The Thermodynamics of Lattice Gas Systems

The present dissertation is an effort to calculate the thermodynamic properties of lattice gases without the use of approximations. In the present dissertation, an exact method--Shift Operator Matrix (SOM) method--is discussed and applied to the calculations of the thermodynamic properties of mono- and multi-layer lattice gas systems. The following are the main points of my work:. 1. Method extensions. The SOM formalism is extended to multilayer systems and a technique is introduced, utilizing symmetry considerations, which reduces the size of the large SOM's encountered. 2. Analytic calculations. The Langmuir's isotherm is obtained analytically for monolayer lattices and the simple BET equation is derived directly from the SOM for multilayer lattice gases. The critical reduced chemical potential, w_{c}, for monolayer and multilayer lattice systems is obtained through ground state analysis. 3. Numerical computations. Thermodynamic functions, correlation functions and response functions are calculated as functions of the reduced temperature and the reduced chemical potential for various interactions such as attractive and repulsive nearest and next-nearest neighbor interactions of gas particles in monolayer systems and for different values of substrate potential in multilayer systems. (a) For monolayers: a V-shaped curve is obtained which describes the relation between the critical temperature vs the next nearest neighbor particle-particle interaction bonding energy; for particles distributed on a cylindrical square -cell lattice, structure parameters, obtained from correlation functions, are used to characterize structures of different phases such as the c(2 x 2) and the p(2 x 1) for non-vanishing temperatures. (b) For bilayers: the total coverage, structure parameters and the coverage of the first layer show that layering transitions occur and depend on the values of substrate potential; the heat capacity signatures show two layering transition critical points, t_ {cl_1} and t_{cl _2}, both of which are less than t _{c}; the heat capacities for different lattice sizes show not only the effect of lattice size on t_{max} for multilayer lattice but also the logarithmic divergence of C _{max}. (c) For trilayers: two substrate potential parameters are needed to determine layering transitions; the conditions for layering transitions for both bilayer and trilayer lattice gases are obtained, which not only agree with the qualitative assertion of mean field theory (65) about the effect of the substrate potential on layering transitions but also give the suitable values of the ratio between the substrate potential and the particle bonding energy at which the layering transitions commence.