Research ArticleCONDENSED MATTER PHYSICS

Mosaics of topological defects in micropatterned liquid crystal textures

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Science Advances  23 Nov 2018:
Vol. 4, no. 11, eaau8064
DOI: 10.1126/sciadv.aau8064

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.

INTRODUCTION

Fabrication of functional micro- and nanopatterns over a large area is one of the notable focuses in the interdisciplinary fields of physics, biotechnology, and material science. Self-organization in soft matter systems such as colloids, block copolymers, and liquid crystals (LCs) has been intensively studied for practical use with emergent functions and as templating materials with new structural properties (14). In the nematic LCs (NLCs), self-organization is achieved through the orientational elasticity, which mediates long-range alignment interactions across the molecules. Locally, the preferred orientation of the molecules is described by the director n, a headless unit vector (5). When the director field is forced by surface anchoring and elasticity in a configuration that cannot be continuously transformed into a uniform state, topological defects appear (6). Although the defects should be removed in LC display applications, their intrinsically deformed orientational order can give rise to optoelectric and elastic effects, facilitating various applications such as templating for colloidal assembly (7, 8), microlens arrays, and photomasks (9, 10). Morphogenesis and dynamics of topological defects also served as an analogy to cosmological models (11), skyrmion lattice formation (12), and atomic interplay in condensed matter (13). Assemblies of point and line defects in the nematic (N) phase exactly reflect physicochemical environments surrounding the LC material. For example, when the colloidal inclusion is placed in the uniform N field, the director field consistently presents the topology of inclusion by balancing surface anchoring and elasticity, resulting in the formation of unique topological defects, which works not only for spherical inclusions (1416) but also for polyhedral (17, 18) and more complex handle bodies (1921), all of which facilitate understanding of surface-related topological behavior in LCs.

For practical use of the LC defects, controlled formation of stable patterns that include point or line defects should be achieved over a large area. For this goal, many studies have been conducted to control self-organization of regular LC patterns using external stimuli such as electric, magnetic, or optical field (2225), chemical interactions (26), and topographical patterning on the surfaces using mechanical scrubbing (27), light-controlled polymerization (28, 29), thermally activated shape shifting (30, 31), or conventional photolithography techniques (3234). In particular, Guo et al. (34) have recently shown that the process based on plasmonic mask photoalignment is a way to produce the desired topological LC defect patterns on demand, although it requires expensive photolithographic tools and does not show reconfigurable characteristics. In contrast to electric or magnetic field controlling the orientation of LC materials in a whole volume, topographical confinement effectively produces more complex and precisely assembled LC structures on a scale of a few micrometers because the confinement can be finely controlled even on the nanometer scale, which provides a rich variety of boundary conditions (35). The strategy using confinement enables the minimization of the total free energy of the LC structure (bulk elasticity and surface anchoring) in the equilibrium state. The local influence of topographic patterns on the alignment can propagate across the entire sample area. However, because of the characteristic long-range order of the N phase on a macroscopic scale, it is still challenging to control LC orientation locally with a micrometer or submicrometer resolution using confinement. This might reduce reproducibility and practical applicability of an LC patterning platform, especially when the LC patterns become more complicated. Thus, it is greatly desirable to design underlying topographic patterns with well-controlled and effective anchoring conditions for a reliable formation of the specific LC structures.

Here, we demonstrate a method to control the N defect patterns with a periodic array of air pockets, shaped and placed by an intaglio pattern of holes in the substrate that is scalable to large areas. The resulting lattice of pillar-like air pockets induces topological frustration, creating a regular texture of topological defects with accompanying spatial variation of the director field. We explore the effects of lattice geometry, cooling rate, and surface treatment to fine-tune the resulting micropatterns and demonstrate the topological rules governing the dislocations and irregularities in the defect lattice. Last, we observe the transformation of the patterned N director into a layered structure during the transition in a smectic A (SmA) phase, leading to a series of undulation instabilities that produce different colorful textures reflecting the order within. The patterns resemble ancient mosaics and kaleidoscopic motifs and are true microscopic analogs to stained glass.

