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Mosaic two-lengthscale quasicrystals


Over the past decade, quasicrystalline order1 has been observed in many soft-matter systems: in dendritic micelles2, in star3 and tetrablock4 terpolymer melts and in diblock copolymer5 and surfactant micelles6. The formation of quasicrystals7,8,9 from such a broad range of ‘soft’ macromolecular micelles suggests that they assemble by a generic mechanism rather than being dependent on the specific chemistry of each system. Indeed, micellar softness has been postulated7 and shown to lead to quasicrystalline order10. Here we theoretically explore this link by studying two-dimensional hard disks decorated with step-like square-shoulder repulsion that mimics, for example, the soft alkyl shell around the aromatic core in dendritic micelles2. We find a family of quasicrystals with 10-, 12-, 18- and 24-fold bond orientational order which originate from mosaics of equilateral and isosceles triangles formed by particles arranged core-to-core and shoulder-to-shoulder. The pair interaction responsible for these phases highlights the role of local packing geometry in generating quasicrystallinity in soft matter, complementing the principles that lead to quasicrystal formation in hard tetrahedra11,12. Based on simple interparticle potentials, quasicrystalline mosaics may well find use in diverse applications ranging from improved image reproduction13 to advanced photonic materials14.

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Figure 1: Real-space structures and diffraction patterns.
Figure 2: Bond orientational order.
Figure 3: Two-lengthscale triangle tilings.


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We thank J. Dolinšek, M. Engel, S. Förster, N. Fujita, M. A. Glaser, C. L. Henley, G. Kahl, R. D. Kamien, Y. Kimura, R. Lifshitz, C. N. Likos, T. C. Lubensky, Y. Sakamoto, M. Schmiedeberg, A. Šiber and A. Takano for discussions. This work was supported by the Japan Society for the Promotion of Science through Grant-in-Aid for Scientific Research (C) (grant number 22540375), by the Slovenian Research Agency (grant number P1-0055) and by the Marie-Curie Initial Training Network COMPLOIDS under FP7-PEOPLE-ITN-2008 (grant number 234810).

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Authors and Affiliations



T.D. and P.Z. conceived the project, T.D. and T.O. performed simulations, and P.Z. proposed the tiling theory. T.D. and P.Z. wrote the manuscript.

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Correspondence to T. Dotera.

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The authors declare no competing financial interests.

Supplementary information

Supplementary Information

This file contains a Supplementary Discussion, comprising Supplementary Text and Data 1-3, Supplementary Tables 1-3, Supplementary Figures 1-36 and additional references. (PDF 6343 kb)

Hexagonal-dodecagonal-decaoctatic transition

This video shows a 31-frame sequence of snapshots and diffraction patterns illustrating the transition from hexagonal crystal to HD12 dodecagonal phase and then to HD18 decaoctatic phase in hard-core/square-shoulder disks induced by cooling accompanied by a moderate 10% decrease of shoulder width. (MOV 11655 kb)

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Dotera, T., Oshiro, T. & Ziherl, P. Mosaic two-lengthscale quasicrystals. Nature 506, 208–211 (2014).

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