Reference fluid, hard-sphere

The first term in this expression, Av0, represents the frequency shift resulting from short range repulsive packing forces. These many-body repulsive forces are in general expected to lead to a non-linear dependence of frequency on density. However, this complex non-linear behavior can be accurately modeled using a hard-sphere reference fluid, with appropriately chosen density, temperature and molecular diameters (see below). 

In order to derive a practical approximation for the repulsive contribution to vibrational frequency shifts the excess chemical potential, A ig, associated with the formation of a hard diatomic of bond length r from two hard spheres at infinite separation in a hard sphere reference fluid is assumed to have the following form. 

This functional form is derived from exact results in the dilute vapor and hydrodynamic solvent limits. The coefficients A, B, C and D used in modeling high density fluids are determined uniquely from the equation of state of the corresponding hard sphere reference system (33,34)- This hard sphere fluid chemical potential model has been shown to accurately reproduce computer simulation results for both homonuclear and heteronuclear hard sphere diatomics in hard sphere fluids up to the freezing point density (35) ... 

In order to use the above expressions for calculating the thermodynamic properties, appropriate expressions for the radial distribution function and for the equation of state for the hard-sphere reference system are required which are given in Appendix A. Fortunately, accurate information for the hard-sphere fluid as well as for the hard-sphere solid is available and this enables the determination of the properties of the coexisting dilute and concentrated phases of colloidal dispersions. 

Dimitrelis, D and Prausnitz, J.M. Comparison of two Hard-Sphere Reference Systems for Perturbation Theories for Mixtures, Fluid Phase Equilibria. Vol. 31. 1986, pp. 1-21. 

If a hard-sphere reference fluid is used, the second-order term has the form... 

Barker and Henderson (10) have given a convenient parametrization of F0(ri, r2) for the hard-sphere reference fluid. 

The only justification of Equation 52 which comes to my mind is to develop a perturbation theory for a fluid of non-spherical molecules using a hard-sphere reference fluid. The diameter of the hard spheres could be... 

The perturbed-hard-ehain (PHC) theory developed by Prausnitz and coworkers in the late 1970s was the first successful application of thermodynamic perturbation theory to polymer systems. Sinee Wertheim s perturbation theory of polymerization was formulated about 10 years later, PHC theory combines results fi om hard-sphere equations of simple liquids with the eoneept of density-dependent external degrees of fi eedom in the Prigogine-Flory-Patterson model for taking into account the chain character of real polymeric fluids. For the hard-sphere reference equation the result derived by Carnahan and Starling was applied, as this expression is a good approximation for low-molecular hard-sphere fluids. For the attractive perturbation term, a modified Alder s fourth-order perturbation result for square-well fluids was chosen. Its constants were refitted to the thermodynamic equilibrium data of pure methane. The final equation of state reads ... 

The background potential, being uniform in this approximation, exerts no forces, so the equilibrium configuration of the molecules is the same as that in a fluid of hard spheres without attraction, at the same density. The probability (o that a test particle will fit at an arbitrary point is then also the same as in such a hard-sphere reference fluid. Then setting u, , = 0 in (5.2),... 

In what follows, attention will be restricted to the case that both s and d molecules have a hard spherical core Eq. (3), equal diameters, and the d molecules will be restricted to having a single association site. For this case, this mixture can be treated as a binary mixture of associating molecules in Wertheim s two-density formalism outlined in Section IV. As done throughout this chapter, we will consider a perturbation neatment with a hard sphere reference fluid. Like previous cases, the challenge is determining the graph sum Ac<">. 

Going beyond the single site case, the theory was recently extended such that the d molecules can have an arbitrary number of association sites [90]. In this approach the interaction between s molecules was also that of the hard sphere reference fluid. To add spherically symmetric attractions (square well, U, etc.) between s molecules, one simply needs to employ the appropriate reference system (square well, LJ, etc.). Work is currently under way to employ this association theory as a model for ion-water solvation. 

These equations provide a convenient and accurate representation of the themrodynamic properties of hard spheres, especially as a reference system in perturbation theories for fluids. 

The high-temperatiire expansion, truncated at first order, reduces to van der Waals equation, when the reference system is a fluid of hard spheres. 

Truncation at the first-order temi is justified when the higher-order tenns can be neglected. Wlien pe higher-order tenns small. One choice exploits the fact that a, which is the mean value of the perturbation over the reference system, provides a strict upper bound for the free energy. This is the basis of a variational approach [78, 79] in which the reference system is approximated as hard spheres, whose diameters are chosen to minimize the upper bound for the free energy. The diameter depends on the temperature as well as the density. The method was applied successfiilly to Lennard-Jones fluids, and a small correction for the softness of the repulsive part of the interaction, which differs from hard spheres, was added to improve the results. 

---