Laser-Induced Target Patterns in the Oscillatory CO Oxidation on Pt

The formation of target patterns in oscillatory CO oxidation on Pt(110) is studied using the controlled creation of temperature heterogeneities on the...
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J. Phys. Chem. B 2004, 108, 14282-14291

Laser-Induced Target Patterns in the Oscillatory CO Oxidation on Pt(110)† Janpeter Wolff,‡ Michael Stich,‡,§ Carsten Beta,‡ and Harm Hinrich Rotermund*,‡ Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany, and Instituto Pluridisciplinar, UniVersidad Complutense de Madrid, Paseo Juan XXIII 1, E-28040 Madrid, Spain ReceiVed: January 14, 2004

The formation of target patterns in oscillatory CO oxidation on Pt(110) is studied using the controlled creation of temperature heterogeneities on the catalyst surface induced by a focused laser beam. Outward and, for the first time, inward traveling target waves are observed as well as target patterns that are spatially confined to the immediate vicinity of the temperature inhomogeneity. The experimental results are compared to theoretical studies of the complex Ginzburg-Landau equation with a nonuniform frequency distribution and to numerical simulations of the Krischer-Eiswirth-Ertl model for the catalytic CO oxidation on Pt(110). Both the complex Ginzburg-Landau equation and the kinetic model need to be extended by a global coupling term to obtain good agreement with the whole range of experimental observations.

1. Introduction Chemical reaction-diffusion systems far from thermal equilibrium are known to have kinetic regimes where complex spatiotemporal behavior can be observed.1 Well-studied reaction-diffusion systems are the Belousov-Zhabotinsky (BZ) reaction or the catalytic oxidation of CO on platinum surfaces, where propagating pulses, rotating spiral waves, and other regular or chaotic spatiotemporal patterns are found.2 While the BZ reaction is an example for a homogeneously catalyzed reaction in the liquid phase, the oxidation reaction of CO on Pt(110) is paradigmatic for the important class of heterogeneous catalytic surface reactions, as pointed out by Ertl and co-workers.3,4 One of the most prominent wave patterns is the target pattern, reported already in 1970 by Zaikin and Zhabotinsky in the first publication on pattern formation in the BZ reaction.5 A target pattern consists of concentric waves that are periodically emitted from a small central region, called pacemaker. Stable pacemakers and target patterns may already be found in uniform reaction-diffusion systems, resulting only from the interplay of nonlinear kinetics and diffusion.6,7 Nevertheless, the majority of target patterns observed in chemical systems is associated with the presence of local heterogeneities. In typical experiments of the BZ reaction, target patterns have a range of operation frequencies, indicating that the oscillation frequency is not uniquely determined by the parameters of the medium, as would be expected for selforganized pacemakers, but depends on local properties of the system. In experiments of the BZ reaction, filtering of the solution and avoiding the production of CO2 decrease the number of observed target patterns. Furthermore, if the activity of a pacemaker is first suppressed by another wave source, which is subsequently removed, the pacemaker often reappears with the same frequency at the same location. Therefore, it is conjectured that many pacemakers in the BZ reaction are formed by small local impurities such as dust particles or gas bubbles, which appear in a random way. The properties of such †

Part of the special issue “Gerhard Ertl Festschrift”. * Corresponding author. E-mail: [email protected]. ‡ Fritz-Haber-Institut der Max-Planck-Gesellschaft. § Instituto Pluridisciplinar, Universidad Complutense de Madrid.

heterogeneous pacemakers and their target wave patterns have been analyzed in a number of studies, mostly in the context of the BZ reaction8-14 or the complex Ginzburg-Landau equation,15-18 a general model for oscillatory media. However, no systematic study of target patterns and pacemakers in the CO oxidation reaction has been reported until now. In the pattern-forming regime of catalytic CO oxidation on Pt(110), the emergence of target patterns is a common feature and was already reported by Jakubith et al. together with the first spatially resolved observations of concentration patterns on the catalytic surface.19 As in other chemical systems, these target patterns are attributed to the presence of local heterogeneities, in particular to defects on the catalytic single-crystal surface.20 The lack of a systematic study of heterogeneous pacemakers in catalytic CO oxidation until today is mostly due to the fact that the nature of these micrometer scale defects and their microscopic details are largely unknown and therefore difficult to control in the experiment. Recently, we developed a new approach to locally influence and control the spatiotemporal dynamics on the catalytic surface using a focused laser beam.21-23 The laser induces a local rise in surface temperature which results in a change of the kinetic parameters at the location of the laser spot, thus creating a local heterogeneity in the medium. The inhomogeneity can act as a pacemaker that starts to emit target waves. This technique is particularly well suited for a systematic study of heterogeneous pacemakers in catalytic CO oxidation since their properties can be easily controlled by focus size and power of the laser beam. In this article, we present both experimental and numerical results on heterogeneous pacemakers created by a laser beam in catalytic CO oxidation. This article is organized as follows. We start with describing our experimental observations of target pattern formation during the CO oxidation on Pt(110) produced by a focused laser beam. Aside from outward and inward traveling target wave patterns that extend over the whole (observable) surface, outward traveling target waves, which remain spatially confined, are described (Section 2). To explain the findings qualitatively, simulations based on the complex Ginzburg-Landau equation (CGLE) are presented in Section 3. To achieve a more detailed agreement, numerical results for

