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.