Barker-Henderson theory

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. 

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. ... 

EoS based on perturbation theory and computer simulation results validity of the Barker-Henderson theory Derivation of the BGY, PY, and HNC integral equations in 2D Fourth virial coefficient Fifth virial coefficient Quantum corrections to the third and fourth virial coefficients quantum corrections to the Helmholtz free energy Results at very high densities EoS based on computer simulation results and on the five first virial coefficients No influence of number of particles for states far from the phase transitions values of RDF... 

The density profile pi z) for the uncharged surface has three different terms according to the contact theorem (73) and according to the expression (28) for the pressure. The first two of them describe the hard-sphere and ion-pairing contributions, respectively. They can be obtained in the framework of the associative version of the Henderson-Abraham-Barker (HAB) theory [53, 54], According to the obtained results [54],... 

We turn now to a discussion of perturbation theories based upon extensions of the Barker-Henderson and Weeks-Chandler-Anderson theories to interaction site potentials. Such theories seek to treat the properties of the fluid as a perturbation about a reference fluid with anisotropic repulsive forces only. The theories have been formulated both explicitly in terms of division of the site-site potential into reference and perturbation potentials (Tildesley, Lombardero et al. )... 

The interactions of molecules can be divided into a repulsive and an attractive part. For the calculation of the repulsive contribution, a reference fluid with no attractive forces is defined and the attractive interactions are treated as a perturbation of the reference system. According to the Barker-Henderson perturbation theory [21], a reference fluid with hard repulsion (Eq. (10.31)) and a temperature-dependent segment diameter di can br applied. For a component i, the following can be found ... 

In the Barker-Henderson [5,35] perturbation theory the potential in Eq. (4.1) is divided into a repulsive branch v (r) and an attractive branch Vg(r)... 

The theoretical studies included in Table 12 are based on calculations of the virial coefficients [195,260,285,294], applications of the Barker-Henderson (BH) [232] or WCA perturbation theories [19,280,288,297,301,303-306,311] and solutions to the Percus-Yevick (PY) and hypemetted-charn equation (HNC) [247-250] and the Bom-Green-Yvon (BGY) [251]... 

The first analytical expression for the equation of state of 2D L-J fluids was given by Henderson [232]. This EoS was based on Monte Carlo computer simulation resxdts and on the Barker-Henderson perturbation theory. The agreement with previous simulations was quite good, except for some ranges close to the critical region. However, it contains nine nonlinear coefficients in its analytical expression, which have been listed only for some temperatures. For that reason, the Henderson EoS is not always applicable and has only been considered here for the temperatures for which the coefficients are known. No comparison with experimental results has been made. 

Other interesting computer simulations for the 2D L-J system are listed in Tables 9-12. Among them we would emphasize the simulation performed by Henderson [232], who analyzed the validity of the Barker-Henderson perturbation theory and obtained an analytical expression for the EoS of the 2D L-J fluids, as was indicated in Sections fVA and IVB. 

Barker-Henderson perturbation theory 67 Barrier heights 179... 

Koyama and Sato ° refined the terms associated with the soft dispersion interactions in eqn [73] by extending the Barker-Henderson theoty for spheres to spherocylinders, and showed that the extended theory precisely describes the concentration dependence of dHjdc for PS samples of low Al in toluene. Oribe and Sato also foimd similar agreements for low molar mass PS in cyclohexane at different temperatures including 0. 

Barker J and Henderson D 1967 Perturbation theory and equation of state for a fluids II. A successful theory of liquids J. Chem. Phys. 47 4714... 

A well-known approximate molecular theory of a fluid at a planar interface is originally due to Helfand, Frisch and Lebowitz [76] and later to Henderson, Abraham and Barker [77] and Perram and White [78]. Consider a binary mixture (A,B) in which one of the species (A) becomes extremely dilute and infinitely large. S.E. [52] show that if the size of species A tends to infinity while the concentration of A tends toward zero, then a consequence of the OZ equation, coupled with the PY equation, is the relation... 

The theory of simple liquids achieved a mature state in the era 1965-1975 (Barker and Henderson, 1976 Hansen and McDonald, 1976). As this mature theory was extended towards molecular liquids, simple molecular cases such as liquid N2 or liquid CCI4 were treated first. But the molecular liquids that were brought within the perimeter of the successful theory were remote extremities compared with the liquids synthesized, poured from bottles or pipes, and used. In addition, the results traditionally sought from molecular theories (Rowlinson and Swinton, 1982) often appear to have shifted to accommodate the limitations of the available theories. Overlooking molecular simulation techniques... 

Henderson, D. and Barker, J. A., Perturbation theories. In D. Henderson (ed.). Physical Chemistry. An Advanced Treatise, pp. 377 12. New York Academic Press (1971). 

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]. 

Table I, in the column headed HSE-VW, shows the results of using Equations 2 through 6 to define the diameters with the HSE method to calculate the properties of an equimolar mixture of LJ fluids. The reference is a pure LJ fluid. Other columns show comparison with the machine-calculated results of Singer and Singer (8) in column MC. The van der Waals (VDW) one-fluid theory (9) and the VDW two-fluid theory (10) are in columns VDW-1 and VDW-2. The GHBL column gives the Grundke, Henderson, Barker, Leonard (GHBL) (11) pertu-bation theory results with each diameter determined by Equation 3.

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