Incorporating elasticity into the thermodynamics and phase diagrams of multi-component systems

Elastic energy plays a critical role in determining phase stability in compositionally complex alloys. However, quantifying elastic contributions in multi-component systems and incorporating them into phase diagram construction remain challenging. In this study, we present a generalized elastic energy formalism tailored for multi-component alloys, which can be directly and efficiently integrated with CALPHAD thermodynamic databases and existing frameworks such as Thermo-Calc (Andersson et al., 2002), Pandat (Cao et al., 2009) or FactSage (Bale et al., 2016). This elasticity formalism can also be introduced as a post-processing layer in open-source software such as pyCALPHAD (Otis and Liu, 2017) and Kawin (Ury et al., 2023) , enabling elastic assessments in multi-component systems.

We apply our framework for constructing the phase diagram of quinary Fe–Mn–Ni–Co–Cu alloy system, utilizing convex hull and Hessian matrix under elastic considerations. Our results reveal that incorporating elastic energy leads to an expansion of both the spinodal region and the miscibility gap. These are governed by the intricate interplay of chemical and elastic driving forces: We found that Mn and Ni contribute strongly to chemical stabilization, while Cu and Co tend to destabilize the alloy, especially at low Mn concentrations. The stabilizing effect of Fe is also pronounced in Mn-deficient regions. Acting as a destabilizing factor, the elastic energy is primarily driven by the presence of Mn, underscoring its multifaceted role in thermodynamic stability. In Mn-rich compositions, Cu markedly reduces the elastic energy contribution. Combined with CALPHAD infrastructures, the current framework offers a practical pathway to improve the predictive accuracy of phase stability and transformations in complex multi-component alloys.