Elsevier

Physics Reports

Volume 425, Issues 2–3, March 2006, Pages 79-194
Physics Reports

Control of waves, patterns and turbulence in chemical systems

https://doi.org/10.1016/j.physrep.2005.11.003Get rights and content

Abstract

We review experimental and theoretical studies on the design and control of spatiotemporal behavior in chemical systems. A wide range of approaches have been pursued to control spatiotemporal dynamics, from periodic forcing of medium excitability to imposing static and dynamic heterogeneities and geometric constraints on the medium to global feedback with and without delays. We focus on the design and control of spatiotemporal dynamics in excitable and oscillatory media. Experimental examples are taken from the Belousov–Zhabotinsky (BZ) reaction and the oxidation reaction of CO on single crystal Pt, which have become paradigmatic chemical systems for studies of spatiotemporal dynamics. We present theoretical characterizations of spatiotemporal dynamics and control based on the complex Ginzburg–Landau equation as well as models of the BZ and CO/Pt reactions. Controlling spatiotemporal dynamics allows the realization of specific modes of behavior or may give rise to completely new types of behavior.

Introduction

Theoretical understanding of physical phenomena opens a way for purposeful manipulation and control of existing systems, as well as to the design of new, artificial systems with desired properties. In the last two decades, detailed experimental evidence of self-organization phenomena in chemical systems has been accumulated. Experiments show that simple reaction mechanisms and elementary interactions can lead to the formation of complex spatiotemporal concentration patterns that are sensitive to changes in the reaction conditions and may undergo complete rearrangement in response to small purposeful perturbations. Engineering of self-organizing chemical systems cannot be based on the same principles as traditional chemical technology. Application of rigid controls would usually destructively interfere with fine interactions between the elements of a system responsible for its self-organization. Instead, spontaneous activity of a system should be steered in a desired direction by applying weak control impulses and imposing various feedbacks. In this manner, transitions between different organization states can be initiated and new forms of collective behavior can be achieved.

The aim of this article is to review current progress in the control of wave patterns and turbulence in reaction–diffusion systems. Our attention shall be focused on systems with local excitable or oscillatory kinetics. These two classes of chemical systems possess a rich variety of wave patterns that do not depend sensitively on the choice of a particular model. The dynamics of individual elements in such media is not chaotic and turbulence (spatiotemporal chaos) can develop only as a result of interactions between the elements.

As examples, two systems—the Belousov–Zhabotinsky (BZ) reaction and the catalytic surface reaction of CO oxidation on platinum—shall be considered. Both of them have been extensively investigated, their mechanisms are well understood and satisfactory theoretical models of their kinetics are available. While belonging to different classes of chemical systems, i.e., to reactions in aqueous solutions and to heterogeneous catalysis, they show strong similarity in the kinds and properties of observed concentration patterns. General aspects of nonlinear wave propagation in reaction–diffusion systems and its control can be well illustrated using these examples. Below in this introductory section, brief descriptions of the two chosen experimental systems are given.

The first homogeneous oscillatory chemical reaction, the iodate-catalyzed decomposition of hydrogen peroxide, was reported by Bray in 1921 [1]. The discovery was met with scepticism by many scientists, however, because it was believed that the oscillatory behavior violated the second law of thermodynamics [2]. About 30 years later, Belousov [3] discovered another oscillatory reaction, the cerium-catalyzed bromate oxidation of citric acid, although the continuing scepticism prevented him from publishing his discovery for a number of years [4]. Zhabotinsky further developed the reaction by substituting the oxidation–reduction indicator ferroin for the cerium catalyst and substituting malonic acid for the citric acid substrate [5]. The result was a robust oscillatory reaction with striking color changes that is now known as the BZ reaction, which continues to be widely studied as a model system in nonlinear dynamics.

Spatiotemporal wave behavior is exhibited in unstirred BZ reaction mixtures, and the features of propagating waves in quasi-two-dimensional (2D) layers of solution were characterized in early studies by Zhabotinsky [6], [7] and Winfree [8], [9]. Spiral waves, which are self-sustaining wave sources that rotate around a phase singularity, were found in the BZ reaction in 1971 [10], [11]. With the advent of digital imaging technology, precise measurements of the spiral structure became possible [12], [13]. Fig. 1 shows an example of a BZ spiral image, where the vertical axis is proportional to the oxidized catalyst concentration in the reaction.

