Evolving, complex topography from combining centers of Gaussian curvature

Liquid crystal elastomers and glasses can have significant shape change determined by their director patterns. Cones deformed from circular director patterns have nontrivial Gaussian curvature localized at tips, curved interfaces, and intersections of interfaces. We employ a generalized metric compatibility condition to characterize two families of interfaces between circular director patterns, hyperbolic and elliptical interfaces, and find that the deformed interfaces are geometrically compatible. We focus on hyperbolic interfaces to design complex topographies and nonisometric origami, including $n$-fold intersections, symmetric and irregular tilings. The large design space of threefold and fourfold tiling is utilized to quantitatively inverse design an array of pixels to display target images. Taken together, our findings provide comprehensive design principles for the design of actuators, displays, and soft robotics in liquid crystal elastomers and glasses.

Figure 1

(a) A director field $\mathbf{n}\left(\mathbf{r}\right)$ making an angle $\alpha$ to the radial vector $\mathbf{r}$ itself at angle $\theta$. (b) A circumference $2\pi r$ contacts by a factor $\lambda$ on heating a circular system of directors, $\alpha =\pi /2$. The in-material radius $r$ extends by ${\lambda }^{-\nu }$, since it is perpendicular to the director, and $\nu$ is the optothermal Poisson ratio. The result is a cone, also visualized in (c). (d) Combination of circular director patterns with interfaces will result in combination of cones; see text.

Figure 2

Compatibility between two constant director patterns. Two constant director patterns (a) can be deformed by (b) the stretch tensors ${\mathbf{U}}_{{\mathbf{n}}_i}$ and (c) follow-up rotations to achieve (d) continuous deformed states. $\mathbf{t}$ and $\tilde{\mathbf{t}}$ are the tangents to reference and deformed interfaces, respectively.

Figure 3

An example of straight-line interface (red) between two symmetrically located circular patterns. (a) Two circular patterns are separated by a straight-line interface (red). (b) The deformed configuration has positive GC at the tips and negative GC along the deformed interface.

Figure 4

Two types of circle-circle interfaces. (a) The centers of the circle systems ${\mathbf{p}}_i$ are the foci of the system of ellipses and hyperbolae that form the two families of possible R-1 connected interfaces. The offset $\mathbf{q}$ is the intersection between a particular hyperbolic interface and the horizontal axis. At the point $\mathbf{x}$, the hyperbola, with tangent $\mathbf{t}$ is seen to bisect the directors ${\mathbf{n}}_1$ and ${\mathbf{n}}_2$ based on systems 1 and 2 of circles, while ellipse with tangent ${\mathbf{t}}^{\prime }$ also passing through $\mathbf{x}$ is seen to bisect ${\mathbf{n}}_1$ and $-{\mathbf{n}}_2$. (b) Hyperbolic interface: The deformed domain (right) is two connected partial cones with the same axis but different heights, deformed from the reference domain (left) by two cone deformations. (c) Elliptical interface: The deformed domain (right) is two connected partial cones with different axes and heights, deformed from the reference domain (left).

Figure 5

Examples of $n$-fold intersection: reference domain and deformed configuration. The red curves are the metric-compatible hyperbolic interfaces between circular director patterns. The threefold intersection (a), fourfold intersection (b), and $n$-fold intersection (c) have topological charges $-1/2$, $-1$, and $-(n-2)/2$ respectively.

Figure 6

(a) The splitting of topological charge $-1$ into $(-1/2)+(-1/2)$. (b) and (c) The splitting of the fourfold intersection into two threefold intersections: (b) reference domain and (c) deformed configuration. (d) Type I two connected threefolds (left) passes the critical fourfold state (middle) and then transforms to Type II (right), as moving ${\mathbf{q}}_1$ closer to ${\mathbf{p}}_1$ but keeping ${\mathbf{q}}_2$, ${\mathbf{q}}_3$, and ${\mathbf{p}}_i$ unchanged.

Figure 7

Examples of topographies with 2D translation symmetry. The unit cells of centers (rightmost) for the reference and deformed states are (a) a square, (b) a rhombus, and (c) a hexagon. Accordingly, the reference (left) and deformed (middle) tilings have (a) fourfold, (b) threefold, and (c) sixfold intersections of their interfaces.

Figure 8

(a) An example of irregular fourfold tiling on the reference domain. (b) The ijth unit cell of the tiling. The four centers and two input offsets ${\mathbf{q}}_2^{ij},{\mathbf{q}}_3^{ij}$ (blue dots) will determine the output offsets ${\mathbf{q}}_1^{ij},{\mathbf{q}}_4^{ij}$ (green dot). (c) The deformed irregular fourfold tiling.

Figure 9

(a) An example of irregular threefold tiling on the reference domain. (b) The ijth unit cell. The four centers and three input offsets ${\mathbf{q}}_1^{ij},{\mathbf{q}}_2^{ij},{\mathbf{q}}_3^{ij}$ (blue dots) will determine the output offset ${\mathbf{q}}_4^{ij}$ (green dot). (c) The deformed irregular threefold tiling.

Figure 10

The schematic of inverse design. The centers (red and black dots) form a square lattice, and the boundary offsets (blue dots) are bisectors. The red dots are designed to be higher than the black dots in the deformed domain by moving ${\mathbf{q}}_1^{ij}$ properly. The positions of ${\mathbf{q}}_1^{ij}$ are determined successively by the target image, from left to right, from bottom to top, as the blue arrows indicate; see text.

Figure 11

(a) Target image. (b) $142\times 37$ pixels designed to illustrate the target image. The red dots represent the centers of circular patterns that are supposed to have larger heights than the black dots in the deformed domain. (c) The design of “C” on the reference domain with light blue curves as the metric-compatible interfaces. The reference domain contains fourfold and Type I or Type II connected threefold intersections. (d) The deformed domain displays “C” as we expect. The contour colors represent the heights.