We study the effects of generalised surface disorder on the monomer-monomer
model of heterogeneous catalysis, where disorder is implemented by allowing
different adsorption rates for each lattice site. By mapping the system in the
reaction-controlled limit onto a kinetic Ising model, we derive the rate
equations for the one and two-spin correlation functions. There is good
agreement between these equations and numerical simulations. We then study the
inclusion of desorption of monomers from the substrate, first by both species
and then by just one, and find exact time-dependent solutions for the one-spin
correlation functions.Comment: LaTex, 19 pages, 1 figure included, requires epsf.st

D. A. Head

D. ben-Avraham

D. Considine

D. Vlachos

G. Forgacs

G. J. Rodgers

J. Evans

J. Zhuo

K. Fichthorn

L. Frachebourg

P. L. Krapivsky

P. Meakin

R. Ziff

English

arXiv

Head, DA

Rodgers, GJ

Brunel University Research Archive

arXiv:cond-mat/9602088 v1   15 Feb 1996Kinetics of Catalysis with Surface DisorderD.A.Head and G.J.RodgersDepartment of Physics, Brunel University, Uxbridge,Middlesex. UB8 3PH, UKAbstractWe study the effects of generalised surface disorder on the mon-omer-monomer model of heterogeneous catalysis, where disorder isimplemented by allowing different adsorption rates for each latticesite. By mapping the system in the reaction-controlled limit onto akinetic Ising model, we derive the rate equations for the one and two-spin correlation functions. There is good agreement between theseequations and numerical simulations. We then study the inclusionof desorption of monomers from the substrate, first by both speciesand then by just one, and find exact time-dependent solutions for theone-spin correlation functions.PACS numbers: 05.70.Ln, 82.65.Jv, 82.20.Mje-mail: David.Head@brunel.ac.uk, G.J.Rodgers@brunel.ac.ukOctober 31, 20061I. INTRODUCTIONDiffusionless surface-reaction models were first introduced by Ziff, Gulariand Barshad [1], who investigated a monomer-dimer reaction correspondingto the chemical reaction 2CO + O2 → 2CO2 on a catalytic surface. A well-studied variant [2, 3, 4] employs the simpler monomer-monomer reaction,described byAgas + SkA−→ AsurfaceBgas + SkB−→ BsurfaceAsurface +BsurfacekR−→ ABgas + 2S, (1)where S denotes an empty site. This process exhibits a kinetic phase whenthere are equal propensities of A and B species, in which the long-time ki-netics become dominated by domain coarsening. Mean-field analysis [5], inwhich every site is taken to be connected to every other site in a ‘completegraph’, demonstrated that finite lattices will always saturate - that is, thelattice will either become full of A’s, or full of B’s, and the process will stop.Krapivsky [6] recently solved the model exactly in the reaction-controlledlimit kR →∞ by mapping the system onto the standard Ising model.Many enhancements to these models have been studied with a view tomore closely modelling actual chemical processes, including nearest neigh-bour excluded adsorption [7] and surface diffusion [4, 8]. However, only re-2cently have the effects of surface disorder been touched upon by Frachebourget.al. [9]. They chose to model a disordered surface by taking a lattice oftwo different types of site, one which favours adsorption by the A-species,and one which favours adsorption by the B’s. They showed numerically thatsuch disorder allows for a reactive equilibrium in two dimensions.In this paper, we extend the analytical method used in [6] to a generalform of surface disorder, based on [9] but allowing for a range of differenttypes of site in the lattice. Furthermore, we also investigate separately theeffects of desorption in the system. All the results presented are for thephysically relevant case of two dimensions.This paper is organised as follows. In section II we define