Phase-field modeling of transition and separation phenomena in continuum thermodynamics

In the framework of continuum thermodynamics a new approach to phase transition and separation phenomena is developed by emphasizing their nonlocal character. The phase-field is regarded as an internal variable and the kinetic or evolution equation is viewed as a constitutive equation of rate type. The second law of thermodynamics is satisfied by virtue of an extra entropy flux which arises from its nonlocal formulation. Such an extra flux is proved to be nonvanishing inside the transition layer, only. Different choices of the state variables distinguish transition form separation models. The former case involves the gradients of the main fields up to the second order, whereas in the latter all gradients up to the fourth order are needed and the total mass of the phase-field is conserved. In both cases, necessary and sufficient restrictions on the constitutive equations are derived from thermodynamics. On this background, some applications to scalar-valued models are developed. A simple model of the temperature-induced first-order transition is derived in connection with a state space involving second-order gradients. Dynamical models of phase separation in a binary fluid mixture are discussed and the classical nonisothermal Cahn-Hilliard system is obtained as a special case of a fourth-gradient model