Ab initio simulations of the surface free energy of TiN(001)

The temperature dependence of the surface free energy of the industrially important TiN(001) system has been investigated by means of an extended two-stage upsampled thermodynamic integration using Langevin dynamics (TU-TILD) methodology, to include the fully anharmonic vibrational contribution, as obtained from ab initio molecular dynamics (AIMD). Inclusion of the fully anharmonic behavior is crucial, since the standard low-temperature quasiharmonic approximation exhibits a severe divergence in the surface free energy due to a high-temperature dynamical instability. The anharmonic vibrations compensate for the quasiharmonic divergence and lead to a modest overall temperature effect on the TiN(001) surface free energy, changing it from around $78\phantom{\rule{0.16em}{0ex}}\mathrm{meV}\phantom{\rule{0.16em}{0ex}}{\AA{}}^{\ensuremath{-}2}$ at 0 K to $73\phantom{\rule{0.16em}{0ex}}\mathrm{meV}\phantom{\rule{0.16em}{0ex}}{\AA{}}^{\ensuremath{-}2}$ at 3000 K. The statistical convergence of the molecular dynamics is facilitated by the use of machine-learning potentials, specifically moment tensor potentials, fitted for TiN(001) at finite temperature. The surface free energy obtained directly from the fitted machine-learning potentials is close to that obtained from the full AIMD simulations.