Abstract
Penrose’s pentagonal P2 quasi-crystal1,2,3,4 is a beautiful, hierarchically organized multiscale structure in which kite- and dart-shaped tiles are arranged into local motifs, such as pentagonal stars, which are in turn arranged into various close-packed superstructural patterns that become increasingly complex at larger length scales. Although certain types of quasi-periodic structure have been observed in hard and soft matter, such structures are difficult to engineer, especially over large areas, because generating the necessary, highly specific interactions between constituent building blocks is challenging. Previously reported soft-matter quasi-crystals of dendrimers5, triblock copolymers6, nanoparticles7 and polymeric micelles8 have been limited to 12- or 18-fold symmetries. Because routes for self-assembling complex colloidal building blocks9,10,11 into low-defect dynamic superstructures remain limited12, alternative methods, such as using optical and directed assembly, are being explored13,14. Holographic laser tweezers15 and optical standing waves16 have been used to hold microspheres in local quasi-crystalline arrangements, and magnetic microspheres of two different sizes have been assembled into local five-fold-symmetric quasi-crystalline arrangements in two dimensions17. But a Penrose quasi-crystal of mobile colloidal tiles has hitherto not been fabricated over large areas. Here we report such a quasi-crystal in two dimensions, created using a highly parallelizable method of lithographic printing and subsequent release of pre-assembled kite- and dart-shaped tiles into a solution–dispersion containing a depletion agent. After release, the positions and orientations of the tiles within the quasi-crystal can fluctuate, and these tiles undergo random, Brownian motion in the monolayer owing to frequent collisions between neighbouring tiles, even after the system reaches equilibrium. Using optical microscopy, we study both the equilibrium fluctuations of the system at high tile densities and also the ‘melting’ of the pattern as the tile density is lowered. At high tile densities we find signatures of a five-fold pentatic liquid quasi-crystalline phase, analogous to a six-fold hexatic liquid crystal. Our fabrication approach is applicable to tiles of different sizes and shapes, and with different initial positions and orientations, enabling the creation of two-dimensional quasi-crystalline systems (and other systems that possess multiscale complexity at high tile densities) beyond those of current self- or directed-assembly methods18,19,20. We anticipate that our approach for generating lithographically pre-assembled monolayers could be extended to create three-dimensional Brownian systems of fluctuating particles with custom-designed shapes through holographic lithography21,22 or stereolithography23.
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Data availability
The data shown in the figures and that support the findings of this study are available from the corresponding author on reasonable request.
References
Penrose, R. The role of aesthetics in pure and applied mathematical research. Bull. Inst. Math. Appl. 10, 266–271 (1974).
Penrose, R. Pentaplexity. A class of non-periodic tilings of the plane. Math. Intell. 2, 32–37 (1979).
Bursill, L. A. & Lin, P. J. Penrose tiling observed in a quasi-crystal. Nature 316, 50–51 (1985).
Levine, D. & Steinhardt, P. J. Quasi-crystals: a new class of ordered structures. Phys. Rev. Lett. 53, 2477–2480 (1984).
Zeng, X. et al. Supramolecular dendritic liquid quasicrystals. Nature 428, 157–160 (2004).
Hayashida, K., Dotera, T., Takano, A. & Matsushita, Y. Polymeric quasicrystal: mesoscopic quasicrystalline tiling in ABC star polymers. Phys. Rev. Lett. 98, 195502 (2007).
Talapin, D. V. et al. Quasicrystalline order in self-assembled binary nanoparticle superlattices. Nature 461, 964–967 (2009).
Fischer, S. et al. Colloidal quasicrystals with 12-fold and 18-fold diffraction symmetry. Proc. Natl Acad. Sci. USA 108, 1810–1814 (2011).
Hoover, M. D. et al. A method for producing non-spherical monodisperse particles using integrated circuit fabrication techniques. J. Aerosol Sci. 21, 569–575 (1990).
Hernandez, C. J. & Mason, T. G. Colloidal alphabet soup: monodisperse dispersions of shape-designed lithoparticles. J. Phys. Chem. C 111, 4477–4480 (2007).
Yang, S.-M., Kim, S.-H., Lim, J.-M. & Yi, G.-R. Synthesis and assembly of structured colloidal particles. J. Mater. Chem. 18, 2177–2190 (2008).
Zeng, C., Chen, Y., Kirschbaum, K., Lambright, K. J. & Jin, R. Emergence of hierarchical structural complexities in nanoparticles and their assembly. Science 354, 1580–1584 (2016).
