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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]
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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 arXiv:cond-mat/0304386v1 [cond-mat.stat-mech] 17 Apr 2003 wihaeotni h nvraiycaso ietdper- states, an directed inactive reactive of between class transitions universality discontinuous the colation), in often states absorbing are to (which transitions continuous studied: been air lhuhGnbr rtracnb eeoe to developed regimes. be can noise-dominated criteria characterize be- Ginzburg (MF) although mean-field by havior, dynamics. governed is nonlinear criticality underlying default, By related the of are bifurcations transitions dif- to phase fo partial where of equation reactions), set reaction-diffusion chemical a (e.g., through equations is first ferential of The one using paradigms. studied framework. nonequi- two effectively be of theoretical challenge can behavior transitions rigorous a librium spatial-temporal provides a the also of Nonetheless, feature development latter nonequilibrium for The for constraints systems. minimiza- detailed-balance energy free or of lack tion the to fundamental part some in also due are differences there However, systems. equilib- rium in phenomena familiar more resembles sometimes ue edn oitrsigcosvrphenomena. crossover tempera- interesting different two to and at flip leading but spin tures dynamics, both exchange with models spin Ising dynamic balance); models; lattice-gases reaction driven or adsorption-desorption in occur which hnmn neulbimpaetransitions. phase equilibrium criti in from phenomena familiar classes universality concepts of and that successfully including finite-size-scaling, can as one such im- techniques models, apply an these For form models subclass. portant lattice-gas systems nonequilibrium particle which interacting of microscopic utilizes proach cl n oilgclsystems. biolog- sociological chemical, physical, and of variety ical, broad a in occur tion aytpso oeulbimpaetastoshave transitions phase nonequilibrium of types Many oeulbimpaetastosadptenforma- pattern and transitions phase Nonequilibrium rsoe ewe enFedadIigCiia eairi Behavior Critical Ising and Mean-Field between Crossover msLbrtr UDE n eateto ahmtc,Iow Mathematics, of Department and (USDOE) Laboratory Ames eairi lcdtdb nlg ihta feulbimI equilibrium of that CO with adsorbed analogy of by teractions. mobility elucidated surface is increasing behavior with f point, crossover ical a demonstrate we analysis, finite-size-scaling we ecieadiatv tts hc iapasabove disappears which states, inactive and reactive tween atc-a oesfrC xdto a xii discontin a exhibit can oxidation CO for models Lattice-gas 5,6 .INTRODUCTION I. n re-iodrtransitions, order-disorder and 4,8 weetediigbek detailed breaks driving the (where nvri¨tHnoe,Clisr -a -06,Hannove D-30167, 3-3a, Callinstr. 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For cal 1,2 d 4 r eieo ai oiiyta sms fe elzdin realized often most is that mobility experiments. hydrodynamic rapid the is In of it mobility, the regime zero considered behavior. or have limited tune of studies regime can theoretical many which rates while parameter diffusion fact, key surface a also provide but with perhaps hand, interactions), (and one mechanism adspecies as- reaction the transitions the On phase with and sociated dynamics models. complex idealized fea- exhibit above they fascinating the many of exhibit real also tures in but observed ingre- catalysis, phenomena essential surface complex the contain elucidate only to motivation not dients Our models mobility. these surface that as is well as re- steps, and action adsorption-desorption incorporate which models, oe,weegs(d)dnt a-hs (adsorbed- gas-phase denote (ads) gas where model, o Ooiaini ihpesr aocl systems. nanoscale high-pressure in oxidation CO crossover of for realization behavior. experimental we crossover on comment quantify also data, We and simulation prediction this of confirm precise studies occur