PHYSICA FA~E~C'IER

Physica A 263 (1999) 329-337

Nonlinear Phenomena in Heterogeneous Catalysis Alexander S. Mikhailov Abteilung Physikalische Chemic, Fritz-Haber-Institut der Max-Planck-Gesellschafl, Faradayweg 4-6, 14195 Berlin, Germany Email: [email protected]

Abstract

The aim of this talk was to review recent progress in theoretical modeling of pattern formation during catalytic chemical reactions on metal surfaces. The attention was focused on two kinds of nonlinear phenomena at different characteristic length scales. The reaction-diffusion models well describe relatively large nonequilibrium structures, such as rotating spiral waves with the wavelength in the range of tens of micrometers. Much smaller submicrometer and nanoscale nonequilibrium patterns appear as a result of attractive interactions between adsorbed molecules and an interplay between the first-order phase transitions and chemical reactions in the system. These short-scale structures are analyzed in the framework of mesoscopic kinetic equations. PACS: 82.65.Jv; 47.54.+r; 05.40.+j; 82.20.Wt Keywords: chemical kinetics, pattern formation, phase transitions

1

Introduction

Heterogeneous catalysis takes place in systems consisting of gas and a metal catalyst. The gas contains molecules t h a t do not react because of a high energetic barrier. These molecules can however adsorb on the metal surface and diffuse on it. When two adsorbed molecules meet, they can react. The products leave the surface and the reaction continues as long as new molecules are supplied to the reaction chamber. In the experiments studying basic mechanisms of surface reactions, single crystals of metals with pure, perfect surfaces are usually employed. Moreover, partial pressures of reactants are maintained low to reduce thermal effects of the reaction. Since surface chemical reactions proceed in monomolecular adsorbates on top of a perfect metal surface, they can be investigated using microscopy methods. Visualization of reaction patterns on the scales from several micrometers to hundreds of micrometers is best achieved using photoemission electron microscopy (PEEM) based on the change of the local work function for electrons in the presence of adsorbed particles. P E E M studies of various surface reactions have revealed t h a t such reactions are often accompanied by the formation of complex spatiotemporal patterns, such as traveling or standing waves, target patterns, rotating spiral waves and spatiotemporal chaos [1-3]. Generally, these 0378-4371/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S0378-4371(98)00523-8

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A. S. Mikhailov / Physica A 263 (1999) 329-337

reactions can be described as active reaction-diffusion systems (see, e.g., [4]). Excitable, oscillatory and bistable surface chemical reactions are known. The patterns imaged by PEEM are essentially macroscopic. Their smallest details have the dimension of a few micrometers which is still ten thousand times larger than the characteristic lattice constant of a metal. To study reaction patterns on shorter scales, other methods have been developed. The most powerful of them is scanning tunneling microscopy (STM) that yields an atomic resolution. Recent technical improvements of this method have allowed to greatly increase its temporal resolution, so that real-time imaging of single diffusing atoms on metal surfaces can now be performed. Today, first observations of surface chemical reactions in real time with an atomic resolution become possible [5, 6]. This opens a new page in experimental studies of chemical kinetics. The STM investigations show that the microscopic behaviour of surface chemical reactions is complex. Reactive adsorbates never represent ideal two dimensional gases where reactants are perfectly mixed by diffusion. Instead, strong spatial correlations between molecules are found and fluctuating nanoscale patterns are observed. Apparently, a fundamental role in these phenomena is played by lateral interactions between adsorbed atoms. These potential interactions are mediated through the metallic substrate. Adsorption changes quantum wave functions of surface electron states and this leads to an effective attraction or repulsion between the adsorbed particles. A different mechanism of lateral interactions is based on elastic deformations induced by adsorbed particles in the substrate. Because of such lateral interactions, adsorbates represent non-ideal systems with significant correlations between particles. The interactions may be so strong that an adsorbate will undergo a first-order condensation phase transition into a dense phase. Hence, these systems are best viewed as representing soft matter [7]. The task of the theory is to study kinetics of chemical reactions in such strongly correlated systems with phase transitions. Below I show three examples of theoretical investigations motivated by the current experimental research. 2

Complex Anisotropy

Diffusion of adsorbed particles on single crystal surfaces is anisotropic. Moreover, adsorption of particles often changes the crystallographic structure of the top substrate layer which, in turn, influences the diffusion anisotropy. This means that the anisotropy in such systems will be state-dependent. If c is the local adsorbate coverage and w is the variable that specifies the local state of the surface, their evolution is determined by two equations:

Ow

When the diffusion coefficients D~ and Dy are constant (or, more generally, when D ~ / D y = c o n s t ) , the anisotropy can be eliminated by rescaling the coordinates. Therefore, any spatial pattern in a medium with simple diffusion anisotropy can be obtained by stretch-

