  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.
