Pleats in crystals on curved surfaces

William T. M. Irvine, Vincenzo Vitelli, Paul M. Chaikin
2010 Nature  
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 2-4 . Here we show that, for more general curved surfaces, curvature may
more » ... 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 . Topological defects have played a crucial role in understanding the order, rigidity and melting of crystals and other phases of matter in two-dimensional flat space 14,15 . On a curved surface (Fig. 1) , these particle-like excitations acquire a new life: they interact not only with each other, but with the curvature of the substrate. In a hexagonal lattice in which every particle has six nearest neighbours (Fig. 1, inset) , there are two types of topological defects (Fig. 2) : disclinations that disrupt orientational order and appear as points of local five-fold or seven-fold symmetry, (pentagons or heptagons, having topological charge 6(2p/6), and dislocations, which disrupt translational order and appear as disclination dipoles (1/2 pairs). That disclinations couple to curvature can be understood intuitively by taking a piece of paper, and adding, or removing, a p/3 wedge to 'make' a disclination, Fig. 2c, d. A host of new discoveries 2,4,7,16 have resulted from studies of these defects on the simplest curved surface: the sphere. With increasing size, the familiar 12-pentagon soccer ball pattern gives way to 'scars', pentagons dressed by strings of dislocations 2-4 . In this Letter, we introduce a different configuration of dislocations, namely, 'pleats'-topologically uncharged grain boundaries with variable spacing that vanish on the surface. We experimentally investigate their interaction with curvature and show when pleats are energetically favoured over undefected crystals or topologically charged disclinations. Apart from experiments on spheres, and bubble bearing paraboloids 17 , the interaction of defects with variable curvature, negative curvature and surfaces of different topologies has remained largely unexplored experimentally and is of growing theoretical interest [8] [9] [10] [11] [18] [19] [20] . The topology of surfaces 21 places a constraint on the total defect charge-for example the net charge on a sphere must be 4p, as exemplified by a soccer ball that has 12 (54p/(p/3)) pentagons dispersed among hexagons. A hemisphere, or disk, requires half the topological a b c d Figure 1 | Colloidal crystals on curved oil-glycerol interfaces. a-d, Fluorescent PMMA particles bound, by image attraction, to oil-glycerol interfaces in the shape of spheres (a), domes (b), waists (c) and barrels (d) (see Supplementary Information section 2 for details of shape). The particles interact via a repulsive screened Coulomb interaction and, on a flat surface, arrange into a hexagonal crystal lattice (a, inset). The oil phase is a mixture of cyclohexyl bromide and dodecane that matches the refractive index of glycerol, allowing us to image particles on highly curved interfaces by confocal microscopy. Because of their topology, crystals on spheres and domes require a net defect charge of 12 3 (2p/6) and 6 3 (2p/6), respectively, whereas waists and barrels require none. While spheres (a) have no boundary, the remaining surfaces (b-d) do, allowing topological defects a choice between boundary and bulk.
doi:10.1038/nature09620 pmid:21164482 fatcat:yn4g7oginjetjcho4fgmuboz74