Geometric Theory of Columnar Phases on Curved Substrates

TL;DR

This paper develops a geometric theory describing how self-assembled columnar structures adapt to curved substrates, revealing that curvature influences their arrangement through geodesic convergence and out-of-plane bending effects.

Contribution

It introduces a novel geometric framework linking substrate curvature to columnar organization, highlighting the role of geodesics and bending as key factors.

Findings

Columns align along geodesics influenced by Gaussian curvature

Out-of-plane bending acts as an effective ordering field

Curvature induces convergence or divergence of column normals

Abstract

We study thin self-assembled columns constrained to lie on a curved, rigid substrate. The curvature presents no local obstruction to equally spaced columns in contrast to curved crystals for which the crystalline bonds are frustrated. Instead, the vanishing compressional strain of the columns implies that their normals lie on geodesics which converge (diverge) in regions of positive (negative) Gaussian curvature, in analogy to the focussing of light rays by a lens. We show that the out of plane bending of the cylinders acts as an effective ordering field.