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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: britz@chem.au.dk 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 Lectur e No tes 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. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the authors and TechBooks using a Springer L A T E X macro package Cover design: design & production,Heidelberg Printed on acid-free paper 2/3141/jl-543210 This book is dedicated to H. H. 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 ex- ample 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, J¨org 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 Some Mathematics: Transport Equations . . . . . . . . . . . . . . . . . . 6 2.2.1 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.2 Diffusion Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.3 Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.4 Migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.5 Total Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.6 Homogeneous Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.7 Heterogeneous Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Normalisation – Making the Variables Dimensionless . . . . . . . . 12 2.4 Some Model Systems and Their Normalisations . . . . . . . . . . . . 14 2.4.1 Potential Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4.2 Constant Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4.3 Linear Sweep Voltammetry (LSV) . . . . . . . . . . . . . . . . . . 25 2.5 Adsorption Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3 Approximations to Derivatives 33 3.1 Approximation Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 Two-Point First Derivative Approximations . . . . . . . . . . . . . . . . 34 3.3 Multi-Point First Derivative Approximations . . . . . . . . . . . . . . . 36 3.4 The Current Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.5 The Current Approximation Function G 39 3.6 High-Order Compact (Hermitian) Current Approximation . . . 39 3.7 Second Derivative Approximations . . . . . . . . . . . . . . . . . . . . . . . . 43 3.8 Derivatives on Unevenly Spaced Points . . . . . . . . . . . . . . . . . . . . 44 3.8.1 Error Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.8.2 A Special Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.8.3 Current Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.8.4 A Specific Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 48 X Contents 4 Ordinary Differential Equations 51 4.1 An Example ode 51 4.2 Local andGlobal Errors 52 4.3 WhatDistinguishestheMethods 52 4.4 EulerMethod 52 4.5 Runge-Kutta, RK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.6 BackwardsImplicit, BI 56 4.7 Trapeziumor MidpointMethod 56 4.8 Backward Differentiation Formula, BDF . . . . . . . . . . . . . . . . . . . 57 4.8.1 Starting BDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.9 Extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.10 Kimble & White, KW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.10.1 Using KW as a Start for BDF . . . . . . . . . . . . . . . . . . . . . 64 4.11 Systems of ode s 65 4.12 Rosenbrock Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.12.1 Application to a Simple Example ODE . . . . . . . . . . . . . . 70 4.12.2 Error Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5 The Explicit Method 73 5.1 The Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.2 Practicalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.3 Chronoamperometry and -Potentiometry . . . . . . . . . . . . . . . . . . 76 5.4 Homogeneous Chemical Reactions (hcr) 77 5.4.1 The Reaction Layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.5 Linear Sweep Voltammetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.5.1 Boundary Condition Handling . . . . . . . . . . . . . . . . . . . . . 81 6 Boundary Conditions 85 6.1 Classification of Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 85 6.2 Single Species: The u-v Device 86 6.2.1 Dirichlet Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.2.2 Derivative Boundary Conditions . . . . . . . . . . . . . . . . . . . . 86 6.3 TwoSpecies 90 6.3.1 Two-Point Derivative Cases . . . . . . . . . . . . . . . . . . . . . . . . 93 6.4 Two Species with Coupled Reactions. U-V 94 6.5 BruteForce 100 6.6 A General Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7 Unequal Intervals 103 7.1 Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 7.1.1 Discretising the Transformed Equation . . . . . . . . . . . . . . 105 7.1.2 The Choice of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 107 7.2 Direct Application of an Arbitrary Grid . . . . . . . . . . . . . . . . . . . 107 7.2.1 Choice of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.3 Concluding Remarks on Unequal Spatial Intervals . . . . . . . . . . 110 [...]... to chemical reactions taking place in the solution, since these do not give rise to a flux of substance Such terms come in later, in the equations relating concentration changes with time to the above components (see (2.15) and Sect 2.2.6) Dieter Britz: Digital Simulation in Electrochemistry, Lect Notes Phys 666, 5–32 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com 6 2 Basic Equations... methods in 1944, applied to several different equation types There is no shortage of mathematical texts on the subject: see, for example, Lapidus and Pinder [350] and Smith [514], two excellent books out of a large number Dieter Britz: Digital Simulation in Electrochemistry, Lect Notes Phys 666, 1–4 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com 2 1 Introduction It should not be imagined... problems What is digital simulation? The term simulation came into wide use with the advent of analog computers, which could produce electrical signals that followed mathematical functions to describe or model a given physical system When digital computers became common, people began to do these simulations digitally and called this digital simulation What sort of systems do we simulate in electrochemistry? ... discrete expressions are obtained simply by extending the diffusion equation by an extra, kinetic term (although practical problems arise, see Chaps 5, 9) The actual form of this depends upon the sort of chemistry taking place In the simplest case, met with in flash photolysis, we have a single substance generated by the flash, then decaying in solution by a first- or second-order reaction; this is represented... applied in simulation will be seen in later chapters The foregoing ignores activity coefficients If these are known, they can be inserted Most often they are taken as unity 2.3 Normalisation – Making the Variables Dimensionless In most simulations, it will be advantageous to transform the given equation variables into dimensionless ones This is done by expressing them each as a 2.3 Normalisation – Making... rightly regarded as the pioneer of digital simulation in electrochemistry, and is still prominent in developments in the field today This has also meant that the box method has become standard practice among electrochemists, while what will here be called the “point” method is more or less standard elsewhere Having experimented with both, the present author favours the point method for the ease with which... Consider Fig 2.1 We imagine a chosen coordinate direction x in a solution volume containing a dissolved substance at concentration c, which may be different at different points – i.e., there may be concentration gradients in the solution We consider a very small area δA on a plane normal to the x-axis Fick’s first equation now says that the net flow of solute (flux fx , in mol s−1 ) crossing the area is proportional... solution to be (practically) stagnant during our experiment, then we must include convective terms in the equations Figure 2.2 shows a plot of concentration against the x-coordinate at a given instant Let x1 be a fixed point along x, with concentration c1 at some time t, and let the solution be moving forward along x with velocity vx , so that after a small time interval δt, concentration c2 (previously... (1.3) and Figure 1.1 shows the resulting grid of points At each drawn point, there is a value of c The digital simulation method now consists of developing rows of c values along x, (usually) one t-step at a time Let us focus on the three filled-circle points ci−1 , ci and ci+1 at time tj One of the various techniques to be described will compute from these three known points a new concentration value ci... must be some sample points very close to the electrode This problem has been overcome only in recent years, first by using unequal intervals, then by the use of dynamic grids, both of which are discussed in Chap 7 12 2 Basic Equations 2.2.7 Heterogeneous Kinetics In real (as opposed to model) electrochemical cells, the net current flowing will often be partly determined by the kinetics of electron transfer . 007 5-8 450 ISBN 3-5 4 0-2 397 9-0 3rd ed. Springer Berlin Heidelberg New York ISBN 3-5 4 0-1 897 9-3 2nd ed. Springer-Verlag Berlin Heidelberg New York ISBN 3-5 4 0-1 056 4-6 1st ed. published as Vol. 23 in. C Denmark Email: britz@ chem.au.dk Dieter Britz, DigitalSimulationinElectrochemistry, Lect. Notes Phys. 666 (Springer, Berlin Heidelberg 2005) , DOI 10.1007/b97996 Library of Congress Control Number: 20059 20592 ISSN. Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does