RESULTS

To fabricate the periodic topological patterns in N field at a scale of a couple of millimeters, we used a sandwich LC cell with a ≈3-μm gap, consisting of a micropatterned silicon (Si) substrate with intaglio holes and a cover glass. We fabricated two different rectangular patterns of square holes (patterns I and II) and a triangular pattern of circular holes (pattern III), where the centers of all holes were positioned 30 μm apart (Fig. 1A). Pattern I is a simple square tiling, and pattern II is a truncated square tiling. Looking at hole positions alone, both form the same lattice, just rotated by 45°. However, the relative orientation of square hole corners differs between the structures, so we tested both patterns to observe the effect of the hole shape on the resulting texture. The gap was filled with 4′-n-octyl-4-cyano-biphenyl (8CB) by capillary action above its isotropic (Iso) to N transition temperature (at ≈40.5°C) and then cooled down to the N phase (at ≈33.5°C). Via capillary action, the LC material only seeps between the cover glass and the bottom surface of patterned Si substrate; thus, pillar-like air cavities are spontaneously generated above the intaglio holes. The surface tension of the air-LC interface makes the corners of the air pillars rounded despite the sharp corners of the square holes. The NLC molecules are parallel to both the underlying Si substrate and the cover glass, imposing planar alignment throughout the cell. The air pillars effectively influence the local distortions of NLC director field when their size is much larger than the characteristic length of the LC material, which is, in our case, ≈1 μm (36). In the same sense, the distance between the air pillars has to be above ≈10 μm to effectively form a regular array of topological defects in the equilibrium state, whereas the surface tension requires the hole diameter to be larger than the length scale of the cell gap for the pillars to form.

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 (neno) ≈ 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.

We tested two different surface treatments of the cover glass. First, we treated the surface with polyimide (PI), imposing a strong degenerate planar anchoring (see Materials and Methods). Second, we left the glass surface untreated, resulting in a weak degenerate planar anchoring (37). Planar degenerate anchoring on the Si substrate and the cover glass confined the LC molecules into an effectively two-dimensional (2D) texture. This means that the defects can be characterized with the winding number, a conserved quantity that measures the amount of in-plane rotation of the director on a circuit around the defect. Just like with the schlieren texture (5), the winding number is easy to read from the polarizing optical microscopy (POM) images. The air pillars impose strong homeotropic surface anchoring, which aligns the director at each pillar into a radial nematic texture with a winding number +1. As the winding number sum over the entire sample tends to cancel out, each pillar stabilizes accompanying topological defects, which arrange into a regular pattern (Fig. 1B, right). In the absence of the pillars, the defects are positioned randomly, attracted to defects of the opposite winding number through elastic interactions, and tend toward annihilation (Fig. 1B, left). Figure 1 (E and G) shows POM images for the square tiling of pattern I, for two surface treatments, revealing the colorful pinwheel-like birefringent optical textures around each defect, and dark outlines of the nonbirefringent air pillars. The color appearance of the pinwheels changes with sample thickness (Fig. 1E) and with temperature (Fig. 1G), moving along the colors of the Michel-Lévy birefringence chart (36), as shown in Fig. 1F. On cooling from the Iso to N phase, the transition occurs from top to bottom of the sample because of the temperature gradient on the heating stage, where the cooling starts from top surface (see fig. S1). Therefore, the effective thickness of the nematic phase increases upon cooling, resulting in a variation of the birefringent colors. Accordingly, the color change triggered by a temperature gradient seems to be similar to the color variation induced by changing the sample thickness. Inserting a 530-nm (λ) wave plate at 45° with respect to the polarizers produces different interference colors along the slow and fast axes associated with ne and no (refractive indices of the LC) of the λ wave plate (Fig. 1D), allowing unambiguous reconstruction of the director and winding numbers of the defects (compare Fig. 1, E and F). We use the λ wave plate imaging in the insets of the subsequent images. In all POM images, the director can be reconstructed by finding regions of the same color and comparing them with director orientations around the air pillar, which are known to be perpendicular to its interface (Fig. 1, C and D). Around the defect, the number of brushes oriented along the polarizer or analyzer direction equals four times the winding number of the defect: The common pinwheel textures with ±1 winding are recognized by their characteristic four brushes.

Figure 2 shows the regular textures of both square patterns, with or without the PI treatment of the cover glass, imaged in two orientations with respect to the crossed polarizers, so that the similarity between the patterns I and II can be directly compared. We observe that with PI treatment of the cover glass, patterns I and II both settle in a structure with −1 defects sitting in the center of each square of four neighboring air pillars that impose +1 winding (Fig. 2, A to D), respectively. Topologically, we obtain a checkerboard pattern of alternating +1 and −1 defects, with the unit side equal to the distance between the air pillars (see dashed colored square outlines in Fig. 2, A and C). As both square patterns induce similar textures, we conclude that the orientation of the square hole with respect to the lattice orientation does not affect the topology of the resulting texture and has only little effect on its geometry.

