Statistical study of vacancy diffusion in TiC and TaC

Density functional theory (DFT) simulations, Metropolis Monte Carlo (MMC), and kinetic Monte Carlo (kMC) simulations were performed to understand the mechanisms of mass diffusion in a group IVB transition metal carbide, TiC, and a group VB transition metal carbide, TaC. The DFT calculations were used to obtain the vacancy formation energy and migration energy of a variety of microstates for off-stoichiometric TiC and TaC. MMC simulations, based on our DFT results, were used to determine the ensemble average of the metal vacancy formation energy and determine the average size metal-vacancy clusters present in these materials. kMC simulations were used to determine the ensemble average of the migration energy barrier as well as understand how the vacancies in these materials, on average, migrate. These collective results show that metal vacancy migration in TiC and TaC are quite different, where Ti vacancies should be surrounded by four to five carbon vacancies whereas, on average, tantalum vacancies are surrounded by one or zero carbon vacancies. However, the carbon vacancies substantially contribute to metal vacancy diffusion in both these materials, as the metal vacancy statistically will have a carbon vacancy near it before and after migration. From these results, we find that the activation energy of metal vacancy diffusion is 7.66 eV in ${\mathrm{Ti}}_{0.995}{\mathrm{C}}_{0.97}$ and 6.41 eV in ${\mathrm{Ta}}_{0.995}{\mathrm{C}}_{0.97}$, which agrees reasonably well with experimentally reported activation energies. These results give further insights into the mechanisms associated with mass diffusion within the transition metal carbide families, an important insight needed to better elucidate high temperature diffusional creep responses which is often difficult to assess experimentally.