Thermodynamic properties of ZrC are calculated up to the melting point (T melt ≈ 3700 K), using density functional theory (DFT) to obtain the fully anharmonic vibrational contribution, and including electronic excitations. A significant improvement is found in comparison to results calculated within the quasiharmonic approximation. The calculated thermal expansion is in better agreement with experiment and the heat capacity reproduces rather closely a CALPHAD estimate. The calculations are presented as an application of a development of the upsampled thermodynamic integration using Langevin dynamics (UP-TILD) approach. This development, referred to here as two-stage upsampled thermodynamic integration using Langevin dynamics (TU-TILD), is the inclusion of tailored interatomic potentials to characterize an intermediate reference state of anharmonic vibrations on a two-stage path of thermodynamic integration between the original DFT quasiharmonic free energy and the fully anharmonic DFT free energy. This approach greatly accelerates the convergence of the calculation, giving a factor of improvement in efficiency of ∼50 in the present case compared to the original UP-TILD approach, and it can be applied to a wide range of materials.

The origin of vacancy ordering in ZrCx is explained considering structure geometry, electronic charge distribution, and atomic bonding features, and linked to stability and volume trends in the vacancy-ordered and -disordered zirconium carbides.

First-principles calculations are used to explore vacancy ordering in zirconium carbide at various stoichiometries as affected by oxygen impurities. Atomic bonding and electronic charge distribution are linked to stability and volume trends as a function of O concentration.

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Zirconium carbide has a wide range of substoichiometry facilitated by varying numbers of carbon vacancies. Most experimental studies consider a solid solution of carbon and vacancies without long-range ordering of vacancies. However, theoretical studies predict several superstructural long-range ordered phases to be stable at low temperatures, and these predictions have been validated by experimental fabrication in some cases. The thermophysical properties of zirconium carbide are, therefore, affected not only by the number of carbon vacancies, but their arrangement, which increases the potential of its use as a tuneable ceramic in various applications in aerospace and nuclear industries. This review summarizes the experimental and theoretical studies exploring the longrange ordered zirconium carbides including the known crystal structures, the mechanism for vacancy ordering, and the presently available information on the thermophysical properties. Explanations for the infrequent experimental observations of ordered zirconium carbides are also discussed considering fabrication temperatures, vacancy diffusion, and the effects of impurities, which may be helpful for future synthesis. While our understanding of vacancy ordering in zirconium carbide has vastly improved in recent years, there remain significant knowledge gaps. Areas where more experimental and theoretical studies are needed for further understanding are highlighted.