Nonlocality and thermodynamic restrictions in phase-field models

This work shows two models of nonlocality for the phase field of non-isothermal phase transitions. In one case the constitutive equations involve the phase field through functionals of the phase field over the region of the body; in the other one the nonlocality is represented by functions of the gradient. The second law is expressed in integral form for the whole region of the body. Upon exploitation of the second law inequality, in the case of functions of the gradient, the evolution equation has to involve the variational derivative of the rescaled free energy, thus supporting an assumption common in the literature.