Abstract
Quasicrystals are intriguing structures that have long-range positional correlations but no periodicity in real space, and typically with rotational symmetries that are ‘forbidden’ in conventional periodic crystals. Here, we present a two-dimensional columnar liquid quasicrystal with dodecagonal symmetry. Unlike previous dodecagonal quasicrystals based on random tiling, a honeycomb structure based on a strictly quasiperiodic tessellation of tiles is observed. The structure consists of dodecagonal clusters made up of triangular, square and trapezoidal cells that are optimal for local packing. To maximize the presence of such dodecagonal clusters, the system abandons periodicity but adopts a quasiperiodic structure that follows strict packing rules. The stability of random-tiling dodecagonal quasicrystals is often attributed to the entropy of disordering when strict tiling rules are broken, at the sacrifice of the long-range positional order. However, our results demonstrate that quasicrystal stability may rest on energy minimization alone, or with only minimal entropic intervention.
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Data availability
Source data are provided with this paper. All data generated or analysed in this study are available in this article and its Supplementary Information. The datasets generated and analysed during the current study are publicly available in the Figshare repository at https://doi.org/10.6084/m9.figshare.21626570 (ref. 48).
References
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).
Steurer, W. & Deloudi, S. Crystallography of Quasicrystals. Concepts, Methods and Structures (Springer-Verlag, 2009).
Penrose, R. The role of aesthetics in pure and applied mathematical research. Bull. Inst. Math. Appl. 10, 266–271 (1974).
Senechal, M. Quasicrystals and Geometry (Cambridge Univ. Press, 1996).
Zeng, X. B. et al. Supramolecular dendritic liquid quasicrystals. Nature 428, 157–160 (2004).
Ishimasa, T., Nissen, H.-U. & Fukano, Y. New ordered state between crystalline and amorphous in Ni–Cr particles. Phys. Rev. Lett. 55, 511–513 (1985).
Chen, H., Li, D. X. & Kuo, K. H. New type of two-dimensional quasicrystal with twelvefold rotational symmetry. Phys. Rev. Lett. 60, 1645–1648 (1988).
Hayashida, K., Dotera, T., Takano, A. & Matsushita, Y. Polymeric quasicrystal: mesoscopic quasicrystalline tiling in ABC star polymers. Phys. Rev. Lett. 98, 195502 (2007).
Zhang, J. W. & Bates, F. S. Dodecagonal quasicrystalline morphology in a poly(styrene-b-isoprene-b-styrene-b-ethylene oxide) tetrablock terpolymer. J. Am. Chem. Soc. 134, 7636–7639 (2012).
Gillard, T. M., Lee, S. W. & Bates, F. S. Dodecagonal quasicrystalline order in a diblock copolymer melt. Proc. Natl Acad. Sci. USA 113, 5167–5172 (2016).
Talapin, D. V. et al. Quasicrystalline order in self-assembled binary nanoparticle superlattices. Nature 461, 964–967 (2009).
Xiao, C. H., Fujita, N., Miyasaka, K., Sakamoto, Y. & Terasaki, O. Dodecagonal tiling in mesoporous silica. Nature 487, 349–353 (2012).
Urgel, J. I. et al. Quasicrystallinity expressed in two-dimensional coordination networks. Nat. Chem. 8, 657–662 (2016).
Yue, K. et al. Geometry induced sequence of nanoscale Frank–Kasper and quasicrystal mesophases in giant surfactants. Proc. Natl Acad. Sci. USA 113, 14195–14200 (2016).
Passens, M. et al. Interface-driven formation of a two-dimensional dodecagonal fullerene quasicrystal. Nat. Comm. 8, 15367 (2017).
Freedman, B. et al. Wave and defect dynamics in nonlinear photonic quasicrystals. Nature 440, 1166–1169 (2006).
Haji-Akbari, A. et al. Disordered, quasicrystalline and crystalline phases of densely packed tetrahedral. Nature 462, 773–777 (2009).
Dotera, T., Oshiro, T. & Ziherl, P. Mosaic two-lengthscale quasicrystals. Nature 506, 208–211 (2014).
