Pair correlation function, interaction site

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 ... 

In this section, we review some of the important formal results in the statistical mechanics of interaction site fluids. These results provide the basis for many of the approximate theories that will be described in Section III, and the calculation of correlation functions to describe the microscopic structure of fluids. We begin with a short review of the theory of the pair correlation function based upon cluster expansions. Although this material is featured in a number of other review articles, we have chosen to include a short account here so that the present article can be reasonably self-contained. Cluster expansion techniques have played an important part in the development of theories of interaction site fluids, and in order to fully grasp the significance of these developments, it is necessary to make contact with the results derived earlier for simple fluids. We will first describe the general cluster expansion theory for fluids, which is directly applicable to rigid nonspherical molecules by a simple addition of orientational coordinates. Next we will focus on the site-site correlation functions and describe the interaction site cluster expansion. After this, we review the calculation of thermodynamic properties from the correlation functions, and then we consider the calculation of the dielectric constant and the Kirkwood orientational correlation parameters. 

They refer to Eq. (3.4.3) as the proper integral equation in view of the fact that the direct correlation function so defined does correspond to the sum of the nodeless diagrams in the interaction site cluster expansion of the total correlation function. Following Lupkowski and Monson, we shall refer to Eq. (3.4.3) as the Chandler-Silbey-Ladanyi (CSL) equation. Interestingly, the component functions have a simple physical interpretation. The elements of Ho correspond to the total correlation functions for pairs of sites at infinite dilution in the molecular solvent. The elements of the sum of Hq and //, (or H ) matrices are the solute-solvent site-site total correlation functions for sites at infinite dilution in the molecular solvent. 

Theories based on the solution to integral equations for the pair correlation functions are now well developed and widely employed in numerical and analytic studies of simple fluids [6]. Further improvements for simple fluids would require better approximations for the bridge functions B r). It has been suggested that these functions can be scaled to the same functional form for different potentials. The extension of integral equation theories to molecular fluids was first accomplished by Chandler and Andersen [30] through the introduction of the site-site direct correlation function c r) between atoms in each molecule and a site-site Omstein-Zemike relation called the reference interaction site... 

It is an essential step to express the pair correlation functions between two molecules in terms of those between interaction sites, which can be accomplished by averaging the functions over orientations fixing the distance between the interaction sites ... 

If go(r), g CrX and g (r) are known exactly, then all three routes should yield the same pressure. Since liquid state integral equation theories are approximate descriptions of pair correlation functions, and not of the effective Hamiltonian or partition function, it is well known that they are thermodynamically inconsistent [5]. This is understandable since each route is sensitive to different parts of the radial distribution function. In particular, g(r) in polymer fluids is controlled at large distance by the correlation hole which scales with the radius of gyration or /N. Thus it is perhaps surprising that the hard core equation-of-state computed from PRISM theory was recently found by Yethiraj et aL [38,39] to become more thermodynamically inconsistent as N increases from the diatomic to polyethylene. The uncertainty in the pressure is manifested in Fig. 7 where the insert shows the equation-of-state of polyethylene computed [38] from PRISM theory for hard core interactions between sites. In this calculation, the hard core diameter d was fixed at 3.90 A in order to maintain agreement with the experimental structure factor in Fig. 5. 

More modem approaches borrow ideas from the liquid state theory of small molecule fluids to develop a theory for polymers. The most popular of these is the polymer reference interaction site model (PRISM) theory " which is based on the RISM theory of Chandler and Andersen. More recent studies include the Kirkwood hierarchy, the Bom-Green-Yvon hierarchy, and the perturbation density functional theory of Kierlik and Rosinbeig. The latter is based on the thermodynamic perturbation theory of Wertheim " where the polymeric system is composed of very sticky spheres that assemble to form chains. For polymer melts all these liquid state approaches are in quantitative agreement with simulations for the pair correlation functions in short chain fluids. With the exception of the PRISM theory, these liquid state theories are in their infancy, and have not been applied to realistic models of polymers. 

In Eqs. (19), (24), and (30), self-consistent PRISM theory is formulated in a general manner to allow for the modeling of polymer mixtures and polymer models containing an arbitrary number of interaction sites. A range of polymers of various complexities have been analyzed using PRISM theory. Piitz et al. [60] studied isotactic and syndiotactic polypropylene, head-to-head syndiotactic polypropylene, poly(ethylene propylene), and polyisobutylene using PRISM theory and MD simulations. In Figs. 8a, 8b we plotted the six independent pair correlation functions between intermolecular sites for isotactic polypropylene (iPP). 

Chandler and Andersen introduced the site-site Omstein-Zemike (SSOZ) integral equation for the radial correlation functions between pairs of interaction sites of polyatomic species [14,27],... 

The details of the pair potential used in the simulations are given in Table I. This consists of an -trans model of the sec-butyl chloride molecule with six moieties. The intermolecular pair potential is then built up with 36 site-site terms per molecular pair. Each site-site term is compost of two parts Lennard-Jones and charge-charge. In this way, chiral discrimination is built in to the potential in a natural way. The phase-space average R-R (or S-S) potential is different from the equivalent in R-S interactions. The algorithm transforms this into dynamical time-correlation functions. 

In the interaction site formalism, the natural pair correlation to consider is the site-site function g y(r) which is a measure of the probability of finding a site y a distance r from a site a on a different molecule, regardless of the orientations of the two molecules. This approach has been developed by Chandler, Ladanyi and Chandler, and by Cummings, Sullivan, and Gray. The development we give here is loosely based on these works. 

An alternative description of a molecular solvent in contact with a solute of arbitrary shape is provided by the 3D generalization of the RfSM theory (3D-RISM) which yields the 3D correlation functions of interaction sites of solvent molecules near the solute. It was first proposed in a general form by Chandler, McCoy, and Singer [22] and recently developed by several authors for various systems by Cortis, Rossky, and Friesner [23] for a one-component dipolar molecular liquid, by Beglov and Roux [24, 25] for water and a number of organic molecules in water, and by Hirata and co-workers for water [26, 27], metal-water [26, 28] and metal oxide-water [31] interfaces, orientationally dependent potentials of mean force between molecular ions in a polar molecular solvent [29], ion pairs in aqueous electrolyte [30], and hydration of hydrophobic and hydrophilic solutes alkanes [32], polar molecule of carbon monoxide [33], simple ions [34], protein [35], amino acids and polypeptides [36, 37]. It should be noted that accurate calculation of the solvation thermodynamics for ionic and polar solutes in a polar molecular liquid requires special corrections to the 3D-RISM equations to eliminate the electrostatic artifacts of the supercell treatment employed in the 3D-RISM approach [30, 34]. 

In Section 9.1.1 we have introduced a stochastic model for the description of surface reaction systems which takes correlations explicitly into account but neglects the energetic interactions between the adsorbed particles as well as between a particle and a metal surface. We have formulated this by master equations upon the assumption that the systems are of the Markovian type. In the model an infinite set of master equations for the distribution functions of the state of the surface and of pairs of surface sites (and so on) arise. This chain of equations cannot be solved analytically. To handle this problem practically this hierarchy was truncated at a certain level. The resulting equations can be solved numerically exactly in a small region and can be connected to a mean-field solution for large distances from a reference point. 

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