Integral equations Percus-Yevick

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... 

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-... 

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... 

Another possible approach solving the equilibrium distribution for an electric double layer is offered by integral equation theories [22]. They are based on approximate relationships between different distribution functions. The two most common theories are Percus-Yevick [23] and Hypernetted Chain approximation (HNQ [24], where the former is a good method for short range interactions and the latter is best for long-range interactions. They were both developed around 1960, but are still used. The correlation between two particles can be divided into two parts, one is the direct influence of particle j on particle i and the other originates from the fact that all other particles correlate with particle j and then influence particle i in precisely... 

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

Wertheim (1963) and Thiele (1963) showed that the equation of state derived by Reiss et al. (1959, 1960) is identical with the equation of state derived by Percus and Yevick (1958), who presented the exact solution of the integral equation for the radial distribution function. Lebowitz et al. (1965) generalized later the validity of this theory for the mixture of simple liquids. The same equation of state was derived by Stillinger (1961), who showed that Eq. (2.1) can be applied to fused salts also if we replace the fused salts by a rigid sphere fluid in which the particle diameter a equals the sum of the cation s and anion s radii. 

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. 

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. 

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... 

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

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... 

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... 

D — and D oo the smooth and generic dimension-dependence of the integrals enables one to interpolate reasonably accurate D = 3 values (rms error 1%) from the dimensional limit results. The interpolated integrals can be used either on their ovm, or in conjunction with an integral equation approximation which sums some subset of the required integrals exactly (such as the hypemetted-chain or Percus-Yevick methods) the combination methods are invariably better than either dimensional interpolation or integral equations alone. Interpolation-corrected Percus-Yevick values can be computed quite easily at arbitrary order however, errors in higher-order values are... 

Once potential parameters have been determined, we can start calculation downward following arrow in the figure. The first key quantity is radial distribution function g(r) which can be calculated by the use of theoretical relation such as Percus-Yevick (PY) or Hypemetted chain (HNC) integral equation. However, these equations are an approximations. Exact values can be obtained by molecular simulation. Ifg(r) is obtained accurately as functions of temperature and pressure, then all the equilibrium properties of fluids and fluid mixtures can be calculated. Moreover, information on fluid structure is contained in g(r) itself. 

A fundamental approach to liquids is provided by the integral equation methods (sometimes called distribution function methods), initiated by Kirkwood and Yvon in the 1930s. As we shall show below, one starts by writing down an exact equation for the molecular distribution function of interest, usually the pair function, and then introduces one or more approximations to solve the problem. These approximations are often motivated by considerations of mathematical simplicity, so that their validity depends on a posteriori agreement with computer simulation or experiment. The theories in question, called YBG (Yvon-Bom-Green), PY (Percus-Yevick), and the HNC (hypemetted chain) approximation, provide the distribution functions directly, and are thus applicable to a wide variety of properties. 

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). 

Two of the classic integral equation approximations for atomic liquids are the PY (Percus-Yevick) and the HNC (hypemetted chain) approximations that use the following closures... 

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... 

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... 

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