The thermodynamics of ordered and disordered carbon vacancy structures in ZrC are investigated using ab initio methods. A density functional theory (DFT) study of low-temperature ordered structure energetics identifies three new structures on the convex hull—${\mathrm{Zr}}_{12}{\mathrm{C}}_{11}$ (C2/m), ${\mathrm{Zr}}_9{\mathrm{C}}_8$ ($\mathrm{R}\overline{3}$), and ${\mathrm{Zr}}_6{\mathrm{C}}_5$ ($\mathrm{P}\overline{1}$)—in addition to structures proposed from previous works—${\mathrm{Zr}}_7{\mathrm{C}}_6$ ($\mathrm{R}\overline{3}$), ${\mathrm{Zr}}_4{\mathrm{C}}_3$ (C2/c), ${\mathrm{Zr}}_3{\mathrm{C}}_2$ (Fddd), and ${\mathrm{Zr}}_2\mathrm{C}$ ($\mathrm{Fd}\overline{3}\mathrm{m}$). The phase diagram of ordered structures and their transition to disorder is examined using grand-canonical Monte Carlo simulations and cluster expansions parametrized on a DFT database of electron cold and vibrational harmonic free energies for a variety of vacancy structures. Most of the ordered structures identified undergo first-order transition to the disordered phase at or below 1000 K, with the exception of ${\mathrm{Zr}}_2\mathrm{C}$, which is second order in transition and stable to moderate temperatures ($\sim 1500\mathrm{K}$) as well as broad carbon-to-metal ratios ($x=0.5\text{–}0.65$). The exceptional stability of ${\mathrm{Zr}}_2\mathrm{C}$ provides a rationale for it being the only conclusively observed ordered structure experimentally. Thermodynamics from the Monte Carlo simulations are combined with higher-temperature free energetics effects, vibrational anharmonicity, and electronic thermal excitation, which are computed from DFT molecular dynamics. These effects are found to be appreciable above 1500 K and are on an equal footing with electron cold and vibrational harmonic effects in governing the stability of vacancies. Computed thermodynamic values are in accord with available experimental thermodynamics of carbon-depleted structures. The equilibrium vacancy concentration derived from the calculations agrees with an existing, experimentally derived thermodynamic model except at low temperatures, where experimental data are limited and the computations suggest lower vacancy concentrations. The results run counter to previous computational studies showing significantly more vacancies than expected from experiments. In summary, this work enhances our understanding of both the phase diagram and vacancy stability of ZrC, ultimately providing data that can be leveraged to guide future experimental and theoretical investigations of vacancy structures and thermodynamics.

Close

PrevNext

View figure in article

• Phase transitions
• Thermodynamics
• Vacancies
• Density functional theory