Verlet-Weis

The pseudo parameters of the HSE theory are derived from an equation of state expanded in powers of 1/kT about a hard-sphere fluid, as is developed by the perturbation theory. Consequently, it is reasonable to expect that procedures for defining optimal diameters for the perturbation theory should work well with the HSE procedure. The first portion of this chapter shows that this is indeed correct. The Verlet-Weis (VW) (5) modification of the Weeks, Chandler, and Anderson (WCA) (6) procedure was used here to determine diameters in a mixture of Len-nard-Jones (LJ) (12-6) fluids. These diameters then were used in the HSE procedure to predict the mixture properties. 

Better agreement between the values calculated using the above equations and the computer simulations is obtained by introducing the correction proposed by Verlet and Weis [32] which forces g(l,r/) to satisfy the consequence of the virial theorem given by Eqn (A3). For values of R > 4, the value of hs(-R) is set equal to unity. 

Verlet and Weis argued that one could expect to develop a satisfactory approximation scheme by retaining in the rotationally-invariant expansion of /i(12) only the terms that appear on a certain minimal basis, which consists of the terms that enter the MSA. [For a w(12) of ideal dipole form, these are just the 5, A, and D terms of (2.62), defined by (2.20)]. The linearization of (2.143) that will ensure the retention of these terms and introduce no others is given by... 

Although the approximation corresponding to (2.144) on the free-energy level was found by Verlet and Weis to be quite accurate for dipolar spheres, (2.144) itself was found to exaggerate the structure of h 2) at the second-nearest-neighbor distance, as discussed in Section III. 

The LIN approximation was first obtained by Verlet and Weis, who linearized the EXP approximation of Andersen, Chandler, and Weeks. This leads to the following approximation for the pair correlation function. 

A general method of predicting the effective molecular diameters and the thermodynamic properties for fluid mix-tures based on the hard-sphere expansion conformal solution theory is developed. The method of Verlet and Weis produces effective hard-sphere diameters for use with this method for those fluids whose intermolecular potentials are known. For fluids with unknown potentials, a new method has been developed for obtaining the effective diameters from isochoric behavior of pure fluids. These methods have been extended to polar fluids by adding a new polar excess function, to account for polar contributions in a mixture. A new set of pseudo parameters has been developed for this purpose. The calculation of thermodynamic properties for several fluid mixtures including CH —C02 has been carried out successfully. 

Verlet and Weis arrived at this same approximation for dipolar liquids by means of a formally different path. They call it the linear approximation (LIN),... 

The second-order, optimized mode expansion of Andersen and Chandler, which proves to be identical to this, has been evaluated for ionic solutions. The results confirm that (104) is a highly accurate approximation for the 1-1 electrol) es except at very low concentrations where our AB2 correlation is important. For dipolar spheres, (104) was evaluated by Verlet and Weis, who were led to its consideration along with the LIN result (99) on somewhat different grounds from ours. It is a reasonably good approximation, but not as good as the Fade result given by Eq. (38). However, if one adds the third-order... 

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