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Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra


All hard, convex shapes are conjectured by Ulam to pack more densely than spheres1, which have a maximum packing fraction of φ = π/√18 ≈ 0.7405. Simple lattice packings of many shapes easily surpass this packing fraction2,3. 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 quasicrystal7,8,9,10, 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 characterized by tetrahedra arranged in densely packed local motifs of pentagonal dipyramids 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.

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Figure 1: Packings of tetrahedra obtained by hand and by computer simulation.
Figure 2: Thermodynamic and structural properties of the hard tetrahedron fluid.
Figure 3: Structural characterization of the hard tetrahedra dodecagonal quasicrystal and its approximant.


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The Air Force Office of Scientific Research supported A.H.-A., P.P.-M. and S.C.G. The National Science Foundation supported A.S.K., A.H.-A. and S.C.G. in the shape-matching analyses that identified local motifs. M.E. was supported by a postdoctoral fellowship of the Deutsche Forschungsgemeinschaft.

Author Contributions A.H.-A. and M.E. performed all simulations and contributed equally to the study. M.E. solved the quasicrystal and approximant structures. A.S.K. performed shape-matching analysis. X.Z., P.P.-M., and R.G.P. proposed and constructed geometric packings. All authors discussed and analysed the results, and contributed to the scientific process. S.C.G., A.H.-A., and M.E. wrote most of the paper, and all authors contributed to refinement of the manuscript. S.C.G. and P.P.-M. conceived and designed the study, and S.C.G. directed the study.

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Correspondence to Sharon C. Glotzer.

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Haji-Akbari, A., Engel, M., Keys, A. et al. Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra. Nature 462, 773–777 (2009).

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