Lattice model of confined pure fluids

In the preceding section, we derived a mean-field equation of state for c on-fined fluids based on the assumptions of 

Within the approach developed in the previous section, neither of these assumptions can be replaced easily by a more realistic one. However, it turns out that, if one abandons the continuous model fluid in favor of a discrete model in whicli molecules are restricted to positions on a rigid lattice, the second of the above assumptions is no longer necessary to derive an analytic expression for the partition function of the fluid. 

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However, one-dimensional confined fluids with purely repulsive interactions can be expected to be only of limited usefulness, especially if one is interested in phase transitions that cannot occur in any one-dimensional system. In treating confined fluids in such a broader context, a key theoretical tool is the one usually referred to as mean-field theory. This powerful theory, by which the key problem of statistical thermodynamics, namely the computation of a partition function, becomes tractable, is introduced in Chapter 4 where we focus primarily on lattice models of confined pure fluids and their binary mixtures. In this chapter the emphasis is on features rendering confined fluids unique among other fluidic systems. One example in this context is the solid-like response of a confined fluid to an applied shear strain despite the absence of any solid-like structure of the fluid phase. 

Here we consider a lattice model of a simple pure confined fluid, that is, a fluid composed of molecules having only translational degrees of freedom. The positions of theses molecules are restricted to M n Uyn sites of a simple cubic lattice of lattice constant f. Each site on the lattice can be occupied by one molecule at most which accounts for the infinitely repulsive hard core of each molecule. In addition to repulsion, pair-wise additive attractive interactions between the molecules exist. They are modeled according to square-well potentials where ff is the depth of the attractive well whose width equals t. 

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