Penetrable-sphere model theory

Had X and y been ordinary rectangular coordinates and m an ordinary positive mass, we would have had — W=p f2m on the trajectory, with p the ordinary momentum and in place of (S.72) and (S.73) we would have had formulae for the action expressing it as an integral of the momentum over the distance 

These formulae for the action in the more familiar system, or the corresponding formulae (S.72) and (5.73) in the system at hand, are analogous to the one-dimensional (3.13) or (S.22), but with one crucial difference due to the difference in dimensionality whereas, from (3.13) or (5.22), the action in a one-dimensional system may be obtained from the potential alone, without first solving the dynamical problem, that is usually not possible in systems of two or more dimensions, where we must also know the trajectory—y(x) or x(y) in this instance. 

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. 

Consider a volume V containing N molecules each at the centre of a sphere of volume Vo- These spheres are freely penetrable and serve only to define the configurational energy. If the volume covered by these spheres is W / ) then the energy (i ) is given by 

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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. 

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... 

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. 

The mass-transfer coefficient in each film is expected to depend upon molecular diffusivity, and this behavior often is represented by a power-law function k . For two-film theory, n = 1 as discussed above [(Eq. (15-62)]. Subsequent theories introduced by Higbie [Trans. AIChE, 31, p. 365 (1935)] and by Dankwerts [Ind. Eng. Chem., 43, pp. 1460-1467 (1951)] allow for surface renewal or penetration of the stagnant film. These theories indicate a 0.5 power-law relationship. Numerous models have been developed since then where 0.5 < n < 1.0 the results depend upon such things as whether the dispersed drop is treated as a rigid sphere, as a sphere with internal circulation, or as oscillating drops. These theories are discussed by Skelland [ Tnterphase Mass Transfer, Chap. 2 in Science and Practice of Liquid-Liquid Extraction, vol. 1, Thornton, ed. (Oxford, 1992)]. 

In this chapter we consider the depletion interaction between two flat plates and between two spherical colloidal particles for different depletants (polymers, small colloidal spheres, rods and plates). First of all we focus on the depletion interaction due to a somewhat hypothetical model depletant, the penetrable hard sphere (phs), to mimic a (ideal) polymer molecule. This model, implicitly introduced by Asakura and Oosawa [1] and considered in detail by Vrij [2], is characterized by the fact that the spheres freely overlap each other but act as hard spheres with diameter a when interacting with a wall or a colloidal particle. The thermodynamic properties of a system of hard spheres plus added penetrable hard spheres have been considered by Widom and Rowlinson [3] and provided much of the inspiration for the theory of phase behavior developed in Chap. 3. 

To summarize, theory and experiment clearly demonstrate that the types of phase equilibria encountered in unmixed colloid-polymer mixtures are rather sensitive to the size ratio q. For sufficiently large ( 0.3) a colloidal gas-liquid phase separation is encountered. For 0.4, the simple model of hard spheres plus penetrable hard spheres fails to accurately describe the phase behaviour of well-defined hard-sphere eolloid plus polymer mixtures. For large -values it is essential to improve the simple description of polymer chains as penetrable hard spheres. 

This theory has been partially confirmed by sedimentation experiment (Langevin and Rondelez, 1978). The value of the slope so far found was —0.50 0.10. We now have some evidence to believe that in the semidilute range of polymer solution the solvent is forced through in orderly fashion around the blob of radius C but still cannot penetrate the interior of the blob. Note that this theory is reminiscent of the pearl necklace model and the hydrodynamic equivalent sphere. 

The measured pore size distribution curves are frequently biased towards the small pore sizes due to the hysteresis effect caused by ink bottle shaped pores with narrow necks accessible to the mercury and wide bodies which are not. Meyer [24] attempted to correct for this using probability theory and this altered the distribution of the large pores considerably. Zgrablich et al. [25] studied the relationship between pores and throats (sites and bonds) based on the co-operative percolation effects of a porous network and developed a model to take account of this relationship. The model was tested for agglomerates of spheres, needles, rods and plates. Zhdanov and Fenelonov [26] described the penetration of mercury into pores in terms of percolation theory. 

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