Composite Dislocations in Smectic Liquid Crystals

Smectic liquid crystals are characterized by layers that have a preferred uniform spacing and vanishing curvature in their ground state. Dislocations in smectics play an important role in phase nucleation, layer reorientation, and dynamics. Typically modeled as possessing one line singularity, the layer structure of a dislocation leads to a diverging compression strain as one approaches the defect center, suggesting a large, elastically determined melted core. However, it has been observed that for large charge dislocations, the defect breaks up into two disclinations [C. E. Williams, Philos. Mag. 32, 313 (1975)]. Here we investigate the topology of the composite core. Because the smectic cannot twist, transformations between different disclination geometries are highly constrained. We demonstrate the geometric route between them and show that despite enjoying precisely the topological rules of the three-dimensional nematic, the additional structure of line disclinations in three-dimensional smectics localizes transitions to higher-order point singularities.

Figure 1

(Left) A composite smectic screw dislocation. The inner (red) core has zero compression, and costs only a finite bending energy. The outer (cyan) shell has zero mean curvature, and costs only a finite compression energy. The structure, with Burgers scalar $b=10$ (in units of the layer spacing), is completed by the two helical (green) $+\frac{1}{2}$ line disclinations at $b$ half-layers (density minima) away from each other. The radius of the core is $ƀ=b/(2\pi )$. (Right) A composite edge dislocation. The structure is composed of a $+\frac{1}{2}/-\frac{1}{2}$ disclination pair, the number of half-layers between them again sets $b=10$. Here the mean curvature does not vanish outside the core, nonetheless disclinations are still found at curvature singularities. Insets: The standard dislocation structures of equivalent Burgers scalars, with diverging core energy.

Figure 2

A composite screw dislocation becoming a composite edge dislocation (with Burgers scalar 4 times the layer spacing). Note that the pincement is made on the top layer, a $-\frac{1}{2}/+\frac{1}{2}$ disclination pair with no net dislocation charge [1, 3] (red triangle and red circle, respectively). The half-layer is indicated as a dotted line to show the pinch. The red and blue circles are the $+\frac{1}{2}$ disclinations.

Figure 3

(Top left) A $+\frac{1}{2}$ disclination line is converted into a $-\frac{1}{2}$ disclination line at a pincement (inset, rear view). (Top right) A toric focal conic domain inserted into an otherwise unperturbed smectic. If we cut this into four by slicing a vertical plane and a perpendicular horizontal plane through the central point defect, the quarter toric domain is exactly the pincement illustrated on the left. (Bottom left) Converting a $+\frac{1}{2}$ disclination to a $-\frac{1}{2}$ disclination without a bend in the disclination line. The monopole is located where the cone attaches to the disclination. (Bottom right) Escaping a $-1$ disclination by attaching two half-cones to the layers.