Grier, D. G. A revolution in optical manipulation. Nature 424, 810–816 (2003).
Bae, W.-G. et al. Scalable multiscale patterned structures inspired by nature: the role of hierarchy. Adv. Mater. 26, 675–700 (2014).
Roichman, Y. & Grier, D. G. Holographic assembly of quasicrystalline photonic heterostructures. Opt. Express 13, 5434–5439 (2005).
Mikhael, J., Roth, J., Helden, L. & Bechinger, C. Archimedean-like tiling on decagonal quasicrystalline surfaces. Nature 454, 501–504 (2008).
Wen, W., Zhang, L. & Sheng, P. Planar magnetic colloidal crystals. Phys. Rev. Lett. 85, 5464–5467 (2000).
Boles, M. A., Engel, M. & Talapin, D. V. Self-assembly of colloidal nanocrystals: from intricate structures to functional materials. Chem. Rev. 116, 11220–11289 (2016).
Niederberger, M. Multiscale nanoparticle assembly: from particulate precise manufacturing to colloidal processing. Adv. Funct. Mater. 27, 1703647 (2017).
Grzelczak, M., Vermant, J., Furst, E. M. & Liz-Marzán, L. M. Directed self-assembly of nanoparticles. ACS Nano 4, 3591–3605 (2010).
Campbell, M., Sharp, D. N., Harrison, M. T., Denning, R. G. & Turberfield, A. J. Fabrication of photonic crystals for the visible spectrum by holographic lithography. Nature 404, 53–56 (2000).
Ullal, C. K. et al. Photonic crystals through holographic lithography: simple cubic, diamond-like, and gyroid-like structures. Appl. Phys. Lett. 84, 5434–5436 (2004).
Kawata, S., Sun, H.-B., Tanaka, T. & Takada, K. Finer features for functional microdevices. Nature 412, 697–698 (2001).
Zhao, K. & Mason, T. G. Directing colloidal self-assembly through roughness-controlled depletion attractions. Phys. Rev. Lett. 99, 268301 (2007).
Zhao, K. & Mason, T. G. Frustrated rotator crystals and glasses of Brownian pentagons. Phys. Rev. Lett. 103, 208302 (2009).
Nelson, D. R., Rubinstein, M. & Spaepen, F. Order in two-dimensional binary random arrays. Philos. Mag. A 46, 105–126 (1982).
Nelson, D. R. Defects and Geometry in Condensed Matter Physics Ch. 2 (Cambridge Univ. Press, New York, 2002).
Zhao, K., Bruinsma, R. & Mason, T. G. Local chiral symmetry breaking in triatic liquid crystals. Nat. Commun. 3, 801 (2012).
Weeks, E. R., Crocker, J. C., Levitt, A. C., Schofield, A. & Weitz, D. A. Three-dimensional direct imaging of structural relaxation near the colloidal glass transition. Science 287, 627–631 (2000).
Chen, K. et al. Low-frequency vibrations of soft colloidal glasses. Phys. Rev. Lett. 105, 025501 (2010).
Wang, P.-Y. & Mason, T. G. Colloidal lock-and-key dimerization reactions of hard annular sector particles controlled by osmotic pressure. J. Am. Chem. Soc. 137, 15308–15314 (2015).
Zhao, K., Bruinsma, R. & Mason, T. G. Entropic crystal–crystal transitions of Brownian squares. Proc. Natl Acad. Sci. USA 108, 2684–2687 (2011).
Zhao, K. & Mason, T. G. Twinning of rhombic colloidal crystals. J. Am. Chem. Soc. 134, 18125–18131 (2012).
Lewandowski, E. P., Bernate, J. A., Tseng, A., Searson, P. C. & Stebe, K. J. Oriented assembly of anisotropic particles by capillary interactions. Soft Matter 5, 886–890 (2009).
Cavallaro, M., Botto, L., Lewandowski, E. P., Wang, M. & Stebe, K. J. Curvature-driven capillary migration and assembly of rod-like particles. Proc. Natl Acad. Sci. USA 108, 20923–20928 (2011).
Wang, P.-Y. & Mason, T. G. Dimer crystallization of chiral proteoids. Phys. Chem. Chem. Phys. 19, 7167–7175 (2017).
Gardner, M. Penrose Tiles to Trapdoor Ciphers (W. H. Freeman, New York, 1989).
