A semi-analytic model of vacancy cluster thermodynamics with applications to transition metal carbides

Abstract

Vacancies are well known to alter a wide range of material properties from electrical conductivity to creep rates. However, there is a dearth of understanding of how vacancy clustering occurs in compounds where vacancy clustering is significant which is highlighted in the B1 structure transition metal carbides. In this study, we present a semi-analytical model for vacancy cluster thermodynamics, which accounts for both isolated vacancies and vacancy clusters by minimizing the total Helmholtz free energy. This approach enables the prediction of equilibrium vacancy concentrations while incorporating configurational entropy and temperature-dependent formation energetics. The model is then applied to two B1-structured transition metal carbides, $\text{TiC}$ and $\text{TaC}$, where the isolated vacancy formation energies and vacancy-cluster binding energies are determined using density functional theory (DFT) calculations. Contributions from vibrational and electronic entropies are also included, which highlights the different bonding in group IVB and VB monocarbides. The results suggest that metal vacancy concentration at thermodynamic equilibrium exhibits a non-Arrhenius behavior since vacancy-cluster formation not only changes the formation energy but also significantly lowers the configurational entropy. This means that the formation of metal vacancies depends on both the temperature and the carbon vacancy concentration, which provides insights into the self-diffusion mechanism in transition metal carbides. Furthermore, the theory presented here can be applied to a wide range of materials to understand the role vacancy clustering in compounds.