Equation Ornstein-Zernike

To discuss briefly the reformulation of the Ornstein-Zernike equation it is most convenient to consider the case of one associating site per molecule, M = 1. A more general derivation can be found, for example, in Ref. 104. The most important ingredient for the following derivation is the associative Mayer function. It characterizes the bonding effects and is... 

There is no other means but to define, if neeessary, the total direct eorrela-tion funetion via the eommon Ornstein-Zernike equation... 

However, the partials h j, and the partials of the direet eorrelation funetion Cij, the latter defined as the subsets of graphs in without bridge points, are related via a Wertheim-type multidensity Ornstein-Zernike equation... 

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. 

These two equations represent the assoeiative analogue of Eq. (14) for the partial one-partiele eavity funetion. It is eonvenient to use equivalent equations eontaining the inhomogeneous total pair eorrelation funetions. Similarly to the theory of inhomogeneous nonassoeiating fluids, this equiva-lenee is established by using the multidensity Ornstein-Zernike equation (68). Eq. (14) then reduees to [35]... 

As in Sec. II, we consider a mixture composed of a dimerizing one-component fluid and a giant hard sphere [21,119]. We begin with the multidensity Ornstein-Zernike equation for the mixture... 

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... 

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... 

To the best of our knowledge, there was only one attempt to consider inhomogeneous fluids adsorbed in disordered porous media [31] before our recent studies [32,33]. Inhomogeneous rephca Ornstein-Zernike equations, complemented by either the Born-Green-Yvon (BGY) or the Lovett-Mou-Buff-Wertheim (LMBW) equation for density profiles, have been proposed to study adsorption of a fluid near a plane boundary of a disordered matrix, which has been assumed uniform in a half-space [31]. However, the theory has not been complemented by any numerical solution. Our main goal is to consider a simple model for adsorption of a simple fluid in confined porous media and to solve it. In this section we follow our previously reported work [32,33]. 

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. 

Our main focus in this chapter has been on the applications of the replica Ornstein-Zernike equations designed by Given and Stell [17-19] for quenched-annealed systems. This theory has been shown to yield interesting results for adsorption of a hard sphere fluid mimicking colloidal suspension, for a system of multiple permeable membranes and for a hard sphere fluid in a matrix of chain molecules. Much room remains to explore even simple quenched-annealed models either in the framework of theoretical approaches or by computer simulation. 

Fig. 4.45 Structure factors versus wavevector for rfPS-PI diblocks in core-contrast matched decane solutions (Gast 1996 McConnell et al. 1994) (a) dPS,W3PI >o( at cote volume fractions of 0.012 (A), 0.02(+), 0.03( ), 0.04 (A) and 0.05 (o) (b) f/PS Pl at core volume fractions of 0.006 (o), 0.013 ( ) and 0.019 (A). The lines are theoretical fits from the self-consistent field interaction potentials and the Rogers-Young closure to the Ornstein-Zernike equation.

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... 

FIG. 7 Radial distribution function of a typical suspension of charged spheres with screened Coulomb interaction. The exact results (open circles) from Monte Carlo computer simulations are compared with the theoretical predictions of the Ornstein-Zernike equation and different closure relations (lines). 

Now, let us look at Fig. 13. Here, the static structure factor of a three-dimensional homogeneous suspension of polystyrene spheres of diameter 94 nm is shown. The particles volume fraction is 0 = 2.0 x 10 4. Experimental data from static light scattering (closed circles) are compared with computer simulation (Monte Carlo) results (symbol x) and theoretical predictions (lines) obtained from the Ornstein-Zernike equation and different closure relations. The computer simulations and the theoretical calculations where carried out assuming that the interaction between the... 

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. 

FIG. 16 Effective pair potential between the colloidal particles in the systems of Fig. 15. In (a) and (b) are shown the cases with n = 0.023 and n = 0.48, respectively. The lines are the results of deconvoluting the radial distribution function using the Ornstein-Zernike equation and three different closure relations HNC, MSA and PY. The closed circles represent the potential of the mean force, which coincides with u(r) at low concentrations. Adapted from Carbajal-Tinoco et al. [42]. 

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],... 

Scattering and turbidity. The non-analytical divergences at critical points result from fluctuations of the order parameter, which can be observed by scattering experiments. The intensity I of single scattering in binary systems is determined by the concentration fluctuations, which in a rather good approximation are described by the Ornstein-Zernike equation,... 

Keywords Replica Ornstein-Zernike equation, partially quenched systems, dipolar fluids,... 

Temperatures are reasonably fit by ihe Ornstein-Zernike equation [curves are fits to Eq. (4.93)]. The scattering intensity is independent of temperature at high 9 > 1/ because... 

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