Numerical solution of the Percus-Yevick equation

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) 

We present here a brief account of the numerical procedure employed for the computations of g(R) which we have used for our illustrations in this book. We start with the integral equation for the function z(R) (see Appendix D) 

We begin the iterative procedure by substituting the initial function 

zi(f ) from (E.3) is substituted in the rhs of (1) to obtain z2(R), and so forth. It turns out that for high densities p, such a procedure does not lead to a convergent solution. Instead, one uses a mixing parameter X, 0 X 1 (Broyles 1960, 1962 Throop and Bearman 1966 Ben-Naim 1972a, b, 1974) so that the (k+ 1 )th input function is constructed from the fcth input and the /cth output, as follows  

In practice, it is found that as p increases, one is compelled to use smaller values of X in (E.4), and large numbers of iterations to get a convergent result. 

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Figure 2.4. Dependence of the pair correlation function g[R) for the Li particles on the number density. The density p is indicated next to each curve in the dimensionless quantity pa3. We choose a — 1 and el kT—0.5 in the Li potential. All the illustrations of g(R) for this book were obtained by numerical solution of the Percus-Yevick equation. See Appendix E for more details.

Fig. 2.6. Dependence of g R) on the density of the system. The corresponding (number) densities are indicated next to each curve. The functions g R) for this illustration were computed by numerical solution of the Percus-Yevick equation for Lennard-Jones particles with parameters cr = 1.0 and ejkT — 0.5 (for details, see Appendix 9-E).

All the computations for these illustrations were performed by a numerical solution of the Percus-Yevick equations. For more details on the numerical procedure, see Appendix 9-E. 

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. 

Bulk phase fluid structure was obtained by solution of the Percus-Yevick equation (W) which is highly accurate for the Lennard-Jones model and is not expected to introduce significant error. This allows the pressure tensors to return bulk phase pressures, computed from the virial route to the equation of state, at the center of a drop of sufficiently large size. Further numerical details are provided in reference 4. 

The details of the derivation of the Percus-Yevick equation and the numerical procedure for its solution are highly technical and will not be presented here. [Details may be found in Appendices 9-D and 9-E and in Ben-Naim (1971c, 1972c,d).] We note, however, that each pairwise function in (6.122) depends only on three coordinates, which we can choose as follows R is the distance between the centers of the two particles, = I — RJ, and is the angle between the vector ii and the direction of Ry = Rj — Rj, measured counterclockwise. The full pair correlation function is thus a function of three variables, g(R, a2) Because of the special symmetry of the pair potential, it is clear that all of the pairwise functions, such as U, y, or g, will be invariant to a rotation of the particle... 

In this chapter, we shall not discuss the methods of obtaining information on molecular distribution functions. There are essentially three sources of information analyzing and interpreting x-ray and neutron diffraction patterns solving integral equations and simulation of the behavior of liquids on a computer. Most of the illustrations for this chapter were done by solving the Percus-Yevick equation. This method, along with some comments on the numerical solution, are described in Appendices B—F. 

A set of equations (15)-(17) represents the background of the so-called second-order or pair theory. If these equations are supplemented by an approximate relation between direct and pair correlation functions the problem becomes complete. Its numerical solution provides not only the density profile but also the pair correlation functions for a nonuniform fluid [55-58]. In the majority of previous studies of inhomogeneous simple fluids, the inhomogeneous Percus-Yevick approximation (PY2) has been used. It reads... 

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