Negative Poisson's ratio and semisoft elasticity of smectic-$C$ liquid-crystal elastomers

Models of smectic-$C$ liquid-crystal elastomers predict that they can display soft elasticity, in which the shape of the elastomer changes at no energy cost. The amplitude of the soft mode and the accompanying shears are dependent on the orientation of the layer normal and the director with respect to the stretch axis. We demonstrate that in some geometries the director is forced to rotate perpendicular to the stretch axis, causing lateral expansion of the sample—a negative Poisson's ratio. Current models do not include the effect of imperfections that must be present in the physical sample. We investigate the effect of a simple model of these imperfections on the soft modes in monodomain smectic-$C$ elastomers in a variety of geometries. When stretching parallel to the layer normal (with imposed strain) the elastomer has a negative stiffness once the director starts to rotate. We show that this is a result of the negative Poisson's ratio in this geometry through a simple scalar model.

Figure 1

(a) The layer normal $\mathbf{k}$ and director $\mathbf{n}$ in the Sm-$C$ phase, and the transverse dipole orientation $\mathbf{p}$ in the Sm-$C^{*}$ phase (polymer chains not shown). (b) The director and layer normals in a pseudo-monodomain.

Figure 2

For the parameter values $r=2$ and ${\theta }_0=0.5$ radians, (a) shows the diagonal components of the deformation matrix, (b) shows the shear components, and (c) shows an illustration of the deformations on the LCE, together with the component of the director perpendicular to the layer normal, $\mathbf{c}$.

Figure 3

The components of the upper triangular deformation matrix for the Sm-$C$ soft mode stretching parallel to the layer normal, with $r=2$ and ${\theta }_0=0.5$ radians. Initially ${\mathbf{k}}_0=\mathbf{x}$ and ${\mathbf{n}}_0=\cos {\theta }_0\mathbf{x}+\sin {\theta }_0\mathbf{y}$.

Figure 4

An illustration of the Sm-$C$ elastomer deformation when stretching parallel to the layer normal. The director (red) moves out into the $z$ direction perpendicular to the stretch axis, maintaining its tilt angle with respect to the layer normal (white) causing the elastomer to expand in the perpendicular direction.

Figure 5

An illustration of the vectors $\widehat{\mathbf{a}}$ and $\widehat{\mathbf{b}}$ used in the numerical calculations.

Figure 6

The orientation of the director and layer normal for each of the elongations considered: (a) perpendicular to the layer normal (note that the director and layer normal are both perpendicular to the $\mathbf{x}$ direction initially), (b) parallel to the layer normal, (c) perpendicular to the director, and (d) at an angle $\psi$ to the layer normal.

Figure 7

The stress and the angle of rotation of a Sm-$C$ elastomer when stretched perpendicular to both the layer normal and the director. The model parameters are $b=60$, $r=2$, ${\theta }_0=0.5$ (radians), and the $\alpha$ and $c$ values shown on the figure. The thick curves (green) are from the more general numerical relaxation, and the black curves are calculated using the decomposition of the deformation matrix explained in the text.

Figure 8

The components of the deformation tensor for $(\alpha ,b,c,{\theta }_0,r)=(0.05,60,1,0.5,2)$ when stretching perpendicular to both the director and the layer normal. Note that the sympathetic shears persist, as the director is unable to complete its $\pi /2$ rotation.

Figure 9

(a) The stress-strain response for a semisoft Sm-$C$ elastomer stretched parallel to the layer normal. The model parameters are $b=60$, $r=2$, and ${\theta }_0=0.5$, and the values of $(\alpha ,c)$ shown in the figure. (b) The corresponding IPRs for the stress-strain curves.

Figure 10

For the parameter values $(\alpha ,c)=(0.05,\infty )$ and $b=60$, $r=2$, and ${\theta }_0=0.5$ radians for stretching parallel to the layer normal, (a) shows the director (thick black) and layer normal rotation (thin red), (b) the shear components, and (c) the diagonal components of the deformation tensor when stretching parallel to the layer normal.

Figure 11

For the scalar model of the negative stress strain curve described in the text, (a) shows the stress-strain curves for $A=0.5,0.75,1$, and (b) the deformation components for $A=1$ for the scalar auxetic model.

Figure 12

(a) The stress-strain curves for stretching perpendicular to the layer normal, and (b) the IPR for various parameters $b=60$, $r=2$, and ${\theta }_0=0.5$ and values of $(\alpha ,c)$ shown in the figure.

Figure 13

When stretching perpendicular to the director, for the parameter values $(\alpha ,c)=(0.05,\infty )$ and $b=60$, $r=2$, and ${\theta }_0=0.5$ radians, (a) shows the director (thick black) and layer normal rotation (thin red), (b) the shear components, and (c) the diagonal components of the deformation tensor when stretching perpendicular to ${\mathbf{n}}_0$.

Figure 14

The free energy calculated numerically (dashed black), and the free energy trajectory of the semisoft mode with continuous director rotation (thick green) when stretching perpendicular to the director. Here $b=60,r=2,{\theta }_0=0.5$, and $(\alpha ,c)$ are shown in the figure.

Figure 15

(a) The stress-strain curves for stretching at an angle of $\psi =0.65$ radians to the layer normal, for $b=60,r=2,{\theta }_0=0.5$ radians, and various parameter values $(\alpha ,c)$, and (b) the Poisson's ratio in this geometry.

Figure 16

For stretching at an angle of $\psi =0.65$ radians to the layer normal, (a) shows the director (thick black) and layer normal (thin red), (b) the shear components of the deformation, and (c) the diagonal components of the deformation for stretching at an angle of $\psi =0.65$ radians to the layer normal, for the case $(\alpha ,c)=(0.05,\infty )$ and $b=60,r=2,{\theta }_0=0.5$.

Figure 17

The average value of ${\lambda }_{yy}$ for 50 domains in a pseudo-monodomain illustrated in Fig. 1 as a function of ${\lambda }_{xx}$ assuming they all experience the same strain and deform independently. Model parameters are $b=60,r=2,{\theta }_0=0.5,c=\infty ,\alpha =0.05$.

Figure 18

(a) An illustration of a discontinuous stress-strain curve for the scalar model described in the text. (b) The free energy as a function of the variable ${\epsilon }_{\mathrm{SM}}$ for fixed total strain values. Here $K_1=10$ and $K_2=1$.