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ing the respective pattern in the isotropic medium. The situation is however different if D~(w)/Dv(w ) ~ coast. Such complex anisotropy cannot be scaled out and leads to interesting dynamical effects [8-10]. Motion of curved reaction fronts in anisotropic systems is described by an eikonal equation with an angle-dependent normal propagation velocity V0(a). Solving this equation, the shape of the expanding front at any time moment can be computed. If the initial front had a circular shape, in systems with a simple diffusion anisotropy it gradually becomes elliptic. In systems with state-dependent anisotropy the growing fronts will however show deviations from an elliptic shape. The analysis of the eikonal equation furthermore reveals that, when the anisotropy variations are sufficiently strong, an expanding front will develop caustics. This happens because the normal propagation direction in anisotropic media does not coincide with the ray direction and therefore the characteristics may intersect. The formation of a caustic is revealed in the appearance of an edge on the expanding front. Thus, a small initial circular front grows, becomes nonelliptic and then develop sharp edges [9]. As a result, strange square-shaped target patterns seen in the experiments [11] are produced.. In excitable media, propagating waves may be broken and have open ends. Therefore, the eikonal approximation should be modified for such systems to include the law of motion of a wave tip. In the framework of the kinematic approximation [4, 12] this is done by introducing the tangential growth velocity G of the tip that depends as G = Go - 7 K on the wave curvature K at the tip. If Go > 0, a fiat broken wave grows and curls. Then the curvature K gradually increases and the growth is slowed down until it is completely terminated once this curvature has reached its critical values Kc = Go/7. In this manner, a steadily rotating spiral wave is formed. In the opposite case Go < 0, a broken wave shrinks and eventually disappears. When diffusion is anisotropic, the tangential tip velocity is a function of the angle, i.e. Go = G0(a). If the anisotropy is simple and can therefore be scaled out, this function should have the same sign for all possible angles a (indeed, a constant scaling transformation cannot convert growth into contraction). Hence, excitable media with simple anisotropy must have the same principal kinds of patterns as the isotropic systems. The situation is different for systems with complex anisotropy where the tangential tip velocity Go(a) may change its sign depending on the propagation direction. This means that there would be an angle a0 such that G0(a0) = 0 and the wave propagating at angles less than (~0 would grow whereas the waves propagating at angles ~ > ~0 would contract. Solving in this case the kinematic evolution equations, we find that an initial flat fragment begins to grow and to curl, tending to form a spiral. Since curling changes the tip propagation direction, increasing its angle (~, this leads to a decrease in the growth velocity until the growth is stopped when the angle c~0 is reached. Thus, a spiral wave does not develop. Instead, a traveling curved wave fragment is formed that steadily propagates through the medium [10]. Such steadily traveling wave fragments are a new kind of patterns that exist in excitable media with complex anisotropy. These patterns were actually obtained in numerical simulations of the FitzHugh-Nagumo model with complex anisotropy and were observed in an experiment with a catalytic surface reaction [10].

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Phase flips a n d s t r i n g s

A characteristic feature of typical experiments with catalytic surface reactions is the presence of a global feedback. Since mixing in the gas phase is fast, the distribution of gaseous reactants in the chamber remains uniform. Reactions in different surface elements consume molecules from the gas phase and thus change their partial pressures. On the other hand, variations of partial pressures influence the adsorption rates and thus control the reaction processes. Hence, in addition to local diffusional coupling between neighbouring elements of the surface, the system would have global coupling through the gas phase. Effects of global coupling are most important when kinetic oscillations take place. The diffusion process can synchronize local oscillations only within a characteristic diffusion length of about a micrometer. Fluctuations and surface defects produce desynchronization of surface oscillations on larger scales. Hence, the very fact that macroscopic oscillations of reaction rates are recorded in surface chemical reactions indicates that a global feedback is in operation there. General effects of global feedbacks in oscillatory systems can be studied [13] in a model of the complex Ginzburg-Landau equation (CGLE) with global coupling: //= (1 + iw)~l - (1 + i~)lul = ~ + (1 + ie)v2~/+ f ( t ) where the global driving force is collectively produced by oscillations in all surface elements, f ( t ) = # e i X ~ e f "(~'t)d~

The global feedback in this model is specified by its intensity # and its characteristic phase shift X. The modulus p and the phase ¢ of the local complex oscillation amplitude can be introduced as ~(r, t) = p(r, t) exp [ - i (wt + ¢(r, t))]. Depending on the phase shift X, global feedbacks can be positive or negative. A negative feedback destabilizes uniform oscillations (see [14]) and breaks them into separate phase domains (similar to the formation of magnetic domains in ferromagnets). Below we consider only negative global feedbacks and focus our attention on the phenomena related to the development of phase flips and strings in this system. When uniform oscillations are modulationally stable (i.e. the condition 1 + e/~ > 0 is satisfied), propagating one-dimensional patterns of phase flips are possible in the presence of a global feedback (similar phase flips exist in systems with external periodic forcing [15]). In such a pattern, the phase changes by 2r within a narrow coordinate interval of width 5x ~ #-1/2. Inside this interval the modulus of the oscillation amplitude is decreased by 5p ~/~. The pattern propagates at a constant velocity, preserving its shape. Traveling phase flips were observed in experiments with catalytic surface reactions [13] and their properties were analyzed in the framework of CGLE with the global feedback [13-14, 16]. Though a global feedback is necessary to maintain a phase flip, strong feedbacks destroy such patterns as first described in Ref. 16. The destruction of a phase flip in a onedimensional system proceeds through the formation of an amplitude defect. When the feedback intensity is increased, the phase flip becomes unstable and the modulus of the oscillation amplitude begins to decrease in its middle. At some moment, the oscillation