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.

The untreated sample contains defects not only at the center of each square but also between each adjacent pair of air pillars (Fig. 2, E to H). This looks like a different structure from the PI-treated sample, but topologically speaking, it is simply the same checkerboard defect pattern on a shorter scale (see dashed colored square outlines in Fig. 2, E and G) and is also topologically equivalent to the checkerboard defect pattern, induced by Murray et al. (38) by means of plasmonic photopatterning. Every other +1 point is not induced by a pillar but is represented by an actual defect in the bulk, as the unit of the topological lattice does not match the unit tile of the pillar lattice. Note the swirl-like appearance of the +1 defects, indicating elastic constant anisotropy (39). The defect density is twice the density for the PI-treated sample. This is attributed to the fact that the strong anchoring of the PI-treated surface makes the free energy penalty of defects high, minimizing their number, while the untreated sample does not impose these constraints. The distinction between patterns I and II again has no effect on the arrangement of the defects.

The large-scale periodic order of the defect lattice depends on the cooling rate. If the cooling from the Iso to the N state is fast, then the order nucleates simultaneously from each of the air pillars, enforced by their strong homeotropic anchoring. The propagating Iso-N interfaces collide at the center of each square of pillars, forming a regular lattice (Fig. 3A and movies S1 and S2). If the cooling is slow, the director has time to relax after each individual collision of the growing Iso-N interface, resulting in a higher-energy metastable state with irregular topological order (Fig. 3B and movie S3). We find partially ordered textures, where different unit tiles of the pattern contain different defects, including intermediate transient states of winding number −2 or higher, which eventually split into −1 defects, and half-integer defects, which are realized as disclination lines spanning from the top to the bottom of the sample, pinned at the corners of the air pillars (Fig. 3, D and E, and movie S4). Profiles of defects with different winding numbers are depicted in Fig. 3C.

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.

Even with irregular placement, the defects are still governed by topological conservation laws. If the defects are localized, then each unit square of four pillars has a neutral winding overall, counting the corners with quarter weight (as four corners make a whole) and edges with half weight (see Fig. 3 for a texture with various examples). Within this restriction, any combination of defects is allowed, but the lattice is mostly periodic, and depending on the surface treatment, the predominant textures are those shown in Fig. 2. Figure 3E shows the director configuration of some of the more complex unit tiles. If a temperature gradient arises during slow cooling (see fig. S1 and movies S3 to S5), then the Iso-N interface is dragged through the lattice of air pillars. Creation of defects during cooling requires Iso islands carrying director winding (evidenced from the color brushes along the interface circumference in Fig. 3C) to pinch off from the Iso region, which is unfavorable if the interface is straight. Consequently, the defects pinch off late and end up displaced from their “home” positions by several lattice sites (Fig. 3F and movie S5). In this situation, the unit squares between each four pillars do not have a zero winding number. Instead, they “exchange” the topological charge through long topological solitons, which can be followed across the lattice, recognized by a uniform color signature (red dashed lines in Fig. 3F). This long-range topologically disordered state can be paralleled with electrostatic charge displacement in conventional dielectric crystals.

When the samples are cooled further from the N phase to the SmA phase, the defect positions, the general director field, and the average colors are roughly preserved, but an additional fine pattern, resembling kaleidoscope images, is revealed (Fig. 4 and movie S6). Thus, the patterning of substrates can also be used to achieve periodic or otherwise arranged smectic textures. The SmA order requires equal layer spacing and disallows elastic deformation of the bend type as a result of the diverging ratio of the bend and splay elastic constant near N-SmA transition temperature (40). Frustration is locally resolved by formation of small facets with locally aligned SmA layers with abrupt angle changes between them. They are visible as a multicolored fan-like pattern with rays parallel to the nematic director (fig. S2), a phenomenon often seen when director geometry enforced by defects and boundary conditions is incompatible with equidistant layers. Under hybrid anchoring, these striped textures are unstable and decompose into focal conic domains (40, 41), but in planar conditions, such as ours, they remain stable (42, 43). While radial +1 nematic defects, present only in the samples without PI treatment, already obey the no-bending condition, they remain mostly unchanged in the SmA phase. The texture of −1 defects, however, is incompatible with the smectic order. The escaped core, seen in nematics, preempts formation of layers, so a singular core is formed, which splits into a pair of −1/2 defects with varying separation, which can be seen in the images. Strong bending required by the negative winding defects creates a fine chaotic pattern of facets. Figure 4 depicts the result of cooling the textures from Fig. 2 across the N-SmA transition at 33.5°C. The +1 defects seen in the untreated sample (Fig. 4, E to H) are composed of fan-like facets, retaining the pinwheel appearance. In this case, the pillar shape is important, because unlike the N state, the SmA phase is sensitive to curvature due to the bending restriction. In pattern I (Fig. 4, E and F), the pinwheel is circular, as it touches the sharp corner of the pillar, while in pattern II (Fig. 4, G and H), four fans of the pinwheel can extend to the entire flat edges of the pillars. As the scale of the smectic layers is orders of magnitude smaller than the scale of the pattern, the layer structure is not perfect, and many dislocations, disclination pairs, and other types of irregularities are expected to be present, giving rise to a fine-textured look of the images that resembles stained glass.

