Generation and performance of special quasirandom structures for studying the elastic properties of random alloys: Application to Al-Ti

The performance of special-quasirandom structures (SQSs) for the description of elastic properties of random alloys was evaluated. A set of system-independent 32-atom-fcc SQS spanning the entire concentration range was generated and used to determine $C_{11}$, $C_{12}$, and $C_{44}$ of binary random substitutional AlTi alloys. The elastic properties of these alloys could be described using the set of SQS with an accuracy comparable to the accuracy achievable by statistical sampling of the configurational space of $3\times 3\times 3$ (108 atom, $C_{44}$) and $4\times 4\times 4$ (256 atom, $C_{11}$ and $C_{12}$) fcc supercells, irrespective of the impurity concentration. The smaller system size makes the proposed SQS ideal candidates for the ab initio determination of the elastic constants of random substitutional alloys. The set of optimized SQS is provided.

Figure 1

(Color online) Generation of the SQS for the ${\text{Al}}_{50}{\text{Ti}}_{50}$ random alloy. The configurational space is sampled by more than $3\times {10}^5$ randomly chosen configurations, which ensures that the quality of the five most favorable SQS candidates (indicated by the red arrow) as measured by Eq. (1) does not vary by more than 5%. The complete configurational space for a given alloy composition spans more than $6\times {10}^8$ structures.

Figure 2

(Color online) (a) $\overline{C_{11}}$, (b) $\overline{C_{12}}$, and (c) $\overline{C_{44}}$ as a function of the Ti fraction within the AlTi solid solution and the dimensions of the supercells used to determine the elastic constants within the standard supercell approach. The corresponding elastic constants determined using the SQS are indicated by triangles. The continuous gray line serves as guide to the eyes only.

Figure 3

(Color online) Average absolute deviation, $\Delta$, of the calculated elastic constants with respect to $\stackrel{̿}{C_{ij}}$, determined using the standard supercell approach (lines) and the SQS (triangles). Continuous lines (filled triangles) correspond to the absolute deviation of the averaged elastic constants $\left(\overline{C_{ij}}\right)$, while dashed lines (open triangles) depict the absolute deviation of the individual elastic constants $\left(C_{ij}\right)$.

Figure 4

(Color online) Effect of randomness on the elastic constants of the AlTi system. The matching of the previously determined elastic constants (Ref. 19) of the intermetallic $\gamma \text{-TiAl}$ and the $\text{D}{0}_{19}{\text{-Ti}}_3\text{Al}$ phases to the $\overline{C_{ij}}$ of the SQS proposed in this study, is described in the text. The dashed line indicates equality between the SQS-derived and intermetallic elastic constants.