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amplitude r# reaches zero at a point inside the flip region. At such a moment the phase is not defined in this point, and its values on both sides of the amplitude defect get switched by 2~, thus eliminating the phase difference. Later on, the modulus p begins to increase and after a while the system approaches a uniform state. In two-dimensional systems, the phase flips correspond to extended objects representing narrow moving strings. Whenever such a string is crossed, the phase is changed by 2~. Inside the string, the modulus of the oscillation amplitude is decreased. A string carl also be broken so that an open end appears. After going along any closed contour that surrounds the tip, the phase is increased by 27r. This means that the oscillation amplitude must vanish in some point at the string tip and a topological phase defect should be sitting there. Our numerical simulations [16] show that the strings can spontaneously break when the intensity of global feedback is increased. If the string is nonuniformly curved, the breakup occurs in the region with the highest curvature. The modulus of the oscillation amplitude decreases there and drops at some point down to zero. This point then splits into two points that correspond to two separate topological defects. In the region between them, the string is destroyed and oscillations become uniform. Thus, the string is broken into two parts with opposite topological defects sitting at their tips. The behaviour of broken strings in oscillatory media with global coupling has much in common with motion of broken waves in excitable media. Such strings can form spiral waves and destruction of spiral waves in such systems proceeds through the string breaking. Phase flips and strings play an important role in the situations when uniform oscillations are modulationally unstable (1 + ~fl < 0). Application of sufficiently strong negative feedbacks stabilizes in this case the uniform oscillations. At lower feedback intensities, intermittent regimes are however found in such systems. In the one-dimensional case [17, 18], these regimes are characterized by cascades of reproducing phase flips on the background of uniform oscillations (see, e.g., Fig. 5 in Ref. 17). Interesting results are obtained in numerical simulations of intermittent regimes in two-dimensional systems [16]. When the system evolution along a central linear cross-section is displayed, such space-time diagrams look similar to the respective diagrams in the one-dimensional system and show reproduction cascades of phase flip pairs. The examination of subsequent time slices, that correspond to such evolution, reveals that the flips actually represent planar sections of string loops. These approximately circular string loops repeatedly originate in the middle of the cells whose boundaries are formed by shocks with an increased oscillation amplitude. They grow, come closer to the boundaries and break there. Clusters of cells with the repeated generation of string loops give rise to turbulent bubbles surrounded by regions with uniform oscillations. MPEG videos of such intermittent turbulent regimes are available at the web site http://www.fhi-berlin.mpg.de/complsys. A different kind of patterns, also possible in oscillatory media with global feedbacks, are standing waves and cellular structures [16-20]. These patterns result from the action of negative global feedbacks on phase turbulence. They become destabilized when the feedback intensity is decreased. First, periodic breathing of these structures develops and then a chaotic spatiotemporal regime sets on. The bifiircation analysis based on the amplitude equations for cellular structures has recently been performed [20]. Similar cellular structures were observed in experiments with catalytic surface reactions [19].

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The role of global feedbacks has also been investigated for excitable media. The action of such feedbacks on rotating spiral waves has been studied [21]. In the systems with long-range inhibition, it was found that such feedback can stabilize traveling localized structures (spots) [22]. Mutual synchronization in systems of globally coupled chaotic oscillators that may correspond to individual microreactors on the catalytic surface has been considered [23-26]. Effects of global feedbacks are discussed in the short review paper [27].

4

Traveling nanoscale structures

The minimal possible size of spatiotemporal patterns in reaction-diffusion systems is determined by the diffusion length, i.e. by the distance passed by a diffusing particle until it enters a reactive collision. For experimentally studied surface chemical reactions this characteristic length is close to a micrometer. Therefore, pattern formation on submicrometer and nanometer scales in these systems is not possible if only diffusion and reactions are present. To explain the appearance of nanoscale nonequilibrium structures, the existence of attractive interactions between adsorbed particles should be taken into account. These interactions can provide cohesion necessary to localize such structures on the scales shorter than the diffusion length. The kinetic equation, that includes the effects of interactions between the adsorbed particles and remains valid when this system undergoes a first-order phase transition, has been proposed [28]. G. Giacomin and J. L. Lebowitz have shown [29] that this equation (without adsorption and desorption terms) yields an exact mean-field kinetic description of phase separation in binary mixtures in the limit of long-range attractive interactions. Statistical fluctuations become significant at nanoscales and should be included into the kinetic description. This can be done in the framework of mesoscopic kinetic equations formulated in terms of continuous fluctuating concentrations [30, 31]. The mesoscopic kinetic equation for adsorbates with potential interactions between particles has been derived from the microscopic master equation for the joint probability distribution [32]. The derivation was based on coarse-graining over lattice areas that included a large number of lattice sites but were still smaller than the radius of interactions between the particles (the small parameter used in this derivation was the inverse number of lattice sites within the circle of the interaction radius). When adsorption, desorption and reactions are absent, the mesoscopic kinetic equation for the fluctuating coverage c is Oc

0

Ot

Or

(~(l-c)c~--~S.(r-r')c(r')dr')

+D 02c ~ + - 21~ c9 (~/2Dc(l_c)f(r,t)) where u(r) is the binary potential of attractive interactions between the particles. The last term in this equation represents the internal noise of diffusion. It includes the parameter Z given by the number of lattice sites per unit surface area. Here f = {f~, fy} is the vector white noise of unit intensity, i.e. < fx(r,t) f~(r',t) > = < fy(r,t) fy(r',t) > :

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5(r - r')5(t - t') and < fx(r,t) fv(r',t) > = 0. This equation is reduced to the CahnHilliard equation in the vicinity of a critical point, when the interaction radius goes to zero and fluctuations are neglected. Such a mesoscopic kinetic equation (with additional terms) has been used to describe the kinetics of first-order phase transition and stochastic nucleation in the presence of thermal desorption and adsorption [32]. In a recent study [33] a similar equation has further been used to theoretically investigate the formation of stationary Turing-like structures in this system when a nonequilibrium reaction was introduced. In contrast to equilibrium systems where thermal relaxation always leads to stationary states, nonequilibrium systems may also show self-oscillations and may support propagation of waves. An interesting question is whether traveling nanoscale structures exist in systems with chemical reactions. We have recently considered [34] a simple model system with two kinds of molecules A and B and an annihilation reaction A + B ~ 0 taking place on the surface. Adsorption, desorption and surface diffusion of molecules A are included and it is assumed that molecules B adsorb on the surface from the gas phase, but do not desorb or diffuse. Attraction between molecules A and A and between molecules A and B is further taken into account. The model is described by two coupled mesoscopic kinetic equations for coverages a and b of both reacting species: ~U(r)~

Oaot = k,pa(1 - a) - kd0e x p , \ k B T ]

