Ornstein-Zernike theory

Levanyuk has considered a problem identical in substance to the introduction of nonlinear terms in Eq. (24). He examined the theory of light scattering at order-disorder transition points according to the formalism of Landau and Lifshitz. In that formalism / g does not vanish, and he concluded that the Ornstein-Zernike theory is inadequate at temperatures sufficiently close to Tj. This, of course, is what has been found here, if m = 3. An actual prediction in this case would require numerical solutions of Eq. (28). 

It is easily seen that this set of equations is in agreement with Eq. (19) in the limit of low densities, where only the first term in Eq. (39) is retained. But, of course, the equations are of interest only in the critical region. One remark on Eq. (39) should be made the virial expansion of is quite different from the virial expansion of G r), and probably converges to zero more quickly than G r). However, does not appear in every term of the expansion and it is not certain that the second moment of is finite at the critical point, which is a necessary condition for the validity of the Ornstein-Zernike theory. 

As was mentioned earlier, Levanyuk has pointed out a possible inaccuracy in the Ornstein-Zernike theory of light scattering in the critical region, but his corrections were not carried to the numerical stage. His equations were essentially only a possible example of a nonlinear equation for G(r), which has already been considered here. 

Richardi J, Fries PH, Fischer R, Rast S, Krienke H Liquid acetone and chloroform a comparison between Monte Carlo simulation, molecular Ornstein-Zernike theory, and site-site Omstein-Zemike theory. Mol Phys 93(6) 925—938, 1998. 

The mere existence of a critical point does not entail a positive r/ in d = 3 as it does in d = 2 we could well have imagined that the Ornstein-Zernike theory, with t = 0 and with m finite at the critical point, might be correct in d = 3. It is therefore of interest that in model fluids with short-ranged forces, which we believe portray faithfully the behaviour of real fluids at their critical points, rj is found to 1 positive, albeit... 

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]... 

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... 

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. 

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. 

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... 

Smoluchowski (1908), Einstein (1910), Ornstein Zernike (1914, 1918). In a textbook on scattering HIGGINS Benoit ([136], Sect. 7.6) consider the fluctuation theory from a different point of view. 

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-... 

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 ... 

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... 

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],... 

This point of interest is brought forward by the RISM approach to the structure of molecular liquids, and a RISM model with HNC closure supports a similar result for the excess chemical potential in terms of atom-atom correlations (Singer and Chandler, 1985 Hirata, 1998). RISM - reference interaction site model - is an acronym that refers to a class of theories for the joint two-atom distributions in molecular liquids. The most basic decision of RISM models is that theories of molecular liquids should focus first on the atom-atom distributions extracted from X-ray and neutron scattering data rather than more complex possibilities this highly practical point was not so obvious in an earlier epoch when models of molecular liquids were scarcely realistic on an atomic scale. That basic decision was encapsulated by invention of a site-site (or atom-atom) Ornstein-Zernike (SSOZ) (Cummings and Stell, 1982) equation that involved intramolecular atom-atom correlations. The original suggestions (Chandler and Andersen, 1972) were sufficiently successful as to support subsequent flamboyant developments, and to be substantially impervious to more fundamental improvements (Chandler et al, 1982). For these reasons a full discussion of the RISM models wouldn t fit here. Fortunately, a devoted exposition of current RISM work is already available (Hirata, 1998). 

In the theory of classical liquids [79],/x<, plus the particle-particle interaction is known as Ornstein-Zernike function. In practice, of course, this quantity is only approximately known. Suitable approximations of will be discussed in section 6. In order to construct such approximate functionals, it is useful to express / c in terms of the full response function x- An exact representation of fxc is readily obtained by solving Eq. (144) for Vi and inserting the result in Eq. (155). Equation (154) then yields... 

We applied the Ornstein-Zernike (OZ) plot to the SAXS data. According to the OZ theory fir the samples, the scattering intensity near - 0 is given by 22 ... 

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]... 

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