Britz / Strutwolf Digital Simulation in Electrochemistry

4th Auflage 2016
ISBN: 978-3-319-30292-8
Verlag: Springer International Publishing
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 500 Seiten, eBook

Reihe: Monographs in Electrochemistry

This book explains how the partial differential equations (pdes) in electroanalytical chemistry can be solved numerically. It guides the reader through the topic in a very didactic way, by first introducing and discussing the basic equations along with some model systems as test cases systematically. Then it outlines basic numerical approximations for derivatives and techniques for the numerical solution of ordinary differential equations. Finally, more complicated methods for approaching the pdes are derived.The authors describe major implicit methods in detail and show how to handle homogeneous chemical reactions, even including coupled and nonlinear cases. On this basis, more advanced techniques are briefly sketched and some of the commercially available programs are discussed. In this way the reader is systematically guided and can learn the tools for approaching his own electrochemical simulation problems. This new fourth edition has been carefully revised, updated and extended compared to the previous edition (Lecture Notes in Physics Vol. 666). It contains new material describing migration effects, as well as arrays of ultramicroelectrodes. It is thus the most comprehensive and didactic introduction to the topic of electrochemical simulation.

Dieter Britz, Ph.D. (Sydney Univ. 1967), Dipl. Comp. Sci. (University of Newcastle, Australia, 1985), Dr. scient (Aarhus Univ., Denmark, 2007).
Dr. Britz has gathered longstanding experience in electrochemistry during research stays all over the world: he worked at the CSIRO, Sydney, on corrosion problems, on inorganic ion exchangers at the University of New York at Buffalo (1967-68), he performed instrumental work at the University of Kentucky, Lexington, USA (1968-70), investigated corrosion and electrosynthesis at the Nuclear Research Centre in Jülich, Germany (1970-75), and performed data analysis of turbulence signals at Newcastle University, Australia (1975-78). In 1978 he accepted the position of Assoc. Professor at Aarhus University in Denmark, from which he retired as Emeritus Assoc. Professor in 2001. In Aarhus, he has worked on a number of projects, focusing on corrosion, electroanalysis and digital simulation.
Jörg Strutwolf received the Diploma and Ph.D. degrees in the Theoretical Chemistry Group, University of Bielefeld, Germany. He has specialized in the investigation of interfacial transport processes by theoretical and experimental methods. His current research interests include the dynamics and reactivity of soft interfaces, the combination of microfluidics and electrochemistry, numerical modelling of transport and reaction phenomena in electrochemistry (mainly in co-operation with Dieter Britz), electrochemistry at the nanoscale, and nanostructuring of interfaces for sensor application. Currently, he is a Visiting Scientist at the University of Tübingen, Germany. He has worked in numerous electrochemistry groups, e.g. at University College London, U.K., University of Warwick, Coventry, U.K., Universitat Rovira i Virgili, Tarragona, Spain, and Tyndall National Institute, Cork, Ireland.

Zielgruppe

Research

Autoren/Hrsg.

