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 10.12.2009   Карта сайта     Language По-русски По-английски
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10.12.2009

Nature 462, 773-777 (10 December 2009) | doi:10.1038/nature08641; Received 5 July 2009; Accepted 3 November 2009



Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra


Amir Haji-Akbari1,6, Michael Engel1,6, Aaron S. Keys1, Xiaoyu Zheng3, Rolfe G. Petschek5, Peter Palffy-Muhoray4 & Sharon C. Glotzer1,2




  1. Department of Chemical Engineering,

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

  3. Department of Mathematical Sciences,

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

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

  6. These authors contributed equally to this work.


Correspondence to: Sharon C. Glotzer1,2 Correspondence and requests for materials should be addressed to S.C.G. (Email: sglotzer@umich.edu).





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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  • Harton Vladislav Vadim  honorary member of ISSC science council

  • Lichtenstain Alexandr Iosif  honorary member of ISSC science council

  • Novikov Dimirtii Leonid  honorary member of ISSC science council

  • Yakushev Mikhail Vasilii  honorary member of ISSC science council

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