the modeland derive the general rate equations for the n-spin correlation functions.In section III, these equations are applied to a model similar to that in [9]and their solutions are compared to numerical simulations. In sections IVand V we include the effects of desorption, first by both species and thenby just one, and derive exact solutions. The conclusions are summarised insection VI.II. RATE EQUATIONSWe consider the surface reaction A+B → 2S on a periodic L×L squarelattice, ignoring the effects of diffusion and desorption. For simplicity, wetake the reaction-controlled limit, where the adsorption of A and B species is3taken to be infinitely fast so that the substrate is always full. The algorithmemployed here is to select a nearest-neighbour (NN) pair at random, checkfor an AB-reaction, and, if so, remove the particles and immediately refillboth sites.With the usual homogeneous model, the probability of filling a site withand A or B is independant of the site chosen - in this model, however, thatprobability is allowed to vary. Specifically, we introduce the site inhomo-geneity matrix Pij , 0 ≤ Pij ≤ 1 ∀i, j, such that the probability of filling thesite (i, j) with an A is given by Pij (or, equivalently, a probability 1− Pij offilling the site with a B).Since in the reaction-controlled limit each site (i, j) has only two possiblestates, we can map this model onto an Ising model with mixed Glauber-Kawasaki dynamics [6], identifying A’s with Sij = +1 and B’s with Sij = −1.The master equation for P (S, t), the probability distribution for the systemto be in the state S = {Sij} at time t, isddtP (S, t) =∑i,j[ Uij(FijS)P (FijS, t)− Uij(S)P (S, t) ]+∑i,j[ Vij(FijFi+1jS)P (FijFi+1jS, t)− Vij(S)P (S, t) ]+∑i,j[Wij(FijFij+1S)P (FijFij+1S, t)−Wij(S)P (S, t) ]. (2)The flip operator Fij acts on the system state vector S by flipping thesign of the Sij component, leaving the remaining components unchanged.4Uij corresponds to Glauber spin-flip dynamics [10], whereas Vij and Wij cor-respond to Kawasaki exchange dynamics. Equation (2) is identical the thehomogeneous case, except that now the full expressions for Uij ,Vij and Wijare given by4τ1Uij = (1− SijSi+1j){1− dij + Sij(1− a+ij)}+ (1− SijSi−1j){1− di−1j + Sij(1− a+i−1j)}+ (1− SijSij+1){1− eij + Sij(1− b+ij)}+ (1− SijSij−1){1− eij−1 + Sij(1− b+ij−1)}, (3)4τ2Vij = (1− SijSi+1j){dij + a−ijSij}, (4)4τ2Wij = (1− SijSij+1){eij + b−ijSij}, (5)where the constant coefficients a±ij ,b±ij , dij and eij are related to the inhomo-geneity matrix Pij,a±ij = Pi+1j ± Pij,b±ij = Pij+1 ± Pij,dij = Pij + Pi+1j − 2PijPi+1j,eij = Pij + Pij+1 − 2PijPij+1. (6)We proceed by deriving the rate equations for the one and two-spin cor-relation functions, where the general n-spin function is given by5〈Si1j1 . . . Sinjn〉 =∑SSi1j1 . . . SinjnP (S, t). (7)Using this and (2), some lengthy but straightforward calculations resultin the following hierarchy of differential equations, using the renormalisedtime scale τ defined by τ−1 = τ−11 + τ−12 , and setting τ1 = τ2,4τddt〈Sij〉 = ∆ij〈Sij〉+ (1− 2Pij)〈Sij{∆ijSij}〉, (8)4τddt〈SijSkl〉 = (∆ij +∆kl)〈SijSkl〉+(1− 2Pij)〈SijSkl{∆ijSij}〉+(1− 2Pkl)〈SijSkl{∆klSkl}〉. (9). . . for |i− k|+ |j − l| > 1Here, ∆ij〈Sij〉 = −4〈Sij〉 + 〈Si+1j〉 + 〈Si−1j〉 + 〈Sij+1〉 + 〈Sij−1〉 is thediscrete Laplacian. For |i−k|+ |j− l| = 1, i.e. for nearest neighbour 2-pointcorrelations, the rate equation has a more complex form. For example,64τddt〈SijSi+1j〉 = (2dij − 8)〈SijSi+1j〉+〈Si−1jSi+1j〉+ 〈SijSi+2j〉+ 〈SijSi+1j+1〉+〈SijSi+1j−1〉+ 〈Sij−1Si+1j〉+ 〈Sij+1Si+1j〉+(1− 2Pij){〈Si−1jSijSi+1j〉 −32〈Si+1j〉}+(1− 2Pi+1j){〈SijSi+1jSi+2j〉 −32〈Sij〉}+2(1− 2dij). (10)In the homogeneous limit Pij →12, the results in [6] are recovered.III. TWO-SITE DISORDERWe now turn to the case where Pij can take just two different values, por q = 1− p, with an equal number of p-sites and q-sites. This