No abstract

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 impurities͒ 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 YBa 2 Cu 3 O 7Ϫ␦ and other materials.

We study the statics and the dynamics of domain patterns in proper hexagonal-orthorhombic ferroelastics; these patterns are of particular interest because they provide a rare physical realization of disclinations in crystals. Both our static and dynamical theories are based entirely on classical, nonlinear elasticity theory; we use the minimal theory consistent with stability, symmetry and ability to explain qualitatively the observed patterns. After scaling, the only parameters of the static theory are a temperature variable and a stiffness variable. For moderate to large stiffness, our static results show nested stars, unnested stars, fans and other nodes, triangular and trapezoidal regions of trapped hexagonal phase, etc observed in electron microscopy of Ta4N and Mg-Cd alloys, and also in lead orthovanadate (which is trigonal-monoclinic); we even find imperfections in some nodes, like those observed. For small stiffness, we find patterns like those observed in the mineral Mg-cordierite. Our dynamical studies of growth and relaxation show the formation of these static patterns, and also transitory structures such as 12-armed bursts, streamers and striations which are also seen experimentally. The major aspects of the growth-relaxation process are quite unlike those in systems with conventional order parameters, for it is inherently nonlocal; for example, the changes from one snapshot to the next are not predictable by inspection.