Disordered, Quasicrystalline and Crystalline Phases of Densely Packed Tetrahedra

TL;DR

This study reveals that hard tetrahedra can form dense, complex structures including a quasicrystal with a packing fraction of 0.8324, surpassing previous known packings, through entropy-driven phase transitions.

Contribution

First demonstration of a quasicrystal formed from hard, non-spherical particles, showing shape and entropy induce complex high-density structures.

Findings

Tetrahedra undergo a first-order phase transition to a dodecagonal quasicrystal.

Quasicrystal packing fraction reaches 0.8324, higher than previous packings.

Disordered states can jam at a packing fraction of 0.7858.

Abstract

All hard, convex shapes are conjectured by Ulam to pack more densely than spheres, which have a maximum packing fraction of {\phi} = {\pi}/\sqrt18 ~ 0.7405. For many shapes, simple lattice packings easily surpass this packing fraction. For regular tetrahedra, this conjecture was shown to be true only very recently; an ordered arrangement was obtained via geometric construction with {\phi} = 0.7786, which was subsequently compressed numerically to {\phi} = 0.7820. Here we show that tetrahedra pack much better than this, and in a completely unexpected way. Following a conceptually different approach, using thermodynamic computer simulations that allow the system to evolve naturally towards high-density states, we observe that a fluid of hard tetrahedra undergoes a first-order phase transition to a dodecagonal quasicrystal, which can be compressed to a packing fraction of {\phi} = 0.8324. By compressing a crystalline approximant of the quasicrystal, the highest packing fraction we obtain is {\phi} = 0.8503. If quasicrystal formation is suppressed, the system remains disordered, jams, and compresses to {\phi} = 0.7858. Jamming and crystallization are both preceded by an entropy-driven transition from a simple fluid of independent tetrahedra to a complex fluid characterized by tetrahedra arranged in densely packed local motifs that form a percolating network at the transition. The quasicrystal that we report represents the first example of a quasicrystal formed from hard or non-spherical particles. Our results demonstrate that particle shape and entropy can produce highly complex, ordered structures.