A phase-field approach to non-isothermal transitions

The purpose of the paper is to set up a phase-field model for first-order transitions. The phase field is identified with the concentration of a phase and hence is subject to the continuity equation with a mass growth due to the progression of the transition. The continuity equation is viewed as a differential constraint. The body, in the transition layer, is regarded as a viscous material with time- and space-dependent fields for the mass density, the (absolute) temperature, and the phase-field. Consistency of the model suggests that gradients up to third order are considered. The thermodynamic restrictions are derived by letting the second law be expressed by the Clausius–Duhem inequality and allowing for an extra-entropy flux which, eventually, proves essential to the whole thermodynamic scheme. Results are obtained by using the Helmholtz and Gibbs free energies. As a check of the model, Clapeyron’s equation is derived by means of the Gibbs free energy. A maximum theorem is proved which shows that the phase field takes values between 0 and 1 if it so does at an initial time.