Three-dimensional formulation of curved nematic shells

TL;DR

This paper explores the transition from three-dimensional to two-dimensional nematic liquid crystal descriptions on curved shells, highlighting conditions where simplifications are valid or lead to significant differences, especially considering activity and curvature effects.

Contribution

It provides a theoretical framework for understanding when a 2D description of nematic shells is appropriate and identifies physical differences caused by curvature and activity.

Findings

Transition from first to second order isotropic-nematic phase change with weak anchoring.

Conditions for valid 2D nematic shell approximation clarified.

Differences between 2D and 3D descriptions in curved geometries identified.

Abstract

In soft matter, the phase of nematic liquid crystals can be made from anisotropic molecules in single component materials, or as a suspension of mesoscopic nematogens. The later offers more versatility in the experimental design of complex shapes, in particular thin curved shells, and is often found in biological systems at multiple scales from cells to tissues. Here, we investigate theoretically the transition from three-dimensional nematics to a two-dimensional description restricted to a tangent plane, using a mean-field approach. We identify a transition from first to second order isotropic-nematic transition in presence of weak tangential anchoring. Then, we clarify the conditions under which a two-dimensional description of thin nematic shells is relevant. Nonetheless, using the example of active nematic stress, we identify physical differences between two- and three-dimensional descriptions in curved geometry. Finally, we construct a thin film approximation of nematohydrodynamics for a nematic shell in contact with a curved substrate. All together, those results show that a tangential restriction of nematic orientation must be used with care in presence of curvature, activity, or weak anchoring boundary conditions.