The simplest spiral wave behavior is rigid rotation of the spiral tip around a small circular core, where every point outside the core undergoes periodic oscillations at the period of rotation of the spiral. Winfree [8] noted in his early studies that the spiral tip may also move, and he called this motion meander. Quantitative studies of the spiral tip showed that the tip motion undergoes a transformation from rigid rotation to motion along epicycle-like patterns as the excitability is varied [14], [15]. Theoretical studies showed that the change from rigid to compound rotation occurs at a subcritical Hopf bifurcation [16], [17]. Fig. 2 shows measurements of the tip motion for three different concentrations of bromate.

Modeling studies of the BZ reaction have relied on a chemical mechanism developed in 1972 by Field, Körös, and Noyes (FKN) [18]. The essential features of the FKN mechanism were captured in a three-variable model called the Oregonator [19], which is comprised of five chemical steps:A+YX+P,X+Y2P,A+X2X+2Z,2XA+P,ZfY,where the variables X, Y, and Z represent the species HBrO2, Br-, and the oxidized catalyst. The constants A and P represent the reactant and product species, BrO3- and HOBr. The autocatalytic generation of X occurs in step (3) after Y has been consumed to a critical concentration in steps (1) and (2). The metal ion catalyst of the reaction is oxidized in the autocatalytic process (3) to generate the oxidized catalyst Z, which resets the oscillation in step (5) by regenerating Y.

A two-variable version of the Oregonator [20] has been widely used to describe the spatiotemporal behavior of the BZ reaction, where diffusion terms are included for the autocatalyst species X and inhibitor species Z. The profiles of these species for a rigidly rotating spiral wave are shown in Fig. 3. The Oregonator can be readily modified to describe various experimental configurations, e.g., a diffusion term is added only for the autocatalyst to describe experiments carried out with the metal ion catalyst immobilized in a gel. For the photosensitive BZ reaction, with Ru(bpy)32+ as the catalyst, a term is added for the photochemical production of the inhibiting species, Br- [21], [22], [23].

Some chemical reactions cannot go on in the gas phase. However, when the molecules are adsorbed on a catalytic metal surface, reactions between them become possible. Such catalytic reactions are broadly used in chemical industry and in environmental technology. In industrial applications, the catalysts are usually porous materials and microparticles. For laboratory experiments, single crystals of metals with perfect surfaces are preferred. The experiments with surface chemical reactions are typically carried under very low pressures of gaseous reactants, thus preventing strong thermal effects and the development of hydrodynamical flows.

Under these conditions, a surface chemical reaction is almost an ideal system for observations of nonequilibrium, reaction-induced wave patterns. The reactants are supplied to the catalytic metal surface from the gas filling the reaction chamber. Under ultra-low pressures, the mean free paths of molecules in the gas are comparable with the linear size of the reaction chamber and mixing of reactants in the gas phase is practically instantaneous. The reaction itself takes place only within a monolayer of molecules adsorbed on the catalyst surface. The molecules diffuse over the surface and react when they collide, making products that go back into the gas phase. The reactants can be supplied in a well-controlled way by dosing gases into the chamber, and the products together with excess reactants are steadily pumped away.

The 2D reaction–diffusion system, formed by adsorbed reactants on the catalytic surface, may show nonequilibrium patterns characteristic for bistable, excitable or oscillatory media. These patterns have characteristic length scales of tens of micrometers and time scales of seconds. They can be observed by using special methods of optical or electron microscopy (see [163]). The photoemission electron microscopy (PEEM) is based on the effect of photoelectron emission from the metal surface under ultraviolet light irradiation. The yield of photoelectrons depends sensitively on the local work function of the substrate which is changed due to the presence of adsorbates. This method produces real-time images of the lateral distribution of adsorbed species on the surface with spatial resolution of about 1μm. Alternatively, ellipsomicroscopy for surface imaging (EMSI) can be used. This method makes use of the fact that, when polarized light is reflected from a surface, its polarization is changed depending on the presence of the adsorbates.

Nonequilibrium pattern formation has been observed in a number of different surface reactions [164]. The most extensively studied surface chemical reaction is the CO oxidation on single platinum crystals. In this reaction, CO molecules react on the catalytic surface with oxygen forming carbon dioxide (CO2) that goes back into the gas phase. This process is of high practical importance, because it is employed in car exhaust catalysts to reduce environment pollution by CO. The net reaction 2CO+O22CO2 follows the Langmuir–Hinshelwood scheme:*+COCOad,2*+O22 Oad,COad+Oad2*+CO2.Here, * denotes a free adsorption site on the catalytic surface. The adsorption of oxygen is dissociative. Adsorbed CO molecules are bound much less strongly to the surface than the oxygen atoms. Therefore, they can desorb and diffuse on the surface. Such processes are negligible for Oad under typical experimental temperatures (below 600 K). At temperatures above 300 K, produced CO2 immediately desorbs into the gas phase, leaving again free space for the adsorbates.

