Mathematical Modeling of Reactive Phase Separation in the System

The bistable behavior of the O2 + H2 reaction on Rh(110) is modified by the presence of coadsorbed potassium. Reaction fronts transporting potassium a...
0 downloads 0 Views 194KB Size
Download PDF
14620

J. Phys. Chem. B 2004, 108, 14620-14626

Mathematical Modeling of Reactive Phase Separation in the System Rh(110)/K/O2 + H2† M. Hinz, S. Gu1 nther, H. Marbach, and R. Imbihl* Institut fu¨r Physikalische Chemie und Elektrochemie, UniVersita¨t HannoVer, Callinstrasse 3-3A, D-30167 HannoVer, Germany ReceiVed: February 13, 2004; In Final Form: May 5, 2004

The bistable behavior of the O2 + H2 reaction on Rh(110) is modified by the presence of coadsorbed potassium. Reaction fronts transporting potassium and the development of stationary Turing-like patterns have been observed. A realistic mathematical model is presented which reproduces qualitatively correct and, to a large part, even quantitatively correct the experimental results. Key factors of the model are the strong chemical affinity between coadsorbed oxygen and potassium, a reduced mobility of potassium on the oxygen covered surface, and a strongly reduced reactivity of oxygen toward hydrogen in the presence of coadsorbed potassium.

1. Introduction Alkali metals are well-known as so-called electronic promoters in a number of important catalytic reactions such as ammonia synthesis via Haber-Bosch or synthetic fuel production via Fischer-Tropsch.1-3 Recent experiments demonstrated that alkali metals not only modify the local electronic properties of the surface but also participate in the spatiotemporal dynamics of a surface reaction. This was shown in the O2 + H2 reaction on a potassium covered Rh(110) surface.4-9 Reaction fronts transporting potassium and stationary concentration patterns were observed in this system. These stationary patterns consisted of K + O condensation islands of irregular shape when they developed via reaction fronts.4 When reducing conditions were chosen as the initial state, also regular Turing-like structures were obtained which bifurcated from a spatially uniform initial state.9 Here we present a realistic mathematical model which reproduces the experimental data qualitatively and to a large part even quantitatively. The basis for the formulation of the model has been the rather extensive characterization of the reaction system with various spectroscopic and spatially resolved techniques, yielding complementary information about the chemical composition of the adlayer, its structural properties, and the reactivity and diffusivity of the relevant species. A spectromicroscopic technique, scanning photoelectron microscopy (SPEM), provided the information about the chemical identity of the imaged species,4,7,8 photoelectron emission microscopy (PEEM) imaged the reaction dynamics via the local work function changes,5,9 and low energy electron microscopy (LEEM) allowed the detection of ordered adlayers and structural changes in the substrate.6 The starting point for the formulation of the model was that a well-established model for the bistable O2 + H2 reaction on Rh(110) already existed.10,11 It soon became evident that the main driving force for the condensation of potassium and oxygen into large K + O coadsorbate islands is the strong chemical affinity of K and O. Accordingly, the K + O coadsorption system on Rh(110) was studied experimentally.12,13 A large number of K + O coadsorption phases was found involving substrate reconstructions of the “missing row” type. The †

Part of the special issue “Gerhard Ertl Festschrift”. * To whom correspondence should be addressed.

presence of strong attractive interactions between the adsorbates meant for the formulation of the mathematical model, that the diffusion of potassium is non-Fickian, since the diffusive flow is determined by the gradient in chemical potential rather than in concentration. 2. Mathematical Model 2.1. Catalytic Water Formation. Catalytic water formation on noble metal surfaces obeys a Langmuir-Hinshelwood (LH) mechanism for which we can formulate the following steps14-16

H2 + 2* h 2Had

(R1)

O2 + 2* h 2Oad

(R2)

Oad + Had f OHad + *

(R3)

OHad + OHad h H2O + Oad + *

(R4)

OHad + Had f H2O + 2*

(R5)

with the asterisk * denoting a vacant adsorption site. In this mechanism, both gases adsorb dissociatively, and the atoms react on the surface to water, which rapidly desorbs. Hydrogen adsorption is strongly inhibited by a layer of adsorbed oxygen.10 This inhibition effect is responsible for the bistable behavior of the reaction system. Reaction fronts initiate transitions between an inactive state, in which the surface is nearly completely covered with oxygen, and an active state, in which the surface is almost adsorbate free.10 2.2. Potassium Diffusion. Under our reaction conditions, potassium does not desorb so that we have only a redistribution of a fixed amount of potassium on the surface. Potassium has a high chemical affinity to oxygen.1-3 The strong attractive interactions between coadsorbed oxygen and potassium cause potassium to migrate from an oxygen freed part of the surface to a still oxygen covered part, as indicated schematically in the energy diagram of Figure 1. The gain in adsorption energy Eb for potassium in the presence of coadsorbed oxygen has been demonstrated in thermal desorption experiments of Rh(110)/ K/O as well as in theoretical calculations.17,18 Experiments, in which an inhomogeneous distribution of potassium was homogenized again by turning off one of the reactants, H2 or O2,

