The Gibbs-thomson equation for a spherical coherent precipitate with applications to nucleation

Abstract

The conditions for interfacial thermodynamic equilibrium form the basis for the derivation of a number of basic equations in materials science, including the various forms of the Gibbs-Thomson equation. The equilibrium conditions pertaining to a curved interface in a two-phase fluid system are well-known. In contrast, the conditions for thermodynamic equilibrium at a curved interface in nonhydrostatically stressed solids have only recently been examined. These conditions can be much different from those at a fluid interface and, as a result, the Gibbs-Thomson equation appropriate to coherent solids is likely to be considerably different from that for fluids. In this paper, the authors first derive the conditions necessary for thermodynamic equilibrium at the precipitate-matrix interface of a coherent spherical precipitate. The authors' derivation of these equilibrium conditions includes a correction to the equilibrium conditions of Johnson and Alexander for a spherical precipitate in an isotropic matrix. They then use these conditions to derive the dependence of the interfacial precipitate and matrix concentrations on precipitate radius (Gibbs-Thomson equation) for a such a precipitate. In addition, these relationships are then used to calculate the critical radius for the nucleation of a coherent misfitting precipitate.