Refinement of the Basic Equations of Phase Field Theory under Nonisothermal Conditions

The paper gives a brief review of thermodynamic theories that take into account the structure of materials and phase interfaces. Among them are classical thermodynamics, surface thermodynamics, thermodynamics with internal variables, and extended thermodynamics. It is shown that thermodynamic phase field theory has common features with both extended thermodynamics and thermodynamics with internal variables, but is not identical to them. Coupled phase field equations for binary systems under nonisothermal conditions are deduced based on classical thermodynamics of irreversible processes. Free energy is expanded in a series in temperature gradients, concentration, and phase variable. The obtained general relations are in agreement with the known ones. The new derivation method allowed refining the basic equations. It is shown that not only models of phase transitions with an infinitely narrow interface but also the theory of a two-phase region can be considered as a limiting variant of the theory. The effective transfer enthalpy and the enthalpy due to the evolution of the phase variable are explicitly given in the equations. It is shown that the general relations are equivalent when the basic equations are derived by any of the thermodynamic potentials. Phase field relations are deduced for a regular solution, and the physical meaning of different summands is clarified. The basic equations of phase field theory are derived with consideration for possible chemical interaction of the components.