Closure relation, 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. 

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... 

When supplemented with a closure relation, Eq. (7) can be solved for h r) and c r). For example, the Percus-Yevick (PY) closure is given by [89]... 

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... 

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... 

To generate the Percus-Yevick (PY) closure relation, we consider the graphical expansion for 02(1 >2) obtained from Eq. (2.1.26). Each graph in this expansion occurs both with and without a/2(l, 2) bond, so we can factor out (1 -l-/2(l,2)) = e2(l,2). This gives... 

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 ... 

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]. 

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