Densest binary sphere packings and phase diagram : revisited

Abstract

We revisit the densest binary sphere packings (DBSP) under the periodic boundary conditions and present an updated phase diagram, including newly found 12 putative densest structures over the $x - \alpha$ plane, where $x$ is the relative concentration and $\alpha$ is the radius ratio of the small and large spheres. To efficiently explore the DBSP, we develop an unbiased random search approach based on both the piling up method to generate initial structures in an unbiased way and the iterative balance method to optimize the volume of a unit cell while keeping the overlap of hard spheres minimized. With those two methods, we have discovered 12 putative DBSP and thereby the phase diagram is updated, while our results are consistent with those of the previous study [Hopkins et al., Phys. Rev. E 85, 021130 (2012)] with a small correction for the case of 12 or fewer spheres in the unit cell. The 5 of the new 12 densest packings are discovered in the small radius range of $0.42 \le \alpha \le 0.50$ where several structures are competitive to each other with respect to packing fraction. Through the exhaustive search, diverse dense packings are discovered and accordingly we find that packing structures achieve high packing fractions by introducing distortion and/or combining a few local dense structural units. Furthermore, we investigate the correspondence of the DBSP with crystals based on the space group. The result shows that many structural units in real crystals, e.g., $\mathrm{LaH_{10}}$ and $\mathrm{SrGe_{2-\delta}}$ being high-pressure phases, can be understood as DBSP. The correspondence implies that the densest sphere packings can be used effectively as structural prototypes for searching complex crystal structures, especially for high-pressure phases.