10.1021/jp0493430 CCC: $27.50 © 2004 American Chemical Society Published on Web 07/07/2004

The System Rh(110)/K/O2 + H2

J. Phys. Chem. B, Vol. 108, No. 38, 2004 14621 2.3. Formulation of the Mathematical Model. In the mathematical description of the reaction system, we neglect intermediates such as OH and we assume that potassium behaves as coadsorbate with oxygen not forming stable bulk compounds such as KOH or K2O. The available experimental results clearly indicate that such bulk compounds are not formed because such a compound would not have been reducible by H2 in the 10-7 and 10-6 mbar range. As the basis for our model, we can therefore use the two-variable model for Rh(110)/O2 + H2, which was shown to accurately reproduce anisotropic front propagation in this system.11 Adding an equation for the diffusive redistribution of potassium, we arrive at a system of three coupled partial differential equations (PDEs)

Figure 1. Schematic potential energy diagram for potassium atoms at the interface reduced/oxygen covered surface. The gain in adsorption energy for potassium atoms moving from the oxygen free surface to the K + O coadsorbate is denoted as Eb, the differently high activation barriers for diffusion on the two surface states are represented as E′K and E′′K.

showed that potassium spreads out rather rapidly on the reduced surface but comparatively slowly on the oxygen covered surface. This difference in mobility is reflected in the energy diagram by differently high activation barriers, E′K and E′′K for K diffusion on the reduced and on the oxygen covered surface, respectively. From the fact that potassium condenses with oxygen into large coadsorption islands, it is clear that the K diffusion cannot be represented as Fickian diffusion. To describe the “uphill" diffusion of potassium, the migration of potassium has to follow the gradient in the chemical potential of potassium, µK

θ˙ K ) ∇

[

DKθK(1 - θK) ∇µK RT

]

(1)

For the chemical potential of adsorbed potassium, we assume the potential of a Langmuirian adsorption layer

µK ) µ0K + RT ln

( )

θK + UK 1 - θK

(2)

to which we add the local potential variation UK, given by the attractive interaction with chemisorbed oxygen as shown in Figure 1

UK ) -(E 0ad + θOEb)

(3)

One arrives at

θ˙ Κ ) ∇(DK∇θK) - ∇

[

DKθK(1 - θK) Eb∇θO RT

]

(4)

The first term on the right side represents Fickian diffusion, whereas the second term, the so-called drift term, is responsible for the “uphill” diffusion into the oxygen covered part of the surface. The activation energy for K diffusion via Arrhenius includes the potassium mobility dependence on the oxygen coverage

DK ) D0K exp

(

)

-(E′K + θO(E′′K - E′K)) RT

(5)

Here E′K represents the activation barrier for the K diffusion on the oxygen free surface and E′′K - E′K stands for the increase of the energetic corrugation on a completely oxygen covered surface, as suggested in Figure 1.

θ˙ Η ) k˜1pH2(1 - θH - βθO)2 + γk˜1pH2(1 - θH)2 - k3θ 2H 2k5θOθH + DH∇2θH (6) θ˙ Ο ) k2pO2(1 - θH - θO)2 - k4θ 2O - k5θOθH + DO∇2θO (7) θ˙ K ) ∇(DK∇θK) - ∇

[

DKθ˜ K(1 - θ˜ K) Eb∇θO RT

]