O(k_~

+Orr

OU)

(1 - a)a~r

- krab

02a

+ D-~r 2 + ~(r, t)

Ob O~ = kbPb(1 -- b) - krab + ((r, t) where the local surface potential is

U(r)=-

f u(r-r')a(r')dr'-

f v(r-r')b(r')dr'

Here pa and Pb are constant partial pressures of both reactants in the gas phase; ~(r, t) and ~(r, t) are internal noises. The linear stability stability analysis of uniform stationary states in this system reveals a Hopf bifurcation with a finite wave number that leads to the growth of periodic wave modes. If the interaction radius r0 is much shorter than the diffusion length LdifI, the critical wavelength can be estimated as )~ ~ ~ . Our numerical simulations of the two-dimensional system have shown that this instability leads to the development of traveling waves. When internal noises are added, the waves become broken and traveling nanoscale fragments are produced. 5

Conclusions

The above examples show that systems with surface chemical reactions have a large potential of nonlinear pattern formation. The possibility of direct observation of such reae-

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tions down to the atomic resolution and a relative simplicity of their reaction mechanisms make them attractive for both experimental and theoretical studies. This research has important practical applications. However, catalytic surface reactions may also be viewed as a model system and thus may play a role similar to that of the classical BelousovZhabotinsky reaction where generic properties of nonlinear pattern formation are found. The results of our theoretical studies may be interesting for some biological applications. The complex state-dependent anisotropy may be perhaps found in the cardiac muscle. The analogies between adsorbates and phase-separating polymer blends suggest that similar mechanisms may produce traveling nanoscale structures in these reacting polymer systems. Such extremely small traveling structures would perfectly fit into the characteristic dimensions of a single biological cell. Once biological applications are mentioned, I want to remark that biochemical enzymic reactions may also be viewed as a variant of heterogeneous catalysis. Our recent theoretical investigations of allosterically regulated enzymic reactions in submicrometer volumes indicate the existence of spontaneous synchronization transitions leading to strong microscopic correlations between catalytic turnover cycles of single enzyme molecules [35-40]. References [1] R. Imbihl and G. Ertl, Chem. Rev. 95, 697 (1995) [2] H. H. Rotermund, Surf. Sci. Rep. 29, 267 (1997) [3] H. H. Rotermund, S. Jakubith, A. von Oertzen, and G. Ertl, Phys. Rev. Lett. 66, 3083 (1991) [4] A. S. Mikhailov, Foundations of Syncrgctics. Vol. 1 (Springer, Berlin, 2nd revised ed. 1994) [5] J. Trost, Z. Zambelli, J. Wintterlin, and G. Ertl, Phys. Rev. B 54, 17850 (1996) [6] J. Wintterlin, J. Trost, S. Renisch, R. Schuster, T. Zambelli, and G. Ertl, Surf. Sci. 394, 159 (1997) [7] A. S. Mikhailov and G. Ertl, Science 264, 223 (1994) [8] A. S. Mikhailov and V. S. Zykov "Spiral waves in weakly excitable media" in Chemical Oscillations and Waves , eds. R. Kapral and K. Showalter (Kluwer, Amsterdam 1995) pp.119-162 [9] A. S. Mikhailov, Phys. Rev. E 49, 5875 (1994) [10] F. Mertens, N. Gottschalk, M. B~ir, M. Eiswirth, A. S. Mikhailov, and R. Imbihl, Phys. Rev. E 51, R5193 (1995) [11] F. Mertens and R. Imbihl, Nature (London) 370, 124 (1994) [12] A. S. Mikhailov, V.A. Davydov, and V. S. Zykov, Physica D 70, 1 (1994) [13] G. Veser, F. Mertens, A. S. Mikhailov, and R. Imbihl, Phys. Rev. Lett. 71,935 (1993) [14] F. Mertens, R. Imbihl, and A. S. Mikhailov, J. Chem. Phys. 99, 8668 (1993) [15] P. Coullet and K. Emilsson, Physica A 88, 190 (1992) [16] D. Battogtokh, A. Preusser, and A. S. Mikhailov, Physica D 106, 327 (1997) [17] F. Mertens, R. Imbihl, and A. S. Mikhailov, J. Chem. Phys. 101, 9903 (1994) [18] D. Battogtokh and A. S. Mikhailov, Physica D 90, 84 (1996) [19] K. C. Rose, D. Battogtokh, A. S. Mikhailov, R. Imbihl, W. Engel, and A. Bradshaw, Phys. Rev. Lett. 76, 3582 (1996)

