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Crossover Between Mean-Field and Ising Critical Behavior in a Lattice-Gas Reaction-Diffusion Model Da-Jiang Liu the Ames Laboratory

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Crossover Between Mean-Field and Ising Critical Behavior in a Lattice-Gas Reaction-Diffusion Model Da-Jiang Liu the Ames Laboratory Physics and Astronomy Publications Physics and Astronomy 2004 Crossover Between Mean-Field and Ising Critical Behavior in a Lattice-Gas Reaction-Diffusion Model Da-Jiang Liu The Ames Laboratory N. Pavlenko Universitat Hannover James W. Evans Iowa State University, [email protected] Follow this and additional works at: http://lib.dr.iastate.edu/physastro_pubs Part of the Chemistry Commons, and the Physics Commons The ompc lete bibliographic information for this item can be found at http://lib.dr.iastate.edu/physastro_pubs/456. For information on how to cite this item, please visit http://lib.dr.iastate.edu/howtocite.html. This Article is brought to you for free and open access by the Physics and Astronomy at Iowa State University Digital Repository. It has been accepted for inclusion in Physics and Astronomy Publications by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Crossover Between Mean-Field and Ising Critical Behavior in a Lattice- Gas Reaction-Diffusion Model Abstract Lattice-gas models for CO oxidation can exhibit a discontinuous nonequilibrium transition between reactive and inactive states, which disappears above a critical CO-desorption rate. Using finite-size-scaling analysis, we demonstrate a crossover from Ising to mean-field behavior at the critical point, with increasing surface mobility of adsorbed CO or with decreasing system size. This behavior is elucidated by analogy with that of equilibrium Ising-type systems with long-range interactions. Keywords critical behavior, lattice-gas reaction-diffusion model, Ising, mean field Disciplines Chemistry | Physics Comments This article is published as Liu, Da-Jiang, N. Pavlenko, and J. W. Evans. "Crossover between mean-field and Ising critical behavior in a lattice-gas reaction-diffusion model." Journal of statistical physics 114, no. 1 (2004): 101-114, doi:10.1023/B:JOSS.0000003105.50683.c6. Posted with permission. Rights The final publication is available at Springer via http://dx.doi.org/10.1023/B:JOSS.0000003105.50683.c6 This article is available at Iowa State University Digital Repository: http://lib.dr.iastate.edu/physastro_pubs/456 Crossover between Mean-Field and Ising Critical Behavior in a Lattice-Gas Reaction-Diffusion Model Da-Jiang Liu Ames Laboratory (USDOE), Iowa State University, Ames, Iowa 50011; e-mail: dajiang@fi.ameslab.gov N. Pavlenko Institut f¨ur Physikalische Chemie und Elektrochemie, Universit¨at Hannover, Callinstr. 3-3a, D-30167, Hannover, Germany J. W. Evans Ames Laboratory (USDOE) and Department of Mathematics, Iowa State University, Ames, Iowa 50011 (Dated: July 25, 2013) Lattice-gas models for CO oxidation can exhibit a discontinuous nonequilibrium transition be- tween reactive and inactive states, which disappears above a critical CO-desorption rate. Using finite-size-scaling analysis, we demonstrate a crossover from Ising to mean-field behavior at the crit- ical point, with increasing surface mobility of adsorbed CO or with decreasing system size. This behavior is elucidated by analogy with that of equilibrium Ising-type systems with long-range in- teractions. I. INTRODUCTION some of these systems, it is possible to derive rigorous reaction-diffusion equations in the hydrodynamic limit of rapid exchange or diffusion.6,10 Nonequilibrium phase transitions and pattern forma- In this paper, we consider specifically surface reaction tion occur in a broad variety of physical, chemical, biolog- 1,2 models, which incorporate adsorption-desorption and re- ical, and sociological systems. Such complex behavior action steps, as well as surface mobility. Our motivation sometimes resembles more familiar phenomena in equilib- is that these models not only contain the essential ingre- rium systems. However, there are also some fundamental dients to elucidate complex phenomena observed in real differences due in part to the lack of free energy minimiza- surface catalysis, but also exhibit many fascinating fea- tion or detailed-balance constraints for nonequilibrium tures of the above idealized models. On the one hand, systems. The latter feature also provides a challenge they exhibit complex dynamics and phase transitions as- for development of a rigorous theoretical framework. sociated with the reaction mechanism (and perhaps with Nonetheless, the spatial-temporal behavior of nonequi- adspecies interactions), but also surface diffusion rates librium transitions can be effectively studied using one of provide a key parameter which can tune behavior. In two paradigms. The first is through a set of partial dif- fact, while many theoretical studies have considered the ferential equations (e.g., reaction-diffusion equation for regime of limited or zero mobility, it is the hydrodynamic chemical reactions), where phase transitions are related 1,2 regime of rapid mobility that is most