Radial distribution function for hard spheres

Using (2.67), we can now rewrite explicitly the form of the radial distribution function for hard spheres at slightly dense concentration ... 

Turning to the results of dynamic simulation experiments. Fig. 16 compares radial distribution functions for hard-sphere, soft-sphere, and LJ glasses, all in comparable states. The compression rates (p dp/dT) for the preparation of the soft-sphere sample, the LJ sample, and the hard-sphere case were 0.024, 0.010, and 0.003 psec respectively, all on a time scale appropriate for LJ argon. Also the densities are all comparable as judged by the reference scales discussed in Section IV.C. If one uses the... 

The factor 2 is included in (2.72) since we compute only half of the intersection of the two spheres. (This corresponds to the volume of revolution of the shaded area in Fig. 2.3.) Using (2.72), we can now write explicitly the form of the radial distribution function for hard spheres at slightly dense concentration ... 

The main change in the SAFT version of Kraska and Gubbins is that they use Lennard-Jones (LJ) spheres for the reference term, rather than hard spheres. The remaining terms are unchanged, except that the radial distribution function used in the calculation of the chain and association contributions in Equations 3.25 and 3.31 is the radial distribution function for LJ spheres rather than hard spheres. Thus, an equation of state for LJ spheres is required. The equation used is that of Kolafa and Nezbeda. The Helmholtz energy for the reference (LJ) system is (for a pure fluid)... 

The next step in the procedure of evaluation of the mixture properties is the evaluation of the pseudo-radial distribution functions for all i — j interactions in the mixture as well as the mean free-path parameter atj for the unlike interaction. It is consistent with the remainder of the procedure to estimate them from mixing rules based upon a rigid-sphere model. Among the many possible mixing rules for the radial distribution function one that has proved successful is based upon the Percus-Yevick equation for the radial distribution function of hard-sphere mixtures (Kestin Wakeham 1980 Vesovic ... 

FIGURE 2.2 Radial distribution functions for (a) a hard sphere fluid, (A) a real gas, (c) a liquid, (li) a crystal. 

For concentrated suspensions of hard spheres, the radial distribution function for the fluid phase is generated from the solution to the Percus-Yevick [37] equation using a Heaviside step function mviltiplied by a nearest neighbor geometric function for a disordered fluid. TTie result is a function for the compressibility derived by Carnahan and Starling [25] ... 

FIGURE llJS Radial distribution function for suspensions of hard spheres (a) in the disordered state and (b) in the ordered state. Taken from Russel [30, pp. 339—340]. Copjni t 1989 by Ceunbridge University Press. Reprinted with the permission of Cambridge University Press. 

First, it is of interest to compare (Fig. 15) the radial distribution functions for an amorphous close-packed assembly of macroscopic (steel) spheres and for the amorphous hard-sphere solid obtained by gradual densification by MD of a 500-sphere system with the usual periodic boundary conditions. Although there are some minor differences for the third peak in g(r), there is little doubt that the mechanical shaking, and the more thermodynamic shaking in the MD experiment, are leading the system toward the same structure. The mechanical shaking experiment can give... 

Fig. IS. Comparison of radial distribution functions for an MD-generated amorphous hard-sphere soUd at a density (po 1.2) very slightly less than the amorphous close-packed density (circles), and a random closed-packed assembly of macroscopic spheres at po —1.22 (histogram). From Refs. 10 and 78.

E. Waisman, The radial distribution function for a fluid of hard spheres at high densities. Mol. Phys. 25, 45 8 (1973). 

Barker and Watts (1969) published a preliminary report on the computations of energy, heat capacity, and the radial distribution function for waterlike particles. The potential function used for these calculations is similar to the one discussed in Section 6.4 however, instead of a smooth switching function, they used a hard-sphere cutoff at 2 A so that the point charges could not approach each other to zero separation. 

Fig. 16.2. Reduced radial distribution functions for TbFcj (left) and GdFcj (right) derived from the transform of the neutron and X-ray scattering data of fig. 16.1. The atomic spacings corresponding to several combinations of hard sphere atoms in contact with radii equal to the Goldschmidt radii are indicated by the arrows. (Gd-Cargill, 1974 Tb-Rhyne et al., 1974a).

