Nematic director fields and topographies of shells of revolution

TL;DR

This paper develops methods to predict and design the shape of nematic sheets transforming into surfaces of revolution with complex topography, using light or heat to induce specific director fields without stretching energy.

Contribution

It introduces analytical solutions and a general integral equation for controlling shape transformation in nematic sheets, including inverse design of director fields for desired surfaces.

Findings

Derived explicit director fields for catenoids and paraboloids.

Established a framework for forward and inverse shape design.

Analyzed the evolution of Gaussian curvature during deformation.

Abstract

We solve the forward and inverse problems associated with the transformation of flat sheets to surfaces of revolution with non-trivial topography, including Gaussian curvature, without a stretch elastic cost. We deal with systems slender enough to have a small bend energy cost. Shape change is induced by light or heat causing contraction along a non-uniform director field in the plane of an initially flat nematic sheet. The forward problem is, given a director distribution, what shape is induced? Along the way, we determine the Gaussian curvature and the evolution with induced mechanical deformation of the director field and of material curves in the surface (proto-radii) that will become radii in the final surface. The inverse problem is, given a target shape, what director field does one need to specify? Analytic examples of director fields are fully calculated that will, for specific deformations, yield catenoids and paraboloids of revolution. The general prescription is given in terms of an integral equation.