Adjacent Empty Sites Research Articles

Chase-escape on the configuration model

Chase-escape is a competitive growth process in which red particles spread to adjacent empty sites according to a rate-λ Poisson process while being chased and consumed by blue particles according to a rate-1 Poisson process. Given a growing sequence of finite graphs, the critical rate λc is the largest value of λ for which red fails to reach a positive fraction of the vertices with high probability. We provide a conjecturally sharp lower bound and an implicit upper bound on λc for supercritical random graphs sampled from the configuration model with independent and identically distributed degrees with finite second moment. We additionally show that the expected number of sites occupied by red undergoes a phase transition and identify the location of this transition.

Adsorbed formate: the key intermediate in the oxidation of formic acid on platinum electrodes

The electrooxidation of formic acid on Pt and other noble metal electrodes proceeds through a dual-path mechanism, composed of a direct path and an indirect path through adsorbed carbon monoxide, a poisoning intermediate. Adsorbed formate had been identified as the reactive intermediate in the direct path. Here we show that actually it is also the intermediate in the indirect path and is, hence, the key reaction intermediate, common to both the direct and indirect paths. Furthermore, it is confirmed that the dehydration of formic acid on Pt electrodes requires adjacent empty sites, and it is demonstrated that the reaction follows an apparently paradoxical electrochemical mechanism, in which an oxidation is immediately followed by a reduction.

Critical behavior of repulsive lineark-mers on triangular lattices

Monte Carlo (MC) simulations and finite-size scaling analysis have been carried out to study the critical behavior in a submonolayer two-dimensional gas of repulsive linear $k$-mers on a triangular lattice at coverage $k∕(2k+1)$. A low-temperature ordered phase, characterized by a repetition of alternating files of adsorbed $k$-mers separated by $k+1$ adjacent empty sites, is separated from the disordered state by an order-disorder phase transition occurring at a finite critical temperature, ${T}_{c}$. The MC technique was combined with the recently reported free-energy minimization criterion approach (FEMCA) [F. Rom\'a et al., Phys. Rev. B 68, 205407 (2003)] to predict the dependence of the critical temperature of the order-disorder transformation. The dependence on $k$ of the transition temperature, ${T}_{c}(k)$, observed in MC is in qualitative agreement with FEMCA. In addition, an accurate determination of the critical exponents has been obtained for adsorbate sizes ranging between $k=1$ and 3. For $k>1$, the results reveal that the system does not belong to the universality class of the two-dimensional Potts model with $q=3$ ($k=1$, monomers). Based on symmetry concepts, we suggested that the behavior observed for $k=1$, 2, and 3 could be generalized to include larger particle sizes $(k\ensuremath{\geqslant}2)$.

Models solvable through the empty-interval method

The most general one dimensional reaction-diffusion model with nearest-neighbor interactions solvable through the empty interval method, and without any restriction on the particle-generation from two adjacent empty sites is studied. It is shown that turning on the reactions which generate particles from two adjacent empty sites, results in a gap in the spectrum of the evolution operator (or equivalently a finite relaxation time).

Universality class of interactingk-mers in a two-dimensional lattice system

Monte Carlo simulations and finite-size scaling analysis have been carried out to study the critical behavior in a submonolayer two-dimensional gas of repulsive linear $k$-mers on a square lattice at half coverage. A $(2k\ifmmode\times\else\texttimes\fi{}2)$ ordered phase, characterized by alternating files of $k$-mers separated by $k$ adjacent empty sites, is separated from the disordered state by a order-disorder phase transition occurring at a finite critical temperature. Based on the strong axial anisotropy of the low-temperature phase for $k\ensuremath{\geqslant}2$, an order parameter measuring the orientation of the particles has been introduced. Taking advantage of its definition, an accurate determination of the critical exponents has been obtained for three adsorbate sizes. Namely, $\ensuremath{\nu}=0.53(1)$, $\ensuremath{\beta}=0.02(1)$, $\ensuremath{\gamma}=1.14(3)$, and $\ensuremath{\alpha}=0.93(3)$ for $k=2$ (dimers); $\ensuremath{\nu}=0.54(1)$, $\ensuremath{\beta}=0.03(1)$, $\ensuremath{\gamma}=1.16(3)$, and $\ensuremath{\alpha}=0.89(3)$ for $k=3$ (trimers); and $\ensuremath{\nu}=0.53(2)$, $\ensuremath{\beta}=0.02(1)$, $\ensuremath{\gamma}=1.14(3)$, and $\ensuremath{\alpha}=0.89(4)$ for $k=4$ (tetramers). In the studied cases, the results reveal that the system does not belong to the universality class of the two-dimensional Ising model ($k=1$, monomers). We pointed out that the breaking of the orientational symmetry characterizing the low-temperature phase for particles occupying more than one site is the main source of this change in the universality class. Consequently, we suggested that the behavior observed for dimers, trimers, and tetramers could be generalized to include larger particle sizes $(k\ensuremath{\geqslant}2)$. Finally, hyperscaling relations have been validated, leading to independent controls and consistency checks of the values of all the critical exponents.

