Chemical Waves and Patterns

640 Pages·1995·18.81 MB·English

by  A. T. Winfree (auth.)

CHEMICAL WAVES AND PATTERNS
Understanding Chemical Reactivity
Volume 10
Series Editor
Paul G. Mezey, University of Saskatchewan, Saskatoon, Canada
Editorial Advisory Board
R. Stephen Berry, University of Chicago, IL, USA
John I. Brauman, Stanford University, CA, USA
A. Welford Castleman, Jr., Pennsylvania State University, PA, USA
Enrico Clementi, IBM Corporation, Kingston, NY, USA
Stephen R. Langhoff, NASA Ames Research Center, Moffet Field, CA, USA
K. Morokuma, Institute for Molecular Science, Okazaki, Japan
Peter J. Rossky, University of Texas at Austin, TX, USA
Zdenek Slanina, Czech Academy of Sciences, Prague, Czech Republic
Donald G. Truhlar, University of Minnesota, Minneapolis, MN, USA
Ivar Ugi, Technische Universitat, MOnchen, Germany
The titles published in this series are listed at the end of this volume.
CHEMICAL WAVES
AND PATTERNS
Edited by
RAYMOND KAPRAL
Department of Chemistry, University of Toronto,
Toronto, Ontario, Canada
and
KENNETH SHOWAL TER
Department of Chemistry, West Virginia University,
Morgantown, WV, U.S.A.
SPRINGER-SCIENCE+BUSINESS MEDIA, BV.
Library of Congress Cataloging-in-Publication Data
Chemieal waves and patterns ! edited by Raymond Kapral and Kenneth
Showalter.
p. em. -- (Understandlng ehemieal reaetivity ; v. 10)
Ine 1u des index.
ISBN 978-94-010-4504-9 ISBN 978-94-011-1156-0 (eBook)
DOI 10.1007/978-94-011-1156-0
1. Wave meehanies. 2. Waves. 3. Vortex motion. 4. Osell lating
ehemieal reactions. 5. Belousov-Zhabotinskli reae!lon. 1. Kapral,
Raymond. II. Showalter, Kenneth. III. Series.
aC174.2.C48 1994
541.3'9--de20 94-16566
ISBN 978-94-010-4504-9
Printed on acid-tree paper
AII Rights Reserved
© 1995 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1995
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
Table of Contents
Introduction vii
PART ONE. SPIRAL WAVES
1. Lingering Mysteries about Organizing Centers in the
Belousov-Zhabotinksy Medium and Its Oregonator Model
A. T. Winfree 3
2. Spiral Wave Dynamics
S. C. Maller and T. Plesser 57
3. A Theory of Rotating Scroll Waves in Excitable Media
J. J. Tyson and J. P. Keener 93
4. Spiral Waves in Weakly Excitable Media
A. S. Mikhailov and V. S. Zykov 119
5. Spiral Meandering
D. Barkley 163
6. Spiral and Target Waves in Finite and Discontinuous Media
J. A. Sepulchre and A. Babloyantz 191
PART TWO. TURING AND TURING-LIKE PATTERNS
7. Turing Patterns: From Myth to Reality
J. Boissonade, E. Dulos, and P. De Kepper 221
8. Onset and Beyond Turing Pattern Formation
Q. Ouyang and H. L. Swinney 269
v
vi TABLE OF CONTENTS
9. The Chemistry behind the First Experimental Chemical
Examples of Turing Patterns
l. Lengyel and I. R. Epstein 297
10. Turing Bifurcations and Pattern Selection
P. Borckmans, G. Dewel, A. De Wit, and D. Walgraef 323
11. The Differential Flow Instabilities
M. Menzinger and A. B. Rovinsky 365
PART THREE. CHEMICAL WAVE DYNAMICS
12. Wave Propagation and Wave Pattern Formation in Non
uniform Reaction-Diffusion Systems
A. M. Zhabotinsky 401
13. Chemical Front Propagation: Initiation and Relative Stability
E. Mori, X. Chu, and J. Ross 419
14. Pattern Formation on Catalytic Surfaces
M. Eiswirth and G. Ertl 447
15. Simple and Complex Reaction-Diffusion Fronts
S. K. Scott and K. Showalter 485
16. Modeling Front Pattern Formation and Intermittent Bursting
Phenomena in the Couette Flow Reactor
A. Arneodo and J. Elezgaray 517
PART FOUR. FLUCTUATIONS AND CHEMICAL WAVES
17. Probabilistic Approach to Chemical Instabilities and Chaos
G. Nicolis, F. Baras, P. Geysermans, and P. Peeters 573
18. Internal Noise, Oscillations, Chaos and Chemical Waves
R. Kapral and X.-G. Wu 609
Index 635
Introduction
The concept of macroscopic waves and patterns developing from chemical
reaction coupling with diffusion was presented, apparently for the first time,
at the Main Meeting of the Deutsche Bunsengesellschaft fur Angewandte
Physikalische Chemie, held in Dresden, Germany from May 21 to 24, 1906.