often realized in to bifurcations of the underlying nonlinear dynamics. experiments.6 By default, criticality is governed by mean-field (MF) be- Our focus here is on first-order phase transitions and havior, although Ginzburg criteria can be developed to 3 associated critical phenomena in models for CO oxida- characterize noise-dominated regimes. The second ap- tion on metal surfaces, behavior analogous to that of proach utilizes microscopic interacting particle systems the ferromagnetic Ising model. The central question is of which nonequilibrium lattice-gas models form an im- “how does critical behavior depend on surface mobility”, portant subclass. For these models, one can successfully anticipating that MF behavior could well apply in the apply techniques such as finite-size-scaling, and concepts hydrodynamic regime, but that a crossover may occur including that of universality classes familiar from critical 4 to non-MF behavior for lower mobility. Through precise phenomena in equilibrium phase transitions. finite-size-scaling (FSS) studies of simulation data, we Many types of nonequilibrium phase transitions have confirm this prediction and quantify crossover behavior. arXiv:cond-mat/0304386v1 [cond-mat.stat-mech] 17 Apr 2003 been studied: continuous transitions to absorbing states4 We also comment on experimental realization of crossover (which are often in the universality class of directed per- for CO oxidation in high-pressure nanoscale systems. colation), discontinuous transitions between reactive and inactive states,5,6 and order-disorder transitions,7 all of which occur in adsorption-desorption or reaction models; II. REACTION MODEL: SPECIFICATION AND driven lattice-gases4,8 (where the driving breaks detailed BEHAVIOR balance); dynamic Ising models with both spin flip and spin exchange dynamics, but at two different tempera- We now describe our lattice-gas reaction-diffusion tures leading to interesting crossover phenomena.9 For model, where gas (ads) denote gas-phase (adsorbed- 2 phase) species. The following steps are implemented and its extension with desorption of CO(ads). The key using a square lattice of adsorption sites with periodic modifications in our model are inclusion of: a) hopping of boundary conditions. (i) CO(gas) adsorbs on single CO(ads); b) NN exclusion of O(ads); and c) finite rather empty sites at rate p = pCO (chosen between 0 and than infinite reaction rate. All these features are impor- 1) and desorbs with rate d. (ii) O2(gas) adsorbs dis- tant for realistic modeling of CO oxidation. However, it sociatively onto a pair of diagonally adjacent empty sites is primarily the first feature [hopping of CO(ads)] which at rate pO2 = 1 − pCO, provided all six neighbors are impacts the critical behavior of the reactive-inactive tran- free of O(ads). This “eight-site” rule reflects strong sition studied in this paper. Furthermore, it is clear nearest-neighbor (NN) O(ads)-O(ads) repulsions. Since that our conclusions about critical behavior would ap- O(ads) is treated as immobile, this adsorption rule en- ply to the ZGB model, modified to include desorption sures that O(ads) never occupy adjacent sites. (iii) ad- and hopping of CO(ads). The second feature [NN exclu- jacent CO(ads) and O(ads) react at rate k (set to unity sion of O(ads)] results in an oxygen poisoning transition here). (iv) CO(ads) hops to adjacent empty sites at rate in the ZBG model5 being replaced by an order-disorder h. This model has also been discussed elsewhere, e.g., transition.7 This does not affect critical behavior of the Refs. 6,7. reactive-inactive transition, with one caveat. For small Conventional kinetic Monte Carlo (KMC) simulations h, NN exclusion does also lead to a loss of the reactive- are used to assess model behavior for finite h. Noting inactive transition (see Sec. IV). The third modification that mobility of CO(ads) is often very high under ultra- to the original ZGB model is that instead of k = ∞, high vacuum conditions, we also perform a direct anal- we choose k = 1, i.e., the reaction rate equals the total ysis of limiting behavior for h = ∞ using a “hybrid” adsorption rate. The choice is of course somewhat arbi- treatment: here the distribution of O(ads) is described trary, but the basic behavior of these models does not within a lattice-gas framework, but one only tracks the change varying the reaction rate from O(1) to ∞. For number of CO(ads) and assumes that they are randomly further discussion of effects of varying k on the steady distributed on sites not occupied by O(ads).6 For very state bifurcation diagram, see Ref. 12. large h and low O-coverages, this is valid. We now briefly review the steady state behavior of this model for an infinite system. First, consider the case of III. CRITICAL POINT DETERMINATION AND finite h< ∞.5,6 For low d, one typically finds a first-order FSS ANALYSIS ∗ transition at some p = pCO = p between a reactive state ∗ with low CO-coverage hθCOi (for p<p ) and an inactive Unlike the Ising model, transitions in reaction model ∗ state with high hθCOi (for p>p ). The transition dis- do not involve simple symmetry-breaking.
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