Soft elasticity in biaxial smectic and smectic-C elastomers

TL;DR

This paper investigates the elastic properties of biaxial smectic elastomers and smectic-C elastomers, revealing their soft elasticity due to spontaneous symmetry breaking through theoretical models.

Contribution

It introduces three related models to analyze the elasticity and phase transitions of biaxial and smectic-C elastomers, highlighting the origin of soft elasticity.

Findings

Emergent phases exhibit soft elasticity with vanishing elastic moduli.

Spontaneous symmetry breaking is key to the softness phenomena.

Different manifestations include absence of restoring forces under specific strains.

Abstract

Ideal (monodomain) smectic-$A$ elastomers crosslinked in the smectic-$A$ phase are simply uniaxial rubbers, provided deformations are small. From these materials smectic-$C$ elastomers are produced by a cooling through the smectic-$A$ to smectic-$C$ phase transition. At least in principle, biaxial smectic elastomers could also be produced via cooling from the smectic-$A$ to a biaxial smectic phase. These phase transitions, respectively from $D_{\infty h}$ to $C_{2h}$ and from $D_{\infty h}$ to $D_{2h}$ symmetry, spontaneously break the rotational symmetry in the smectic planes. We study the above transitions and the elasticity of the smectic-$C$ and biaxial phases in three different but related models: Landau-like phenomenological models as functions of the Cauchy--Saint-Laurent strain tensor for both the biaxial and the smectic-$C$ phases and a detailed model, including contributions from the elastic network, smectic layer compression, and smectic-$C$ tilt for the smectic-$C$ phase as a function of both strain and the $c$-director. We show that the emergent phases exhibit soft elasticity characterized by the vanishing of certain elastic moduli. We analyze in some detail the role of spontaneous symmetry breaking as the origin of soft elasticity and we discuss different manifestations of softness like the absence of restoring forces under certain shears and extensional strains.