Integral equations Ornstein-Zernike equation

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 term (r ) represents the direct part and the integral term represents the influence of the particle at r on that at r as mediated by a third particle. Equation (41) is called the Ornstein-Zernike equation and is really a definition of the direct correlation function c(r). Note that the Fourier transform of the Ornstein-Zernike equation gives... 

Baxter (1968b) showed that the Ornstein-Zernike equation could, for some simple potentials, be written as two one-dimensional integral equations coupled by a function q(r). In the PY approximation for hard spheres, for instance, the q(r) functions are easily solved, and the direct-correlation function c(r) and the other thermodynamic properties can be obtained analytically. The pair-correlation function g(r) is derived from q(r) through numerical solution of the integral equation which governs g(r) for which a method proposed by Perram (1975) is especially useful. Baxter s method can also be used in the numerical solution of more complicated integral equations such as the hypernetted-chain (HNC) approximation in real space, avoiding the need to take Fourier transforms. An equivalent set of relations to Baxter s equations was derived earlier by Wertheim (1964). 

Most integral equations are based on the Ornstein-Zernike (OZ) equation [3-5]. The idea behind the OZ equation is to divide the total correlation function h ri2) iiito a direct correlation function (DCF) c r 12) that describes the fact that molecules 1 and 2 can be directly correlated, and an indirect correlation function 7( 12), that describes the correlation of molecule 1 with the other molecules that are also correlated with molecule 2. At low densities, when only direct correlations are possible, 7(r) = 0. At higher densities, where only triplet correlations are possible, we can write... 

In Sec. 3 our presentation is focused on the most important results obtained by different authors in the framework of the rephca Ornstein-Zernike (ROZ) integral equations and by simulations of simple fluids in microporous matrices. For illustrative purposes, we discuss some original results obtained recently in our laboratory. Those allow us to show the application of the ROZ equations to the structure and thermodynamics of fluids adsorbed in disordered porous media. In particular, we present a solution of the ROZ equations for a hard sphere mixture that is highly asymmetric by size, adsorbed in a matrix of hard spheres. This example is relevant in describing the structure of colloidal dispersions in a disordered microporous medium. On the other hand, we present some of the results for the adsorption of a hard sphere fluid in a disordered medium of spherical permeable membranes. The theory developed for the description of this model agrees well with computer simulation data. Finally, in this section we demonstrate the applications of the ROZ theory and present simulation data for adsorption of a hard sphere fluid in a matrix of short chain molecules. This example serves to show the relevance of the theory of Wertheim to chemical association for a set of problems focused on adsorption of fluids and mixtures in disordered microporous matrices prepared by polymerization of species. 

Considerable progress has been made in the last decade in the development of more analytical methods for studying the structural and thermodynamic properties of liquids. One particularly successful theoretical approach is. based on an Ornstein-Zernike type integral equation for determining the solvent structure of polar liquids as well as the solvation of solutes.Although the theory provides a powerful tool for elucidating the structure of liquids in... 

Scattering and Disorder. For structure close to random disorder the SAXS frequently exhibits a broad shoulder that is alternatively called liquid scattering ([206] [86], p. 50) or long-period peak . Let us consider disordered, concentrated systems. A poor theory like the one of Porod [18] is not consistent with respect to disorder, as it divides the volume into equal lots before starting to model the process. He concludes that statistical population (of the lots) does not lead to correlation. Better is the theory of Hosemann [158,211], His distorted structure does not pre-define any lots, and consequently it is able to describe (discrete) liquid scattering. The problems of liquid scattering have been studied since the early days of statistical physics. To-date several approximations and some analytical solutions are known. Most frequently applied [201,212-216] is the Percus-Yevick [217] approximation of the Ornstein-Zernike integral equation. The approximation offers a simple descrip-... 

Let us discuss now a usual theoretical method employed in the calculation of the radial distribution function. The basic equation obeyed by g(r) is the integral equation, introduced by Ornstein and Zernike in 1914 [25,32]... 

The so-called product reactant Ornstein-Zernike approach (PROZA) for these systems was developed by Kalyuzhnyi, Stell, Blum, and others [46-54], The theory is based on Wertheim s multidensity Ornstein-Zernike (WOZ) integral equation formalism [55] and yields the monomer-monomer pair correlation functions, from which the thermodynamic properties of the model fluid can be obtained. Based on the MSA closure an analytical theory has been developed which yields good agreement with computer simulations for short polyelectrolyte chains [44, 56], The theory has been recently compared with experimental data for the osmotic pressure by Zhang and coworkers [57], In the present paper we also show some preliminary results for an extension of this model in which the solvent is now treated explicitly as a separate species. In this first calculation the solvent molecules are modelled as two fused charged hard spheres of unequal radii as shown in Fig. 1 [45],... 

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]... 

Replica Ornstein-Zernike integral equation approach... 

This equation is the Ornstein-Zernike (OZ) equation and gives the mathematical definition of c(r 2) with the indirect effect being expressed as a convolution integral of h and c. By Fourier transformation, one obtains... 

At the core of any integral equation approach we have the (exact) Ornstein-Zernike (OZ) equation [300] relating the total correlation function(s) of a given fluid to the so-called direct correlation function(s). For the replicated system at hand, the OZ equation is that of a multicomponent mixture [30],... 

Relation (D.18) is often referred to as the Percus-Yevick approximation. If we use (D.18) in the Ornstein-Zernike relation, we get an integral equation for y... 

The SSOZ equation is an exact integral equation relating the two site-site correlation functions h y(r) and c y(r). Strictly speaking, it is simply a definition of the site-site direct correlation function c,y(r). This fact is made clearest in the derivation by H iye and Stell. Here they consider the Ornstein-Zernike equation for an equimolar mixture of two species a and y. For each atom... 

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... 

To sketch briefly the integral equation methodology,178181 we again focus on a one-component atomic system. The direct correlation function, c(r), can be related to the function g(r) — 1 = h(r) by the Ornstein-Zernike equation170 178... 

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. 

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

Here we illustrate the solvation formalism by integral equation calculations for binary mixtures described by the Lennard-Jones model (see Tables 8.1 and 8.2), and based on the Percus-Yevick approximation for the solution of the Ornstein-Zernike equations (Hansen and McDonald 1986) according to the approach proposed by McGuigan and Monson (McGuigan and Monson 1990). We focus on the solute-induced effects on the microstructure and the thermodynamic properties of infinitely dilute solutions of pyrene in carbon dioxide and Ne in Xe along the... 

A second, entirely different class of new polymer integral equation theories have been developed by Lipson and co-workers, Eu and Gan, " and Attard based on the site-site version of the Born-Green-Yvon (BGY) equation. The earliest work in this direction was apparently by Whittington and Dunfield, but they addressed only a special aspect of the isolated polymer problem (dilute solution). The central quantity in the BGY approaches is the formally exact expressions that relate two and three (or more) intramolecular and intermolecular distribution functions. The generalized site-site Ornstein-Zernike equations and direct correlation functions do not enter. In the BGY schemes the closure approximation(s) enter as approximate relations between the two- and three-body distribution functions supplemented with exact normalization and asymptotic conditions. In the recent BGY work of Taylor and Lipson a four-point distribution function also enters. 

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