Geometry of Thin Nematic Elastomer Sheets

A thin sheet of nematic elastomer attains 3D configurations depending on the nematic director field upon heating. In this Letter, we describe the intrinsic geometry of such a sheet and derive an expression for the metric induced by general nematic director fields. Furthermore, we investigate the reverse problem of constructing a director field that induces a specified 2D geometry. We provide an explicit recipe for how to construct any surface of revolution using this method. Finally, we show that by inscribing a director field gradient across the sheet’s thickness, one can obtain a nontrivial hyperbolic reference curvature tensor, which together with the prescription of a reference metric allows dictation of actual configurations for a thin sheet of nematic elastomer.

Figure 1

Inscription of geometry of a nematic elastomer sheet (left-to-right). The director field can be imposed by substrate pretreatment or by applying an electromagnetic field. After cross-linking, we are left with an elastic plate, within which is a nematic liquid crystal with a prescribed director field. Upon heating, the material switches to the disordered state and the resulting sheet locally expands or contracts by a factor $\sqrt{\alpha }$ in the direction parallel to the director field and by a factor $\sqrt{{\alpha }^{-{\nu }_t}}$ in the perpendicular direction. This (in general) results in a non-Euclidean two-dimensional metric, which causes out-of-plane deformations if the sheet is thin enough.

Figure 2

Director field (top) and resulting surface of revolution (bottom) for $\alpha =2$ and ${\nu }_t=1$, for spherical ($\mathrm{\Omega }(\xi )=\sqrt{2}/\cosh \xi$), pseudospherical ($\mathrm{\Omega }(\xi )=\sqrt{1/2}/\cos \xi$), and toroidal ($\mathrm{\Omega }(\xi )=\sqrt{2}/1+(1+\sqrt{5}/2){\sin }^2\xi$) surfaces. The $\sqrt{2}$ factors were introduced in order to maximize the size of the domain, given that $\mathrm{\Omega }(\xi )$ must, by Eq. (8), lie between $\sqrt{1/2}$ and $\sqrt{2}$. The curved boundaries of the planar domains at the top panel become longitudes of the resulting surfaces upon heating.

Figure 3

Top—director field on the top (red) and bottom (green) surfaces, in all of which $⟨\theta ⟩$ is the same (hence, inscribing the same metric); however, $\mathrm{\Delta }\theta$ differs (hence, inscribing different curvature tensors). Center—the resulting curvature field dictated (red and green mark the two principal curvatures). By applying different $\mathrm{\Delta }\theta$ fields, we can “activate” the curvature tensor wherever its directions match the surface we wish to design. Bottom—the resulting surfaces. All surfaces are exact isometries of the inscribed metric (and of each other); however, their curvature tensors best fit the inscribed ones, and are therefore energetically preferable among isometries due to their lower bending energy.