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[20] D. Lima, D. Battogtokh, A. S. Mikhailov, P. Borckmans, and G. Dewel, Europhys. Lett. 42, 631 (1998) [21] V. S. Zykov, A. S. Mikhailov, and S. Miiller, Phys. Rev. Lett., 78, 3398 (1997) [22] K. Krischer and A. S. Mikhailov, Phys. Rev. Lett., 73, 3165 (1994) [23] N. Khrustova, G. Veser, A. S. Mikhailov, and R. Imbihl, Phys. Rev. Lett. 'i'5, 3564 (1995) [24] N. Khrustova, A. S. Mikhailov, and R. Imbihl, J. Chem. Phys. 107, 2096 (1997) [25] D. Zanette and A. S. Mikhailov, Phys. Rev. E 57, 276 (1998) [26] D. Zanette and A. S. Mikhailov, Phys. Rev. E 58, 872 (1998) [27] D. Battogtokh, M. Hildebrand, K. Krischer, and A. S. Mikhailov, Phys. Rep. 288, 435 (1997) [28] A. S. Mikhailov and G. Ertl, Chem. Phys. Lett. 238, 104 (1995); 267, 400 (1997) [29] G. Giacomin and J. L. Lebowitz, Phys. Rev. Lett. 76, 1094 (1996) [30] A. S. Mikhailov, Phys. Rep. 184, 307 (1989) [31] A. S. Mikhailov and A. Yu. Loskutov, Foundations of Synergetics. Vol. 2 (Springer, Berlin, 2nd revised ed. 1996) [32] M. Hildebrand and A. S. Mikhailov, J. Phys. Chem. 100, 19089 (1996) [33] M. Hildebrand, A. S. Mikhailov, and G. Ertl, Phys. Rev. E 58, No. 5 (1998) [34] M. Hildebrand, A. S. Mikhailov, and G. Ertl, Phys. Rev. Lett. 81, 2602 (1998) [35] B. Hess and A. S. Mikhailov, Science 264, 223 (1994) [36] B. Hess and A. S. Mikhailov, J. Theor. Biol. 176, 181 (1995) [37] B. Hess and A. S. Mikhailov, Biophys. Chem. 58, 365 (1996) [38] A. S. Mikhailov and B. Hess, J. Phys. Chem. 100, 19059 (1996) [39] P. Stange, A. S. Mikhailov, and B. Hess, Biophys. Chem. 72, 73 (1998) [40] P. Stange, A. S. Mikhailov, and B. Hess, J. Phys. Chem. B 102, 6273 (1998)

Physica A 263 (1999) 329-337

Nonlinear Phenomena in Heterogeneous Catalysis Alexander S. Mikhailov Abteilung Physikalische Chemic, Fritz-Haber-Institut der Max-Planck-Gesellschafl, Faradayweg 4-6, 14195 Berlin, Germany Email: [email protected]

Abstract

The aim of this talk was to review recent progress in theoretical modeling of pattern formation during catalytic chemical reactions on metal surfaces. The attention was focused on two kinds of nonlinear phenomena at different characteristic length scales. The reaction-diffusion models well describe relatively large nonequilibrium structures, such as rotating spiral waves with the wavelength in the range of tens of micrometers. Much smaller submicrometer and nanoscale nonequilibrium patterns appear as a result of attractive interactions between adsorbed molecules and an interplay between the first-order phase transitions and chemical reactions in the system. These short-scale structures are analyzed in the framework of mesoscopic kinetic equations. PACS: 82.65.Jv; 47.54.+r; 05.40.+j; 82.20.Wt Keywords: chemical kinetics, pattern formation, phase transitions

1

Introduction

Heterogeneous catalysis takes place in systems consisting of gas and a metal catalyst. The gas contains molecules t h a t do not react because of a high energetic barrier. These molecules can however adsorb on the metal surface and diffuse on it. When two adsorbed molecules meet, they can react. The products leave the surface and the reaction continues as long as new molecules are supplied to the reaction chamber. In the experiments studying basic mechanisms of surface reactions, single crystals of metals with pure, perfect surfaces are usually employed. Moreover, partial pressures of reactants are maintained low to reduce thermal effects of the reaction. Since surface chemical reactions proceed in monomolecular adsorbates on top of a perfect metal surface, they can be investigated using microscopy methods. Visualization of reaction patterns on the scales from several micrometers to hundreds of micrometers is best achieved using photoemission electron microscopy (PEEM) based on the change of the local work function for electrons in the presence of adsorbed particles. P E E M studies of various surface reactions have revealed t h a t such reactions are often accompanied by the formation of complex spatiotemporal patterns, such as traveling or standing waves, target patterns, rotating spiral waves and spatiotemporal chaos [1-3]. Generally, these 0378-4371/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S0378-4371(98)00523-8

330

A. S. Mikhailov / Physica A 263 (1999) 329-337

reactions can be described as active reaction-diffusion systems (see, e.g., [4]). Excitable, oscillatory and bistable surface chemical reactions are known. The patterns imaged by PEEM are essentially macroscopic. Their smallest details have the dimension of a few micrometers which is still ten thousand times larger than the characteristic lattice constant of a metal. To study reaction patterns on shorter scales, other methods have been developed. The most powerful of them is scanning tunneling microscopy (STM) that yields an atomic resolution. Recent technical improvements of this method have allowed to greatly increase its temporal resolution, so that real-time imaging of single diffusing atoms on metal surfaces can now be performed. Today, first observations of surface chemical reactions in real time with an atomic resolution become possible [5, 6]. This opens a new page in experimental studies of chemical kinetics. The STM investigations show that the microscopic behaviour of surface chemical reactions is complex. Reactive adsorbates never represent ideal two dimensional gases where reactants are perfectly mixed by diffusion. Instead, strong spatial correlations between molecules are found and fluctuating nanoscale patterns are observed. Apparently, a fundamental role in these phenomena is played by lateral interactions between adsorbed atoms. These potential interactions are mediated through the metallic substrate. Adsorption changes quantum wave functions of surface electron states and this leads to an effective attraction or repulsion between the adsorbed particles. A different mechanism of lateral interactions is based on elastic deformations induced by adsorbed particles in the substrate. Because of such lateral interactions, adsorbates represent non-ideal systems with significant correlations between particles. The interactions may be so strong that an adsorbate will undergo a first-order condensation phase transition into a dense phase. Hence, these systems are best viewed as representing soft matter [7]. The task of the theory is to study kinetics of chemical reactions in such strongly correlated systems with phase transitions. Below I show three examples of theoretical investigations motivated by the current experimental research. 2

Complex Anisotropy

Diffusion of adsorbed particles on single crystal surfaces is anisotropic. Moreover, adsorption of particles often changes the crystallographic structure of the top substrate layer which, in turn, influences the diffusion anisotropy. This means that the anisotropy in such systems will be state-dependent. If c is the local adsorbate coverage and w is the variable that specifies the local state of the surface, their evolution is determined by two equations:

