Exact results for kinetics of catalytic reactions

Abstract

The kinetics of an irreversible catalytic reaction on a substrate of arbitrary dimension is examined. In the limit of infinitesimal reaction rate (reaction-controlled limit), we solve the dimer-dimer surface reaction model (or voter model) exactly in arbitrary dimension D. The density of reactive interfaces is found to exhibit a power law decay for D < 2 and a slow logarithmic decay in two dimensions. We discuss the relevance of these results for the monomer-monomer surface reaction model. PACS numbers: 05.40.+j, 68.10.Jy, 82.20.Mj In modeling heterogeneous catalysis [1], the monomermonomer surface reaction model plays an important role, at least from the theoretical point of view since an appealing simplicity of this model allows one to examine several issues analytically. In particular, investigations of the monomer-monomer model clarified the role of fluctuations [2][3][4][5][6][7][8], interfacial roughening [9], diffusion of the adsorbants [10], and surface disorder [11]. In the simplest situation (no diffusion, no disorder, etc.), it was found that single-species clusters grow with time when the dimensionality D of the substrate is sufficiently small, D ≤ 2. However, the details of the coarsening like the decay rate of the density of reactive interfaces remain uncertain in two dimensions, -simulations [5,12,13] revealed a very slow decay which could be logarithmic or power law with a small exponent. In this paper, we clarify these questions by computing analytically kinetic characteristics of an idealized version of the monomer-monomer model, the voter model. We then expand these results and perform numerical simulations for the full model.The monomer-monomer surface reaction process can be schematically represented by the following kinetic steps:A and B particles impinge upon a surface, with respective rates k A and k B , and adsorb onto vacant sites V to form a monolayer of adsorbed particles, A V and B V . Nearest-neighbor pairs of dissimilar adsorbed particles, A V B V , react and desorb with rate k r , leaving behind two vacancies. For k A = k B , the adsorption imbalance leads to the quick saturation of the surface by the majority species. For k A = k B and for dimensions D ≤ 2, there is a fluctuation-induced coarsening of the surface into growing A and B adsorbed islands. This nontrivial case of equal adsorption rates will be considered in the following. Furthermore, in theoretical analysis we will restrict ourselves to the reaction-controlled limit, k r ≪ k A = k B , which was found to provide qualitatively the same behavior as the general case [4,6] but more amenable to theoretical treatment. In the reaction-controlled limit, the substrate quickly becomes completely covered and then stays covered forever, since in units of the typical, i.e. adsorption, time interval unoccupied sites are refilled instantaneously. The kinetics of the monomer-monomer surface reaction model is conveniently described by a mapping onto the Ising model with mixed zero-temperature voter dynamics and infinite-temperature...

Nonequilibrium Phase Transition in a Model for Social Influence

“…However, the coarsening process is different from the one of a multicomponent system with nonconserved order parameter, where the interfacial density, corresponding to the density of active bonds, decays as t 21͞2 . Here n a ͑t͒ decays much more slowly, similar to what occurs in the two-dimensional voter model [9], and in a model for catalytic reactions [10].…”

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“…In **This behavior is similar to that observed in the voter model (21), in which each unit randomly selects one of its neighbors and adopts that neighbor's current state. In the one-dimensional voter model, the system evolves to a consensus in a time t Ϸ N 2 (22). In contrast, a unit evolving in accordance with the majority rule will not switch its state to match one particular neighbor.…”

Solution of voter model dynamics on annealed small-world networks

“…(4) and (5) coincide with the equations for the one-dimensional voter model [13]. In order to study the dynamics of the voter model on the Watts-Strogatz network we must average Eqs.…”

“…Starting from a disordered initial condition, the model follows a simple dynamical evolution: at each time step one site is selected at random and set equal to one of its nearest neighbors, chosen at random in its turn. On regular lattices, in d = 1 and d = 2 the model converges to an ordered state with all variables having the same value, whereas for d ≥ 3 the system reaches a disordered stationary state [12,13]. The voter model on complete graphs has been considered recently [14,15].…”