In this study, we revisit the densest binary sphere packings (DBSPs) under periodic boundary conditions and present an updated phase diagram, including newly found 12 putative DBSPs over the $x-\alpha$ plane, where
$x$ is the relative concentration and $\alpha$ is the radius ratio of the small and large spheres. You can download this paper
For $0.20 \le \alpha \le 0.40$, the exhaustive search is conducted for 11 radius ratios of $0.20$, $0.22$, $0.24$, $0.26$, $0.28$, $0.30$, $0.32$, $0.34$, $0.36$, $0.38$, and $0.40$. In order to confirm that the (12-1) structure appears on the phase diagram, the XY12, XY11, and (22-1) structures are re-optimized at $\alpha = 0.203$. The number of spheres in the unit cell is set between 6 and 24. As a result, we have identified the new six putative DBSPs: the XY12, (22-1), (12-1), (20-1), (8-2), and the (2-1) structures.
For $0.420 \le \alpha \le 0.500$, the exhaustive search is conducted for 17 radius ratios of $0.420$, $0.425$, $0.430$, $0.435$, $0.440$, $0.445$, $0.450$, $0.455$, $0.460$, $0.465$, $0.470$, $0.475$, $0.480$, $0.485$, $0.490$, $0.495$, and $0.500$. In order to confirm that the (9-4) structure appears on the phase diagram, the (2-2), (10-4), AuTe2, and (9-4) structures are reoptimized at $\alpha = 0.481$, $0.482$, and $0.483$. The number of spheres in the unit cell is set between 12 and 32. As a result, we have identified 5 new putative DBSPs: the (14-5), (16-4), (8-4), (10-4) , and (9-4) structures.
For $0.52 \le \alpha \le 0.64$, the exhaustive search is conducted for 7 radius ratios of $0.52$, $0.54$, $0.56$, $0.58$, $0.60$, $0.62$, and $0.64$. The number of spheres in the unit cell is set between 6 and 24. As a result, we have identified one new putative densest packing: the (12-6) structure.
The XYn structure is defined as DBSP in which large spheres X constitute the fcc densest structure and small spheres penetrate into the tetrahedral and octahedral sites constituted by X. If the XY n are excluded, there are 21 putative DBSP on the phase diagram. Most of those structures are named as ($m$-$n$) structure. A ($m$-$n$) structure contains $m$ small spheres and $n$ large spheres. The phase diagram is shown in this page, and you can download the three-dimensional data of the DBSPs.
DBSPs and phase diagram
When the highest packing fraction is achieved by phase separation into densest binary packing R and the fcc densest packing constituting of small spheres, only the symbol of the densest binary packing R is plotted.
On the other hands, when the highest packing fraction is achieved by phase separation into two densest binary packings, those two symbols are plotted together.
The three-dimensional data of the DBSP for the phase diagram are downloadable