(8)

max with k˜1 ) k1exp(-δθK) and θ˜ K ) θK/θmax ) 0.22. The K , θK first two equations, which are practically identical to the equations in ref 11 for Rh(110)/O2 + H2, describe the LH mechanism for catalytic water formation, as expressed in steps R1-R5. However, we neglect step R4 which contributes insignificantly under our reaction conditions, and we furthermore assume that the addition of the first proton to oxygen in step R3 is rate-limiting for the water formation. The first two terms in eq 6 describe H2 adsorption at regular surface sites (k˜1) and at defect sites (γ k˜1), i.e., at sites where oxygen does not inhibit H2 adsorption. The following terms represent H2 desorption (k3), water formation (k5), and the surface diffusion of adsorbed hydrogen (DH). Similarly, for oxygen, the first term in eq 7 describes oxygen adsorption (k2), the term with k4 stands for oxygen desorption, the term with k5 represents water formation, and finally, also oxygen diffusion (DO) is included. The third equation contains just the redistribution of potassium by diffusion according to the spatial distribution of adsorbed oxygen as discussed above. The coupling between the first two equations and the third equation is established first by oxygen driving the redistribution of potassium and second by coadsorbed potassium reducing strongly the reactivity of adsorbed oxygen toward hydrogen.12 This reduced reactivity, which has been well studied in titration experiments, is described in the equations by an exponential decrease of the hydrogen sticking coefficient with K coverage according to k˜1 ) k1exp(-δθK) with k1 representing hydrogen adsorption on the unpromoted surface. In the model, we do not distinguish between different adsorption sites and the equations ensure that θH + θO e 1. Assuming pairwise interactions of the adsorbates, a symmetrical drift term in which coadsorbed K influences O diffusion should also be included in eq 7. Simulations with this additional term were carried out, but since the effect of this term was very small, the term was dropped again. In contrast to K diffusion, we thus have Fickian diffusion for adsorbed oxygen and hydrogen. In the original model for the unpromoted surface site, blocking effects for the diffusion of these adsobates had been taken into account and they were in fact essential for reproducing the parameter-dependent anisotropy of the front propagation,

14622 J. Phys. Chem. B, Vol. 108, No. 38, 2004

Hinz et al.

TABLE 1: Temperature Independent Constants k1 k2 Eb β γ δ

2.186 × 106 ML s-1 mbar-1 0.546 × 106 ML s-1 mbar-1 15 kJ/mol 1.1 4 × 10-4 6

[19] [19] fit [11] [11] fit

TABLE 2: Temperature Dependent Constants k3 k4 k5

νi [s-1]

Ei[kJ/mol]

ref

1013

72 205 90

20 21 22

1012 1013

TABLE 3: Diffusion Parameters DH DO DK a

D0A [cm2/s]

EA[kJ/mol]

ref

4.0 × 10-3 2.5 × 10 4.0 × 10-1

18 120 a)

11 and 23 11 24 + fit

E′K ) 37.8kJ/mol, E′′K ) 84.8kJ/mol.

but for the 1D-simulations conducted here, these effects turned out to be negligible.11 2.4. Choice of Constants. The constants used in the simulation are summarized in Tables 1-3. For the temperaturedependent constants, the usual Arrhenius dependence ki ) νi exp(-Ei/RT), (i ) 3-5) was taken, which was assumed to also hold for the diffusion constants DA (A ) H, O, K) such that DA ) D0A exp(-EA/RT). The activation energy EK for potassium diffusion depends on the oxygen coverage as expressed in eq 5. Nearly all constants, with exception of the diffusion parameters could be taken from the experiment. The constants of the O2 + H2 subsystem were all taken from ref 11 because there they were shown to describe the unpromoted system rather well. The constant δ representing the deactivation of chemisorbed oxygen by coadsorbed potassium was determined by fitting the experimentally observed slowing down of the front propagation with progressive K enrichment.8 Quite generally, few experimental data are available for diffusion of adsorbates on metal surfaces.23 For the diffusion of hydrogen and oxygen on Rh(110), the data in ref 11 for Rh(110)/O2 + H2 were taken, which were obtained by fitting the front propagation in the [11h0] direction. For the very fast diffusion of potassium on the adsorbate free surface, no experimental data were available and therefore, the data for Ru(0001)/K were taken for D0K and E′K.24 For potassium diffusion on the oxygen covered Rh(110) surface, the kinetic parameters were determined by fitting the following experiment depicted in Figure 2. After reaction fronts in the O2 + H2 reaction on Rh(110)/K had created large K + O coadsorbate islands, H2 was turned off. The homogenization process of potassium on the completely oxygen covered Rh(110) surface was followed by K concentration profiles recorded with scanning photoelectron microscopy (SPEM).17 The measured K profiles together with the fits are displayed in Figure 2. The value for the increase of the energetic corrugation on a completely oxygen covered surface, E′′K - E′K, was thus determined to be 47 kJ/mol. 2.5. Calculation Procedure. For the numerical integration of the reaction-diffusion model, the routine D03PCF of the NAG Fortran Library (version 19) was used. D03PCF integrates systems of parabolic partial differential equations in one space variable. For the spatial discretisation, finite differences are employed, and the method of lines is used to reduce the PDEs

