Modified embedded-atom method used to derive interatomic potentials for defects and phase formation in the W-C system

An interatomic potential for the W-C system has been derived based on the second-nearest-neighbor modified embedded-atom method scheme. The potential parameters were constructed by fitting to the experimental information on the formation energy of interstitial carbon atoms, the migration energy of carbon atoms in body-centered-cubic (bcc) tungsten and lattice parameters, bulk modulus, and cohesive energy of $B$${}_{h}$ W${}_{2}$C carbide. We demonstrate that such potential can not only reproduce the behavior of carbon atoms in bcc tungsten, but also be employed for modeling energetics and structural properties of tungsten carbides. Moreover, this potential is used to study the stability of multicarbon-multivacancy (${C}_{n}{V}_{m}$) clusters in bcc tungsten. The binding energy and configurations of ${C}_{n}{V}_{m}$ clusters that were not attainable with existing potentials or identified previously via $ab$ $initio$ methods are proposed. Results show that carbon atoms can be strongly trapped by vacancies and interstitials in bcc W. Such bonding between carbon and vacancies lead to the formation of ${C}_{n}{V}_{m}$ complexes and influence the physical properties of metallic tungsten. Besides, a positive binding energy of C-C cluster and C-SIA clusters was found, and both clusters form a hexagonal WC-like local structure. This result provides a plausible explanation for the experimentally observed formation of hexagonal WC (not W${}_{2}$C) in bcc W.