Elastic moduli of a smectic membrane: a rod-level scaling analysis

TL;DR

This paper develops a microscopic variational theory to calculate the elastic moduli of rod-based smectic membranes, revealing significant differences from bulk nematic fluids and emphasizing the importance of considering distinct elastic constants.

Contribution

It introduces a detailed microscopic approach to determine elastic moduli of rods in smectic membranes, highlighting differences from bulk nematic phases and challenging the one-constant approximation.

Findings

Elastic moduli are significantly different from bulk nematic fluids.

Splay elasticity is much larger, curvature elasticity is smaller in membranes.

The one-constant approximation is inadequate for modeling smectic membranes.

Abstract

Chiral rodlike colloids exposed to strong depletion attraction may self-assemble into chiral membranes whose twisted director field differs from that of a 3D bulk chiral nematic. We formulate a simple microscopic variational theory to determine the elastic moduli of rods assembled into a bi-dimensional smectic membrane. The approach is based on a simple Onsager-Straley theory for a non-uniform director field that we apply to describe rod twist within the membrane. A microscopic approach enables a detailed estimate of the individual Frank elastic moduli (splay, twist and bend) as well as the twist penetration depth of the smectic membrane in relation to the rod density and shape. We find that the elastic moduli are distinctly different from those of a bulk nematic fluid, with the splay elasticity being much larger and the curvature elasticity much smaller than for rods assembled in a three-dimensional nematic fluid. We argue that the use of the simplistic one-constant approximation in which all moduli are assumed to be of equal magnitude is not appropriate for modelling the structure-property relation of smectic membranes.