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.

Figure 1

Schematic picture showing the function mapping an arbitrary configuration ($\vec{\sigma }$) to a value of the order parameters ($\vec{\eta }$).

Figure 2

Conventional cell of bcc showing two sublattices.

Figure 3

Schematic of the space spanned by the sublattice compositions of Fig. 2. The figure shows the four orderings commensurate with this supercell, as well as the reorientation of the space defined by Eqs. (2) and (3).

Figure 4

Sublattices in the conventional fcc crystal structure, labeled 1–4.

Figure 5

Crystal structure of the $L{1}_2$ ($A_{3}B$) ordering on the fcc crystal structure. This ordering has four symmetrically equivalent translational variants.

Figure 6

Schematic of the domain of ${\eta }_1,{\eta }_2,$ and ${\eta }_3$ values spanned by real configurations at a composition of ${\eta }_0$ = 0.25. Perfect $L{1}_2$-type orderings are formed at the corners of the tetrahedron. The disordered phase with the same composition is mapped onto the origin of this domain.

Figure 7

Variants of the $(2\times 1)$ ordering on the triangular lattice. The primitive cell required to describe each ordering is highlighted in blue, while the decoration of each variant in the mutually commensurate cell is highlighted in orange.

Figure 8

Reference cell that can accommodate all the translational and orientational variants of the $2\times 1$ ordering on the triangular lattice. The cell contains four sublattices that are labeled as shown in the figure.

Figure 9

Schematic of the ${\eta }_1,{\eta }_2,{\eta }_3$ domain for the triangular lattice at a composition (${\eta }_0$) of 0.5. Each row ordering is formed at the corners of the octahedron. The disordered phase is mapped onto the origin.

Figure 10

Phase diagram and free energies for a model Hamiltonian on the triangular lattice with $\frac{J_{NN}}{J_{NNN}}=10$. The temperature and free energies are scaled based on the nearest-neighbor interaction. Free energies are mirrored across ${\eta }_1=0$ due to the symmetry of the domain.

Figure 11

Ni-rich part of the phase diagram for the Ni-Al binary alloy calculated from Monte Carlo simulations on a cluster expansion Hamiltonian parameterized from first principles [2].

Figure 12

Free-energy surfaces and sections calculated from biased Monte Carlo sampling on a cluster expansion Hamiltonian parameterized from first-principles calculations. The ensemble averages were calculated at 600 K. (a) Free energy as a function of aluminum composition and the lower-dimensional order $\mathrm{parameter}(\xi$) calculated from Monte Carlo at 600 K. The minimum free-energy path of Eq. (36) is outlined as a black dashed line. The points on the common tangent across composition are shown in gray. The extreme sections corresponding to the disordered (purple) and the ordered (orange) phases are also highlighted. (b) Free energy as a function of composition, minimized along the order parameter as defined by Eq. (36). The integrated data from Monte Carlo calculations is shown in light gray, with the green spline drawn on top as a guide to the eye. The common tangent is shown as the dashed gray line. (c) Sections of the free-energy surface for a disordered (purple) and ordered (orange) phase. The integrated free energies from Monte Carlo are shown as the lighter colored points, with the spline fits drawn as a guide to the eye.