10.1021/jp0498015 CCC: $27.50 © 2004 American Chemical Society Published on Web 05/29/2004

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Figure 1. (a) Snapshot showing a developed target pattern emanating from a temperature inhomogeneity marked by the arrow. Experimental conditions: pCO ) 1.0 × 10-4 mbar, pO2 ) 3.0 × 10-4 mbar, T ) 541 K, laser power Lp ) 1 W, imaged area 0.9 × 1.1 mm. (b) Space-time diagram (7.3 s) taken along the dashed line in (a). (c) Snapshot showing an inward traveling target pattern. The arrow again marks the position of the laser spot. pCO ) 9.1 × 10-5 mbar, pO2 ) 3.0 × 10-4 mbar, T ) 548 K, laser power Lp ) 3 W, imaged area 1.3 × 1.1 mm. (d) Space-time diagram (12.6 s) taken along the dashed line in (c). The black arrow marks the time, when the CO partial pressure was reduced by about 1% to the value given in (c). The vertical white line is a guide to the eye to demonstrate that the waves are indeed traveling inward albeit quite fast. (e) Snapshots showing one cycle of a spatially confined outward traveling target pattern. pCO ) 1.0 × 10-4 mbar, pO2 ) 3.0 × 10-4 mbar, T ) 548 K, laser power Lp ) 3 W, imaged area 0.5 × 0.5 mm. (f) Space-time diagram (12.9 s) taken along the line marked in the first frame of (e).

the Krischer-Eiswirth-Ertl model of CO oxidation are reported in Section 4. The article ends with a discussion of the obtained results (Section 5). 2. Experiments 2.1. The Experimental Setup. The experiments were performed in an ultrahigh-vacuum chamber equipped with a lowenergy electron diffraction (LEED) system, an Ar-ion sputtering gun, and sample heating (from the backside) via a halogen lamp. The 10-mm diameter Pt(110) single-crystal sample was prepared by repeated cycles of Ar-ion sputtering and oxygen treatment at 570 K and subsequent annealing up to 1000 K. Adsorbate concentration patterns on the surface of the sample were imaged using ellipsomicroscopy for surface imaging (EMSI).24 The reaction proceeds via Oad + COad f CO2v, whereby, under the experimental conditions chosen, dark areas in the images reflect high concentrations of adsorbed oxygen, while on the brighter areas the adsorbed CO dominates. To achieve local heating of the surface, the beam of an Arion laser was focused onto a spot with a diameter of about 50 µm. Typical maximum temperature increases, at the center of the heated spot, were of the order of 10 K for 1 W of laser power. The maximum power used during the repeated experiments was 3 W. For the results presented below, conditions (partial pressures and temperature) were chosen such that the reaction showed oscillatory behavior, in particular either uniform oscillations or wave patterns. 2.2. Experimental Results. Depending on the experimental conditions, we observed three different types of target patterns emanating from the artificially created defect s the temperature heterogeneity created by the laser spot. One type was found under conditions where the undisturbed reaction developed

Figure 2. Frequency of the uniform, globally coupled oscillations at different sample temperatures. Experimental conditions: pCO ) 1 × 10-4 mbar, pO2 ) 3.0 × 10-4 mbar.

irregular wave patterns; two additional types appeared when the reaction was undergoing uniform, globally coupled oscillations. (See Supporting Information.) Examples and spacetime diagrams for all three types are shown in Figure 1. Also the effect of a temperature rise on the frequency of oscillations was analyzed. Figure 2 shows frequency data collected from experiments for a constant setting of the partial pressures with variation of the reaction temperature. Figure 1(a) shows a target pattern that develops starting from the location of the laser spot amidst an irregular wave pattern that already existed on the sample. This target pattern consists of outward traveling waves and it grows slowly, thus superseding the other patterns. Figure 1(b) shows a space-time diagram taken along the dashed line in Figure 1(a). The two dark horizontal zones at the top and bottom can be identifiedsby comparison with Figure 1(a) s as the borders of the target