Zhao, K. & Mason, T. G. Shape-designed frustration by local polymorphism in a near-equilibrium colloidal glass. Proc. Natl Acad. Sci. USA 112, 12063–12068 (2015).
Caspar, D. L. D. & Fontano, E. Five-fold symmetry in crystalline quasicrystal lattices. Proc. Natl Acad. Sci. USA 93, 14271–14278 (1996).
Dotera, T., Oshiro, T. & Ziherl, P. Mosaic two-lengthscale quasicrystals. Nature 506, 208–211 (2014).
Steinhardt, P. J. & Jeong, H.-C. A simpler approach to Penrose tiling with implications for quasicrystal formation. Nature 382, 431–433 (1996).
Dontabhaktuni, J., Ravnik, M. & Zumer, S. Quasicrystalline tilings with nematic colloidal platelets. Proc. Natl Acad. Sci. USA 111, 2464–2469 (2014).
Kosterlitz, J. M. & Thouless, D. J. Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. Chem. 6, 1181–1203 (1973).
Steurer, W. & Deloudi, S. Crystallography of Quasi-Crystals (Springer, Berlin, 2009).
Socolar, J. Phason strain in quasicrystals. J. Phys. Colloques 47, 217–226 (1986).
Li, F. H., Pan, G. Z., Huang, D. X., Hashimoto, H. & Yokota, Y. Phason-strain identification for quasicrystals by high-resolution electron microscopy. Ultramicroscopy 45, 299–305 (1992).
Gummelt, P. Penrose tilings as coverings of congruent decagons. Geom. Dedicata 62, 1–17 (1996).
Saitoh, K., Tsuda, K., Tanaka, M., Kaneko, K. & Tsai, A. P. Structural study of an Al72Ni20Co8 decagonal quasicrystal using the high-angle annular dark-field method. Jpn. J. Appl. Phys. 36, L1400–L1402 (1997).
Acknowledgements
We thank C. M. Knobler for discussions and UCLA for financial support. We acknowledge the use of the Integrated Systems Nanofabrication Cleanroom at the California NanoSystems Institute at UCLA.
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T.G.M. conceived the experimental approach and supervised the project. P.-Y.W. and T.G.M. designed the experiments. P.-Y.W. performed the experiments. P.-Y.W. and T.G.M. analysed the data. P.-Y.W. and T.G.M. wrote the manuscript.
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Extended data figures and tables
Extended Data Fig. 1 Detailed sequence of steps for fabricating and releasing a dense litho-PAM of mobile shape-designed tiles.
The tiles (purple), which are composed of cross-linked polymeric SU-8 photoresist, are released after exposure and development from the glass wafer (black hatching) using an RSD (blue) that contains a release agent (TMAH) to dissolve the Omnicoat release layer (green), a stabilizing agent (SDS) that adsorbs onto released tiles and prevents their aggregation, and a depletion agent (nanoscale polystyrene spheres) that strongly inhibits the released tiles from leaving the monolayer as a consequence of Brownian excitations (steps are shown as side views).
Extended Data Fig. 2 Examples of organized superstructural sets of motifs extending over different length scales in the P2 quasi-crystal before and after release.
a, Regular centre-filled pentagonal superstructural set of one central and five outer wheel motifs (each wheel motif consists of ten kites (filled blue) and five darts (filled red)) before release (0 h, leftmost) and after release (6 h, 12 h, 24 h, 36 h and 48 h, left to right). b, Regular decagonal superstructural set of ten wheel motifs before release (0 h, leftmost) and after release (12 h, 24 h and 48 h, left to right). c, Regular icosagonal (that is, a 20-sided polygon) superstructural set of 20 wheel motifs before release (0 h, left) and after release (48 h, right). Scale bar (20 μm) shown in a is the same for all images.
Extended Data Fig. 3 Restructuring of Penrose kite tiles in a P2 quasi-crystal after release.
a–d, Before release; e–h, after release. a, Kite tiles are separated by post-processing of the micrograph image from Fig. 2a. Scale bar, 20 μm. b, Effective scattering pattern, given by the Fourier transform intensity of a, showing ten rays extending from the centre to high scattering wavenumbers q. c, Central region of b, magnified by a factor of about six, revealing Bragg peaks at intermediate and low q, corresponding to large distances. d, Central region of c, magnified by a factor of about two, revealing Bragg peaks at very low q associated with superstructural ordering of motifs of kite tiles over large length scales. e, Brownian fluctuations near equilibrium (48 h after release) have caused kite tiles to deviate from the original perfect quasi-crystal order seen in the unreleased structure; kites are separated from Fig. 2e. f, Fourier transform intensity of e. Rays have broadened azimuthally as a consequence of Brownian fluctuations. g, h, Close-ups of f over the same q ranges as for c and d, respectively, showing the smearing of Bragg peaks into ten-fold azimuthal intensity modulations, reminiscent of liquid-crystalline materials, indicating a retention of quasi-crystalline orientational order. Intensity colour scale is the same as in Fig. 2k.
