Modelling Electroanalytical Experiments by the Integral Equation Method (eBook)

Preface 8
Acknowledgements 10
Acronyms 12
Contents 14
1 Introduction 19
References 24
2 Basic Assumptions and Equations of ElectroanalyticalModels 26
2.1 General Assumptions 26
2.2 Equations of Transport in Electrolytes 28
2.3 Equations of Transport in Non-electrolytes 31
2.4 Spatial Domains, Their Dimensionality,and Coordinate Systems 31
2.5 Diffusion Fields 34
2.6 Convection-Diffusion Fields 35
2.6.1 Dropping Mercury Electrode 35
2.6.2 Rotating Disk Electrode 37
2.6.3 Channel and Tubular Electrodes 39
2.7 Equations for Reaction Kinetics 41
2.8 The Effect of Homogeneous Reactions on Transport PDEs 48
2.9 The Effect of Heterogeneous Reactions on Boundary Conditions and Governing Equations for Localised Species 49
2.10 The Effect of Reactions on Initial Conditions 52
2.11 Electroanalytical Methods 53
2.12 Anomalous Diffusion 55
2.13 Uncompensated Ohmic Drop and Double Layer Charging 57
References 59
3 Mathematical Preliminaries 65
3.1 Integral Equations: Basic Concepts and Definitions 65
3.2 The Laplace Transformation 70
3.3 The Fourier Transformation 72
3.4 The Hankel Transformation 73
References 74
4 Models Independent of Spatial Coordinates 77
4.1 Derivation of the IEs 78
4.2 Literature Examples 90
4.3 The Relative Merits: ODEs vs. IEs 92
References 93
5 Models Involving One-Dimensional Diffusion 95
5.1 Derivation of the IEs 95
5.2 Diffusion in Semi-Infinite Spatial Domains 105
5.2.1 Concentration–Production Rate Relationships 106
5.2.2 Literature Examples 111
5.2.2.1 Planar Diffusion 111
5.2.2.2 Spherical Diffusion 114
5.2.2.3 Cylindrical Diffusion 115
5.3 Diffusion in Finite Spatial Domains 115
5.3.1 Concentration–Production Rate Relationships 117
5.3.2 Literature Examples 128
5.3.2.1 Planar Diffusion 128
5.3.2.2 Spherical Diffusion 130
5.3.2.3 Cylindrical Diffusion 130
5.4 Anomalous Diffusion 131
5.4.1 Concentration–Production Rate Relationship 131
5.4.2 Literature Examples 133
5.5 Benefits from Using the IE Method 133
References 134
6 Models Involving One-Dimensional Convection-Diffusion 142
6.1 Derivation of the IEs 142
6.2 Concentration–Production Rate Relationships 143
6.2.1 Constant Convection Velocity 143
6.2.2 Dropping Mercury Electrode 145
6.2.3 Rotating Disk Electrode 147
6.2.4 Channel and Tubular Electrodes 150
6.3 Literature Examples 153
6.3.1 Expanding Plane 154
6.3.2 Rotating Disk Electrode 154
6.3.3 Tubular Electrodes 155
References 155
7 Models Involving Two- and Three-Dimensional Diffusion 157
7.1 Derivation of the IEs 157
7.2 The Mirkin and Bard Approach 162
7.3 The Boundary Integral Method 165
7.4 Literature Examples 166
References 167
8 Models Involving Transport Coupled with HomogeneousReactions 170
8.1 Derivation of the IEs 170
8.2 A Single Reaction–Transport PDE, First-Order Homogeneous Reactions 174
8.3 Literature Examples, Transport Kernels Multiplied by exp[-kj(t-?)] 181
8.3.1 Planar Diffusion, Semi-Infinite Spatial Domain 181
8.3.2 Spherical Diffusion, Semi-Infinite Spatial Domain 185
8.3.3 Cylindrical Diffusion, Semi-Infinite Spatial Domain 186
8.3.4 Planar Diffusion, Finite Spatial Domain 186
8.3.5 Expanding Plane 187
8.3.6 Tubular Electrodes 189
8.3.7 Two- and Three-Dimensional Diffusion 190
8.4 Coupled Reaction–Planar Diffusion PDEs, First-Order Homogeneous Reactions 190
8.5 Transport with Nonlinear Homogeneous Reaction Kinetics 196
