Elastic energy of multi-component solid solutions and strain origins of phase stability in high-entropy alloys

The elastic energy of mixing for multi-component solid solutions is derived by generalizing Eshelby's sphere-in-hole model for binary alloys. By surveying the dependence of the elastic energy on chemical composition and lattice misfit, we propose a lattice strain coefficient {\lambda}*. Applying to several high-entropy alloys and superalloys, we found that most solid solution alloys are stable when {\lambda}*<0.16, analogous to the Hume-Rothery atomic-size rule for binary alloys. We also reveal that the polydispersity index {\delta}, frequently used for describing strain in multi-component alloys, is directly related to the elastic energy (e) with e=q{\delta}^2, q being an elastic constant. Furthermore, the effects of (i) the number and (ii) the atomic-size distribution of constituting elements on the phase stability of high-entropy alloys were quantified. The present derivations open for richer considerations of elastic effects in high-entropy alloys, offering immediate support for quantitative assessments of their thermodynamic properties and studying related strengthening mechanisms.