Tailoring on Rotational Symmetry of Liquid Crystal Domain Lattices

Rotational symmetry is ubiquitous in nature. However, self‐assemblies of soft condensed matter such as liquid crystals (LCs) are entropy‐driven, making the tailoring of rotational symmetry challenging. Here, an approach is proposed to control the rotational symmetry of LC domain lattices based on the anchoring condition predesign and the orientational‐order inheritance during the nematic‐smectic phase transition. By this means, periodic and quasiperiodic LC textures with Ci symmetry (i = 2–6) determined by the preset alignment lattices are realized, and verified by symmetries of diffraction patterns. Topological analysis is carried out to disclose distinct evolutions of orders between disclination line textures and defect wall textures of different rotational symmetries. The influences of anchoring conditions, phase transitions, and mechanical stress on the self‐assembly of LCs, as well as the underlying mechanisms and dynamics, are investigated. This work realizes controllable rotational symmetry for large‐area self‐organized LCs and brings new insights to soft condensed matter.


Introduction
Rotational symmetry is a ubiquitous order in nature, from rigid molecules, mineral crystals, snowflakes, and corollas to honeycomb arrays in beehives. [1]The self-organization of blockcopolymers, [2,3] colloids, [4,5] and liquid crystals (LCs) [6,7] can spontaneously form aggregations featuring distinct rotational symmetry and thus mimic the above natural systems.Self-assemblies are usually entropy-driven processes accompanied by the emergence of random frustrations and deformations, as well as various topological defects.Investigating the underlying mechanisms, dynamics, and large-area ordered architecting has kept being a frontier in soft condensed matter physics.Among these building blocks, LCs have attracted specific attention due to their DOI: 10.1002/apxr.202300127richness in phases and topological defect species. [8]Moreover, the LC-ordered systems exhibit excellent stimuliresponsiveness.
Nematic (N) LC is an anisotropic fluid that possesses one-dimensional (1D) long-range orientational order.The director is usually adopted to depict the preferred direction of LC molecules. [9]n a real LC medium, the directors usually spatially vary and even discontinue because of physical confinements, [10,11] phase transitions, [12] or external field disturbances, [13] thus leading to various textures with diverse topological defects.The equilibrium state is attributed to the accurate balance of the elastic energy, the anchoring energy, and external stimuli.During the past decade, many efforts have been made to enhance the controllability of LC self-organization, which may enable promising applications such as optical vortex array generation, [14,15] grating, [16,17] and microlens array fabrication. [18,19]Mechanical scrubbing [20] and chemically modified [21] surfaces are used to address the locations of LC domains and corresponding topological defects.3D artificial reliefs, such as microchannels [22] and micropillars, [23,24] are adopted to geometrically confine the LC deformation and thus further enhance the controllability.With these methods, square and hexagonal domain lattices are commonly generated.Via reasonably preprogramming the anchoring condition of the alignment layer, the rotational symmetry is further extended to the triangular lattice. [25][32] Due to the thermal motion and fluidity of LCs, the obtained textures often deviate from or even violate the original design.In addition, the rotational symmetry is commonly restricted to triangular, square, and hexagonal lattices due to the relatively low free energy, hindering a direct and systematic comparison among LC textures of different rotational symmetries.Thus, it is an urgent task to establish a universal technique to freely tailor the rotational symmetry of LC domain lattices and deeply explore the underlying mechanisms and dynamics.
In this work, we present the symmetry of photopatterned alignment in a hybrid cell to guide the rotational symmetry of generated LC domain lattices.On the basis of the inheritance of orientational order across the N-smectic A (SmA) phase transition, disclination lines are totally suppressed and transform to defect walls that rigorously follow the predesigned alignment boundaries.By this means, LC textures of rotational symmetry consistent with predesigned are obtained.We systematically verify LC domain lattices with two-fold to six-fold rotational symmetry (C i , i = 2-6).Among them, C 5 symmetry depicts a quasiperiodic lattice with only rotational symmetry but a lack of translational symmetry.It increases the free energy of the LC system, and layer deformation caused by stress in the SmA phase must be introduced to reach the domain lattice of C 5 symmetry.We sequence different rotationally symmetric lattices according to their free energy and investigate the influences of anchoring conditions, phase transitions, and mechanical stress on the selfassembly of LCs.This work drastically enhances the controllability of LC self-organization and extends our knowledge of soft condensed matter, which may supply a promising platform for advanced photonic devices such as diffraction gratings, lasers, and multiplexing optics.