8.5.1 Equilibrium State Assumption 197
8.5.2 Steady State Assumption 198
8.5.3 The Gerischer Approximation 201
8.5.4 Conversion to IDEs 203
References 205
9 Models Involving Distributed and Localised Species 211
9.1 Derivation of the IEs or IDEs 212
9.2 Literature Examples 227
9.2.1 One-Reaction Schemes 227
9.2.2 Two-Reaction Schemes 229
9.2.3 Three-Reaction Schemes 233
9.2.4 More Complicated Reaction Schemes 236
References 238
10 Models Involving Additional Complications 245
10.1 Uncompensated Ohmic Drop and Double Layer Charging 245
10.1.1 Ohmic Drop Only 246
10.1.2 Double Layer Charging Only 247
10.1.3 General Case 249
10.1.4 Literature Examples 252
10.2 Electric Migration Effects 255
References 256
11 Analytical Solution Methods 260
11.1 Analytical Methods for One-Dimensional IEs or IDEs 260
11.1.1 Integral Solutions 261
11.1.2 Power Series Solutions 267
11.1.3 Exponential Series Solutions 268
11.1.4 The Method of Successive Approximations 272
11.1.5 Application of Steady State Approximations 272
11.2 Analytical Methods for Two-Dimensional IEs 275
References 275
12 Numerical Solution Methods 280
12.1 Numerical Methods for One-Dimensional Volterra IEs 280
12.1.1 The Step Function and the Huber Methods 284
12.1.1.1 Discretisation 285
12.1.1.2 Solution at Internal Grid Nodes 287
12.1.1.3 Initial Solutions 288
12.1.1.4 Calculation of the Moment Integrals of the Kernels 291
12.1.2 The Adaptive Huber Method 295
12.1.2.1 Error Control 295
12.1.2.2 Performance 301
12.1.3 Methods Based on Discrete Differintegration 304
12.1.4 Approximate (Degenerate) Kernel Method 307
12.1.5 The Analog Method 309
12.2 Numerical Methods for Multidimensional Volterra IEs 309
References 310
Appendices 316
A Solution–Flux Relationships for One-Dimensional Diffusion Equations 317
A.1 Planar Diffusion 318
A.1.1 Semi-infinite Spatial Domain 320
A.1.2 Finite Spatial Domain, Permeable Second Boundary 321
A.1.3 Finite Spatial Domain, ImpermeableSecond Boundary 323
A.2 Spherical Diffusion 325
A.2.1 Semi-infinite Spatial Domain 327
A.2.2 Finite Spatial Domain, Permeable Second Boundary 329
A.2.3 Finite Spatial Domain, ImpermeableSecond Boundary 332
A.3 Cylindrical Diffusion 334
A.3.1 Semi-infinite Spatial Domain 337
A.3.2 Finite Spatial Domain, Permeable Second Boundary 338
A.3.3 Finite Spatial Domain, ImpermeableSecond Boundary 341
References 343
B Solution–Flux Relationships for One-Dimensional Convection-Diffusion Equations 344
B.1 Constant Convection Velocity 344
B.2 Expanding Plane 348
B.3 Rotating Disk Electrode 349
B.4 Channel and Tubular Electrodes 355
References 358
C Solution–Flux Relationships for Multidimensional Diffusion Equations (Mirkin and Bard Approach) 359
C.1 Diffusion to Infinite Parallel Band(s) on a Plane 360
C.2 Diffusion to Disk/Ring(s) on a Plane 363
References 366
D Solution–Flux Relationships for Multidimensional Diffusion Equations (the BIM) 367
References 371
E Solution–Flux Relationships for One-Dimensional Reaction-Diffusion Equations 372
E.1 A Single Transient Linear Reaction-Diffusion PDE 372
E.2 A Single Steady State Linear Reaction-Diffusion ODE 376
E.3 A System of Coupled Transient Linear Reaction-Diffusion PDEs 379
E.4 A Single Steady State Reaction-Diffusion ODE for Planar Diffusion and an mth-Order Reaction 385
References 388
Glossary 389
List of Symbols 391
About the Author 398
About the Editor 399
Index 400