Britz, Dieter

Strutwolf, Jörg

Inhaltsverzeichnis

1;Preface;8
1.1;References;9
2;Contents;10
3;1 Introduction;17
3.1;References;20
4;2 Basic Equations;21
4.1;2.1 General;21
4.2;2.2 Some Mathematics: Transport Equations;22
4.2.1;2.2.1 Diffusion;22
4.2.2;2.2.2 Diffusion Current;24
4.2.3;2.2.3 Convection;24
4.2.4;2.2.4 Migration;25
4.2.5;2.2.5 Total Transport Equation;26
4.2.6;2.2.6 Homogeneous Kinetics;26
4.2.7;2.2.7 Heterogeneous Kinetics;28
4.3;2.3 Normalisation: Making the Variables Dimensionless;29
4.4;2.4 Some Model Systems and Their Normalisations;31
4.4.1;2.4.1 Potential Steps;31
4.4.1.1;2.4.1.1 Cottrell System;32
4.4.1.2;2.4.1.2 Potential Step, Reversible System;35
4.4.1.3;2.4.1.3 Potential Step, Quasi- and Irreversible System;37
4.4.1.4;2.4.1.4 Potential Step, Homogeneous Chemical Reactions;38
4.4.2;2.4.2 Constant Current;42
4.4.3;2.4.3 Linear Sweep Voltammetry;43
4.5;2.5 Adsorption Kinetics;47
4.6;References;51
5;3 Approximations to Derivatives;54
5.1;3.1 Approximation Order;54
5.2;3.2 Two-Point First Derivative Approximations;55
5.3;3.3 Multi-Point First Derivative Approximations;57
5.4;3.4 The Current Approximation;60
5.5;3.5 The Current Approximation Function G;60
5.5.1;3.5.1 Unequal Intervals;61
5.6;3.6 High-Order Compact (Hermitian) Current Approximation;61
5.7;3.7 Second Derivative Approximations;65
5.8;3.8 Derivatives on Unevenly Spaced Points;66
5.8.1;3.8.1 Error Orders;69
5.8.2;3.8.2 A Special Case;70
5.8.3;3.8.3 Current Approximation;70
5.8.4;3.8.4 An Example;71
5.9;3.9 The Fornberg Algorithm;72
5.10;References;73
6;4 Ordinary Differential Equations;75
6.1;4.1 An Example ode;76
6.2;4.2 Local and Global Errors;76
6.3;4.3 What Distinguishes the Methods;76
6.4;4.4 Euler Method;77
6.5;4.5 Runge–Kutta (RK);78
6.6;4.6 Backwards Implicit (BI);80
6.7;4.7 Trapezium Method;81
6.8;4.8 Backward Differentiation Formula (BDF);82
6.8.1;4.8.1 Starting BDF;83
6.8.1.1;4.8.1.1 Time Shifts;84
6.8.1.2;4.8.1.2 Testing the Starting Protocols;85
6.9;4.9 Extrapolation;86
6.10;4.10 Kimble and White (KW);87
6.10.1;4.10.1 Using KW as a Start for BDF;90
6.11;4.11 Systems of odes;91
6.12;4.12 Rosenbrock Methods;94
6.12.1;4.12.1 Application to a Simple Example ODE;97
6.12.2;4.12.2 Error Estimates;97
6.13;4.13 Padé Approximants;98
6.14;References;99
7;5 The Explicit Method;102
7.1;5.1 The Discretisation;102
7.2;5.2 Practicalities;103
7.3;5.3 Chronoamperometry and -Potentiometry;105
7.4;5.4 Homogeneous Chemical Reactions (hcr);106
7.4.1;5.4.1 The Reaction Layer;108
7.5;5.5 Linear Sweep Voltammetry;109
7.5.1;5.5.1 Boundary Condition Handling;111
7.6;References;112
8;6 Boundary Conditions;114
8.1;6.1 Classification of Boundary Conditions;114
8.2;6.2 Single Species: The u-v Device;115
8.2.1;6.2.1 Dirichlet Condition;115
8.2.2;6.2.2 Derivative Boundary Conditions;115
8.3;6.3 Two Species;119
8.3.1;6.3.1 Two-Point Derivative Cases;123
8.4;6.4 Two Species with Coupled Reactions: U-V;124
8.5;6.5 Brute Force;130
8.6;6.6 A General Formalism;132
8.7;References;133
9;7 Unequal Intervals;135
9.1;7.1 Transformation;136
9.1.1;7.1.1 Discretising the Transformed Equation;138
9.1.2;7.1.2 Choice of Transformation Parameters;139
9.2;7.2 Direct Application of an Arbitrary Grid;140
9.2.1;7.2.1 Choice of Parameters;143
9.2.2;7.2.2 Current and C0 Approximations;144
9.3;7.3 Concluding Remarks on Unequal Spatial Intervals;144
9.4;7.4 Unequal Time Intervals;145
9.4.1;7.4.1 Implementation of Exponentially Increasing Time Intervals;146
9.5;7.5 Adaptive Interval Changes;147
9.5.1;7.5.1 Spatial Interval Adaptation;147
9.5.2;7.5.2 Time Interval Adaptation;151
9.6;References;152
10;8 The Commonly Used Implicit Methods;157
10.1;8.1 The Laasonen Method or BI;159
10.2;8.2 The Crank–Nicolson Method, CN;160
