Ornstein-Zernike form

As we discussed in Section II.B, site-site correlation functions provide a very useful formalism for describing the structure of fluids modeled with interaction site potentials. In this formalism, information equivalent to g l,2) is obtained from the set of site-site correlation functions and intramolecular correlation functions. For this reason, a great deal of effort has been put into the development of integral equation theories for these correlation functions. The seminal contribution in this area was made by Chandler and Andersen, who sought to write an integral equation of the Ornstein-Zernike form in which the set of site-site total correlation functions were related to a set of site-site direct correlation functions. Their equation has the form... 

The integrals are over the full two-dimensional volume F. For the classical contribution to the free energy /3/d([p]) the Ramakrishnan-Yussouff functional has been used in the form recently introduced by Ebner et al. [314] which is known to reproduce accurately the phase diagram of the Lennard-Jones system in three dimensions. In the classical part of the free energy functional, as an input the Ornstein-Zernike direct correlation function for the hard disc fluid is required. For the DFT calculations reported, the accurate and convenient analytic form due to Rosenfeld [315] has been used for this quantity. 

The direct correlation function c is the sum of all graphs in h with no nodal points. The cluster expansions for the correlation functions were first obtained and analyzed in detail by Madden and Glandt [15,16]. However, the exact equations for the correlation functions, which have been called the replica Ornstein-Zernike (ROZ) equations, have been derived by Given and Stell [17-19]. These equations, for a one-component fluid in a one-component matrix, have the following form... 

However, before proceeding with the description of simulation data, we would like to comment the theoretical background. Similarly to the previous example, in order to obtain the pair correlation function of matrix spheres we solve the common Ornstein-Zernike equation complemented by the PY closure. Next, we would like to consider the adsorption of a hard sphere fluid in a microporous environment provided by a disordered matrix of permeable species. The fluid to be adsorbed is considered at density pj = pj-Of. The equilibrium between an adsorbed fluid and its bulk counterpart (i.e., in the absence of the matrix) occurs at constant chemical potential. However, in the theoretical procedure we need to choose the value for the fluid density first, and calculate the chemical potential afterwards. The ROZ equations, (22) and (23), are applied to decribe the fluid-matrix and fluid-fluid correlations. These correlations are considered by using the PY closure, such that the ROZ equations take the Madden-Glandt form as in the previous example. The structural properties in terms of the pair correlation functions (the fluid-matrix function is of special interest for models with permeabihty) cannot represent the only issue to investigate. Moreover, to perform comparisons of the structure under different conditions we need to calculate the adsorption isotherms pf jSpf). The chemical potential of a... 

Different techniques can be applied to calculate the effective interaction in a form of the potential of mean force. Because we are looking for W (r) which in the following will be used in computer simulations, the mostly suitable is the integral equation theory (IET) approach. The advantage of IET lies in a fact that in many most important applications this approach leads to an analytical equation for W (r). The IET technique is based on the Ornstein-Zernike (OZ) equation [33]... 

This is the Ornstein-Zernike equation. It is an exact integral equation relating the two 2-particle correlation functions li2(l,2) and C2(l,2). It is possible to motivate this equation form purely physical arguments the idea is to interpret the total correlation function li2(l,2) as the sum of all possible direct correlations, thus C2(l, 2) is termed the direct correlation function. We imagine that 112(1,2) is the sum of the direct correlation between 1 and 2 (that is 2(1,2)), and all chains of direct correlations via a third, fourth etc., particle. The weakness of this heuristic derivation is that we do not know how to write down an expression for 2(1,2). The great advantage of the formal... 

The Ornstein-Zernike equation can be written in a general form applicable... 

This is a Lorentzian function of q or equivalently of sin (0/2) of width 2qo. This is called the Ornstein-Zernike Approximation. It is based on the following assumptions (a) the direct correlation function is short-ranged, (b) C(q) can be expanded in a power series in q, and (c) for small values of q all terms 0(<73) can be ignored. Assumption (c) restricts our attention to small q and thereby to large distances R. Fourier inversion of Eq. (10.7.16) then gives the asymptotic form of G(R) for large R as... 

The scattering intensity in bulk contrast can be calculated easily in the Ornstein-Zernike approximation for all lattice [15, 90-92] and Ginzburg-Landau models. In the limit of wave vector q < q — n/a, one obtains in all cases the Teubner-Strey form... 

Thus, for < q the film scattering intensity decays as /q, which differs significantly from the usual Ornstein- Zernike behavior. The 1 q behavior can be observed in its pure form only for systems with q 1, i.e., for strongly swollen microemuisions. In all other cases,... 

For liquids in contact with a face of a periodic crystal, the correlation functions can be represented as Steele s expansion into sums of Fourier components periodic on the surface lattice [11, 12]. The Ornstein-Zernike (OZ) integral equation then reduces to a linear matrix equation for the expansion coefficients dependent on the distance to the surface [13, 14, 15]. This approach, however, is not very convenient since the surface symmetry entirely determines a particular form of the periodic functional basis. 

Within the equivalent monomer approximation scheme, each monomer in the linear chain is constructed from one or more spherically symmetric Interaction sites A, B, C, and so forth. The generalized Ornstein-Zernike-like matrix equations of Chandler and Andersen can be conveniently written in Fourier transform space in the general form... 

Allnatt (1964) showed that an expression equivalent to the Ornstein-Zernike relation can be written in matrix form as... 

I have already mentioned the limit X = (cr/L) /h with L < a/2 when the system should consist of dipolar dumbbells. In the absence of a solvent, the asymptotic form of the direct correlation function [defined through the Ornstein-Zernike (OZ) equation] for this system is given by (Rasaiah and Lee, 1985a)... 

The Ornstein-Zernike equation with Percus-Yevick approximation has a closed-form solution for the case of a hard-sphere potential. In the Fourier transform domain, the solution to the Ornstein-Zernike equation is... 

The surface elevations bear a (formal) resemblance to (bulk) density fluctuations of a fluid near the critical point, which also have a large range correlation, and for which Ornstein and Zernike found the same form as (3.6). 

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