Smectic elastomer membranes

We present a model for smectic elastomer membranes which includes elastic and liquid-crystalline degrees of freedom. Based on our model, we determined the qualitative phase diagram of a smectic elastomer membrane using mean-field theory. This phase diagram is found to comprise five phases, viz., smectic-$A$-flat, smectic-$A$-crumpled, smectic-$C$-flat, smectic-$C$-crumpled, and smectic-$C$-tubule phases, where in the latter phase, the membrane is flat in the direction of mesogenic tilt and crumpled in the perpendicular direction. The transitions between adjacent phases are second-order phase transitions. We study in some detail the elasticity of the smectic-$C$-flat and the smectic-$C$-tubule phases which are associated with a spontaneous breaking of in-plane rotational symmetry. As a consequence of the Goldstone theorem, these phases exhibit soft elasticity characterized by the vanishing of in-plane shear moduli.

Figure 1

Sketch of the cross section of (a) $\mathrm{Sm}A$ and (b) $\mathrm{Sm}C$ elastomer membranes. The membranes consist of a few smectic layers such that their height can be neglected in comparison to their lateral extension.

Figure 2

Schematic phase diagram of a smectic elastomer membrane in mean field theory. There are five phases, viz., $\mathrm{Sm}A$-flat, $\mathrm{Sm}C$-flat, $\mathrm{Sm}A$-crumpled, $\mathrm{Sm}C$-crumpled, and $\mathrm{Sm}C$-tubule phases separated by second-order phase transitions (solid lines). The five second-order lines meet at the origin, which, therefore, is a pentacritical point.