Omstein-Zernike Equation

Hie topological reductions and resummations described in Section 6 permit us to express the thermodynamic functions of a model system in terms of series of integrals. For many systems of interest, such as ionic solutions in the 1 A/concentration range, these series [Eqs. (138)to(140)]do not converge fast enough to be convenient as approximation methods. In such cases it is often useful to perform additional resummations, or to express the cluster series as an integral equation. 

In this section we shall develop expressions for the pair correlation function gafc(l,2) and its Ursell function hab(l,2) = gab(l,2)-l. From Eq. (113), the cluster expansion of Aab(l, 2) is found to be (cf. Fig. 15) 

We also introduce the direct correlation function Ca (l, 2), which is defined by the Ornstein-Zernike integral equation  

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The integral equation theory consists in obtaining the pair correlation function g(r) by solving the set of equations formed by (1) the Omstein-Zernike equation (OZ) (21) and (2) a closure relation [76, 80] that involves the effective pair potential weff(r). Once the pair correlation function is obtained, some thermodynamic properties then may be calculated. When the three-body forces are explicitly taken into account, the excess internal energy and the virial pressure, previously defined by Eqs. (4) and (5) have to be, extended respectively [112, 119] so that... 

One of the important methods of developing a model for the potential energy of the liquid system is that based on integral equations. These include the Kirkwood integral equation, the Born-Green-Yvon equation, and the Omstein-Zernike equation. The last approach leads to the definition of the direct correlation function c(fi2). It is the approach which is most frequently used and is the one which is considered here. 

The integral equations themselves are obtained by substituting the various approximations to Cab into the Omstein-Zernike equation [Eq. (142)]. Substitution of Eq. (156) gives the HNC integral equation... 

Because V2 comprises a solvation contribution (associated with the solvent s local density perturbation) and a compressibility-driven contribntion (associated with the propagation of the pertnrbation), it also offers a way to identify the short-and long-ranged contributions to this mechanical partial molecular property, and consequently, to discriminate the solvation process from the compressibility-driven phenomenon. In fact, after invoking the Omstein-Zernike equation (Hansen and McDonald 1986), we can split the TCFIs into their direct and indirect contributions (Chialvo et al. 1999), that is. 

In Eq. (2.5X p is the number density of molecules and rf is the position vector site a on molecule i. It can be seen from Eq. (Z that gay(r) is the intermolecular radial distribution function for sites a and y on diflerent molecules. Chandler and Andersen generalized the Omstein-Zernike equation to reflect the fact that in molecular liquids, unlike monatomic I uids, there exist intramolecular correlations. This generalized Omstein-Zernike-like, or RISM, equation has the form [6, 8]... 

Off-lattice models 16 Order parameters 8, 19, 21,46, 86 distribution 19, 20 Omstein-Zernike equation 217 Ostwald s rule 51... 

For completeness, note that several researchers have exploited the well-developed analytical theories of the stmcture of fluids to model percolation in mixtures of interacting particles. By proposing various extensions of the multicomponent Omstein-Zernike equation, coupled with connectivity definitions from continuum percolation theories, simplified analytical expressions are derived for the percolation threshold of a composite system subjected to interparticle and medium-induced interactions. However, to date, simulations dominate the study of dynamic percolation. 

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