Shape Morphing of Planar Liquid Crystal Elastomers

We consider planar liquid crystal elastomers: two-dimensional objects made of anisotropic responsive materials that remain flat when stimulated, however change their planar shape. We derive a closed form, analytical solution based on the implicit linearity featured by this subclass of deformations. Our solution provides the nematic director field on an arbitrary domain starting with two initial director curves. We discuss the different gauge choices for this problem and the inclusion of disclinations in the nematic order. Finally, we propose several applications and useful design principles based on this theoretical framework.

Figure 1

Planar liquid crystal elastomer (PLCE) deformations of a circular domain. The elongations along (by factor ${\lambda }_1$) and perpendicular (${\lambda }_2$) to the director field deform the sheet and change the shape of its boundary, without buckling out of the plane (regardless of the sheet’s thickness).

Figure 2

Solving the PLCE director field. (a) A domain and two curves that intersect orthogonally are set in the laboratory $xy$ coordinates; the curves will become director and director-perpendicular integral curves. (b) We fix the gauge functions $\alpha (u,v_0)$, $\beta (u_0,v)$ by choosing a parametrization of the initial curves. The geodesic curvatures further fix the gauge functions $r(u)$, $t(v)$. (c) This sets the Goursat initial value problem, whose (d) solution in the $uv$ coordinates is given by the expression (9). (e) Finally, we map the resolved director field back to the $xy$ coordinates, and restrict it to the desired domain.

Figure 3

Geodesic curvature changes of the director field integral curves. (a) the nematic director is everywhere positively bent and positively splayed. The geodesic curvatures of both sets of integral curves are everywhere nonzero. (b) here, the bend $b$ changes sign twice while $s$ remains positive; the change of signs occurs along $v$ curves, a hallmark of LPCEs. (c) similarly, $s$ could only change sign across $u$ curves.

Figure 4

A PLCE designed to change the font weight of a text. The initial $u$ curves are chosen to run along the letters’ backbones, while the $v$ curves are chosen to be straight lines. Actuation makes the font either lighter (${\lambda }_2<{\lambda }_1$) or bolder (${\lambda }_1<{\lambda }_2$), without buckling out of the plane.