Landau theory of structures in tetragonal-orthorhombic ferroelastics

A Landau expansion of the elastic energy in the strains is used to study two-dimensional structures in tetragonal-orthorhombic ferroelastics with constraints. Local energy minima are found with respect to the components of the displacement and so the strains satisfy the compatibility relation; this interdependence of the strains, combined with the constraints, can give rise to a subtle frustration. Extraordinarily, a complex energy surface with many bulk metastable states results purely from boundary conditions, without bulk inhomogeneities (such as impurities) of any sort. Some settings require twin walls in only one set of tetragonal 110-type planes; only two variants appear, and the dilatational and shear strains are localized near the surface. Tip splitting can occur when twin walls collide with fixed boundaries. Other settings require both 110 and $11¯0$ walls and so all four variants appear. The structures resulting from collisions of the two twin families are so complex that the ground state of a large system cannot be found with confidence. Strange walls appear between variants with identical deviatoric strain. The dilatational and shear strains are large also in the bulk. Walls wobble, bow, and bend counterintuitively, and pairs sometimes pinch in. Study of the rotation is shown to be essential for understanding some aspects of the structures, particularly collisions of orthogonal twin bands. The ferroelastic-ferromagnet analogy is found to be misleading in important respects. Tip splitting, pinching-in, wall wobbling, and other phenomena are seen in electron microscopy of ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ and other materials.