Omstein-Zemike equations

The second tenu in the Omstein-Zemike equation is a convolution integral. Substituting for h r) in the integrand, followed by repeated iteration, shows that h(r) is the sum of convolutions of c-fiinctions or bonds containing one or more c-fiinctions in series. Representing this graphically with c(r) = o-o, we see that... 

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 compressibility equation can also be written in tenns of the direct correlation fiinction. Taking the Fourier transfomi of the Omstein-Zemike equation... 

From the Omstein-Zemike equation in Fourier space one finds that... 

In the limit k = (a/i) /i with L < all, the system should consist of dipolar dumb-bells. The asymptotic fonn of the direct correlation fiinction (defined tln-ough the Omstein-Zemike equation) for this system (in the absence of a solvent) is given by... 

Differentiation of the Fourier transformed Omstein-Zemike equation with respect to T(temperature) gives the following expression. 

The growth of orientational correlations and the slow down of collective orientational dynamics were subsequently investigated using a soft potential [106]. Allen and Warren (AW) studied a system consisting of N = 8000 particles of ellipsoids of revolution, interacting with a version of the Gay-Beme potential, GB (3, 5, 1, 3), originally proposed by Berardi et al. [107]. AW computed the direct correlation function, c( 1, 2), in the isotropic phase near the I-N transition. The direct correlation function is defined through the Omstein-Zemike equation [108]... 

For Eq. (1) to yield the Omstein-Zemike equation, it is necessary that dr be finite, contrary to Eq. (42). Green also showed... 

L. Blum, Invariant expansion II The Omstein-Zemike equation for nonspherical molecules and an extended solution to the mean spherical model, /. Chem. Phys. 57,1862-1869 (1972). 

Fig. 17. Graphical representation of the Omstein-Zemike equation. h2 bonds on black p circles and terminal white labeled 1 circles.

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. " ... 

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). 

Blum, L. and A. J. TorrueUa. 1972. Invariant expansion for 2-body correlations—Thermodynamic functions, scattering, and Omstein-Zemike equation. Journal of Chemical Physics. 56,303. 

Henderson, D., F. F. Abraham, and J. A. Barker. 1976. The Omstein-Zemike equation for a fluid in contact with a surface. Molecular Physics An International Journal at the Interface between Chemistry and Physics 31, no. 4 1291. doi 10.1080/00268977600101021. 

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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