Figure 2. Diffusional spreading of a large potassium island on the oxygen covered Rh(110) surface. The diagram displays calibrated K concentration profiles together with the simulated profiles (s. text). The fitting of the profiles was used to determine E′′K. The initial K distribution was created by reaction fronts forcing the potassium into large K + O condensation islands. H2 was turned off at t ) 0 s. Experimental conditions: θK ≈ 0.1 ML, T ≈ 533 K, pO2 ) 1.3 × 10-7 mbar. From ref 17.

to a system of ordinary differential equations (ODEs). The resulting system is solved via a backward differential formula method. In our case, zeroflux boundary conditions have been chosen

|

|

∂θA ∂θA ) ) 0, A ) H, O, K ∂x x)0 ∂x x)L

(9)

with L being the length of the integration interval. 3. Simulation Results 3.1. Front Propagation. With coadsorbed potassium, reaction fronts still initiate transitions between the unreactive oxygen covered and the reactive oxygen free state of the surface, but the motion of the front is now coupled to the mass transport of potassium.4-8 Potassium is redistributed from the oxygen freed to the still oxygen covered part of the surface. In the front region potassium is enriched. The following experiment was conducted such that, starting from a spatially uniform initial state with θK ≈ 0.10 and θO ≈ 1, pH2 was increased until a reduction front nucleated. The progressive enrichment of potassium in the front region leads to time-dependent front profiles and a continuous slowing down of the front. This is demonstrated in Figure 3, parts a and b. The diagram in Figure 3a displays only the K concentration profiles but the position of the reduction front in the corresponding steplike oxygen profiles practically coincides with the steep rise of the K concentration in the profiles. The simulation with the realistic model reproduces the development of the profiles and of the front velocity nearly quantitatively, as demonstrated by Figure 3, parts c and d. The

The System Rh(110)/K/O2 + H2

J. Phys. Chem. B, Vol. 108, No. 38, 2004 14623

Figure 3. Slowing down of reduction fronts in the system due to progressive enrichment of potassium in the front region. The diagrams in (a) and (b) display the experimental K concentration profiles together with the time dependent variation of the front velocity. The corresponding simulations are shown in (c) and (d). Experimental conditions: θK ≈ 0.1, T ) 623 K, pO2 ) 1.3 × 10-7 mbar, pH2 ) 1.32 × 10-7 mbar. Experimental data from ref 8.

fitting of the simulated to the experimentally observed decrease of the front velocity with time was used to determine the inhibition parameter δ resulting in δ ) 6. The profiles in parts a and c show that the initially peak-like narrow enrichment broadens rapidly with time, accompanied by a strong reduction of the front velocity. Clearly, with coadsorbed potassium, the system is no longer bistable. The measured profiles are transients and at the end a stationary pattern is formed. The slowing down of the reaction front with progressively increasing K concentration in the still oxygen covered part is apparently caused by the decreasing reactivity of oxygen toward hydrogen in the K + O phase. As demonstrated in titration experiments, the reactivity of oxygen decreases dramatically in the presence of coadsorbed potassium. This is evidenced by an increase of the total H2 exposure required for the complete reactive removal of oxygen by roughly 2 orders of magnitude at θK ) 0.22 compared to the K-free oxygen adlayer.12 This drastic effect is reflected in the mathematical model by an exponential decrease of the H2 sticking coefficient with θK. One of the first questions to be discussed was of course where the K enrichment in the front region originated from. Kinetically, this effect could be explained by the slowing down of diffusing K atoms as soon as they invade the oxygen covered part of the surface. Alternatively, one might consider a thermodynamic stabilization by a different chemical environment in the front region, for example, by a substantial OH coverage in this region. This question could very clearly be answered by an experiment in which a reduction front was stopped by a pH2 decrease.8 The start-stop experiment reproduced in Figure 4a demonstrates that the K concentration profile flattens in the front region after halting the front because the potassium has now enough time to spread out on the oxygen covered surface. The profile sharpens again when front propagation is reinitiated. The corresponding simulation displayed in Figure 4b reproduces the experiment. The K enrichment in the front region is therefore clearly due to the slow kinetics of K migration on the oxygen covered surface.

