Developed Smectics: When Exact Solutions Agree

In the limit where the bending modulus vanishes, we construct layer configurations with arbitrary dislocation textures by exploiting a connection between uniformly spaced layers in two dimensions and developable surfaces in three dimensions. We then show how these focal textures can be used to construct layer configurations with finite bending modulus.

Figure 1

We construct the two-dimensional layers by taking level sets of a piecewise developable (Gaussian curvature $K=0$) surface which makes a constant angle with the $\widehat{\mathbf{z}}$ direction. From back left to close right, the surface is made of two intersecting planes which end and attach to pieces of cone which necessarily intersect another set of parallel planes on a parabola.

Figure 2

Comparison of focal and BPS constructions: the dashed lines show the layers from the focal construction in Fig. 1 for the layers on one side of (a) one or (b) two dislocations. The solid curves are level sets of $\varphi =y-u(x,y)$, where $u$ is determined by BPS evolution, starting with the first layer of the focal construction, for $\lambda =0.05$ and $\lambda =0.1$. The background is shaded according to the compression energy of the asymptotic BPS solution, Eq. (3), for $\lambda =0.05$. The parabolic focal line is $y=\pm (x^2/b-b/4)$; we use the vertical offset in the BPS solution [10, 17]. (c) The focal construction for two dislocations, built by attaching two single dislocations as in Fig. 1. Note that now some of the focal lines arise from the intersection of cones with cones and are pieces of hyperbolas, not parabolas.

Figure 3

Two edge dislocations at arbitrary ($x_i$, $y_i$). The dashed curves are the focal layers, and the solid, gray curves are BPS evolution with $\lambda =0.05$. Parabolic (red and orange, dark shade) and hyperbolic (green, light shade) cusps of the focal construction are also shown. The thick solid lines indicates the division between upward and downward evolving BPS solutions.