14284 J. Phys. Chem. B, Vol. 108, No. 38, 2004 pattern. The small dark lines in the middle mark the position of the laser spot. Due to the use of background subtraction, the heated spot appears dark as the subtracted image is very bright at that position. The space-time diagram clearly shows the slow growing of the target pattern. The frequency of the oscillations close to the center is about 2.6 Hz, while the oscillations outside the target pattern are 0.3 Hz slower. Therefore, the artificially created target pattern grows and “devours” the other patterns. As the center emits more waves per unit time and as colliding waves annihilate each other, every wave emanating from the artificial defect will make it roughly 1/eighth wavelength farther into the territory of the other patterns (assuming constant propagation speeds) before meeting a wave front traveling in the opposite direction and annihilating again. Thus, a propagation speed for the edge of the target pattern can be measured. It is determined to be about 1.3 µm s-1 (in the direction of fast diffusion). The elliptical shape of the target pattern s due to anisotropic diffusion s and the relatively small velocity of the waves indicate that the propagation of waves is associated with the diffusion of species on the surface. The patterns shown in Figure 1(c-f) are found at the borders of a wide parameter region, where uniform oscillations are stable even though the laser spot is creating a defect. This is probably due to global coupling through the gas phase. Although an increased sample temperature is usually associated with a higher oscillation frequency for uniform, globally coupled oscillations (as shown in Figure 2), the combination of global coupling and local heating does not necessarily lead to the formation of a pacemaker at the heated spot. This will be discussed in more detail in Section 3.2. Here, we report the first observation of inward traveling target waves during the CO oxidation. They can only be found in a narrow parameter range where the region for stable global oscillations ends toward low CO partial pressure. Figure 1(c) shows a snapshot of such a pattern. Figure 1(d) displays the corresponding space-time diagram, that includes the point in time, marked by a black arrow, when the CO partial pressure was reduced by 1% to pCO ) 9.1 × 10-5 mbar. Shortly after this reduction ( 0. All experiments discussed in this article show positive wave dispersion. In such a system, any local frequency increase leads to an extended wave pattern and the heterogeneity therefore constitutes a pacemaker. Accordingly, any local frequency decrease gives rise to a strictly localized wave pattern and the heterogeneity represents a waVe sink.18 For simple cases, k can be determined

V)

Ω1 - Ω2 k 1 + k2

(4)

If the values for the frequencies and wavenumbers are extracted from the experiment, eq 4 predicts the interface velocity to be V ) 1.1 µm s-1, while its experimentally determined value is V ) 1.3 µm s-1. The difference may be due to the anharmonicity of the waves in the CO oxidation system, to a slow drift of the system, and to the fact that the collision zone is a rather broad area making the determination of frequencies and wavelengths of colliding waves difficult. Nevertheless, the good agreement between the two results underlines the significance of the CGLE for describing oscillatory systems even far from the Hopf bifurcation. 3.2. The CGLE including Global Coupling. As discussed in Section 2, the target patterns shown in Figure 1(c-f) are found close to a parameter regime where uniform oscillations, globally coupled through the gas phase, are stable. Such a situation can be modeled by adding an appropriate feedback term to the CGLE.27 The resulting globally coupled CGLE (gcCGLE) reads

∂A ) (1 - iω(x))A - (1 + iR)| A|2A + (1 + iβ)∇2A - f (A) ∂t (5) The feedback term f(A) is given by

h f(A) ) µeiχA

(6)

where A h ) (1/L)∫L0 (ReA + iImA)dx denotes the spatial average of the complex oscillation amplitude, L is the system size, µ determines the coupling strength, and χ characterizes the phase shift between the amplitude and its spatial average. The local frequency shift is again given by eq 2. Pattern formation in the gcCGLE in the presence of a heterogeneity has already been considered before,27,28 and some relevant results are reviewed here. The frequency of globally coupled uniform oscillations in a system without heterogeneity is determined by

Ωg ) Ω0 + µ(sin χ - R cos χ)

(7)

For vanishing global coupling, µ ) 0, this reduces to the frequency Ω0 of uniform oscillations in the standard CGLE. Note that the globally coupled uniform oscillations (below referred to as global oscillations) may be slower or faster than the standard, locally coupled uniform CGLE oscillations (below referred to as uniform or locally coupled oscillations). Another important issue is the stability of the global oscillations. Using

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the phase dynamics approximation,28 it can be shown that global oscillations are unstable with respect to smooth phase perturbations if

cos(χ - tan-1 R) > 0

(8)

An important feature of the dynamics of the gcCGLE is that global oscillations are able to suppress the emission of waves from a heterogeneity.27 This means that the local frequency shift should exceed a critical value in order for the heterogeneity to emit waves. Below the critical value, the impact of the heterogeneity on the system is strictly local; the local oscillations in the area of the heterogeneity have the same frequency as the global oscillations and the heterogeneity just leads to a local shift of the oscillation phase. Such patterns do not constitute pacemakers but represent wave sinks.18 In this way, pacemaker activity can be effectively suppressed in large parameter regions of the gcCGLE. In the CO oxidation experiments, stable global oscillations are found in a wide parameter region where the laser beam does not produce any extended wave pattern (see Section 2). It can be concluded that in such cases the laser beam does not represent a pacemaker but a wave sink. However, the brightness of the laser beam prevents the observation of the local dynamics at the laser spot, and the details of the wave sink patterns cannot be directly investigated. If instead of global oscillations extended wave patterns are present in the whole medium, the averaged amplitude A h is small (if the system is sufficiently large) and the feedback effectively breaks down. Then, the frequency of the waves is approximately described by the dispersion relation (eq 3) of the standard CGLE. Therefore, there are two typical results of simulations of the gcCGLE with heterogeneity: global oscillations suppressing any extended wave pattern and extended target patterns formed by expanding waves in the effective absence of global feedback. However, in the experiments displayed in Figure 1(c-f), the presence of both global oscillations and waves is clearly observed. Therefore, we look for target pattern solutions of the gcCGLE where the influence of global oscillations is still detectable. As stated in Section 2, Figure 1(c), we observe an inward traveling target pattern which is associated with an effective frequency decrease in a relatively large region around the laser spot. The space-time diagram in Figure 1(d) reveals that the target wave pattern is composed of two types of waves: longwavelength waves found far from the laser spot, and shortwavelength waves present close to the center. There are two ways to create such a pattern in the simulation. The first possibility is to assume that at the laser spot the oscillation frequency is locally decreased, i.e., that ∆ω < 0. The second possibility consists of assuming that the global oscillations are interrupted at the heterogeneity and the local dynamics is governed by standard uniform CGLE oscillations of a lower frequency. In the following, simulations for both possibilities are shown. First, a circular region of the medium is subjected to a small frequency decrease ∆ω < 0, while the surrounding region remains in high-frequency global oscillations. Therefore, almost the whole medium serves as a pacemaker for the low-frequency region close to the location of the laser beam, and circular inward traveling waves are formed. In Figure 4, a corresponding two-dimensional simulation is displayed. Since the twodimensional system is chosen to be rectangular, the region which emits the waves is effectively restricted to those parts of the system which are located farthest from the center. This means that the edges of the system serve more as wave sources than