When CO oxidation takes place on perfect surfaces of single Pt crystals, its properties strongly depend on what crystallographic plane is used to run the reaction. On the Pt(1 1 1) surface, the reaction shows only bistability between the mainly oxygen covered, reactive state and the nonreactive CO covered state. This bistability is due to asymmetric inhibition of adsorption. CO molecules can still adsorb on the platinum surface covered by oxygen. In contrast to this, when the surface is completely covered by CO, adsorption of oxygen is not possible and the reaction is poisoned. A two-variable kinetic model, describing bistability and front propagation in the CO oxidation on Pt(1 1 1) was constructed by Bär et al. [175].

When CO oxidation proceeds on the Pt(1 1 0) crystallographic plane, its dynamics is much more rich. Depending on temperature and partial pressures of gaseous reactants, it can show not only bistability, but also oscillations and excitable kinetics. The dynamical behavior becomes complex, because the reaction is then accompanied by structural changes of the metal surface [164]. If platinum atoms in the top layer of metal are ordered as in the bulk of the metal, this would correspond to the 1×1 arrangement. This is indeed the case for the Pt(1 1 1) crystallographic plane. The atoms occupying an open Pt(1 1 0) are however undergoing spontaneous rearrangement into a 1×2 “missing row” geometry through a process known as surface reconstruction.

If CO molecules are adsorbed on the reconstructed Pt(1 1 0) surface and their coverage is high enough, the surface reconstruction becomes lifted and the surface atoms acquire the 1×1 geometry that corresponds to the bulk of the metal. This is an adsorbate-induced phase transition which takes place above a certain critical adsorbate coverage. Such structural surface phase transitions, which are usually much slower than the characteristic reaction rates, can affect the reaction. As a result of an additional inertial feedback, oscillations and excitability become possible.

A similar adsorbate-induced structural phase transition occurs when the reaction takes place on the Pt(1 0 0) plane. In this case, the reconstructed state of the open surface has the “hex” geometry. The CO oxidation reaction on this surface is also accompanied by kinetic oscillations, which are however irregular [164].

Already the first experimental PEEM observations of pattern formation in the CO oxidation reaction on Pt(1 1 0) have revealed a large variety of patterns, including rotating spiral waves, target patterns, standing waves and irregular wave regimes [165], [166]. Spiral waves, observed in the excitable regime (Fig. 4), were usually pinned by material defects of the surface and were therefore characterized by a distribution of rotation periods [167]. However, free spiral waves, that drifted over the surface as a result of intrinsic meandering or reaction parameter variations, could also be seen.

In some parameter regions inside the excitable kinetics domain, solitary traveling wave fragments were observed. Such wave fragments were not always annihilating in front collisions—reflection collisions and fusion events were also observed. These effects, as well the development of standing waves, could be related to the formation of subsurface oxygen in the CO oxidation reaction [168], [169], [170]. The adsorbed oxygen can penetrate from the surface into the next underlying layers of the metal, thus creating an oxygen depot. Later on, such subsurface oxygen can come back to the surface and participate in the oxidation reaction.

The general Langmuir–Hinshelwood scheme corresponds to the mathematical model formulated by Ziff et al. [171]. This abstract model does not, however, take into account the dissociative adsorption of oxygen on platinum and the asymmetry of inhibition, which are important for understanding of the reaction bistability. The first theoretical model taking into account surface reconstruction was proposed [172] for the CO oxidation on Pt(1 0 0). It was formulated in terms of four coupled ordinary differential equations describing the variation of the adsorbate coverages on the hex and 1×1 phases of Pt(1 0 0) and the phase transition between the two substrate phases. The phenomenological four-variable model was reformulated in terms of statistical physics of first-order phase transitions by Andrade et al. [173], [174]. Stochastic lattice models of this reaction have also been constructed [176], [177], [178] (for a review on stochastic modeling of surface reactions, see [179]).

