In this paper we propose a vector function x () as an alternative to the scalar function g () with which the free energy of anisotropic surfaces is usually described. The vector function is chosen such that its component on the unit normal represents the tendency of the surface to minimize free energy by contraction; the components of x off the normal represents the tendency of the surface to minimize free energy by rotation. By elementary operations one obtains from x both the tensile and torque components of the thermodynamic force to be balanced at any exposed edge of a planar surface. The x-vectors for surfaces of various orientations on a given solid are, in fact, radius vectors of the corresponding Gibbs-Wulff form for the shape of the equilibrium body. It is therefore suggested that the Giggs-Wulff form itself is preferable to the gamma plot for representing the free energy of solid surfaces. To demonstrate the incisiveness of the vector formulation we consider equilibrium at the edges and corners of facetted crystals and at the junctions of three and four grains or phases. In particular, analysis of the point junction of four isotropic phases is simplified, while solution of the anisotropic case is achieved for the first time. We conclude that the configurations of interfaces which satisfy equilibrium at the point junction of four crystals are discrete and fixed in space, allowing now freedom in the orientations of any element . . .
The chemical potential at a curved or faceted surface is proportional to the surface divergence of the previously-defined vector function x, which represents the energetics of both isotropic and anisotropic surfaces in a convenient manner. This relationship is shown to be a generalized reformulation of the equations of Gibbs, Thomspon, Herring and Johnson, but leads in addition to new alternative formulas in which certain underlying symmetries become evident. Applications of the vector formalism include analytic proof of the Wulff theorem, distribution of torque on a facet, formulation of the anisotropic Plateau problem, and the equilibrium shapes of particles at surfaces and grain boundaries.