Conditions of compatibility for the solid-liquid interface

The seminal theory of singular surfaces propounded by Hadamard and Thomas is examined within the context of the dynamics of a solid-liquid interface. It is shown that most of the hypotheses upon which Clapeyron's equation is based can be weakened and two generalized versions of it are derived: with and without curvature effects. The remaining part of the paper is mainly focused on the interface conditions for the classical Stefan problem. The counterpart of Clapeyron's equation for such a problem will give an explicit expression for the supercooling temperature without recourse to linearization procedures. Furthermore, a decay law for the latent heat of melting is given which shows, in an explicit way, its complex dependence upon the curvature and the normal speed of the interface. Finally, a transport equation for the interface temperature is derived and a qualitative solution of a simplified version of it is given for the particular case in which the jump in the Helmoltz free energies of the bulk phases is a conserved quantity throughout the field.