Oscillations and excitable behavior in the CO oxidation reaction on Pt(1 1 0) are well described by the Krischer–Eiswirth–Ertl (KEE) model [180]. This phenomenological mean-field model is formulated in terms of three variables, u, v and w. The first of them, u, is the local CO coverage, i.e., the fraction of CO adsorption sites on the metal surface that are occupied by adsorbed CO molecules. The second variable, v, is the local oxygen coverage. The third variable, wspecifies the local fraction of the surface area occupied by the nonreconstructed 1×1 structural phase. The kinetic equations of the KEE model areut=k1sCOpCO(1-u3)-k2u-k3uv+D2u,vt=k4pO2[sO,1×1w+sO,1×2(1-w)](1-u-v)2-k3uv,wt=k511+exp((u0-u)/δu)-w.The first term in Eq. (9) describes the process of CO adsorption. Here, k1 is the adsorption rate constant, sCO is the sticking coefficient for CO molecules, and pCO is the partial pressure of CO in the gas phase. Instead of being proportional to the fraction of free CO adsorption sites, 1-u, the adsorption rate includes the factor 1-u3. This is done to take phenomenologically into account that the formation of a physical adsorbed state of a CO molecule is preceded by the appearance of a weakly bound precursor state of this molecule at the Pt surface. The second and the third terms in this equation describe desorption of CO and its reaction with adsorbed oxygen molecules. The last term takes into account surface diffusion of adsorbed CO molecules. It was introduced into the model in a subsequent publication by Falcke et al. [181] where traveling waves in this system have been first studied.

Eq. (10) describes the kinetics of adsorbed oxygen. The first term is the adsorption rate depending on the partial pressure pO2 of oxygen molecules. This rate is proportional to (1-u-v)2, which is the square of the fraction of available oxygen adsorption sites. Because adsorption is dissociative, each O2 molecule needs two free adsorption sites on the surface. Moreover, it is taken into account here that adsorbed CO inhibits oxygen adsorption. Therefore, O2 molecules can adsorb only on the sites which are both free of CO and oxygen. The last term in this equation corresponds to the chemical reaction. Under typical experimental temperatures, oxygen is strongly bound to the surface and, therefore, its desorption and surface diffusion can be neglected. The oxygen sticking coefficient depends on the structural state of the surface. It takes values sO,1×1 for the nonreconstructed surface with the 1×1 geometry and sO,1×2 for the reconstructed surface.

Eq. (11) is a phenomenological mean-field description of the phase transition kinetics. The surface free of CO molecules is in the reconstructed 1×2 phase, while the surface completely covered by CO is in the nonreconstructed 1×1 phase. At intermediate CO coverages, a mosaic of microscopic domains of both structural phases occupy the surface. The characteristic sizes of such domain are, however, on the nanometer scale and cannot be resolved in the above mean-field micrometer-scale description. Here, it is simply assumed that, at a fixed CO coverage u, the local fraction w of the surface area in the nonreconstructed phase tends to approach w¯(u)=[1+exp((u0-u)/δu)]-1. Note that, in the original KEE model the function w¯(u) was chosen in the piecewise-linear form; the approximation by a continuous step function was later suggested in Ref. [168]. The variable w approaches the equilibrium value w¯(u) at rate constant k5.

The kinetics of surface reconstruction is slow. First, adsorption of CO takes place on the reconstructed surface (w0), but then the reconstruction becomes slowly lifted and variable w increases. Because the sticking coefficient sO,1×1 of CO on the nonreconstructed surface is however smaller and CO is consumed in the reaction, its coverage decreases. This leads to a reverse structural phase transition. The repetition of such cycles gives rise to kinetic oscillations. Their period is essentially determined by the rate constant k5.

The KEE model can be extended to take into account surface facetting [180] and subsurface oxygen formation [168], [169], which become however important only under certain conditions. For simplicity, anisotropy of diffusion is neglected here. The model parameters have been determined by using the data of various independent measurements and can be found in Ref. [180]. The phase diagram of the KEE model at temperature T=540K is shown in Fig. 5. The dash curve “h” represents the boundary of the Hopf bifurcation, the solid lines “sn” indicate the boundary of the saddle-node bifurcation and the dash-dot line marked as “sniper” corresponds to the saddle-node bifurcation on a limit cycle (saddle node with infinite period). Oscillations are found inside the dashed region in this diagram. They arise through a supercritical Hopf bifurcation and are approximately harmonical near the bifurcation boundary “h”.

When diffusion of CO molecules is taken into account, the state with uniform oscillations can become unstable with respect to phase modulation (the Benjamin–Feir instability), leading to spontaneous development of turbulence. This instability was found by direct numerical determination of the coefficients of the complex Ginzburg–Landau equation for the KEE model along the Hopf bifurcation boundary [182]. Such spontaneous development of turbulence under oscillatory conditions, starting from a uniform state, is indeed observed in the experiments.

