Time evolution of tetragonal-orthorhombic ferroelastics

We study numerically the time evolution of two-dimensional (2D) domain patterns in proper tetragonal-orthorhombic (T-O) ferroelastics. Our equations of motion are derived from classical elasticity theory, augmented by nonlinear and strain-gradient terms. Our results differ from those found by other dynamical methods. We study first the growth of the 2D nucleus resulting from homogeneous nucleation events. The later shape of the nucleus is largely independent of how it was nucleated. In soft systems, the nucleus forms a flowerlike pattern. In stiff systems, which seem to be more realistic, it forms an X shape with twinned arms in the 110 and $1\ifmmode\bar\else\textasciimacron\fi{}10$ directions. Second, we study the relaxation that follows completion of the phase transition; at these times, the T phase has disappeared and both O variants are present, separated by walls preferentially in 110-type planes. We observe a variety of coarsening mechanisms, most of them counterintuitive. Our patterns are strikingly similar to those observed in transmission electron microscopy of the improper T-O ferroelastic ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7}.$