Figure 4. Inward traveling target wave pattern in the gcCGLE. The frequency distribution ω(x) is shown in (a) and the wave pattern (ReA at t ) 200) is displayed in (b). The space-time diagrams of ReA for cross sections through the center of the system for the time interval ∆t ) 100 are shown in (c) and (d). In (c), the cross section connects one system edge with the opposite one, in (d), the cross section is parallel to the system boundary. The parameters are ∆ω ) -0.4, σ ) 10, ω ) 0, R ) -0.5, β ) 0, µ ) 0.5, χ ) 0.65π, and the dimensions of the system are Lx ) Ly ) 200.

other regions, Figure 4(b). Therefore, two different space-time diagrams for ReA are shown: one where the dynamics is displayed along a line connecting one corner of the system with the opposite corner, Figure 4(c), and another one where the line through the center is drawn parallel to the system boundary, Figure 4(d). Both space-time diagrams show qualitatively the same behavior, namely, that the wave pattern is composed of two types of waves. Far from the center, where the local frequency difference is negligibly small, |ω(x) - ω| , ∆ω, waves with a large wavelength are found. Close to the center and in particular where the frequency is locally decreased, the wavelength is much shorter. Theoretical studies for heterogeneous pacemakers with a piecewise-constant frequency distribution have shown that the wavenumber inside a wave source of target waves generally is not zero (it is exactly zero only in the center of a symmetric wave source) but increases toward the boundary of the wave source.18 Since here the wave source is formed by the entire area sufficiently far away from the laser spot, waves with a large wavelength can be found in large parts of the medium. Waves with a small wavelength are then found in the area close to the heterogeneity, i.e., in the area with decreased local oscillation frequency. The second possibility to create inward traveling waves in the gcCGLE consists of assuming that the heterogeneity locally interrupts the global coupling. Therefore, the heterogeneity is assumed not to change the frequency but the feedback term, such that the system dynamics is governed by

∂A ) (1 - iω)A - (1 + iR)| A|2A + (1 + iβ)∇2A - f(A, x) ∂t (9) where the feedback term f(A, x) is given by

(

h 1 - exp f(A, x) ) µeiχA

[

])

-(x - x0)2 2σ2

) f(A)g(x) (10)

To yield inward traveling waves in such a system, two conditions have to be fulfilled. First, the frequency of global

Oscillatory CO Oxidation on Pt(110)

Figure 5. Inward traveling target wave pattern in the gcCGLE. The space-dependent contribution g(x) to the feedback term, ranging from 0 (black) to 1 (white), is shown in (a), the wave pattern (ReA at t ) 200) is displayed in (b), and the space-time diagram of ReA for a cross section through the center of the system parallel to the system boundary is shown for the time interval ∆t ) 200. The parameters are σ ) 15, ω ) 0, R ) 1, β ) 2, µ ) 0.3, χ ) π, and the dimensions of the system are Lx ) Ly ) 125.

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Figure 6. Spatially confined target wave pattern in the gcCGLE. The frequency distribution ω(x) is shown in (a), the wave pattern (ReA at t ) 75) is displayed in (b), and the space-time diagram of ReA for a cross section through the center of the system parallel to the system boundary is shown in (c) for the time interval ∆t ) 100. The parameters are ∆ω ) 0.8, σ ) 3, ω ) 0.5, R ) 1, β ) 2, µ ) 0.6, χ ) 1.7π, and the dimensions of the system are Lx ) Ly ) 125.

