Correlation function Ornstein-Zernike

Again as in Sections III. A and III. B we use the simplest Ornstein-Zernike correlation function... 

The correlation length of the polymer solution can be estimated in the static light scattering. Here, we use the Ornstein-Zernike correlation function goz(r) to find f experimentally. This correlation function is often used for the correlation (Ap(r)Ap(O))/p of the density fluctuation Ap(r) = p(r) — (p) in the semidilute polymer solution. The function is expressed as... 

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. 

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... 

The multidensity Ornstein-Zernike equation (70) and the self-consistency relation (71) actually describe a nonuniform system. To solve these equations numerically for inhomogeneous fluids one needs only an appropriate generalization of the Lowett-Mou-Buff-Wertheim equation (14). Such a generalization, employing the concept of the partial correlation function has been considered in Refs. 34,35. 

Eq. (101) is the multidensity Ornstein-Zernike equation for the bulk, one-component dimerizing fluid. Eqs. (102) and (103) are the associative analog of the singlet equation (31). The last equation of the set, Eq. (104), describes the correlations between two giant particles and may be important for theories of colloid dispersions. The partial correlation functions yield three... 

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... 

The correlation functions of the partly quenched system satisfy a set of replica Ornstein-Zernike equations (21)-(23). Each of them is a 2 x 2 matrix equation for the model in question. As in previous studies of ionic systems (see, e.g.. Refs. 69, 70), we denote the long-range terms of the pair correlation functions in ROZ equations by qij. Here we apply a linearized theory and assume that the long-range terms of the direct correlation functions are equal to the Coulomb potentials which are given by Eqs. (53)-(55). This assumption represents the mean spherical approximation for the model in question. Most importantly, (r) = 0 as mentioned before, the particles from different replicas do not interact. However, q]f r) 7 0 these functions describe screening effects of the ion-ion interactions between ions from different replicas mediated by the presence of charged obstacles, i.e., via the matrix. The functions q j (r) need to be obtained to apply them for proper renormalization of the ROZ equations for systems made of nonpoint ions. 

The PRISM (Polymer-Reference-Interaction-Site model) theory is an extension of the Ornstein-Zernike equation to molecular systems [20-22]. It connects the total correlation function h(r)=g(r) 1, where g(r) is the pair correlation function, with the direct correlation function c(r) and intramolecular correlation functions (co r)). For a primitive model of a polyelectrolyte solution with polymer chains and counterions only, there are three different relevant correlation functions the monomer-monomer, the counterion-counterion, and the monomer-counterion correlation function [23, 24]. Neglecting chain end effects and considering all monomers as equivalent, we obtain the following three PRISM equations for a homogeneous and isotropic system in Fourier space ... 

Deeper insight into the consequences of counterion condensation is gained by an effective monomer-monomer and counterion-counterion potential, respectively. The idea is to reduce the multicomponent system (macromolecules + counterions) to effective one-component systems (macromolecules or counterions, respectively). We define the simplified model in such a way that the effective potential between the counterions or monomers, respectively, of the new system yields exactly the same correlation function (gcc, gmm) as found in the multicomponent case at the same density. Starting from the correlation function gcc -respectively gmm-of the multi-component model we calculate an effective direct correlation function cefy via the one-component Ornstein-Zernike equation. An effective potential is then obtained from the RLWC closures of the one- and multicomponent models [24]. For low and moderate densities the effective potential is well approximated by... 

The correlation functions play an essential role in the static description of homogenous classical liquids whose particles are taken to interact through an effective pair potential. The starting point of the liquid-state theory, in terms of correlation functions, is the well-known Ornstein-Zernike equation [25]. The total correlation function h r) defined in Section II is actually a sum of two contributions that is illustrated by the following relationship... 

Assuming the pair potential known, the radial distribution function for two-dimensional systems can be calculated using the two-dimensional version of the Ornstein-Zernike equation, Eq. (22), and one of the closure relations. Although Eq. (22) does not relate one to one the radial distribution function with the pair potential, one might attempt to invert the procedure to get u(r) from the experimental values for g(r). Thus, by taking the Fourier-Bessel (FB) transform [43,44] of Eq. (22) an expression for c(k) is obtained in terms of the FB transform of the measured total correlation function, i.e. 

Most modern theories are based on the equation of Ornstein and Zernike (OZ). This equation was developed eighty years ago to describe light scattering in a fluid. Its utility in the theory of fluids was realized much later, about thirty-five years ago. The OZ equation is obtained by defining the total correlation function,... 

As is briefly described in the Introduction, an exact equation referred to as the Ornstein-Zernike equation, which relates h(r, r ) with another correlation function called the direct correlation function c(r, r/), can be derived from the grand canonical partition function by means of the functional derivatives. Our theory to describe the molecular recognition starts from the Ornstein-Zernike equation generalized to a solution of polyatomic molecules, or the molecular Ornstein-Zernike (MOZ) equation [12],... 

Finally, introduce c(r,r ), the Ornstein-Zernike direct correlation function defined by x (r, r ), and derive... 

Equation (5) is the familiar Ornstein-Zernike relation and hereafter we denote the direct correlation function C2(r) as c(r). Based on (1) and (4) with (2), one can discuss freezing transition of one-component liquids. " For an S-component mixture, which is specified by S density fields, nj(r) j = l,- -,5), we have, instead of (4),... 

In real space the correlation QiQj) of phonon amplitudes at the sites i and j is described by the Ornstein-Zernike function... 

The classification and topological reduction of the diagrams for pair correlation functions leads to the Wertheim modification of the Ornstein-Zernike (WOZ) equations [9],... 

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],... 

In earlier discussion, the total correlation function h r]2) was defined to be ii n) 1- This quantity is a measure of the total influence molecule 1 has on molecule 2 at a distance rn- Ornstein and Zernike [16] proposed that this influence could be considered as composed of two parts, a direct part and an indirect part. The direct part, which measures the direct influence of molecule 1 on molecule 2, is given by c(ri2). The indirect part is the influence propagated by molecule 1 on molecule 3, which then affects molecule 2 either directly or indirectly through other molecules. The indirect effect is weighted by the density and averaged over all positions of molecule 3. As a result one may write... 

Fig. 2.8 The pair correlation function g r) for a fluid composed of hard spheres at a packing fraction of = 0.49 calculated as a function of distance, r, using the Ornstein-Zernike equation with the direct correlation function given by equations (2.6.6) and (2.6.10). The data shown as ( ) are from a Monte Carlo calculation.

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],... 

B. Ornstein-Zernike Like Equations for Site-Site Correlation Functions... 

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... 

Each of the symbols H, W, and C is a matrix of site-site correlation functions. If these are real space functions, then a convolution is implied by the matrix multiplication, however it is more usual to write site-site Ornstein-Zernike (SSOZ) equation in k-space, where... 

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... 

In some respects, this approach is very attractive since, if the spherical harmonic expansions of the correlation functions are sufficiently rapidly convergent, the approximate solution of the Ornstein-Zernike equation for a molecular fluid can be placed upon essentially the same footing as that for a simple atomic fluid. The question of convergence of the spherical harmonic expansions turns out to be the key issue in determining the efficacy of the approach, so it is worthwhile to review briefly the available evidence on this question. Most of the work on this problem has concerned the spherical harmonic expansion of (1,2) for linear molecules. This work was pioneered by Streett and Tildesley, who showed how it was possible to write the spherical harmonic expansion coefficients as ensemble averages obtainable from a Monte Carlo or molecular dynamics simulation via... 

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... 

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