10.3;8.3 Solving the Implicit System;161
10.4;8.4 Using Four-Point Spatial Second Derivatives;163
10.5;8.5 Improvements on CN and Laasonen;166
10.5.1;8.5.1 Damping the CN Oscillations;168
10.5.1.1;8.5.1.1 First-Interval Subdivision;168
10.5.1.2;8.5.1.2 Initial BI Step(s);169
10.5.1.3;8.5.1.3 Averaging and Extrapolation;170
10.5.1.4;8.5.1.4 Singularity Correction;171
10.5.1.5;8.5.1.5 Recommendations;171
10.5.2;8.5.2 Making Laasonen More Accurate;171
10.5.2.1;8.5.2.1 BDF;172
10.5.2.2;8.5.2.2 Extrapolation;174
10.6;8.6 Homogeneous Chemical Reactions;175
10.6.1;8.6.1 Nonlinear Equations;175
10.6.1.1;8.6.1.1 Linearising Squared Concentration Terms;176
10.6.1.2;8.6.1.2 Linearising the Product of Concentrations of Two Species;177
10.6.1.3;8.6.1.3 An Example Case: Linearising;177
10.6.1.4;8.6.1.4 An Example Case: Nonlinear;179
10.6.2;8.6.2 Coupled Equations;182
10.7;References;184
11;9 Other Methods;189
11.1;9.1 The Box Method;189
11.2;9.2 Improvements on Standard Methods;193
11.2.1;9.2.1 The Kimble and White Method;193
11.2.2;9.2.2 Multi-Point Second Spatial Derivatives;195
11.2.3;9.2.3 DuFort–Frankel;197
11.2.4;9.2.4 Saul'yev;198
11.2.5;9.2.5 Hopscotch;201
11.2.6;9.2.6 Runge–Kutta;203
11.2.7;9.2.7 Hermitian Methods;204
11.2.7.1;9.2.7.1 Numerov/Douglas;204
11.2.7.2;9.2.7.2 Hermitian Current Approximation;207
11.2.7.3;9.2.7.3 Method of Wu and White;209
11.3;9.3 MOL and DAE;210
11.4;9.4 The Rosenbrock Method;212
11.4.1;9.4.1 An Example, the Birk–Perone System;215
11.5;9.5 FEM, BEM, FVM and FAM (Briefly);218
11.6;9.6 Orthogonal Collocation (OC);219
11.6.1;9.6.1 Current Calculation with OC;226
11.6.2;9.6.2 A Numerical Example;226
11.7;9.7 Eigenvalue–Eigenvector Method;228
11.8;9.8 Integral Equation Method;231
11.9;9.9 The Network Method;232
11.10;9.10 Treanor Method;233
11.11;9.11 Monte Carlo Method;233
11.12;References;234
12;10 Adsorption;247
12.1;10.1 Transport and Isotherm Limited Adsorption;248
12.2;10.2 Adsorption Rate Limited Adsorption;250
12.3;References;250
13;11 Effects Due to Uncompensated Resistance and Capacitance;253
13.1;11.1 Boundary Conditions;255
13.1.1;11.1.1 An Example;257
13.2;References;260
14;12 Two (and Three) Dimensions;262
14.1;12.1 Theories;263
14.1.1;12.1.1 The Ultramicrodisk Electrode;263
14.1.1.1;12.1.1.1 Ranges of Applicability;269
14.1.1.2;12.1.1.2 LSV;270
14.1.2;12.1.2 Other UMEs;271
14.1.3;12.1.3 Some Relations;273
14.2;12.2 Simulations;274
14.3;12.3 Simulating the UMDE;276
14.3.1;12.3.1 Methods of Solution;277
14.3.1.1;12.3.1.1 Hopscotch;277
14.3.1.2;12.3.1.2 ADI;277
14.3.1.3;12.3.1.3 Some Other Methods;278
14.3.2;12.3.2 Direct Discretisation;279
14.3.3;12.3.3 Discretisation in the Mapped Space;287
14.3.3.1;12.3.3.1 Some Transformations;288
14.3.3.2;12.3.3.2 Inversion of the Transformations;292
14.3.3.3;12.3.3.3 The Diffusion Equation in the Mapped Spaces;294
14.3.3.4;12.3.3.4 Current Integration in Conformal Coordinates;295
14.3.4;12.3.4 Band Electrodes;296
14.3.4.1;12.3.4.1 Choice of ?max;297
14.3.4.2;12.3.4.2 Discretisation;297
14.3.4.3;12.3.4.3 Unequal Intervals in the Mapped Space;299
14.3.5;12.3.5 A Remark on the Boundary Conditions;299
14.4;12.4 Three-Dimensional Simulations;300
14.4.1;12.4.1 Square and Rectangular UMEs;300
14.4.1.1;12.4.1.1 Discretisation;304
14.4.2;12.4.2 The Grid;307
14.5;12.5 Ultramicroelectrode Arrays;308
14.5.1;12.5.1 Regular Arrays of UMDEs;309
14.5.1.1;12.5.1.1 The Diffusion Domain Approach;311
14.5.1.2;12.5.1.2 An Example;313
14.5.2;12.5.2 Arrays of UMBEs;317
14.5.3;12.5.3 Elevated UMBEs;321
14.5.4;12.5.4 Dual Electrode Systems;325
14.6;References;330
15;13 Migrational Effects;349
15.1;13.1 Theory;350
15.2;13.2 Simulations;352
15.3;13.3 Time Development of a Liquid Junction;352
15.3.1;13.3.1 Normalisation;353
15.4;13.4 RPC Example;362
