Continuum Thermodynamics and Phase-Field Models

Phase transitions between two phases are modelled as space regions where a phase-field changes smoothly. The two phases are separated by a thin transition layer, the so-called diffuse interface. All thermodynamic quantities are allowed to vary inside this layer, including the pressure and the mass density. A thermodynamic approach is developed by allowing for the nonlocal character of the continuum. It is based on an extra entropy flux which is proved to be non vanishing inside the transition layer, only. The phase-field is regarded as an internal variable and the kinetic or evolution equation is viewed as a constitutive equation of rate type. Necessary and sufficient restrictions placed by thermodynamics are derived for the constitutive equations and, furthermore, a general form of the evolution equation for the phase-field is obtained within the schemes of both a non-conserved and a conserved phase-field. Based on the thermodynamic restrictions, a phase-field model for the ice-water transition is established which allows for superheating and undercooling. A model ruling the liquid-vapor phase transition is also provided which accounts for both temperature and pressure variations during the evaporation process. The explicit expression of the Gibbs free enthalpy, the Clausius-Clapeyron formula and the customary form of the vapor pressure curve are recovered.