Modern aspects of electrochemistry no 32 conway

MODERN ASPECTS OF
ELECTROCHEMISTRY
No. 32


LIST OF CONTRIBUTORS
THOMAS Z. FAHIDY
Department of Chemical Engineering
University of Waterloo
Waterloo, Ontario, Canada, N2L 3G1

VIJAY MODI
Department of Mechanical Engineering
Columbia University
New York, New York 10027

KATHARINA KRISCHER
Abt. Physikalische Chemie
Fritz-Haber-Institut der Max-PlanckGesellschaft
Berlin, Germany D-14195

ASHOK K. VIJH
Electrochemistry Section
Hydro-Quebec Research Institute
Varennes, Quebec, Canada J3X 1S1

ANDRZEJ LASIA
Department of Chemistry
University of Sherbrooke
Sherbrooke, Quebec, Canada J1K 2R1

MARK C. LEFEBVRE
Department of Chemistry
University of Ottawa
Ottawa, Ontario, Canada K1N 6N5

ALAN C. WEST
Department of Chemical Engineering
and Applied Chemistry
Columbia University
New York, New York 10027
J. DELIANG YANG
Department of Chemical Engineering
and Applied Chemistry
Columbia University
New York, New York 10027

A Continuation Order Plan is available for this series. A continuation order will bring delivery
of each new volume immediately upon publication. Volumes are billed only upon actual
shipment. For further information please contact the publisher.


MODERN ASPECTS OF
ELECTROCHEMISTRY
No. 32
Edited by

B. E. CONWAY
University of Ottawa
Ottawa, Ontario, Canada

J. O’M. BOCKRIS
Molecular Green Technology
College Station, Texas

and

RALPH E. WHITE
University of South Carolina, Columbia
Columbia, South Carolina

KLUWER ACADEMIC PUBLISHERS
NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW


eBook ISBN

0-306-46916-2

Print ISBN

0-306-45964-7

©2002 Kluwer Academic Publishers
New York, Boston, Dordrecht, London, Moscow

All rights reserved

No part of this eBook may be reproduced or transmitted in any form or by any means, electronic,
mechanical, recording, or otherwise, without written consent from the Publisher

Created in the United States of America

Visit Kluwer Online at:
and Kluwer's eBookstore at:





Preface
This volume is composed of six chapters covering both fundamental and
applied electrochemistry, as in previous monographs in this series.
The first chapter, by Krischer, provides a detailed analysis of oscillatory processes that arise in the kinetics of certain electrode processes, for
example, in active to passive transitions involving oxide films and in H 2
and small organic molecule oxidations. The origin of such periodic
phenomena in electrochemistry has remained obscure for some time.
(Why are steady states not simply attained?) The author gives a thorough
and mathematical treatment of the conditions required for onset of oscillations, including, it is important to note, coupling with resistive elements
of experimental circuits and diffusion. Her review encompasses broader
aspects of periodic phenomena such as those currently being considered
in theories of transition between “order” and “chaos,” part of a new
paradigm in biology and cosmology.
Lasia, in the second chapter, offers a much-needed comprehensive
treatment of ac impedance (“impedance spectroscopy”) as applied to the
study of kinetics and mechanisms of electrode processes. He starts out
with the elements and fundamentals of the subject and develops case
studies for treatment of progressively more complex processes involving
coupling between activation and diffusion-controlled faradaic reactions,
also including pseudo-capacitative elements in parallel relations with the
ubiquitous double-layer capacitance. An extension to the study of electrochemical sorption of hydrogen into host cathode metals is also usefully

given. In a forthcoming volume, a second part of this review will be
published, covering practical applications, for example, in corrosion,
industrial electrolytic processes and battery electrochemistry.
Continuing on the fundamental side, Lefebvre, in Chapter 3, revisits
the problem of the significance of stoichiometric numbers in analysis of
mechanisms of multistep electrode processes. He considers both forward
v


