Classical Relaxation Phenomenology (eBook)

Ian Hodge received his PhD in physical chemistry from Purdue University and studied in the departments of chemistry of the University of Aberdeen in the UK and McGill University in Montreal. He was a member of the research staff at the B F Goodrich Research Laboratories and later joined the Research Laboratory of Eastman Kodak. He then taught introductory physics at the Rochester Institute of Technology before retiring from there. He has almost 50 publications to his name, of which 12 have more than 100 citations, and an h-index of 28.

Preface 6
Acknowledgments 8
Contents 9
About the Author 14
Part I: Mathematics 15
Chapter 1: Mathematical Functions and Techniques 16
1.1 Gamma and Related Functions (https://dlmf.nist.gov/5) 16
1.2 Error Function (https://dlmf.nist.gov/7) 17
1.3 Exponential Integrals (https://dlmf.nist.gov/6) 18
1.4 Hypergeometric Function (https://dlmf.nist.gov/15) 18
1.5 Confluent Hypergeometric Function (https://dlmf.nist.gov/13) 19
1.6 Williams-Watt Function 20
1.7 Bessel Functions (https://dlmf.nist.gov/10) 20
1.8 Orthogonal Polynomials (https://dlmf.nist.gov/18) 21
1.8.1 Legendre (https://dlmf.nist.gov/14.4) 22
1.8.2 Laguerre (https://dlmf.nist.gov/18.4) 23
1.8.3 Hermite (https://dlmf.nist.gov/18.4) 23
1.9 Sinc Function 24
1.10 Airy Function (https://dlmf.nist.gov/9) 25
1.11 Struve Function (https://dlmf.nist.gov/11) 25
1.12 Matrices and Determinants (https://dlmf.nist.gov/1.3) 25
1.13 Jacobeans (https://dlmf.nist.gov/1.5#vi) 28
1.14 Vectors (https://dlmf.nist.gov/1.6) 30
References 36
Chapter 2: Complex Variables and Functions 37
2.1 Complex Numbers 37
2.2 Complex Functions 38
2.2.1 Cauchy Riemann Conditions 45
2.2.2 Complex Integration and Cauchy Formulae 46
2.2.3 Residue Theorem 46
2.2.4 Hilbert Transforms, Crossing Relations, and Kronig-Kramer Relations 48
2.2.5 Plemelj Formulae 51
2.2.6 Analytical Continuation 52
2.3 Transforms 53
2.3.1 Laplace 53
2.3.2 Fourier 56
2.3.3 Z 58
2.3.4 Mellin 59
References 59
Chapter 3: Other Functions and Relations 60
3.1 Heaviside and Dirac Delta Functions 60
3.2 Green Functions 61
3.3 Schwartz Inequality, Parseval Relation, and Bandwidth Duration Principle 62
3.4 Decay Functions and Distributions 66
3.5 Underdamping and Overdamping 70
3.6 Response Functions for Time Derivative Excitations 73
3.7 Computing g(ln?) from Frequency Domain Relaxation Functions 74
References 80
Chapter 4: Elementary Statistics 81
4.1 Probability Distribution Functions 81
4.1.1 Gaussian 81
4.1.2 Binomial 83
4.1.3 Poisson 83
4.1.4 Exponential 84
4.1.5 Weibull 84
4.1.6 Chi-Squared 84
4.1.7 F 85
4.1.8 Student t 86
4.2 Student t-Test 86
4.3 Regression Fits 87
References 90
Chapter 5: Relaxation Functions 91
5.1 Single Relaxation Time 91
5.2 Logarithmic Gaussian 93
5.3 Fuoss-Kirkwood 94
5.4 Cole-Cole 95
5.5 Davidson-Cole 98
5.6 Glarum Model 100
5.7 Havriliak-Negami 104
5.8 Williams-Watt 106
5.9 Boltzmann Superposition 108
5.10 Relaxation and Retardation Processes 109
5.11 Relaxation in the Temperature Domain 114
5.12 Thermorheological Complexity 116
References 117
Part II: Electrical Relaxation 118
Chapter 6: Introduction to Electrical Relaxation 119
6.1 Introduction 119
6.1.1 Nomenclature 119
6.1.2 Relaxation of Polarization 120
6.2 Electromagnetism 120
6.2.1 Units 120
6.2.2 Electromagnetic Quantities 122
6.2.3 Electrostatics 123
Point Charge (Coulomb´s Law) 124
Long Thin Rod with Uniform Linear Charge Density 124
Flat Insulating Plate 124
Flat Conducting Plate 125
Two Parallel Insulating Flat Plates 125
Two Parallel Conducting Flat Plates 125
Concentric Conducting Cylinders 126
Concentric Conducting Spheres 126
Isolated Sphere 127
6.2.4 Electrodynamics 127
6.2.5 Maxwell´s Equations 128
6.2.6 Electromagnetic Waves 131
6.2.7 Local Electric Fields 134
6.2.8 Circuits 135
Simple Circuits 135
Resistances in Series and in Parallel 135
Capacitances in Series and in Parallel 135
Inductances in Series and in Parallel 136
Combined Series and Parallel Elements 137
AC Circuits 137
Resistances 138
Capacitances 138
