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) CeO2 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) CeO2 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) CeO2 from 0.78 to 0.64 J/m2. The energy of (110) CeO2 dropped from 1.19 J/m2 at 0 K to 0.92 J/m2 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.