Pair correlation function Percus-Yevick equation

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... 

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.

Figure 2.10 The pair correlation function for g R ) for liquid argon (at 84.25 K and 0.71 atm) with R = RI3.S. The dotted curve is experimental values provided by N.S. Gingrich (to which the author is very grateful). The solid curve is a solution of the Percus-Yevick equation with parameters rr = 3.5A e//c7" = 1.39 and per3 = 0.85 (for details see Appendix E). The theoretical and experimental curves are almost indistinguishable.

The pair correlation functions for this system were calculated by solving the Percus-Yevick equations as described in section 2.9, and Appendices D and E. 

The history of the search for an integral equation for the pair correlation function is quite long. It probably started with Kirkwood (1935), followed by Yvon (1935, 1958), Born and Green (1946), and many others. For a summary of these efforts, see Hill (1956), Fisher (1964), Rushbrooke (1968), Munster (1969), and Hansen and McDonald (1976). Most of the earlier works used the superposition approximation to obtain an integral equation for the pair correlation function. It was in 1958 that Percus and Yevick developed an integral equation that did not include explicitly the assumption of superposition, i.e., pairwise additivity of the higher order potentials of mean force. The Percus-Yevick (PY) equation was found most useful in the study of both pure liquids as well as mixtures of liquids. 

The Percus-Yevick (PY) equation for spherical particles in three dimensions was found to be very useful for the study of the pair correlation function. For the purposes of the present 2-D system, the corresponding PY equation may be obtained most directly from the Ornstein-Zernike relation ... 

An approximate version of the Percus-Yevick ( Y) equation has been applied for the pair potential based on the Bjerrum model [Ben-Naim (1970)]. The pair correlation function is written in... 

To conclude this section, I would like to add two sets of results on the pair correlation function (PCF) between two simple solutes in an L/ solvent. These results may or may not be relevant to the problem of Hhard-sphere (particles labeled A) solutes in an LJ solvent (particles labeled B) with varying strength of the energy parameter sbb- All the calculations for this and the subsequent demonstration were done by solving the Percus-Yevick integral equations with the following molecular... 

The microscopic static property that is usually of primary concern is the pair correlation function g(r). To calculate g(r), each molecule in turn is imagined to be at the center of a series of concentric spheres. The number of molecules in each spherical shell is divided by the volume of that shell, with attention being paid to periodic image locations of molecules outside the box. The results are averaged over all the molecules, and then over many time steps. The pair correlation function is usually calculated in this way only for distances less than the range of the potential r,. Verlet has provided a method for extending g(r) beyond r, using the direct correlation function determined from the Percus-Yevick equation. ... 

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... 

Fig. 8.23. The spatial pair correlation functions for a two-component system in two dimensions. The computations were carried out by solving the four Percus-Yevick equations (8.150) with the parameters listed in (8.153). [For more details see appendix 9-E.]...

The adaptation of the Percus-Yevick approximation starts with the three Omstein-Zemike equations which relate h , h, and hi, to the set of direct functions Cu, c.h> and Cm, in a homogeneous binary mixture of molecules a and b, which have hard cores but otherwise unspecified pair potentials. The limit is now taken in which the radius of the hard core of b becomes infinite and its concentration goes almost to zero, so that the system comprises a fluid of a molecules in contact with the flat wall of the one remaining b molecule. Only two Omstein-Zemike equations remain, one for h and one for the molecule-wall correlation, These are solved by using the Percus-Yevick approximation,... 

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