15.5;13.5 Copper Deposition on an RDE;368
15.5.1;13.5.1 Note on Normalisations;373
15.6;References;375
16;14 Convection;378
16.1;14.1 Some Fluid Dynamics;378
16.1.1;14.1.1 Layer Relations;382
16.2;14.2 Electrodes in Flow Systems;383
16.3;14.3 Simulations;384
16.4;14.4 A Simple Example: The Band Electrode in a Channel Flow;385
16.5;14.5 Normalisations;386
16.6;References;390
17;15 Performance;398
17.1;15.1 Convergence;398
17.2;15.2 Consistency;400
17.3;15.3 Stability;401
17.3.1;15.3.1 Heuristic Method;402
17.3.2;15.3.2 Von Neumann Stability Analysis;403
17.3.3;15.3.3 Matrix Stability Analysis;405
17.3.3.1;15.3.3.1 Using Eigenvalues;406
17.3.3.2;15.3.3.2 Using the Matrix Norm;411
17.3.4;15.3.4 Some Special Cases;412
17.4;15.4 The Stability Function;412
17.5;15.5 Accuracy Order;415
17.5.1;15.5.1 Order Determination;415
17.6;15.6 Sensitivity Analysis;418
17.7;15.7 Accuracy, Efficiency and Choice;418
17.7.1;15.7.1 Determining Accuracy;422
17.8;15.8 Two- (and Three-)Dimensional Problems;423
17.9;15.9 Summary of Methods;423
17.10;References;425
18;16 Programming;430
18.1;16.1 Language and Style;430
18.2;16.2 Debugging;431
18.3;16.3 Libraries;433
18.4;References;433
19;17 Simulation Packages;435
19.1;17.1 Kinetic Compilers;438
19.2;17.2 Parameter Estimation;439
19.3;References;441
20;Erratum to: Digital Simulation in Electrochemistry;447
21;A Tables and Formulae;449
21.1;A.1 First Derivative Approximations;449
21.2;A.2 Current Approximations;449
21.3;A.3 Second Derivative Approximations;449
21.4;A.4 Unequal Intervals;450
21.5;A.5 Jacobi Roots for Orthogonal Collocation;454
21.6;A.6 Rosenbrock Constants;455
21.7;References;456
22;B Transforming the Diffusion Equation into Curvilinear Coordinates;457
22.1;B.1 Introduction;457
22.2;B.2 A Simple Example: Cartesian to Cylindrical;458
22.3;B.3 Transformations for the Band Electrode;460
22.3.1;B.3.1 Cartesian to MWA;460
22.3.2;B.3.2 Extension to VB;460
22.4;B.4 Disk Electrode Transformations;461
22.5;B.5 Transforming the Current;462
22.6;References;463
23;C Some Mathematical Proofs;464
23.1;C.1 Consistency of the Sequential Method;464
23.2;C.2 The Feldberg Start for BDF;465
23.3;C.3 Similarity of the Exponential Expansion and Transformation Functions;471
23.4;References;473
24;D Finding ?max;474
24.1;References;476
25;E Procedure and Program Examples;477
25.1;E.1 Example Modules;477
25.1.1;E.1.1 Module STUFF;477
25.1.2;E.1.2 Module ROSTUFF;478
25.2;E.2 Procedures;479
25.2.1;E.2.1 File Names Routine;479
25.2.2;E.2.2 The Error Functions;480
25.2.3;E.2.3 Current Approximations;480
25.2.4;E.2.4 Matrix Inversion;480
25.2.5;E.2.5 MINMAX;481
25.2.6;E.2.6 EE_FAC;481
25.2.7;E.2.7 DAMPED_EXPANSION;481
25.2.8;E.2.8 SV_FAC;482
25.2.9;E.2.9 Gradient Routine FORNBERG and FORN;482
25.2.10;E.2.10 Current Integration on an Unequally Gridded Surface;482
25.2.11;E.2.11 Reference Fluxes and Errors;482
25.2.12;E.2.12 JCOBI;483
25.2.13;E.2.13 I1I2;483
25.3;E.3 Example Programs;483
25.3.1;E.3.1 Program COTT_EX;483
25.3.2;E.3.2 Program CHRONO_EX;484
25.3.3;E.3.3 Program CV_EX;484
25.3.4;E.3.4 Program COTT_CN;485
25.3.5;E.3.5 Program CHRONO_CN;486
25.3.6;E.3.6 Program CHRONO_CN_HERM;486
25.3.7;E.3.7 Program LSV_CN;487
25.3.8;E.3.8 Program COTT_EXTRAP;487
25.3.9;E.3.9 A Nonlinear System: Programs for the Birk/Perone Reaction;487
25.3.10;E.3.10 EC Reaction, Cyclic Voltammetry: CV_EC;488
25.3.11;E.3.11 CV of the EC' Reaction: Program CV_CAT;489
25.3.12;E.3.12 LSV Simulation with iR Drop and Capacitance: Program LSV4IRC;489
25.3.13;E.3.13 Program UMDE_DIRECT;490
25.3.14;E.3.14 Program UMDE_VB;490
25.3.15;E.3.15 Program UMDE_ARRAY;490
25.3.16;E.3.16 Programs LIQU_JUNC, RPC, CURDE;491
25.3.17;E.3.17 Program CHANNEL_BAND;491
25.4;References;491
26;Index;493

