Thermodynamic Relationships for Bulk Crystalline and Liquid Phases in the Phase-Field Crystal Model

TL;DR

This paper establishes thermodynamic relationships for the phase-field crystal (PFC) model, linking free energy to state variables, and validates a phase coexistence procedure, applying it to a bcc Fe model to analyze stability and parameterization.

Contribution

It derives thermodynamic relationships for the PFC model based on classical thermodynamics and applies them to determine solid-liquid coexistence, validating the common-tangent method within the PFC framework.

Findings

The phase coexistence procedure aligns with traditional methods.

The EOF-PFC model does not predict stable bcc structures with positive vacancy densities.

An alternative PFC parameterization is suggested based on direct correlation function shifts.

Abstract

We present thermodynamic relationships between the free energy of the phase-field crystal (PFC) model and thermodynamic state variables for bulk phases under hydrostatic pressure. This relationship is derived based on the thermodynamic formalism for crystalline solids of Larch\'e and Cahn [Larch\'e and Cahn, Acta Metallurgica, Vol. 21, 1051 (1973)]. We apply the relationship to examine the thermodynamic processes associated with varying the input parameters of the PFC model: temperature, lattice spacing, and the average value of the PFC order parameter, $\bar{n}$. The equilibrium conditions between bulk crystalline solid and liquid phases 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, justifying the use of the common-tangent construction to determine solid-liquid phase coexistence in the PFC model. Finally, we apply the procedure to an eighth-order-fit (EOF) PFC model that has been parameterized to body-centered-cubic ($bcc$) Fe [Jaatinen et al., Physical Review E 80, 031602 (2009)] to demonstrate the procedure as well as to develop physical intuition about the PFC input parameters. 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 $\bar{n}$.