Journal Article

Pleats in crystals on curved surfaces

Vitelli, Vincenzo, Chaikin, Paul M., Irvine, William T. M.

2010

Overview

Pleating crystals A hexagonal lattice can be persuaded to tile a curved surface by the introduction of alternative shapes or 'topological defects' — such as heptagons and pentagons — as in the well-known 'buckyball' with its 20 hexagons and 12 pentagons. This paper reports a previously unknown type of defect that accommodates curvature in the same way as fabric pleats. Defects of this type, uncharged grain boundaries that vanish on the surface, can be observed on the negatively curved surfaces of stretched colloidal crystals. These findings will facilitate the exploration of general theories of defects in curved spaces, the engineering of curved structures and novel methods for soft lithography and directed self-assembly. Hexagons can easily tile a flat surface, but not a curved one. Defects with topological charge (such as heptagons and pentagons) make it easier to tile curved surfaces, such as soccer balls. Here, a new type of defect is reported that accommodates curvature in the same way as fabric pleats. The appearance of such defects on the negatively curved surfaces of stretched colloidal crystals are observed. The results will facilitate the exploration of general theories of defects in curved spaces, the engineering of curved structures and novel methods for soft lithography and directed self-assembly. 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 domes 1 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 sphere 2 , 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 spaces 5 , 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 lithography 12 and directed self-assembly 13 .

• Publisher: Nature Publishing Group UK,Nature Publishing Group
• Subject: 639/301/930/328/1978 / 639/638/541/961 / Algebraic topology / Composition; defects and impurities / Condensed matter: structure, mechanical and thermal properties / Crystal defects / Crystal structure / Crystallography / Crystals / Curvature / Curved / Dislocations / Evaluation / Exact sciences and technology / Glycerin / Glycerol / Humanities and Social Sciences / letter / Mathematical analysis / multidisciplinary / Nanotechnology / Physics / Properties / Science / Science (multidisciplinary) / Self assembly / Solid surfaces and solid-solid interfaces / Structure / Surface structure and topography / Surfaces and interfaces; thin films and whiskers (structure and nonelectronic properties) / Topology