vi

Preface

and backward directions of multi- (two or three) electron-transfer reactions (e.g., as in Al deposition), and the participation of the associated
intermediates. This chapter illustrates the complexity of interpretations of
determined stoichiometric numbers and the limitations that arise in their
application to mechanism analysis.
Chapter 4 by Vijh is on the environmentally related topic of electroosmotic dewatering of clays. This subject encompasses interfacial electrochemical and colloid science, and has important applications in
washing clay and sand, the treatment of ores and tailings, and dewatering
of brown coal and peat, as well as in dealing with liquors and wastes from
the electroplating and metal-finishing industries. Geotechnical applications also arise, for example, in the stabilization of soils in locations where
mudslides occur. Electrochemistry is involved through the high-area
double layers at colloid interfaces and in the provision of the high voltages
at the electrodes that drive the processes of electro-osmosis involved in
the “dewatering” phenomenon.
Magnetic effects in electrolytic processes have always held a special
if somewhat distant interest for electrochemists. In Chapter 5, by Fahidy,
an excellent account is given of the fundamentals of this topic and its
applications, through magnetohydrodynamics, to electrodeposition and
corrosion. Also treated is the basis of the electrolytic Hall effect, which is

essential for understanding how electrohydrodynamic forces act on moving ions in a magnetic field.
In industrial electrolytic processes, including metal electrodeposition
and preparation reactions, mass transfer and fluid flow are usually of
central importance, especially in scaleup from laboratory-scale experimentation. In the final chapter of this volume, West and co-authors give
the essential aspects of computer analysis and modeling of such processes
in terms of fluid dynamics and mass transfer.
B. E. Conway, University of Ottawa
J. O’M. Bockris, Molecular Green Technology
Ralph E. White, University of South Carolina


Contents
Chapter 1
PRINCIPLES OF TEMPORAL AND SPATIAL PATTERN
FORMATION IN ELECTROCHEMICAL SYSTEMS
Katharina Krischer
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
II. Principles of Temporal Pattern Formation . . . . . . . . . . . . . . . . . . 6
1. Bistability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2. Simple Periodic Oscillations of Type I: Negative
Differential Resistance Oscillators . . . . . . . . . . . . . . . . . . . . 12
3. Simple Periodic Oscillations of Type II: Hidden
Negative Differential Resistance Oscillators . . . . . . . . . . . . 25
4. Mixed-Mode Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
III. Principles of Spatial Pattern Formation . . . . . . . . . . . . . . . . . . . 71
1. Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2. Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
IV. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

vii


viii

Contents

Chapter 2
ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY AND ITS
APPLICATIONS
Andrzej Lasia
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Response of Electrical Circuits . . . . . . . . . . . . . . . . . . . . .
2. Impedance of Electrical Circuits . . . . . . . . . . . . . . . . . . . .
3. Interpretation of Complex Plane and Bode Plots . . . . . . . .
II. Impedance Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. ac Bridges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Lissajous Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3. Phase-Sensitive Detection . . . . . . . . . . . . . . . . . . . . . . . . .
4. Frequency Response Analyzers . . . . . . . . . . . . . . . . . . . . .
5. Fast Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . .
III. Impedance of Faradaic Reactions in the Presence of
Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. The Ideally Polarizable Electrode . . . . . . . . . . . . . . . . . . .
2. Semi-Infinite Linear Diffusion . . . . . . . . . . . . . . . . . . . . . .
3. Spherical Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4. Cylindrical Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5. Disk Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6. Finite-Length Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . .
7. Analysis of Impedance Data in the Case of Semi-Infinite
Diffusion: Determination of Kinetic Parameters . . . . . . . .
IV. Impedance of a Faradaic Reaction Involving Adsorption
of Reacting Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Faradaic Reaction Involving One Adsorbed Species . . . . .
2. Impedance Plots in the Case of One Adsorbed Species . . .
3. Faradaic Impedance in the Case Involving Two Adsorbed
Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4. Impedance Plots in the Case of Two Adsorbed Species . . .
5. Faradaic Impedance for a Process Involving Three or
More Adsorbed Species . . . . . . . . . . . . . . . . . . . . . . . . . . .
V. Impedance of Solid Electrodes . . . . . . . . . . . . . . . . . . . . . . . .
1. Frequency Dispersion and Electrode Roughness . . . . . . . .
2. Constant Phase Element . . . . . . . . . . . . . . . . . . . . . . . . . . .

143
144
148
154
156
156
157
157
160
162
167
167
167

174
175
177
178
182
187
188
191
196
199
199
201
201
202


ix

Contents

VI.

VII.

