Penetrable-sphere model applications

The surface tension and surface excess energy for the penetrable-sphere model in the mean-field approximation. 

The integral equation for p(z) has been solved numerically and the resulting p(z) and z used to calculate from (5.99) and a from (S.lOl) and from (5.104), that is, via the energy route and via the equation common to the virial and correlation function routes. Both equations lead to the same value of a whidi, with is shown in Fig. 5.8. 

The mean-field approximation used throi ihout this sectk n is exact for a system of infinite dimensionality, and over-emphasizes the propensity of the model to separate into two phases for systems of fimte dimensionality. Thus the critical temperature for infinite dimensionality, and so also for the mean-field approximation for all dimensions, whidi is fcT°/e = 1/e, is an upper bound for for systems of finite dimensionality. For a one-dimensional system there is no phase separation, for hm become zero. Simflarly, the mean-field approximation underestimates the thidoiess of the surface layer between the fluid phases in a three-dimensional system. Nevertheless its high degree of internal consistency, and qualitative correctness make it a useful approximation for systems three (or more) dimensions. 

Tractable models may be used to test approximations whose effect on the calculation of the properties of realistic systems is difficult to assess. We illustrate this point by using the results of the last section to test several versions of the van der Waals or density-gradient theory of Chapters 3 and 4. This theory, even in its most general form, is to be thought of as a set of approximations (smallness of p (z), constancy of the 

The free-energy density of the homogeneous fluid can be found in the mean-field approximation by integrating the Gibbs-Helmholtz equation, since we know the energy density (p), (5.82). 

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Now we turn to the penetrable-sphere models of fluids and treat them as we have done the lattice-gas models and the model of attracting hard spheres but there are some respects in which we shall be able to go beyond the mean-field approximation, and so find the limits of its applicability. 

In a lattice gas the molecules move continuously but are subject to discrete potentials. The density profile p(z) and other properties of the system can be found only at fixed points separated by the lattice spacing, as in Figs 5.2 and 5.5. The penetrable-sphere model" is a true continuum model which has much of the tractabOity of the lattice gps, shares with it a symmetry similar to the hole-partide symmetry of (5.32), but differs from it in that p(z) etc. can now be calculated for all values of z. In this and die following sections we describe the model briefly and use it to iDustrate the application of some of the results of the last chapter. 

The model of diffusion of hard spheres is applicable to interpret self-diffusion in liquids which behave according to the van der Waals physical interaction model (56). This might be the case for simple dense fluids at high temperature, T Tg, but it is an oversimplified model for the real diffusion of small organic penetrants in polymers. The functional relationships derived in the model of hard-spheres have been reinterpreted over course of the time, leading to a series of more sophisticated free-volume diffusion models. Some of these models are presented briefly below. 

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