Statics and dynamics of domain patterns in hexagonal-orthorhombic ferroelastics

Abstract

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.

Ferroelastic dynamics and strain compatibility

“…Figure 1 shows the T R, the square to rectangular or SR, and other lattice transitions. (While it is true 16,17 that these correspond to 2D projections/analogs of hexagonal to orthorhombic, and tetragonal to orthorhombic lattice transitions, respectively, we will reserve this 3D terminology for full 3D analyses, elsewhere.) The symmetry-adapted non-OP compressional strain e 1 = 1 2 (∆ x u x + ∆ y u y ), whereas the OP are the 'deviatoric' ε 2 = 1 2 (∆ x u x − ∆ y u y ) and shear strain ε 3 = 1 2 (∆ x u y + ∆ y u x ).…”

“…Section IV discusses the equivalent inhomogeneous oscillator description of the BG dynamics. Section V deals with its T DGL truncation, also derived in Appendix C from truncated displacement dynamics 16,17 . Section VI presents the compatibility kernels for other 2D symmetries.…”

“…Yet other models considered displacement accelerations equated to displacement-gradients of the free energy as forces; or strain T DGL equations coupled to vacancy field dynamics 15 . Finally the Lagrange-Rayleigh procedure 5,6,26 has recently been applied 16,17 in 2D, yielding an underdamped dynamics for the displacement, that is truncated to overdamped equations, that are seemingly different from the T DGL form.…”

“…Dynamics plays a central role in proper ferroelastic transitions 2,5,6,7,8,9,10,11,12,13,14,15,16,17 . As noted, these materials undergo diffusionless, displacive transitions, with strain (components) as the order parameter, and develop complex microstructures in their dynamical evolution, finally forming spatially varying, multiscale 'textures' or strain patterns.…”

“…Textured improper ferroelastics include materials of technological importance such as superconducting cuprates 18 and colossal magnetoresistance (CMR) manganites 19 . Many dynamical models have been invoked to follow aspects of (proper) ferroelastic pattern formation 5,6,7,8,9,10,11,12,13,14,15,16,17 such as nucleated twin-front propagation; width-length scaling of twin dimensions 20,21 ; tweed 22,23,24 ; stress effects; elastic domain misfits; and acoustic noise generation 25 .…”