VIII.
IX.

3. Fractal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4. Porous Electrode Model . . . . . . . . . . . . . . . . . . . . . . . . . . .
5. Generalized Warburg Element . . . . . . . . . . . . . . . . . . . . . .

Conditions for “Good” Impedances . . . . . . . . . . . . . . . . . . . . .
1. Linearity, Causality, Stability, Finiteness . . . . . . . . . . . . . .
2. Kramers–Kronig Transforms . . . . . . . . . . . . . . . . . . . . . . .
3. Nonstationary Impedances . . . . . . . . . . . . . . . . . . . . . . . . .
Modeling of Experimental Data . . . . . . . . . . . . . . . . . . . . . . .
1. Selection of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. CNLS Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Instrumental Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

207
210
223
224
224
226
230
231
231
235
239
242
242

Chapter 3
ESTABLISHING THE LINK BETWEEN MULTISTEP
ELECTROCHEMICAL REACTION MECHANISMS AND
EXPERIMENTAL TAFEL SLOPES
Mark C. Lefebvre

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
1. Structure of this Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
II. Chemical Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
III. Simple One-Step, One-Electron Electrochemical Kinetics . . . 255
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
2. Energetics of the Electrochemical Transition State at
Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
3. Electrochemical Reaction under Polarization. . . . . . . . . . . 259
4. The Symmetry Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
5. Double-Layer Considerations . . . . . . . . . . . . . . . . . . . . . . . 264
6. Rate Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
IV Sequence of Consecutive Electrochemical Reactions
Involving a Single Rate-Determining Step . . . . . . . . . . . . . . . 266
1. Reaction Schemes and Intermediates . . . . . . . . . . . . . . . . . 266
2. Underlying Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 268


Contents

x

3. Steady-State and Quasi-Equilibrium Treatments . . . . . . . .
4. Rate Equation for Consecutive Electrochemical
Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5. Adsorption of Intermediates . . . . . . . . . . . . . . . . . . . . . . . .
6. Validity of the Quasi-Equilibrium Approximation . . . . . . .
V. Modifications to the Consecutive Electrochemical
Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Chemical Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3. Multielectron Transfers . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4. Combination or Dissociation as a Rate-Limiting Step . . . .
–1
VI. Tafel Slopes Greater than 118 mV dec . . . . . . . . . . . . . . . . .
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Stoichiometric Number . . . . . . . . . . . . . . . . . . . . . . . . . . .
3. Reaction Mechanisms Involving a Stoichiometric
Number Greater than 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4. Prior Dissociation, Forward Reaction Direction . . . . . . . .
5. Prior Dissociation, Reverse Reaction Direction . . . . . . . . .
6. Following Combination Step . . . . . . . . . . . . . . . . . . . . . . .
7. Electron Number Coefficients . . . . . . . . . . . . . . . . . . . . . .
VII. Application to the Processes of Aluminum Deposition
and Dissolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VIII. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

269
272
274
275
281
281
282
282
283
285
285
285

287
289
291
292
293
294
297
299
299

Chapter 4
ELECTRO-OSMOTIC DEWATERING OF CLAYS, SOILS, AND
SUSPENSIONS
Ashok K. Vijh
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
II. An Outline of Electro-Osmotic Dewatering . . . . . . . . . . . . . .
III. Phenomenological Equations . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Components of Current and Flux During Electro-Osmotic
Dewatering with or without Pressure . . . . . . . . . . . . . . . . .

301
303
306
308


Contents

IV.
V.

VI.
VII.

xi

2. Connection of Electro-Osmosis to Other Electrokinetic
Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
The Electrochemical Approach to Electro-Osmotic
Dewatering: Helmholtz–Smoluchowski Relation . . . . . . . . . . 311
Electro-Osmotic Dewatering: Some Experimental Aspects. . . 315
1. Electro-Osmotic Dewatering under Interrupted Direct
Current Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
2. Electro-Osmotic Dewatering under Galvanic
Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
3. Dewatering Efficiency in Terms of Liters per Ampere-Hour
(or Liters per Watt-Hour at Constant Voltage) . . . . . . . . . . . 324
4. High Voltages Needed for Dewatering Al-Kaolinite and the
Aluminum Electrode Effect . . . . . . . . . . . . . . . . . . . . . . . . . 325
5. Electro-Osmotic Dewatering at Low Applied Voltages . . . . 326
6. Components of Voltage in an Electro-Osmotic Cell . . . . . . 327
Applications of Electro-Osmotic Dewatering . . . . . . . . . . . . 328
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