Through (FSS) mobility. may lower the crossover finite-size-scaling for in a behavior apply non-MF that to well but could regime, behavior hydrodynamic mobility”, is MF of surface question that on that central depend anticipating behavior The to critical analogous does model. “how behavior Ising oxida- ferromagnetic surfaces, CO the for metal models on in tion phenomena critical associated oeo hs ytm,i spsil odrv rigorous diffusion. limit derive or to hydrodynamic exchange possible rapid the of is in it equations systems, reaction-diffusion these of some I ECINMDL PCFCTO AND SPECIFICATION MODEL: REACTION II. nti ae,w osdrseiclysraereaction surface specifically consider we paper, this In enwdsrb u atc-a reaction-diffusion lattice-gas our describe now We u ou eei nfis-re hs rniin and transitions phase first-order on is here focus Our o sn oma-edbhvo ttecrit- the at behavior mean-field to Ising rom igtp ytm ihln-ag in- long-range with systems sing-type rtclC-eopinrt.Using rate. CO-desorption critical a rwt eraigsse ie This size. system decreasing with or tt nvriy ms oa50011 Iowa Ames, University, State a osnnqiiru rniinbe- transition nonequilibrium uous 6 ,Germany r, BEHAVIOR Lattice-Gas a n 6,10 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 ). The transition dis- do not involve simple symmetry-breaking. As in the appears as d increases above some critical value dc(h). liquid-vapor phase separation problem, one needs to lo- Second, consider the hybrid model with h = ∞.6,7 Here cate the critical point in a two parameter space, i.e., (p, d) one finds a region of bistability with both reactive and for the reaction model. However here one also has the ∗ inactive states (the discontinuous transition at p = p for disadvantage that computationally efficient techniques finite h corresponding to the equistability point). Bista- for equilibrium systems (e.g., histogram-reweighting and bility disappears at a cusp bifurcation upon increasing cluster algorithms) do not apply. Thus, when using nu- d above some dc(∞). A coherent picture for both cases merical techniques such as FSS, careful analysis of sim- of infinite and finite h comes from the observation that ulation data is necessary. Below, we briefly describe our decreasing h decreases the degree of metastability or hys- procedure. teresis in the system. There are several reasonable ways to define the effec- ∗ Next, we describe how behavior changes for finite L. tive transition point p = pL for finite L. One natural For the reaction model with finite h, the discontinuous choice is the point where the change in the relevant or- transition in the hθCOi versus pCO mentioned above is der parameter (i.e., the CO coverage hθCOi) is greatest. rounded for finite L, but becomes sharper as L increases. All choices should converge to the same value as L di- The trend is analogous to behavior for the equilibrium verges (for fixed d < dc), and should yield the same Ising model in finite systems. In the hybrid model with critical exponents in the FSS analysis described below. h = ∞, there is also a smooth transition in the hθCOi For a given d < dc, we measure CO-coverage hθCOi ver- versus pCO for finite L. This reflects the feature that sus p for two different system sizes, say L and 2L. We for L < ∞, the system can make noise-induced jumps find that a convenient definition of p∗ is the point where ∗L ∗ between the low hθCOi reactive and high hθCOi inac- the curves cross, i.e., where hθCO(pL)iL = hθCO(pL)i2L. tive states, and that the relative weight of these states The technique is similar in its underlying motivation to changes smoothly with pCO. This transition also becomes a method for equilibrium first-order transitions by Borgs sharper with increasing L, occurring at the equistability and Janke.13 point for L → ∞. See Ref. 11 for details. After determining the transition pressure p∗ for each Finally, to place our study in a broader context, we d value, one can study the critical behavior using the note that our CO oxidation model corresponds to a mod- usual FSS techniques. For example, the quantity χL ≡ 5 2 2 2 ified version of the Ziff-Gulari-Barshad (ZGB) model, L (hθCOi − hθCOi ), which is related to susceptibility in 3 equilibrium systems, is assumed to have the following behavior 2.20
