Penetrable-sphere models

We saw following (5.60) that it was the repulsion between unlike molecules in the model that was responsible for the form of (5.60), so it is that which is now also responsible for the form of (5.65). In the two-component (primitive) version of the penetrable-sphere model, which we treat in 5.S-5.7, we see the equivalent of (5.65) again in (5.133), where it has a similar physical origin. The negative mass m in that case is (in dimensionless form) -l/2(s+2), with s the dimensionality of the potential see (5.134) below. 

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. 

Fia. 5.7. Sketch of the phase diagram of the penetrable-sphere model. The fiill curve is the orthobaric line and the dashed curve is the line of symmetry. 

At non-zero temperatures the surface tension can be calculated only in the mean-field approximation, after solving the integral equation for p(z). This has been done numerically for the penetrable-sphere model, and analytically for a penetrable-cube model, in which the spheres are replaced by oriented cubes of volume Uo. The effective density or(z) is then given by an equation simpler than (5.83), viz. 

Thus z here retains at all temperatures its low-temperature limit of Integration of the integral equation for p(z) shows that for the penetrable-sphere model, z falls from i at zero temperature to >/(3/20) = 0 387 at the critical point. 

Fio. S.8. The surface tension and surface excess energy for the penetrable-sphere model in the mean-field approximation. 

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. 

In transcribing these results into those for the penetrable-sphere model we identify p for that model with p and obtain Ae surface tension for the penetrable-sphere model without knowing explidtly the function p(z) ... 

We have seen in 4.8 the difficulty of reconciling the arbitrariness of the surface of tension, z defined medianically (or quasi-thermodynamically) by (2.89) and the planar limit of the surface of tension of a drop, introduced through the thermodynamic arguments of 2.4—arguments which fix its position with respect to the equimolar surface, R. One way of analysing problems of this kind is by exact calculations for a model system. The mean-field treatment of the penetrable-sphere model is not exact, but, as we have seen, it becomes so in the two limits of (1) infinite dimensionality at all temperatures, and (2) zero temperature for all dimensions. Here we examine the three-dimensional spherical drop (and bubble) in the mean-field approximation and show that the results resolve some of the difficulties of 4.8. 

Here 2. is the height of the equimolar dividing surface, and D is a measure of the thickness. Althoi a hyperbolic tangent arises naturally for the penetrable-sphere model (S S.S) and in the van der Waals theory of a system near its gas-liquid critical point (S 9.1) its use here is purely empirical indeed, we shall see in S 7.S that an exponential decay of p(z) at large values of 2 -2. is not correct for a Lennard-Jones potential that runs to r=9c. This equation is, however, a convenient one since it can be fitted to experimental points by inverting it to give... 

We see that (7.14) is formally the same equation for the orthobaric densities as was obtained for the penetrable-sphere model in the mean-field approximation, namely (5.91), and is the analogue of (5.31) for the lattice-gas. Like (5.91), it has the solution... 

The modified van der Waals theory can hself be regarded as an approximation to the theory based on the direct correlation function. We saw in 5.6 (Fig. S.IO) that this theory and its derivatives leads to too high a surface tension for the penetrable-sphere model. Here also its surface tension is larger than the results found by computer simulation, and it is clearly less accurate than the other theories, between which it is hard to discriminate on this basis. The surface tension is not sensitive to the details of how g or c has been approximated, provided that these functions satisfy the constraints set out above. 

In 5.6 we studied the interface in a symmetric mixture model—the two-component, primitive version of the penetrable-sphere model— which we treated in a mean-field approximation. Thm is a three-component version of that model due to Guerrero e( of., with which we may illustrate many of the ideas of this section. Again, like molecules do not interact, while unlike molecules repel each other as hard spheres. If the components are a, 6, c, then in any homogeneous phase, in mean-field approximation, the densities p., etc., and activities etc., all in units of tlte volume Vo the exdusion sphere, are related by ... 

Ultrasoft repulsive interactions (without the long-range Coulomb part) have been extensively studied, both for its theoretical aspects and as a description of a soft matter system. Studies reveal two distinct behaviors. Some ultrasoft potentials supports stacked configurations, where two or more particles collapse, even though no true attractive interactions come into play [51, 52]. This behavior leads to a peak in a correlation function around r = 0. To this class of potentials belongs the penetrable sphere model [53]. The Gaussian core model [54], on the other hand, represents the class of soft particles unable to support stacked configurations. 

The simulation results for hard-sphere particles with the same diameter yield a profile with more defined structure. The penetrable sphere model captures only qualitatively these features. 

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