Omstein-Zemike relation

The direct correlation fimction c(r) of a homogeneous fluid is related to the pair correlation fimction tiirough the Omstein-Zemike relation... 

The solution detennines c(r) inside the hard core from which g(r) outside this core is obtained via the Omstein-Zemike relation. For hard spheres, the approximation is identical to tire PY approximation. Analytic solutions have been obtained for hard spheres, charged hard spheres, dipolar hard spheres and for particles interacting witli the Yukawa potential. The MS approximation for point charges (charged hard spheres in the limit of zero size) yields the Debye-Fluckel limiting law distribution fiinction. 

Theories based on the solution to integral equations for the pair correlation fiinctions are now well developed and widely employed in numerical and analytic studies of simple fluids [6]. Furtlier improvements for simple fluids would require better approximations for the bridge fiinctions B(r). It has been suggested that these fiinctions can be scaled to the same fiinctional fomi for different potentials. The extension of integral equation theories to molecular fluids was first accomplished by Chandler and Andersen [30] through the introduction of the site-site direct correlation fiinction c r) between atoms in each molecule and a site-site Omstein-Zemike relation called the reference interaction site... 

To calculate Rpp(t), the two-particle direct correlation function c 2(q) is required which is obtained from the nearly analytical expression given by Baus and Colot for a 2-D system [177]. The static structure factor S(q) has been calculated from the two-particle direct correlation function through the well-known Omstein-Zemike relation [21]. 

Benoit and Benmouna [18] applied the Omstein-Zemike relation [19], first proposed for simple liquids, to polymer solutions. According to it, Qj(i)jt(2) can be expressed as follows ... 

Combining tliis witli the Omstein-Zemike equation, we have two equations and tluee unknowns h(r),c(r) and B(r) for a given pair potential u r). The problem then is to calculate or approximate the bridge fiinctions for which there is no simple general relation, although some progress for particular classes of systems has been made recently. 

From the many tools provided by statistical mechanics for determining the EOS [36, 173, 186-188] we consider first integral equation theories for the pair correlation function gxp(ra,rp) of spherical ions which relates the density of ion / at location rp to that of a at ra. In most theories gafi(ra,rp) enters in the form of the total correlation function hxp(rx,rp) = gxp(rx,rp) — 1. The Omstein-Zemike (OZ) equation splits up hap(rx,rp) into the direct correlation function cap(ra, rp) for pair interactions plus an indirect term that reflects these interactions mediated by all other particles y ... 

The adaptation of the Percus-Yevick approximation starts with the three Omstein-Zemike equations which relate h , h, and hi, to the set of direct functions Cu, c.h> and Cm, in a homogeneous binary mixture of molecules a and b, which have hard cores but otherwise unspecified pair potentials. The limit is now taken in which the radius of the hard core of b becomes infinite and its concentration goes almost to zero, so that the system comprises a fluid of a molecules in contact with the flat wall of the one remaining b molecule. Only two Omstein-Zemike equations remain, one for h and one for the molecule-wall correlation, These are solved by using the Percus-Yevick approximation,... 

Fortunately, in the classical statistical mechanics of fluids there is a general way out of this problem, which is based on the powerful concept of the direct correlation function [22, 148]. This key mathematical object was introduced by Omstein and Zemike [178] one hundred years ago to deal with the classical fluctuations of the fluid density near the critical point. The so-called Omstein-Zemike equation at the pair level (OZ2) relates the direct correlation function between a pair of atoms cfR ) to the total correlation function through the integral... 

In combination with Eq. (12) the above relation can be put in form similar to the Omstein-Zemike equation,... 

Statistical mechanics offers many techniques for the simplification of problems involving solute-solvent interactions. These usually involve the introduction of a suitable analytical model for the solvent and yield solutions which relate the observed macroscopic properties to the micro.scopic properties of the solvent and solute. The starting point is the Omstein-Zemike equation for simple (one-component) fluids. 

Let us consider now behaviour of the gas-liquid system near the critical point. It reveals rather interesting effect called the critical opalescence, that is strong increase of the light scattering. Its analogs are known also in other physical systems in the vicinity of phase transitions. In the beginning of our century Einstein and Smoluchowski expressed an idea, that the opalescence phenomenon is related to the density (order parameter) fluctuations in the system. More consistent theory was presented later by Omstein and Zemike [23], who for the first time introduced a concept of the intermediate order as the spatial correlation in the density fluctuations. Later Zemike [24] has applied this idea to the lattice systems. 

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