4. Realistic Reaction Model oscillations must be higher than the frequency of locally coupled (uniform) oscillations. According to eq 7, for a given R there is an interval of χ-values where this is fulfilled. Second, global oscillations are present outside the heterogeneity and are stable. This condition is described by eq 8. Figure 5 shows a twodimensional simulation for parameter values that meet both criteria. Although far from the heterogeneity the wavenumber is practically zero, the simulation clearly demonstrates the presence of inward traveling waves close to and inside the heterogeneity. As described in Section 2, a localized target pattern formed by outward traveling waves is experimentally found, see Figure 1(e,f). Stable global oscillations are present in the entire observable medium, i.e., in the whole system with the possible exception of the location of the laser beam. Therefore, the heterogeneity created by the laser beam almost constitutes a wave sink. However, a small-amplitude oxygen wave is periodically emitted from the central region when the medium is in the CO-covered state. As the global oscillation reaches the oxygen-covered phase, the small-amplitude wave is erased, and therefore the wave pattern remains confined to the vicinity of the central region. Such a behavior, however, could not be reproduced exactly in the one- or two-dimensional globally coupled CGLE. Nevertheless, similar dynamics which leads to localized target patterns has been found and is presented in Figure 6. There, the localized frequency shift ∆ω is chosen to be relatively large. It is known for the standard CGLE that large frequency shifts can create Eckhaus-unstable waves that disappear in phase slips.18 For smaller ∆ω, the gcCGLE shows the typical behavior of a wave sink for this choice of µ and χ, i.e., global oscillations suppress any wave pattern of extension larger than the heterogeneity. However, for the large frequency shift chosen here, the heterogeneity emits waves which propagate a short distance before they decay, as displayed in the space-time diagram in Figure 6(c). Since the properties of the emitted waves are controlled by the size of the heterogeneity and the frequency shift, the frequency of the emitted waves is unrelated to the frequency of global oscillations.

In this section, we present results from numerical simulations of a realistic reaction model for catalytic CO oxidation on Pt(110) single-crystal surfaces. We model the effect of the laser by introducing a space-dependent temperature profile that shows a local increase in temperature at the location of the laser. Taking into account the temperature dependence of the reaction parameters via a simple Arrhenius-type expression, the temperature inhomogeneity induces a local change of the reaction dynamics. First, we introduce the model and analyze the influence of a temperature rise on the dynamics of a single element in the oscillatory regime. We then present results from numerical simulations exemplifying the effect of a temperature inhomogeneity on the spatiotemporal dynamics of the system. Finally, we extend the reaction model by taking into account global coupling through the gas phase which significantly affects the formation of patterns induced by the presence of a temperature inhomogeneity. 4.1. The Model. In this work, we use a three-variable realistic reaction model (Krischer-Eiswirth-Ertl model) that has been well established over the past decade for computing the dynamical behavior of catalytic CO oxidation on Pt(110) single-crystal surfaces.29-32 It consists of a system of three coupled partial differential equations for the variables u, V, and w representing the CO coverage, the oxygen coverage, and the fraction of the surface found in the nonreconstructed 1 × 1 phase, respectively,

∂tu ) k1sCO pCO(1 - u3) - k2u - k3uV + D∇2u

(11)

∂tV ) k4 pO2[sO,1×1w + sO,1×2(1 - w)](1 - u - V)2 - k3uV (12)

(

∂tw ) k5

1

1 + exp

(

)

u0 - u δu

-w

)

(13)

The model accounts for the most significant physical processes that are known to determine the dynamics of the reaction: adsorption of CO and oxygen on the catalytic surface, desorption

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TABLE 1: Model Parameters k1 k2 ν2 E2 k3 ν3 E3 k4 k5 ν5 E5 sCO sO,1×1

3.14 × 105 s-1 mbar-1 ) ν2 exp(-E2/kBT) 2 × 1016 s-1 38 kcal mol-1 ) ν3 exp(-E3/kBT) 3 × 106 s-1 10 kcal mol-1 5.86 × 105 s-1 mbar-1 ) ν5 exp(-E5/kBT) 1056.1 s-1 7 kcal mol-1 1.0 0.6

sO,1×2

0.4

u0, δu

0.35, 0.05

D D0 ED

) D0 exp(-ED/kBT) 3.56 × 106 µm2 s-1 12.3 kcal mol-1

impingement rate of CO desorption rate of CO reaction rate impingement rate of O2 phase transition rate CO sticking coefficient oxygen sticking coefficient on the 1 × 1 phase oxygen sticking coefficient on the 1 × 2 phase parameters for the structural phase transition CO diffusion coefficient

and surface diffusion of CO, reaction between the two adsorbed species, and an adsorbate-driven structural change of the platinum surface. More intricate effects such as the roughening of the catalytic surface or the formation of subsurface oxygen,33 that do not significantly affect the overall dynamics of the system, are not included here. For the rate constants of CO desorption, CO diffusion, surface reaction, and surface structural phase transition we assume a simple Arrhenius-type temperature dependence. If parameters are chosen appropriately, the model can show excitable as well as oscillatory behavior. The model parameters used for the present investigation can be found in Table 1 or, if they are changed for different simulations, are specified together with the results below. In general, they are chosen such that the reaction exhibits oscillatory dynamics and, with increasing temperature, approaches a Hopf bifurcation point beyond which its dynamics becomes excitable. We assume that the impact of the laser on the dynamics of the reaction is mostly due to a local rise of surface temperature at the location of the laser spot. This spatial temperature inhomogeneity affects the dynamics of the model by a local change of the above-mentioned temperature-dependent rate constants. We take the temperature profile to be stationary23 showing a heterogeneity in the middle of the domain that is of Gaussian shape,

T ) T0 + ∆T exp

[

]

-(x - x0)2 2σ2

(14)

The center of the Gauss profile is chosen to coincide with the middle of the domain, x0 ) 400 µm. The width σ and the base surface temperature T0 are set to 12 µm and 543.47 K, respectively, unless stated otherwise. To get an overall understanding of how the increased temperature inside the laser spot influences the local dynamics of the system, Figure 7 shows amplitude and frequency of oscillations in the CO coverage u as a function of temperature for a single oscillator. With increasing temperature the frequency rises almost linearly, which is in qualitative agreement with the experimental result displayed in Figure 2. The amplitude, on the other hand, becomes smaller with increasing temperature. Slightly above T ) 545 K, the amplitude goes down to zero and the reaction no longer shows oscillatory behavior. Close to the onset of oscillations where the amplitude is small, oscillations are approximately harmonic, as is expected for a supercritical Hopf bifurcation.

