Partitioning lattice strain theory

Since we observe that the theory embodied in Equations (9)-(ll) is in good agreement with partitioning behavior of ions of different charge into several major silicate phases (Figures 4-6), the question now is how to develop the framework of lattice strain theory into a quantitative model. Specifically, we need to know how and... 

The problem of determination of the partition function Z(k, N) for the iV-link chain having the fc-step primitive path was at first solved in Ref. [17] for the case a = c by application of rather complicated combinatorial methods. The generalization of the method proposed in Ref. [17] for the case c> a was performed in Refs. [19,23] by means of matrix methods which allow one to determine the value Z(k,N) numerically for the isotropic lattice of obstacles. The basic ideas of the paper [17] were used in Ref. [19] for investigation of the influence of topological effects in the problem of rubber elasticity of polymer networks. The dependence of the strain x on the relative deformation A for the uniaxial tension Ax = Xy = 1/Va, kz = A calculated in this paper is presented in Fig. 6 in Moon-ey-Rivlin coordinates (t/t0, A ), where r0 = vT/V0(k — 1/A2) represents the classical elasticity law [13]. (The direct Edwards approach to this problem was used in Ref. [26].) Within the framework of the theory proposed, the swelling properties of polymer networks were investigated in Refs. [19, 23] and the t(A)-dependence for the partially swollen gels was obtained [23]. In these papers, it was shown that the theory presented can be applied to a quantitative description of the experimental data. 

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. 

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