Center for Soft Matter Research, Department of Physics, New York University, 4 Washington Place, New York, New York 10003, USA.
Nature. 2010 Dec 16;468(7326):947-51. doi: 10.1038/nature09620.
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 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. 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. 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 and directed self-assembly.
正六边形可以轻易铺满一个平面，但无法铺满曲面。引入七边形和五边形（具有拓扑电荷的缺陷）会使曲面更容易平铺；例如，基于巴克明斯特·富勒的测地线穹顶的足球恰好有 12 个五边形（正电荷）。在平面上总是形成正六边形晶体的相互作用粒子在球面上表现出引人入胜的瘢痕缺陷模式。在这里，我们表明，对于更一般的曲面，褶皱可以使曲率松弛：在表面上消失的无电荷位错线（拓扑偶极子），它们起着与织物褶皱相同的作用。我们在具有空间变化的正曲率和负曲率的表面上实验性地研究了晶体的有序性。在拉伸以产生负曲率的圆柱形毛细桥上，我们观察到一系列转变——与我们的能量计算一致——从无缺陷到孤立位错，随后位错增殖并形成褶皱；最后，出现了瘢痕和孤立的七边形（以前未见）。这种对曲率下晶体有序性的精细控制将能够探索弯曲空间中缺陷的一般理论。从实际的角度来看，有可能设计出具有曲率的结构（例如腰部纳米管和拱形建筑），并开发出用于软光刻和定向自组装的新方法。
