Theory of strain phase equilibria and diagrams

Chemical phase equilibria and phase diagrams are well established and have served as indispensable guides for designing materials through chemical tuning. However, a general thermodynamic theory for strain phase equilibria remains elusive despite extensive studies which have revealed dramatic impacts of mechanical strain on the stability of phases and domain states in solid-state materials. Here, we establish the thermodynamic theory of strain equilibria and a general framework for efficiently constructing multidomain and multiphase diagrams of arbitrarily strained solids under incoherent conditions. As examples, we obtain temperature-strain phase diagrams of ferroelectric PbTiO₃, strongly correlated VO₂, and unconventional ferroelectric Hf_0.5Zr_0.5O₂. We reveal the analogs of the Gibbs phase rules, multiple- and multi-critical points, common-tangent construction, and level rule in strain equilibria to those in the familiar chemical phase equilibria. Our strain equilibria theory offers a powerful framework for predicting the thermodynamic stability of structural phases and domains in strained solids and for guiding the strain engineering of their functional properties.