Generalization of the Gibbs-Thomson equation

The Gibbs-Thomson equation relates the chemical potential of the vapor in equilibrium with a spherical drop to the radius and isotropic surface free energy of the drop. It is shown that this equation has a simple generalization to the case of arbitrary anisotropic surface free energy. This general form of the Gibbs-Thomson equation provides a simple and direct connection between the size and shape of a crystal in equilibrium with its vapor, the chemical potential of the vapor in equilibrium with the crystal and the detailed behavior of the specific surface free energy, γ, as a function of surface orientation. Herring1) has given two equations which relate the chemical potential of the vapor in equilibrium with a smoothly curved surface element or a facet on a given body of arbitrary (i.e., not necessarily equilibrium) shape to the detailed behavior of γ as a function of surface orientation and to the size and geometry of the given body. It is shown that Herring's equation for a smoothly curved surface element can be made to take a considerably simpler form by comparing the geometry of the given body to that of an equilibrium body; if the given body does in fact have the equilibrium shape the still simpler general Gibbs-Thomson equation is obtained. Herring's equation for a facet is also shown to yield the generalized Gibbs-Thomson equation when the given body is an equilibrium body.