Barker-Henderson diameter

The division of the potential into reference and perturbation terms is carried out according to the procedure described earlier and the values of the Barker-Henderson diameters are given in Table I. The Helmholtz free energy of the system is obtained by using Eq. [22], in which the upper limit of the integration is so chosen that raising it further does not change... 

For the potentials for which the repulsive part is not infinitely steep, the reference potential can be reduced to a system of hard spheres by introducing an effective diameter, the Barker—Henderson diameter. This... 

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 case of polyethylene, we observed in the previous section that to be consistent with the ermerimental structure factor the hard core diameter should be fixed at d = 3.90 A at 430 K. With this added constraint we therefore have only a single adjustable parameter since d is related to Ou and e through the Barker-Henderson relation, Eq. (4.3) or WCA equation, Eq. (4.5). 

At this stage, we encounter the difficulty of describing the repulsive system defined in Eq. (131), which is nol well known. In order fo circumvent this problem, we map the repulsive system on a simpler mixture of hard spheres with appropriate diameter. The key step is then to choose the hard sphere diameters such that the error brought about by this mapping is minimized. In a one component fluid, a suifable choice is fo defermine the diameter according to the Barker-Henderson prescription [300]. Here we just consider the straightforward extension, so that the hard sphere diameter of species i is calculated as ... 

The only parameter that has not yet been defined in (1) is the rigid-sphere diameter d. Barker and Henderson define d such that... 

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.

The second line in Eq. (37) is derived straightforwardly by referring to Eq. (30). The reference HS system has identical local density distribution p r) as the LJ system and the diameter of the HS can be estimated by using BH method (Barker and Henderson, 1967)... 

Fischer and Methfessel start by dividing u(r) into two parts, Uo(r), as in (7.7), and the perturbation term U](r) = u(r)-Uo(r). This separation gives two integrals on the right-hand side of (7.2), in the second of which they put g == 1 that is, they use a mean-field approximation for the attractive part of the potential, u,(r). In the first integral they replace g by go, the distribution function for the reference system whose potential is Uq. The function go is, in turn, approximated by g), where the diameter of the hard spheres, dh is given by the prescription of Barker and Henderson, ... 

---