Topologically close-packed phases in binary transition-metal compounds: matching high-throughput ab initio calculations to an empirical structure map

In steels and single-crystal superalloys the control of the formation of topologically close-packed (TCP) phases is critical for the performance of the material. The structural stability of TCP phases in multi-component transition-metal alloys may be rationalised in terms of the average valence-electron count $\bar{N}$ and the composition-dependent relative volume-difference $\overline{\Delta V/V}$. We elucidate the interplay of these factors by comparing density-functional theory calculations to an empirical structure map based on experimental data. In particular, we calculate the heat of formation for the TCP phases A15, C14, C15, C36, $\chi$, $\mu$, and $\sigma$ for all possible binary occupations of the Wyckoff positions. We discuss the isovalent systems V/Nb-Ta to highlight the role of atomic-size difference and observe the expected stabilisation of C14/C15/C36/$\mu$ by $\overline{\Delta V/V}$ at $\Delta N=0$ in V-Ta. In the systems V/Nb-Re, we focus on the well-known trend of A15$- \sigma - \chi$ stability with increasing $\bar{N}$ and show that the influence of $\overline{\Delta V/V}$ is too weak to stabilise C14/C15/C36/$\mu$ in Nb-Re. As an example for a significant influence of both $\bar{N}$ and $\overline{\Delta V/V}$, we also consider the systems Cr/Mo-Co. Here the sequence A15$- \sigma - \chi$ is observed in both systems but in Mo-Co the large size-mismatch stabilises C14/C15/C36/$\mu$. We also include V/Nb-Co that cover the entire valence range of TCP stability and also show the stabilisation of C14/C15/C36/$\mu$. Moreover, the combination of a large volume difference with a large mismatch in valence-electron count reduces the stability of the A15/$\sigma$/$\chi$ phases in Nb-Co as compared to V-Co. By comparison to non-magnetic calculations we also find that magnetism is of minor importance for the structural stability of TCP phases in Cr/Mo-Co and in V/Nb-Co.