γ/ν 1/ν χL = L χ˜Ising (d − dc)L , (1) 2.10 h i if the transition belongs to the Ising universality class. L
Here ν = 1 and γ =7/4 are the critical exponents for the R 2.00 correlation length and susceptibility respectively (in 2D). In contrast, for the MF universality class, one has14,15 1.90 D/2 D/2 χL = L χ˜MF (d − dc)L , (2) h i 1.80 where D = 2 is the spatial dimension of the system so 0.051 0.052 0.053 0.054 0.055 that LD is the volume (or area in our case) of the system. d Defining RL ≡ χ2L/χL, in either case, one has
γ˜ FIG. 1: Ratio, RL, of mean-square fluctuation amplitudes RL(dc) → 2 , as L → ∞ (3) for systems of size 2L and L at equistability points for the hybrid model. L ranges from 8, 16, 32 to 64 with increasing where the size scaling exponentγ ˜ is D/2 = 1 for the MF steepness. The dotted line plot RL = 2, the MF prediction. universality class, and γ/ν =7/4 for the Ising universal- ity class. Since one does not know the value of d a priori, a c TABLE I: Effective critical point and critical exponent for convenient way to determine both the critical point and fluctuations of θCO for the hybrid model obtained from finite- exponent is by finding the crossing point of RL(d) and size-scaling analysis. R2L(d), so that L dc pc γ˜L L L γ˜L RL dc = R2L dc =2 . (4) 8 0.05304(6) 0.4138(1) 0.948(2) 16 0.05275(5) 0.41343(8) 0.968(4) γ˜L can be considered as the effective critical exponent for L 32 0.05264(3) 0.41330(6) 0.981(5) finite systems (see Sec. IV). Note that in determining dc andγ ˜L, we need simulations of system of linear sizes L, 2L, and 4L. Figure 1 shows RL versus d for the hybrid model. The 4.0 FSS argument above predicts that RL for different L’s will cross at (dc, Rc) where Rc = 2 for MF criticality, and 7/4 3.5 Rc = 2 ≈ 3.364 for Ising (in 2D) criticality. Figure 1 clearly shows MF behavior for system up to L = 128. Note that in the limit of L → ∞, RL is a step function L 3.0
16 R with RL = 4 when d < dc, and RL = 1 when d > dc. Table I lists the effective critical exponent obtained using the above procedure for the hybrid model. All es- 2.5 timates forγ ˜L are close to unity, consistent with the pre- 14 diction of MF universality class whereγ ˜ = 1. Extrap- 2.0 olation to L = ∞ assuming a 1/L2 finite size correction (consistent with MF criticality) gives d = 0.05258(5) 0.016 0.018 0.020 0.022 0.024 c d and pc = 0.41327(5). Assuming a 1/L finite size cor- rection forγ ˜L givesγ ˜ = 0.995(6). It is possible that corrections to scaling are described by other exponents. FIG. 2: Ratio, RL, of mean-square fluctuations amplitudes However, our analysis shows that the range of system for systems of size 2L and L at equistability points for the sizes is large enough so that the value of dc and the con- reaction model with finite CO diffusion model h = 1. L ranges sistency with MF behavior is not dependent on the form from 8, 16, and 32 with increasing steepness. The dotted line 7/4 of the corrections. Note that MF behavior occurs despite shows RL = 2 , the Ising prediction. the presence of spatial correlations in the distribution of O(ads) due to limited mobility of O(ads) and interactions between O(ads).7 precludes definitive convergence of FSS results for either Figure 2 shows RL versus d for the reaction model with the hybrid model (Fig. 1 or finite h (Fig. 2. However, finite h = 1. The crossing point appears to approach 27/4 comparing these cases reveals very different behavior, as L increases, consistent with the prediction of Ising uni- and we argue that the above assignment of universality versality. Limitations in analysis of larger system sizes classes is quite reasonable and natural. 4
TABLE II: Variation of the critical desorption rate dc(h) with 2.0 2.0 CO hop rate h. Results are obtained from FSS scaling using system sizes L =16, 32, and 64. 1.8 (a) 1.8 (b) Ising
