We perform molecular dynamics simulations to compress binary hard spheres into jammed packings as a function of the compression rate $R$, size ratio $alpha$, and number fraction $x_S$ of small particles to determine the connection between the glass-forming ability (GFA) and packing efficiency in bulk metallic glasses (BMGs). We define the GFA by measuring the critical compression rate $R_c$, below which jammed hard-sphere packings begin to form random crystal structures with defects. We find that for systems with $alpha gtrsim 0.8$ that do not de-mix, $R_c$ decreases strongly with $Delta phi_J$, as $R_c sim exp(-1/Delta phi_J^2)$, where $Delta phi_J$ is the difference between the average packing fraction of the amorphous packings and random crystal structures at $R_c$. Systems with $alpha lesssim 0.8$ partially de-mix, which promotes crystallization, but we still find a strong correlation between $R_c$ and $Delta phi_J$. We show that known metal-metal BMGs occur in the regions of the $alpha$ and $x_S$ parameter space with the lowest values of $R_c$ for binary hard spheres. Our results emphasize that maximizing GFA in binary systems involves two competing effects: minimizing $alpha$ to increase packing efficiency, while maximizing $alpha$ to prevent de-mixing.
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