Simple Lattice Gas Model

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]... 

In a preliminary approach the swelling dependence of the mixbg contribution to the free enthalpy was simulated by a simple lattice gas model that roughly took into account the different volumes available to the network chains at different stages of swelling and disinterspersion. This approach yields an additional contribution Gc< >(v2) to the elastic chemical potential where... 

The Flory-Huggins theory is in fact nothing more than a two-component polymer version of the simple lattice gas model introduced in section 2. We divide the free energy into an entropic part, which is assumed to take the simplest perfect gas form, while the enthalpic part is estimated using a typical mean-field assumption. 

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 ... 

We now sketch a simple deterministic lattice gas model of diffusion that becomes exactly Lorentz invariant in the continuum limit. We follow Toffoli ([toff89], [tofiSOb]) and Smith [smithm90]. 

All applications of the lattice-gas model to liquid-liquid interfaces have been based upon a three-dimensional, typically simple cubic lattice. Each lattice site is occupied by one of a variety of particles. In the simplest case the system contains two kinds of solvent molecules, and the interactions are restricted to nearest neighbors. If we label the two types of solvents molecules S and Sj, the interaction is specified by a symmetrical 2x2 matrix w, where each element specifies the interaction between two neighboring molecules of type 5, and Sj. Whether the system separates into two phases or forms a homogeneous mixture, depends on the relative strength of the cross-interaction W]2 with respect to the self-inter-action terms w, and W22, which can be expressed through the combination ... 

We first consider the simplest system consisting of two pure, immiscible solvents. Within the lattice-gas model the energetics of the system on a particular lattice are governed by the single parameter w [see Eq. (1)], which determines the structure of the interface and the particle profiles. The results presented in this section are for a simple cubic lattice. 

These results have been initially considered as evidence for specific ion adsorption at ITIES [71,72]. Its origin was ascribed to extensive ion pair formation between ions in the aqueous phase and ions in the organic phase [71] [cf. Eq. (20)], or to a penetration into the interfacial region [72]. The former model, which has been considered in this context earlier [60], allows one to interpret the enhanced capacity in terms of Eq. (22). Pereira et al. (74) presented more experimental data demonstrating the effect of electrolytes and proposed a simple model, which is based on the lattice-gas model of the liquid liquid interface [23]. Theoretical calculations showed that ion pairing can lead to an increase in the stored... 

Lattice gas models are simple to construct, but the gross approximations that they involve mean that their predictions must be treated with care. There are no long-range interactions in the model, which is unrealistic for real molecules the short-range interactions are effectively hard-sphere, and the assumption that collisions lead to a 90° deflection in the direction of movement of both particles is very drastic. At the level of the individual molecule then, such a simulation can probably tell us nothing. However, at the macroscopic level such models have value, especially if a triangular or hexagonal lattice is used so that three-body collisions are allowed. 

Phase separations and boundaries may occur in many systems in alloys, in ferromagnetic substances, in solutions. The basic mechanism can be understood within a simple model, which is known as the lattice gas model. We present a simple version adapted to liquid-liquid interfaces. 

Although the simple mean-field expression (Eqn (7.10)) for a lattice-gas model has been used to understand intercalation systems qualitatively... 

Several groups have developed cellular automata models for particular reaction-diffusion systems. In particular, the Belousov-Zhabotinsky oscillating reaction has been examined in a number of studies.84-86 Attention has also been directed at the A + B —> C reaction, using both lattice-gas models 87-90 and a generalized Margolus diffusion approach.91 We developed a simple, direct cellular automaton model92 for hard-sphere bimolecular chemical reactions of the form... 

In many physically important cases of localized adsorption, each adatom of the compact monolayer covers effectively n > 1 adsorption sites [3.87-3.89, 3.98, 3.122, 3.191, 3.214, 3.261]. Such a multisite or 1/n adsorption can be caused by a crystallographic Me-S misfit, i.e., the adatom diameter exceeds the distance between two neighboring adsorption sites, and/or by a partial charge of adatoms (A < 1 in eq. (3.2)), i.e., a partly ionic character of the Meads-S bond. The theoretical treatment of a /n adsorption differs from the description of the 1/1 adsorption by a simple Ising model. It implies the so-called hard-core lattice gas models with different approximations [3.214, 3.262-3.266]. Generally, these theoretical approaches can only be applied far away from the critical conditions for a first order phase transition. In addition, Monte Carlo simulations are a reliable tool for obtaining valuable information on both the shape of isotherms and the critical conditions of a 1/n adsorption [3.214, 3.265-3.267]. 

The results shown in Figures 2 and 4 are intuitively obvious, and reflect the well known fact that the critical temperature in the system depends primarily on the strength of molecular interactions. In particular, in the lattice gas models the maximum of Tc is reached for the system of particles characterized by a" corresponding to the highest interaction between adsorbed particles. This can be readily demonstrated by considering the prediction of a very simple mean-field theory in the Bragg-Williams approximation. 

The observed changes in the experimentally determined critical temperatures with the dimensional incompatibility between adsorbate and adsorbent provide a direct evidence that the corrugation of the gas - solid interaction potential plays a very important role in determining the properties of adsorbed films. These findings support also the view that theoretical approaches based on the lattice gas models can appropriately represent adsorption of simple gases on crystals at low temperatures. 

It is then of great importance to develop simple models capable of describing the energetic topography on the basis of a few parameters and to study the effects of these parameters on several surface processes, with the hope that, in such a process, methods to obtain the relevant parameters from experimental data will be envisaged. These models can be of two kinds continuum models or lattice-gas models. The former are more suited to mobile adsorption, generally physisorption, and then more closely related to the surface energetic characterization problem, whereas the latter are more suited to locaHzed adsorption (e.g., chemisorption). 

We have used a lattice-gas model to analyze preferential flow mechanisms in porous media. This numerical approach provides information that most experimental methods are presently unable to supply. The need for simple macroscopic models of preferential flow oriented us to test the kinematic wave approach. The comparison of the lattice gas solution with the kinematic approximation has shown the limitations of the latter approach and the possible ways of ameliorating it. 

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