correspondsto the model given in [9] with equal fluxes of A and B species, ǫ = |p − 12|and c− = c+ =12, using the notation given there.Since p + q = 1, the global dynamics of the system must be unchangedunder the transformation (p, q) → (1 − p, 1 − q) = (q, p). This symmetrymeans that the system cannot favour one state over the other, and so theaverage of 〈Sij〉 taken over the entire L× L lattice,1L2∑i,j〈Sij〉, will alwaystend to zero in the L → ∞ limit. An important consequence of this is thatif a finite system always saturates, then it does so with equal probability7of saturating either to every site being +1, or every site being -1, and so〈Sij〉|t=∞ = 0 ∀i, j, regardless of whatever Pij may be. If a reactive steady-state occurs - that is, if the average saturation time diverges at least as fastas eL2[5] - then it should be expected that 〈Sij〉 may be non-zero for t→∞(if p 6= 12). It is the purpose of this section to apply the rate equations derivedin section II to predict the equilibrium value of 〈Sij〉 on p-sites in any suchnon-trivial steady state.Although the concentrations of p-sites and q-sites are equal, differentarrangements of the sites can dramatically alter the long-time dynamics ofthe system. For instance, choosing to split the lattice into two alternatingc(2 × 2) sublattices, with one sublattice full of p-sites and the other full ofq-sites, results in a system with no non-trivial steady states for p 6= 0 or 1.Since saturation always occurs, 〈Sij〉t=∞ = 0 on either type of site.A more informative model can be constructed by randomly arranging thep and q sites. This allows for regions of p-sites, which will all tend to be fixedinto the same state, and regions of q-sites, which will all tend to be fixed intothe other state, to ‘pin’ the dynamics into a reactive equilibrium. Althoughexact analysis of this model is obviously impossible, a useful approximationcan be made by assuming that every site is surrounded by exactly 2 p-sitesand 2 q-sites. It is then possible to write down (8) and (10) for the two sortsof site, 〈Sij〉p and 〈Sij〉q, and the various two-point functions.To obtain a closed set of equations, further approximations must be made8to reduce the three-point functions in (10) to one and two point functions.The obvious choice is〈SijSklSmn〉 ≈ 〈SijSkl〉〈SklSmn〉, (11)but this has the unwanted side-effect that 〈Sij〉p + 〈Sij〉q 6= 0, somethingwhich cannot be true since p + q = 1. To restore the required symmetry wemust also include an alternative three-point approximation,〈SijSklSmn〉 ≈ 〈Sij〉〈SklSmn〉. (12)For greater clarity, we denote the one-spin correlation function〈Sij〉p = −〈Sij〉q by yp, the two-spin correlation function between two NNp-sites (or, equivalently, two NN q-sites) by zpp, and use zpq for the two-pointfunction between nearest neighbour p and q sites. Setting τ = 1, we can nowobtain a closed set of equations,2ddtyp = −2yp + (1− 2p)(zpq + zpp − 2), (13)4ddtzpp = (4pq − 8)zpp − 3(1− 2p)yp +{(1− 2p)yp + 3zpp}(zpp + zpq) + (2− 4pq), (14)4ddtzpq = 4pq − (4pq + 6)zpq + 3(1− 2p)yp + 3zpq(zpp + zpq). (15)The most constructive way to test the validity of this analysis is to com-pare the value of yp at equilibrium, as predicted by iterating eqns.(13–15),9to numerical simulations. In these simulations, the sites are initially ran-domly filled with +1’s or −1’s, so the corresponding initial conditions forthe iteration procedure are yp|t=0 = zpp|t=0 = zpq|t=0 = 0. The results arecompared in fig.1., where the simulation results compare favourably with theapproximate analysis, the agreement improving for larger values of p.Note that even when p = 1, yp does not tend to +1, either in the theoryor in the numerical work. This is because it is possible to have a jammedstate where, for instance, a q-site surrounded by 4 p-sites may initially startat +1 but be unable to