A sequence of PEEM images showing the spontaneous development of chemical turbulence from a uniformly oxygen covered surface state is displayed in Fig. 6. A characteristic property of such turbulence is the spontaneous creation of irregular wave fronts and multiple fragments of rotating spiral waves. The spiral waves repeatedly undergo breakups, leading to the formation of new spiral fragments at different locations. Such spatiotemporal chaos is found in a wide range of temperatures for an appropriate choice of the partial pressures of gases in the chamber [183]. The statistical analysis of experimental data (including the statistics of topological defects) reveals that this chemical turbulence is similar to the amplitude turbulence in the complex Ginzburg–Landau equation [184]. Therefore, the CO oxidation reaction on Pt(1 1 0) provides a unique opportunity to experimentally investigate the generic behavior described by this equation.

In studies of the KEE model under excitable conditions, its reduced version with two equations is often used. In this model, the fast variable v, describing the oxygen coverage, is adiabatically eliminated [181]. The effective two-component model is further simplified by replacing a complicated nonlinear function by its piecewise-linear approximation [175]. Thus, it is transformed to a variant of the activator–inhibitor FitzHugh–Nagumo model.

Approximate analytical solutions for solitary excitation pulses and pulse trains in the KEE model have been constructed [181]. Numerical simulations have shown that rotating spiral waves in this model can spontaneously break up, leading to a state of spatiotemporal chaos [185]. An example of such instability of spiral waves is shown in Fig. 7. Statistical properties of such chemical turbulence in excitable media have also been investigated [186]. These results are important in the context of the general theory of pattern formation and spiral wave instabilities in excitable media. Spontaneous breakup of spirals, similar to the process shown in Fig. 7, has been indeed found in the BZ reaction [187]. However, this instability of spiral waves has not been so far observed in the experiments with CO oxidation on Pt(1 1 0). This may be due to the fact that, under excitable conditions, most of the spiral waves are pinned by material impurities and therefore are not subject to this instability, characteristic for freely rotating spirals. Moreover, the rotation frequency of pinned spirals is much higher than that of the freely rotating spirals and, in the course of time, such free spirals (and also turbulence, if it has started to develop) would be ousted by these rapid wave sources. The experiments on control of turbulence in the CO oxidation reaction, which are described later in the review, have been always performed under oscillatory conditions where turbulence spontaneously developed through an instability of the uniform state.

Section snippets

Controlling wave behavior in excitable media

We have seen how chemical waves give rise to target patterns and spiral waves in the 2D media of the BZ reaction and the surface oxidation reaction of CO on Pt. We now examine the manipulation and control of spatiotemporal behavior in 2D and 3D systems by geometrical constraints, medium inhomogeneities, and spatiotemporal perturbations. Our focus in this section is on excitable media; oscillatory media will be discussed Section 3.

Control of waves in oscillatory media

Application of periodic forcing and global feedbacks to oscillatory media allows to create new kinds of wave patterns and to control them by variation of the forcing or feedback parameters. In this Chapter, we shall consider only such media where, without forcing or feedback, uniform oscillations are stable and chemical turbulence is absent. This is characteristic for the experiments with the oscillatory BZ reaction where turbulence does not spontaneously develop. For surface chemical

Control of chemical turbulence

In oscillatory media considered in the previous section, uniform oscillations and waves were intrinsically stable. Therefore, external forcing and global feedback could be used in such systems only to modify the properties of waves and/or create new wave patterns. If the medium intrinsically finds itself in the state of turbulence (spatiotemporal chaos), the primary question is whether and under what conditions this turbulence can be suppressed. Furthermore, periodic forcing and global feedback

Conclusions

We have presented an overview of experimental and theoretical studies of control of spatiotemporal dynamics in chemical systems. Many different approaches for controlling such behavior have been pursued. Some, such as those based on feedback techniques, take advantage of the inherent sensitivity to perturbations displayed by nonlinear dynamical systems, and very small perturbations are typically sufficient to significantly alter the behavior or stabilize a particular desired state. Global

Acknowledgements

We are grateful to G. Ertl for stimulating discussions and wish to thank our friends and collaborators H.H. Rotermund, M. Bertram and V. Zykov for their valuable contributions. During the work on this review, one of us (K.S.) was financially supported by the Alexander von Humbold Foundation (Germany) through a Humboldt research award. A. S. M. acknowledges financial support from the German Science Agency (DFG) in the framework of the Cooperative Research Programme “Complex Nonlinear Processes”

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