Percus Yevick equation

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

Wertheim M S 1964 Analytic solution of the Percus-Yevick equation J. Math. Phys. 5 643... 

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

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. 

Percus-Yevick Equation Applied to a Lennard-Jones Fluid ... 

An efficient method of solving the Percus-Yevick and related equations is described. The method is applied to a Lennard-Jones fluid, and the solutions obtained are discussed. It is shown that the Percus-Yevick equation predicts a phase change with critical density close to 0.27 and with a critical temperature which is dependent upon the range at which the Lennard-Jones potential is truncated. At the phase change the compressibility becomes infinite although the virial equation of state (foes not show this behavior. Outside the critical region the PY equation is at least two-valued for all densities in the range (0, 0.6). 

On this way we arrive at Bom-Green-Ivon, Percus-Yevick and hyperchain equations [5, 9], all having a general form (x,Vx,n,T) = 0. These non-linear integro-differential equations are close with respect to the joint correlation function, and Percus-Yevick equation gives the best approximation amongst known at present. An important point is that the accuracy of... 

Note that the binary HMSA [60] scheme gives the solute-solvent radial distribution function only in a limited range of solute-solvent size ratio. It fails to provide a proper description for such a large variation in size. Thus, here the solute-solvent radial distribution function has been calculated by employing the well-known Weeks-Chandler-Anderson (WCA) perturbation scheme [118], which requires the solution of the Percus-Yevick equation for the binary mixtures [119]. 

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 ... 

Ben-Naim (1972b, c) has examined hydrophobic association using statistical mechanical theories of the liquid state, e.g. the Percus-Yevick equations. He has also examined quantitative aspects of solvophobic interactions between solutes using solubility data for ethane and methane. The changes in thermodynamic parameters can be calculated when two methane molecules approach to a separation of, 1-533 x 10-8 cm, the C—C distance in ethane, and the solvophobic quantities 8SI/i, s 2 and 8SiS2 can be calculated. In water (solvophobic = hydrophobic) 5si/i is more negative than in other solvents and decreases as the temperature rises both 8s iH%... 

Throop, G. J., and R. J. Bearman Radial Distribution Functions for Binary Fluid Mixtures of Lennard-Jones Molecules Calculated from the Percus-Yevick Equation. J. Chem. Phys. 44, 1423—1444 (1966). 

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,... 

The KBIs for the LJ fluids can be calculated with eqn (1) using for the radial distribution function gy the Percus-Yevick equation. The KBIs obtained in this manner by Kojima, Kato and Nomura" have been employed. 

Using the hard sphere adhesive state equation proposed by Baxter (16), it is possible to calculate the demixing line due to interactions. This state equation corresponds to the exact solution of the Percus-Yevick equation in the case of an hard sphere potential with an infinitively thin attractive square well. In our calculation we assum that the range of the potential is short in comparison to the size of the particles (in fact less than 10 %). 

The following table gives the distribution function g r) for liquid argon at 85 K. The Lennard-Jones parameters for argon are a = 350 pm and = 118K. Estimate the Lennard-Jones potential at each value of r, and then use the Percus-Yevick equation to calculate the direct correlation function c(r) (equation (2.6.5)). Plot g(r) and c(r) against r. 

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. 

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.

Finally we show two illustrations of g(R) for real liquids, first, figure 2.10 for liquid argon (drawn as a function of the reduced distance R = R/3.5). Clearly the general behavior is similar to the LJ fluid. It is also shown in the figure that the theoretical curve, obtained from the solution of the Percus-Yevick equation, is almost indistinguishable from the experiment curve. 

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.

Note that the volume density of closed pack spheres is about rfcpK, 0.74. The choice of t] = 0.45, which is about 6/10 of the maximum density, was chosen for convenience. In fact even at these densities converging of the Percus-Yevick equation is quite slow (see also Appendix E). 

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. 

We present here a derivation of the Percus-Yevick equation based on the material of Appendices B and C. As in Appendix C, we consider a system in an external potential In the present case, the external potential is produced by a particle (identical to the other particles of the system) fixed at R0 ... 

By linearizing the exponential with respect to h(r) — c(r), one obtains an alternative closure, called the Percus-Yevick equation,164170... 

Both closures have been employed for determining the distribution functions for liquids 182 the Percus-Yevick equation tends to yield better results for nonpolar systems, while the hypemetted-chain equation (with the appropriate renormalization of long-range interactions)113,183 is found to be more appropriate for polar and ionic liquids.184... 

Fig. 12. The radial distribution function for argon calculated from (5) using the Lennard-Jones 12-6 potential and for the rigid-sphere diameter. The circles and squares are the results of Monte Carlo calculations and calculations based on the Percus-Yevick equation, respectively. All curves were calculated at a temperature of 327.5°K and a density of 0.67 g/cm .

Two radial distribution functions, as calculated from (5) (using the L-J (12-6) potential and d = t/j), are shown in Figs. 12 and 13. In both figures the circles and squares are the results of Monte Carlo calculations and calculations based upon the Percus-Yevick " equation, respectively. Both the radial distribution functions were calculated at a temperature of 327.5°K. In Figs. 12 and 13 the densities used in the calculation were 0.67 and 1.86 g/cm , respectively. The pressures predicted by these radial distribution functions are given in Table V, where MC and PY refer to the Monte Carlo and Percus-Yevick calculations, respectively. Calc, refers to the calculations based on (5). 

Edwards R.F. Chaiken, Detonation Calculations with a Percus-Yevick Equation of State , BuMines-RI-7905, Washington (1974)... 

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