Lattice-gas models two-component

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... 

One usually distinguishes two types of lattice models. The first type may be called lattice-gas models. In this case, the number of molecules in the system is less than the number of available sites. In other words, there are vacant sites. The second type of lattice models may be called lattice fluids. In this case, all lattice sites are filled exactly by the molecular components in the system the system is considered to be incompressible. It is easily shown that a two-component incompressible lattice fluid model can be mapped on a one-component lattice gas one. In other words, it is possible to interpret vacant sites to be occupied by a ghost ... 

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. 

However, when po is maintained constant, a linear van t Hoff plot can no longer be extracted for the compositional variable. This illustrates the fact that w is a composite variable which, as such, does not represent an actual participant in the equilibrium in eq. (23). Two types of models are used to approach these real components (i) a defect-chemistry model, based on solid-state equilibria among variously charged and clustered defects, and (ii) a lattice-gas model, based on structural features of the oxygen sublattice and statistical thermodynamics. 

The two-component lattice-gas model we treat here provides our first example of an interface of composition defined by two independently varying densities, as in a two-component version of the general, many-component van der Waals theory of Chapter 3. 

Fig. 5.3. Coeiistence curve and crhical point for the model two-component lattice gas in the Pa tb-fdane. The lines are tielines, connecting coexisting phases, and the dot is the critical point.

Trappeniers NJ, Schouten JA, Ten Seldam CA (1970) Gas-gas equilibrium and the two-component lattice-gas model. Chem Phys Lett 5(9) 541-545... 

Let us consider a two-component A-B alloy (within the lattice gas model) whose primitive unit cell consists of v crystal lattice sites. Taking into account the many-body atomic interactions of arbitrary orders and radii of action, the Hamiltonian H of it can be written in the following form ... 

On the basis of the DLP model, the solid is expected to be an ideal solution for systems in which both components have similar lattice constants. This case is true for the AlAs-GaAs system (Figure 5), in which the solid-phase composition equals the gas-phase composition. Compounds in which the two components have very different behavior will show highly nonlinear composition variations, and miscibility gaps may occur (90, 91). As an example, Figure 5 also shows that the solid-phase composition of In As Sb is a nonlinear function of the gas-phase composition. 

The two-term crossover Landau model has been successfully applied to the description of the near-critical thermodynamic properties of various systems, that are physically very different the 3-dimensional lattice gas (Ising model) [25], one-component fluids near the vapor-liquid critical point [3, 20], binary liquid mixtures near the consolute point [20, 26], aqueous and nonaqueous ionic solutions [20, 27, 28], and polymer solutions [24]. 

We have considered mixtures of two components in w hich one can exchange into the gas phase, or into other liquids or solids. The degree to which a species exchanges depends on its concentration in the solution. Such processes can be described by thermodynamic models, which do not describe the microscopic details, or statistical mechanical models such as the lattice model. Chapters 27 and 28 treat dimerization and other binding processes in more detail. 

The one-component lattice gas of 5.3 may also be treated in the Bethe-Guggenheim approximation, which is a generalization and improvement upon, the simple mean-field theory. The latter follows from the former in Uie limit of large c and small e. The resulting mean-field theory is then necessarily thermodynamically consistent, because the Bethe-Guggenheim approximation is consistent for all c and e. In the present two-component model, in which the only interactions are infinitely strong repulsions, there is no simplification we can make beyond the Bethe-Guggenheim approximation and still retain thermodynamic consistency there is no parameter e, and, while the coordination number c is at our disposal, there is no limit to whidi we can usefully take h. 

A second line of improvement of the original van der Waals approximation, which can also he tested on the penetrable-sphere model. is the substitution of a two-density for a one-density theory, as set out in H 3.3 and as applied to the two-component lattice gas in S 5.4. 

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