Figure 4. Start-stop-experiment demonstrating that the enrichment of potassium in the front region is due to kinetics. (a) Experiment. Starting from a homogeneous K distribution on a O-covered surface a reduction front was first initiated (t ) 0 s) and then stopped by decreasing pH2 to 0.66 × 10-7 mbar at t ) 815 s. Increasing pH2 again to 1.09 × 10-7 mbar caused the front to propagate again. Experimental conditions: θK ) 0.08, T ≈ 533 K, pO2 ) 1.3 × 10-7 mbar. Concentration profiles were taken at t ) 189, 369, 815, 1004, and 1225 s. From ref 8. (b) Simulation with the same parameters and the same temporal sequence as in the experiment.

14624 J. Phys. Chem. B, Vol. 108, No. 38, 2004

Hinz et al.

Figure 6. Bifurcation diagram obtained by numerical integration showing the parameter ranges in which oxidation and reduction fronts are observed. The dashed line marks the equistability of the two states given by zero front velocity. The parameter range in which the system is monostable has only been determined for the oxygen covered state of the surface but not for the reduced oxygen free state. Simulation parameters: θK ) 0.08, pO2 ) 1.3 × 10-7 mbar.

Figure 5. K profiles demonstrating the reversal of front propagation through a parameter change. (a) Experiment. A reduction front propagating from left to right had been stopped by decreasing pH2 (t1). Further decrease of pH2 to 0.66 × 10-7 mbar caused the front to move in the opposite direction (t2 - t4). A second reversal of the front propagation direction (t5) was initiated by increasing pH2 to 0.76 × 10-7 mbar. Experimental conditions: (a) θK ) 0.08, T ≈ 632 K, pO2 ) 1.3 × 10-7 mbar. From ref 8. (b) Simulation with the same parameters and the same temporal sequence as in the experiment.

The time dependent change of the front profile and of the front velocity indicates that the reaction system with potassium is no longer bistable, and the fronts we observe are, in fact, transients in the formation of a stationary pattern. Nevertheless, the system still bears some features reminiscent of true bistability. Above 570 K, the direction of front propagation can be reversed by decreasing pH2.8 The reduction front is thus transformed into an oxidation front, as demonstrated by the experimental K concentration profiles in Figure 5a. The simulation reproduces the experimentally measured K profiles quite well, as demonstrated by Figure 5b. Stopping the front (t ) 4251 s) leaves a high K concentration in the front region, but this concentration decreases substantially when the reduction front is turned into an oxidation front (t2 - t4). The front profile sharpens again when, by a second parameter change, the reduction front is reestablished (t5). The practically quantitative reproduction of the measured K profiles by the simulation shows that the simplified description of the K diffusion through eq 8 works rather well. The pH2, T-parameter range, in which reduction fronts or oxidation fronts are obtained in the 1D simulations, is displayed in Figure 6. One notes that below 570 K the parameter region for an oxidation front becomes very narrow in the pH2-parameter space. This temperature is exactly the temperature limit found in the experiments below which the propagation of the reduction fronts could not be reversed by a pH2 decrease.8 The asymmetry with respect to the differently broad existence ranges below 570 K for the oxidation and reduction front,

respectively, was also present in the bifurcation diagram of the unpromoted system.10 The bifurcation diagram in Figure 6 shows, that the system behavior changes markedly at ≈550-570 K. This change, which is also seen in the experiments, can be attributed to the fact that above this temperature potassium becomes mobile on the oxygen covered surface. The best demonstration for this change in the system behavior are the qualitatively different K concentration profiles we observed above and below this threshold temperature. The comparison of the two PEEM images in Figure 7, parts a and d, demonstrates that at 545 K potassium is enriched only in a roughly 10 µm wide zone at the front, whereas at T ) 650 K, the zone of a high K concentration extends far into the still oxygen covered surface area.5,8 In addition, we note that the contrast of the PEEM image has been reversed. At high T, the oxygen free area appears darker in PEEM than the oxygen covered area. The corresponding K concentration profiles recorded with SPEM in Figure 7, parts b and e, reveal that at low T potassium it is only partially redistributed by the reduction front, whereas at high temperature the reduced surface area is nearly completely depleted of potassium. The simulations in Figure 7, parts c and f, reproduce this behavior very nicely with the only difference being that the simulation predicts for low temperature a stronger K enrichment in the front region as it was experimentally observed. The contrast reversal at high T is explained by the fact that, at high temperature, a substantial K coverage is present on the oxygen covered surface, which reduces the work function of the K + O coadsorbate below that of the clean surface.5,8 3.2. Stationary Patterns. With progressive slowing down of the reduction front, a stationary concentration pattern finally resulted in which large K + O coadsorption islands with a size up to the mm range were created.4,8 Since these islands typically were of irregular shape and since they did not develop smoothly out of a homogeneous initial state, the definition of a Turing structure at first sight did not seem to apply.25,26 Nevertheless, these patterns were only stable under reaction conditions and so the question was, whether the absence of a periodic pattern was not just a consequence of a too slow ordering process caused by the low mobility of potassium on the oxygen covered surface. To verify this hypothesis the experimental procedure was modified and instead of starting from oxidizing conditions the

