Lattice gases

The Ising model is isomorphic with the lattice gas and with the nearest-neighbour model for a binary alloy, enabling the solution for one to be transcribed into solutions for the others. The tlnee problems are thus essentially one and the same problem, which emphasizes the importance of the Ising model in developing our understanding not only of ferromagnets but other systems as well. 

The relationship between tlie lattice gas and the Ising model follows from the observation that the cell occupation number... 

The specific heat along the critical isochore hence has the same smgularity as (5 P /5T )p for a lattice gas. 

The relationship between tlie lattice gas and the Ising model is also transparent in the alternative fomuilation of the problem, in temis of the number of down spins [i] and pairs of nearest-neighbour down spins [ii]. For a given degree of site occupation [i]. 

Our discussion shows that the Ising model, lattice gas and binary alloy are related and present one and the same statistical mechanical problem. The solution to one provides, by means of the transcription tables, the solution to the others. Flistorically, however, they were developed independently before the analogy between the models was recognized. 

Onsager s solution to the 2D Ising model in zero field (H= 0) is one of the most celebrated results in theoretical chemistry [105] it is the first example of critical exponents. Also, the solution for the Ising model can be mapped onto the lattice gas, binary alloy and a host of other systems that have Hamiltonians that are isomorphic to the Ising model Hamiltonian. 

Lee T D and Yang C N 1952 Statistical theory of equations of state and phase transitions II. Lattice gas and Ising models Phys. Rev. 87 410... 

Diffraction is not limited to periodic structures [1]. Non-periodic imperfections such as defects or vibrations, as well as sample-size or domain effects, are inevitable in practice but do not cause much difSculty or can be taken into account when studying the ordered part of a structure. Some other forms of disorder can also be handled quite well in their own right, such as lattice-gas disorder in which a given site in the unit cell is randomly occupied with less than 100% probability. At surfaces, lattice-gas disorder is very connnon when atoms or molecules are adsorbed on a substrate. The local adsorption structure in the given site can be studied in detail. 

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... 

McNamara G R and Zanetti G 1998 Use of the Boltzmann equation to simulate lattice-gas automata Phys. Rev. Lett. 61 2332... 

Doolen G D (ed.) 1990 Lattice Gas Methods for Partial Differential Equations (Redwood City, CA Addison-Wesley)... 

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]... 

Wetting phenomena on irregularly rough surfaces have not been studied so far. It seems quite reasonable to use computer simulation methods for this purpose. Of course, such computer simulation would be very expensive as the finite size of the simulation cells would require appropriate averaging over different spatial distributions of surface inhomogeneities. Nevertheless, with modern fast computers and using multispin coding techniques such calculations can be efficiently carried out for lattice gas systems. 

Recently, Vigil and Willmore [67] have reported mean field and lattice gas studies of the oscillatory dynamics of a variant of the ZGB model. In this example oscillations are also introduced, allowing the reversible adsorption of inert species. Furthermore, Sander and Ghaisas [69] have very recently reported simulations for the oxidation of CO on Pt in the presence of two forms of oxygen, namely chemisorbed atomic O and oxidized metal surface. These species, which are expected to be present for reaction under atmospheric pressure, are relevant for the onset of oscillatory behavior [69]. 

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. 

---