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The Percus-Yevick integral equation

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. [Pg.312]

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  [Pg.312]

Consider the singlet density at Rx in the presence and in the absence of r/r. Clearly, we have (using the notation of Appendix C) [Pg.312]

Both the vertical and the slashed lines can be read as given that. On the lhs of (D.2) and (D.3), we have the singlet density given and external potential r/r, whereas on the rhs, we have the conditional density given a particle at Rq. [Pg.312]

This particular expansion does not prove to be useful. The reason, as explained in Appendix B, is that a first-order Taylor expansion is expected to be useful when the increment here, i/r, is small. For instance, in equation (B.28) of Appendix B, if x is very large, we cannot expect that a first-order Taylor expansion will lead to a good approximation. In (D.4), ijr replaces x (of the one-dimensional example). Since if/iR1) — oo as R1 — R0, the increment cannot be considered to be small.  [Pg.313]


This is the Percus-Yevick integral equation for y. Once a solution for y is obtained, one can calculate gfrom (D.17). [Pg.315]

M. S. Wertheim, Exact solution ot the Percus-Yevick integral equation for hard spheres, Ihys. Rev. Lett. 10, 321-323 (1963). [Pg.83]

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... [Pg.542]

Preu, H., Zradba, A., Rast, S., Kunz, W., Hardy, E. H., and Zeidler, M. D. (1999). Small angle neutron scattering of DjO-Brij 35 and DjO-alcohol-Brij 35 solutions and their modelling using the Percus-Yevick integral equation. Phys. Chem. Chem. Phys., 1, 3321-3329. [Pg.181]

However, in certain cases where these integral equations yield analytical solutions, the method will be interesting to chemical engineers. For instance, Wertheim (2,3) and Thiele (4) have solved the Percus-Yevick (PY) equation for the hard-sphere potential, where... [Pg.13]

FIG. 10 Theoretical adsorption isotherms for a Lermard-Jones fluid on a perfectly flat surfiice at high temperatures [213]. is defined in Eq. (35), and the adsorption isotherm in Eq. (23). H = Henderson equation of state [232], PY = Percus-Yevick integral equation [210], RO = Reddy-O Shea equation of state [279], Eq. (28), CM = Cuadros-Mulero equation of state [288], Eq. (29). [Pg.481]

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... [Pg.478]

Scattering and Disorder. For structure close to random disorder the SAXS frequently exhibits a broad shoulder that is alternatively called liquid scattering ([206] [86], p. 50) or long-period peak . Let us consider disordered, concentrated systems. A poor theory like the one of Porod [18] is not consistent with respect to disorder, as it divides the volume into equal lots before starting to model the process. He concludes that statistical population (of the lots) does not lead to correlation. Better is the theory of Hosemann [158,211], His distorted structure does not pre-define any lots, and consequently it is able to describe (discrete) liquid scattering. The problems of liquid scattering have been studied since the early days of statistical physics. To-date several approximations and some analytical solutions are known. Most frequently applied [201,212-216] is the Percus-Yevick [217] approximation of the Ornstein-Zernike integral equation. The approximation offers a simple descrip-... [Pg.186]

A number of approximate integral equations for the radial distribution function g(r) of fluids have been proposed in recent years. Two particularly useful approximations are the Percus-Yevick (PY)1,2 and the Convolution Hypernetted Chain (CHNC)3-4 equations. In this paper an efficient numerical method of solving these equations is described and the results obtained bv applying the method to the PY equation are discussed. A later paper will describe the behavior of the... [Pg.28]

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. [Pg.21]

Relation (D.18) is often referred to as the Percus-Yevick approximation. If we use (D.18) in the Ornstein-Zernike relation, we get an integral equation for y... [Pg.315]

Since this chapter was submitted in January, 1962, a number of investigations dealing with the hard sphere fluid have appeared, in particular, those based on the Percus-Yevick (P.Y.) integral equation. This equation is the simplest and, on the basis of... [Pg.285]

As a last physical approach we mention, but do not further consider, the scaled-particle-theory (SPT) which was developed about the same time as the Percus-Yevick theory. It gives good results for the thermodynamic properties of hard molecules (spheres or convex molecules). It is not a complete theory (in contrast to the integral equation and perturbation theories) since it does not yield the molecular distribution functions (although they can be obtained for some finite range of intermolecular separations). [Pg.461]

Here we illustrate the solvation formalism by integral equation calculations for binary mixtures described by the Lennard-Jones model (see Tables 8.1 and 8.2), and based on the Percus-Yevick approximation for the solution of the Ornstein-Zernike equations (Hansen and McDonald 1986) according to the approach proposed by McGuigan and Monson (McGuigan and Monson 1990). We focus on the solute-induced effects on the microstructure and the thermodynamic properties of infinitely dilute solutions of pyrene in carbon dioxide and Ne in Xe along the... [Pg.200]

The intra-chain pair density functions obtained from both the bulk simulation and the continuous unperturbed chains were used as input to the polymer-RISM integral equation for estimating the intermolecular pair distribution function g(r) (using a soft-Percus Yevick closure). We found that PRISM underpredicts the first peak in g(r), while also overpredicting the steefmess of the rise to the first peak. [Pg.274]


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