Partitioning of solutes in multiphase $\mathrm{Ti}–\mathrm{Al}$ alloys

First-principles calculations based on a plane-wave pseudopotential method, as implemented in the VASP code, are presented for the formation energies of several transition-metal and non-transition-metal dopants in $\mathrm{Ti}–\mathrm{Al}$ alloys. Substitution for either Ti or Al in $\gamma \text{‐}\mathrm{Ti}\mathrm{Al}$, ${\alpha }_2\text{‐}{\mathrm{Ti}}_3\mathrm{Al}$, ${\mathrm{Ti}}_2\mathrm{Al}\mathrm{C}$, and ${\mathrm{Ti}}_3\mathrm{Al}\mathrm{C}$ are considered. Calculated (zero-temperature) defect formation energies exhibit clear trends as a function of the periodic-table column of transition metal solutes. Early transition metals in TiAl prefer the Ti sublattice, but this preference gradually shifts to the Al sublattice for late transition metals; the Ti sublattice is preferred by all transition metal solutes in ${\mathrm{Ti}}_3\mathrm{Al}$. Partitioning of solutes to ${\mathrm{Ti}}_3\mathrm{Al}$ is predicted for mid-period transition elements, and to TiAl for early and late transition elements. A simple Ising model treatment demonstrates the plausibility of these trends, which are in excellent overall agreement with experiment. The influence of temperature on formation energies is examined with a cluster expansion for the binary TiAl alloys and a low temperature expansion for dilute ternary alloys. Results for Nb-doped alloys provide insight into the relative sensitivity of solute partitioning to individual contributions to the free energy. Whereas the calculated formation energy of Nb (substitution) at zero temperature favors partitioning to ${\alpha }_2\text{‐}{\mathrm{Ti}}_3\mathrm{Al}$, temperature-dependent contributions to the formation free energy, evaluated at $1075\mathrm{K}$, favor partitioning to $\gamma \text{‐}\mathrm{Ti}\mathrm{Al}$, in agreement with experiment.

Figure 1

Difference between the formation energies for solute substitution on the Al sublattice and on the Ti sublattice of TiAl (filled circles) and ${\mathrm{Ti}}_3\mathrm{Al}$ (filled diamonds). Results are shown for selected solutes in the transition metal series IIIB–VIIIB. The nontransition elements Si and B (not represented in the plot) strongly favor the Al sublattice in both phases. The abscissa value is shifted slightly to enable members of the same group to be distinguished, e.g., Cr: 5.9, Mo: 6.0, W: 6.1. The trends in the data indicate that the Al sublattice in TiAl is increasingly preferred for the later transition groups. The crossover between Ti preference and Al preference occurs roughly between groups VB and VIB. Transition metal solutes in ${\mathrm{Ti}}_3\mathrm{Al}$ all prefer the Ti sublattice.

Figure 2

Difference between the formation energies for solute substitution in ${\mathrm{Ti}}_3\mathrm{Al}$ and TiAl for selected solutes in the transition metal series IIIB–VIIIB. The formation energy that corresponds to the preferred sublattice (cf. Fig. 1) for each phase is employed. Apart from a few early (Y) and late (Mn,Ru) transition metals, a distinct preference for the ${\mathrm{Ti}}_3\mathrm{Al}$ phase is observed.

Figure 3

The difference between the formation energies for solute substitution in the preferred carbide (${\mathrm{Ti}}_2\mathrm{Al}\mathrm{C}$ or ${\mathrm{Ti}}_3\mathrm{Al}\mathrm{C}$) and in the aluminide host (either ${\mathrm{Ti}}_3\mathrm{Al}$ or TiAl, whichever has the lower energy).

Figure 4

The predictions of solute formation energies based on simplified Ising model, Eq. (12). Panel (a) is to be compared with Fig. 1 and panel (b) with Fig. 2.

Figure 5

The predicted (open circles) and measured (filled circles) partitioning of transition-metal solutes in $\gamma \text{‐}\mathrm{Ti}\mathrm{Al}$ with composition ${\left({\mathrm{Ti}}_{51}{\mathrm{Al}}_{49}\right)}_{99}{M}_1$ at $1473\mathrm{K}$. The ordinate represents the fraction of solutes that reside on the Ti sublattice.

Figure 6

Predicted (filled circles) and measured (diamonds) ratio of solute concentration in the $\gamma$ phase to that in the ${\alpha }_2$ phase.