Extended Data Fig. 4 Restructuring of Penrose dart tiles in a P2 quasi-crystal after release.
a–d, Before release; e–h, after release. a, Dart tiles are separated by post-processing of the micrograph image from Fig. 2a. Scale bar, 20 μm. b, Effective scattering pattern, given by the Fourier transform intensity of a, showing ten rays extending from the centre to high scattering wavenumbers q. c, Central region of b, magnified by a factor of about six, revealing Bragg peaks at intermediate and low q, corresponding to large distances. d, Central region of c, magnified by a factor of about two, revealing Bragg peaks at very low q associated with superstructural ordering of motifs of dart tiles over large length scales. e, Brownian fluctuations near equilibrium (48 h after release) have caused dart tiles to deviate from the original perfect quasi-crystal order seen in the unreleased structure; darts are separated from Fig. 2e. f, Fourier transform intensity of e. Rays have broadened azimuthally as a consequence of Brownian fluctuations. g, h, Close-ups of f over the same q ranges as for c and d, respectively, showing the smearing of Bragg peaks into ten-fold azimuthal intensity modulations, reminiscent of liquid-crystalline materials, indicating a retention of quasi-crystalline orientational order. Intensity colour scale is the same as in Fig. 2k.
Extended Data Fig. 5 Tracking anisotropic, bounded Brownian fluctuations of darts in a PSDM.
a, Filled and thresholded optical micrographs of darts in individual video frames (frame number in upper right) are overlayed with blue pentagons with vertices at the centroids of the darts (see Methods, Supplementary Video 4). Red arrows at the vertices of the blue pentagon denote the pointing directions of the darts. The shapes of the blue pentagons fluctuate over time and at any given instant can deviate substantially from a regular pentagon as a consequence of Brownian excitations of the P2 system. Actual time between frames is 720 s. Scale bar, 3 μm. b, Trajectories of the centroids of five darts in a PSDM over a duration of 32 h, after correcting for a slight long-time drift of the entire motif. The time-average position of each dart is denoted by a plus symbol overlaid on each trajectory; the centre of the PSDM is given by crossed box symbol at the centre. These trajectories have non-circular shapes, indicating that the bounded Brownian motion of the darts is anisotropic, reflecting the local five-fold time-averaged quasi-crystal symmetry of the motif. Considering an ensemble average over all five darts, standard deviations of step size distributions projected along directions between the centre of the motif and the time-averaged centroids of the five darts are 1.3 ± 0.2 times less than standard deviations of step size distributions projected perpendicular to these five directions. This small yet detectable anisotropy in the bounded diffusion of the darts reflects the underlying time-averaged five-fold symmetry of their local quasi-crystal environment. Scale bar, 3 μm. c, d, Normalized probability distributions of the calculated area APSDM (c) and internal angles βPSDM (d) of the fluctuating blue pentagons in Supplementary Video 4 and a. Black lines, fits using a log-normal distribution (c) and a Gaussian distribution (d); see Methods for functional forms and fit parameters.
Extended Data Fig. 6 Melting of PSDMs in an unconfined P2 quasi-crystal.
The area fraction of PSDMs ϕA,DM (red darts in Fig. 4a–c) decays to zero at larger distances d, towards the direction where the quasi-crystal is not confined (right). The motif melting front moves from right to left over time: black circles, 4 h after release; orange diamonds, 24 h after release; purple triangles, 48 h after release. Fits are using a Fermi-like function (see Methods); fit parameters are given in Extended Data Table 1.
Supplementary information
Supplementary Information
This file contains Supplementary Methods, Supplementary Discussion, and Supplementary References
Video 1
Release kinetics of Penrose kite and dart tiles forming a fluctuating Brownian P2 quasi-crystal imaged using optical transmission microscopy. Tiles of SU-8 at ϕA = 0.78 are attached initially to a solid Omnicoat release layer on a glass wafer and, after adding a basic aqueous solution–dispersion containing a depletion agent, become mobile yet remain in a monolayer as this release layer dissolves. Kinetics of release are first order (see Fig. 1). Scale: kite’s longer edge = 9.1 µm. Acquisition: 1 frame every 720 s. Acquisition duration: 32 h (i.e. 160 frames). Video playback: duration 16 s at 10 FPS.
