Abstract
The most striking feature of conventional quasicrystals is their non-traditional symmetry characterized by icosahedral, dodecagonal, decagonal or octagonal axes1,2,3,4,5,6. The symmetry and the aperiodicity of these materials stem from an irrational ratio of two or more length scales controlling their structure, the best-known examples being the Penrose7,8 and the Ammann–Beenker9,10 tiling as two-dimensional models related to the golden and the silver mean, respectively. Surprisingly, no other metallic-mean tilings have been discovered so far. Here we propose a self-similar bronze-mean hexagonal pattern, which may be viewed as a projection of a higher-dimensional periodic lattice with a Koch-like snowflake projection window. We use numerical simulations to demonstrate that a disordered variant11 of this quasicrystal can be materialized in soft polymeric colloidal particles with a core–shell architecture12,13,14,15,16,17. Moreover, by varying the geometry of the pattern we generate a continuous sequence of structures, which provide an alternative interpretation of quasicrystalline approximants observed in several metal–silicon alloys18.
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References
Levine, D. & Steinhardt, P. J. Quasicrystals: a new class of ordered structures. Phys. Rev. Lett. 53, 2477–2480 (1984).
Shechtman, D., Blech, I., Gratias, D. & Cahn, J. W. Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951–1953 (1984).
Ishimasa, T., Nissen, H.-U. & Fukao, Y. New ordered state between crystalline and amorphous in Ni-Cr particles. Phys. Rev. Lett. 55, 511–513 (1985).
Tsai, A. P., Inoue, A. & Masumoto, T. A stable quasicrystal in Al–Cu–Fe system. Jpn. J. Appl. Phys. 26, L1505–L1507 (1987).
Janssen, T., Chapuis, G. & de Boissieu, M. Aperiodic Crystals: From Modulated Phases to Quasicrystals (Oxford Univ. Press, 2007).
Steurer, W. & Deloudi, S. Crystallography of Quasicrystals: Concepts, Methods and Structures (Springer, 2009).
Penrose, R. The role of aesthetics in pure and applied mathematical research. Bull. Inst. Math. Appl. 10, 266–271 (1974).
Mackay, A. Crystallography and the Penrose pattern. Physica A 114, 609–613 (1982).
Grünbaum, B. & Shephard, G. C. Tilings and Patterns (Freeman, 1987).
Beenker, F. P. M. Algebraic theory of non-periodic tilings of the plane by two simple building blocks: a square and a rhombus. TH-Report Vol. 82-WSK04 1–64 (Eindhoven University of Technology, 1982).
Oxborrow, M. & Henley, C. L. Random square-triangle tilings: a model for twelvefold-symmetric quasicrystals. Phys. Rev. B 48, 6966–6998 (1993).
Zeng, X. et al. Supramolecular dendritic liquid quasicrystals. Nature 428, 157–160 (2004).
Chanpuriya, S. et al. Cornucopia of nanoscale ordered phases in sphere-forming tetrablock terpolymers. ACS Nano 10, 4961–4972 (2016).
Fischer, S. et al. Colloidal quasicrystals with 12-fold and 18-fold diffraction symmetry. Proc. Natl Acad. Sci. USA 108, 1810–1814 (2011).
Iacovella, C. R., Keys, A. S. & Glotzer, S. C. Self-assembly of soft-matter quasicrystals and their approximants. Proc. Natl. Acad. Sci. USA 108, 20935–20940 (2011).
Engel, M. & Trebin, H.-R. Self-assembly of monatomic complex crystals and quasicrystals with a double-well interaction potential. Phys. Rev. Lett. 98, 225505 (2007).
Dotera, T., Oshiro, T. & Ziherl, P. Mosaic two-lengthscale quasicrystals. Nature 506, 208–211 (2014).
Ishimasa, T. Dodecagonal quasicrystals still in progress. Isr. J. Chem. 51, 1216–1225 (2011).
Gumbs, G. & Ali, M. K. Dynamical maps, Cantor spectra, and localization for Fibonacci and related quasiperiodic lattices. Phys. Rev. Lett. 60, 1081–1084 (1988).
Suzuki, T.-K. & Dotera, T. Dynamical systems for quasiperiodic chains and new self-similar polynomials. J. Phys. A: Math. Gen. 26, 6101–6113 (1993).
Buitrago, A. R. Polygons, diagonals, and the bronze mean. Nexus Netw. J. 9, 321–326 (2008).
Stampfli, P. Dodecagonal quasiperiodic lattice in two dimensions. Helv. Phys. Acta 59, 1260–1263 (1986).
Niizeki, K. A step toward an MLD classification of selfsimilar quasilattices. Prog. Theor. Phys. 128, 629–691 (2012).
Socolar, J. E. S., Lubensky, T. C. & Steinhardt, P. J. Phonons, phasons, and dislocations in quasicrystals. Phys. Rev. B 34, 3345–3360 (1986).
Dotera, T. & Steinhardt, P. J. Ising-like transition and phason unlocking in icosahedral quasicrystals. Phys. Rev. Lett. 72, 1670–1673 (1994).
Iga, H., Mihalkovič, M. & Ishimasa, T. Approximant of dodecagonal quasicrystal formed in MnSiV alloy. Philos. Mag. 91, 2624–2633 (2011).
Ishimasa, T., Iwami, S., Sakaguchi, N., Oota, R. & Mihalkovič, M. Phason space analysis and structure modelling of 100 Å-scale dodecagonal quasicrystal in Mn-based alloy. Philos. Mag. 95, 3745–3767 (2015).
Förster, S. et al. Observation and structure determination of an oxide quasicrystal approximant. Phys. Rev. Lett. 117, 095501 (2016).
Ye, X. et al. Quasicrystalline nanocrystal superlattice with partial matching rules. Nat. Mater. 16, 214–219 (2017).
Sasisekharan, V., Baranidharan, S., Balagurusamy, V. S. K., Srinivasan, A. & Gopal, E. S. R. Non-periodic tilings in 2-dimensions with 4, 6, 8, 10 and 12-fold symmetries. Pramana J. Phys. 33, 405–420 (1989).
Acknowledgements
This project has been supported by the Japan Society for the Promotion of Science through Grant-in-Aid for Scientific (B) (No. 16H04037), and has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 642774. The authors acknowledge the financial support from the Slovenian Research Agency (research core funding No. P1-0055).
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T.D. and P.Z. conceived the project and proposed the tiling theory. S.B. and T.D. performed simulations and higher-dimensional analysis, and T.D. and P.Z. wrote the manuscript.
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Dotera, T., Bekku, S. & Ziherl, P. Bronze-mean hexagonal quasicrystal. Nature Mater 16, 987–992 (2017). https://doi.org/10.1038/nmat4963
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DOI: https://doi.org/10.1038/nmat4963