Surface energy and surface tension

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Abstract
Some thermodynamic properties of surface regions are defined. The interdependence of these properties, the effects of particle size, etc. are examined.Results of computer summations of Lennard–Jones (6–12) interaction energies of atoms at lattice sites of face-centered cubic arrays are presented. "Shells" of lattice sites are described by reduced lattice vectors of constant magnitude. Such shells contain (1), 6, 8, 12,24, or 48 atoms.Sublimation energies of atoms in shells of clusters containing from 1 to 1337 atoms are compared with the sublimation energy per atom of an infinite cluster ε0. Mean sublimation energies of clusters of 13 and 1337 atoms are, respectively, about 0.36ε0 and 0.83ε0.Specific surface energies of clusters show relatively little variation with particle size; the specific surface energy of a cluster of 13 atoms is about 0.82 of the specific surface energies of clusters containing from 1337 → ∞ atoms.It is shown that two bulk phases and the interphase between them can grow in extent from suitable sources of mass, etc. while the nature and state of each of the three regions remain constant. Thus the extensive thermodynamic variables of each of the three regions can behave (mathematically) as homogeneous functions of first degree in one another. If the three regions consist of a single common component, under equilibrium conditions the Gibbs potentials per unit mass of the three regions must be the same. Accordingly if the Helmholtz potential of the surface region exceeds that of, say, the condensed bulk phase by As, per gram, the displacement mechanical growth potential, the pv potential per gram, must be less than that of the condensed bulk phase by As. If the pv potential of the bulk phase is zero, then As + (pv = 0 and for a flat interface As = γσ where γ is the surface tension and a the area per gram of the flat surface region.The above ideas underlie Gibb's treatment of surface regions. Such treatments cannot be applied, literally, to systems consisting of only a few atoms or molecules. Thus surface tension in the sense of a negative tangential surface stress of relatively large magnitude is a property of systems of very many molecules (bulk systems) in a state of complete thermodynamic equilibrium.The surface energies of crystals of molecular dimensions are reasonably meaningful and are only a little less than those of laboratory scale crystals. However, the "surface tensions" of such crystals are largely meaningless. The magnitudes of any negative stresses in the surface regions of such minute crystals are probably very much less than those corresponding to the true surface tension of large crystals; further, the surface stresses of these minute crystals are not closely related to excess surface energies, excess surface Helmholtz free energies, etc.

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