Optimizing solid solution and σ phase formation: Predictive criteria for Mo-containing high-entropy alloys

Abstract

This study establishes a novel set of criteria for predicting the formation of single-phase solid solutions and sigma precipitates in Mo-containing high-entropy alloys (HEAs). Under both equilibrium and non-equilibrium conditions, the formation conditions for face-centered cubic (FCC), body-centered cubic (BCC) solid solutions and sigma phases were comprehensively assessed. These criteria greatly reduce experimental effort by offering a streamlined approach for creating high-performance Mo-containing HEAs and coatings. The criteria were established utilizing thermodynamic factors including mixing entropy (Delta S-mix), mixing enthalpy (Delta H-mix), atomic radius differential (delta), valence electron concentration (VEC), paired sigma-forming elements (PSFE), and average electron holes (N-a). Single-phase solid solutions are formed with the following thermodynamic parameters: 6.81 <= Delta S-mix <= 14.91 J.K-1 mol(-1), -6.5 <= Delta H-mix <= 1.4 kJ.mol(-1), 0 <= delta <= 11.12 (equilibrium condition); 2.43 <= Delta S-mix <= 15.09 J.K-1.mol(-1), -16.60 <= Delta H-mix <= -0.62 kJ.mol(-1), 0 <= delta <= 12.36 (non-equilibrium condition). The creation of the sigma phase exhibits dual-segment behavior. Under equilibrium conditions, the predictive criteria are as: 1.39 <= delta <= 8.51, 5.79 <= VEC <= 8.24, PSFE >= 46 %, and 2.40 <= Na <= 4.86; For non-equilibrium conditions, two distinct segments are identified: 1.21 <= delta <= 3.26, 6.12 <= VEC <= 8.93, PSFE >= 42, 1.73 <= Na <= 4.54, or 6.90 <= delta <= 9.45, 7.64 <= VEC <= 8.23, PSFE >= 66 %, 2.75 <= Na <= 3.02. These criteria advance the creation of high-performance materials by acting as useful recommendations for forecasting and designing Mo-containing HEAs.