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Pleats in crystals on curved surfaces

Abstract

Hexagons can easily tile a flat surface, but not a curved one. Introducing heptagons and pentagons (defects with topological charge) makes it easier to tile curved surfaces; for example, soccer balls based on the geodesic domes1 of Buckminster Fuller have exactly 12 pentagons (positive charges). Interacting particles that invariably form hexagonal crystals on a plane exhibit fascinating scarred defect patterns on a sphere2,3,4. Here we show that, for more general curved surfaces, curvature may be relaxed by pleats: uncharged lines of dislocations (topological dipoles) that vanish on the surface and play the same role as fabric pleats. We experimentally investigate crystal order on surfaces with spatially varying positive and negative curvature. On cylindrical capillary bridges, stretched to produce negative curvature, we observe a sequence of transitions—consistent with our energetic calculations—from no defects to isolated dislocations, which subsequently proliferate and organize into pleats; finally, scars and isolated heptagons (previously unseen) appear. This fine control of crystal order with curvature will enable explorations of general theories of defects in curved spaces5,6,7,8,9,10,11. From a practical viewpoint, it may be possible to engineer structures with curvature (such as waisted nanotubes and vaulted architecture) and to develop novel methods for soft lithography12 and directed self-assembly13.

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Figure 1: Colloidal crystals on curved oil–glycerol interfaces.
Figure 2: Disclinations and pleats in a hexagonal lattice.
Figure 3: Topological charge on domes and waists.
Figure 4: Pleating and disclination unbinding on a stretched capillary bridge.
Figure 5: Disclination and polarization charge on the surfaces of waists, domes and barrels.

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Acknowledgements

We acknowledge discussions with M. Bowick, A. Grosberg, S. Sacanna and A. M. Turner. W.T.M.I. acknowledges guidance in particle synthesis from A. D. Hollingsworth and M. T. Elsesser. W.T.M.I. acknowledges support from Rhodia and the English Speaking Union. P.M.C. acknowledges support from MRSEC DMR-0820341 and NASA NNX08AK04G. W.T.M.I. and V.V. acknowledge hospitality from Stichting FOM and the Aspen Center for Physics.

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Authors

Contributions

P.M.C. and W.T.M.I. initiated and designed research. W.T.M.I. designed and performed experiments, analysed data and synthesized colloidal particles. W.T.M.I., P.M.C and V.V. interpreted data. V.V. and W.T.M.I. performed elasticity calculations. W.T.M.I. and P.M.C. wrote the manuscript.

Corresponding authors

Correspondence to William T. M. Irvine or Paul M. Chaikin.

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The authors declare no competing financial interests.

Supplementary information

Supplementary Information.

The file contains Supplementary Materials comprising: 1 Cut-out disclinations; 2 Capillary bridge shape; 3 Energetics of dislocations and pleats on curved capillary bridges and 4Topological charge in a discrete hexagonal lattice. The file also contains Supplementary Figures and legends and information on Supplementary Movies 1 and 2. (PDF 1485 kb)

Supplementary Movie 1

The movie shows Raw confocal data for an experiment in which the authors pulled on a quasi-cylindrical capillary bridge. (MOV 14804 kb)

Supplementary Movie 2

The movie shows the reconstructed crystal structure from the confocal data of supplementary movie 1. (MOV 16347 kb)

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Irvine, W., Vitelli, V. & Chaikin, P. Pleats in crystals on curved surfaces. Nature 468, 947–951 (2010). https://doi.org/10.1038/nature09620

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