Extrinsic curvature, geometric optics, and lamellar order on curved substrates

When thermal energies are weak, two-dimensional lamellar structures confined on a curved substrate display complex patterns arising from the competition between layer bending and compression in the presence of geometric constraints. We present broad design principles to engineer the geometry of the underlying substrate so that a desired lamellar pattern can be obtained by self-assembly. Two distinct physical effects are identified as key factors that contribute to the interaction between the shape of the underlying surface and the resulting lamellar morphology. The first is a local ordering field for the direction of each individual layer, which tends to minimize its curvature with respect to the three-dimensional embedding. The second is a nonlocal effect controlled by the intrinsic geometry of the surface that forces the normals to the (nearly incompressible) layers to lie on geodesics, leading to caustic formation as in optics. As a result, different surface morphologies with predominantly positive or negative Gaussian curvature can act as converging or diverging lenses, respectively. By combining these ingredients, as one would with different optical elements, complex lamellar morphologies can be obtained. This smectic optometry enables the manipulation of lamellar configurations for the design of materials.

Figure 1

(Color online) From left to right: “pin stripe,” “rugby stripe,” and “barber pole” textures on a cylinder. These can be constructed from straight lines on a plane by identifying two sides of a rectangle (arrows). Though the layers are always geodesics, the layers of the left cylinder (pin stripe) are also straight. When $K=0$, the layers preferentially lie along the principle direction of zero curvature.

Figure 2

(Color online) From left to right: “equally spaced geodesics with sharp bend,” “equally spaced lines with no bend singularities,” and “geodesics with vanishing extrinsic bending” textures on a cone. These are all constructed by taking a pattern on a flat sheet, as could have been done for the cylindrical texture of Fig. 1. A wedge angle is removed as shown. On the left cone, the layers have equal spacing everywhere but for the line of dislocations along the seam and the layers have some normal curvature. In the center, the layers are equally spaced but there is both geodesic and normal bending. The right-hand cone has vanishing bending energy of all kinds, but the layer spacing is incorrect almost everywhere. The dashed lines on the cone to the left indicate the layer normals.

Figure 3

Left: turning angles $\Delta {\theta }_i$ defined on a triangle. Right: a square with analogous turning angles formed from two uniformly spaced layers (solid) and two normals (dashed). The normals are geodesics, so ${\kappa }_g=0$ along them.

Figure 4

(Color online) Layers (solid lines), normals (dashed lines), and geodesic curvature (shading) for a bump with aspect ratio $h_0/R=3$ projected onto the $xy$ plane. The Gaussian curvature vanishes on the circle (depicted in red online). A scale bar of length $R$ has been provided. With the constraint of equally spaced layers along the normals, the curvature induces the formation of a grain boundary (heavy dashed line) that extends infinitely far to the right of the bump center. The apparent unequal layer spacings in the figure are an artifact of the projection. The inset shows the bump, in perspective, overlaid with a square grid. Note that the grid lines are not equally spaced in this case.

Figure 5

(Color online) A converging lens: layers (solid lines, only upper half-plane shown) and normals (dashed lines, only lower half-plane shown) for a bump with $h=h_0\left[\left(x^2+y^2\right)/R\right]\text{exp}\left[-{\left(r/R\right)}^2/2\right]$. The ratio $h_0/R$ was set to unity. A scale bar of length $R$ has been provided. All quantities exhibit mirror symmetry about the horizontal midline. Boundaries between regions of Gaussian curvature with different signs are delineated in red (bold). Grain boundaries, the analogs of caustics in geometrical optics, are shown as bold dashed lines. Note the focusing of the normal lines onto the grain boundary in the lower half-plane. All lengths are measured in units of the bump width $R$. The surface is shown in perspective in the inset.

Figure 6

(Color online) Layers (solid lines, only upper half-plane shown) and normals (dashed lines, only lower half-plane shown) for a saddle bump with $h\left(x,y\right)=0.6\left[\left(x^2-y^2\right)/R\right]\text{exp}\left[-{\left(r/R\right)}^2/2\right]$. A scale bar of length $R$ has been provided. Boundaries between regions of Gaussian curvature with different signs are delineated in red (bold). Grain boundaries are shown as heavy dashed lines. The surface is shown in perspective in the inset.

Figure 7

(Color online) A diverging lens: layers (solid lines, only upper half-plane shown) and normals (dashed lines, only lower half-plane shown) for a bump with $h\left(x,y\right)=0.55\left(xy/R\right)\text{exp}\left[-{\left(r/R\right)}^2/2\right]$. A scale bar of length $R$ has been provided. Boundaries between regions of Gaussian curvature with different signs are delineated in red (bold). Grain boundaries are shown as heavy dashed lines. The surface itself is shown in the inset.

Figure 8

(Color online) Smectic optometry: a compound “lens” built from the bumps described in Fig. 5, 7. Layers are again denoted by solid lines and layer normals by (light) dashed lines. A scale bar of length $R$ has been provided. The grain boundary that appears due to the convergence of the left-hand bump ends at the diverging bump on the right. A second pair of caustics/grain boundaries appear to the right of this bump. Only one is shown, as the heavy dashed line.

Figure 9

(Color online) This texture of involutes generalized to a curved surface has both uniform layer spacing and minimizes the normal curvature along the circle $K=0$. Note that the layers bend tightly to be perpendicular to this circle, leading to a diverging curvature. To the right is a top view: a projection onto the $\left(x,y\right)$ plane.