Ornstein-Zernike formalism

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],... 

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... 

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... 

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). 

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... 

Subsequently, the model has been extended [37, 38] to the case of associated electrolytes by using a recent model for associating electrolytes[39]. Unlike the classic chemical model of the ion pair the effect of the pairing association is included in the computation of the MSA screening parameter F. Simple formulas for the thermodynamic excess properties have been obtained in terms of this parameter when a new EXP approximation is used. The new formalism based on closures of the Wertheim-Ornstein-Zernike equation (WOZ)[40, 41 does accommodate all association mechanisms (coulombic, covalent and solvation) in one single association parameter, the association constant. The treatment now includes the fraction of particles that are bonded, which is obtained by imposing the chemical equilibrium mass action law. This formalism was shown to be very successful for ionic systems, both in the HNC approximation and MSA [42, 43, 44, 45, 46, 47]. 

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. 

Finally, we mention an interesting recent study by Chandler that extended the Gaussian field-theoretic model of Li and Kardar to treat atomic and polymeric fluids. Remarkably, the atomic PY and MSA theories were derived from a Gaussian field-theoretic formalism without explicit use of the Ornstein-Zernike relation or direct correlation function concept. In addition, based on an additional preaveraging approximation, analytic PRISM theory was recovered for hard-core thread chain model fluids. Nonperturbative applications of this field-theoretic approach to polymer liquids where the chains have nonzero thickness and/or attractive forces requires numerical work that, to the best of our knowledge, has not yet been pursued. 

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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