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Classical Relaxation Phenomenology
Ian M. Hodge
Verlag Springer-Verlag, 2019
ISBN 9783030024598 , 257 Seiten
Format PDF, OL
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Geräte
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
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