Gas-lattice model

In various forms, lattice-gas models permeate statistical mechanics. Consider a lattice in which each site has two states. If we interpret the states as full or empty , we have a lattice-gas model, and an obvious model for an intercalation compound. If the states are spin up and spin down , we have an Ising model for a magnetic system if the states are Atom A and Atom B , we have a model for a binary alloy. Many different approximation techniques have been derived for such models, and many lattices and interactions have been considered. 

One complication in applying these models to intercalation compounds is in treating the dissociation of the intercalated atom into ions and electrons. The chemical potential can be written as a sum of contributions from ions and electrons, according to 

LGs can also serve as powerful alternatives to PDEs themselves in modeling physical systems. The distinction is an important one. It must be remembered, however, that not all PDEs (and perhaps not all physical systems see chapter 12) are amenable to a LG simulation. Moreover, even if a candidate PDE is selected for simulation by a LG. there is no currently known cookbook recipe allowing a researcher to go from the PDE to a LG description (or vice versa). Nonetheless, by their very nature, LGs lend themselves to modeling any partial differential equation (PDE) for which the underlying physical basis for its construction involves a large number of particles with local interactions [wolf86c]. 

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Salsburg Z W, Jacobson J D, Fickett W and Wood W W 1959 Application of the Monte Carlo method to the lattice gas model. Two dimensional triangular lattice J. Chem. Phys. 30 65-72... 

A number of theoretical models have been proposed to describe the phase behavior of polymer—supercritical fluid systems, eg, the SAET and LEHB equations of state, and mean-field lattice gas models (67—69). Many examples of polymer—supercritical fluid systems are discussed ia the Hterature (1,3). 

To illustrate the complexity of the phase behavior in a more compact way it is instructive to employ a mean-field lattice-gas model. The relative simplicity of the grand potential... 

FIG. 22 Coexistence curves for the lattice-gas model, (a) bulk (------) chemically... 

The behavior of an adsorbate on a single patch of size L has been represented by the familiar two-dimensional lattice gas model Hamiltonian with the added term resulting from the presence of a boundary field ... 

To present briefly the different possible scenarios for the growth of multilayer films on a homogeneous surface, it is very convenient to use a simple lattice gas model language [168]. Assuming that the surface is a two-dimensional square lattice of sites and that also the entire space above the surface is divided into small elements, forming a cubic lattice such that each of the cells can be occupied by one adsorbate particle at the most, the Hamiltonian of the system can be written as [168,169]... 

A lattice gas model with adsorbate-induced surface reconstructions has also very recently been proposed by Kusovkov et al. [73]. This model also exhibits a rich oscillatory behavior. 

FIG. 16 Variation of the steady-state rate of production, Pcoj, with Pco in the NO + CO lattice gas model with NO desorption (rate d o = 0.5), and CO desorption at various rates (shown). The inset shows the reaction rate measured experimentally at 410 K. (From Ref. 81.)... 

E. Albano. Displacement of inactive phases by the reactive regime in a lattice gas model for a dimer-monomer irreversible surface reaction. Phys Rev E 55 7144-7152, 1997. 

In this review we put less emphasis on the physics and chemistry of surface processes, for which we refer the reader to recent reviews of adsorption-desorption kinetics which are contained in two books [2,3] with chapters by the present authors where further references to earher work can be found. These articles also discuss relevant experimental techniques employed in the study of surface kinetics and appropriate methods of data analysis. Here we give details of how to set up models under basically two different kinetic conditions, namely (/) when the adsorbate remains in quasi-equihbrium during the relevant processes, in which case nonequilibrium thermodynamics provides the needed framework, and (n) when surface nonequilibrium effects become important and nonequilibrium statistical mechanics becomes the appropriate vehicle. For both approaches we will restrict ourselves to systems for which appropriate lattice gas models can be set up. Further associated theoretical reviews are by Lombardo and Bell [4] with emphasis on Monte Carlo simulations, by Brivio and Grimley [5] on dynamics, and by Persson [6] on the lattice gas model. 

This review is structured as follows. In the next section we present the theory for adsorbates that remain in quasi-equilibrium throughout the desorption process, in which case a few macroscopic variables, namely the partial coverages 0, and their rate equations are needed. We introduce the lattice gas model and discuss results ranging from non-interacting adsorbates to systems with multiple interactions, treated essentially exactly with the transfer matrix method, in Sec. II. Examples of the accuracy possible in the modehng of experimental data using this theory, from our own work, are presented for such diverse systems as multilayers of alkali metals on metals, competitive desorption of tellurium from tungsten, and dissociative... 

