Extrinsic Curvature, Geometric Optics, and Lamellar Order on Curved Substrates

TL;DR

This paper explores how the geometry of curved substrates influences lamellar pattern formation, revealing principles to engineer desired structures through local and non-local geometric effects, with potential applications in material design.

Contribution

It introduces a framework combining local curvature minimization and geodesic normal alignment to control lamellar morphologies on curved surfaces.

Findings

Surface curvature directs lamellar alignment and pattern formation.

Geodesic constraints lead to lens-like effects in lamellar structures.

Complex morphologies can be engineered by combining geometric effects.

Abstract

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.