Fig. 4: Stability of high-entropy carbides (HECs).
From: Settling the matter of the role of vibrations in the stability of high-entropy carbides
a Temperature-dependent vibrational free energy Fvib of \(\left({{{{{{{\rm{HfNbTaTi}}}}}}}}R\right){{{{{{{\rm{C}}}}}}}}\) (HEC-R) with TPOCC = 2473 K. b Changes in Fvib of the HECs at 1500 K when force constant disorder (avg. Φ) and mass disorder (avg. m) are removed. A temperature of 1500 K was chosen as entropy changes are constant in this regime (see d). c Vibrational free energy ΔFvib,f and d entropy changes ΔSvib,f for the formation reaction of the HECs from binary reactants along with configurational entropy contributions ΔSconf. Solid and dashed lines represent values obtained from the Automatic Phonon and GIBBS Libraries (APL and AGL), respectively. Insets: comparison of vibrational and electronic contributions ΔFelec,f/ΔSelec,f (dotted lines). e Fvib from APL and AGL as a function of (average) cation mass m at 1500 K. f Gibbs free energy changes ΔGf for the formation reaction of the HECs (solid lines). ΔGf without ΔFvib,f is shown as dashed lines. Shaded areas symbolize energies at which the HECs are unstable. g APL vibrational free energy changes ΔFvib,d for the predicted decomposition reactions of the HECs compared with contributions from Sconf. h Gibbs free energy changes ΔGd for the decomposition reaction of the HECs (solid lines). ΔGd without ΔFvib,d is shown as dashed lines. Shaded areas symbolize energies at which the HECs are unstable.
