Dense Regular Packings of Irregular Nonconvex Particles

We present a new numerical scheme to study systems of nonconvex, irregular, and punctured particles in an efficient manner. We employ this method to analyze regular packings of odd-shaped bodies, both from a nanoparticle and from a computational geometry perspective. Besides determining close-packed structures for 17 irregular shapes, we confirm several conjectures for the packings of a large set of 142 convex polyhedra and extend upon these. We also prove that we have obtained the densest packing for both rhombicuboctahedra and rhombic enneacontrahedra and we have improved upon the packing of enneagons and truncated tetrahedra.

Figure 1

The dimer crystal structure for truncated tetrahedra that achieves a packing fraction ${\varphi }_{\mathrm{LB}}=0.988\dots$. (a) The dimer formed by two truncated tetrahedra (blue and red; dark and light gray, respectively). (b) A piece of the crystal structure this unit cell generates; only 7 periodic images are shown. The viewpoint is such that the dimers in this crystal are pointing out of the paper, with the top triangle [as given in (a)] facing the reader.

Figure 2

Four nonconvex shapes and three of their associated crystal structures. (a) A side and bottom view of the model (3850 triangles) for a colloidal cap. (b) A piece of the double-braided structure formed by these caps. There are four particles in the unit cell, i.e., the caps form a quadrumer, each has been indicated with a different color. (c) The centrosymmetric dimer formed by two Szilassi polyhedra (blue and red; dark and light gray, respectively) that achieves the densest-known packing in relation to its unit cell (gray box). We do not show the crystal this generates, since it is difficult to make out individual particles in it even when they are color coded. (d) A centrosymmetric tetrapod dimer (red, blue) and the associated unit cell (gray box). (e) A piece of the crystal structure formed by this dimer. (f) The densest-known packing for great stellated dodecahedra, again the structure is a dimer lattice as indicated by the red and blue color coding.

Figure 3

A crystal structure obtained for octapod-shaped particles, which have an additional soft interaction term. (a) The model of the octapod used in this simulation. (b) Side view of the candidate crystal structure, which is a simple cubic phase. Note that the octapods touch each other at the tips.

Figure 4

Four representations of convex particles and the densest-known packing for enneagons. A rhombicuboctahedron (a) and rhombic enneaconrahedron (b). For both we have proved that its Bravais lattice packs densest. (c) The snub cube, which is not centrally symmetric, yet it achieves its densest-known packing in a Bravais lattice. (d) An enneagon. (e) The centrosymmetric-dimer (blue, red) lattice that achieves the densest packing for enneagons.