Dotera, T., Bekku, S. & Ziherl, P. Bronze-mean hexagonal quasicrystal. Nat. Mat. 16, 987 (2017).
Barkan, K., Diamant, H. & Lifshitz, R. Stability of quasicrystals composed of soft isotropic particles. Phys. Rev. B 83, 172201 (2011).
Ungar, G. & Zeng, X. B. Frank–Kasper, quasicrystalline and related phases in liquid crystals. Soft Matter 1, 95–106 (2005).
Baake, M., Klitzing, R. & Schlottmann, M. Fractally shaped acceptance domains of quasiperiodic square-triangle tilings with dedecagonal symmetry. Phys. A 191, 554 (1992).
Oxborrow, M. & Henley, C. L. Random square-triangle tilings: a model for twelvefold-symmetric quasicrystals. Phys. Rev. B 48, 6966–6998 (1993).
Zhang, R. B., Zeng, X. B. & Ungar, G. Direct AFM observation of individual micelles, tile decorations and tiling rules of a dodecagonal liquid quasicrystal. J. Phys. Condens. Matter 29, 414022 (2017).
Wang, P. Y. & Mason, T. G. A Brownian quasi-crystal of pre-assembled colloidal Penrose tiles. Nature 561, 94–99 (2018).
Dontabhaktuni, J., Ravnik, M. & Zumer, S. Quasicrystalline tilings with nematic colloidal platelets. Proc. Natl Acad. Sci. USA 111, 2464–2469 (2014).
Senyuk, B., Liu, Q., Bililign, E., Nystrom, P. D. & Smalyukh, I. I. Geometry-guided colloidal interactions and self-tiling of elastic dipoles formed by truncated pyramid particles in liquid crystals. Phys. Rev. E 91, 040501(R) (2015).
Reinhardt, A., Schreck, J. S., Romano, F. & Doye, J. P. K. Self-assembly of two-dimensional binary quasicrystals: a possible route to a DNA quasicrystal. J. Phys. Condens. Matter 29, 014006 (2017).
Nova, E. G., Wong, C. K., Llombart, P. & Doye, J. P. K. How to design an icosahedral quasicrystal through directional bonding. Nature 596, 367–371 (2021).
Ungar, G. et al. Self-assembly at different length scales: polyphilic star-branched liquid crystals and miktoarm star copolymers. Adv. Funct. Mater. 21, 1296–1323 (2011).
Tschierske, C. et al. Complex tiling patterns in liquid crystals. Interface Focus 2, 669–680 (2012).
Chen, B., Zeng, X. B., Baumeister, U., Ungar, G. & Tschierske, C. Liquid crystalline networks composed of pentagonal, square, and triangular cylinders. Science 307, 96–99 (2005).
Liu, F. et al. The triangular cylinder phase: a new mode of self-assembly in liquid crystalline soft matter. J. Am. Chem. Soc. 129, 9578–9579 (2007).
Chen, B. et al. Liquid crystals with complex superstructures. Angew. Chem. Int. Ed. 116, 4721–4725 (2004).
Ungar, G. et al. GISAXS in the study of supramolecular and hybrid liquid crystals. J. Phys. Conf. Ser. 247, 012032 (2010).
Liu, F. et al. The trapezoidal cylinder phase: a new mode of self-assembly in liquid–crystalline soft matter. J. Am. Chem. Soc. 130, 9666–9667 (2008).
Takakura, H., Shiono, M., Sato, T. J., Yamamoto, A. & Tsai, A. P. Ab initio structure determination of icosahedral Zn–Mg–Ho quasicrystals by density modification method. Phys. Rev. Lett. 86, 236–239 (2001).
Takakura, H., Gómez, C. P., Yamamoto, A., de Boissieu, M. & Tsai, A. P. Atomic structure of the binary icosahedral Yb–Cd quasicrystal. Nat. Mater. 6, 58–63 (2007).
Gähler F. in Crystallography of Dodecagonal Quasicrystals in Quasicrystalline Materials (eds Janot C. & Dubois J. M.) 272–284 (World Scientific 1988).