Ow

When the diffusion coefficients D~ and Dy are constant (or, more generally, when D ~ / D y = c o n s t ) , the anisotropy can be eliminated by rescaling the coordinates. Therefore, any spatial pattern in a medium with simple diffusion anisotropy can be obtained by stretch-

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ing the respective pattern in the isotropic medium. The situation is however different if D~(w)/Dv(w ) ~ coast. Such complex anisotropy cannot be scaled out and leads to interesting dynamical effects [8-10]. Motion of curved reaction fronts in anisotropic systems is described by an eikonal equation with an angle-dependent normal propagation velocity V0(a). Solving this equation, the shape of the expanding front at any time moment can be computed. If the initial front had a circular shape, in systems with a simple diffusion anisotropy it gradually becomes elliptic. In systems with state-dependent anisotropy the growing fronts will however show deviations from an elliptic shape. The analysis of the eikonal equation furthermore reveals that, when the anisotropy variations are sufficiently strong, an expanding front will develop caustics. This happens because the normal propagation direction in anisotropic media does not coincide with the ray direction and therefore the characteristics may intersect. The formation of a caustic is revealed in the appearance of an edge on the expanding front. Thus, a small initial circular front grows, becomes nonelliptic and then develop sharp edges [9]. As a result, strange square-shaped target patterns seen in the experiments [11] are produced.. In excitable media, propagating waves may be broken and have open ends. Therefore, the eikonal approximation should be modified for such systems to include the law of motion of a wave tip. In the framework of the kinematic approximation [4, 12] this is done by introducing the tangential growth velocity G of the tip that depends as G = Go - 7 K on the wave curvature K at the tip. If Go > 0, a fiat broken wave grows and curls. Then the curvature K gradually increases and the growth is slowed down until it is completely terminated once this curvature has reached its critical values Kc = Go/7. In this manner, a steadily rotating spiral wave is formed. In the opposite case Go < 0, a broken wave shrinks and eventually disappears. When diffusion is anisotropic, the tangential tip velocity is a function of the angle, i.e. Go = G0(a). If the anisotropy is simple and can therefore be scaled out, this function should have the same sign for all possible angles a (indeed, a constant scaling transformation cannot convert growth into contraction). Hence, excitable media with simple anisotropy must have the same principal kinds of patterns as the isotropic systems. The situation is different for systems with complex anisotropy where the tangential tip velocity Go(a) may change its sign depending on the propagation direction. This means that there would be an angle a0 such that G0(a0) = 0 and the wave propagating at angles less than (~0 would grow whereas the waves propagating at angles ~ > ~0 would contract. Solving in this case the kinematic evolution equations, we find that an initial flat fragment begins to grow and to curl, tending to form a spiral. Since curling changes the tip propagation direction, increasing its angle (~, this leads to a decrease in the growth velocity until the growth is stopped when the angle c~0 is reached. Thus, a spiral wave does not develop. Instead, a traveling curved wave fragment is formed that steadily propagates through the medium [10]. Such steadily traveling wave fragments are a new kind of patterns that exist in excitable media with complex anisotropy. These patterns were actually obtained in numerical simulations of the FitzHugh-Nagumo model with complex anisotropy and were observed in an experiment with a catalytic surface reaction [10].

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Phase flips a n d s t r i n g s

A characteristic feature of typical experiments with catalytic surface reactions is the presence of a global feedback. Since mixing in the gas phase is fast, the distribution of gaseous reactants in the chamber remains uniform. Reactions in different surface elements consume molecules from the gas phase and thus change their partial pressures. On the other hand, variations of partial pressures influence the adsorption rates and thus control the reaction processes. Hence, in addition to local diffusional coupling between neighbouring elements of the surface, the system would have global coupling through the gas phase. Effects of global coupling are most important when kinetic oscillations take place. The diffusion process can synchronize local oscillations only within a characteristic diffusion length of about a micrometer. Fluctuations and surface defects produce desynchronization of surface oscillations on larger scales. Hence, the very fact that macroscopic oscillations of reaction rates are recorded in surface chemical reactions indicates that a global feedback is in operation there. General effects of global feedbacks in oscillatory systems can be studied [13] in a model of the complex Ginzburg-Landau equation (CGLE) with global coupling: //= (1 + iw)~l - (1 + i~)lul = ~ + (1 + ie)v2~/+ f ( t ) where the global driving force is collectively produced by oscillations in all surface elements, f ( t ) = # e i X ~ e f "(~'t)d~

The global feedback in this model is specified by its intensity # and its characteristic phase shift X. The modulus p and the phase ¢ of the local complex oscillation amplitude can be introduced as ~(r, t) = p(r, t) exp [ - i (wt + ¢(r, t))]. Depending on the phase shift X, global feedbacks can be positive or negative. A negative feedback destabilizes uniform oscillations (see [14]) and breaks them into separate phase domains (similar to the formation of magnetic domains in ferromagnets). Below we consider only negative global feedbacks and focus our attention on the phenomena related to the development of phase flips and strings in this system. When uniform oscillations are modulationally stable (i.e. the condition 1 + e/~ > 0 is satisfied), propagating one-dimensional patterns of phase flips are possible in the presence of a global feedback (similar phase flips exist in systems with external periodic forcing [15]). In such a pattern, the phase changes by 2r within a narrow coordinate interval of width 5x ~ #-1/2. Inside this interval the modulus of the oscillation amplitude is decreased by 5p ~/~. The pattern propagates at a constant velocity, preserving its shape. Traveling phase flips were observed in experiments with catalytic surface reactions [13] and their properties were analyzed in the framework of CGLE with the global feedback [13-14, 16]. Though a global feedback is necessary to maintain a phase flip, strong feedbacks destroy such patterns as first described in Ref. 16. The destruction of a phase flip in a onedimensional system proceeds through the formation of an amplitude defect. When the feedback intensity is increased, the phase flip becomes unstable and the modulus of the oscillation amplitude begins to decrease in its middle. At some moment, the oscillation

