Pleats in crystals on curved surfaces

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