Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Extrinsic Curvature, Geometric Optics, and Lamellar Order on Curved
  Substrates   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.

17
The categories of this paper are: q-fin
30
The categories of this paper are: q-alg
8
The categories of this paper are: hep-ex
35
The categories of this paper are: mtrl-th
21
The categories of this paper are: adap-org
6
The categories of this paper are: astro-ph
15
The categories of this paper are: nucl-ex
24
The categories of this paper are: dg-ga
23
The categories of this paper are: solv-int
11
The categories of this paper are: hep-lat
2
The categories of this paper are: math
16
The categories of this paper are: stat
1
The categories of this paper are: cs
27
The categories of this paper are: comp-gas
5
The categories of this paper are: gr-qc
0
The categories of this paper are: hep-ph