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 meanfield 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. In a reduced description, the strains at Landau minima induce temperature-dependent, clock-like $\mathbb{Z}_{N_V +1}$ hamiltonians, with $N_{OP}$-component strain-pseudospin vectors ${\vec S}$ pointing to $N_V + 1$ discrete values (including zero). We study elastic texturing in five such first-order structural transitions through a local meanfield approximation of their pseudospin hamiltonians, that include the powerlaw interactions. As a prototype, we consider the two-variant square/rectangle transition, with a one-component, pseudospin taking $N_V +1 =3$ values of $S= 0, \pm 1$, as in a generalized Blume-Capel model. We then consider transitions with two-component ($N_{OP} = 2$) pseudospins: the equilateral to centred-rectangle ($N_V =3$); the square to oblique polygon ($N_V =4$); the triangle to oblique ($N_V =6$) transitions; and finally the 3D cubic to tetragonal transition ($N_V =3$). The local meanfield solutions in 2D and 3D yield oriented domain-walls 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.