Percus 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... 

Baxter R J 1970 Ornstein Zernike relation and Percus-Yevick approximation for fluid mixtures J. Chem. Phys. 52 4559... 

Cummings P T and Stell G 1984 Statistical mechanical models of chemical reactions analytic solution of models of A + S AS in the Percus-Yevick approximation Mol. Phys. 51 253... 

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... 

However, it is known that the direct correlation functions have an exact long-range asymptotic form, arising due to intramolecular correlations in clusters formed via the association mechanism. This asymptotics is not included in the Percus-Yevick approximation. Other common liquid state approximations also do not provide correct asymptotic behavior of Ca ir). 

The EMSA requires the degree of dimerization A as an input parameter. This is quite disappointing. However, it ehminates the deficiency of the Percus-Yevick approximation, Eq. (38). The EMSA represents a simpHfied version, to obtain an analytic solution, of a more sophisticated site-site extended mean spherical approximation (SSEMSA) [67-69]. The results of the aforementioned closures can be used as an input for subsequent calculations of the structure of nonuniform associating fluids. 

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... 

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... 

Typical forms of the radial distribution function are shown in Fig. 38 for a liquid of hard core and of Lennard—Jones spheres (using the Percus— Yevick approximation) [447, 449] and Fig. 44 for carbon tetrachloride [452a]. Significant departures from unity are evident over considerable distances. The successive maxima and minima in g(r) correspond to essentially contact packing, but with small-scale orientational variation and to significant voids or large-scale orientational variation in the liquid structure, respectively. Such factors influence the relative location of reactants within a solvent and make the incorporation of the potential of mean force a necessity. 

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... 

From the various possible closures, the mean spherical approximation (MSA) [189] has found particularly wide attention in phase equilibrium calculations of ionic fluids. The Percus-Yevick (PY) closure is unsatisfactory for long-range potentials [173, 187, 190]. The hypemetted chain approximation (HNC), widely used in electrolyte thermodynamics [168, 173], leads to an increasing instability of the numerical algorithm as the phase boundary is approached [191]. There seems to be no decisive relation between the location of this numerical instability and phase transition lines [192-194]. Attempts were made to extrapolate phase transition lines from results far away, where the HNC is soluble [81, 194]. 

The behavior of VACF and of D in one-dimensional systems are, therefore, of special interest. The transverse current mode of course does not exist here, and the decay of the longitudinal current mode (related to the dynamic structure factor by a trivial time differentiation) is sufficiently fast to preclude the existence of any "dangerous" long-time tail. Actually, Jepsen [181] was the first to derive die closed-form expression for the VACF and the diffusion coeffident for hard rods. His study showed that in the long time VACF decays as 1/f3, in contrast to the t d 2 dependence reported for the two and three dimensions. Lebowitz and Percus [182] studied the short-time behavior of VACF and made an exponential approximation for VACF [i.e, Cv(f) = e 2 ], for the short times. Haus and Raveche [183] carried out the extensive molecular dynamic simulations to study relaxation of an initially ordered array in one dimension. This study also investigated the 1/f3 behavior of VACF. However, none of the above studies provides a physical explanation of the 1/f3 dependence of VACF at long times, of the type that exists for two and three dimensions. 

A very popular closure relation is the Percus-Yevick (PY) approximation [39]. For a generic potential u r), this approximation assumes that... 

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... 

In fairness to Percus and Yevick, the PY approximation was not obtained by linearizing Eq. (27). However, it is convenient to take this view here. 

One more relation is required to achieve closure, i.e., to determine the two types of correlation functions. The most commonly used relations are the Percus-Yevick (PY) and the hypernetted chain (HNC) approximations [47-49]. From graph or diagram expansion of the total correlation function in powers of the density n(r) and resummation, an exact relation between the total and direct correlation functions is obtained, namely... 

The Percus-Yevick (PY) approximation which is obtained by dropping bridge and so-called parallel diagrams, reads... 

An approximate functional for Ap, is derived in Matubayasi et al. [15,16] and Takahashi et al. [19]. The functional is constructed by adopting the Percus-Yevick-type approximation in the unfavorable region of the solute-solvent interaction and the hypernetted-chain-type approximation in the favorable region. Ap, is then given... 

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). 

The equations described earlier contain two unknown functions, h(r) and c(r). Therefore, they are not closed without another equation that relates the two functions. Several approximations have been proposed for the closure relations HNC, PY, MSA, etc. [12]. The HNC closure can be obtained from the diagramatic expansion of the pair correlation functions in terms of density by discarding a set of diagrams called bridge diagrams, which have multifold integrals. It should be noted that the terms kept in the HNC closure relation still include those up to the infinite orders of the density. Alternatively, the relation has been derived from the linear response of a free energy functional to the density fluctuation created by a molecule fixed in the space within the Percus trick. The HNC closure relation reads... 

The relation Eq. (6.56) is key to Percus s derivation of the Percus-Yevick approximation (Percus, 1964). We consider the case where the inhomogeneity is produced by the location of a distinguished particle of type v at the origin, and inquire about the surrounding fluid. The Kirkwood-Salsburg formula Eq. (6.24), p. 130, offers the interpretation of the left side of Eq. (6.56) ... 

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