Low-cost Vibrational Free Energies in Solid Solutions with Machine Learning Force Fields

The rational design of alloys and solid solutions relies on accurate computational predictions of phase diagrams. The cluster expansion method with effective cluster interactions fitted to energies from first principles calculations has proven to be a valuable tool for studying disordered crystals. However, the effects of vibrational entropy are commonly neglected due to their additional computational cost. Here, we devise a method for including vibrational free energy in cluster expansions at a very low computational cost by fitting a machine learning force field (MLFF) to the relaxation trajectories already available from the cluster expansion construction. We demonstrate our method for two (pseudo)binary systems, Na1-xKxCl and Ag1-xPd_x, for which accurate phonon dispersions and vibrational free energies are derived from the MLFF. For both systems, inclusion of vibrational effects results in significantly better agreement with miscibility gaps in experimental phase diagrams. This methodology can allow routine inclusion of vibrational effects in calculated compositional phase diagrams, and thus more accurate predictions of properties and stability for materials mixtures.