Characterization of soft stripe-domain deformations in $\text{Sm-}C$ and $\text{Sm-}{C}^{\ast }$ liquid-crystal elastomers

The neoclassical model of $\text{Sm-}C$ (and $\text{Sm-}{C}^{\ast }$) elastomers developed by Warner and Adams predicts a class of “soft” (zero energy) deformations. We find and describe the full set of stripe domains—laminate structures in which the laminates alternate between two different deformations—that can form between pairs of these soft deformations. All the stripe domains fall into two classes, one in which the smectic layers are not bent at the interfaces, but for which—in the $\text{Sm-}{C}^{\ast }$ case—the interfaces are charged, and one in which the smectic layers are bent but the interfaces are never charged. Striped deformations significantly enhance the softness of the macroscopic elastic response.

Figure 1

(Color online) A $\text{Sm-}C$ elastomer. liquid crystal (LC).

Figure 2

(Color online) Top: a small region of a $\text{Sm-}C$ elastomer after the formation of the first category of stripe domain. The region was cuboidal before deformation and has undergone different soft deformations on either side of the orange boundary plane. The smectic layers, which deform affinely, pass through the boundary without being bent. This figure was drawn with the material parameters $r=25$ and $\theta =0.6$. Bottom: a top view of a square region of a smectic layer that passes through the boundary, which is shown as an (orange) dashed line. The ovals represent the projection of the liquid-crystal rods in the plane and the arrows show the projection of the liquid-crystal director in the plane. The component of the LC director in the smectic plane, which we call $\hat{\mathbf{c}}$, is also shown in white on the top figure.

Figure 3

(Color online) Top: a small region of a sample of $\text{Sm-}C$ elastomer after the formation of the second type of stripe domain. Before deformation the sample was cuboidal. In this type of stripe domain, the smectic layers are bent at the boundary. This figure was drawn with the material parameters $r=25$ and $\theta =0.6$. Bottom: the two halves of the region on top each viewed down its smectic layer normal. The red ovals are the components of the LC rods in the LC layer plane and the black arrows are the components of the LC director in the LC layer plane. The component of the LC director in the smectic plane, which we call $\hat{\mathbf{c}}$, is also shown in white on the top figure.

Figure 4

Primed axes defined after deformation.

Figure 5

Left: a block of nematic elastomer in the reference configuration split into two regions. Middle: uniaxial stretches are applied in the two regions. The boundary is stretched to the same degree by both stretches provided the boundary line bisects the two axes; however, the material still rips at the boundary. Right: a body rotation of the two regions can restore material continuity because the boundary was stretched by the same amount under both deformations. The strains in the third direction (out of the page) also match because this direction is perpendicular to both axes, so the interfacial plane normal is in the same plane as the two axes and bisects them.

Figure 6

The same construction as Fig. 5 but for smectic elastomers. Top left: a slice through a block of $\text{Sm-}C$ elastomer in the reference state. The smectic planes are shown as dotted lines. The slice is at an angle to the planes so the planes are not, in general, either perpendicular or parallel to the slice. Middle: two uniaxial stretches are applied in the two regions. As in the nematic case the boundary must bisect the axes, but in this case the axes must also make equal angles with the dotted lines to ensure the smectic director makes the preferred tilt angle with the layer normal. As before this rips the material. Top right: since the boundary is stretched to the same degree by both stretches material continuity can be restored by body rotations. The resulting structure has the layers bent at the boundary. Bottom: the same construction again but with the other possible bisecting boundary. In this case the layers are not bent at the boundary.

Figure 7

A different deformation is applied on each side of a plane boundary. A stripe domain is simply many repetitions of this structure so that the deformation alternates in stripes separated by parallel plane boundaries.

Figure 8

A sample in the (layered) reference state is divided into two regions and different (but compatible) soft deformations are applied in each region. The deformations move the boundary between the regions and the layers and introduce an LC director in both regions.

Figure 9

Shear angle across the boundary for both classes of stripe domain as a function of $b$ plotted with $r=25$ and $\theta =0.6$; the maximum shear angle is $\text{tan}\left(\zeta \right)=\left(r-1\right)\text{sin}\left(2\theta \right)/\left[\left(r-1\right){\text{cos}}^2\left(\theta \right)+1\right]$ and occurs at $b=\pi /2$.