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amplitude r# reaches zero at a point inside the flip region. At such a moment the phase is not defined in this point, and its values on both sides of the amplitude defect get switched by 2~, thus eliminating the phase difference. Later on, the modulus p begins to increase and after a while the system approaches a uniform state. In two-dimensional systems, the phase flips correspond to extended objects representing narrow moving strings. Whenever such a string is crossed, the phase is changed by 2~. Inside the string, the modulus of the oscillation amplitude is decreased. A string carl also be broken so that an open end appears. After going along any closed contour that surrounds the tip, the phase is increased by 27r. This means that the oscillation amplitude must vanish in some point at the string tip and a topological phase defect should be sitting there. Our numerical simulations [16] show that the strings can spontaneously break when the intensity of global feedback is increased. If the string is nonuniformly curved, the breakup occurs in the region with the highest curvature. The modulus of the oscillation amplitude decreases there and drops at some point down to zero. This point then splits into two points that correspond to two separate topological defects. In the region between them, the string is destroyed and oscillations become uniform. Thus, the string is broken into two parts with opposite topological defects sitting at their tips. The behaviour of broken strings in oscillatory media with global coupling has much in common with motion of broken waves in excitable media. Such strings can form spiral waves and destruction of spiral waves in such systems proceeds through the string breaking. Phase flips and strings play an important role in the situations when uniform oscillations are modulationally unstable (1 + ~fl < 0). Application of sufficiently strong negative feedbacks stabilizes in this case the uniform oscillations. At lower feedback intensities, intermittent regimes are however found in such systems. In the one-dimensional case [17, 18], these regimes are characterized by cascades of reproducing phase flips on the background of uniform oscillations (see, e.g., Fig. 5 in Ref. 17). Interesting results are obtained in numerical simulations of intermittent regimes in two-dimensional systems [16]. When the system evolution along a central linear cross-section is displayed, such space-time diagrams look similar to the respective diagrams in the one-dimensional system and show reproduction cascades of phase flip pairs. The examination of subsequent time slices, that correspond to such evolution, reveals that the flips actually represent planar sections of string loops. These approximately circular string loops repeatedly originate in the middle of the cells whose boundaries are formed by shocks with an increased oscillation amplitude. They grow, come closer to the boundaries and break there. Clusters of cells with the repeated generation of string loops give rise to turbulent bubbles surrounded by regions with uniform oscillations. MPEG videos of such intermittent turbulent regimes are available at the web site http://www.fhi-berlin.mpg.de/complsys. A different kind of patterns, also possible in oscillatory media with global feedbacks, are standing waves and cellular structures [16-20]. These patterns result from the action of negative global feedbacks on phase turbulence. They become destabilized when the feedback intensity is decreased. First, periodic breathing of these structures develops and then a chaotic spatiotemporal regime sets on. The bifiircation analysis based on the amplitude equations for cellular structures has recently been performed [20]. Similar cellular structures were observed in experiments with catalytic surface reactions [19].

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The role of global feedbacks has also been investigated for excitable media. The action of such feedbacks on rotating spiral waves has been studied [21]. In the systems with long-range inhibition, it was found that such feedback can stabilize traveling localized structures (spots) [22]. Mutual synchronization in systems of globally coupled chaotic oscillators that may correspond to individual microreactors on the catalytic surface has been considered [23-26]. Effects of global feedbacks are discussed in the short review paper [27].

4

Traveling nanoscale structures

The minimal possible size of spatiotemporal patterns in reaction-diffusion systems is determined by the diffusion length, i.e. by the distance passed by a diffusing particle until it enters a reactive collision. For experimentally studied surface chemical reactions this characteristic length is close to a micrometer. Therefore, pattern formation on submicrometer and nanometer scales in these systems is not possible if only diffusion and reactions are present. To explain the appearance of nanoscale nonequilibrium structures, the existence of attractive interactions between adsorbed particles should be taken into account. These interactions can provide cohesion necessary to localize such structures on the scales shorter than the diffusion length. The kinetic equation, that includes the effects of interactions between the adsorbed particles and remains valid when this system undergoes a first-order phase transition, has been proposed [28]. G. Giacomin and J. L. Lebowitz have shown [29] that this equation (without adsorption and desorption terms) yields an exact mean-field kinetic description of phase separation in binary mixtures in the limit of long-range attractive interactions. Statistical fluctuations become significant at nanoscales and should be included into the kinetic description. This can be done in the framework of mesoscopic kinetic equations formulated in terms of continuous fluctuating concentrations [30, 31]. The mesoscopic kinetic equation for adsorbates with potential interactions between particles has been derived from the microscopic master equation for the joint probability distribution [32]. The derivation was based on coarse-graining over lattice areas that included a large number of lattice sites but were still smaller than the radius of interactions between the particles (the small parameter used in this derivation was the inverse number of lattice sites within the circle of the interaction radius). When adsorption, desorption and reactions are absent, the mesoscopic kinetic equation for the fluctuating coverage c is Oc

0

Ot

Or

(~(l-c)c~--~S.(r-r')c(r')dr')

+D 02c ~ + - 21~ c9 (~/2Dc(l_c)f(r,t)) where u(r) is the binary potential of attractive interactions between the particles. The last term in this equation represents the internal noise of diffusion. It includes the parameter Z given by the number of lattice sites per unit surface area. Here f = {f~, fy} is the vector white noise of unit intensity, i.e. < fx(r,t) f~(r',t) > = < fy(r,t) fy(r',t) > :

