Unraveling symmetric hierarchy in solid-state reactions of tungsten-based refractory metal carbides through first-principles calculations

Refractory metal carbides, usually fabricated via solid state reactions, require precise control of their reactants and temperature, especially when they enter a complex compositional space, like high-entropy (multi-component) carbides. In this work, the solid-state reactions of tungsten based refractory metal carbides M-W-C (M = Ti, Zr, Hf, V, Nb, Ta) are systematically studied through first-principles thermodynamic calculations and percolation simulations, and its relationship with symmetric principles is unraveled. Symmetric hierarchy is defined by the group-subgroup Bärnighausen tree and the gap in their space group number. It suggests MC and WC/W₂C are two possible reactants to form the highest symmetric M-W-C ternary carbides, and indicates the larger the gap in their space group number, the harder the reactions. From the symmetric hierarchy, we found the reaction path from MC to M-W-C ternary carbides is the most probable, supported by the Gibbs reaction free energy. Carbon percolation within the metal framework plays another role in the solid-state reactions of tungsten based refractory metal carbides. It reveals the phase transition from M₆W₆C to M₃W₃C undergoes a transient M₂W₂C. The success in predicting the phase relationship of M-W-C ternary system offers a new paradigm for the design and synthesis of high-entropy carbides, nitrides, and oxides.