Theoretical model for the morphotropic phase boundary in lead zirconate–lead titanate solid solution

Abstract

A statistical model is proposed to address the problem of two-phase coexistence near the morphotropic phase boundary (MPB) in Pb(${\mathrm{Zr}}_{1\mathrm{-}\mathit{x}}$ ${\mathrm{Ti}}_{\mathit{x}}$)${\mathrm{O}}_3$ solid-solution series. Functional forms for the molar fractions of tetragonal and rhombohedral phases inside the coexistence region are obtained, which may be used to replace the lever rule to describe the phase mixing in a complete binary solid-solution series without solubility gap. The model predicts that the width of this coexistence region is inversely proportional to the volume of each element in the statistical ensemble. In addition, the shift of the MPB composition from the composition of equal molar fraction of the two coexisting phases is found to be proportional to the width of the coexistence region. Several existing controversial experimental observations can be reconciled by this model.