Thermodynamic relationships for homogeneous crystalline and liquid phases in the phase-field crystal model

We present thermodynamic relationships between the free energy density of the phase-field crystal (PFC) model and thermodynamic state variables that correspond to the model input parameters: temperature, lattice spacing, and an average value of the PFC order parameter, n¯. These relationships, derived for homogeneous phases under hydrostatic and nonhydrostatic stresses, are based on the thermodynamic formalism for crystalline solids of Larché and Cahn (1973). These relationships provide clear thermodynamic descriptions of the physical processes that are associated with changing PFC input parameters, and demonstrate that a crystalline phase from the PFC model can be considered a network of lattices occupied by atoms and vacancies, as described by Larché and Cahn. The equilibrium conditions between a crystalline phase and a liquid phase are imposed on the thermodynamic relationships for the PFC model to obtain a procedure for determining solid-liquid phase coexistence. The resulting procedure is found to be in agreement with the method commonly used in the PFC community. Finally, we apply the procedure to an eighth-order-fit (EOF) PFC model that has been parameterized to body-centered-cubic (bcc) iron (Jaatinen et al., 2009) to verify the applicability of the procedure. We demonstrate that the EOF-PFC model parameterization does not predict stable bcc structures with positive vacancy densities. This result suggests an alternative parameterization of the PFC model which requires the primary peak position of the two-body direct correlation function to shift as a function of n¯.