Thermodynamic stability of Pd-Ru alloy nanoparticles: combination of density functional theory calculations, supervised learning, and Wang-Landau sampling.

Solid-solution alloy nanoparticles (NPs) comprising Pd and Ru, which are immiscible in the bulk state, have been synthesised and show excellent catalytic performance. To date, most studies have evaluated the stability of alloy NPs at 0 K only. Because the thermodynamic stability of Pd-Ru alloy NPs may differ from that of the alloy in the bulk state, the stable configuration of the NPs must be evaluated under a finite temperature. Such stability evaluations are critical for developing the durable NPs as catalysts. Therefore, the thermodynamic stability of Pd-Ru alloy NPs was analysed using density functional theory (DFT), supervised learning (SL), and Wang-Landau sampling. We calculated the excess energy of Pd-Ru alloy NPs, which depends on their composition, structure, NP size, adatom type, and defects, and applied SL to all models. The excess energies of the Pd-Ru alloy NPs expressed by structural information, such as the surface-to-volume ratio, correlated with those calculated using DFT. Wang-Landau sampling based on the energy estimated by SL gave the thermodynamic stability of Pd-Ru alloy NPs with a stable configuration under a finite temperature. The solid-solution atomic configuration was subdivided into partially mixed configurations in the surface layer or in the core of the NPs, which is different from the bulk state. The partially mixed configuration was determined by the overall composition and surface properties. The findings from the combined method could contribute to a better understanding of the alloy-NP stability and their application in catalysis.