Microstructure from ferroelastic transitions using strain pseudospin clock models in two and three dimensions: A local mean-field analysis

We show how microstructure can arise in first-order ferroelastic structural transitions, in two and three spatial dimensions, through a local mean-field approximation of their pseudospin Hamiltonians, that include anisotropic elastic interactions. Such transitions have symmetry-selected physical strains as their ${N}_{\text{OP}}$-component order parameters, with Landau free energies that have a single zero-strain ``austenite'' minimum at high temperatures, and spontaneous-strain ``martensite'' minima of ${N}_{V}$ structural variants at low temperatures. The total free energy also has gradient terms, and power-law anisotropic effective interactions, induced by ``no-dislocation'' St Venant compatibility constraints. In a reduced description, the strains at Landau minima induce temperature dependent, clocklike ${\mathbb{Z}}_{{N}_{V}+1}$ Hamiltonians, with ${N}_{\text{OP}}$-component strain-pseudospin vectors $\stackrel{P\vec}{S}$ pointing to ${N}_{V}+1$ discrete values (including zero). We study elastic texturing in five such first-order structural transitions through a local mean-field approximation of their pseudospin Hamiltonians, that include the power-law interactions. As a prototype, we consider the two-variant square/rectangle transition, with a one-component pseudospin taking ${N}_{V}+1=3$ values of $S=0,\ifmmode\pm\else\textpm\fi{}1$, as in a generalized Blume-Capel model. We then consider transitions with two-component $({N}_{\text{OP}}=2)$ pseudospins: the equilateral to centered rectangle $({N}_{V}=3)$; the square to oblique polygon $({N}_{V}=4)$; the triangle to oblique $({N}_{V}=6)$ transitions; and finally the three-dimensional (3D) cubic to tetragonal transition $({N}_{V}=3)$. The local mean-field solutions in two-dimensional and 3D yield oriented domain-wall patterns as from continuous-variable strain dynamics, showing the discrete-variable models capture the essential ferroelastic texturings. Other related Hamiltonians illustrate that structural transitions in materials science can be the source of interesting spin models in statistical mechanics.