Abstract
Recent advances in surface-patterning techniques of liquid crystals have enabled the precise creation of topological defects, which promise a variety of emergent applications. However, the manipulation and application of these defects remain limited. Here, we harness the moiré effect to engineer topological defects in patterned nematic liquid crystal cells. Specifically, we combine simulation and experiment to examine a nematic cell confined between two substrates of periodic surface anchoring patterns; by rotating one surface against the other, we observe a rich variety of highly tunable, novel topological defects. These defects are shown to guide the three-dimensional self-assembly of colloids, which can conversely impact defects by preventing the self-annihilation of loop-defects through jamming. Finally, we demonstrate that certain nematic moiré cells can engender arbitrary shapes represented by defect regions. As such, the proposed simple twist method enables the design and tuning of mesoscopic structures in liquid crystals, facilitating applications including defect-directed self-assembly, material transport, micro-reactors, photonic devices, and anti-counterfeiting materials.
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Introduction
Liquid crystals (LCs) consist of rod- or disk-like molecules, which can self-assemble into well-defined mesoscopic structures with long-range orientational order1. This ordering can be locally frustrated due to topological reasons, leading to regions called topological defects2,3,4. These regions are shown to segregate foreign molecules and particles in the system, leading to defect-based applications in, for example, directed self-assembly of molecules and colloids5, photonic devices6, biosensing7, and material transport8,9,10. The capability of engineering defects in LCs is important for the abovementioned applications. Existing defect manipulation methods include magnetic and electric field actuation9,11,12, optical control13,14, active stresses15,16, curvature imposed by boundaries17,18,19, patterned substrates20,21,22,23,24,25,26, and chemical interactions5,\(\Psi={12}^{\circ}\) when \(H/L=0.3\) (C), \(H/L=0.7\) (D) and \(H/L=0.9\) (E) (colored by angle \(\beta\)). In (E), the 1st group of disclinations is near the top surface, and the 2nd group defect is generated close to the bottom surface. F Defect state diagram in terms of \(H/L\) and \(\Psi\), blue dotted line for \({\Psi=12}^{\circ}\) and red dotted line for \(H/L=0.43\) (Fig. 2E, F). G Periodic, helical topological structure of the C-state (colored by angle \(\beta\)). H Topological structure of the S-state and W-state (colored by angle \(\alpha\)). I–K Corresponding POM images for the S-, C-, and W-states at \(\Psi\, \approx \,{12}^{\circ}\) showing good agreement with the simulation. Using confocal microscopy to scan the cell from top to bottom, we observe L disclination lines (the first group) close to the top substrate and M the disclination lines (the second group) appearing near the bottom substrate. N, O Two 3D rotation views of the sample to visualize the two groups at the same time. \({z}_{c}\) is the spacing between the two groups in the scanning process, \({z}_{c}=40{{{{{\rm{\mu }}}}}}{{{{{\rm{m}}}}}}\). N When the two groups of disclinations are visualized from the cell top at the same time, the bright lines are the first group disclination lines, close to the top substrate. O Viewing the sample from the cell bottom. Both groups of lines are shown due to the scattering effect, and the second group of lines close to the bottom substrate is in gray because of lower contrast. P 3D colloidal assembly in the W-state. Scale bar: 50 μm.
