Soft elasticity in biaxial smectic and smectic-$C$ elastomers

Ideal (monodomain) smectic-$A$ elastomers cross-linked 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.

Figure 1

Sample distortion and rotations of the Frank director $\mathbf{n}$, the uniaxial anisotropy axis $\mathbf{e}\equiv {\mathbf{e}}_z$, and the layer normal $\mathbf{N}$ in a transition from a (a) $\mathrm{Sm}A$ to (b) a biaxial and (c) a sheared $\mathrm{Sm}C$ elastomer. In part (c) we chose the geometry, i.e., the coordinate system in target space, so that the smectic layers do not rotate.

Figure 2

Soft shears in biaxial elastomers. The force $\mathbf{f}$ exerted across an infinitesimal surface element $d\mathbf{S}$ is $f_i={\sigma }_{ij}d{S}_j$. There are no restoring forces for (a) external forces along $\pm {\stackrel{\mathrm{̃}}{\mathbf{e}}}_x$ applied to opposing surfaces with normal along $\pm {\stackrel{\mathrm{̃}}{\mathbf{e}}}_y$, and (b) for external forces along $\pm {\stackrel{\mathrm{̃}}{\mathbf{e}}}_y$ applied to surfaces with normal along $\pm {\stackrel{\mathrm{̃}}{\mathbf{e}}}_x$.

Figure 3

Schematic plot (arbitrary units) of the engineering stress ${\sigma }_{yy}^{\mathrm{eng}}$ vs the deformation ${\Lambda }_{yy}^{\prime }$ for a soft biaxial elastomer with an equilibrium order in the $xy$-plane along $x$ and positive anisotropy, $r_{\perp }>0$. Up to a critical deformation ${\Lambda }_{0yy}^{\prime }=\sqrt{r_{\perp }}$ the sample responds to the deformation merely by rotating $\stackrel{\mathrm{̃}}{\mathbf{c}}$ and consequently the stress is zero. Above ${\Lambda }_{0yy}^{\prime }$ the sample stretches along the new direction $y$ of the $c$-director and ${\sigma }_{yy}^{\mathrm{eng}}$ grows linearly for small ${\Lambda }_{yy}^{\prime }-{\Lambda }_{0yy}^{\prime }$.

Figure 4

Soft distortions of $\mathrm{Sm}C$ elastomers. There are no restoring forces (a) for external forces in the $xz$-plane along $\pm {\tilde{\mathbf{e}}}_2$ applied to opposing surfaces with normal along $\pm {\tilde{\mathbf{e}}}_y$, and (b) for external forces along $\pm {\tilde{\mathbf{e}}}_y$ applied to surfaces with normal along $\pm {\tilde{e}}_2$.

Figure 5

Schematic representation of distortions in the $xz$-plane induced by the $\mathrm{Sm}A$-to-$\mathrm{Sm}C$ transition: (a) undistorted $\mathrm{Sm}A$ phase, (b) $\mathrm{Sm}C$ phase with a symmetric deformation tensor $\underset{̱}{\underset{̱}{\Lambda }}$, (c) $\mathrm{Sm}C$ phase with ${\Lambda }_{xz}>0$ and ${\Lambda }_{zx}=0$, and (d) $\mathrm{Sm}C$ phase with ${\Lambda }_{xz}=0$ and ${\Lambda }_{zx}>0$.

Figure 6

(Color online) Effect of deformations ${\underset{̱}{\underset{̱}{\Lambda }}}^{\prime }$ as given in Eq. (5.52) for a series of values of $\vartheta$ between (a) and (f) $\pi ∕2$ with $\vartheta$ increased in steps of $\pi ∕10$. In the process a parallelepiped-shaped sample with initial shear in the $xz$-plane is transformed into a parallelepiped with shear in the $yz$-plane that appears as if the original parallelepiped had been rotated by $\pi ∕2$ about the $z$-axis.