Diverse densest ternary sphere packings

Abstract

The exploration of the densest spheres packings is a fundamental problem in mathematics and a wide variety of sciences including materials science. We present our exhaustive computational exploration of the densest ternary sphere packings (DTSPs) for 451 radius ratios and 436 compositions on top of our previous study [Koshoji and Ozaki, Phys. Rev. E 104, 024101 (2021)]. The unbiased exploration by a random structure searching method discovers diverse 22 putative DTSPs, and thereby 60 putative DTSPs are identified in total including the 38 DTSPs discussed by the previous study. Some of the discovered DTSPs are well-ordered, for example, the medium spheres in the (9-7-3) structure are placed in a straight line with comprising the unit cell, and the DTSP has the Pm-3m symmetry if the structural distortion is corrected. At a considerable number of radius ratios, the highest packing fractions are achieved by the phase separations consisting of only the FCC and/or the putative densest binary sphere packings (DBSPs) for all compositions. The trend is becoming more evident as the small and medium spheres are getting larger, which suggests either the binary packings are actually the densest packings or that the dense ternary packings have unit cells larger than those in this study. On the other hand, the number of the DTSPs increase as the particle size of small spheres gets small. The structural diversity indicates that many unknown DTSPs may hide in a very narrow range of radius ratio where the size of small spheres is smaller due to the competition with respect to the packing fractions of many structural candidates. Finally, we discuss the correspondence of the DTSPs with real crystals based on the space group. Our study suggests that the diverse structures of DTSPs can be effectively used as structural prototypes for searching multinary crystal structures.

Mathematical Crystal Chemistry

“…However, the structure converges to an optimal solution by gradually reducing the maximum ∆x max and ∆A max , even though there is the discontinuity. The effectiveness of linear relaxations of inequality constraints has already been shown in the previous studies on the densest sphere packings as iterative-balance method [34][35][36]: The method enables the structures to reach a local optima precisely enough to calculate packing fractions. The inequality constraints are widely approximated by the logarithmic barrier functions [37], but the advantage of the linear potentials consists of the two folds: One is that the computational cost is the lowest and the other is that the potentials can impose hard penalties only when the constraints are not satisfied.…”