We report first-principles phase diagram calculations for the binary systems HfC–TiC, TiC–ZrC, and HfC–ZrC. Formation energies for superstructures of various bulk compositions were computed with a plane-wave pseudopotential method. They in turn were used as a basis for fitting cluster expansion Hamiltonians, both with and without approximations for excess vibrational free energies. Significant miscibility gaps are predicted for the systems TiC–ZrC and HfC–TiC, with consolute temperatures in excess of 2000 K. The HfC–ZrC system is predicted to be completely miscibile down to 185 K. Reductions in consolute temperature due to excess vibrational free energy are estimated to be $\sim 7\%$, $\sim 20\%$, and $\sim 0\%$, for HfC–TiC, TiC–ZrC, and HfC–ZrC, respectively. Predicted miscibility gaps are symmetric for HfC–ZrC, almost symmetric for HfC–TiC and asymmetric for TiC–ZrC.

• Received 27 March 2009

DOI:https://doi.org/10.1103/PhysRevB.80.134112