Closure approximation Percus-Yevick

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. 

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. 

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... 

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]. 

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

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 AMSA closure for the electroneutrality sum problem (subscript s) is the same as for the associative Percus-Yevick (APY) approximation, [25]... 

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... 

For separations outside the hard core, the direct correlation functions have to be approximated. Classic closure approximations recently applied to QA models axe the Percus-Yevick (PY) closure [301], the mean spherical approximation (MSA) [302], and the hypernetted chain (HNC) closure [30]. None of these relations, when formulated for the replicated system, contains any coupling between different species, and wc can directly proceed to the limit n — 0. The PY closure then implies... 

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 ... 

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... 

The model athermal blend is defined [59,62] as the hypothetical limit of vanishing interchain attractive potentials relative to the thermal energy, i.e., Pvmm-W = 0- For this situation the atomic site-site Percus-Yevick closure approximation of Eq. (2.7) is employed where the subscripts now refer to the spedes type. The constant volume athermal blend is of theoretical interest since it isolates the purely entropic packing effects. However, as emphasized by several workers [2,62,63,67], the athermal reference blend is not an adequate model of any real phase separating system. Its primary importance is as a reference system for the theories of thermally-induced phase separation discussed in Sect 8. 

Closure approximations to the PRISM equation are generally developed via an analogy with atomic liquids. Three of the common closures for hard spheres are the Percus-Yevick (PY), hypemetted chain (HNC), and Martynov-Sarkisov (MS) closures. It has been shown that the PY closure is the most accurate of the three, and in fact the HNC and MS closures have either no solution or unphysical solutions at low densities. The PY closure is given by. 

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... 

The basic approximation in the Percus-Yevick closure is that the direct correlation function is short range. In Fig. 7 we can observe that indeed C(r) for both the bead-spring model and polyethylene are short range approaching zero on a scale of 5 A. However, C(r) from self-consistent PRISM theory is even shorter range and... 

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