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Dieter Britz
Digital Simulation
in Electrochemistry
Third Completely Revised and Extended Edition
With Supplementary Electronic Material
123
Author
DieterBritz
KemiskInstitut
˚ArhusUniversitet
8000˚ArhusC
Denmark
Email:[email protected]
DieterBritz,DigitalSimulationinElectrochemistry,
Lect.NotesPhys.666(Springer,BerlinHeidelberg2005),DOI10.1007/b97996
LibraryofCongressControlNumber:2005920592
ISSN0075-8450
ISBN3-540-23979-03rded.SpringerBerlinHeidelbergNewYork
ISBN3-540-18979-32nded.Springer-VerlagBerlinHeidelbergNewYork
ISBN3-540-10564-61sted.publishedasVol.23inLectureNotesinChemistry
Springer-VerlagBerlinHeidelbergNewYork
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,reproductiononmicrofilmorinanyotherway,and
storageindatabanks.Duplicationofthispublicationorpartsthereofispermittedonly
undertheprovisionsoftheGermanCopyrightLawofSeptember9,1965,initscurrent
version, and permission for use must always be obtained from Springer. Violations are
liabletoprosecutionundertheGermanCopyrightLaw.
SpringerisapartofSpringerScience+BusinessMedia
springeronline.com
©Springer-VerlagBerlinHeidelberg2005
PrintedinGermany
Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt
fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
Typesetting:bytheauthorsandTechBooksusingaSpringerLATEXmacropackage
Coverdesign:design&production,Heidelberg
Printedonacid-freepaper
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
inFortran90/95,whichallowsamuchclearerprogrammingstylethanearlier
Fortran versions.
Intheyearssincethe2ndEdition,muchhashappenedinelectrochemical
digital simulation. Problems that ten years ago seemed insurmountable have
beensolved,suchasthethinreactionlayerformedbyveryfasthomogeneous
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
verifiedasfaraspossible.Whilesomeerrorsmightremain,theyarehopefully
very few.
I have a debt of gratitude to a number of people who have checked the
manuscriptordiscussedproblemswithme.MywifeSandrapolishedmyEng-
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. Lesl(cid:3)aw
Bieniasz, Ole Østerby, Jo¨rg Strutwolf and Thomas Britz. I thank the various
editorsatSpringerfortheirsupportandpatience.IfIhaveleftanybodyout,
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 and Global Errors................................. 52
4.3 What Distinguishes the Methods ......................... 52
4.4 Euler Method.......................................... 52
4.5 Runge-Kutta, RK ...................................... 54
4.6 Backwards Implicit, BI.................................. 56
4.7 Trapezium or Midpoint Method .......................... 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 odes ........................................ 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 Two Species ........................................... 90
6.3.1 Two-Point Derivative Cases........................ 93
6.4 Two Species with Coupled Reactions. U-V ................ 94
6.5 Brute Force............................................ 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