Next, we rotate the top substrate counterclockwise with respect to the origin of the \({xy}\) plane by an angle \(\Psi\) while fixing the bottom substrate (Fig. 1B). Upon the rotation operation, the bulk nematic is frustrated by the mismatched preferred orientations of the two substrates (Fig. 1A, B). We introduce \({\theta }^{{{{{{\rm{t}}}}}}}\left(x,y\right)\) and \({\theta }^{{{{{{\rm{b}}}}}}}\left(x,y\right)\equiv \theta \left(x\right)\) to represent the preferred orientations of the top and bottom surfaces at coordinates (x, y), respectively. The preferred orientation angle difference, defined as \({\Delta \theta \left(x,y\right)=\theta }^{{{{{{\rm{t}}}}}}}\left(x,y\right)-{\theta }^{{{{{{\rm{b}}}}}}}\left(x,y\right)\), will locally distort the nematic (Supplementary Material 2.2). Because of the preferred angle mismatch between the two substrates, there is a pointwise twist deformation in the bulk achiral nematic. The handedness of the twist is determined by the acute angle the two preferred orientations make. When this angle transitions from acute to obtuse, there will be a twist reversal, at which the bulk nematic is frustrated. Therefore, we expect that disclinations will emerge where the two preferred orientation angles are orthogonal, i.e., \(\Delta \theta=m\pi /2\) with \(m=\pm 1,\pm 3,\ldots .\)44,45 The contour lines of \(\Delta \theta=m\pi /2\) appear as equispaced, parallel lines in the xy plane (Fig. S2).
To elucidate the nematic structure in the above patterned system, we further perform continuum simulations (Materials and Methods) by varying cell gaps \(H\) while fixing pattern period \(L\) and rotation angle \(\Psi={12}^{\circ}\). For simplification, the one-elastic-constant assumption is applied in our simulations, and a comparison with elastic constants of real materials is discussed in Supplementary Information 6.2.4 and Fig. S37. Interestingly, our simulations uncover three different types of defect structures (Fig. 1C−E, Movie S1, Supplementary Material 2.4). When \(H/L < 0.3\), straight line defects (S-state) emerge, appearing as parallel and equispaced lines within the midplane of the cell; their locations and directions match well with the theoretical prediction (Fig. 1C, Fig. S2). The schematics of different local profiles of disclination lines are given in Fig. S3. When \(0.3 \, < \, H/L \, < \, 0.8\), curved line defects (C-state) appear as equispaced, aligned curves, each of which resembles a 3D helix and appears as a wavy line viewed in the z-direction (Fig. 1D, Fig. S4). The handedness of the C-state defects can be reversed by changing the rotation direction (Supplementary Information 2.4 and Fig. S5). When \(H/L \, > \, 0.8\), web-like crossing lines (W-state) emerge, consisting of two groups of equispaced, parallel defect lines located near the two surfaces, exhibiting different orientations, which appear like a web when viewed from the z-direction (Fig. 1E, Fig. S4). The transition of the disclinations from straight to curling shape as cell thickness increases can also be understood by making an analogy to the Peach–Koehler force of dislocations in solids under external stresses46. Note that as soon as the angle \(\Psi\) starts to deviate from \({0}^{\circ}\), the disclination lines in the above states come from infinity one by one with their distance decreasing with increasing \(\Psi\).
Our subsequent experiments confirmed all three predicted defect structures at different cell gaps using a photopatterning system (Fig. 1I−K, Materials and Methods). Photo-alignment is a versatile tool to pattern confining substrates and stabilize both singular and nonsingular defects in LC cells44,45,47. The out-of-plane anchoring strength is on the order of \({ \sim 10}^{-3}\,{{{{{\rm{J}}}}}}/{{{{{{\rm{m}}}}}}}^{2}\) and the in-plane anchoring energy is on the order of \({ \sim 10}^{-4}\,{{{{{\rm{J}}}}}}/{{{{{{\rm{m}}}}}}}^{2}\) (strong anchoring)48. Based on the simulated director fields, we apply the Jones-matrix approach to obtain simulated polarized optical microscope (POM) images (Supplementary Material 6.2.5). The POM images of the three defect states also agree with the experiments. These POM images consist of parallel arrays of dark grains, the size and aspect ratio of which are different among the three defect states (Fig. S6). By contrast, isotropic moirés constructed from the cosinusoidal grating patterns appear as a similar spatial distribution of white grains (Figs. 1B, 2A and Fig. S6). The Jones-matrix approach is good for thin samples (Fig. S6A, B), while for thicker ones where focusing, ray deflection, and oblique rays become relevant (Fig. S6C), the director field cannot be seen clearly (Supplementary Information 6.2.5). These POM patterns are insensitive to the choice of the orientation of the cross polarizers (Movie S2). Although the separation distance between neighboring arrays of grains is the same for the two types of moiré patterns, the grain density in the nematic moiré is twice as dense as that in the isotropic moiré (Fig. S6). To probe the 3D defect structure in the W-state predicted by the simulation, laser scanning confocal fluorescence microscopy was conducted (Fig. 1L–O). Near the top substrate, there are four disclination lines (the first group) along the same direction, which are bright lines inside thick dark bands (Fig. 1L). With scanning along the z-axis, another four thin, gray disclination lines (the second group) appear near the bottom substrate in another orientation (Fig. 1M). Our configuration is similar to the two-layer configuration observed in ref. 49 by confocal microscopy. Therefore, when viewed from the z-direction (Fig. 1M), the two groups of defects appear as a web-like structure, confirming the simulation prediction (Movie S3). 3D views of the web-like structure are presented in Fig. 1N, O (Movie S4).