Critical behavior of repulsive lineark-mers on square lattices at half coverage: Theory and Monte Carlo simulations

Monte Carlo (MC) simulations have been used to study the critical behavior of repulsive linear k-mers on square lattices at 50% coverage. A $(2k\ifmmode\times\else\texttimes\fi{}2)$ ordered phase, characterized by a repetition of alternating files of adsorbed k-mers separated by k adjacent empty sites, was found. This ordered phase is separated from the disordered state by a order-disorder phase transition occurring at a critical temperature ${T}_{c},$ which presents an intriguing dependence with the size k of the adsorbed molecules. In addition, two analytical techniques were combined with Monte Carlo simulations to predict the critical temperature of the order-disorder transformation. The first is based on a detailed mean-field approximation (DMFA), considering the exact interactions between the k-mers belonging to a region $\ensuremath{\gamma}$ (the cluster) and a mean-field interaction with the rest outside $\ensuremath{\gamma}.$ Different sizes for $\ensuremath{\gamma}(k\ifmmode\times\else\texttimes\fi{}1,k\ifmmode\times\else\texttimes\fi{}2,2k\ifmmode\times\else\texttimes\fi{}1,2k\ifmmode\times\else\texttimes\fi{}2)$ were used in the calculations in order to discuss its influence in the determination of ${T}_{c}.$ The second approach is based on a free energy minimization criterion (FEMCA). The dependence on k of the transition temperature ${T}_{c}(k)$ observed in MC is in remarkable qualitative agreement with DMFA and FEMCA. Both allow us to interpret the physical meaning of the mechanisms underlying the observed transitions.

Relaxation effects in random sequential adsorption: application to chemisorption systems

We consider the effect of slow relaxation processes in two models for irreversible chemisorption which incorporate superlattice ordering of the adsorbed layer on a square lattice of adsorption sites. The first model involves adsorption of monomers onto single empty sites, and we the second involves dissociative adsorption of dimers onto diagonally adjacent empty sites, where adsorption in both models occurs subject to the constraint that no adjacent pairs of occupied sites can be created. These adsorption processes produce adlayers comprised of disordered ‘checkerboard’ or c(2×2) domains, adsorption continuing until a non-equilibrium ‘jammed’ state is reached where no further space is available for adsorption. We focus on characterizing the evolution from this jammed state as a result of slow surface diffusion, or slow desorption (or reaction) processes. These relaxation processes open up further adsorption sites which are immediately filled in the presence of continued adsorption. In all cases, evolution involves Lifshitz–Cahn–Allen-type coarsening of the c(2×2) domains.

CO-oxidation model with superlattice ordering of adsorbed oxygen. I. Steady-state bifurcations

We analyze a model for CO oxidation on surfaces which incorporates both rapid diffusion of adsorbed CO, and superlattice ordering of adsorbed immobile oxygen on a square lattice of adsorption sites. The superlattice ordering derives from an “eight-site adsorption rule,” wherein diatomic oxygen adsorbs dissociatively on diagonally adjacent empty sites, provided that none of the six additional neighboring sites are occupied by oxygen. A “hybrid” formalism is applied to implement the model. Highly mobile adsorbed CO is assumed randomly distributed on sites not occupied by oxygen (which is justified if one neglects CO–CO and CO–O adspecies interactions), and is thus treated within a mean-field framework. In contrast, the distribution of immobile adsorbed oxygen is treated within a lattice–gas framework. Exact master equations are presented for the model, together with some exact relationships for the coverages and reaction rate. A precise description of steady-state bifurcation behavior is provided utilizing both conventional and “constant-coverage ensemble” Monte Carlo simulations. This behavior is compared with predictions of a suitable analytic pair approximation derived from the master equations. The model exhibits the expected bistability, i.e., coexistence of highly reactive and relatively inactive states, which disappears at a cusp bifurcation. In addition, we show that the oxygen superlattice ordering produces a symmetry-breaking transition, and associated coarsening phenomena, not present in conventional Ziff–Gulari–Barshad-type reaction models.

