Percus-Yevick method

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... 

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... 

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). 

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... 

Standard approximate methods, e.g., the Percus-Yevick or hyper-chain approximations, are applicable for systems with the Gibbs distribution and are based on the distinctive Boltzmann factor like exp —U r)/ ksT)), where U(r) is the potential energy of interacting particles. The basic kinetic equation (2.3.53) has nothing to do with the Gibbs distribution. The only approximate method neutral with respect to the ensemble averaging is the Kirkwood approximation [76, 77, 87]. 

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. 

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... 

In the calculation, a model of the averaged structure factor for a hard-sphere (HS) interaction potential, S(g) is used [47, 48], which considers the Gaussian distribution of the interaction radius cr for individual monodisperse systems for polydispersity m, and a Percus-Yevick (PY) closure relation to solve Omstein-Zernike (OZ) equation. The detailed theoretical description on the method has been reported elsewhere [49-51]. 

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. 

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. ... 

From a theoretical point of view the hypernetted chain (HNC) and Percus-Yevick (PY) equations are better approximations. Although both can be solved only by numerical methods, they offer the opportunity to study any model potential if appropriate computer facilities are available. 

A different method of studying the same problem is by applying the set of four Percus-Yevick equations for a two-component system. These can be written by a simple generalization of the one-component equation ... 

At least in the case of liquid simple metals, a knowledge of the effective pan-potentials describing the interaction between the ions in the liquid metal can also be utilized to calculate g(r) and A K). The most common such method involves the assumption of a hard-sphere potential in the Percus-Yevick (PY) equation its solution provides the hard-sphere structure factor, /4hs( C). (See Ashcroft and Lekner 1966.) The two parameters that must be provided for a calculation of Ahs( ) are the hard-sphere diameter, a, and the packing fraction, x. It is found that j = 0.45 for most liquid metals at temperatures just above their melting points. A hard-sphere solution of the PY equation has also been obtained for binary liquid metal alloys, and provides estimates of the three partial structure factors describing the alloy structure (Ashcroft and Langreth 1967). To the extent that the hard-sphere approximation appears to be valid for the liquid R s, pair potentials should dominate these metals also, at least at short distances. 

Two other methods have used different approximations to the direct correlation function. One is an adaptation of the Percus-Yevick equations for a homogeneous binary mixture and the other is based on the functional expansions of 4.5. 

The Percus-Yevick approximation uses C =g[l-exp(M,// 7)] to obtain the physical quantities of a homogeneous fluid. Before examining this and the Camahan-Starling (CS) approximation for hard spheres, some manipulations for one-dimensional (ID) rods is presented to get a feel for the methods. 

The closure approximation is the fundamental statistical mechanical approximation in PRISM theory. Determining the appropriate closure depends on the form of the potentials as well as the system parameters such as temperature and pressure [6]. The standard Percus-Yevick (PY) closure has been found to work well for repulsive force potentials in small molecule and macromolecular systems. The PY closure for atomic liquids can be derived using Percus method [79, 80] of a perturbative expansion of the density functional or by Stell s [8] graph summation method. The pair and direct correlation functions in PY theory are given by... 

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