High-entropy or multi-principal component ultra-high temperature ceramics, including the rocksalt structured MC_1-x (where the cation M is an equiatomic or non-equiatomic mixture of metallic elements including Ti, Zr, Hf, Nb, and Ta, and the anion C is carbon), have recently generated significant interest due to their potential improved or tuneable properties such as melting point, hardness, ductility, and oxidation resistance. The single metallic element carbides, such as zirconium carbide, are known to have a wide range of stoichiometry facilitated by significant numbers of carbon vacancies (up to around 50% of the carbon atoms), although until now there are no systematic investigations varying carbon stoichiometry in multi-principal cation carbides.
 
This work uses first-principles calculations to explore multiatomic mixing (binary, ternary, quaternary, and quinary) on the metallic element lattice at different carbon stoichiometries. Special Quasirandom Structures (SQS) were generated for each composition, to provide a supercell approximation of random mixing. Density functional theory (DFT) calculations were used to obtain the ground state energy and structural and electronic properties. The atomic bonding is investigated and trends are identified by considering the local atomic environments and cluster configurations.
 
Individually, all the MC_1-x carbides have a tendency to form long-range vacancy-ordered phases at low temperature, although these are extremely challenging to experimentally synthesise. In single cation transition metal carbides (e.g. ZrC_1-x, HfC_1-x), the short- and long-range carbon vacancy ordering is driven by differing local bonding surrounding different vacancy cluster configurations. However, similar but slightly different trends are observed in group IV and group V transition metal carbides.
 
Using the DFT calculations above, this work explores the effect of the multi-principal cation lattice on the carbon vacancy ordering on the anion lattice, at absolute zero and considering finite temperature effects. Miscible and immiscible mixtures are identified, where approximate temperatures of miscibility gaps are determined using the Bragg-Williams configurational entropy approximation for ideal mixtures. The tendency for phase separation is compared in equivalent composition multi-cation vacancy-disordered and vacancy-ordered carbides. Approximate order-disorder transition temperatures are obtained for mixed-cation substoichiometric carbides and compared with the miscibility gaps in the same system. On this basis, the possibility of synthesising certain vacancy-ordered multi-principal cation compounds is revealed, alongside the required range of synthesis temperatures.