Self-consistent molecular field for packing

The quantities K R) describe occupancy transformations fully involving the solution neighborhood of the observation volume. These coefficients are known only approximately. Building on the preceding discussion, however, we can go further to develop a self-consistent molecular field theory for packing problems in classical liquids. We discuss here specifically the one component hard-sphere fluid this discussion follows Pratt and Ashbaugh (2003). 

Here v = 4TTi /3 is the volume of the observation sphere, b = 1. Because of the explicit factors of p in Eq. (7.26), y will approach the thermodynamic excess activity, y, when R is macroscopically large. The integrals of Eq. (7.29) 

This molecular field /3 ]yjp(r) describes the effect of the exterior solution on solvent molecules within the observation volume. We will adopt the convention that the molecular field /3 mf( ) be zero at the center of the observation volume. This convention resolves a spatially uniform, additive contribution to /3 ]y(p(r) that would otherwise be ambiguous, and with this convention the Lagrange multiplier y in this equation may still be recognized as the excess activity as in Eq. (7.29) in the large R limit. The molecular field together with the Lagrange 

Example results are shown in Eigs. 7.10-7.14 (Pratt and Ashbaugh, 2003). The self-consistent molecular field was obtained iteratively, including an update of t y performing additional few-body simulations to evaluate the work associated with turning on the molecular field using thermodynamic integration  

In addition to achieving a uniform density across the observation volume, the self-consistent molecular field also nearly achieves thermodynamic consistency for the chemical potential. With the choice of an additive constant which makes in the deepest interior of the observation volume, In y should 

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Pratt, L. R. Ashbaugh, H. S., Self-consistent molecular field theory for packing in classical liquids, Phys. Rev. E 2003, 68, 021505... 

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