Digital Simulation in Electrochemistry

131 Pages·1981·3.106 MB·English

by  Dieter Britz (auth.)

Lecture Notes in
Chemistry
Edited by G. Berthier M. J. S. Dewar H. Fischer
K. Fukui G. G. Hall H. Hartmann H. H. Jaffe J. Jortner
W. Kutzelnigg K. Ruedenberg E. Scrocco
23
Dieter Britz
Digital Simulation
in Electrochemistry
Springer-Verlag
Berlin Heidelberg GmbH 1981
Author
Dieter Britz
Department of Chemistry
Aarhus University
DK-8000 Aarhus C
ISBN 978-3-540-10564-0 ISBN 978-3-662-21819-8 (eBook)
DOI 10.1007/978-3-662-21819-8
This work is subject to copyright. All rights are reserved, whether the whole
or part of the material is concerned, specifically those of translation, re
printing, re-use of illustrations, broadcasting, reproduction by photocopying
machine or similar means, and storage in data banks. Under § 54 of the
German Copyright Law where copies are made for other than private use,
a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-verlag Berlin Haideiberg 1981
Originally published by Springer-Verlag Berlin Haideiberg New York in 1981
Softcoverreprint of the hardcover 1st edition 1981
2152/3140-543210
This book is dedicated
to my friend and teacher
Henry H. Bauer
PREFACE
This book is the result of frustration. When I first became interested in digi
tal simulation in 1967 (I didn't know the name then), there were no texts to tell
one the how of it. This has not changed greatly since then; it is significant that
just about all publications about the technique refer to a chapter by Feldberg in an
electrochemical series, written in 1969. When I ran a course on the method recently,
it became evident that this chapter is not enough for the raw beginner. Neither does
he/she get much help from the mathematical textbooks which, at best, leave the special
electrochemical aspects (if not a lot else) to one's imagination. This book, then,
is written for practical digital simulators who do not have a friend who will tell
them how to do it.
The beauty of the digital approach is that one can separate out various dynamic
processes taking place simultaneously. I have structured the book in this way. The
major computing usually lies in the diffusion of substance, while the major program
ming effort (and preparatory paper work) goes into the boundary conditions. These
are treated separately.
It is an unfortunate accident of history, in my view, that electrochemists have
tended to use what I term the explicit box method; I believe I make quite clear in
the book that much can be gained by joining the majority of non-electrochemical
users of finite differences, in using the point methods and especially the time
saving and more accurate implicit formulae. I am aware that this could, in a reader
ship expecting details of the (hitherto) orthodox procedure, cause some unease, but
I feel that I cannot support orthodoxy wholeheartedly in this respect.
My treatment of the Laplace equation is very skimpy and more or less meant only
as a whetter of appetites. If a reader needs to solve these systems, chapter 8 may
help to put some fairly naYve programs on their feet quickly; in the longer run it
will be necessary to go into the mathematical literature to arrive at more sophisti
cated and economical procedures.
The decision whether or not to present actual programs was not an easy one: for
one thing, I myself hate trying to understand other people's programs, no matter how
transparent they might be; for another, I had to assume roughly what the reader might
want. Evidently, I chose to give a few examples. A program that is known to work will
often answer all sorts of unforeseen questions as well as provide a basic framework
on which to build other, new, programs. I hope they, and the book for that matter,
will prove useful.
I gratefully acknowledge the helpful criticism by several of my colleagues and
my wife, the painstaking typing and checking by Lisbeth Heilesen and Springer-Verlag's
readiness to produce the book in this form whose price should make it accessible to
students.
