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Dieter Britz Digital Simulation in Electrochemistry Third Completely Revised and Extended Edition With Supplementary Electronic Material 123 Author Dieter Britz Kemisk Institut Arhus˚ Universitet 8000 Arhus˚ C Denmark Email: [email protected] Dieter Britz, DigitalSimulationinElectrochemistry, Lect. Notes Phys. 666 (Springer, Berlin Heidelberg 2005), DOI 10.1007/b97996 Library of Congress Control Number: 2005920592 ISSN 0075-8450 ISBN 3-540-23979-0 3rd ed. Springer Berlin Heidelberg New York ISBN 3-540-18979-3 2nd ed. Springer-Verlag Berlin Heidelberg New York ISBN 3-540-10564-6 1st ed. published as Vol. 23 in Lecture Notes in Chemistry Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. 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Bauer, teacher and friend Preface This book is an extensive revision of the earlier 2nd Edition with the same title, of 1988. The book has been rewritten in, I hope, a much more didac- tic manner. Subjects such as discretisations or methods for solving ordinary differential equations are prepared carefully in early chapters, and assumed in later chapters, so that there is clearer focus on the methods for partial differential equations. There are many new examples, and all programs are in Fortran 90/95, which allows a much clearer programming style than earlier Fortran versions. In the years since the 2nd Edition, much has happened in electrochemical digital simulation. Problems that ten years ago seemed insurmountable have been solved, such as the thin reaction layer formed by very fast homogeneous reactions, or sets of coupled reactions. Two-dimensional simulations are now commonplace, and with the help of unequal intervals, conformal maps and sparse matrix methods, these too can be solved within a reasonable time. Techniques have been developed that make simulation much more efficient, so that accurate results can be achieved in a short computing time. Stable higher-order methods have been adapted to the electrochemical context. The book is accompanied (on the webpage www.springerlink.com/ openurl.asp?genre=issue&issn=1616-6361&volume=666) by a number of example procedures and programs, all in Fortran 90/95. These have all been verified as far as possible. While some errors might remain, they are hopefully very few. I have a debt of gratitude to a number of people who have checked the manuscript or discussed problems with me. My wife Sandra polished my Eng- lish style and helped with some of the mathematics, and Tom Koch Sven- nesen checked many of the mathematical equations. Others I have consulted for advice of various kinds are Professor Dr. Bertel Kastening, Drs. Leslaw Bieniasz, Ole Østerby, Jörg Strutwolf and Thomas Britz. I thank the various editors at Springer for their support and patience. If I have left anybody out, I apologize. As is customary to say (and true), any errors remaining in the book cannot be blamed on anybody but myself. Arhus,˚ Dieter Britz February 2005 Contents 1 Introduction .............................................. 1 2 Basic Equations .......................................... 5 2.1 General............................................... 5 2.2 SomeMathematics:TransportEquations.................. 6 2.2.1 Diffusion........................................ 6 2.2.2 DiffusionCurrent................................. 7 2.2.3 Convection...................................... 8 2.2.4 Migration....................................... 9 2.2.5 Total Transport Equation . 10 2.2.6 HomogeneousKinetics............................ 10 2.2.7 HeterogeneousKinetics ........................... 12 2.3 Normalisation – Making the Variables Dimensionless . 12 2.4 SomeModelSystemsandTheirNormalisations ............ 14 2.4.1 PotentialSteps................................... 14 2.4.2 ConstantCurrent ................................ 24 2.4.3 LinearSweepVoltammetry(LSV).................. 25 2.5 AdsorptionKinetics .................................... 28 3 Approximations to Derivatives ............................ 33 3.1 ApproximationOrder................................... 33 3.2 Two-PointFirstDerivativeApproximations................ 34 3.3 Multi-PointFirstDerivativeApproximations............... 36 3.4 TheCurrentApproximation............................. 38 3.5 The Current Approximation Function G ................... 39 3.6 High-Order Compact (Hermitian) Current Approximation . 39 3.7 SecondDerivativeApproximations........................ 43 3.8 DerivativesonUnevenlySpacedPoints.................... 44 3.8.1 ErrorOrders..................................... 47 3.8.2 ASpecialCase................................... 48 3.8.3 CurrentApproximation........................... 48 3.8.4 ASpecificApproximation......................... 48 X Contents 4 Ordinary Differential Equations........................... 51 4.1 An Example ode ......................................

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