Percus-Yevick equation approximation

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-... 

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... 

In Eq. (9), the correlation functions for macroparticles do not appear on the RHS of this equation. Thus, applying a particular closure for the macroparticle correlations need not relate to the closures used for the correlation functions of the suspending fluid on the RHS of Eq. (9). This means, that using the Percus-Yevick (PY) approximation to describe the correlations between macrospheres... 

Abbreviations HNC, hypernetted chain equation MC, Monte Carlo simulation MD, molecular dynamics simulation MSA, mean spherical approximation PY, Percus-Yevick equation ChM, chemical model. 

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. 

To complete our theoretical treatment of liquids, we need a procedure to calculate g(r). It so happens that there is no exact equation for g(r), but there are several accurate approximate equations. Four equations that have had some success for fluids are the Kirkwood equation, the Born-Green-Yvon equation, the Percus-Yevick equation and the Hypernetted chain equation (McQuarrie, 1976). All four of these equations are integral equations for g(r) in terms of u(r). 

If we substitute this approximation into the Ornstein-Zernike equation, then we obtain the Percus-Yevick equation, Eq. (39). 

Mitchell et al. (1977) showed that the zeroth- and second-moment conditions follow from the asymptotic form of the direct-correlation function [Eq. (33)]. The MS, HNC, RHNC, and Debye Huckel limiting law approximations satisfy both these conditions the Percus-Yevick equation does not obey the second-moment condition and is less useful for electrolytes than it is for uncharged systems. 

Figure 8.8 (a) The evolution of packing fraction of spheres under cyclic shear. Inset A schematic of the apparatus used to measure the packing, (b) The radial correlation function g(r) develops sharp peaks as the system compacts. The inset shows that g(r) of the initial packing is described approximately by the Percus-Yevick equation. (Continued)... 

The general equation can be further reduced to the case of infinite dilution limit, a binary mixmre, ionic solutions, and so on. These equations are supplemented by closure relations such as the Percus-Yevick (PY) and hypernetted chain (HNC) approximations. 

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... 

In the limit of zero association, x — 0 the latter equation reduces to the adsorption isotherm of hard spheres, evaluated within the singlet Percus-Yevick approximation, whereas for xx 1 (i-S- the limit of complete association) one obtains the adsorption isotherm of tangent dimers... 

To solve the replica OZ equations, they must be completed by closure relations. Several closures have been tested against computer simulations for various models of fluids adsorbed in disordered porous media. In particular, common Percus-Yevick (PY) and hypernetted chain approximations have been applied [20]. Eq. (21) for the matrix correlations can be solved using any approximation. However, it has been shown by Given and Stell [17-19] that the PY closure for the fluid-fluid correlations simplifies the ROZ equation, the blocking effects of the matrix structure are neglected in this... 

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... 

Tc. The two power-law exponents are not independent but depend on a single parameter, the so-called critical exponent X, which is specific for a given interaction potential (e.g., hard spheres). Actually, the interaction potential enters the MCT equations only indirectly via the structure factor S(q), which fixes the nonlinear coupling in the generalized oscillator equation. It is important to note that the MCT exponents are not universal in contrast to those of second-order phase transitions. In the case of hard spheres, for example, S(q) can be calculated via the Percus-Yevick approximation [26], and the full time and -dependence of < >(q. f) were obtained. As an example, Fig. 10 shows the susceptibility spectra of the hard-sphere system at a particular q. Note that temperature cannot be defined in the hard-sphere system instead, the packing fraction cp is used as a parameter. Above the critical packing fraction 0), which corresponds to T < Tc in systems where T exists, the a-process is absent (frozen) and only the fast dynamics is present. At cp < tpc the a-peak and the concomitant susceptibility minimum shift to lower frequencies with increasing cp, so that the closer cp is to the critical value fast dynamics can be identified (curve c in Fig. 10). 

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 Figure 5.27 a curve calculated from Equation 5.216 is compared with the predictions of other studies. The dotted line is calculated by means of the Henderson theory. The theoretical curve calculated by Kjellander and Sarmatf for ( ) = 0.357 and h>2 by using the anisotropic Percus-Yevick approximation is shown by the dashed line the crosses represent grand canonical Monte Carlo simulation results due to Karlstrom. We proceed now with separate descriptions of solvation, depletion, and colloid structural forces. 

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... 

Here we have included a density factor of in the Fourier transforms of the site-site correlation functions h (r) and which is more convenient for mixture calculations. is the interamolecular distance between sites a and rj. For mixtures, = 0 when sites a and t] are in molecules of different species. The SSOZ equation simply relates the total correlation functions h (k) to the direct correlation functions c, (/c). A second relation is required to obtain a closed system of equations. Two of the commonly used closure relations are the Percus-Yevick (PY) and hypernetted chain (HNC) approximations. The PY closure is ... 

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