Figure 7. Temperature dependence of amplitude (top) and frequency (bottom) of oscillations in the CO coverage. The two branches in the left half of the amplitude graph represent the maximum (upper branch) and minimum (lower branch) values of CO coverage that are reached during one oscillation cycle. Partial pressures of the reactants are pCO ) 4.55 × 10-5 mbar and pO2 ) 1.1 × 10-4 mbar.

The numerical simulations presented in the following are performed for a one-dimensional system of length L ) 800 µm. We use a second-order finite difference scheme for the spatial discretization on a grid of equidistant gridpoints with ∆x ) 4 µm and impose periodic boundary conditions. An explicit Euler scheme with a fixed time step of ∆t ) 0.001 s is employed for the time discretization. 4.2. Numerical Simulations. In a first series of simulations, we investigate the change of spatiotemporal dynamics as a result of varying CO partial pressure, which enters the model through eq 11 and is one of the global system parameters. Results for four different CO partial pressures are shown in Figure 8, where pCO is decreased from (a) to (d). While Figure 8(a) shows a transient in the temporal evolution of the system immediately after the laser was switched on, Figures 8(b-d) show asymptotic states that are reached after long integration times. For the simulations shown in Figures 8(b-d), the maximal temperature in the laser spot is increased such that the dynamics at the center of the temperature heterogeneity is excitable. In agreement with the results shown in Figure 7, the oscillation frequency inside the heterogeneity is increased with respect to the rest of the medium. As can be clearly seen in the space-time diagram in Figure 8(a), the central region of increased frequency becomes a pacemaker that starts to send out waves which propagate into the medium. An outward traveling target pattern is formed that grows in the course of time until, eventually, it entrains the whole system. This situation corresponds to the experimental result displayed in Figure 1(a,b), where a target pattern of outward traveling wave trains is established in the medium and grows slowly at the expense of the surrounding patterns with lower oscillation frequency. If the CO partial pressure is slightly lowered compared to the conditions chosen for the previous simulation, a completely different behavior is encountered as shown in Figure 8(b). For this choice of parameters, the system does not settle down to a stable regular pattern. Instead, waves that are sent out from the temperature heterogeneity evolve in a disordered way due to irregular series of phase slips. Of particular interest in this

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Figure 9. Inward and outward traveling target patterns under global gas-phase coupling. In both cases, ∆T ) 2.5 K, pO2 ) 8 × 10-5 mbar, γ ) 6 s-1, and χ ) 1 × 10-9 mbar m-1. The other parameters are (a) T0 ) 543.47 K, pCO ) 4 × 10-5 mbar, σ ) 12 µm, and (b) T0 ) 542 K, pCO ) 3.635 × 10-5 mbar, σ ) 40 µm. The displayed time interval is 50 s. In (b), a zoom from x ) 200 µm to x ) 600 µm is shown.

Figure 8. Target patterns induced by a temperature inhomogeneity at different CO partial pressures pCO ) 4.55 × 10-5 mbar (a), 4.48 × 10-5 mbar (b), 4.44 × 10-5 mbar (c), and 4.3255 × 10-5 mbar (d). Other parameters are ∆T ) 0.2 K for (a), ∆T ) 2.5 K for (b), (c), and (d), and pO2 ) 1.1 × 10-4 mbar, T0 ) 543.47 K in all cases. The displayed time interval is 150 s for (a), (b), and (c), and 75 s for (d). While (a) shows the transient immediately after the laser was switched on, the other space-time diagrams show asymptotic states that are reached after long integration times.

parameter regime is the repeated occurrence of long transients of inward traveling target patterns. An interval of time, where a transient with these irregular inward moving wave trains is observed, can be seen in the space-time diagram in Figure 8(b). Several features of this pattern are in good agreement with the inward traveling target pattern reported from the experiment, that is displayed in Figure 1(c,d). Here, the wave trains also show a slightly irregular, distorted shape, and their speed decreases as they approach the temperature inhomogeneity. However, in computations with the model eqs (11)-(14), this pattern was observed as a transient and we were not able to stabilize it for much longer time intervals. If the CO partial pressure is decreased further, the spatiotemporal evolution becomes even more irregular and a state reminiscent of intermittent turbulence34 is established in the system, see Figure 8(c). Besides, for even lower values of pCO close to the border of the oscillatory regime, there exists a small interval in CO partial pressure where regular outward traveling target patterns can be found as displayed in Figure 8(d). Both situations shown in Figure 8(c) and (d) were not observed in the experiment. Note that this scenario of different patterns for varying CO partial pressure remains qualitatively similar over a broad range of oxygen partial pressures. Besides, even in two spatial dimensions, the overall behavior seems to be similar, as was indicated by a number of simulations that we carried out on a two-dimensional system.