Results and Discussion
We use a hybrid cell with a gap h = 8.0 μm to provide a hybrid boundary condition.A thin PDMS layer is coated on the superstrate to perform the vertical alignment, while the substrate covered with the photoalignment agent SD1 [33] gives planar anchoring.The infiltrated LC 8CB orients toward the guidance of adjacent alignments.We record various rotational symmetries into the SD1 alignment lattices using a multistep partly overlapping photoalignment technique. [34]The alignment lattices are summarized in Figure 1.Rectangular (Figure 1a), diamond (Figure 1b), triangular (Figure 1c), square (Figure 1d), and hexagonal (Figure 1e) lattices, which have C 2 , C 3 , C 4 , and C 6 symmetry, respectively, are in the periodic family.All periodic lattices are composed of single polygon species with a side length of L. Dürer's penciling is applied to generate a quasiperiodic lattice with five-fold (C 5 ) rotational symmetry (Figure 1f), which is composed of both pentagons and diamonds.
When the infiltrated 8CB is annealed rapidly (−1°C min −1 ) from the isotropic phase, the LC directors adjacent to the sub-strate follow the guidance of patterned planar alignment and gradually turn to a uniform vertical alignment approaching the superstrate.The resultant textures are observed under a polarized optical microscope.Figure 2 shows micrographs of samples with different periodic lattices.A +1 singularity (red dot) is set in the center of each repeatable alignment unit.Figure 2a presents a rectangular lattice with length 2L = 60 μm and width L = 30 μm.When temperature T decreases below the clearing point (40.5 °C), unidirectional disclination lines are observed (upper image in Figure 2b, and the white dashed boxes correspond to the alignment area of the top row).After several thermal cycles across the N-SmA phase transition (at a rate of 0.1°C min −1 within the temperature range of 33.3 to 33.7 °C), all disclination lines transform to defect walls (middle image in Figure 2b), which exhibit C 2 symmetry consistent with the rectangular alignment lattice.The director fields of these two textures are discussed detailed in our previous work. [35]The transformation is attributed to ordered toric focal conic domains (TFCDs, bottom image in Figure 2b) being more favorable in energy minimization in the SmA phase.Each TFCD consists of a family of smectic layers shaped as nested tori, with a circular defect line at the center of the tori and a vertical cusp line in the center of each unit (as shown in the lower right inset of Figure 2b).Such configuration will effectively reduce the total free energy of the SmA LC system. [36,37]Moreover, defect walls can be considered as a direct inherence of orientational order from ordered TFCDs.We investigate the influence of the aspect ratio of rectangular lattices on texture generation (Figure S1, Supporting Information).With increasing aspect ratio, the transformation becomes more and more difficult.When the aspect ratio reaches 4, this transformation is interrupted by the unerasable disclination lines.
Figure 2c,e,g,i reveal cases for diamond, triangular, square, and hexagonal alignment lattices.All edge lengths are set as L = 30 μm.LC textures with C 2 (Figure 2d), C 3 (Figure 2f), C 4 (Figure 2h), and C 6 (Figure 2j) symmetries are demonstrated.The defect wall textures exhibit exactly the same rotational symmetries as the preset alignment lattices.For TFCDs array with different rotational symmetries, the area occupied by all complete TFCDs is calculated in a certain area, and the deformed FCDs are not included (Figure S2, Supporting Information).By analyzing the area proportion of different TFCDs in the SmA phase, one can compare their free energy.The minimum area proportion for LC texture with C 2 symmetry is attributed to the formation of TFCD is restricted by the width of the rectangular unit lattice.The ratio of deformed FCDs increases significantly along with the increasing aspect ratio.While LC textures with C 6 and C 4 symmetries exhibit larger area proportions than other cases, making their transformations from disclination lines to defect walls much more facilitated.The perfect rotational invariance of TFCD and the gradient refractive index change caused by its specific director field endow TFCDs with the function of focusing light as a microlens array. [19,38]Therefore, when a light beam passes through such a TFCD array, it is focused by a sample similar to a microlens array.Imaging performances of generated TFCDs are studied as well (Figure S3, Supporting Information), and the resultant image array also exhibits a rotational symmetry consistent with the predesign.