The approach considered is that first proposed by Enskog himself (Chapman Cowling 1952), who suggested that a pseudo-radial distribution function g for a real gas, to replace the function for hard spheres at contact, and a consistent effective hard-sphere diameter could be obtained from the equation of state for the real gas. Specifically, he proposed that the radial distribution function could be obtained from the equation of state for the real gas by replacing the pressure of the real gas by the thermal pressure Pt through the equation... 

The essence of the application of these equations to real fluids consists of three parts first the replacement of the hard-sphere results for the pure gas viscosity and the interaction viscosity by the values for the real fluid system second, the evaluation of a pseudo-radial distribution function for each binary interaction to replace the hard-sphere equivalent at contact and, finally, the selection of a molecular size parameter for each binary interaction to account for the mean free-path shortening in the dense gas. 

Many of the equilibrium properties of such systems can be obtained through the two-body reduced coordinate distribution function and the radial distribution function, defined in Eqs. (27.6-5) and (27.6-7). There are a number of theories that are used to calculate approximate radial distribution functions for liquids, using classical statistical mechanics. Some of the theories involve approximate integral equations. Others are perturbation theories similar to quantum mechanical perturbation theory (see Section 19.3). These theories take a hard-sphere fluid or other fluid with purely repulsive forces as a zero-order system and consider the attractive part of the forces to be a perturbation. ... 

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. 

Here ji(qa) is the spherical Bessel function of order l,g(a) is the radial distribution function at contact, and f = /fSmn/Anpo2g a) is the Enskog mean free time between collisions. The transport coefficients in the above expressions are given only by their Enskog values that is, only collisional contributions are retained. Since it is only in dense fluids that the Enskog values represents the important contributions to transport coefficient, the above expressions are reasonable only for dense hard-sphere fluids. Earlier Alley, Alder, and Yip [32] have done molecular dynamics simulations to determine the wavenumber-dependent transport coefficients that should be used in hard-sphere generalized hydrodynamic equations. They have shown that for intermediate values of q, the wavenumber-dependent transport coefficients are well-approximated by their collisional contributions. This implies that Eqs. (20)-(23) are even more realistic as q and z are increased. 

The more obvious and consistent deviations from the hard sphere theory occur, at the low density values, due to the effects of attractive forces in the real system. We can attempt to correct for these effects using a method described previously (27-30) for the analysis of angular momentum correlation times in supercritical CFjj and CFjj mixtures with argon and neon. We replace the hard sphere radial distribution function at contact hs with a function gp (0) which uses the more realistic... 

Inspection of Eq. 7 reveals that the molecular interference function, s(x), can be derived from the ratio of the total cross-section to the fitted IAM function, when the first square bracketed factor has been accounted for. A widely used model of the liquid state assumes that the molecules in liquids and amorphous materials may be described by a hard-sphere (HS) radial distribution function (RDF). This correctly predicts the exclusion property of the intermolecular force at intermolecular separations below some critical dimension, identified with the sphere diameter in the HS model. The packing fraction, 17, is proportional for a monatomic species to the bulk density, p. The variation of r(x) on 17 is reproduced in Fig. 14, taken from the work of Pavlyukhin [29],... 

In order to use the above expressions for calculating the thermodynamic properties, appropriate expressions for the radial distribution function and for the equation of state for the hard-sphere reference system are required which are given in Appendix A. Fortunately, accurate information for the hard-sphere fluid as well as for the hard-sphere solid is available and this enables the determination of the properties of the coexisting dilute and concentrated phases of colloidal dispersions. 

In order to determine the thermodynamic properties by means of the perturbation theory, the thermodynamic properties of the reference system are needed. Here, the expressions for the equation of state and the radial distribution function of a system of hard spheres are included for both the fluid and solid reference states. A face-centred-eubic arrangement of the particles at closest packing is assumed for the solid phase. 

Smith and Henderson [31] have derived an analytical expression for the radial distribution function of a hard-sphere fluid by solving the Percus— Yevick equation. The expression for the radial distribution function is given by ... 

FIG. 8 Radial distribution function of a system of hard-spheres with volume fraction = 0.45. Exact Monte Carlo results (open circles) are compared with the theoretical predictions of the Ornstein-Zernike equation for different closure relations (lines). The inset shows in more detail the accuracy of the different approximations in predicting the value of g(r) at contact. 

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