Robert Luther, Director of the Physical Chemistry Laboratory in Leipzig,
read his paper on the discovery and analysis of propagating reaction-diffusion
fronts in autocatalytic chemical reactions [1, 2]. He presented an equation for
the velocity of these new waves, V = a(KDC)1/2, and asserted that they
might have features in common with propagating action potentials in nerve
cell axons. During the discussion period, a skeptic in the audience voiced his
objections to this notion. It was none other than the great physical chemist
Walther Nernst, who believed that nerve impulse propagation was far too rapid
to be akin to the propagating fronts. He was also not willing to accept Luther's
wave velocity equation without a derivation. Luther stood his ground, saying
his equation was "a simple consequence of the corresponding differential
equation." He described several different autocatalytic reactions that exhibit
propagating fronts (recommending gelling the solution to prevent convection)
and even presented a demonstration: the autocatalytic permanganate oxidation
of oxalate was carried out in a test tube with the image of the front projected
onto a screen for the audience.
We now know that reaction-diffusion waves and nerve impulse propaga
tion are related by the similarity of their underlying mathematical descrip
tions. The seminal studies of Fisher [3] and Kolmogorov et al. [4] some 30
years later put Luther's ideas on a firm theoretical footing and generalized
the concept of propagating fronts to include such phenomena as the spread of
an advantageous gene through a population.
The beginning of modem research on pattern formation is usually associat
ed with the paper 'The Chemical Basis of Morphogenesis' published in 1952
by Alan Turing [5], the renowned - and enigmatic - British mathematician.
This paper stands today as the most influential work on self-organization in
far-from-equilibrium systems and arguably ranks among the most important
scientific contributions of the century. It not only spawned several decades of
intense research activity on spatio-temporal behavior in physical and chem
ical systems but also provided the basis for modem theories of biological
R. Kapral and K. Showalter (eds.), Chemical Waves and Patterns, vii-x.
viii INTRODUCTION
patterns and form [6]. Turing's work set the stage for the influential studies
of the Brussels school in the 1960s and 1970s [7].
A year before Turing's famous paper, experiments carried out by B. P.
Belousov [8] in the Soviet Union led to the discovery of a chemical reaction
that has become one of the most studied and celebrated chemical systems of
modem science. Belousov found a reaction that exhibited robust oscillatory
behavior [9], one of the essential ingredients for the experimental realization
of the pattern formation predicted by Turing. Another chemist from the Soviet
Union, A. M. Zhabotinsky, carried out extensive studies of the new system and
developed a modified version that is widely studied today [10]. This reaction
in its various forms is now called the Belousov-Zhabotinsky or BZ reaction.
Although Turing patterns have not been found in the BZ reaction, its beautiful
target and spiral patterns have been the subject of hundreds of experimental
and theoretical investigations. It has become an invaluable model system for
excitable media, providing deep insights into the properties of such media in
biological systems [11].
Our knowledge of chemical instabilities and pattern formation processes
has undergone explosive growth in the past decade. In the late 1970s and early
1980s discoveries related to deterministic chaos, especially the bifurcation
scenarios leading to its onset or destruction, fueled a large number of inves
tigations of the temporal structure of far-from-equilibrium reacting systems.
Well-stirred chemical systems have provided some of the most beautiful and
clear-cut examples of these low-dimensional phenomena. For most systems
in nature, however, diffusion is unable to maintain spatial homogeneity and
spatial degrees of freedom must be taken into account. This fact has prompted
a renewed interest in the spatio-temporal dynamics of reacting systems.
Advances in this area have been driven largely by experiment: the abili
ty to perform high-precision measurements under controlled conditions has
allowed one to carry out quantitative studies of the detailed structure and
dynamics of chemical waves and patterns. The development of continuously
fed gel reactors has played a central role in recent advances. Some 40 years
after Turing patterns were theoretically predicted, they were observed in a gel
reactor [12]. This observation, together with a series of elegant experiments
carried out in Bordeaux, Austin, and Brandeis, stimulated many theoretical
developments, and currently there is a strong interplay between theory and
experiment.
The deeper understanding of the nonlinear dynamics of chemically react
ing systems, gained through earlier studies of the temporal dynamics, has
allowed the design of chemical systems that display a bewildering variety
of chemical patterns. The phenomenology and origin of these patterns is
being investigated actively today. This book presents a survey of some of the
major components of research being carried out in this field. It is divided into
INTRODUCTION ix
four main sections: Spiral Waves, Turing and Turing-Like Patterns, Chemical
Wave Dynamics, and Fluctuations and Chemical Waves. Both experimental
and theoretical contributions have been selected to illustrate the phenomena
and their interpretations.