The System Rh(110)/K/O2 + H2

Figure 7. Temperature dependence of front propagation: (a) PEEMimage at T ) 545 K, pH2 ) 7.84 × 10-6 mbar, pO2 ) 2.6 × 10-6 mbar; (b) SPEM-profile at T ) 533K, pH2 ) 1.75 × 10-7 mbar, pO2 ) 1.3 × 10-7 mbar; (c) simulation of the potassium profile (same parameters as (b)); (d) PEEM-image at T ) 650 K, pH2 ) 3.92 × 10-6 mbar, pO2 ) 2.6 × 10-6 mbar; (e) SPEM-profile at T ) 623 K, pH2 ) 1.51 × 10-7 mbar, pO2 ) 1.3 × 10-7 mbar; (f) simulation of the potassium profile (same parameters as (e)). Experimental results from ref 5.

experiment was started from reducing conditions ensuring in this way a high mobility of potassium. By decreasing pH2, in fact a nearly periodic pattern developed, which bifurcated from the homogeneous state, as demonstrated in Figure 8a.9,17 The bright area in the PEEM images can be attributed to the K + O coadsorbate phase, as potassium is known to strongly reduce the work function. The intrinsic wavelength in the pattern of Figure 8a is around 50 µm. In simulations in which we start with oxidizing conditions, we observe, similar to the experiment, the formation of large K + O coadsorption islands via propagating reduction fronts, but no periodic pattern evolves.4,8 If, however, we start in the simulation from reducing conditions decreasing pH2 to a certain critical value, then a periodic concentration pattern develops, which is reproduced in Figure 8b. The spatial periodicity is 90 µm and thus close to the value in the experiment. High coverage K + O coadsorption islands and nearly adsorbate free surface alternate in this pattern, as observed in the experiment.9,17 The same diagram also displays the rate of water formation and, as expected from the reduced reactivity of oxygen, the K + O islands exhibit a lower rate of H2O formation than the surrounding area. Coadsorbed potassium in this reaction system thus acts as a catalytic poison and not as a promoter. However, we note that if we compare this state with the state of potassium being uniformly distributed over the surface, that the inhomo-

J. Phys. Chem. B, Vol. 108, No. 38, 2004 14625

Figure 8. Development of stationary Turing-like structures due to reactive phase separation. (a) Left image: PEEM images showing the development of stationary patterns during the transition from reducing to oxidizing conditions (initial potassium concentration θK ) 0.08). Experimental conditions: T ) 518 K, pH2 ) 2.2 × 10-6 mbar, pO2 varied from ) 1 × 10-6 mbar to 2 × 10-6 mbar. Right image: PEEM image of a stationary concentration pattern recorded 840 s after adjusting reaction conditions. Experimental conditions: T ) 565 K, pO2 ) 2 × 10-7 mbar, pH2 ) 1.8 × 10-7 mbar. (b) 1D-simulation. Top: stationary concentration pattern. Bottom: spatial variation of the local reaction rate. Model parameters pH2 ) 2.2 × 10-6 mbar, pO2 ) 2.0 × 10-6 mbar, T ) 518 K. Initial conditions: θH ) 0.0, θO ) 0.2, θK ) 0.1, with a 20 µm wide adsorbate free region in the center of a 1200 µm wide interval (figure shows 400 µm wide section). From ref 9.

geneous state exhibits a higher catalytic activity than the uniform state. Pattern formation in this system enhances the catalytic activity because the catalytic poison is concentrated in a part of the surface with the remaining area still being active. For the specific pH2, pO2 parameter range of Figure 8, the T range for the stationary patterns is roughly 60 K wide with the spatial periodicity increasing with temperature. The wavelength grows if the diffusion of potassium and/or hydrogen is made faster, whereas a variation of the oxygen diffusion coefficient was found to have no discernible effect on the lateral periodicity. 4. Discussion The findings that the stationary patterns bifurcate from a spatially uniform state and that they possess an intrinsic wavelength, at first sight seems to clearly point toward a Turing structure. In the original mechanism of a Turing structure the different diffusion constants of activator and inhibitor, are essential for the pattern, whereas energetic interactions between the reacting particles were not introduced.25,26 In our case, however, the energetic interactions are essential because without them no pattern would develop. One could of course still denote the patterns discussed here as Turing-like patterns but such a term would be imprecise. A better description would be reactive