The evolution of topological defects during the texture transformation vividly reveals the order variation of LCs.Therefore, topological analysis is introduced to better understand the order evolution of different symmetric lattices.Topologies of two different N states (disclination lines and defect walls) of lattices with C 2 , C 3 , C 4, and C 6 symmetries are shown in Figure 3.According to micrographs of textures with disclination lines in Figure 2, a pair of disclination lines emerge in each unit, connected by the preset +1 singularity, which decomposes into a pair of +1/2 defects (red triangles).Each disclination line links adjacent +1/2 and −1/2 defects (blue triangles).For the C 2 and C 4 symmetry conditions (Figure 3a,c,g), two −1/2 defects appear at the intersection of adjacent units.While for cases of C 3 (Figure 3e) and C 6 (Figure 3i) symmetry, four and one −1/2 defects appear at one intersection, respectively.The above phenomena reveal the rotational symmetry-induced restriction on the generation of disclination lines.The topology in each unit is depicted by a sum of all included defect types, which is {(−1/2-+1/2) × 2} for all the above cases.After transforming to defect wall textures, the topology changes to {+1; −1/4 × 4; 0 × 4} for rectangular and square lattices (Figure 3b,h), which means that each unit contains one +1 point defect, four −1/4 (blue quarter disk) point defects, and four defect walls (blue line), which are equivalent to the uniform state in topology.It is {+1; −1/6 × 2; −1/3 × 2; 0 × 4} for the diamond lattice (Figure 3d), {+1; −1/3 × 3; 0 × 3} for the triangular lattice (Figure 3f) and {+1; −1/6 × 6; 0 × 6} for the hexagonal lattice (Figure 3j).Notably, the sum of topological charge in all units always remains zero for two different types of N textures.The defect wall textures are also restricted by the rotational symmetries.
A red laser ( = 633 nm) passes through a polarizer and is then normally incident onto samples placed on a heating stage.A screen is fixed behind to receive the diffraction patterns (Figure S4, Supporting Information).Based on the above analysis, the director distributions in defect wall textures are schematically illustrated in the upper images in We further extend this technique to the formation of quasiperiodic LC textures.Figure 5a exhibits a C 5 symmetry Dürer's penciling composed of pentagons and diamond-shaped gaps.A +1 singularity (red dot) is set in the center of each alignment polygon.When cooling from the isotropic state to the N phase, disclination lines arise, as expected.Unlike periodic lattices, disclination lines here are not unidirectionally orientated (Figure 5b).The disordered disclination lines are attributed to the complexity of the alignment pattern.Even after several thermal cycles, disordered disclination lines still coexist with defect walls (Figure S5, Supporting Information).Fortunately, mechanical stress at the N-SmA phase transition point helps to completely transform the disclination lines into defect walls.After cooling to the SmA phase, larger TFCDs appear in pentagonal regions, and smaller TFCDs appear in the diamond gaps, which together form a C 5 symmetry lattice.Topological analysis of the two different N states is also carried out.For the disclination line textures, both the density of ±1/2 defects and their connecting types increase (Figure 5c).Therefore, much energy is required to overcome the barrier to reach the perfect defect wall texture.The topology of the defect walls is depicted as {+1; −1/5 × 5; 0 × 5} in a pentagonal lattice and {+1; −1/10 × 2; −2/5 × 2; 0 × 4} in a diamond lattice (Figure 5d).The sums of topological charge remain zero.The diffraction pattern exhibits a perfect ten-fold rotational symmetry, matching well with the C 5 symmetry design (Figure 5e). [39]A quasiperiodic texture with C 8 symmetry is also generated (Figure S6, Supporting Information).In this case, some defect walls deviate from the designed location due to the increased complexity in symmetry.
Alignment lattices are preset via photopatterning to guide the self-assembly of LCs.When turning to SmA phase, LC tends to form TFCDs matching the patterned alignments; after heating back to the N phase, defect wall textures, which are perfectly consistent with the predesigned rotational symmetry of alignment lattices, are generated due to the inheritance of orientational order.The free energy of textures with defect walls is higher than that of textures with disclination lines, indicating the latter case  is more stable than the former one.Thanks to topological protection, the transformation between the two cases involves a global director field change, leading to a high energy barrier.As a result, texture with defect walls (a topologically protected nonsingular localized structure) can exist stably as a metastable state (Figure S7 and Movie S1 (Supporting Information), taking LC texture of C 4 symmetry as an example).We find that the order of rotational symmetry intensively affects the transformation between the disclination line texture and defect wall texture.C 6 , C 4 , C 2 , and C 3 support direct transformation during thermodynamic processes, while the difficulty increases gradually.The C 5 case only occurred with the assistance of mechanical stressing.Even with mechanical stressing, disclination lines cannot be completely eliminated for the C 8 case.This is attributed to the distinct free energies caused by different topological defects associated with rotational symmetry.Notably, the confinement (especially the ratio between sample thickness and unit size) also affects the predesign of LC textures.8CB is adopted here for proof demonstration, and the LC can be replaced to meet the requirements of room temperature applications.

