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Random phase approximation for the non-uniform Yukawa fluid

Mean-field density functional theory can be used to estimate the free energy of non-uniform fluids. The second functional derivative with respect to density of the free energy is related to the direct correlation function of the fluid and, in principle, this can be inverted to find an improved approximation for the pair correlation function and hence the free energy, the so-called ‘random phase approximation’. If the repulsive molecular interaction is approximated by the local density approximation and the attractive interaction is assumed to be of the Yukawa form, the problem reduces to that of finding the eigenvalues of Schrödinger-like equations, which, for certain models (such as the ‘Φ4 model’), can be done analytically in the planar case. The relationship between this approach and field theoretical treatment of the vapour-liquid interface is discussed. The ultraviolet divergence of the expression can be eliminated by separating the first term in the expansion, although quantitative results still depend on the behaviour of the attractive potential in the repulsive core. In the case of a spherical droplet of radius R, correction terms to the free energy involving lnR appear due to (i) cluster translational invariance, (ii) the unstable mode corresponding to droplet growth, and (iii) capillary waves. The net effect of these terms is to modify the classical expression for the nucleation rate by a factor proportional to R4/3.