Approximations for the direct correlation function

Rearranging this, we have the PY approximation for the direct correlation function... 

The very idea of introducing direct correlation fiinctions is that they are short-ranged, essentially zero beyond distances larger than the range of the interaction pair potential. This suggests the following approximations for the direct correlation functions when they are at distances greater than contact separations... 

Note that the R-MMSA and R-MPY/HTA approximations for the direct correlation functions are now deterministic, that is, uncoupled from the determination of the full pair correlation functions gv,M (r). However, for the most complex R-MPY closure the direct correlation functions are still self-consistently linked to the full radial distribution functions and tail potentials. 

The next step is to find suitable approximations for the direct correlation functions (the closure relations for the OZ equation). To this end, the particle-ion correlations have been treated by using the HNC approximation ... 

For hard spheres, the PY approximation yields an analytic solution [15-17]. The result for the direct correlation function is... 

This another popular closure [43] deals with spherical particles fluids that interact through an infinite repulsive potential at short range u(r) = +oo for r mean-spherical approximation (MSA) is formulated in terms of an ansatz for the direct correlation function. In this approach, c(r) is supposed to be... 

General expressions for the flexocoefiicients of nematic liquid crystals have been obtained in terms of the direct correlation function using the powerful density functional approach. These expressions have been used to obtain some interesting numerical results using the Perkus-Yevic approximation for the pair correlation function. The results from the density functional theory have also been used in computer simulations of flexoelectricity using model bent-core molecules interacting via the Gay-Berne potential. Alternative general expressions for the flexocoefiicients have... 

This equation is the derivative of the first part of (5.90), so we see that the condition of symmetry for this model leads, via the mean-field approximation for die direct correlation function, to the general integral equation for the profile (4.52). 

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. 

Integral Equation and Eield-Theoretic Approaches In addition to theories based on the direct analytical extension of the PB or DH equation, PB results are often compared with statistical-mechanical approaches based on integral equation or density functional methods. We mention only a few of the most recent theoretical developments. Among the more popular are the mean spherical approximation (MSA) and the hyper-netted chain (HNC) equation. Kjellander and Marcelja have developed an anisotropic HNC approximation that treats the double layer near a flat charged surface as a series of discrete layers.Attard, Mitchell and Ninham have used a Debye-Hiickel closure for the direct correlation function to obtain an analytical extension (in terms of elliptic integrals) to the PB equation for the planar double layer. Both of these approaches, which do not include finite volume corrections, treat the fluctuation potential in a manner similar to the MPB theory of Outhwaite. 

The correlation functions of the partly quenched system satisfy a set of replica Ornstein-Zernike equations (21)-(23). Each of them is a 2 x 2 matrix equation for the model in question. As in previous studies of ionic systems (see, e.g.. Refs. 69, 70), we denote the long-range terms of the pair correlation functions in ROZ equations by qij. Here we apply a linearized theory and assume that the long-range terms of the direct correlation functions are equal to the Coulomb potentials which are given by Eqs. (53)-(55). This assumption represents the mean spherical approximation for the model in question. Most importantly, (r) = 0 as mentioned before, the particles from different replicas do not interact. However, q]f r) 7 0 these functions describe screening effects of the ion-ion interactions between ions from different replicas mediated by the presence of charged obstacles, i.e., via the matrix. The functions q j (r) need to be obtained to apply them for proper renormalization of the ROZ equations for systems made of nonpoint ions. 

There are several ways of obtaining functionals for nonideal systems. In most cases the free energy functional is expressed as the sum of an ideal gas term, a hard-sphere term, and a term due to attractive forces. Below, I present a scheme by which approximate expression for the free energy functional may be obtained. This approach relies on the relationship between the free energy functional and the direct correlation function. Because the direct correlation functions are defined through functional derivatives of the excess free energy functional, that is,... 

There have been several attempts to treat the RPM on an analogous basis. To this end, Leote de Carvalho and Evans [281] used the GMSA, Lee and Fisher [283] used the GDH, and Weiss and Schroer [239,280,284] examined several DH-based models that approximate the direct correlation function or the pair correlation function. In some cases the results depended significantly on details of the approximations. In total, none of these studies, whatever theory used, gave evidence that Nqi may be significantly smaller than observed for simple nonionic fluids. Rather the opposite seems to be the case. From this perspective, the experimental results for some ionic systems remain a mystery. 

An extension of the HNC approach is the reference hypemetted chain (RHNC) approximation [52-54]. In this approach, studying a system preliminarily requires that we determine the properties of a reference fluid (RF). Usually, it has to be assumed that the properties of the actual system are closed to those of the RF. For example, the interaction potential of the system under study can be written as the sum of the RF potential and of a small perturbation, namely, u r) = wRF(r) + Au(r). Correspondingly, one has y(r) = yKF(r) + Ay(r), and the direct correlation function is given by... 

If the structure is decided by an effective potential ues (r), it was demonstrated in the mean spherical approximation (MSA) that the direct correlation function c(r) should rapidly approach — pMerr (r) for large r (see Section IE). According to Reatto and Tau [131], this relationship, which is asymptotically exact for large distance and low density, holds quite well when the long-range dispersion term of the AS potential, - Cg / r6, and the AT triple-dipole potential, < m3 (r) > (8n/3)vp/r6, are considered, so that the direct correlation function reads... 

Such a distinction is made here for two reasons a) In the cases where the theoretical considerations lead to a given vxc directly, the parent functional is not known, b) In the process of developing approximations to the exchange-correlation functional, it is frequently the case that the functional is tested on electron densities obtained with a potential corresponding to another exchange-correlation potential i.e. not self-consistently. 

In the mean spherical approximation (MSA) treatment of the ion association in aqueous solutions, the linearity of the relative permittivity and of the hydrated cation diameters with the electrolyte concentration was taken into account and a good fit of the experimental activity and osmotic coefficient was obtained [72-75]. The MSA model was elaborated on the basis of cluster expansion considerations involving the direct correlation function the treatment can deal with the many-body interaction term and with a screening parameter and proved expedient for the interpretation of experimental results concerning inorganic electrolyte solutions [67,75-77]. 

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