In a recent paper [11] this approach has been generalized to deal with reactions at surfaces, notably dissociation of molecules. A lattice gas model is employed for homonuclear molecules with both atoms and molecules present on the surface, also accounting for lateral interactions between all species. In a series of model calculations equilibrium properties, such as heats of adsorption, are discussed, and the role of dissociation disequilibrium on the time evolution of an adsorbate during temperature-programmed desorption is examined. This approach is adaptable to more complicated systems, provided the individual species remain in local equilibrium, allowing of course for dissociation and reaction disequilibria. 

Most microscopic theories of adsorption and desorption are based on the lattice gas model. One assumes that the surface of a sohd can be divided into two-dimensional cells, labelled i, for which one introduces microscopic variables Hi = 1 or 0, depending on whether cell i is occupied by an adsorbed gas particle or not. (The connection with magnetic systems is made by a transformation to spin variables cr, = 2n, — 1.) In its simplest form a lattice gas model is restricted to the submonolayer regime and to gas-solid systems in which the surface structure and the adsorption sites do not change as a function of coverage. To introduce the dynamics of the system one writes down a model Hamiltonian which, for the simplest system of a one-component adsorbate with one adsorption site per unit cell, is... 

FIG. 6 (a) Atomic desorption rates calculated with the two-site lattice gas model... 

If it cannot be guaranteed that the adsorbate remains in local equilibrium during its time evolution, then a set of macroscopic variables is not sufficient and an approach based on nonequihbrium statistical mechanics involving time-dependent distribution functions must be invoked. The kinetic lattice gas model is an example of such a theory [56]. It is derived from a Markovian master equation, but is not totally microscopic in that it is based on a phenomenological Hamiltonian. We demonstrate this approach... 

In this section we review a generahzation of the kinetic lattice gas model when surface reconstruction takes place upon adsorption and desorption. 

In the standard lattice gas model of adsorption we assume that the surface of the solid remains inert, providing adsorption sites. This implies that the state of the surface before adsorption and after desorption is the same. This is not the case if the surface reconstructs or lifts the reconstruction upon adsorption. Such a situation we want to describe. We introduce occupation numbers for the surface = 0 or 1, depending on whether the surface... 

For adsorbates out of local equilibrium, an analytic approach to the kinetic lattice gas model is a powerful theoretical tool by which, in addition to numerical results, explicit formulas can be obtained to elucidate the underlying physics. This allows one to extract simplified pictures of and approximations to complicated processes, as shown above with precursor-mediated adsorption as an example. This task of theory is increasingly overlooked with the trend to using cheaper computer power for numerical simulations. Unfortunately, many of the simulations of adsorbate kinetics are based on unnecessarily oversimplified assumptions (for example, constant sticking coefficients, constant prefactors etc.) which rarely are spelled out because the physics has been introduced in terms of a set of computational instructions rather than formulating the theory rigorously, e.g., based on a master equation. 

K. Chen, C. Ebner, C. Jayaprakash, R. Pandit. Microemulsions in oil-water-surfactant mixtures Systematics of a lattice-gas model. Phys Rev A 55 6240, 1988. 

The relaxation of a thermodynamic system to an equilibrium configuration can be conveniently described by a master equation [47]. The probability of finding a system in a specific state increases by the incoming jump from adjacent states, and decreases by the outgoing jump from this state to the others. From now on we shall be specific for the lattice-gas model of crystal growth, described in the previous section. At the time t the system will be found in the state. S/ with a probability density t), and its evolution... 

As long as the condition (13) is satisfied, any choice of the transition probability is possible. For the lattice-gas model with the Hamiltonian (2), a simple choice is the following ... 

Let us consider a binary system A — B. Its particular configuration is determined by a set of occupation indices rji, where rji = 1 if the site i is occupied by an atom of the type A, and r = 0 otherwise. This form corresponds to the lattice gas model. The effective concentration-independent Islng Hamiltonian reads... 

Lattice gases are micro-level rule-based simulations of macro-level fluid behavior. Lattice-gas models provide a powerful new tool in modeling real fluid behavior ([doolenQO], [doolenQl]). The idea is to reproduce the desired macroscopic behavior of a fluid by modeling the underlying microscopic dynamics. 

Chapter 9 provides an introductory discussion of a research area that is rapidly growing in importance lattice gases. Lattice gases, which are discretized models of continuous fluids, represent an early success of CA modeling techniques. The chapter begins with a short primer on continuum fluid dynamics and proceeds with a discussion of CA lattice gas models. One of the most important results is the observation that, under certain constraints, the macroscopic behavior of CA models exactly reproduces that predicted by the Navier-Stokes equations. 

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