Evolution of the dense packings of spherotetrahedral particles: from ideal tetrahedra to spheres

Abstract

Particle shape plays a crucial role in determining packing characteristics. Real particles in nature usually have rounded corners. In this work, we systematically investigate the rounded corner effect on the dense packings of spherotetrahedral particles. The evolution of dense packing structure as the particle shape continuously deforms from a regular tetrahedron to a sphere is investigated, starting both from the regular tetrahedron and the sphere packings. The dimer crystal and the quasicrystal approximant are used as initial configurations, as well as the two densest sphere packing structures. We characterize the evolution of spherotetrahedron packings from the ideal tetrahedron (s = 0) to the sphere (s = 1) via a single roundness parameter s. The evolution can be partitioned into seven regions according to the shape variation of the packing unit cell. Interestingly, a peak of the packing density Φ is first observed at s ≈ 0.16 in the Φ-s curves where the tetrahedra have small rounded corners. The maximum density of the deformed quasicrystal approximant family (Φ ≈ 0.8763) is slightly larger than that of the deformed dimer crystal family (Φ ≈ 0.8704), and both of them exceed the densest known packing of ideal tetrahedra (Φ ≈ 0.8563).

Figure 1

(a) The definition of a spherotetrahedron in two-dimensional schematic diagram. (b) Numerical models of four spherotetrahedra with the roundness ratio s of (1) 0.16, (2) 0.5, (3) 0.84 and (4) 0.98. (c) The definition of the local parameter δ_c in two-dimensional schematic diagram. If two particles i and j have overlaps when they regrow shape corners, particle j is translated along O_iO_j until no overlap exists between them regardless of other overlapping pairs of particles. The translation vector is O_jO_j′.

Figure 2

(a) The packing density Φ for the dense packing of spherotetrahedra as a function of s. (b) The length ||a_i|| (i = 1, 2, 3) of the three lattice vectors a_i. (c) The cosine of the angles θ_ij (i < j = 1, 2, 3) between the three a_i. Grey vertical dash lines partition the s domain into seven regions.

Figure 3. Illustration of three dense evolution families with periodic boundaries.

(a) Deformed DC at s = 0.26 with four particles participating in two inverted pairs per unit cell. (b) Deformed FCC at s = 0.78 with one particle located on a vertex of the unit cell and the other three sited on the centers of three cell surfaces approximately. (c) Deformed HCP at s = 0.88 with two particles per unit cell forming analogue of hexagonal system.

Figure 4

(a) The relationship between the local parameter δ_c and N/V_Cell for the def DC, def FCC, and def HCP families. The seven regions partitioned by the grey vertical dash lines are consistent with that in Fig. 2. (b) The packing density Φ for the dense packing of spherotetrahedra of the def DC and the def QA families as a function of s.

Figure 5

(a) The relationship between the local parameter δ_c and the global parameter N/V_Cell for the def DC and def QA families at small roundness ratios. (b) The relationship between δ_c and the roundness ratio s for the def DC and def QA families.