Thermodynamically consistent Cahn-Hilliard and Allen-Cahn models in elastic solids

The goal of this paper is to derive again the generalized Cahn-Hilliard and Allen-Cahn models in deformable continua introduced previously by E. Fried and M. E. Gurtin on the basis of a microforce balance. We use a~different approach based on the second law in the form of the entropy principle according to I. Müller and I. S. Liu which leads to the evaluation of the entropy inequality with multipliers.
    Both approaches provide the same systems of field equations. In particular, our differential equation for the multiplier associated with the balance law for the order parameter turns out to be identical with the Fried-Gurtin microforce balance.