Potential functions hard-core

The calculations that have been carried out [56] indicate that the approximations discussed above lead to very good thermodynamic functions overall and a remarkably accurate critical point and coexistence curve. The critical density and temperature predicted by the theory agree with the simulation results to about 0.6%. Of course, dealing with the Yukawa potential allows certain analytical simplifications in implementing this approach. However, a similar approach can be applied to other similar potentials that consist of a hard core with an attractive tail. It should also be pointed out that the idea of using the requirement of self-consistency to yield a closed theory is pertinent not only to the realm of simple fluids, but also has proved to be a powerful tool in the study of a system of spins with continuous symmetry [57,58] and of a site-diluted or random-field Ising model [59,60]. 

Fig. 8.2 The bond energy per atom of the four-fold, three-fold, two-fold and one-fold coordinated lattices as a function of the number of valence electrons per atom, /V, for the cases of degrees of normalized hardness of the potential, 3- -> 2- - 1 -fold coordination of the (8-N) rule.

Typical forms of the radial distribution function are shown in Fig. 38 for a liquid of hard core and of Lennard—Jones spheres (using the Percus— Yevick approximation) [447, 449] and Fig. 44 for carbon tetrachloride [452a]. Significant departures from unity are evident over considerable distances. The successive maxima and minima in g(r) correspond to essentially contact packing, but with small-scale orientational variation and to significant voids or large-scale orientational variation in the liquid structure, respectively. Such factors influence the relative location of reactants within a solvent and make the incorporation of the potential of mean force a necessity. 

Consider the simple case where the radial distribution function in the fluid is zero for radii less than a cut-off value determined by the size of the hard core of the solute, and one beyond that value. Calculate the value of the parameter a appearing in the equation of state Eq. (4.1) for a potential of the form cr , where c is a constant and n is an integer. An example is the Lennard-Jones potential where = 6 for the long-ranged attractive interaction. What happens if n <37 Explain what happens physically to resolve this problem. See Widom (1963) for a discussion of the issue of thermodynamic consistency when constructing van der Waals and related approximations. 

Whether these requirements can be met depends on the model considered and on the closure relation involved for the calculation of the correlation functions. Examples for which Eq. (7.54) has actually been used pertain to the class of simple QA systems, that is, QA systems with no rotational degree of freedom where the interaction potentials contain a spherical hard-core contribution plus (at most) an attractive perturbation. For such sj stems, the free energy has been calculated on the basis of correlation functions in the mean sphericfxl approximation (or an optimized random-phase approximation) [114, 298). 

Of course, the above considerations may not be relevant to the problem at hand, since in solving the OZ equation, the important functions are y(l, 2), its Fourier transform and B(l,2). ° It is to be expected that y(l,2) will vary less quickly between different orientations and will be continuous even for hard core potentials. Thus, its expansion in spherical harmonics should be better behaved than that of gf(l,2). Computer simulation cannot be used to obtain y(l,2) but Lado has presented some evidence based upon his solution of the RHNC approximation for a hard diatomic fluid using a spherically averaged bridge function that the convergence is good. Nevertheless, the results he presents are, in our view, for a rather short diatomic bond length and may not be conclusive. 

The general formulation of density-functional theory for molecular fluids has been described by Smithline et al. They have applied a simplified version of the theory to the freezing of hard core diatomic fluids. A suitable starting point for such theories for rigid nonspherical molecules is the following expression for the grand potential... 

Let us plot the logarithmic values of the shear viscosities calculated by the PNM model, divided by the dilute gas values (which vary as the square root of temperature 7 as a function of ejkT, e being the depth of the Lennard-Jones potential well. The results are given in Fig. 1 for a reduced density of na = 0.818, a being the hard core diameter. The experimental points are well fitted by two approximately straight lines. We observe that Eyring s formula for shear viscosity at constant density... 

The solution determines c r) inside the hard core from which g(r) outside this core is obtained via the Omstein-Zemike relation. For hard spheres, the approximation is identical to the PY approximation. Analytic solutions have been obtained for hard spheres, charged hard spheres, dipolar hard spheres and for particles interacting with the Yukawa potential. The MS approximation for point charges (charged hard spheres in the limit of zero size) yields the Debye-Huckel limiting law distribution function. 

Although one might question the applicability of this kind of perturbation theory to hard-core interactions (where the potentials become infinite), we note here that the term gd(r) [V(r) — Vj(r)] never diverges since (i) do < d so divergences due to the actual hard core in V are subtracted off by the hard core of the effective potential, Vj, and (ii) the pair-correlation function, gd vanishes when r < d. 

All our discussion of theories of polymer surface tensions up to now has relied ultimately on the van der Waals theory in conjunction with some equation of state. There is, however, another possible approach, which starts out from a principle of corresponding states, as was first applied to polymers by Patterson and Rastogi (1970). We start by assuming spherical molecules interacting via an attractive potential, which is described by the depth of the energy minimum (s ) and a hard core or impenetrable radius (r ). The partition function is written in terms of variables scaled by the characteristic energy and length scale and thus all physical properties can be described in terms of these scaled variables. Clear corollaries with the equation of state... 

In an interacting system, each polymer chain will have a distribution function obe5dng equation (4.3.2), each with its own function Ufr) whose form can be obtained if the positions of all the other poljmier chains are known. The resulting system of coupled differential equations, one for each chain in the system, is obviously completely intractable as it stands. However, we can make progress if we make a mean-field assumption - we assume that every polymer chain of the same chemical type experiences an average, mean-field potential U f). This potential has two parts. The first part arises simply from the hard core potential which prevents two segments occupying the same... 

Hence, the KBIs in Equation 2.70 are all negative, and are due to the repulsive part of the hard core potential function Equation 2.57. 

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