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Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra

Nature volume 462, pages 773–777 (2009)Cite this article

Abstract

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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Acknowledgements

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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Author notes

  1. Amir Haji-Akbari and Michael Engel: These authors contributed equally to this work.

Authors and Affiliations

  1. Department of Chemical Engineering,,

    Amir Haji-Akbari, Michael Engel, Aaron S. Keys & Sharon C. Glotzer

  2. Department of Materials Science and Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA,

    Sharon C. Glotzer

  3. Department of Mathematical Sciences,,

    Xiaoyu Zheng

  4. Liquid Crystal Institute, Kent State University, Kent, Ohio 44242, USA ,

    Peter Palffy-Muhoray

  5. Department of Physics, Case Western Reserve University, Cleveland, Ohio 44106, USA,

    Rolfe G. Petschek

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  1. Amir Haji-Akbari
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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). https://doi.org/10.1038/nature08641

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