4.3. Numerical Simulations including Global Coupling. Numerical simulations of the model given by eqs (11)-(14) show both outward and inward traveling target waves, the two main types of target patterns that were also found in the experiment. However, the experimental results presented in Section 2, together with the theoretical predictions from Section 3, suggest that the influence of global coupling through the gas phase has to be taken into account for a more thorough explanation of the observed phenomena. The effects of intrinsic global gas-phase coupling on the dynamics of catalytic CO oxidation on Pt(110) have already been studied in some detail in the framework of the above-introduced three-variable model without temperature heterogeneity.32,35-37 In the following, we extend the model eqs (11)-(14) by an additional equation for the temporal evolution of CO partial pressure that takes into account global gas-phase coupling,

∂pCO ) γ(p0CO - pCO) - χ ∂t

∫L [ k1sCO pCO(1 - u3) - k2u] dx

(15)

For a detailed derivation of this equation and the meaning of the coupling parameters γ and χ, see, e.g., the appendix of ref 32. Also with global coupling, both outward and inward traveling target patterns can be observed. Figure 9 displays two spacetime diagrams produced from numerical simulations of the extended model eqs (11)-(15), where an example of each, an outward (a) and an inward (b), traveling target pattern is shown. In comparison to the previous results, we notice as the most significant effect of global coupling that the target patterns are confined to the vicinity of the temperature heterogeneity and do not spread out farther into the medium. As in Figure 8(bd), the maximal temperature inside the laser spot is such that dynamics at the center of the temperature heterogeneity are excitable. The two cases differ in the choice of CO partial pressure as well as in the base surface temperature. Moreover, the laser profiles were carefully adapted to enhance the observed phenomena. Outward traveling waves, as shown in Figure 9(a), spread over a narrow zone at the outer edge of the laser profile where temperature is only slightly increased with respect to T0. The temperature inhomogeneity sends out an oxygen wave that travels only a short distance into the medium and then merges with the oxygen phase of the overall oscillation of the system.

14290 J. Phys. Chem. B, Vol. 108, No. 38, 2004 In (a), we enlarged this zone by extending the Gaussian temperature profile with linear slopes at the outer edges (0.4 mK/µm, rising from x ) 200 µm and x ) 600 µm toward x0). The resulting spatially confined outward traveling target pattern corresponds to the experimental case shown in Figure 1(e,f). In the experiment, the system performs uniform globally coupled oscillations throughout the medium. In the simulation, however, the uniform oscillations do not persist in the immediate vicinity of the temperature heterogeneity. Here, only the oxygen wave sent out by the laser spot can be observed. Without global coupling, inward traveling target waves were only observed as transients, cf. Figure 8(b). If global coupling is introduced, they can be stabilized as shown in Figure 9(b). Similar to the outward traveling target pattern in the presence of global coupling, the inward going waves are confined to a small region close to the laser spot. They actually appear inside the temperature heterogeneity along the temperature slope that rises toward the center of the laser profile. To obtain a more pronounced illustration of this effect, we have chosen a larger width of the temperature profile for the simulation shown in (b). Although the agreement with the experimental results in Figure 1(c,d) is merely qualitative, we emphasize that global coupling can stabilize inward moving target waves that otherwise could be observed as transients only. 5. Discussion In this article, we studied, both experimentally and theoretically, the emergence of different types of target patterns in oscillatory CO oxidation on Pt(110) caused by the presence of a temperature inhomogeneity. These patterns can be explained and reproduced by numerical simulations of a modified complex Ginzburg-Landau equation and the realistic Krischer-EiswirthErtl model of catalytic CO oxidation. At the location where the laser beam hits the surface, the system temperature is increased. This leads to a change in the kinetics of the reaction resulting in higher oscillation frequencies for a moderate increase in temperature and in a transition from oscillatory to the excitable dynamics inside the laser spot for strongly increased temperature. However, even if the local dynamics is excitable at the center of the laser spot, the continuous temperature profile created by the laser beam will always produce a zone where the medium is oscillatory and the oscillation frequency is increased. Since the brightness of the laser spot prevents the direct observation of the local dynamics inside the laser spot, we do not know exactly whether the spatial profile of the increase of the oscillation frequency has a circular (if the medium is oscillatory in the center) or annular shape (if the medium is excitable in the center). In any case, the laser spot constitutes an effective heterogeneity in the system and forms either a pacemaker or a wave sink, depending on the global and local properties. The first kind of pattern is found when the system already shows wave patterns, so that no globally coupled uniform oscillations are present. The laser beam gives rise to an extended target pattern formed by outward traveling waves which entrain the whole system. Such target patterns are well-known to exist in the catalytic CO oxidation and are usually attributed to microscopic defects on the catalytic surface. Here, this pattern is artificially created in a controlled way. A similar approach has been used by Petrov et al.38 who reported an outward traveling target pattern created by focusing a light beam onto a small region of the photosensitive BZ reaction. The second type of pattern has not been described before in catalytic CO oxidation and consists of inward traveling wave

