Letter | Published:

Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra

Nature volume 462, pages 773777 (10 December 2009) | Download Citation

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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References

  1. 1.

    The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems 135 (Norton, 2001)

  2. 2.

    & Densest lattice packings of 3-polytopes. Comput. Geom. 16, 157–186 (2000)

  3. 3.

    , , & Unusually dense crystal packing of ellipsoids. Phys. Rev. Lett. 92, 255506 (2004)

  4. 4.

    A dense packing of regular tetrahedra. Discrete Comput. Geom. 40, 214–240 (2008)

  5. 5.

    & Dense packings of the Platonic and Archimedean solids. Nature 460, 876–879 (2009)

  6. 6.

    & Dense packings of polyhedra: Platonic and Archimedean solids. Phys. Rev. E 80, 041104 (2009)

  7. 7.

    et al. Supramolecular dendritic liquid quasicrystals. Nature 428, 157–160 (2004)

  8. 8.

    , , & Polymeric quasicrystal: mesoscopic quasicrystalline tiling in ABC star polymers. Phys. Rev. Lett. 98, 195502 (2007)

  9. 9.

    et al. Quasicrystalline order in self-assembled binary nanoparticle superlattices. Nature 461, 964–967 (2009)

  10. 10.

    , & A dodecagonal quasicrystalline chalcogenide. Angew. Chem. Int. Ed. 37, 1384–1386 (1998)

  11. 11.

    Historical overview of the Kepler conjecture. Discrete Comput. Geom. 36, 5–20 (2006)

  12. 12.

    A proof of the Kepler conjecture. Ann. Math. 162, 1065–1185 (2005)

  13. 13.

    & Anisotropy of building blocks and their assembly into complex structures. Nature Mater. 6, 567–572 (2007)

  14. 14.

    , , , & Spontaneous self-assembly of CdTe nanocrystals into free-floating sheets. Science 314, 274–278 (2006)

  15. 15.

    The effect of shape on the interaction of colloidal particles. Ann. NY Acad. Sci. 51, 627–659 (1949)

  16. 16.

    in Phase Transformations in Solids (eds Smoluchowski, R., Mayer, J. E. & Weyl, W. A.) 67 (Wiley, 1951)

  17. 17.

    & Tracing the phase boundaries of hard spherocylinders. J. Chem. Phys. 106, 666–687 (1997)

  18. 18.

    & Phase diagram of the hard biaxial ellipsoid fluid. J. Chem. Phys. 106, 6681–6688 (1997)

  19. 19.

    & Phase-behavior of disk-like hard-core mesogens. Phys. Rev. A 45, 5632–5648 (1992)

  20. 20.

    , & Phase behavior in colloidal hard perfect tetragonal parallelepipeds. J. Chem. Phys. 128, 044909 (2009)

  21. 21.

    & Packing, tiling and covering with tetrahedra. Proc. Natl Acad. Sci. USA 103, 10612–10617 (2006)

  22. 22.

    & The hard tetrahedron fluid: a model for the structure of water. Mol. Phys. 84, 421–434 (1994)

  23. 23.

    & Complex alloy structures regarded as sphere packings. 1. Definitions and basic principles. Acta Crystallogr. 11, 184–190 (1958)

  24. 24.

    Tetrahedral symmetry in nematic liquid crystals. Phys. Rev. E 52, 702–717 (1995)

  25. 25.

    Crystallography of quasiperiodic crystals. Acta Crystallogr. A 52, 509–560 (1996)

  26. 26.

    & Random square-triangle tilings—a model for twelvefold-symmetrical quasi-crystals. Phys. Rev. B 48, 6966–6998 (1993)

  27. 27.

    & Solid-phase structures of the Dzugutov pair potential. Phys. Rev. E 61, 6845–6857 (2000)

  28. 28.

    & How do quasicrystals grow? Phys. Rev. Lett. 99, 235503 (2007)

  29. 29.

    , , & Archimedean-like tiling on decagonal quasicrystalline surfaces. Nature 454, 501–504 (2008)

  30. 30.

    Structural phase transitions from and to the quasicrystalline state. Acta Crystallogr. A 61, 28–38 (2005)

  31. 31.

    & Self-assembly of complex crystals and quasicrystals with a double-well interaction potential. Phys. Rev. Lett. 98, 225505 (2007)

  32. 32.

    , & A dense periodic packing of tetrahedra with a small repeating unit. Preprint at 〈〉 (2009)

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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.

Author information

Author notes

    • Amir Haji-Akbari
    •  & Michael Engel

    These authors contributed equally to this work.

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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Correspondence to Sharon C. Glotzer.

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    This file contains Supplementary Figures S1-S8 with Legends and Supplementary Notes and Data.

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https://doi.org/10.1038/nature08641

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