Some Theorems on the Free Energies of Crystal Surfaces

TLDR

Although the interpretation of experiments on small particle shapes and surface etching usually involves kinetic rather than equilibrium considerations, knowledge of the relative free energies of different shapes or surface configurations may provide a useful perspective. This paper presents theorems on these relative free energies derived from the Wulff construction and establishes relations between atomic models of crystal surfaces and the surface free energy function used in that construction. The authors formulate a condition for the thermodynamic stability of a flat crystal face against hill‑and‑valley formation and use it to analyze equilibrium shapes. The study classifies equilibrium shapes of crystals and anisotropic media, showing that crystals can be smoothly rounded or sharp‑cornered, and notes that curvature‑dependent limitations of the theorems are unlikely to be serious for most real substances but may affect certain idealized models.

Abstract

Although the interpretation of experiments in such fields as the shapes of small particles and the thermal etching of surfaces usually involves problems of kinetics rather than mere equilibrium considerations, it is suggested that a knowledge of the relative free energies of different shapes or surface configurations may provide a useful perspective. This paper presents some theorems on these relative free energies which follow from the Wulff construction for the equilibrium shape of a small particle, and some relations between atomic models of crystal surfaces and the surface free energy function used in this construction. Equilibrium shapes of crystals and of noncrystalline anisotropic media are classified, and it is pointed out that the possibilities for crystals include smoothly rounded as well as sharp-cornered forms. The condition is formulated for thermodynamic stability of a flat crystal face with respect to formation of a hill-and-valley structure. A discussion is presented of the limitations on the applicability of the results imposed by the dependence of surface free energy on curvature; and it is concluded that these limitations are not likely to be serious for most real substances, though they are serious for certain idealized theoretical models.