Dieter Britz
Arhus, December 1980
CONTENTS
page
~~~!?~!:!:_!: INTRODUCTION 1
References • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 3
~~~e~!:E_~: BASIC EQUATIONS 5
2.1 General ••••••••••••••••••••••••••••••••••••••••••••••••• 5
2.2 Diffusion ••••••••••••••••••••••••••••••••••••••••••••••• 5
2.2.1 Diffusion current ••••••••••••••••••••••••••••••••••••••• 7
2.3 Convection ••••••••••••••••••••••.••••••••••••••••••••••• 8
2.4 Migration ••••••••••••••••••••.•••••••••••••••••••••••••• 9
2.5 Total transport equation •••••.•••••••••••••••••••••••••• 9
2.6 Homogeneous kinetics •••••••••••••••••••••••••••••••••••• 10
2.7 Heterogeneous kinetics •••.•••••••••••••••••.•••••••••••• 11
2.8 The Nernst diffusion layer and dimensionless variables •• 12
References •••••••••••••••••••••••••••••••.•••..•••••.••• 16
~~~!?~!:!:_~: DIFFUSIONAL TRANSPORT - DIGITALLY 17
3.0 Introduction • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • 17
3.1 Finite-difference expressions of derivatives............ 17
3.1.1 The box method • • • • • • • • • • • . • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 18
3.1.2 Discretisation by common sense •••••••••••••••••••••••••• 20
3.1.3 Taylor-series developments •••••••••••••••••••••••••••••• 22
3.1.4 The point method •••••••••••.••••••••••••••••••.••••••••• 23
3.1.5 The point method with a l&x grid shift ••••••••.••••••••. 25
3.2 Spherical systems ••••••••••••••••••••••••••••••••••••••• 26
3.2 .1 Box method • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 26
3.2.2 Point method •• • • • • • . • • • • • • •• • • •• • • • • • • • • • •• • • • . • • • • • . • • • 28
3.3 The factor A • • • . • • • • • • • • • • • • • • • • • • • • • • • • . • • • • . • • • • .. .. .. 28
3.3.1 A simulation "by hand" with A too large •••.•.•.•••.••••. 29
3.3.2 What happens to errors? ••••••••••••••••••••••••••••••••• 30
3.4 A simple program (almost) ••••••••••••••••••••••••••••••• 31
3.5 How far out into solution do we go? ••••••••••••••••••••. 32
References • • • • . • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 33
S~~E!!:E_~: HANDLING OF BOUNDARY PROBLEMS 34
4.1 Controlled current •••••••••••.•••••••••••••••••••••••••• 34
4.2 Inert wall boundaries ••••••••••••••••••••••••••••••••••• 35
4.3 The electrode potential at controlled current ••••••.•••• 36
4.4 Controlled-potential systems ••••••••.••••••••••••••••••• 37
4.4.1 Reversible systems ••••••••••••••••••••••••••••••••..•••• 37
4.4.2 Quasi- or irreversible systems •••••••••••••••.•••••••••• 39
4.5 Adsorption kinetics •.••••••••••••••••••••••••••••••••••• 41
References ••.••••••••••••••••••••••••••••••••••••••••••• 43
~~~e~!:E_~: IMPLICIT TECHNIQUES AND OTHER COMPLICATIONS 44
5.1 A symmetrical failure: the Richardson scheme •••••••••••• 44
5.2 A symmetrical success: DuFort and Frankel ••••••••••••••• 45
5.3 Degree of implicitness- general concepts ••••••••••••••• 45
5.4 The Crank-Nicolson method ••••••••••••••••••••••••••••••• 48
5.4.1 How to implement CN ••••••••••••••••.•••••••••••••••••••• 48
5.4.2 Crank-Nicolson: why? •••••••••••••••••••••••••••••••••••• 52
5.5 Unequal intervals ••••••••••••••••••••••••••••••••••••••• 52
VIII
5.5.1 How to implement unequal intervals •••••••••••••••••••••• 54
5.5.2 Development of the digital form for unequal intervals ••• 54
5.6 Diffusion in 2 dimensions ••••••••••••••••••••••••••••••• 58
