Strain phase separation: Formation of ferroelastic domain structures

Phase decomposition is a well-known process leading to the formation of two-phase mixtures. Here we show that a strain imposed on a ferroelastic crystal promotes the formation of mixed phases and domains, i.e., strain phase separation with local strains determined by a common tangent construction on the free energy versus strain curves. It is demonstrated that a domain structure can be understood using the concepts of domain/phase rule, lever rule, and coherent and incoherent strain phase separation, in a complete analogy to phase decomposition. The proposed strain phase separation model is validated using phase-field simulations and experimental observations of $\mathrm{PbTi}{\mathrm{O}}_3$ and $\mathrm{BiFe}{\mathrm{O}}_3$ thin films as examples. The proposed model provides a simple tool to guide and design domain structures of ferroelastic systems.

Figure 1

Schematics of composition and strain phase separation processes. (a) and (b) Common tangent constructions for (a) phase decomposition and (b) strain phase separation. (c) Overall and local deformation in a 1D system. (d) and (e) Coexistence of two domains with different (d) normal strains and (e) shear strains.

Figure 2

Strain domain diagrams of $\mathrm{PbTi}{\mathrm{O}}_3$. (a) Strain-temperature domain diagram. (b) Free energy as a function of strain at different temperatures. (c) Strain-strain domain diagram at room temperature 20 °C. (d) Energy surfaces of $a_1,a_2$, and $c$ domains. The green triangle denotes the common tangent plane of the three energy surfaces.

Figure 3

Phase stability analysis of $\mathrm{BiFe}{\mathrm{O}}_3$. (a) Energy contours under stress-free boundary conditions. (b) Total free energy of the different phases under thin film boundary conditions. Common tangent of the two curves is plotted, and the shaded region represents the results from the phase-field simulations. (c) Order parameter evolution as a function of strain. (d) Volume fraction of the T-like phases from the phase-field simulations. The order parameters in (a) and (c) have SI units, i.e., $\mathrm{C}/{\mathrm{m}}^2$ and rad for $P_i$ and ${\theta }_i$, respectively.

Figure 4

Domain structures from [(a) and (b)] the phase-field simulations and [(c) and (d)] PFM measurements. (a) Three-dimensional domain structures under a compressive strain of −4.0%. (b) Corresponding domain structures in the $x_1{x}_2$ plane. The colors are assigned based on the polarization directions. (c) Topography of a mixed-phase BFO film with thickness of $\sim 130\mathrm{nm}$. (d) Corresponding in-plane PFM phase image.