Document Type
Article
Publication Date
12-10-2009
Abstract
All hard, convex shapes are conjectured by Ulam to pack more densely than spheres, which have a maximum packing fraction of Ø /18 0.7405. Simple lattice packings of many shapes 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 Ø= 0.7786 (ref. 4), which was subsequently compressed numerically to Ø= 0.7820 (ref. 5), while compressing with different initial conditions led to Ø= 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, which can be compressed to a packing fraction of Ø= 0.8324. By compressing a crystalline approximant of the quasicrystal, the highest packing fraction we obtain is Ø= 0.8503. If quasicrystal formation is suppressed, the system remains disordered, jams and compresses to Ø= 0.7858. Jamming and crystallization are both preceded by an entropy-driven transition from a simpe 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.
Keywords
density, Envisat-1, Monte Carlo method, liquid, article, computer simulation, compression, crystal structure, crystallization, thermodynamics, tetrahedra, sphere, priority journal, phase transition, particle size, lattice dynamics, geometry, entropy
Publication Title
Nature
Rights
© 2009 Macmillan Publishers Limited. All rights reserved. Since 2003, ownership of copyright in original research articles remains with the Authors*, and provided that, when reproducing the Contribution or extracts from it, the Authors acknowledge first and reference publication in the Journal, the Authors retain the following non-exclusive rights: To post a copy of the Contribution as accepted for publication after peer review (in Word or Tex format) on the Author's own web site, or the Author's institutional repository, or the Author's funding body's archive, six months after publication of the printed or online edition of the Journal, provided that they also link to the Journal article on NPG's web site (eg through the DOI). URL - http://www.nature.com/nature/journal/v462/n7274/full/nature08641.html, DOI - 10.1038/nature08641
Recommended Citation
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