Barker-Henderson approach

As we have already pointed out, the theoretical basis of free energy calculations were laid a long time ago [1,4,5], but, quite understandably, had to wait for sufficient computational capabilities to be applied to molecular systems of interest to the chemist, the physicist, and the biologist. In the meantime, these calculations were the domain of analytical theories. The most useful in practice were perturbation theories of dense liquids. In the Barker-Henderson theory [13], the reference state was chosen to be a hard-sphere fluid. The subsequent Weeks-Chandler-Andersen theory [14] differed from the Barker-Henderson approach by dividing the intermolecular potential such that its unperturbed and perturbed parts were associated with repulsive and attractive forces, respectively. This division yields slower variation of the perturbation term with intermolecular separation and, consequently, faster convergence of the perturbation series than the division employed by Barker and Henderson. 

Barker-Henderson approach, see [204], Open circle shows the critical point of the Asakura-Oosawa model, and the closed circle is the critical point of the model with s = 0.625. Adapted from Zausch et al. [204]... 

The evidence available suggests that the two approaches are about equally accurate, although the approach based on site-site correlation functions is more readily generalized to the treatment of multipolar interactions as well as to the effect of the attractive forces upon the structure and free energy at moderate and low density. In addition to the efforts made at extending the WCA theory to interaction site fluids, the Barker-Henderson theory has also been extended to these systems by Lombardero, Abascal, Lago and their co-workers. ... 

The application of this approach to the hard-sphere system was presented by Ree and Hoover in a footnote to their paper on the hard-sphere phase diagram. They made a calculation where they used Eq. (2.27) for the solid phase and an accurate equation of state for the fluid phase to obtain results that are in very close agreement with their results from MC simulations. The LJD theory in combination with perturbation theory for the liquid state free energy has been applied to the calculation of solid-fluid equilibrium for the Lennard-Jones 12-6 potential by Henderson and Barker [138] and by Mansoori and Canfield [139]. Ross has applied a similar approch to the exp-6 potential. A similar approach was used for square well potentials by Young [140]. More recent applications have been made to nonspherical molecules [100,141] and mixtures [101,108,109,142]. 

The properties of a concentrated, disordered dispersion may be described by perturbation theory. In the approach of Barker and Henderson,the pair potential is assumed to be of the form... 

Points 1 and 2 can be incorporated straightforwardly using the ideas presented earlier in this chapter. For example, we could use the analytic Percus-Yevick equations of state for hard spheres (Eqs. 47a and b) or the Carnahan-Starling equation of state (Eq. 49) for p. Furthermore, we could use the hard-sphere radial distribution function obtained numerically from one of the integral equations or even that calculated from computer simulation. Points 3 and 4 are less straightforward and represent contributions that were made around 1970 by Barker and Henderson (1976) and by Weeks, Chandler and Andersen (1971). The results of these two approaches are comparable and are illustrated in Figs. 10 and 11 and Table 3. 

A great deal of systematic information has been developed from model systemswhich can be qualitatively applicable to real systems. A series of definitive reviews of the Monte Carlo method and results on model systems have been prepared by Wood. The dynamics approach was Initially characterized in the series of papers by Alder, Wainwright and coworkers. A ccmprehenslve review of liquid state theory was recently published by Barker and Henderson. ... 

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