Wolff et al. trains. Inward traveling waves are rarely seen in experiments of reaction-diffusion systems. The only evidence of such patterns until now has been reported by Vanag and Epstein in the oscillatory regime of the BZ-AOT system,39 motivating studies of inward traveling spirals, see Nicola et al. in this issue. In the CO oxidation system, the inward traveling target patterns are observed close to a parameter region where oscillations, globally coupled through the gas phase, are stable. Therefore, global coupling has been taken into account in the simulations. It should be noted that Middya and Luss described inward and outward traveling target patterns for a two-component reactiondiffusion model with global coupling.40 There, however, the existence and stability of the waves relies on the circular symmetry of the spatial domain and on the boundary conditions. No heterogeneity is present there, and the waves are created by carefully chosen initial conditions. Consequently, the properties of their target waves are entirely determined by the properties of the medium and not by an externally controllable frequency shift. Within the globally coupled CGLE, there are several ways to create inward traveling target waves, even if the wave dispersion is positive (as in the experiment). The most simple way is to assume that the laser beam locally decreases the oscillation frequency. However, this mechanism is unlikely to take place in the experiment, since there is clear evidence that the oscillation frequency increases with temperature. The second possibility is that the laser beam heats the surface and therefore increases the oscillation frequency but simultaneously interrupts the gas-phase coupling. Then, the oscillation frequency at the laser spot is determined by the local properties only. In this case, the local oscillation frequency has contributions from the usual (locally coupled) oscillations and the local heating. There is a parameter range in the gcCGLE, where the locally coupled oscillations in the region of the laser spot (despite the local frequency increase due to heating) are slower than the globally coupled oscillations outside of the laser spot such that the frequency is effectively decreased at the location of the laser. As a result, inward traveling target waves are indeed found. The realistic Krischer-Eiswirth-Ertl model already shows inward traveling target waves in the absence of global gas-phase coupling, cf. Figure 8(b). Although they occur as transients and are not asymptotically stable, they are clearly present over large time intervals of 10-100 oscillation cycles which suggests that they may correspond to an experimentally observable state. If global coupling is introduced into the kinetic model, the inward traveling target patterns can be stabilized but then become locally confined to the vicinity of the laser spot, see Figure 9(b). Inward traveling wave patterns are found when the dynamics of the system in the center of the heterogeneity is excitable. Therefore, a direct correspondence to patterns found in the CGLE (where the whole medium is oscillatory) cannot be expected. Nevertheless, the different interpretations offered by the gcCGLE and the Krischer-Eiswirth-Ertl model do not contradict but complement one another s both models provide the effective frequency difference which is the essential building block for the creation of a pacemaker. Besides these two cases, we experimentally found a third kind of target pattern that consists of spatially confined outward traveling waves. To the knowledge of the authors, such kind of target patterns have not been reported before in reactiondiffusion systems. The following scenario is observed. At the laser spot, the surface is mainly CO-covered, leading to the formation of oxygen waves which propagate outward and subsequently merge with the overall global oscillation of the

Oscillatory CO Oxidation on Pt(110) medium. Compared to the global oscillation, the oxygen wave has much smaller width and amplitude. Although the gcCGLE allows the formation of locally confined outward traveling target waves (Figure 6), we believe that the underlying mechanism is different from the one observed in the experiment, since the gcCGLE only provides harmonic oscillations and waves. In the realistic model, spatially confined outward traveling target patterns have been found if global coupling was taken into account, see Figure 9(a). Here, the maximal temperature increase is such that the dynamics at the center of the laser spot shows excitable behavior. The difference to the experiment is that the emitted oxygen wave has a large amplitude and width, converting into the global oscillation. Thus, we achieve good agreement of the computational results with the experimental observations if global coupling is taken into account. Moreover, the numerical results from the realistic reaction model suggest that the dynamics at the center of the temperature inhomogeneity is excitable. However, due to the brightness of the laser spot, this conjecture cannot be verified in our experimental setting. In summary, this article presents a systematic study of target patterns observed in oscillatory CO oxidation on Pt(110), created by controlled local heating of the catalyst surface using a focused laser beam. We discuss three different types of target patterns, two of which have not been reported before in catalytic CO oxidation. The experimental observations are interpreted by theoretical studies in the framework of a modified complex Ginzburg-Landau equation and supported by numerical simulations of the realistic KrischerEiswirth-Ertl model. Acknowledgment. Financial support of the Deutsche Forschungsgemeinschaft (SFB 555) is gratefully acknowledged. Supporting Information Available: Three short videos which correspond directly to the displayed snapshots in Figure 1, parts a, c, and e. These illustrate the different types of target patterns emanating from an artificially created defect in the form of a temperature heterogeneity created by the laser spot. When irregular wave patterns were dominant without the laser spot, the spot would create outward-traveling waves, which were slowly growing and thereby superseding the other patterns (see Figure 1a.avi). When the system was undergoing uniform global oscillations, inward- and outward-traveling waves have been found emanating from the laser spot (see Figure 1c.avi and Figure 1e.avi, respectively). The experimental conditions for these movies are identical to those given in the caption for Figure 1, parts a, c, and e within the paper. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Cross, M. C.; Hohenberg, P. C. ReV. Mod. Phys. 1993, 65, 8511112.

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