Chapter 5
THE EFFECT OF MAGNETIC FIELDS ON ELECTROCHEMICAL
PROCESSES
Thomas Z. Fahidy

I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
II. Magnetic Field Effects on Electrolyte Behavior . . . . . . . . . . 336
1. The Hall Effect in Electrolytes . . . . . . . . . . . . . . . . . . . . . 336
2. Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
3. Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
4. Properties of Chemical Equilibrium . . . . . . . . . . . . . . . . . 337
5. Magnetic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
III. Magnetic Field Effects on Surface Morphology . . . . . . . . . . 339
1. Cathode Deposits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
2. Deposits Formed in Anodic Dissolution . . . . . . . . . . . . . . 340


xii

Contents

IV. The Magnetic Field Effect on Electrode Reaction Kinetics . .340
V. The Magnetic Field Effect on Ionic Mass Transport . . . . . . . . 341
VI. Magnetic Field Effects in Environmental
Electrochemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
1. Areas of Importance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
2. Magnetic Field Effects on Corrosion Rates . . . . . . . . . . . . 343
3. Miscellaneous Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
VII. Microscale Behavior: Application of Boltzmann
Equation-Based Transport Models . . . . . . . . . . . . . . . . . . . . . 344
VIII. Application of the Model of Slightly Ionized Plasmas . . . . . .346
IX. Macroscale Behavior: Application of MHD Theory . . . . . . .346
1. Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
2. The Application of MHD Theory to Mass Transport . . . . .347

3. Magnetoelectrolytic Mass Transport in a Magnetic Field
Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
4. Profitability of Magnetoelectrolytic Processes: The MHD
View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
X. Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1
Chapter 6
ANALYSIS OF MASS TRANSFER AND FLUID FLOW FOR
ELECTROCHEMICAL PROCESSES
J. Deliang Yang, Vijay Modi, and Alan C. West
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
II. Literature Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
III. Computational Fluid Dynamics. . . . . . . . . . . . . . . . . . . . . . . . 362
1. CFD Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
2. Grid Generation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
IV. Mass Transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
1. Finite-Volume Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . .370
2. Two-Dimensional Formulation on an Orthogonal Grid . . . 372
3. Validation of Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 375


Contents

xiii

V. Examples of Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
1. Copper Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
2. Mass Transfer during Unstable Flows Generated by a
Blocking Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

VI. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
Cumulative Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
Cumulative Title Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415


This page intentionally left blank


MODERN ASPECTS OF
ELECTROCHEMISTRY
No. 32


This page intentionally left blank


1

Principles of Temporal and Spatial Pattern
Formation in Electrochemical Systems
Katharina Krischer
Abt. Physikalische Chemie, Fritz-Haber-Institut der Max-Planck-Gesellschaft,
Faradayweg 4 - 6, D-14195 Berlin, Germany

I . INTRODUCTION
The vast body of literature on electrochemical oscillations has revealed a

quite surprising fact: dynamic instabilities, manifesting themselves, for
example, in bistable or oscillatory reaction rates, occur in nearly every
electrochemical reaction under appropriate conditions. An impressive
compilation of all the relevant papers up to 1993 can be found in a review
article by Hudson and Tsotsis.¹ This finding naturally raises the question
of whether there are common principles governing pattern formation in
electrochemical systems. In other words, are there universal mechanisms
leading to self-organization phenomena in systems with completely different chemical compositions, and thus also distinct rate laws?

Modern Aspects of Electrochemistry, Number 32, edited by B. E. Conway et al. Kluwer
Academic / Plenum Publishers, New York, 1999.
1


