Strain pseudospins with power-law interactions: Glassy textures of a cooled coupled-map lattice

We consider a spin-1 model of strain pseudospins $S(\stackrel{P\vec}{r})=0,\ifmmode\pm\else\textpm\fi{}1$ that arise from a triple-well Landau free energy for a square/rectangle or ``austenite-martensite'' structural transformation of a two-dimensional lattice. The pseudospin model has elastic-compatibility-induced power-law anisotropic (PLA) interactions and no quenched disorder. The iteratively solved local mean-field equations for $⟨S(\stackrel{P\vec}{r},t)⟩$ form a temperature-dependent PLA-coupled nonlinear-map lattice, where $t$ is the iteration ``time.'' On cooling at a constant rate, the excess entropy shows a weak roll-off near a temperature $T={T}_{g}$ and a sharper elbow at a lower ${T}^{\ensuremath{\ast}}$, just above a Kauzmann-type ${T}_{K}$ where the excess entropy would have become negative. The crossover temperatures ${T}_{g},{T}^{\ensuremath{\ast}}$ decrease logarithmically with cooling rate and mark stability changes in spatiotemporal attractors of the cooled PLA-coupled map. Three phases in $⟨S(\stackrel{P\vec}{r},t)⟩$ are found, with textures of the martensitic-variant domain walls as ``inherent structures.'' There is a high-temperature $(T>{T}_{g})$ fine scale phase of feathery domain walls and an intermediate temperature $({T}_{g}>T>{T}^{\ensuremath{\ast}})$ phase of mazelike domain walls, with both showing square-wave oscillations as predominantly period-two attractors but with minority-frequency subharmonic clusters. Finally, there is a low-temperature freezing $({T}^{\ensuremath{\ast}}>T)$ to a static fixed point or period-one attractor of coarse, irregular bidiagonal twins, as in a strain glass. A Haar-wavelet analysis is used to identify the local attractor dynamics. A central result is that dynamically heterogeneous and mobile low-strain droplets act as catalysts, and can form correlated chains or transient ``catalytic corrals'' to incubate an emerging local texture. The hotspot lifetime vanishes linearly in $T\ensuremath{-}{T}_{K}$, suggesting that ${T}_{K}$ is a dynamic spinodal limit for generating the ``austenitic'' catalyst, the disappearance of which drives a trapping into one of many bidiagonal glassy states. The model has relevance to martensitic or complex-oxide textures, coupled-map lattices, and configurational-glass transitions.

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