h 0.5 1 2 4 10 20 ∞ 1.6 1.6 dc(h) 0.012 0.019 0.027 0.035 0.042 0.047 0.0527 L
~ γ 1.4 1.4 L=32 1.2 1.2 L=16 From the slope of RL at the crossing point and the MF θ scaling formχ ˜[(d − dc)L ] in Eqs. (1) and (2) one can 1.0 L=8 1.0 also estimate the exponent related to the rounding of the critical region due to finite sizes. Using results in Fig. 1 0.8 0.8 0.0 0.5 1.0 1.5 2.0 0.1 1.0 10.0 100.0 we find that θ =1.0(1) for the largest system sizes, con- 1/h L/h sistent with the MF value θ = D/2 = 1. Uncertainties in the data of Fig. 2 are too large to obtain accurate es- timate of θ, but they are consistent with the Ising value FIG. 3: Effective critical exponentγ ˜ versus h. See text for 1/ν = 1. In two dimensions, θ can not be used to distin- details. guish between the MF and Ising universality class. We note one previous study by Tom´eand Dickman17 Thus, to summarize our results for the critical point of the ZGB lattice-gas model5 for CO oxidation (O ad- 2 in our reaction model with k = 1, we find that the crit- sorption on adjacent sites, infinitely fast reaction, no CO ical desorption rate d increases from d = 0 for (small) diffusion) modified to include CO-desorption. They mea- c c h = h to d = 0.0527 for h = ∞. In contrast, for the sure the shift of the critical point with system size and t c − ZGB model (no NN exclusion of O) with k = ∞, modi- found that d (L) − d (∞) ∼ L λ where λ = 1 consis- c c fied to include CO desorption and CO mobility, one finds tent with results of the two-dimensional Ising model. In- that d = 0.0417,19 for h = 0 increasing to d = 2/3 for deed, using a similar method, we found that λ ≈ 1 for c c h = ∞.12 Further refining this ZGB-type model to in- h = 1 and λ ≈ 2 for the hybrid model.18 This result corporate k = 1 (rather than k = ∞), one finds that is consistent with our conclusion regarding universality d = 0.0320 for h = 0 increasing to d = 0.142 for classes. However, care must be taken with the interpre- c c h = ∞.21. tation of the shift exponent, λ, because it is sensitive to the boundary conditions (which were chosen to be Of more central interest to this paper is the variation periodic in both the above studies) in the case of MF with h of the (effective) critical exponents. Upon in- universality where hyperscaling is violated.16 For exam- creasing the CO diffusion rate, h, how does one crossover ple, choosing free boundary conditions (which modifies from Ising criticality (applying for finite h>ht) to the adsorption and reaction processes near the boundaries) MF criticality of the hybrid model (which corresponds to could lead to modified λ for the hybrid model. first taking the limit of h → ∞). A related question is: for finite (possibly large) h, how does critical behavior depend on finite system size? In the literature on crossover studies, effective expo- IV. CROSSOVER BEHAVIOR nents are usually defined as the slope of the log-log plot of the measured critical quantities versus deviation from Table II shows the variation with h of the critical point the critical point, e.g., γeff ≡−d ln χ/d ln |d − dc|. A dif- dc for the CO poisoning transition obtained using FSS. ferent definition, which is more amenable to numerical The h = ∞ value is taken from the hybrid model. By simulations is to fix the parameters at the critical point, analogy with equilibrium studies, discussed below and in while changing the system size L. The effective (reduced) Ref. 14, we assume that the shift of dc(h) away from the exponent can then be defined asγ ˜eff ≡ d ln χ/d ln L. In limit of h → ∞ scales as 1/h (although there could be principle, this derivative (or its finite difference approxi- logarithmic corrections). mation) should be evaluated at the critical point for the As an interesting aside, we note that extrapolation be- infinite system, dc. In practice, dc is unknown a pri- havior to the regime of small h suggests that dc(h) van- ori, and estimation by extrapolation produces additional ishes at h = ht ≈ 0.2. Thus, for small 0
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