2

Katharina Krischer

In general terms, the occurrence of self-organization phenomena is
tied to two conditions: The system has to be far from thermodynamic
equilibrium, and appropriate feedback mechanisms have to be present.
Obviously, in electrochemical experiments, the first condition is
almost always fulfilled. However, the requirement of appropriate feedback
mechanisms (i.e., appropriate nonlinear evolution laws) seems to constitute a severe restriction on the possible reaction mechanisms that give rise
to pattern formation. From this point of view, it is astonishing that nearly
all electrochemical systems exhibit dynamic instabilities.
The progress achieved in understanding oscillatory behavior in various electrochemical systems during the past decade has brought a common
framework to light. In most systems the occurrence of dynamic instabilities is linked to the interplay of electrode kinetics, transport processes
occurring in the electrolyte, and the electrical circuit. Only the first one of
these “ingredients” of the oscillation mechanism depends on the elementary reaction steps occurring at the interface. In contrast, the other two are

determined by potentiostatic or galvanostatic control and cell design.
Thus, any attempt to understand the physical origin of the instability has
to take into consideration the complete electrochemical system. The
interfacial phenomena themselves have only limited effect on whether the
system has dynamic instability.
The important role that electric circuit plays in dynamic instabilities
was recognized a long time ago, and lately since the famous and muchcited review article by Wojtowicz² appeared, electrochemical oscillators
have been divided into two categories: those described by “chemical
models” and those described by “electrical models.” In systems belonging
to the latter class, self-organization phenomena arise owing to the properties of all elements of the circuit, while in chemical models they are caused
exclusively by the properties of the electrode/electrolyte interface. Thus,
chemical models result from mass balance only, and hence the variables
are the concentrations of the reacting species and possibly the state of the
electrode. Electrochemical oscillators belonging to this class can be described in exactly the same framework as oscillations in heterogeneous
catalysis *: The rate equations are derived from transport processes of the

*Exceptions here seem to be some metal dissolution reactions; see e.g., Ref. 3.


Principles of Temporal and Spatial Pattern Formation

3

reactants to (and possibly also from) the electrode; adsorption, reaction,
and desorption steps; as well as possible changes in the state of the
electrode surface. The electrode potential is kept fixed and thus reduces
to a parameter.
Frequently encountered mechanisms leading to the spontaneous occurrence of time-dependent reaction rates in heterogeneous catalysis can
be found in other review articles. 4 – 7 However, the relevance of chemical
models for explaining the physical mechanism of electrochemical oscillators has been overestimated for a long time, and the hope of learning

much from heterogeneous catalysis about how oscillations arise in electrochemical systems has not been fulfilled. In most cases where chemical
models were proposed in order to explain the temporal behavior of the
system, they were later proved wrong. At present, there are only a few
systems in which experimental results strongly suggest that the instability
is of chemical origin. Among these are iron dissolution from nitric acid 8 ,9
and the electrodeposition of Zn. 10 Still debated is whether also the oxidation of Si in acid fluoride solution falls into this category.11–14 However,
with one possible exception, 3 the reaction mechanisms are not yet understood for any of the possible candidates in this class.
In electrical models, the instability results from the interaction of the
characteristics of the electrode/electrolyte interface (i.e., the faradaic
impedance) with the additional “external” elements of the electric circuit
(i.e., electrode capacity, electrolyte) and the control device (i.e., a potentiostat or galvanostat). Consequently, the differential equations governing
the temporal evolution of these systems are derived from charge as well
as mass balance. The double-layer potential thus constitutes a variable
evolving in time. All oscillators belonging to this class (which, as it seems,
is the overwhelming majority of the electrochemical oscillators) possess
a negative real faradaic impedance in a certain parameter region. That a
negative slope in the stationary current-potential curve can destabilize an
electrochemical system had been known for many decades. 8 However, it
is only recently and mainly due to work by Koper, partly in collaboration
with Sluyters and Gaspard, that the interplay of this electrical instability
with slow reaction steps or transport processes has been elaborated.15–17
Through this work, a simple model that accounts for surprisingly many
electrochemical oscillators revealed an unexpected wealth of different
dynamic behaviors. Meanwhile, a consistent picture of why and when


