A thermodynamic approach to non-isothermal phase-field evolution in continuum physics

Phase transitions between two phases are modelled as space regions where an order parameter, or phase field, changes smoothly. A thermodynamic approach is developed by allowing for the nonlocal character of the continuum. The phase field is regarded as an internal variable and the kinetic or evolution equation is viewed as a constitutive equation. Along with the other constitutive equations, the unknown evolution equation is required to satisfy the second law of thermodynamics. Necessary and sufficient restrictions placed by thermodynamics are derived for the constitutive equations and, furthermore, a general form of the evolution equation for the order parameter is obtained within the schemes of a non-conserved or a conserved phase field. Based on the thermodynamic restrictions, a model for the ice–water transition is established which allows for superheating and undercooling. A model is also provided for the transition in superconducting materials.