Percus-Yevick closure

Obviously, Eqs. (101-103) are exact. However, their solution requires closures. The associative Percus-Yevick closure to Eq. (101) has been given by Eq. (72) the associative Percus-Yevick closure to Eq. (103) reads... 

We apply the singlet theory for the density profile by using Eqs. (101) and (103) to describe the behavior of associating fluids close to a crystalline surface [120-122], First, we solve the multidensity OZ equation with the Percus-Yevick closure for the bulk partial correlation functions, and next calculate the total correlation function via Eq. (68) and the direct correlation function from Eq. (69). The bulk total direct correlation function is used next as an input to the singlet Percus-Yevick or singlet hypernetted chain equation, (6) or (7), to obtain the density profiles. The same approach can be used to study adsorption on crystalline surfaces as well as in pores with walls of crystalline symmetry. 

Eqs. (22) and (23), together with closures (38)-(40), represent a complete ROZ-HNC problem for the numerical solution. The Percus-Yevick closure is given similarly to Eq. (33) however, in addition, the blocking term in the fluid-fluid direct correlations is neglected, = 0. 

The intra-chain pair density functions obtained from both the bulk simulation and the continuous unperturbed chains were used as input to the polymer-RISM integral equation for estimating the intermolecular pair distribution function g(r) (using a soft-Percus Yevick closure). We found that PRISM underpredicts the first peak in g(r), while also overpredicting the steefmess of the rise to the first peak. 

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. 

Grayce and Schweizer," based on graph-theoretic and hueristic arguments, suggested a modified form for the solvation potential ( PY style ) that is in the spirit of the Percus-Yevick closure... 

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... 

One approach of this category is to solve the integral equations using the Percus-Yevick closure for the system of adhesive hard sphere (AHS) mixtures (17-22). An adhesive hard sphere is a hard sphere that has attractive sites at surface. The attractive interaction on these attractive sites is infinitely strong and infinitesimally short ranged. The Percus-Yevick closime yields an analytical solution for such systems. The adhesive attraction, which resembles the chemical bonding, is used to build up chains by employing the proper connectivity constraints. 

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... 

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... 

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... 

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 ... 

By linearizing the exponential with respect to h(r) — c(r), one obtains an alternative closure, called the Percus-Yevick equation,164170... 

Both closures have been employed for determining the distribution functions for liquids 182 the Percus-Yevick equation tends to yield better results for nonpolar systems, while the hypemetted-chain equation (with the appropriate renormalization of long-range interactions)113,183 is found to be more appropriate for polar and ionic liquids.184... 

---