5.6.1 Square point-grid ••••••••••••••••••••••••••••••••••••••• 58
5.6.2 Non-square grids •••••••••••••••••••••••••••••••••••••••• 59
References ••.•••••.•••••••.••••••••••.•.•.•.•.••.•••••.• 61
ACCURACY AND CHOICE 62
6.1 What does (should) accuracy mean? ••••••••••••••••••••••• 62
6.1.1 We demand an error in computed current! ••••••••••••••••• 62
6.1.2 Can we improve on the current approximation? •••••••••••• 64
6.1.3 Cumulative effects •••••••••••••••••••••••••••••••••••••• 64
6.2 Fortuitous accuracy and fudge factors ••••••••••••••••••• 65
6.2.1 The l8t shift ••.•••••••••••••••••••••••••••••••••••••••• 65
6.2.2 Arbitrary parameters and asymmetries •••••••••••••••••••• 66
6.2.3 The end-element expression •••••••••••••••••••••••••••••• 66
6.3 Accuracy comparisons •••••••••••••••••••••••••••••••••••• 68
6. 3 .1 Convergence ••••••••••••••••••••••••••••••••••••••••••••• 69
6. 3. 2 A. -dependence •••••••••••••••••••••••••••••••••••••••••••• 70
6.3.3 Box- or point- ? ........................................ . 71
6.4 Recommendations ••••••••••••••••••••••••••••••••••••••••• 71
6.5 Analytical, semi-analytical or digital? ••••••••••••••••• 72
6. 6 Checking •••••••••••••••••••••••••••••••••••••••••••••••• 73
References ••.••••••.••.•...••.•••••..•.•.••.•••.•..••.•• 73
NON-DIFFUSIONAL CONCENTRATION CHANGES 74
7.1 Homogeneous chemical reactions •••••••••••••••••••••••••• 74
7 .1.1 Digitising an EC reaction ••••••••••••••••••••••••••••••• 74
7.1.2 Normalisation ••••••••••••••••••••••••••••••••••••••••• , • 76
7.1.3 Problems with A. and K ••••••••••••••••••••••••••••••••••• 77
7.2 Convective transport •••••••••••••••••••••••••••••••••••• 78
7.2.1 The differential equations •••••••••••••••••••••••••••••• 78
7.2.2 The digitised convection term- point method ••..•••••••• 79
7.2.3 The digitised convection term- box method •••••••••••••• 80
References ••....•.•...•...••.•..•...•.••.••..•.•.•••.•.. 80
THE LAPLACE EQUATION AND OTHER STEADY-STATE SYSTEMS 81
8.1 Mathematical formulation of potential distribution 81
8.2 Digital solution by finite differences •••••••••••••••••• 83
8.2.1 Digi tisation ••••••••••••••••••••.••••••••••••••••••••••• 83
8.2.2 A simple procedure •••••••••••••••••••••••••••••••••••••• 84
8.2.3 Boundaries •••••••••••••••••••••••••••••••••••••••••••••• 84
8.3 Speed-up devices •••••••••••••••••••••••••••••••••••••••• 85
8.3.1 Using new values immediately •••••••••••••••••••••••••••• 85
8.3.2 Overrelaxation •••••••••••••••••••••••••••••••••••••••••• 85
8.4 When do we stop? •••••••••••••••••••••••••••••••••••••••• 87
8.5 Steady-state transport •••••••••••••••••••••••••••••••••• 88
References .••••.•.•...••••••••...••••••..•••.••.•••••••• 89
~~~e!~~-~: PROGRAMMING EXAMPLES 90
9.0 Introductory remarks •••••••••••••••••••••••••••••••••••• 90
9.1 Potential-step experiment ••••••••••••••••••••••••••••••• 92
9. 2 Cyclic vol tammetry • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 96
9.3 Dropping mercury electrode •••••••••••••••••••••••••••••• 100
9.4 Adsorption kinetics ••••••••••••••••••••••••••••••••••••• 107
9.5 The RDE by steady-state ••••••••••••••••••••••••••••••••• 112
References • • • • • • • • • • . . • • . . • • . . • • • • . • • • • • • • • • • • • • • • • • • • • • 114
Index 115
SYMBOLS USED
a attraction coefficient in Frumkin isotherm; coefficient in equations.
