Vibrations for Free: Fast Calculations of Vibrational Free Energies in Solid Solutions with Machine Learned Force Fields

Alloys and solid solutions form an integral part of modern-day technology ranging from structural materials to semiconductors and catalysts. Recent years have seen a renewed interest in compositionally complex materials with the advent of high entropy alloys and compounds [1]. First principles calculations of their compositional phase diagrams can significantly aid the design and understanding of these complex materials.<br/><br/>Compositional phase diagrams are typically calculated from first principles using cluster expansion methods, in which effective cluster interactions are fitted to internal energies of a set of representative configurations calculated from density functional theory (DFT). Configurational entropy is then incorporated through Monte Carlo simulations at longer length scales. Despite calculations showing that vibrational entropic contributions are important [2], these effects are largely neglected due to their computational cost, and when included they are typically based on approximate bond-length-bond-strength relations [3].<br/><br/>Here, we introduce a method for fast calculations of vibrational free energies from machine learned force fields (MLFF). For cluster expansion construction, a large series of structural relaxations are performed, but only the final energies are used. Here, we use the energies, forces and stresses already available from these geometry relaxations to train an MLFF for a given solid solution, which is benchmarked against DFT lattice dynamics simulations. With this force field in hand, we perform lattice dynamics calculations and extract vibrational free energies for each configuration from which we construct a temperature dependent cluster expansion model.<br/><br/>We apply our method for simple solid solutions and demonstrate predictions of phase diagrams in significantly improved agreement with experiments. This paves the way for easy calculations of vibrational free energies of solid solutions allowing for more realistic predictions of materials thermodynamics from first principles [4].<br/><br/>[1] A. Ferrari, F. Körmann, M. Asta, J. Neugebauer, <i>Nat. Comput. Sci.</i>, 2023, <b>3</b>, 221-229<br/>[2] A. van de Walle, G. Ceder, <i>Rev. Mod. Phys.</i>, 2002, <b>74</b>, 11-45<br/>[3] W. Shao, S. Liu, J. Llorca, <i>Comput. Mater. Sci.</i>, 2023, <b>217</b>, 111898<br/>[4] K. Tolborg, J. Klarbring, A. Ganose, A. Walsh, <i>Digital Discovery</i>, 2022, <b>1</b>, 586-595