Temperature dependence of (111) and (110) ceria surface energy

High-temperature properties of ceria surfaces are important for many applications. Here, we report the temperature dependencies of surface energy for (111) and (110) ${\mathrm{CeO}}₂$ obtained in the framework of the extended two-stage up-sampled thermodynamic integration using Langevin dynamics. The method was used together with machine-learning potentials called moment tensor potentials (MTPs), which were fitted to the results of the ab initio molecular dynamics calculations for (111) and (110) ${\mathrm{CeO}}₂$ at different temperatures. The parameters of MTP training and fitting were tested, and the optimal algorithm for the ceria systems was proposed. We found that the temperature increases from 0 to 2100 K led to the decrease of the Helmholtz free energy of (111) ${\mathrm{CeO}}₂$ from 0.78 to 0.64 $\mathrm{J}/{\mathrm{m}}²$. The energy of (110) ${\mathrm{CeO}}₂$ dropped from 1.19 $\mathrm{J}/{\mathrm{m}}²$ at 0 K to 0.92 $\mathrm{J}/{\mathrm{m}}²$ at 1800 K. We show that it is important to consider anharmonicity, as simple consideration of volume expansion gives the wrong temperature dependencies of the surface energies.