change, since if all 4 NN p-sites get fixed into a +1state before they have reacted with the central q-site, then the q-site willnever be able to react and so it will stay as +1 for all time, despite the factthat it has Pij = 0.IV. INHOMOGENEOUS DESORPTIONWe now turn to an enhanced model studied by Fichthorn, Gulari andZiff [11], which introduces noise into the system in the form of the desorp-tion of A and B species from the substrate. They demonstrated numerically,later confirmed by mean-field analysis [12], that even a small desorptionrate induces steady-state reactivity onto finite lattices. In our version of themodel, sites vacated by desorption are refilled by an A or a B as defined bythe inhomogeneity matrix, which we now call Qij . Qij differs from Pij inthat now it only applies to sites refilled after desorption - sites vacated after10an A +B → 2S reaction have an equal chance of being refilled either by anA or by a B. Thus, the reaction kinetics alone are the same as the usual ho-mogeneous model, and the Uij ,Vij and Wij operators without the desorptiontake their simpler form found by setting Pij =12in (3–5). Explicitly,8τ1Uij(S) = 4− Sij(Si+1j + Si−1j + Sij+1 + Sij−1), (16)8τ2Vij(S) = 1− SijSi+1j , (17)8τ2Wij(S) = 1− SijSij+1. (18)To include inhomogeneous desorption within this formulation, we replaceUij with Udij ,Udij = Uij +12τ3{1 + Sij(1− 2Qij)}, (19)where, as in [6], we introduce a renormalised time scale τ and the spin-flipparameter γ, defined by1τ=1τ1+1τ2+1τ3, (20)γ = 1− τ/τ3. (21)The one point spin-correlation rate equation can now be recalculatedusing (2) and (16–19),114τddt〈Sij〉 = γ∆ij〈Sij〉 − 4(1− γ){〈Sij〉+ (1− 2Qij)}. (22)This can be solved by using a generating function, G(X, Y, t), defined interms of the time-dependant one-spin correlation function 〈Sij〉,G(X, Y, t) =∞∑i=−∞∞∑j=−∞X iY j〈Sij〉. (23)Combining this with (22) gives rise to a differential equation for G,∂G∂t=Gτ{γ4(X +1X+ Y +1Y)− 1}−1τ3∞∑i,j=−∞X iY j(1− 2Qij). (24)Noting that, except for G(X,Y,t), the right hand side of (24) is inde-pendent of time, it is not difficult to derive an explicitly time-dependentexpression for 〈Sij〉 in terms of its initial state, σij = 〈Sij〉|t=0,〈Sij〉 = e−t/τ∞∑k,l=−∞σklIi−k(γt2τ)Ij−l(γt2τ)−1τ3∞∑k,l=−∞(1− 2Qkl)∫ t0e−t′/τIi−k(γt′2τ)Ij−l(γt′2τ)dt′, (25)where Ii(t) is the ith order modified Bessel function. In the special case∆ijQij = 0 it is possible to to rewrite the second term on the right hand sideof (25) as12−14τ3∞∑k,l=−∞(1− 2Qkl) {fi−k+1 j−l + fi−k−1 j−l + fi−k j−l+1 + fi−k j−l−1} ,(26)where for clarity we have introducedfij(t) =∫ t0e−t′/τIi(γt′2τ)Ij(γt′2τ)dt′, (27)which obeys the identityfi+1j + fi−1j + fij+1 + fij−1 =4γfij −4τγδi0δj0+4τγe−t/τIi(γt2τ)Ij(γt2τ), (28)with δij the usual Kro¨necker Delta. Substituting (28) into (25) and (26)results in an exact expression,〈Sij〉 = 2Qij − 1 + e−t/τ∞∑k,l=−∞(1− 2Qkl + σkl)Ii−k(γt2τ)Ij−l(γt2τ). (29)So when ∆ijQij = 0, 〈Sij〉 → 2Qij − 1 exponentially as t→ ∞, again inagreement with the homogeneous result of [11]. With desorption, jammingis no longer possible and so now 〈Sij〉 → 1 when Qij = 1. Although thisfinal solution is exact, it is hard to see what physical applications a mixedhomo/inhomogeneous model such as this one may have.13V. INHOMOGENEOUS ONE-SPECIES DESORPTIONWhilst investigating the monomer-dimer model, Ziff, Gulari and Bar-shad [1] briefly discussed the additional feature of allowing just the monomersto desorb. Physically, this corresponds to the reaction 2CO +O2 → 2CO2where only the CO can desorb from the substrate, which is a good approxi-mation for this reaction at the usual operating temperatures.To apply a similar principle to our monomer-monomer model, we extendthe analysis in section IV to allow for the desorption of A-species only, withthe inhomogeneity matrix Qij only applying to sites vacated after