Inductances 139
Parallel Resistance and Capacitance 139
Series Resistance and Capacitance 141
Experimental Factors 142
Cable Effects 142
Electrode Polarization 143
References 144
Chapter 7: Dielectric Relaxation 146
7.1 Frequency Domain 146
7.1.1 Dipole Rotation 146
7.1.2 Hopping Ions 151
7.2 Resonance Absorption 151
7.3 Time Domain 152
7.4 Temperature Domain 153
7.5 Equivalent Circuits 154
7.6 Interfacial Polarization 155
7.7 Maxwell-Wagner Polarization 156
References 158
Chapter 8: Conductivity Relaxation 159
8.1 General Aspects 159
8.2 Distribution of Conductivity Relaxation Times 162
8.3 Resonance Absorption Contribution 163
8.4 Constant Phase Element Analysis 163
References 163
Chapter 9: Examples 165
9.1 Dielectric Relaxation of Water 165
9.1.1 Equilibrium Liquid Water 165
9.1.2 Supercooled Water 166
9.1.3 Water of Hydration 169
9.2 Conductivity Relaxation in Sodium ?-Alumina 173
9.3 Complex Impedance Plane Analysis of Electrode Polarization in Sintered ?-Alumina 174
9.4 Electrode Polarization and Conductivity Relaxation in the Frequency Domain 176
9.5 Complex Impedance Plane Analysis of Atmosphere Dependent Electrode Effects in KHF2 177
9.6 Intergranular Effects in Polycrystalline Electrolytes 179
9.7 Intergranular Cracking 179
9.7.1 Lower Frequency (Intergranular) Relaxation in Cracked Sample 180
9.7.2 Higher frequency (Intragranular) Relaxation in Cracked Sample 180
9.8 Intergranular Gas Adsorption 181
9.9 Estimation of ?0 182
9.10 Analyses in the Complex Resistivity Plane 182
9.11 Modulus and Resistivity Spectra 183
9.12 Complex Admittance Applied to Polycrystalline Electrolytes and Electrode Phenomena 183
References 184
Part III: Structural Relaxation 185
Chapter 10: Thermodynamics 186
10.1 Elementary Thermodynamics 186
10.1.1 Nomenclature 186
10.1.2 Temperature Scales 187
10.1.3 Quantity of Material 187
10.1.4 Gas Laws and the Zeroth Law of Thermodynamics 187
10.1.5 Heat, Work, and the First Law of Thermodynamics 189
10.1.6 Entropy and the Second Law of Thermodynamics 189
10.1.7 Heat Capacity 190
10.1.8 Debye Heat Capacity and the Third Law of Thermodynamics 191
10.2 Thermodynamic Functions 193
10.2.1 Entropy S 193
10.2.2 Internal Energy U 193
10.2.3 Enthalpy H 193
10.2.4 Free Energies A and G 193
10.2.5 Chemical Potential ? 194
10.2.6 Internal Pressure 194
10.2.7 Derivative Properties 195
10.3 Maxwell Relations 195
10.4 Fluctuations 196
10.5 Ergodicity and the Deborah Number 197
10.6 Ehrenfest Classification of Phase Transitions 198
References 200
Chapter 11: Structural Relaxation 201
11.1 Supercooled Liquids and Fragility 201
11.1.1 Adam-Gibbs Model 203
11.2 Glassy State Relaxation 206
11.2.1 Secondary Relaxations 209
11.3 The Glass Transition 209
11.3.1 Introduction 209
11.3.2 Glass Transition Temperature 209
11.3.3 Thermodynamic Aspects of the Glass Transition 211
11.3.4 Kinetics of the Glass Transition 214
11.4 Heat Capacity 217
11.5 Sub-Tg Annealing Endotherms 220
11.6 TNM Parameters 222
11.7 SH Parameters 222
References 224
Correction to: Classical Relaxation Phenomenology 227
Appendix A: Laplace Transforms 228
Appendix B: Elementary Results 230
Solution of a Quadratic Equation 230
Solution of a Cubic Equation 230
Arithmetic and Geometric Series 231
Full and Partial Derivatives 232
Differentiation of Definite Integrals 233
Integration by Parts 233
Binomial Expansions 233
Partial Fractions 233
Coordinate Systems in Three Dimensions 234
Appendix C: Resolution of Two Debye Peaks of Equal Amplitude 236
Appendix D: Resolution of Two Debye Peaks of Unequal Amplitude 238
Appendix E: Cole-Cole Complex Plane Plot 239
Appendix F: Dirac Delta Distribution Function for a Single Relaxation Time 242
Appendix G: Derivation of M* for a Debye Relaxation with No Additional Separate Conductivity 245
Appendix H: Matlab/GNU Octave Code for Debye Relaxation with Additional Separate Conductivity ?0 247
Appendix I: Derivation of Debye Dielectric Expression from Equivalent Circuit 249
Appendix J: Simplified Derivation of the Van der Waal Potential 250
Author Index 252
Subject Index 254