弯曲衬底上的外曲率、几何光学和层状序
Extrinsic Curvature, Geometric Optics, and Lamellar Order on Curved Substrates
摘要：
当热能较弱时，限制在弯曲基底上的二维层状结构显示出复杂的图案，这是由于存在几何约束时层弯曲和压缩之间的竞争而产生的。我们提出了广泛的设计原则，工程师的几何形状的底层基板，以便可以通过自组装获得所需的层状图案。两个不同的物理效应被确定为关键因素，有助于底层表面的形状和由此产生的层状形态之间的相互作用。第一个是一个局部排序字段的方向，每个单独的层，往往最小化其曲率相对于三维嵌入。第二种是由表面的固有几何形状控制的非局部效应，其迫使（几乎不可压缩的）层的法线位于测地线上，导致光学中的焦散形成。因此，主要具有正或负高斯曲率的不同表面形态可以分别充当会聚或发散透镜。通过组合这些成分，如将与不同的光学元件，可以获得复杂的层状形态。这种近晶验光使得能够操纵用于设计新型材料的层状配置。
英文摘要：
When thermal energies are weak, two dimensional lamellar structures confined on a curved substrate display complex patterns arising from the competition between layer bending and compression in the presence of geometric constraints. We present broad design principles to engineer the geometry of the underlying substrate so that a desired lamellar pattern can be obtained by self-assembly. Two distinct physical effects are identified as key factors that contribute to the interaction between the shape of the underlying surface and the resulting lamellar morphology. The first is a local ordering field for the direction of each individual layer which tends to minimize its curvature with respect to the three-dimensional embedding. The second is a non-local effect controlled by the intrinsic geometry of the surface that forces the normals to the (nearly incompressible) layers to lie on geodesics, leading to caustic formation as in optics. As a result, different surface morphologies with predominantly positive or negative Gaussian curvature can act as converging or diverging lenses respectively. By combining these ingredients, as one would with different optical elements, complex lamellar morphologies can be obtained. This smectic optometry enables the manipulation of lamellar configurations for the design of novel materials.
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