14626 J. Phys. Chem. B, Vol. 108, No. 38, 2004 phase separation because the formation of the large K + O coadsorption islands is driven by attractive interactions between potassium and oxygen. A phase separation already occurs of course in simple systems without reaction and just attractive interactions but the size of the islands formed there would be microscopic. The coupling with the reaction driving the system far from the thermodynamic equilibrium is responsible for the macroscopic size of the coadsorbate islands. This has been demonstrated in simulations with a general model in which the effect of attractive interactions had been incorporated, whereas the influence of the promoter/poison on the catalytic reaction steps had been dropped completely.9 Experimental examples for reactive phase separation were found in polymer blends and two-component Langmuir layers.27-31 In the realistic model, we have neglected details of the surface chemistry such as the formation of an OH species and the observation of an oxide-like high coverage K + O phase with shifted Rh 3d core level peaks in photoelectron spectroscopy.8 We likewise neglected the formation of ordered K + O phases, whose presence under pattern forming reaction conditions has been verified with LEEM.6 The presence of ordered phases presumably influences the surface diffusion of the mobile species and the reactivity of oxygen.1 The nearly quantitative reproduction of the experimental results with the present model suggests, that these simplifications probably have only a minor influence on the system behavior. Aside from the present system, the effect of alkali metal condensation has only been studied so far with microstructured Rh(110)/Pt surfaces and with alloyed Rh/Pt surfaces.32 In the NO + H2 reaction on Rh(110)/K, the system preserved its excitability in the presence of potassium, and pulse trains transporting potassium were observed.33 A certain analogy to some aspects of the potassium system can be found in a quite different field, in solid-state physics. Due to distortions of the lattice, impurities in a material are attracted to dislocations and, if an external force causes these dislocations to move, the impurities are dragged along in a force field influencing the motion of the dislocation.34 This behavior is reminiscent of the K enrichment in the front region found here. How far this analogy really holds and what consequences can be drawn from it need to be shown in further studies. The most important applications of the present results clearly are in the field of heterogeneous catalysis. In a number of important industrial catalytic reactions alkali metals are used as promoters but the demonstration of pattern forming effects in such systems will first of all be an experimental task. 5. Conclusions The effect of alkali condensation into stationary coadsorbate islands under reaction conditions has been successfully simulated with a realistic model. The enrichment of potassium in the front region, the time-dependent changes in the K concentration profiles and in the front velocity and formation of stationary K + O islands could all be modeled nearly quantitatively. The formation of the Turing-like stationary patterns can be described as reactive phase separation because energetic interactions between the coadsorbed particles were shown to be the main driving force. The concept of reactive phase separation appears applicable to a broad class of systems including promotors and

