Barker—Henderson perturbation theory

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 ... 

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... 

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... 

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. 

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). 

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 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]. 

Relation (3.183) is useful whenever we know the free energy of the unperturbed system and when the perturbation energy is small compared with kT. It is clear that if we take more terms in the expansion (3.179), we end up with integrals involving higher-order molecular distribution functions. Therefore, such an expansion is useful only for the cases discussed in this section. For a recent review on the application of perturbation theories to liquids see Barker and Henderson (1972). 

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 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... 

Modern use of perturbation theory stems from the quantitative success of Barker and Henderson in 1967 in calculating ffie structure and thermodynamic properties of a homogeneous Lennard-Jones fluid. A few years later Toxvaerd" extended this work to the liquid-gas surface. It is based, not on (7.7), but on the division u(r) = Uo(r) + Ui(r) where... 

An alternative is the application of so-called perturbation theories (e.g.. Barker and Henderson [9], Weeks et al. [10]). The main assumption here is that the residual (the difference from an ideal gas state) part of the Helmholtz energy of a system A" (and thereby also the system pressure) can be written as the sum of different contributions, whereas the main contributions are covered by the Helmholtz energy of a chosen reference system Contributions to the Helmholtz energy that are not covered by the reference system are crmsidered as perturbations and are described by... 

Fig. 10. Densities of coexisting phases for a fluid obeying the 6-12 potential according to a statistical mechanical perturbation theory developed by Barker and Henderson. The points are a mixture of machine calculations and actual experimental data (from Barker and Henderson, 1967).

Barker, J.A, and Henderson, D., 1967, Perturbation Theory and Equation of State for Fluid. II. A Successful Theory of Liquids, J. Chem. Phys., 47 4714. 

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 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]... 

Monomer contribution As an alternative to the mentioned TPTl versions, we will attempt to describe the thermodynamics and structme of the binary reference mixture of monomers by means of a second order perturbation theory, based on an analytical solution of the Mean Spherical Approximation (MSA) of simple fluids [299,317]. In this theory, the free energy of the mixture of monomers is described perturbatively in terms of the properties of an auxiliary fluid which contains only repulsive interactions. We therefore split the full Lennard—clones potential, Vy(r) into repulsive and perturbative contributions as suggested by Barker and Henderson [300], so that the repulsive potential, contains all of the positive part of the Lennard—Jones potential, while the perturbation, wf contains all of the negative region ... 

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