Symmetry-adapted order parameters and free energies for solids undergoing order-disorder phase transitions

Accurate thermodynamic descriptions are a key ingredient to kinetic theories that describe the mesoscale evolution of a solid undergoing ordering or decomposition reactions. We introduce a general approach to identify order parameters for order-disorder reactions and to calculate first-principles free-energy surfaces as a function of these order parameters. The symmetry of the disordered phase is used to formulate order parameters as linear combinations of sublattice compositions of a reference supercell. The order parameters can distinguish the disordered phase from the symmetrically equivalent variants of a particular ordered phase. A thermodynamic formalism is then developed to rigorously define a coarse-grained free energy as a function of order parameters. Bias potentials are added to the potential energy to enable sampling of the unstable regions within the order parameter domain. Monte Carlo sampling in the biased ensemble is combined with free-energy integration to calculate high-temperature free energies. We illustrate the approach by analyzing the free energies of order-disorder transitions on a two-dimensional triangular lattice and in the technologically important Ni-Al alloy.