Omstein-Zemike approximation

The Omstein-Zemike approximation also exists in the relationship between S(h) and S(0),... 

Here d in the coefficient l/(2d) is again the dimensionality of the space. The coeflbrient of the first-derivative term in the expansion (9.30) vanishes, by the spherical symmetry of c(r) and h(r). We shall refer to (9.30) as the Omstein-Zemike approximation. 

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. 

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... 

Integral equation theories are widely used in the theoretical study of liquids. There are two broad classes of integral equation theories those based on the Bom-Green-Yvon (BGY) hierarchy and those based on the Omstein-Zemike (OZ) equation [88]. Although the formalism is exact in both classes, it is generally easier to fashion approximations in the case of the OZ-equation-based approach, and this type of theory has therefore been more popular. Surprisingly, the BGY approach has never been implemented for nonuniform polymers, and this section is therefore restricted to a discussion of the OZ-equation-based approach. 

Here Si(k) is the equilibrium structure factor of the solvent, which we approximate by the Omstein-Zemike expression, given by (87)... 

There exist pre-transition effects in the isotropic phase heralding the I-N phase transition. Such pre-transition effects, which are consistent with the weakly first-order nature of the I-N transition, can be attributed to the development of short-range orientational order, which can be characterized by a position-dependent local orientational order parameter Q(r), where all component indices have been omitted [2]. In the Landau approximation, the spatial correlation function < G(0)G(r) > has the Omstein-Zemike form < G(0)G(r) exp(—r/ )/r, where is the coherence length or the second-rank orientational correlation length. The coherence length is temperature-dependent and the Landau-de Gennes theory predicts... 

Usually, h r) represented by eq 2.36 is called the Omstein-Zemike form. It shows that h r) decays faster with r as becomes smaller. Thus, may be taken as a measure of the range of r in which density fluctuations are effectively correlated. It is called the correlation length for density fluctuation [17]. In general, eq 2.33 holds only for small k so that h r) for r < is no longer the Zemike-Omstein form. This means that characterizes the decay rate of h r) in the tail region (r > (). To calculate h r) valid over the entire range of r (hence S k) or H k) valid for all k) we have to face a very difficult problem. An approximate solution by Benoit and Benmouna is described in Section 2.4. 

In conjunction with the Omstein-Zemike equation [Eq. (142)], the MSA defines an integral equation that has been solved exactly for a number of systems. For hard spheres, Eq. (168) is the same as the hard-sphere PY approximation, which has been solved by Thiele and Wertheim. For point charges, the MSA is equivalent to the DH approximation. Solutions have also been found for charged hard spheres of equal and disparate diameters, dipolar hard spheres, hard spheres with a Yukawa tail, charged hard spheres in a uniform neutralizing background,and hard nonspherical molecules with general electrostatic interactions. " ... 

A further improvement in the mentioned approach has been achieved by Richardi et al. by coupling the molecular Omstein-Zemike theory with a self-consistent mean-field approximation in order to take the polarizability into account. For the previously-mentioned solvents (acetone, acetonitrile and chloroform), the calculated values are in excellent agreement with experimental data, showing the cmcial role of taking into account polarizability contributions for polar polarizable aprotic solvents. 

The main issue here is to achieve the unambiguous separation between solvation and compressibility-driven phenomena, based on the formal splitting of the total correlation functions into their corresponding direct and indirect contributions (Chialvo and Cummings 1994,1995) according to the Omstein-Zemike equation (Hansen and McDonald 1986), and then use the derived rigorous expressions as zeroth-order approximations, for example, reference systems, in the subsequent perturbation expansion of the composition-dependent thermodynamics properties of multicomponent dilute fluid mixtures (vide infra Section 8.3). 

H) Near the spinodal, the pair correlation g r) becomes of long range and is well approximated by an Omstein-Zemike form ... 

Several studies have been directed at elucidating the limits of applicability of the mean-field approximation for mixtures of polymers. At small scattering wave vectors (0 0) Eqs. (7.25) and (7.26) reduce to the well-known Omstein-Zemike form [115],... 

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,... 

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