Introduction.- Basic Equations.- Approximations to Derivatives.- Ordinary Differential Equations.- The Explicit Method.- Boundary Conditions.- Arbitrary Intervals.- The Commonly Used Implicit Methods.- Other Methods.- Adsorption.- Uncompensated Resistance and Capacitance.- Two-Dimensional Systems.- Migration.- Convection.- Performance.- Programming.- Simulation Packages.- Appendices: Some Mathematical Proofs.- Useful Procedures.- Example Programs.

Über Autor(innen)

Dieter Britz, Ph.D. (Sydney Univ. 1967), Dipl. Comp. Sci. (University of Newcastle, Australia, 1985), Dr. scient (Aarhus Univ., Denmark, 2007). Dr. Britz has gathered longstanding experience in electrochemistry during research stays all over the world: he worked at the CSIRO, Sydney, on corrosion problems, on inorganic ion exchangers at the University of New York at Buffalo (1967-68), he performed instrumental work at the University of Kentucky, Lexington, USA (1968-70), investigated corrosion and electrosynthesis at the Nuclear Research Centre in Jülich, Germany (1970-75), and performed data analysis of turbulence signals at Newcastle University, Australia (1975-78). In 1978 he accepted the position of Assoc. Professor at Aarhus University in Denmark, from which he retired as Emeritus Assoc. Professor in 2001. In Aarhus, he has worked on a number of projects, focusing on corrosion, electroanalysis and digital simulation. Jörg Strutwolf received the Diploma and Ph.D. degrees in the Theoretical Chemistry Group, University of Bielefeld, Germany. He has specialized in the investigation of interfacial transport processes by theoretical and experimental methods. His current research interests include the dynamics and reactivity of soft interfaces, the combination of microfluidics and electrochemistry, numerical modelling of transport and reaction phenomena in electrochemistry (mainly in co-operation with Dieter Britz), electrochemistry at the nanoscale, and nanostructuring of interfaces for sensor application. Currently, he is a Visiting Scientist at the University of Tübingen, Germany. He has worked in numerous electrochemistry groups, e.g. at University College London, U.K., University of Warwick, Coventry, U.K., Universitat Rovira i Virgili, Tarragona, Spain, and Tyndall National Institute, Cork, Ireland.