Curvature generation in nematic surfaces

In recent years there has been a growing interest in the study of shape formation using modern responsive materials that can be preprogrammed to undergo spatially inhomogeneous local deformations. In particular, nematic liquid crystalline solids offer exciting possibilities in this context. Considerable recent progress has been made in achieving a variety of shape transitions in thin sheets of nematic solids by engineering isolated points of concentrated Gaussian curvature using topological defects in the nematic director field across textured surfaces. In this paper, we consider ways of achieving shape transitions in thin sheets of nematic glass by generation of nonlocalized Gaussian curvature in the absence of topological defects in the director field. We show how one can blueprint any desired Gaussian curvature in a thin nematic sheet by controlling the nematic alignment angle across the surface and highlight specific patterns which present feasible initial targets for experimental verification of the theory.

Figure 1

Director fields of the form $\mathit{n}=cos\psi \left(x_2\right){\widehat{\mathit{e}}}^1+sin\psi \left(x_2\right){\widehat{\mathit{e}}}^2$ on the unit square for different choices of the function $\psi =\psi \left(x_2\right)$. The resulting Gaussian curvature $K$ upon stimulation is also indicated.

Figure 2

The nematic pattern obtained by shifting a circular arc along a fixed direction generates constant positive Gaussian curvature $K>0$. The nematic pattern obtained by shifting a tractrix along its axis generates constant negative Gaussian curvature $K<0$.

Figure 3

Orthogonal duality of director fields. The director fields that were discovered to generate constant Gaussian curvatures of opposite sign are seen to be equivalent to orthogonally dual patterns. Note that every tractrix intersects every circle at right angles.

Figure 4

An initially flat nematic sheet forming part of a sphere upon stimulation.

Figure 5

Blueprinting prescribed positive Gaussian curvature $K>0$ on a rectangular sheet of arbitrary dimensions whose edges are indicated by the dashed lines. The Gaussian curvature is well defined everywhere except along the solid horizontal lines where the distinct arcs join.

Figure 6

Blueprinting prescribed negative Gaussian curvature $K<0$ on a rectangular sheet of arbitrary dimensions whose edges are indicated by the dashed lines. The Gaussian curvature is well defined everywhere except along the solid horizontal lines.

Figure 7

Loxodromic spiral director fields of the form $\mathit{n}=cos\alpha {\widehat{\mathit{e}}}_{\theta }+sin\alpha {\widehat{\mathit{e}}}_{\varphi }$ on a sphere of radius $R$ for the cases $\alpha =0,\pi /4,\pi /2$ and the Gaussian curvature generated upon stimulation. Note that $\alpha$ is the angle that the directors make with the meridian lines.