4

Katharina Krischer

electrochemical systems oscillate has been developed, which naturally
leads to a further division of electrical models into two subcategories,
according to the nature of the slow feedback process. 18 –20
However, in spatially extended systems, self-organization in time is,
in general, accompanied by pattern formation in space. An understanding
of dynamic behavior is not possible without taking into account the spatial
degrees of freedom. The first hint that spatial structures may develop at
the electrode/electrolyte interface goes back as far as 1844, 21 and intense
research on wave phenomena in electrochemical systems was carried out
during the first half of this century. It was initiated by Ostwald, who
recognized a close relation between electrochemical waves and nerve
impulse propagation.22 As we know today, at a certain level nerve impulse
propagation and chemical waves are in fact described by mathematically
equivalent equations.23 Bonhoeffer and Franck continued Ostwald’s
work.24 –31 They mainly studied activation pulses on iron wires. In view of
the fact that the modern concepts of nonlinear dynamics were not yet
developed, these remarkable experiments were ahead of their time. Progress in understanding spatial pattern formation had to await the formulation of a theory for temporal self-organization and the development of
new techniques that imaged the electrode/electrolyte interface. Both have
been achieved only recently. Thus, at present, we are in a position to
elucidate the basic mechanism of spatial pattern formation.
The aim of this chapter is to provide a concise discussion of the
current understanding of basic principles governing temporal and spatial
behavior in electrochemical systems exhibiting dynamic instabilities. The
emphasis is on deriving a coherent picture of the theoretical description.
In doing so, a hierarchy is built up of models that successively describe
more complex behavior, starting with bistability in spatially uniform
systems and ending with complex spatiotemporal dynamics. Only electrical models are considered. Furthermore, experimental examples were
chosen for detailed discussion only where the relation between experiment
and model is unambiguous.
Such an approach is neither compatible with a compilation of the

different dynamic behaviors found in one system, because for different
dynamic regimes different levels of the theoretical description are adequate, nor can it cover all the different systems that exhibit instabilities.
Readers interested in an overview of oscillating systems are referred to
the exhaustive review article by Hudson and Tsotsis¹ or an even more
recent review article, which is not as comprehensive, by Fahidy and Gu. 32


Principles of Temporal and Spatial Pattern Formation

5

Another contemporary and noteworthy review article by Koper follows
yet another concept. 20 Koper first stresses the importance of the electric
circuit by evaluating, in a rigorous way, the stability of electrochemical
systems by frequency response methods. He then thoroughly discusses the
dynamics of selected examples, including some semiconductor systems,
which are not included in this chapter, with special emphasis on how they
relate to the frequency response theory.
The organization of this chapter can be summarized as follows:
Section II treats temporal models, which are adequate whenever the
system is uniformly parallel to the electrode. It starts with a description of
how bistable behavior arises in electrochemical systems. This constitutes
a comparatively old result. It is, however, thoroughly explained because
it forms the basis of the rest of the chapter. In Sections II.2 and II.3, two
distinct mechanisms are discussed that give rise to simple periodic behavior. In each case, a prototype model is first introduced, which can be
regarded as a minimal model exhibiting the essential features of oscillators
of the respective class. Then, experimental examples that follow this
model are reviewed. Since the second type of oscillators is more complex,
the analysis of the experimental examples includes a discussion of more
realistic models as well as an analysis of the connection of the individual

terms of these “physical models” with the terms in the prototype model.
Section II.3 summarizes the extent of our knowledge for more complex
oscillations, which typically arise in any system belonging to one of these
two categories for certain parameter values.
Section III deals with spatial phenomena. The current state of theoretical description is given in Section III.1, and experimental results are
compiled in Section III.2. The organization of these two parts is analogous
to Section II, that is, first waves in bistable media are discussed and then
pattern formation in oscillatory media. Because the investigations of
spatial self-organization are still in their infancy, not all theoretical predictions have yet been experimentally verified, and many experiments
cannot yet be understood in terms of the underlying physical mechanisms.
Hence this section represents a first approach toward a coherent understanding of spatial structures, and a series of open questions is listed at the
end.
In the final section, a summary of what has been achieved so far in
the understanding of electrochemical self-organization is given.


6

Katharina Krischer

II. PRINCIPLES OF TEMPORAL PATTERN FORMATION
1. Bistability
As mentioned in the introduction, the electrical nature of a majority of
electrochemical oscillators turns out to be decisive for the occurrence of
dynamic instabilities. Hence any description of dynamic behavior has to take
into consideration all elements of the electric circuit. A useful starting point
for investigating the dynamic behavior of electrochemical systems is the
equivalent circuit of an electrochemical cell as reproduced in Fig. 1. The
parallel connection between the capacitor and the faradaic impedance accounts for the two current pathways through the electrode/electrolyte interface: the faradaic and the capacitive “routes.” The ohmic resistor in series with
this interface circuit comprises the electrolyte resistance between working and

