Scaled free energies, power-law potentials, strain pseudospins, and quasiuniversality for first-order structural transitions

We consider ferroelastic first-order phase transitions with N_OP order-parameter strains entering Landau free energies as invariant polynomials that have N_V structural-variant Landau minima. The total free energy includes (seemingly innocuous) harmonic terms, in the n=6- N_OP nonorder-parameter strains. Four three-dimensional (3D) transitions are considered, tetragonal/orthorhombic, cubic/tetragonal, cubic/trigonal, and cubic/orthorhombic unit-cell distortions, with, respectively, N_OP =1, 2, 3, and 2; and N_V =2, 3, 4, and 6. Five two-dimensional (2D) transitions are also considered, as simpler examples. Following Barsch and Krumhansl, we scale the free energy to absorb most material-dependent elastic coefficients into an overall prefactor, by scaling in an overall elastic energy density; a dimensionless temperature variable; and the spontaneous-strain magnitude at transition λ 1. To leading order in λ the scaled Landau minima become material independent, in a kind of "quasiuniversality. " The scaled minima in N_OP -dimensional order-parameter space, fall at the center and at the N_V corners, of a transition-specific polyhedron inscribed in a sphere, whose radius is unity at transition. The "polyhedra" for the four 3D transitions are, respectively, a line, a triangle, a tetrahedron, and a hexagon. We minimize the n terms harmonic in the nonorder-parameter strains, by substituting solutions of the "no dislocation" St Venant compatibility constraints, and explicitly obtain power-law anisotropic, order-parameter interactions, for all transitions. In a reduced discrete-variable description, the competing minima of the Landau free energies induce unit-magnitude pseudospin vectors, with N_V +1 values, pointing to the polyhedra corners and the (zero-value) center. The total scaled free energies then become Z_NV+1 clocklike pseudospin Hamiltonians, with temperature-dependent local Landau terms, nearest-neighbor Ginzburg couplings, and power-law St Venant interactions that drive the elastic domain-wall texturing. The scaled free energies can be used in relaxational or underdamped dynamic simulations to study ferroelastic strain textures and their dynamical evolution pathways. The pseudospin models can similarly be studied via local meanfield treatments and Monte Carlo simulations.