Lifting map for ordered surfaces

TL;DR

This paper introduces a general method to lift order tensors from flat to curved surfaces, facilitating the modeling of functionalized materials with surface order, exemplified through nematic shells and molecular dynamics.

Contribution

It proposes a novel, general approach for representing surface order tensors by lifting from flat to curved geometries, applied specifically to nematic shells.

Findings

Method successfully models order on curved surfaces.

Application to nematic shells offers new insights.

Aligns with molecular dynamics experiment results.

Abstract

When a material surface is functionalized so as to acquire some type of order, functionalization of which soft condensed matter systems have recently provided many interesting examples, the modeller faces an alternative. Either the order is described on the curved, physical surface where it belongs, or it is described on a flat surface that is unrolled as pre-image of the physical surface under a suitable height function. This paper proposes a general method that pursues the latter avenue by lifting whatever order tensor is deemed appropriate from a flat to a curved surface. To produce a specific application, we specialize this method to nematic shells, for which it also provides a simple, but convincing interpretation of the outcomes of some molecular-dynamics experiments on ellipsoidal shells.