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-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 Z(NV+1) Hamiltonians, with N-OP-component strain-pseudospin vectors (S) over right arrow pointing to Z(NV+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 Z(NV+1)=3 values of S=0, +/- 1, as in a generalized Blume-Capel model. We then consider transitions with two-component (NOP=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.