Shock profiles for the asymmetric simple exclusion process in one dimension

The asymmetric simple exclusion process (ASEP) on a one-dimensional lattice is a system of particles which jump at rates $p$ and $1-p$ (here $p>1/2$) to adjacent empty sites on their right and left respectively. The system is described on suitable macroscopic spatial and temporal scales by the inviscid Burgers' equation; the latter has shock solutions with a discontinuous jump from left density $\rho_-$ to right density $\rho_+$, $\rho_-<\rho_+$, which travel with velocity $(2p-1)(1-\rho_+-\rho_-)$. In the microscopic system we may track the shock position by introducing a second class particle, which is attracted to and travels with the shock. In this paper we obtain the time invariant measure for this shock solution in the ASEP, as seen from such a particle. The mean density at lattice site $n$, measured from this particle, approaches $\rho_{\pm}$ at an exponential rate as $n\to\pm\infty$, with a characteristic length which becomes independent of $p$ when $p/(1-p)>\sqrt{\rho_+(1-\rho_-)/\rho_-(1-\rho_+)}$. For a special value of the asymmetry, given by $p/(1-p)=\rho_+(1-\rho_-)/\rho_-(1-\rho_+)$, the measure is Bernoulli, with density $\rho_-$ on the left and $\rho_+$ on the right. In the weakly asymmetric limit, $2p-1\to0$, the microscopic width of the shock diverges as $(2p-1)^{-1}$. The stationary measure is then essentially a superposition of Bernoulli measures, corresponding to a convolution of a density profile described by the viscous Burgers equation with a well-defined distribution for the location of the second class particle.

Power law decay of correlations in stationary nonequilibrium lattice gases with conservative dynamics

We report computer simulations and high-temperature approximations of the pair correlation in a stationary nonequilibrium system, a lattice gas subject to a strong uniform driving field E. The dynamics of the system is given by hoppings of particles to adjacent empty sites with rates biased for jumps in the direction of E. We study the anisotropic short-distance behavior as well as the long-distance decay properties of the two-point correlations along the principal axes. The simulations as well as the (approximate) expansion in ..beta.. strongly suggest that the correlations in this system have a power law decay, r/sup -D/ for dimensions D = 2 and 3, even at high temperatures.

Nonequilibrium percolative c(2×2) ordering: Oxygen on Pd(100)

Dissociative adsorption of oxygen on certain (100) metal surfaces has been modeled as random dimer adsorption onto diagonally adjacent empty sites of a square lattice subject to the additional constraint that all six neighboring sites must be empty (the 8-site model). Here we adapt this model to analyze the nonequilibrium c(2×2) ordering recently observed for oxygen on Pd(100) at coverages up to saturation (&amp;gt;1/4 monolayer), under conditions of low temperature and high pressure where effects of diffusive mobility can be ignored. We do, however, propose that adsorption could be followed immediately by short range transient mobility to dissipate excess energy. We first show how exact master equations for this model can be used to obtain analytic expressions for various local quantities of interest: the probability of an empty 8-site configuration (which determines the sticking coefficient), the c(2×2) island edge or domain boundary densities, etc. They also provide a characterization of, e.g., the asymptotic decay of spatial correlations. Near-percolating (percolative) c(2×2) ordering is readily observed in computer simulations of the saturation state. Through a simple extension of the physical model, we provide a framework for analysis of the large scale characteristics of this ordering via correlated polychromatic percolation theory. Corresponding scaling relations and some real space renormalization group analysis are described. Simulation results for average sizes, the effective dimension, and perimeter length to size ratios, of c(2×2) islands, are also presented.