Spiral waves are some of the most ubiquitous chemical patterns observed
in nature and in the laboratory. Both excitable and oscillatory chemical media
are capable of supporting such waves and spiral waves were observed in
early experiments on the BZ reaction. Chemical excitable and oscillatory
media provide paradigms for more complex biological systems; for instance,
since both cardiac and nerve tissue are excitable a knowledge of such wave
dynamics is crucial for an understanding of these biological systems. The
chapters in Spiral Waves explore the structure and dynamics of spiral waves in
two and three dimensions and provide a theoretical basis for their description.
In spite of the interest generated by Turing's paper over the past forty years,
Turing patterns have not been unambiguously identified as being responsi
ble for biological pattern formation; however, their observation in chemical
systems has triggered further explorations of the possible types of chemi
cal patterns generated by reaction-diffusion and reaction-flow instabilities.
The chapters in Turing and Turing-Like Patterns describe the observations
of Turing patterns, give the theoretical basis for Turing pattern formation
and selection, present the mechanisms responsible for their appearance and
show how Turing's original scheme can be generalized. The resulting new
'zoology' of chemical patterns along with their theoretical descriptions have
given us new insights into the spatio-temporal dynamics of chemical systems.
The diverse phenomena described in this section follow from such general
chemical principles that they shall surely find application in other fields like
biology, where the complexity of the systems has precluded detailed studies.
Chemical wave fronts separate regions of differing chemical concentration
in a reacting system and constitute the basic elements responsible for the
formation of chemical patterns. Such wave fronts need not be simple and can
undergo instabilities leading to complex structures that rival those seen in
hydrodynamics. Chapters in Chemical Wave Dynamics describe studies of
wave propagation in both homogeneous and inhomogeneous chemical media
and consider the mechanisms responsible for the formation of complicated
interfacial structure in chemical waves.
At the most basic level, the pattern formation processes seen in chemi
cal systems must arise from the reactive and nonreactive collisions that the
molecules comprising the system undergo. Because of the molecular nature of
chemical systems, fluctuations in the concentrations are always present. Often
fluctuations can be neglected, for example, in systems close to equilibrium, far
removed from phase transition points. However, the very instabilities that are
responsible for pattern formation provide circumstances in which such fluc-
x INTRODUCTION
tuations cannot be ignored. The studies in Fluctuations and Chemical Waves
indicate how macroscopic reaction-diffusion descriptions can be extended to
more microscopic scales in order to investigate the role of fluctuations on
chemical pattern formation, especially near chemical chaos.
The focus of this book has been intentionally restricted to pattern forma
tion processes in chemical systems; however, the phenomena and theoretical
tools used in their description transcend this particular class of systems. This
overview of recent advances in chemical pattern formation should also be
of use in related fields such as biology and hydrodynamics where similar
phenomena arise.
References
1. Luther, R., Elektrochem. 12(32), 596 (1906).
2. Arnold, R., Showalter, K., and Tyson, 1. 1.,1. Chem. Ed. 64, 740 (1987); Showalter, K.
and Tyson, 1. 1., 1. Chem. Ed. 64, 742 (1987).
3. Fisher, R. A., Ann. Eugenics 7, 355 (1937).
4. Kolmogorov, A., Petrovsky, I., and Piscounoff, N., Bull. Univ. Moscow, Ser. Int., Sect. A
1, 1 (1937).
5. Turing, A. M., Phil. Trans. R. Soc. London B 237,37 (1952).
6. Murray, 1. D., Mathematical Biology (Springer-Verlag, Berlin, 1989).
7. Nicolis, G. and Prigogine, I., Self-Organization in Nonequilibrium Systems (Wiley, New
York, 1977).
8. Belousov B. P., in Autowave Processes in Systems with Diffusion, edited by M. T.
Grecova (USSR Acad. Sci., Gorky, 1981); translation by Maloney, P. G. and Field, R. J.,
in Oscillations and Traveling Waves in Chemical Systems, edited by R. 1. Field and M.
Burger (Wiley-Interscience, New York, 1985).
9. Another oscillatory chemical reaction, the Bray reaction (Bray, W. c., 1. Am. Chem. Soc.
43, 1262 (1921», was discovered 30 years earlier; however, bubbles from its reaction
product disrupt spatial behavior.
10. Zhabotinsky, A. M., Biojizika 9, 306 (1964); Zhabotinsky, A. M., in Oscillations and
Traveling Waves in Chemical Systems, edited by R. 1. Field and M. Burger (Wiley
Interscience, New York, 1985).
11. Winfree, A. T., The Geometry of Biological Time (Springer-Verlag, New York, 1980).
12. Castets, v., Dulos, E., Boissonade, J., and De Kepper, P., Phys. Rev. Lett. 64,2953 (1990).