Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra

A surprise package One of the simplest shapes for which the densest packing arrangement remains unresolved is the regular tetrahedron — despite much theoretical, computational and experimental effort. Using a novel approach involving thermodynamic computer simulations that allow the system to evolve naturally towards high-density states, Sharon Glotzer and colleagues have worked out the densest ordered packing yet for tetrahedra, a configuration with a packing fraction of 0.8324. Unexpectedly, the structure is a dodecagonal quasicrystal, the first example of a quasicrystal formed from hard particles or from non-spherical building blocks. All hard, convex shapes pack more densely than spheres, although for tetrahedra this was demonstrated only very recently. Here, tetrahedra are shown to pack even more densely than previously thought. Thermodynamic computer simulations allow the system to evolve naturally towards high-density states, showing that a fluid of hard tetrahedra undergoes a first-order phase transition to a dodecagonal quasicrystal, and yielding the highest packing fractions yet observed for tetrahedra. All hard, convex shapes are conjectured by Ulam to pack more densely than spheres 1 , which have a maximum packing fraction of φ = π/√18 ≈ 0.7405. Simple lattice packings of many shapes easily surpass this packing fraction 2 , 3 . For regular tetrahedra, this conjecture was shown to be true only very recently; an ordered arrangement was obtained via geometric construction with φ = 0.7786 (ref. 4 ), which was subsequently compressed numerically to φ = 0.7820 (ref. 5 ), while compressing with different initial conditions led to φ = 0.8230 (ref. 6 ). Here we show that tetrahedra pack even more densely, 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 7 , 8 , 9 , 10 , which can be compressed to a packing fraction of φ = 0.8324. By compressing a crystalline approximant of the quasicrystal, the highest packing fraction we obtain is φ = 0.8503. If quasicrystal formation is suppressed, the system remains disordered, jams and compresses to φ = 0.7858. Jamming and crystallization are both preceded by an entropy-driven transition from a simple fluid of independent tetrahedra to a complex fluid characteriz