Percus-Yevick equation for hard spheres

Wertheim M S 1963 Exact solution of the Percus-Yevick equation for hard spheres Phys. Rev. Lett. 10 321... 

Baxter R J 1968 Percus-Yevick equation for hard spheres with surface adhesion J. Chem. Phys. 49 2770... 

It is well known that the function y R) is a monotonically decreasing function of R in the range 0 < R < cr. We write here the form of y(R) from the exact solution of the Percus-Yevick equation for hard spheres in the region 0 < R < a ... 

M. S. Wertheim, Exact solution ot the Percus-Yevick integral equation for hard spheres, Ihys. Rev. Lett. 10, 321-323 (1963). 

Mandel, F., Bearman, F.J. and Bearman, M.Y., 1970, Numerical solutions of the Percus-Yevick equation for the Lennard-Jones (6-12) and hard-sphere potentials, J. Chem. Phts., 52 3315. 

Percus-Yevick Equation for the Hard-Sphere Potentials, J. Chem. 

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 ... 

Lebowitz, J. L. 1964. Exact solution of generalized Percus-Yevick equation for a mixture of hard spheres. Phys. Rev. A 133 895. 

We will describe integral equation approximations for the two-particle correlation fiinctions. There is no single approximation that is equally good for all interatomic potentials in the 3D world, but the solutions for a few important models can be obtained analytically. These include the Percus-Yevick (PY) approximation [27, 28] for hard spheres and the mean spherical (MS) approximation for charged hard spheres, for hard spheres with point dipoles and for atoms interacting with a Yukawa potential. Numerical solutions for other approximations, such as the hypemetted chain (EfNC) approximation for charged systems, are readily obtained by fast Fourier transfonn methods... 

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 ... 

Figure 2.3. Radial distribution function hs( ) for suspensions of hard spheres in the disordered state at various volume fractions

A A eoe 1 i theoretical static stnicture fracture from the Percus-Yevick equation of state for hard spheres at < = 0.563,... 

Estimate the packing fraction for a hard-sphere liquid with a density of 21.25 atoms nm and a hard-sphere diameter of 350 pm. Use this result to calculate the Percus-Yevick product for the system at 85 K using the Carnahan-Starling equation of state (equation (2.9.11)). 

In order to complete the MSA estimate of Iny,- one must add the hard-sphere contribution, which accounts for the fact that work must be done to introduce the ions as hard spheres into the solution. It is obtained from the Percus-Yevick model for non-interacting hard spheres. For the case that all ions (spheres) have the same radius, the result is (see equation (3.9.22))... 

X — term, defined by Equation 77, in the Percus-Yevick expression for the pressure of a hard-sphere mixture... 

Andersen, H. C. and D. Chandler. 1972. Optimized cluster expansions for classical fluids. 1. General theory and variational formulation of the mean spherical model and hard sphere Percus-Yevick equations. Journal of Chemical Physics. 57, 1918. 

Fig. 4. The radial distribution for a hard-sphere fluid as calculated from the Percus-Yevick equation by Throop and Bearman (1965). The dashed lines represent the function y(r) = exp [u(r)]g(r). Notice that y(r) is a continuous function at r = a. (McQuarrie, 1976)...

Points 1 and 2 can be incorporated straightforwardly using the ideas presented earlier in this chapter. For example, we could use the analytic Percus-Yevick equations of state for hard spheres (Eqs. 47a and b) or the Carnahan-Starling equation of state (Eq. 49) for p. Furthermore, we could use the hard-sphere radial distribution function obtained numerically from one of the integral equations or even that calculated from computer simulation. Points 3 and 4 are less straightforward and represent contributions that were made around 1970 by Barker and Henderson (1976) and by Weeks, Chandler and Andersen (1971). The results of these two approaches are comparable and are illustrated in Figs. 10 and 11 and Table 3. 

Closure approximations to the PRISM equation are generally developed via an analogy with atomic liquids. Three of the common closures for hard spheres are the Percus-Yevick (PY), hypemetted chain (HNC), and Martynov-Sarkisov (MS) closures. It has been shown that the PY closure is the most accurate of the three, and in fact the HNC and MS closures have either no solution or unphysical solutions at low densities. The PY closure is given by. 

One approach of this category is to solve the integral equations using the Percus-Yevick closure for the system of adhesive hard sphere (AHS) mixtures (17-22). An adhesive hard sphere is a hard sphere that has attractive sites at surface. The attractive interaction on these attractive sites is infinitely strong and infinitesimally short ranged. The Percus-Yevick closime yields an analytical solution for such systems. The adhesive attraction, which resembles the chemical bonding, is used to build up chains by employing the proper connectivity constraints. 

In the limit of zero association, x — 0 the latter equation reduces to the adsorption isotherm of hard spheres, evaluated within the singlet Percus-Yevick approximation, whereas for xx 1 (i-S- the limit of complete association) one obtains the adsorption isotherm of tangent dimers... 

Tc. The two power-law exponents are not independent but depend on a single parameter, the so-called critical exponent X, which is specific for a given interaction potential (e.g., hard spheres). Actually, the interaction potential enters the MCT equations only indirectly via the structure factor S(q), which fixes the nonlinear coupling in the generalized oscillator equation. It is important to note that the MCT exponents are not universal in contrast to those of second-order phase transitions. In the case of hard spheres, for example, S(q) can be calculated via the Percus-Yevick approximation [26], and the full time and -dependence of < >(q. f) were obtained. As an example, Fig. 10 shows the susceptibility spectra of the hard-sphere system at a particular q. Note that temperature cannot be defined in the hard-sphere system instead, the packing fraction cp is used as a parameter. Above the critical packing fraction 0), which corresponds to T < Tc in systems where T exists, the a-process is absent (frozen) and only the fast dynamics is present. At cp < tpc the a-peak and the concomitant susceptibility minimum shift to lower frequencies with increasing cp, so that the closer cp is to the critical value fast dynamics can be identified (curve c in Fig. 10). 

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] ... 

The exact solution of the Percus-Yevick (PY) equation is known for a one-component system of hard spheres (Wertheim 1963 Thiele 1963) and for mixtures of hard spheres (Lebowitz 1964). Numerical solutions of the PY equation (for Lennard-Jones particles) have been carried out by many authors, e.g., Broyles (1960, 1961), Broyles et al. (1962), Throop and Bearman (1966), Baxter (1967), Watts (1968), Mandel et al. (1970), Grundke and Henderson (1972a, b)... 

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