First-principles phase diagram calculations for the HfC–TiC, ZrC–TiC, and HfC–ZrC solid solutions

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 ~7%, ~20%, and ~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.