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5(r - r')5(t - t') and < fx(r,t) fv(r',t) > = 0. This equation is reduced to the CahnHilliard equation in the vicinity of a critical point, when the interaction radius goes to zero and fluctuations are neglected. Such a mesoscopic kinetic equation (with additional terms) has been used to describe the kinetics of first-order phase transition and stochastic nucleation in the presence of thermal desorption and adsorption [32]. In a recent study [33] a similar equation has further been used to theoretically investigate the formation of stationary Turing-like structures in this system when a nonequilibrium reaction was introduced. In contrast to equilibrium systems where thermal relaxation always leads to stationary states, nonequilibrium systems may also show self-oscillations and may support propagation of waves. An interesting question is whether traveling nanoscale structures exist in systems with chemical reactions. We have recently considered [34] a simple model system with two kinds of molecules A and B and an annihilation reaction A + B ~ 0 taking place on the surface. Adsorption, desorption and surface diffusion of molecules A are included and it is assumed that molecules B adsorb on the surface from the gas phase, but do not desorb or diffuse. Attraction between molecules A and A and between molecules A and B is further taken into account. The model is described by two coupled mesoscopic kinetic equations for coverages a and b of both reacting species: ~U(r)~

Oaot = k,pa(1 - a) - kd0e x p , \ k B T ]

O(k_~

+Orr

OU)

(1 - a)a~r

- krab

02a

+ D-~r 2 + ~(r, t)

Ob O~ = kbPb(1 -- b) - krab + ((r, t) where the local surface potential is

U(r)=-

f u(r-r')a(r')dr'-

f v(r-r')b(r')dr'

Here pa and Pb are constant partial pressures of both reactants in the gas phase; ~(r, t) and ~(r, t) are internal noises. The linear stability stability analysis of uniform stationary states in this system reveals a Hopf bifurcation with a finite wave number that leads to the growth of periodic wave modes. If the interaction radius r0 is much shorter than the diffusion length LdifI, the critical wavelength can be estimated as )~ ~ ~ . Our numerical simulations of the two-dimensional system have shown that this instability leads to the development of traveling waves. When internal noises are added, the waves become broken and traveling nanoscale fragments are produced. 5

Conclusions

The above examples show that systems with surface chemical reactions have a large potential of nonlinear pattern formation. The possibility of direct observation of such reae-

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tions down to the atomic resolution and a relative simplicity of their reaction mechanisms make them attractive for both experimental and theoretical studies. This research has important practical applications. However, catalytic surface reactions may also be viewed as a model system and thus may play a role similar to that of the classical BelousovZhabotinsky reaction where generic properties of nonlinear pattern formation are found. The results of our theoretical studies may be interesting for some biological applications. The complex state-dependent anisotropy may be perhaps found in the cardiac muscle. The analogies between adsorbates and phase-separating polymer blends suggest that similar mechanisms may produce traveling nanoscale structures in these reacting polymer systems. Such extremely small traveling structures would perfectly fit into the characteristic dimensions of a single biological cell. Once biological applications are mentioned, I want to remark that biochemical enzymic reactions may also be viewed as a variant of heterogeneous catalysis. Our recent theoretical investigations of allosterically regulated enzymic reactions in submicrometer volumes indicate the existence of spontaneous synchronization transitions leading to strong microscopic correlations between catalytic turnover cycles of single enzyme molecules [35-40]. References [1] R. Imbihl and G. Ertl, Chem. Rev. 95, 697 (1995) [2] H. H. Rotermund, Surf. Sci. Rep. 29, 267 (1997) [3] H. H. Rotermund, S. Jakubith, A. von Oertzen, and G. Ertl, Phys. Rev. Lett. 66, 3083 (1991) [4] A. S. Mikhailov, Foundations of Syncrgctics. Vol. 1 (Springer, Berlin, 2nd revised ed. 1994) [5] J. Trost, Z. Zambelli, J. Wintterlin, and G. Ertl, Phys. Rev. B 54, 17850 (1996) [6] J. Wintterlin, J. Trost, S. Renisch, R. Schuster, T. Zambelli, and G. Ertl, Surf. Sci. 394, 159 (1997) [7] A. S. Mikhailov and G. Ertl, Science 264, 223 (1994) [8] A. S. Mikhailov and V. S. Zykov "Spiral waves in weakly excitable media" in Chemical Oscillations and Waves , eds. R. Kapral and K. Showalter (Kluwer, Amsterdam 1995) pp.119-162 [9] A. S. Mikhailov, Phys. Rev. E 49, 5875 (1994) [10] F. Mertens, N. Gottschalk, M. B~ir, M. Eiswirth, A. S. Mikhailov, and R. Imbihl, Phys. Rev. E 51, R5193 (1995) [11] F. Mertens and R. Imbihl, Nature (London) 370, 124 (1994) [12] A. S. Mikhailov, V.A. Davydov, and V. S. Zykov, Physica D 70, 1 (1994) [13] G. Veser, F. Mertens, A. S. Mikhailov, and R. Imbihl, Phys. Rev. Lett. 71,935 (1993) [14] F. Mertens, R. Imbihl, and A. S. Mikhailov, J. Chem. Phys. 99, 8668 (1993) [15] P. Coullet and K. Emilsson, Physica A 88, 190 (1992) [16] D. Battogtokh, A. Preusser, and A. S. Mikhailov, Physica D 106, 327 (1997) [17] F. Mertens, R. Imbihl, and A. S. Mikhailov, J. Chem. Phys. 101, 9903 (1994) [18] D. Battogtokh and A. S. Mikhailov, Physica D 90, 84 (1996) [19] K. C. Rose, D. Battogtokh, A. S. Mikhailov, R. Imbihl, W. Engel, and A. Bradshaw, Phys. Rev. Lett. 76, 3582 (1996)

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