Density-functional theory of the surface tension of simple liquid metals

Abstract

A general density-functional formalism for an inhomogeneous liquid metal is described. The free energy is expressed in terms of two densities: electronic and ionic. In the limit of small spatial density variations of arbitrary wave number, the electron density can be eliminated exactly and the free energy expressed as a functional of the ionic density alone, although the electronic degrees of freedom are still implicitly present. This formalism is applied within the gradient approximation to calculate the surface tension and surface widths of a number of simple liquid metals. The free-energy density is evaluated within the structural expansion in conjunction with a hard-sphere variational approximation to treat the liquid structure, and the gradient coefficient is calculated using a mean-spherical-like approximation to estimate the direct correlation function. This first-principles calculation thus has as input only a parameter describing the pseudopotential and the bulk liquid density. Results for the surface tension of the alkali metals are in excellent agreement with experiment. The temperature derivative of the surface tension is calculated to within a factor of 3 of experiment and probably within experimental error bars. Results for Al and Zn lead to a surface tension which is considerably larger than experiment. Surface widths are in all instances computed to be quite narrow but in reasonable agreement with available experiment. It is argued that discrepancies for the polyvalent metals arise not from the theory itself but rather from difficulty in calculating parameters of the theory from first principles. To verify this, a simple scaling form for surface tension is proposed, motivated by the density-functional theory but in which the relevant parameters are estimated in terms of the liquid density and melting temperature. Agreement with experiment for the scaling expression is at least as good as existing empirical expressions. Finally, a formalism is described which permits, in principle, calculation of both the electronic and ionic singlet density near the liquid surface. Both one-body and two-body forces are obtained, although no numerical evaluations of these are presented. The possible relevance of these to liquid surface structure is briefly discussed.