Zeng, X. B. & Ungar, G. Inflation rules of square-triangle tilings: from approximants to dodecagonal liquid quasicrystals. Phil. Mag. 86, 1093–1103 (2006).
Förster, S., Meinel, K., Hammer, R., Trautmann, M. & Widdra, W. Quasicrystalline structure formation in a classical crystalline thin-film system. Nature 502, 215–218 (2013).
Lifshitz, R. & Diamant, H. Soft quasicrystals—why are they stable? Phil. Mag. 87, 3021–3030 (2007).
Poppe, M., Chen, C., Poppe, S., Liu, F. & Tschierske, C. A periodic dodecagonal supertiling by self-assembly of star-shaped molecules in the liquid crystalline state. Commun. Chem. 3, 70 (2020).
Engel, M., Damasceno, P. F., Phillips, C. L. & Glotzer, S. C. Computational self-assembly of a one-component icosahedral quasicrystal. Nat. Mater. 14, 109–116 (2015).
Zeng, X. B. et al. Complex multicolor tilings and critical phenomena in tetraphilic liquid crystals. Science 331, 1302–1306 (2011).
Glettner, B. et al. Liquid–crystalline kagome. Angew. Chem. Int. Ed. 47, 9063–9066 (2008).
Liu, F. et al. Arrays of giant octagonal and square cylinders by liquid crystalline self-assembly of X-shaped polyphilic molecules. Nat. Commun. 3, 1104 (2012).
Zeng, X. et al. Data for a columnar liquid quasicrystal with a honeycomb structure that consists of triangular, square and trapezoidal cells. Figshare https://doi.org/10.6084/m9.figshare.21626570 (2023).
Acknowledgements
For support with the experiments we thank O. Shebanova and N. Terrill at station I22, Diamond Light Source, O. Bikondoa and P. Thompson at XMaS beamline (BM28), ESRF, and S. Sasaki and H. Masunaga at BL40B2, Spring-8. Financial support is acknowledged from EPSRC (EP-P002250 and EP-T003294), DFG (436494874-RTG 2670), the 111 Project 2.0 of China (BP0618008) and National Natural Science Foundation of China (92156013, 21761132033, 21374086).
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C.T. and G.U. conceived and directed the project. B.G., U.B. and B.C., under the supervision of C.T., synthesized the compound and carried out the differential scanning calorimetry, polarized optical microscopy and initial X-ray diffraction characterizations. X.Z. and F.L., supervised by G.U., carried out the powder and GIXRD experiments, and data analysis that led to reconstruction of the electron density maps. All the authors contributed to the construction of the structural model of the CLQC phase. Simulation of the CLQC diffraction pattern was carried out by X.Z. X.Z. wrote the manuscript with written contributions from all the co-authors.
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Nature Chemistry thanks Marianne Imperor-Clerc, Shiki Yagai, Slobodan Zumer and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
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Supplementary Information
Supplementary Sections 1 (Synthesis and analytical data), 2 (Simulation of diffraction intensities of CLQC) and 3 (Additional data), Figs. 1–13 and Tables 1–5.
Supplementary Data 1
DSC source data for supplementary Figure 4.
Supplementary Data 2
SAXS source data for supplementary Figure 7.
Supplementary Data 3
WAXS source data for supplementary Figure 7.
Supplementary Data 4
SAXS source data for supplementary Figure 10.
Supplementary Data 5
SAXS experimental and fitted data for supplementary Figure 12.
Supplementary Data 6
Peak widths and standard errors source data for supplementary Figure 13.
Source data
Fig. 3a
SAXS source data of p4gmL and CLQC phases for Fig. 3a.
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Zeng, X., Glettner, B., Baumeister, U. et al. A columnar liquid quasicrystal with a honeycomb structure that consists of triangular, square and trapezoidal cells. Nat. Chem. 15, 625–632 (2023). https://doi.org/10.1038/s41557-023-01166-5
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DOI: https://doi.org/10.1038/s41557-023-01166-5
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