 61–66 phase separation, 104 –105 Rouse model, 678– 680 spherical domains, 715– 725 unsteady state mass balance, viscometry, 595 –600

upper critical solution temperature (UCST), 95–96 example, 133 Flory –Huggins, 98 –99 Vand equation, cell suspensions, viscosity, 581 van der Waals gas, 75 van Laar model activity coefficients, 161 volume fractions, 159 vs. Flory–Huggins model, 85 velocity Maxwell-Boltzmann distribution, 557 vector, 359 virial expansion, real gases, 75 viscoelastic loss tangent, torsion pendulum, 424, 468 torsion pendulum, 467–468 time constant, 363, 367 viscoelasticity, nonlinear, 469–470 viscometry, capillary, 595–600 viscosity cell suspensions, Vand equation, 581 complex, forced vibration experiments, 415 complex, oscillatory experiments, 465–466 dilute suspension, Einstein’s eq., 580–581, 604 dynamic, complex, 674 intrinsic, Einstein model, 582, 604 intrinsic, MW scaling law, 583, 604 relative, 581, 585 vs. MW, scaling law, 404 vs. terminal relaxation time, 404 viscosity, zero-shear activation energy, 392 Doolittle equation, 389–391, 395 fractional exponent, 475 relaxation time distribution, 404 solutions, 394–395 temperature dependence, 458–459 vs. relaxation modulus, 389– 390, 404, 406 viscous dissipation, Newtonian fluids, 580–581 viscous stress tensor, volume-averaged, 580 Vogel’s equation, shift factor, 394

Index Voigt element Laplace transforms, 379 –381 rate-of-strain, 379 –383 strain, convolution theorem, 381 with mass, torsion pendulum, 420 –425 Voigt model, 374 –377 Boltzmann integral vs. integrating factor, 454 –455 creep, creep recovery, 376 –377 dielectric relaxation, 695 –696 electrical analog, 375 force balance, 374 –375 general solution, 375 stress relaxation, 374, 456 Voigt –Kelvin model, 374 creep, 430 –431 stress relaxation, 456 –457 volume sphere, partially filled, 596 with order parameters, 173 volume-averaged viscous stress tensor, 580 volume continuity at Tg, 39 at Tg, electric field, 153 –155 composition dependence of Tg, 18 –20 Ehrenfest equation, 11–12, 36– 37 Ehrenfest equation, electric field, 154 –155 volume fraction of crystallinity Avrami equation, 290– 296 Avrami exponent, 297 –299 density, 340 volume fraction Flory– Huggins lattice, 83, 634 monomers and r-mers, 106, 112 of polymer vs. MW at UCST, 99 trimers and hexamers, 132 van Laar model, 159 volume of mixing, exothermic energetics, 128 –131 volume relaxation, below Tg, 8– 10 volume vs. temperature glycerol, thermal expansion coefficient, 342 nonlinear least squares analysis, 342 –349 poly(vinylamine), CoCl2(H2O)6, in benzaldehyde, 346– 349 polystyrene in glycerol, 342 volume, crystallographic unit cell, 340

797

volume, dependence on pressure, 30 temperature, 22 temperature and pressure, 12 volume, effect of electric field, 153,158 volume, linear additivity for mixtures, 18– 19 volume, partial molar in electric field, 146, 158, 160 zero-field, 161 Vrentas and Duda, free volume theory of diffusion, 55–60 water methanol, Gibbs free energy, E-field, 163 methyl acetate, Gibbs free energy, E-field, 162 wave equation, hyperbolic, force balance, 421 wave function Schro¨dinger equation, 652–653 3-d orbital, 219 WAXD data, isotactic polypropylene, 324–325 measurements, density, 340–341 weight fraction distribution anionic, 512, 515 free radical, 503 most probable, 489–490 weight loss, via TGA, 345 weight-average degree of polymerization, 496–498, 504, 512, 517–520 diffusion coefficient, 68 –69 molecular weight, 545 Wiener –Khinchin-Einstein theorem, spectral density, 674 WLF constants, free volume, 391 WLF equation linear least squares analysis, 387–388 shift factor, 387– 388 universality, 388–389, 393 WLF shift factor activation energy, 392 compositional dependence, 394–397 reference temp., 393 theoretical justification, 389–391 viscosity, linear least squares, 458–460

