Cluster-model-embedded first-principles study of thermodynamic stability and elastic properties in (Zr, Ti)C_x carbides

Multi-component carbide ceramics have garnered significant attention as ultra-high-temperature structural materials due to their exceptionally high melting points and excellent mechanical properties. In this work, we systematically investigate the synergistic effects of C vacancies and Ti alloying on the thermodynamic stability and elastic behavior of (Zr, Ti)C_x carbides using first-principles calculations. Specific cluster structural models of [C-M₆](C,□)₅ (M = Zr/Ti, □ = vacancy) were constructed by considering the local chemical short-range orders of elemental distribution and the ordering of vacancies on C sublattice, which were then employed as inputs for first-principles calculations. The results reveal that the introduction of C vacancies decreases the free energy at high temperatures and enhances the thermodynamic stability, whereas Ti substitution for Zr tends to reduce stability. Notably, the ternary carbide Zr₅Ti₁C₅ ([C-Zr₅Ti₁](C,□)₅) with an equimolar ratio of Ti-to-vacancy ratio of exhibits an superior high-temperature thermodynamic stability. Analysis of entropy contributions indicates that both vacancies and Ti addition primarily alter the free energy by modifying the lattice vibration modes, an effect dominated by the vibrational entropy. These two types of defects weaken the M-C bond strength, resulting in reduced binding energy and Young’s modulus. Furthermore, this synergistic effect considerably lowers the critical temperature required to stabilize the single-phase solid solution structure in multi-component carbides, which is attributed to a decrease in mixing enthalpy and an increase in configurational entropy caused by vacancies. The cluster-model-embedded first-principles approach offers valuable insight for designing high-performance carbides in complex ceramic systems.