Computing Anharmonic Free Energies in Solids with Machine-Learning Interatomic Potentials

Accurate free energies are essential for understanding phase stability and constructing phase diagrams, yet harmonic and quasiharmonic approximations become increasingly inaccurate at elevated temperatures, while thermodynamic integration remains computationally demanding. Here, we establish an efficient workflow for calculating anharmonic free energies by combining nonequilibrium approaches with purpose-trained machine-learning interatomic potentials. The workflow is validated using fcc Ag, hcp Ru, and bcc Mo, and it accurately captures anharmonic contributions. It is then applied to the Fe-C system to construct temperature- and composition-dependent phase diagrams under syngas conditions. We show that inclusion of finite-temperature vibrational free energies qualitatively alters phase stability, stabilizing θ-Fe₃C at elevated temperatures consistent with experiments. Moreover, anharmonic effects significantly modify the stability window of χ-Fe₅C₂ relative to harmonic predictions, owing to stronger anharmonic phonon softening that enhances its vibrational entropy. Overall, this framework provides a robust route for anharmonic free energy calculations and phase diagram prediction in solids.