798

Index

work term, due to electric field, 142 –143 work, mechanical, 417 x-ray data, isotactic polypropylene, 324 –325 x-ray measurements, density, 340 –341 zero-field chemical potential, 148 –149 partial molar volume, 161 zero-shear viscosity activation energy, 392 concentrated solutions, 394 –395 fractional exponent, 475

relaxation time distribution, 404 temperature dependence, 458–459 vs. MW, scaling law, 404 vs. relaxation modulus, 389– 390, 404, 406 vs. terminal relaxation time, 404 WLF shift factor, 389–391 Ziegler –Natta catalysts, 250 zinc acetate, P4VP, 13C NMR vs. DSC, 741–743 zinc laurate, P4VP, 13C NMR vs. DSC, 742–743 z-transform, moments-generating function, 496–498, 502–504, 511–512

Postface

THE BIRTH OF A FAMILY My father Alphonse, who served with the 5th Armored Division in Europe during World War II, told me that the history of the Belfiore family began in approximately 1850 when my great grandfather Antonio, an illegitimate child of Prince Ranieri, was placed on the steps of a church as an infant in the commune di Calabritto, Province di Avellino (Italy). Calabritto is 470 meters above sea level, slightly south of the intersection of SS91 and SS7 (SS is an abbreviation for strada statale), east of Naples and Salerno, southeast of Benevento, and 40 km east from the city of Avellino. The courts gave this illegitimate child of Prince Ranieri the name Antonio Belfiore (the family name translates to beautiful flowers). Antonio married Filomena Pecorara (born 1857) in the 1870s, and they had five children. Filomena and her four sons (Lorenzo, Vito, Alphonse, and Donato) immigrated to northeastern New Jersey (the Calabrittani), but Antonio and his daughter Giovannina remained in Italy, near Calabritto. My grandfather, Lorenzo (born 1880; died 1956) was the eldest of Antonio’s and Filomena’s four sons. Lorenzo married Filomena Mattia from Calabritto, shortly after the turn of the century, and they raised two sons and three daughters in the US. Lorenzo’s older son, Antonio (my uncle: born 1907; died 1970), married but did not have children. Lorenzo’s younger son, Alphonse (my father: born May 8th, 1909; died July 9th, 1985) married Olivia DeVito (my mother: born June 18th, 1911; died November 24th, 1982) on May 19th, 1946, at 4:30pm in Saint Rose of Lima’s church on Orange Street, Newark, NJ (see photos below of beloved Mom and Dad), and they had one daughter (my sister Lorraine Francis: born February 5th, 1947) and me. Hence, the royalty of Prince Ranieri’s genetic code was imparted to the author of this book. Unfortunately, my great grandfather’s hometown of Calabritto, Italy, was devastated by a 6.8-magnitude earthquake (terremoto) at 7:35pm on Sunday, November 23rd, 1980 (i.e., one day after my 26th birthday). The epicenter of the quake was in San Angelo dei Lombardi, and its duration was approximately one minute, leaving 2375 dead and approximately 280,000 homeless.

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The author’s father Alphonse was awarded the purple heart on March 25th, 1954 (while the author adopted the fetal position), for wounds received in Germany on November 29th, 1944. Here is a direct quote from the original letter written by the author’s father and mailed to his cousin in a Red Cross envelope, dated September 24th, 1944 from Luxembourg; “I don’t know how long this rest period will last, or where we will go from here (i.e., Luxembourg). However, I hope that where ever we go will help bring an end to this war.” The author (sitting in

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the third row from the bottom, 6th from the left) graduated from First Avenue School, 8th grade, on June 19th, 1968, in Newark, New Jersey. The poet, Michael Berardi, whose haiku verses appear on the first page of each chapter, is standing one row below the top, second from the right.