Densest ternary sphere packings

Abstract

We present our exhaustive exploration of the densest ternary sphere packings (DTSPs) for 45 radius ratios and 237 kinds of compositions, which is a packing problem of three kinds of hard spheres with different radii, under periodic boundary conditions by a random structure searching method. To efficiently explore DTSPs we further develop the searching method based on the pilingup and iterative balance methods [Koshoji et al., Phys. Rev. E 103, 023307 (2021)]. The unbiased exploration identifies diverse 38 putative DTSPs appearing on phase diagrams in which 37 DTSPs of them are discovered in the study. The structural trend of DTSPs changes depending especially on the radius of small spheres. In case that the radius of small spheres is relatively small, structures of many DTSPs can be understood as derivatives of densest binary sphere packings (DBSPs), while characteristic structures specific to the ternary system emerge as the radius of small spheres becomes larger. In addition to DTSPs, we reveal a lot of semi-DTSPs (SDTSPs) which are obtained by excluding DBSPs in the calculation of phase diagram, and investigate the correspondence of DTSPs and SDTSPs with real crystals based on the space group, showing a considerable correspondence of SDTSPs having high symmetries with real crystals including Cu2GaSr and ThCr2Si2 structures. Our study suggests that the diverse structures of DBSPs, DTSPs, and SDTSPs can be effectively used as structural prototypes for searching complex crystal structures.

Diverse densest ternary sphere packings

“…While some studies had discussed the dense ternary sphere packings [11][12][13][14][15], the putative densest ternary sphere packings (DTSPs) had not been predicted until recent years; the densest phase diagrams for the ternary systems had not been constructed. However, we successfully found the 37 putative DTSPs with construction of phase diagrams in 2021 [16]. The result showed that when the radius ratios of small and medium spheres are small, the DTSPs tend to be complex because the spheres comprise clusters in many cases and the clusters are placed in relatively small voids consisting of large spheres.…”

“…For example, the crystal of the LaH 10 , which is one of the superhydrides synthesized under high pressure, corresponds to the XY 10 structure that is one of the DBSPs. On the other hand, the study [16] indicated that the correspondence of the DTSPs with crystals seems to be exceptional. In fact, for most of DTSPs, it is difficult to understand the structural property from the perspectives of chemical bonds and/or local polyhedrons which are typically found in coordination structures to transition elements.…”

“…The discovery of 37 putative DTSPs has been realized by an algorithm we have developed [10,16]. The algorithm is based on two methods: The piling-up method and the iterative balance method [10,16].…”

“…The discovery of 37 putative DTSPs has been realized by an algorithm we have developed [10,16]. The algorithm is based on two methods: The piling-up method and the iterative balance method [10,16]. The pilingup method randomly generates multilayered initial struc-tures, which can be smoothly optimized to packing structures, by randomly choosing the number of spheres on each layer while avoiding a large overlap between spheres.…”

“…The iterative balance method optimizes initial structures to packing structures by minimizing the volume of unit cells and the overlaps between spheres simultaneously by the steepest descent method under pressure with the hard sphere potential. The combination of the two methods enables us to explore a vast configuration space and to accurately determine the packing fraction, resulting in the discovery of the 12 putative DBSPs [10] and 37 putative DTSPs [16]. The success indicates that this methodology can be applied to explore unknown DSPs for multi-sized sphere and/or higher dimensional packings.…”