Pair correlation function, interaction site fluids

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. 

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... 

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. 

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