A graphical method for constructing coherent phase diagrams

The equilibrium conditions for a stressed two-phase coherent solid are employed to develop a graphical construction for coherent equilibrium. We show that in certain cases when the stress is generated by a misfit between the precipitate and matrix, the equilibrium coherent phase compositions can be found drawing parallel tangents to the molar free energy curves of the phases measured in the absence of stress. Total energy diagrams and phase stability diagrams can be constructed to determine the combinations of phase compositions and volume fractions which yield unstable, metastable, and stable states. Through these constructions it is possible to show that elastic stress can result in the appearance of cusps on a total energy plot, indicating the presence of an nonanalytic total free energy of the system as a function of alloy composition. Finally, we show how a stable and metastable coherent phase diagram can be determined qualitatively using these constructions.