desorption.Thus, the flip-exchange operators are unchanged from (16–18), but now wereplace Uij withUdij = Uij +1−Qij2τ3(1 + Sij) . (30)Furthermore, Qij is also taken to be a constant matrix, Qij = q ∀i, j. Therate equation for the one-spin correlation function (22) is now4τddt〈Sij〉 = ∆ij〈Sij〉 − γ(1− q)(1 + 〈Sij〉). (31)The definitions of τ and γ have now altered from the previous case,1τ=1τ1+1τ2, (32)14γ =4ττ3. (33)Applying the same generating function (23) results in a new partial dif-ferential equation for G(X, Y, t) acting on an L× L lattice,∂G∂t=Gτ{14(X +1X+ Y +1Y)− 1}−1− qτ3(G+ L2). (34)Continuing as before, an explicit time-dependent expression for 〈Sij〉 isreached,〈Sij〉 = e−t(1τ+1−qτ3)L/2∑k,l=−L/2σklIi−k(t2τ)Ij−l(t2τ)−(1− q)L2τ3∫ t0e−t′(1τ+1−qτ3)Ii(t′2τ)Ij(t′2τ)dt′. (35)VI. CONCLUSIONS AND DISCUSSIONWe have introduced a methodology for dealing with the effects of gener-alised surface disorder on the monomer-monomer reaction processA+B → 2S, by mapping the system in the reaction-controlled limit ontoan Ising Model. The two-dimensional rate equations were derived, includingthe very concise one-spin correlation equation (8), and used to study thespecial case of two-site disorder. Here, it was found that the global systemdynamics are sensitive to the choice of layout of the two different types of15site. Catalysts consisting of two different molecules arranged in a regularmanner, such as on two alternating c(2× 2) sublattices, allow for no reactiveequilibrium and will always saturate on finite lattices. Choosing to randomlyarrange the sites, however, allowing compact clusters of the same site, wasshown to produce a reactive steady-state. Analysis based on the rate equa-tions was used to predict the concentration of A’s and B’s on the differenttypes of site, showing reasonable agreement between theory and simulationdespite the rather crude approximations involved in the analysis.The model was then extended to include desorption from the substrate,either by one or both species, and was solved exactly in both cases.Extending this work to dimensions other than d = 2 is straightforwardonce the mapping onto the Ising model has been achieved. Indeed, the rateequations for d = 1 can been immediately seen from those given here (8–10).We have focused on d = 2 since the most useful physical application is ofsurface catalysis.It should be noted that the definition of inhomogeneity we chose to employhere is only one of many ways of modelling surface disorder. For instance,requiring that each site be ‘hit’ a different number of times before adsorbinga particle, or assigning a quenched random ‘energy’ to each site and alwaysadsorbing the particles onto the vacant site with the lowest energy, are justtwo alternative possibilities. We intend to study some of these in future work.16References[1] R.Ziff, E.Gulari and Y.Barshad, Phys.Rev.Lett 56 2553(1986).[2] R.Ziff and K.Fichthorn, Phys.Rev.B 34 2038(1986).[3] P.Meakin and D.Scalapino, J.Chem.Phys. 87 731(1987).[4] J.Evans and T.Ray, Phys.Rev.E 47 1018(1993).[5] D.ben-Avraham, S.Redner, D.B.Considine and P.Meakin, J.Phys.A 23L613(1990).[6] P.L.Krapivsky, Phys.Rev.A 45 1067(1992).[7] J.Zhuo, S.Redner and H.Park, J.Phys A 26 4197(1993).[8] D.Vlachos, L.Schmidt and R.Aris, J.Chem.Phys 93 8306(1990).[9] L.Frachebourg, P.L.Krapivsky and S.Redner, Phys.Rev.Lett 752891(1995).[10] G.Forgacs, D.Mukamel and R.A.Pelcovits, Phys.Rev.B 30 205(1984).[11] K.Fichthorn, E.Gulari and R.Ziff, Phys.Rev.Lett 63 1527(1989)[12] D.Considine, S.Redner and H.Takayasu, Phys.Rev.Lett 63 2857(1989).17Figure 1. Plot of p vs. yp|t=∞. The line gives the values predicted bythe rate equations. Numerical simulation results are plotted as crosses. Thesimulations were performed on a 200 × 200 lattice, and averaged over 100runs.1819

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