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.

In pattern III, we created a triangular lattice to promote the creation of half-integer defects. As the unit tile is now an equilateral triangle spanned by only three pillars, the requirement for a neutral winding number demands a single vertical −1/2 disclination line in each of the unit tiles, inducing a dual honeycomb lattice of defects (Fig. 5A). The threefold symmetry of the disclination line director matches nicely with the same symmetry of the lattice. Under POM, it looks like a tiling of three types of rhombic tiles (Fig. 5B), which is preserved during transition into the SmA phase, along with the general structure (Fig. 5C). The fast cooling procedure to the N phase shows a similar growth and collision of the Iso-N interface (Fig. 5D). Subsequent cooling into the SmA phase straightens out the layers in each rhombic facet, creating a very ordered structure. Because of its hyperbolic nature, the −1/2 defect is a point of frustration for the SmA order; it has a scar-like appearance under depolarized light (Fig. 5C), and it shows a lot of fine structure between crossed polarizers (Fig. 5D).

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.

DISCUSSION

The structures seen in the planar samples with patterned substrates can be understood as regular planar LC cells with an external potential. The empty planar cell acts as a “vacuum,” and its ground state is a homogeneous director structure with no defects. The schlieren texture is a vacuum fluctuation, a transient mixture of topologically charged particles that decay back into the vacuum state. The air pillars, stabilized by a pattern of holes, not only stabilize the defects through topological charge conservation but also impose a periodic order on them, analogous to what a semiconductor lattice does to electrons.

All observed structures are completely consistent with a 2D description of the director field, because due to the degenerate anchoring, the director lies parallel to the substrate throughout most of the sample. Wherever the anchoring condition is obeyed exactly, the anchoring strength is irrelevant as there is no competition between elasticity and anchoring. The exceptions are −1 and +1 defects, which are actually nonsingular umbilical defects where the director escapes from the 2D plane toward the sample normal (44). The defects span the thickness of the sample vertically, terminating at two “boojums”—singular surface defects—where the anchoring condition is violated (see Fig. 3C). This configuration is more stable compared with pairs of half-integer defects or unescaped singular integer defects for thick enough samples, such as ours (38, 44). Application of PI thus increases the cost of the integer defects by increasing the anchoring strength, which explains the preference for structures with fewer defects, although they are still cheaper than the singular half-integer defects. However, even with weak anchoring, faster spatial variation of the director means more distortion and thus greater elastic cost, so we conclude that the high-density defect structures observed when the substrate is untreated are locally metastable states, reached because of specific propagation of the phase fronts during a cooling process (Fig. 3 and movies S4 and S5). Defects of higher topological charges may appear transiently during cooling (movie S3), because every Iso island that collapses to a point carries a winding consistent with the number of dark brushes along its perimeter that were propagated with the traveling interface. Defects with winding larger than |1| are unstable and eventually split into smaller defects, as demonstrated in Fig. 3 (B and C).

Coupling existing features of micropatterned surfaces with strong optical response and tunability of LCs can expand the usability of these materials to new fields. LCs are ubiquitous in display technology, optics, and photonics and were also demonstrated to be useful as sensors (45). These applications may benefit from the structure imposed by the patterning. Fabrication of the patterned surface is not difficult, regular textures are readily assembled even without special glass and substrate treatment, and the air pillars set in place spontaneously when the LC cell is filled, making the preparation of samples easier and to scale to fast serial production. Unlike in the case of self-assembled inclusions, the lattice size, hole shape, and anchoring condition may be freely selected according to needs and are fixed. With fast cooling, the pattern is exceptionally regular and reproducible. The 2D topological rules, as well as qualitative behavior of defects under different cooling rates, can be applied to any lattice and can serve to accurately predict the structure without the need for additional research.

