  The critical cooling rate $\mathcal{R}_c$, below which liquids crystallize
upon cooling, characterizes the glass-forming ability (GFA) of the system.
While pure metals are typically poor glass formers with $\mathcal
{R}_c>10^{12}\, {\rm K/s}$, specific multi-component alloys can form bulk
metallic glasses (BMGs) even at cooling rates below $\mathcal {R}\sim 1\, {\rm
K/s}$. Conventional wisdom asserts that metal alloys with three or more
components are better glass formers (with smaller ${\cal R}_c$) than binary
alloys. However, there is currently no theoretical framework that provides
quantitative predictions for $\mathcal{R}_c$ for multi-component alloys. We
perform simulations of ternary hard-sphere systems, which have been shown to be
accurate models for the glass-forming ability of BMGs, to understand the roles
of geometric frustration and demixing in determining $\mathcal {R}_c$.
Specifically, we compress ternary hard sphere mixtures into jammed packings and
measure the critical compression rate, below which the system crystallizes, as
a function of the diameter ratios $\sigma_B/\sigma_A$ and $\sigma_C/\sigma_A$
and number fractions $x_A$, $x_B$, and $x_C$. We find two distinct regimes for
the GFA in parameter space for ternary hard spheres. When the diameter ratios
are close to $1$, such that the largest ($A$) and smallest ($C$) species are
well-mixed, the GFA of ternary systems is no better than that of the optimal
binary glass former. However, when $\sigma_C/\sigma_A \lesssim 0.8$ is below
the demixing threshold for binary systems, adding a third component $B$ with
$\sigma_C < \sigma_B < \sigma_A$ increases the GFA of the system by preventing
demixing of $A$ and $C$. Analysis of the available data from experimental
studies indicates that most ternary BMGs are below the binary demixing
threshold with $\sigma_C/\sigma_A < 0.8$.