2
A area, m •
-1
b adsorption parameter, L mol ; equation coefficient.
B dimensionless adsorption parameter.
c concentration, mol L-1, or also mol m-3
cb bulk concentration, mol L-1.
ci concentration at point i.
c! new concentration.
1
Co, co dimensionless concentration (of species x) at x 0.
X
d diffusion coefficient ratio, e.g. DRed/Dox·
2 -1
D diffusion coefficient, m s
E potential, V.
f flux, mol s -1 •
F Faraday constant, coulomb mol-l.
g concentration gradient ac/ax, or ac;ax.
go g at x = 0.
g0i g0 for species i.
h discretisation interval ox (e.g.).
X
i index, mostly for discretised x-axis.
I adsorption isotherm function.
j index, mostly for discretised t-axis.
k kinetic parameter, various dimensions.
K dimensionless kinetic parameter; also non-dimensional parameter in
adsorption kinetics (p. 110).
n number of space intervals (also of electrons transferred).
Ox substance in oxidised state.
p dimensionless electrode potential,
= RnFT ( E - Eo ).
(mercury drop) radius, m.
(cont'd)
X
R dimensionless radius.
0
Red substance in reduced state.
s sphericity
t time, s.
T dimensionless time.
-1
vx convection velocity, m s , in (e.g.) x-direction.
vx dimensionless convection velocity.
X x-axis distance, m.
X dimensionless x-axis distance.
y, y as for x, X, but y-axis.
z
z, as for x, X, but z-axis.
a transfer coefficient.
a coefficient in several places.
0 or 01 diffusion-layer (Nernst) thickness, m; defined in various ways.
r electrode surface coverage, mol m- 2
e electrode fractional coverage.
2
X stability parameter D ~t/ox •
T time of an experiment, s.
w rotation rate, rad s -1 •
Chapter l
INTRODUCTION
This book is about the application of digital simulation (DS) to electrochemical
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 which might 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 electrochemis
try? Most commonly it is electrochemical transport problems that we find difficult to
solve, in all but a few model systems - when things get more complicated, as they do
in real electrochemical cells, problems may not be solvable algebraically, yet we
still want answers.
Most commonly, the basic equation we need to solve is the diffusion equation,
relating concentration c to time t and distance x from the electrode surface, given
the diffusion coefficient:
2
ol..£
(l.l)
al
This is Fick's second diffusion equation. Technically, it is a partial second-order
differential equation, of parabolic type. In fact, mostly it will be only the skele
ton of the actual equation one needs to solve - there will usually be such complica
tions as convection (solution moving) and chemical reactions taking place in the so
lution, which will act to cause concentration changes in addition to diffusion itself.
Numerical solution may then be the only way we can get numbers from such equations,
hence digital simulation.
The actual numerical technique employed in digital simulation is that of finite
differences, and this is much older than the digital computer. It dates back at least
to l9il (Richardson)1; in 1924 Schmidt used a graphical variant2, and Courant, Fried
richs and Lewy3 described what we now take to be the essentials of the method in 1928;
Emmons4 wrote a detailed description of finite difference methods, applied to several
different equation types, in 1944. There is no shortage of mathematical texts on the
. 5-10
subJect
It should not be imagined that the technique became used only when digital com
puters appeared; engineers certainly used it long before that time, and were not
afraid to spend hours with pencil and paper. Emmons casually mentions4 that one fluid
flow problem took him 36 hours! Not surprisingly, it was during this early pre-com
puter era that most of the theoretical groundwork was done, as well as refinements
worked out to make the work easier - those early stalwarts wanted their answers as
quickly as possible, and they wanted them correct the first time through!