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In steels and single-crystal superalloys the control of the formation oftopologically close-packed (TCP) phases is critical for the performance of thematerial. The structural stability of TCP phases in multi-componenttransition-metal alloys may be rationalised in terms of the averagevalence-electron count $\bar{N}$ and the composition-dependent relativevolume-difference $\overline{\Delta V/V}$. We elucidate the interplay of thesefactors by comparing density-functional theory calculations to an empiricalstructure map based on experimental data. In particular, we calculate the heatof formation for the TCP phases A15, C14, C15, C36, $\chi$, $\mu$, and $\sigma$for all possible binary occupations of the Wyckoff positions. We discuss theisovalent systems V/Nb-Ta to highlight the role of atomic-size difference andobserve the expected stabilisation of C14/C15/C36/$\mu$ by $\overline{\DeltaV/V}$ at $\Delta N=0$ in V-Ta. In the systems V/Nb-Re, we focus on thewell-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 stabiliseC14/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-Cothe large size-mismatch stabilises C14/C15/C36/$\mu$. We also include V/Nb-Cothat cover the entire valence range of TCP stability and also show thestabilisation of C14/C15/C36/$\mu$. Moreover, the combination of a large volumedifference with a large mismatch in valence-electron count reduces thestability of the A15/$\sigma$/$\chi$ phases in Nb-Co as compared to V-Co. Bycomparison to non-magnetic calculations we also find that magnetism is of minorimportance for the structural stability of TCP phases in Cr/Mo-Co and inV/Nb-Co.

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