  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.