reference electrodes and possible additional ohmic resistors in the external
circuit. The voltage drops across the interface and the series resistance are
kept constant, which is generally achieved by means of a potentiostat.
The current balance of the equivalent circuit readily leads to the
general differential equation for that kind of circuitry:
(1)
where C is the (double-layer) capacitance per unit area, A is the area of
the capacitor (electrode), and Re is the sum of all (external) ohmic
resistances in series to the working electrode. φD L denotes the potential
drop across the double layer, and U is the externally applied voltage. The
two terms on the left-hand side (lhs) arise from the two current pathways

Figure 1. General equivalent circuit
of an electrochemical cell with double-layer capacitance C, faradaic impedance ZF , series resistance (comprising ohmic cell resistance and external resistances) R e . U is the externally fixed voltage and φDL the potential drop across the double layer.


Principles of Temporal and Spatial Pattern Formation

7

through the electrode; the right-hand side (rhs) represents the current
through R e , which is equal to the total current through the cell.
The steady states of Eq. (1) [i.e., the solutions of Eq. (1) with dφDL/ dt
= 0] can be easily obtained graphically by plotting the characteristics of
the external circuit, the load line, and the current-potential characteristics
of the electrode/electrolyte interface in one graph [Fig. 2(a)]. Obviously,
intersections of both curves are steady states or fixed points of the system,
and from Fig. 2(a) it becomes immediately clear that whenever the
interfacial characteristic is N-shaped, Eq. (1) possesses three stationary
states in a certain range of U and Re.

When analyzing the stability of the steady states of the circuit, one
finds that a steady state, φ 0DL , is stable, unless
(2)
where ZF denotes the (zero frequency) faradaic impedance. These conditions have been known for a long time and are often discussed in the
literature. 8,15,30,33 The first inequality expresses the fact that any unstable
stationary state has to lie on a branch with a negative differential resistance
(NDR) of the current-potential curve. The second inequality implies that
a steady state can only become unstable if the ohmic resistance of the
circuit is larger than the absolute value of the faradaic impedance of the
reaction.
Applying these stability criteria to the situation shown in Fig. 2(a), it
becomes apparent that whenever the middle branch is unstable (i.e., R e >
|Z F |), there are three fixed points, the two outer ones being necessarily
stable (because they lie on a branch with a positive slope). Hence a small
perturbation of the middle steady state will drive the system, depending
on the direction in which the perturbation occurred, to one of the outer
stationary states. This bistability manifests itself in a hysteresis when the
external voltage is varied [Fig. 2(b)]. The border of the bistable region is
formed by saddle-node bifurcations. In general, a bifurcation occurs if the
dynamic behavior of the system changes qualitatively. This is, for example, the case if the number or the stability of stationary states or oscillatory
solutions changes. At a saddle-node bifurcation, the first of these cases
applies: two stationary states merge at the bifurcation point, disappearing
when the parameter is changed in one direction and separating when it is
changed in the other direction. In Fig. 2(a), a saddle-node bifurcation


8

Katharina Krischer

Figure 2. (a) N-shaped current-potential curve and load lines (resulting
from the external circuit) for three different values of the external voltage U.
The intersections of both curves are
steady states. The two outer load lines
mark the border of the bistable regime.
(b) Bistable region in a current vs.
external voltage plot referring to the
situation shown in (a). (c) Location of
the saddle-node bifurcation (separating monostable and bistable regions)
in the U/R e parameter plane. (sn =
saddle-node bifurcation)

corresponds to the degenerate case where the load line and current-potential characteristic coincide at two points, and the number of fixed points
changes from one to three.
For a given N-shaped current-potential characteristic, there are two
parameters that determine the bistable region, R e and U. In the U/R e
parameter diagram, this region becomes broader while shifting toward
larger values of U for increasing Re , irrespective of the electrochemical
reaction [Fig. 2(c)]. Below we will see that this feature is also encountered
in all more complicated electrical models that describe simple or complex
oscillatory behavior since all of them require an N-shaped polarization curve.
At this point, it appears to be useful to compile the assumptions that
have been implicitly made and which enable us to describe the dynamics
of the system by the evolution equation of the double-layer potential only.