Hinz et al. poisons on catalytic surfaces as well as alloyed bimetallic catalytic surfaces where one of the metallic constituents is mobile and has a tendency to form stable compounds with one of the catalytic reactant. Acknowledgment. The authors are indebted to I. G. Kevrekidis, Princeton, for many fruitful and stimulating discussions. References and Notes (1) Kiskinova, M. Poisoning and Promotion in Catalysis Based on Surface Science Concepts. In Studies in Surface Science and Catalysis 70; Delmon, B., Yates, J. T., Eds.; Elsevier: Amsterdam, 1992. (2) Physics and Chemistry of Alkali Metal Adsorption; Bonzel, H. P., Bradshaw, A. M., Ertl, G., Eds.; In Material Science Monographs 67; Elsevier: Amsterdam, 1989. (3) Diehl, R. D.; McGrath, R. Physics of Covered Solid Surfaces-I. Adsorbed Layers on Surfaces; Bonzel, H. P., Ed.; In Landolt-Bornstein Numerical Data and Functional Relationships in Science and Technology; Springer: Berlin, 2001; Vol. 111/42, pp 131-177. Diehl, R. D.; McGrath, R. Surf. Sci. Rep. 1996, 23, 43. (4) Marbach, H.; Gu¨nther, S.; Luerssen, B.; Gregoratti, L.; Kiskinova, M.; Imbihl, R. Catal. Lett. 2002, 83, 161. (5) Marbach, H.; Hinz, M.; Gu¨nther, S.; Gregoratti, L.; Kiskinova, M.; Imbihl, R. Chem. Phys. Lett. 2002, 364, 207. (6) Marbach, H.; Lilienkamp, G.; Wei, H.; Gu¨nther, S.; Suchorski, Y.; Imbihl, R. Phys. Chem. Chem. Phys. 2003, 5, 2730. (7) Gu¨nther, S.; Marbach, H.; Imbihl, R.; Gregoratti, L.; Barinov, A.; Kiskinova, M. Surf. ReV. Lett. 2002, 9, 751. (8) Marbach, H.; Gu¨nther, S.; Neubrand, T.; Hoyer, R.; Gregoratti, L.; Kiskinova, M.; Imbihl, R. J. Phys. Chem., submitted. (9) De Decker, Y.; Marbach, H.; Hinz, M.; Gu¨nther, S.; Kiskinova, M.; Mikhailov, A. S.; Imbihl, R. Phys. ReV. Lett. 2004, 92, 198305. De Decker, Y.; Mikhailov, A. S. J. Phys. Chem., in press. (10) Mertens, F.; Imbihl, R. Chem. Phys. Lett. 1995, 242, 221. (11) Makeev, A.; Imbihl, R. J. Chem. Phys. 2000, 113, 3854. (12) Gu¨nther, S.; Marbach, H.; Imbihl, R.; Baraldi, A.; Lizzit, S.; Kiskinova, M. J. Chem. Phys. in press. (13) Esch, F. in preparation. (14) Norton, P. R. In The Chemical Physics of Solid Surfaces and Heterogeneous Catalysis; Elsevier: Amsterdam, 1982; Vol. 4, p 27. (15) Vo¨lkening, S.; Bedu¨rftig, K.; Jacobi, K.; Wintterlin, J.; Ertl, G. Phys. ReV. Lett. 1999, 83, 2672. (16) Anton, A. B.; Cadogan, D. C. Surf. Sci. 1990, 239, L548. (17) Marbach, H. Thesis, Hannover, 2002. (18) Mavrikakis, M. in preparation. (19) Schwarz, E.; Lenz, J.; Wohlgemuth, H.; Christmann, K. Vacuum 1990, 41, 167. Ehsasi, M.; Christmann, K. Surf. Sci. 1988, 194, 172. (20) Kreuzer, H. J.; Pangher, Z.; Payne, S. H.; Nichtl-Pecher, W.; Hammer, L.; Mu¨ller, K. Surf. Sci. 1994, 303, 1. Payne, S. H.; Kreuzer, H. J.; Frie, W.; Hammer, L. Surf. Sci. 1999, 421, 279. (21) Comelli, G.; Dhanak, V. R.; Kiskinova, M.; Pangher, N.; Paolucci, G.; Prince, K. C.; Rosei, R. Surf. Sci. 1992, 260, 7. (22) Wagner M. L.; Schmidt, L. D. J. Chem. Phys. 1995, 99, 805. (23) Seebauer, E. G.; Allen, C. E. Prog. Surf. Sci. 1995, 49, 265. Mann, S. S.; Seto, T.; Barnes, C. J.; King, D. A. Surf. Sci. 1992, 261, 155. (24) Westre, E. D.; Brown, D. E.; Kutzner, J.; George, S. M. Surf. Sci. 1993, 294, 185. (25) Turing, A. M. Philos. Trans. R. Soc. B 1952, 237, 37. (26) Kapral, R., Showalter, K., Eds.; Chemical WaVes and Patterns; Kluwer: Dordrecht, The Netherlands, 1994. (27) Glotzer, S. C.; Di Marzio, E. A.; Muthukumar, M. Phys. ReV. Lett. 1995, 74, 2034. (28) Motoyama, M.; Ohta, T. J. Phys. Soc. Jpn. 1997, 66, 2715. (29) Tran-Cong, Q.; Harada, A. Phys. ReV. Lett. 1996, 76, 1162. (30) Tabe, Y.; Yokoyama, H. Langmuir 1995, 11, 4609. (31) Reigada, R.; Sagues, F.; Mikhailov, A. S. Phys. ReV. Lett. 2002, 89, 038301. (32) Gu¨nther, S.; Marbach, H.; Hoyer, R.; Imbihl, R.; Gregoratti, L.; Barinov, A.; Kiskinova, M. J. Chem. Phys. 2002, 117, 2923. (33) Marbach, H.; Gu¨nther, S.; Neubrand, T.; Imbihl, R. Chem. Phys. Lett., in press. (34) Mendelev, M. I.; Srolovitz, D. J. Acta mater. 2001, 49, 2843.