Conclusion
We propose a strategy for tailoring the rotational symmetry of LC domains based on alignment-lattice predesign and the inheritance of orientational order during the N-SmA phase transition.Defect wall textures with C 2 -C 6 symmetries are demonstrated, which match well with the preset alignment lattices.Topological analysis reveals the distinct evolution of order between disclination line textures and defect wall textures of different rotational symmetries.This work enhances the controllability of LC selfassembly and enriches the knowledge of self-organized ordered systems.It may pave the way for advanced applications of topological defects and release unprecedented applications.

Experimental Section
Materials: 8CB (NCLCP, China) exhibits an isotropic-N phase transition at 40.5 °C and an N-SmA phase transition at 33.5 °C.The photoalignment agent SD1 (NCLCP, China) was dissolved in dimethylformamide (DMF) at a concentration of 0.3 wt.%.Polydimethylsiloxane (PDMS, Dow Corning, USA) was used as a homeotropic alignment layer.
Sample Fabrication: Ultrasonic bathing and UV-ozone cleaning were adopted successively to treat the glass substrates.The substrates were spin-coated with SD1 and cured at 100 °C for 10 min.Then, the substrate coated with the SD1 layer was exposed under a dynamic polarization microlithography system (Digi Optron-120, NCLCP, China) to record the desired planar alignment lattices.A thin PDMS layer mixed with a small amount of initiator was spin-coated onto the other substrate and baked at 120 °C for 20 min for curing.Subsequently, two pieces of substrates were separated by UV glue doped with spacers to form a hybrid cell.8CB was then injected into cells by capillary action at 65 °C and then cooled to the N and SmA phases successively using a heating stage (LTS 120, Linkam, UK).
Characterization: A polarized optical microscope (50i, Nikon, Japan) with a pair of crossed polarizers was utilized for microscopy characterization, and all micrographs were captured by a CCD camera (DS-Ri1, Nikon, Japan).Diffraction patterns were recorded by a digital camera (EOS M, Canon, Japan).

Figure 1 .
Figure 1.The rotational symmetry of different alignment lattices.a) Rectangular and b) diamond lattices with C 2 symmetry; c) triangular lattice with C 3 symmetry; d) square lattice with C 4 symmetry; e) hexagonal lattice with C 6 symmetry; f) quasiperiodic lattice with C 5 symmetry composed of pentagons and diamonds.

Figure 2 .
Figure 2. Evolution of LC textures guided by different alignment lattices across the N-S phase transition.Alignment singularities of s = +1 (red dots) assembled in a) rectangular, c) diamond, e) triangular, g) square, and i) hexagonal lattices.Textural evolution in thermal cycling for b) rectangular, d) diamond, f) triangular, h) square, and j) hexagonal lattices.The white dashed boxes correspond to the alignment area of the top row.The lower right inset in b) exhibits the layered configuration and director distribution corresponding to a single TFCD.The white arrows denote the directions of crossed polarizers.The scale bars indicate 30 μm for all micrographs.

Figure 4 .
Diffraction patterns clearly present the periodicity and rotational symmetry of different lattices.The corresponding diffraction parameters are consistent with the predesign, further confirming the C 2 , C 3 , C 4 , and C 6 symmetry.When the samples are cooled to SmA phase, the diffraction patterns remain unchanged, verifying the inheritance of orientational order across the N-SmA phase transition.

Figure 3 .
Figure 3. Topological analysis of the two different types of N textures (disclination lines and defect walls) of lattices with different rotational symmetry.a,b) Rectangular lattice.c,d) Diamond lattice.e,f) Triangular lattice.g,h) Square lattice.i,j) Hexagonal lattice.

Figure 4 .
Figure 4. Schematic illustration of director distributions in defect wall textures and corresponding diffraction patterns with a) rectangular, b) diamond, c) triangular, d) square, and e) hexagonal lattices.

Figure 5 .
Figure 5. Textures and topological analysis of a quasiperiodic LC lattice with C 5 symmetry.a) The quasiperiodic lattice of +1 alignment singularities.b) Textural evolution in thermal recycling and mechanical stress process.Topological analysis of the two different N states with c) disclination lines and d) defect walls.e) Schematic illustration of the director distribution in the defect wall texture and corresponding diffraction pattern.The white arrows denote the directions of crossed polarizers.The scale bar indicates 30 μm for all micrographs.