Mosaics of topological defects in micropatterned liquid crystal textures

Abstract

Topological defects in the orientational order that appear in thin slabs of a nematic liquid crystal, as seen in the standard schlieren texture, behave as a random quasi-two-dimensional system with strong optical birefringence. We present an approach to creating and controlling the defects using air pillars, trapped by micropatterned holes in the silicon substrate. The defects are stabilized and positioned by the arrayed air pillars into regular two-dimensional lattices. We explore the effects of hole shape, lattice symmetry, and surface treatment on the resulting lattices of defects and explain their arrangements by application of topological rules. Last, we show the formation of detailed kaleidoscopic textures after the system is cooled down across the nematic-smectic A phase transition, frustrating the defects and surrounding structures with the equal-layer spacing condition of the smectic phase.

Fig. 1. Substrate patterns and the resulting nematic textures.

(A) Schematic sketches and scanning electron microscopy images of patterned Si substrates, etched with three patterns of holes: simple square, truncated square, and triangular lattice. (B) 8CB in a planar LC cell forms schlieren textures outside the topographic patterns (left), but the air pillars anchored by the holes generate perfectly periodic textures (right). P, polarizer; A, analyzer. (C) Dark brushes in the polarized image mark places where the director lies in the direction of either a polarizer or an analyzer, and defects are found at intersections of these brushes. Only black and reflected interference colors can be distinguished using this approach. (D) With addition of the λ wave plate, both diagonal directions gain different colors, which helps to identify the director texture unambiguously. (E) Texture under a cover glass treated with PI, exhibiting different colors at cell thicknesses from 2 to 5 μm, in 1-μm increments. At all thicknesses, the director texture (marked in the third panel) and defect winding numbers (marked in the last panel) are qualitatively the same. (F) Michel-Lévy birefringence chart, relating the optical path difference between ordinary and extraordinary polarization to the color of transmitted light. The retardation Δ is calculated by multiplying the sample thickness and its birefringence (n_e − n_o) ≈ 0.13 (36). Black dotted lines mark the approximate values of birefringence for images in (E). Note that color balance correction of the camera influences the exact appearance. (G) Texture assembled without PI treatment, imaged at different temperatures with an inserted λ wave plate. Increasing the temperature lowers the birefringence and thus changes the color appearance, while the director remains unchanged. The slow axis of the wave plate direction is marked at 45° to the polarizers. The last two panels mark the director and the defect windings, respectively. Scale bars, 20 μm.

Fig. 2. POM images of the periodic pinwheel-like topological defect arrays in the N phase of 8CB.

(A to D) Lattice of hyperbolic defects with winding number −1 sitting in each square of air pillars in PI-treated LC cells with pattern I (A and B) and pattern II (C and D) sets of intaglio square holes. (E to H) Pattern I (E and F) and pattern II (G and H) LC cells without PI treatment show a denser lattice of defects, with −1 defects between each pair of pillars and +1 defects in the middle of each square of pillars. The insets show the POM images with an inserted λ wave plate. The orientation of LC molecules is related to the observed colors, as schematically depicted in an inset to (A), (C), (E), and (G). The bottom row shows the same structures as the top row, but rotated by 45° with respect to the polarizer directions. This simplifies the pairwise comparison of (B) to (C), (A) to (D), (E) to (H), and (G) to (F), showing that patterns I and II, which only differ in the shape of the hole, produce the exact same defect topology. Scale bars, 20 μm.

Fig. 3. Evolution of the topological order under different cooling rates.

(A) PI-treated LC cell under fast cooling (10 K/min) across Iso-N transition; simultaneous collision of growing nematic domains results in a perfectly ordered lattice. (B) Slow cooling rate (1 K/min) allows time for relaxation of partially merged domains and results in displaced defects. Note the appearance of transient −2 defects, which eventually split into stable −1 defects. (C) Isolated occurrences of defects of different windings with marked director field, with the number of brushes equal to twice the winding amount. Middle inset shows how the dark brushes are dragged by the N-Iso interface during cooling, and the resulting defect is determined by the number of dark brushes. Integer-strength defects are nonsingular and involve escape of the director into the third dimension (inset below). (D) Topologically disordered state with zero winding number unit squares. Red dashed squares mark the squares with a single −1 defect in the middle, yellow-colored are the squares with a −1 defect at the edge and a −1/2 vertical disclination line next to a pillar, and dashed green ones mark the squares with two −1 defects at the edge. (E) Schematic representation of different tiles from the above POM image with added director field annotation to aid understanding of the textures. (F) A slow cooling (1 K/min) Iso-N sequence with a temperature gradient. Dragging a phase interface across the sample dislocates defects by several unit squares, dragging topological solitons (dashed lines) behind. Unit squares are not topologically neutral in this case. (G) Fast cooling (10 K/min) of a PI-untreated sample across both Iso-N and N-SmA transitions, showing the emergence of a kaleidoscopic pattern when smectic layers are formed. The orientational order of the N phase is qualitatively preserved under this transition (see insets). Scale bars, 20 μm.

Fig. 4. POM images of the kaleidoscopic textures in the SmA phase of 8CB with the same hole patterns and anchoring conditions as in Fig. 2.

(A to D) The textures in PI-treated LC cells segment into facets with fan-like smectic order, which meet with abrupt angle changes. The hyperbolic −1 defect is a junction of four facets and has fine structure due to incompatibility with the no-bending condition, sometimes visibly splitting into pairs of −1/2 defects (encircled). (E to H) LC cells without PI treatment contain +1 defects, which retain a pinwheel appearance. In pattern I, the pinwheel only touches the pillars, while in pattern II, the fans connect to the facing edges of the square holes. The −1 defects are centers of the disorder-looking region of the smectic between the fan-like facets. The insets show the POM images with an inserted λ wave plate. Observe that the overall color hues remain similar to those in the nematic phase, presented in Fig. 2, as the birefringence and large-scale orientational order remain almost unchanged. Scale bars, 20 μm.

Fig. 5. Defect arrangements and cooling transition of 8CB in pattern III.

(A) The triangular lattice of pillars induces −1/2 defects in the middle of each unit triangle. A schematic director field texture is added to show how the triangular lattice of air pillars stabilizes the half-integer defect lines. (B) POM image reveals a tiling of differently colored rhombic facets, which retain their structure during N-SmA transition. (C) The SmA structure is similar to the N structure, but with scar-like appearance of defects. (D) Fast cooling sequence (10 K/min) of Iso-N-SmA transition. Growth of N domains is uniform, forming the defects when they collide. The transition into SmA straightens out the layers in each of the rhombic tiles, with a small frustrated disordered state around each −1/2 defect. Scale bars, 20 μm.