Effect of disorder on the dilute equilibrium vacancy concentrations of multicomponent crystalline solids

We develop statistical mechanical methods to predict the thermodynamic properties of dilute vacancies in multicomponent solids from first principles. The approach relies on a coarse-graining procedure to predict dilute vacancy concentrations with Monte Carlo simulations in alloys exhibiting varying degrees of short- and long-range order. We apply this approach to a study of vacancies in hcp based Ti-Al binary alloys and find a strong dependence of the equilibrium vacancy concentration on the Al concentration and the degree of long-range order, especially at low temperature.

Figure 1

Ternary free-energy diagram with a schematic of the chemical potentials and the zero vacancy chemical potential.

Figure 2

Crystal structure schematics of the ${\text{DO}}_{19} \left({\alpha }_2\right)$ ordered phase, including a three-dimensional representation and a projection view down the $c$ axis.

Figure 3

DFT (blue diamonds) and cluster expansion predicted (pink circles) formation energies for the Ti-Al binary system as a function of Al concentration. Blue lines denote the convex hull and correspond to two-phase regions.

Figure 4

Convergence test data for the vacancy formation energy in pure hcp Ti as a function of supercell size and shape.

Figure 5

Comparison of (a) Al-Va and (b) Al-Al pair cluster relative energies as calculated with DFT (blue diamonds) and predicted with the cluster expansion (pink circles).

Figure 6

Calculated temperature-concentration phase diagram for the Ti-Al binary system. Triangles represent points along the predicted phase boundary. Blue (cooling) and red (heating) lines are lines of constant chemical potential.

Figure 7

Gibbs free energy curves for the hcp based Ti-Al binary at different temperatures: (a) 600 K (blue squares) and (b) 1600 K (red diamonds). The free energies are calculated using pure hcp Ti and fcc Al as references.

Figure 8

The dependence of the exchange chemical potentials ${\tilde{\mu }}_{\text{Al}}$ and ${\tilde{\mu }}_{\text{Va}}$ on each other and on the Al concentration when ${\mu }_{\text{Va}}=0$, calculated using approximations that are valid when $x_{\text{Va}}$ is very small. The data are shown at three different temperatures: 600 K (blue squares), 1100 K (purple circles), and 1600 K (red diamonds).

Figure 9

Comparison of vacancy concentration as a function of Al concentration obtained by the coarse-grained Monte Carlo method (red diamonds) and the full ternary Monte Carlo simulations (black squares) at 1600 K.

Figure 10

Vacancy concentration as a function of aluminum concentration at different temperatures: 600 K (blue squares), 1100 K (purple circles), and 1600 K (red diamonds).

Figure 11

Short-range order around a vacancy and Al in the hcp Ti-Al system. Images (a) and (b) illustrate the collection of nearest neighbors 1–8 around a vacancy and aluminum, respectively, in hcp. The size of the balls correspond to the strength of the Va-Al and Al-Al interaction (absolute value) in pure titanium. In (c) and (d), the corresponding Al shell concentration around a vacancy and aluminum, respectively, is plotted as a function of the average Al concentration as calculated with the full ternary Monte Carlo simulations at 1600 K. The colors in all images consistently correspond to a specific nearest-neighbor shell, as detailed in the plot legends (e.g., orange is always the fourth nearest neighbor).