Video 2
Equilibrium fluctuations of a 15 kite-5 dart pentagonal flower in a Brownian P2 quasi-crystal. A central pentagonal-star-kite motif (PSKM) is surrounded by five darts and ten outermost kites, forming a flower. The internal PSKM rotates collectively by intermittent hopping as the system is driven by entropic Brownian excitations (evident during playback times between 3 s and 8 s). These entropic fluctuations break the initially achiral symmetry of the flower motifs. Scale: kite’s longer edge = 9.1 µm. Acquisition: 1 frame every 720 s. Acquisition duration: 32 h (i.e. 160 frames). Video playback: duration 16 s at 10 FPS.
Video 3
Equilibrium fluctuations of a 10 kite-5 dart pentagonal wheel in a Brownian P2 quasi-crystal. A central pentagonal-star-dart motif (PSDM) is surrounded by ten kites, forming a wheel. Because the PSDM has a highly corrugated exterior, it is much more sterically inhibited with regard to collective rotations as compared to the pentagonal-star-kite motif. Entropic Brownian fluctuations cause local chiral symmetry breaking of the initially achiral wheel motif. Scale: kite’s longer edge = 9.1 µm. Acquisition: 1 frame every 720 s. Acquisition duration: 32 h (i.e. 160 frames). Video playback: duration 16 s at 10 FPS.
Video 4
Tracking of equilibrium configurational fluctuations of a pentagonal star-dart motif in a Brownian P2 quasi-crystal. The positions and orientations of the five darts in the central pentagonal star-dart motif (PSDM) in Video 3 have been tracked using a custom-programmed routine (see Methods). Blue lines connect the centroids of the darts in filled and thresholded images, yielding a fluctuating pentagonal shape. Red arrows at the vertices of the blue pentagon indicate pointing directions of the constituent darts. Scale, acquisition duration, and video playback: same as Video 3.
Video 5
Collective rotational hopping dynamics of a pentagonal star-kite motif in a fluctuating Brownian P2 quasi-crystal. At ϕA ≈ 0.78, starting at t ≈ 15.3 h after release, the PSKM collectively rotates 36 deg clockwise (CW), 36 deg counterclockwise (CCW), 36 deg CCW, and 36 deg CW back to its original orientation (Fig. 3b). Such collective hopping fluctuations driven by Brownian excitations are a form of heterogeneous dynamics as particles explore available microstates. Scale: kite’s longer edge = 9.1 µm. Acquisition: 1 frame every 1080 s. Acquisition duration: 75 h (i.e. 250 frames). Video playback: duration 25 s at 10 FPS.
Video 6
Observed sound waves in an equilibrium fluctuating Brownian P2 quasi-crystal of hard Penrose kite and dart tiles. Mobile P2 kite and dart tiles exhibit sound wave (i.e. phonon) propagation and scattering phenomena. In-plane interactions between all tiles are nearly hard, and the entire system is driven entropically at room temperature (T = 298 K). Sound wavelets appear to have a limited coherence distance as a consequence of significant scattering and viscous damping. Scale: kite’s longer edge = 9.1 µm. Acquisition: 1 frame every 60 s. Acquisition duration: 7 h (i.e. 425 frames). Video playback: duration 14 s at 30 FPS.
Video 7
Melting a fluctuating Brownian P2 quasi-crystal that is partially unconfined. We lithographically remove one confining wall and then release P2 tiles. After release, unconfined Brownian tiles migrate diffusively towards the empty space (on the right, beyond the field of view), creating a gradient in ϕA, which becomes smaller towards the right (see Fig. 4). When ϕA drops below a critical value, kite and dart motifs melt. Contrast has been enhanced using levels. Scale: kite’s longer edge = 9.1 µm. Acquisition: 1 frame every 180 s. Acquisition duration: 46 h (i.e. 920 frames). Video playback: duration 61 s at 15 FPS.
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Wang, PY., Mason, T.G. A Brownian quasi-crystal of pre-assembled colloidal Penrose tiles. Nature 561, 94–99 (2018). https://doi.org/10.1038/s41586-018-0464-9
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DOI: https://doi.org/10.1038/s41586-018-0464-9
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