CONCLUSIONS

Here, we demonstrated optical behavior, topological rules, and transitional properties of 8CB material in LC cells with patterned substrates. A periodic lattice of square-shaped and round holes on the scale of few tens of micrometers was etched into the Si substrate in periodic square and triangular lattices, serving as a pattern to impose structure on the nematic order. In both N and SmA states, the samples exhibit stable pinwheel textures of vibrant colors, which depend on temperature and cell thickness. When cooled to the SmA state, the regular structure is preserved, with added kaleidoscopic appearance, similar to stained glass windows. The topological rules based on the conservation of the winding number govern the defect arrangements and set rules for different modes of disorder: local and global migration and distribution of topological charge.

With the theoretical rules and the manufacturing process established, more elaborate structures can be designed, according to desired effects. Both the bottom substrate and the cover glass could be patterned, possibly with a shifted, rotated, incommensurable, or geometrically complementary design. Different lattice symmetries can be used, including heterogeneous patterns with distinct holes. In addition to the presented zero-field state, symmetry breaking through electric field and flow can be investigated as phenomena that can be exploited in optoelectronic and microfluidic devices. Possible LC-based enhancements of existing technologies suggest promising pathways for future research and a potential to eventual everyday applications.

MATERIALS AND METHODS

Experimental design and characterization

Patterned substrates (patterns I, II, and III) were fabricated on (100) Si wafers with conventional photolithography and reactive ion etching techniques. The patterned wafers were chemically cleaned by ultrasonication in a mixture of dimethylformamide and methanol to remove organic/inorganic impurities, followed by rinsing several times with deionized water and, lastly, exposing to oxygen plasma for 5 min. The cleaned Si substrates promote strong degenerate planar alignment of LC molecules with surface anchoring energy Wa > 10−4 J/m2. Cover glass slides were treated via the same cleaning method, and the bare glass provided weak degenerate planar alignment of LC molecules (Wa ≈ 10−5 J/m2). For part of the experiments, the glass slides were spin-coated with the PI material planar alignment-induced PI (PAPI) PIA-5550-02A, JNC Corporation) to induce strong degenerate planar anchoring on the LC material (Wa ≈ 10−3 J/m2). The chemically treated glass substrates were further soft-baked at 90°C for 100 s and then cured at 200°C for 2 hours. To assembly 2D confinement cells with different strengths of surface anchoring, the PI-coated and pristine glass slides were put on selected patterns with a few-micrometer gaps using silica beads. Most experiments were conducted with LC cells having a 3-μm gap. The LC material, 8CB (SYNTHON Chemicals), was injected in the cells via capillary force in its Iso state (T > 40.5°C). The samples were cooled to the N phase at a rate of 10 K/min as fast cooling and 1 K/min as slow cooling. Optical textures of the LC material were investigated by POM (Nikon Eclipse LV100 POL) with a λ wave plate. Temperature was controlled with a heating stage (Linkam LTS420) and a temperature controller (Linkam TMS94).

SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/11/eaau8064/DC1

Fig. S1. Photography of temperature gradient–heating stage.

Fig. S2. Large-area uniformity of micropatterned SmA texture.

Movie S1. Fast Iso-N cooling of 8CB in pattern I with PI surface treatment.

Movie S2. Fast Iso-N cooling of 8CB in pattern I without PI surface treatment.

Movie S3. Slow Iso-N cooling of 8CB in pattern I with PI surface treatment with a temperature gradient, example I.

Movie S4. Slow Iso-N cooling of 8CB in pattern I with PI surface treatment with a temperature gradient, example II.

Movie S5. Slow irregular Iso-N cooling of 8CB in pattern II with PI surface treatment with horizontal and vertical temperature gradients.

Movie S6. Slow N-SmA cooling of 8CB in pattern I with PI surface treatment.

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.

REFERENCES AND NOTES

Acknowledgments: We thank M. Ambrožič and S. Kralj for helpful discussions. Funding: This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) nos. 2017M3C1A3013923 and 2017R1E1A1A01072798. The research was additionally supported by the Slovenian Research Agency (ARRS) through research core funding nos. P1-0055 (to U.T.) and P1-0099 (to S.Č.). Author contributions: D.K.Y. and U.T. designed and led the research. D.S.K. performed the experiments. S.Č. performed the theoretical analysis of the system. All authors analyzed experimental data and wrote the manuscript. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.
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