Inhomogeneous fluids statistical mechanics

L. Blum, D. Henderson, Statistical Mechanics of Electrolytes at Interfaces, in Fundamentals of Inhomogeneous Fluids, D. Henderson. Ed. Marcel Dekker (1992) chapter 6 (various statistical approaches discussed). 

Let dA be a surface element at point r within the inhomogeneous fluid, and let Y = 0. The region of fluid on the side of the surface away from which the vector dA is pointing is called region 1, and that on the other side, region 2 (Fig. 4.1). The stress across the surface element comprises two terms, a kinetic term from the momentum of the molecules, and a potential term from the intermolecular forces. Tlie first is isotropic and, because of the separability of the translational motion in classical statistical mechanics, has the same value as in a perfect gas, namely kTp(r)1, where 1 is the unit tensor. It is the second term that introduces the arbitrariness into the definition of p(r) for there is no unique way of specifying which intermolecular forces contribute to the... 

Ideas that go back to van der Waals [217, 218] and Lord Rayleigh [219] on inhomogeneous systems were applied by Cahn and Hilliard [216] to the interface problem. In inhomogeneous fluids, the Helmholtz free energy is a functional of the component density distributions. Although exact formal expressions for this functional have been derived [220,221] from statistical mechanics, they are impractical without approximation [222]. In the gradient approximation, this functional has been expressed as the sum of two contributions one is a function of the local composition and the other is a function of the local composition derivatives [216, 223, 224]. The free energy for a binary system is postulated to have the form ... 

Stillinger, F.H. and Buff, F.P., 1962, Equilibrium statistical mechanics of inhomogeneous fluids, J. Chem. Phys., 37 1. 

Statistical mechanics of inhomogeneous fluids. Proceedings of the Royal Society of London Series A-Mathematical, Physical and Engineering Sciences, 379(1776), 231-46. 

In parallel to these experimental approaches, much progress has been made in theoretical studies of liquid surfaces. Advances in the statistical mechanics of inhomogeneous fluids " have contributed significantly to our understanding of the molecular structure of liquid/solid, liquid/liquid, and liquid/vapor interfaces. However, the mathematical complexity because of losing the spherical symmetry of the bulk has limited the application to mainly calculating a small number of properties (such as density profile, surface tension, and molecular orientation) of neat inhomogeneous liquids. ... 

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