A 1D cosinusoidal geometric moiré lattice showing the (±1, ∓1)-moiré period \(T\) and (±1, ±1)-moiré \(T^{\prime}\). B Simulation and experimental results of the nematic moiré at \(\Psi={11}^{\circ}\), \({21}^{\circ}\), \({34}^{\circ}\) and \({45}^{\circ}\). Disclinations in the simulation are colored by twist angle \(\beta\). \(T\) is the spacing distance for neighboring defect curves, and \(\omega\) is their tilting angle with respect to the \(x\) axis. The defect helical structure diameter \({A}_{{xy}}\) and pitch \({T}^{*}\) are introduced to characterize defect shapes. C The periodicity and orientation of the colloidal assembly by the defects can be tuned in the nematic moiré. D Moiré period \(T/L\) as a function of rotation angle \(\Psi\). E Moiré tilting angle \(\omega\) as a function of rotation angle \(\Psi\). The insert is the W-state defect. Crossings are denoted by gray lines, connected by pseudolines (black dashed lines). F \({T}^{*}/{T{{\hbox{'}}}}\) and \({A}_{{xy}}/L\) as functions of \(\Psi\) while fixing \(H/L=0.43\). G \({T}^{*}/{T{{\hbox{'}}}}\) and \({A}_{{xy}}/T\) as functions of \(H/L\) at \(\Psi={12}^{\circ}\). Scale bar: 50 μm. Source data are provided as a Source Data file.
We further perform simulations to construct a defect state diagram by varying the gap-to-pattern ratio \(H/L\) and the rotation angle \(\Psi\) (Fig. 1F). Consistent with the cell gap effect, we find that the three defect states appear roughly in different ranges of \(H/L\) in the diagram (Fig. 1C–E). This state diagram can be understood by considering the relative elastic energy costs of different defect states (Supplementary Material 2.3). When the cell gap is narrow (small \(H/L\)), the surface anchoring effect is important. The bulk nematic field represented by the orientation angle \(\theta \left(x,y,z\right)\) adopts surface-preferred orientations. The anchoring conflict between the two surfaces will cause a transition from \({\theta \left(x,y,z\right) \, \approx \, \theta }^{{{{{{\rm{b}}}}}}}\left(x,y\right)\) near the bottom surface to \(\theta \left(x,y,z\right) \, \approx \, {\theta }^{{{{{{\rm{t}}}}}}}\left(x,y\right)\) near the top surface (Fig. 1C, F). This transition can be a smooth twist in most areas. However, in certain regions where the angle difference \(\Delta \theta \left(x,y\right)\) becomes the maximum value \(\pi /2\), twist-winding disclinations are expected to appear (Supplementary Material 2.2). This explains the agreement between the prediction and thin-cell results. When the two substrates are away from each other (large \(H/L\)), the surface patterning effect becomes relatively weak and the bulk nematic dominates. Consequently, the system chooses to minimize its elastic free energy by aligning the nematic uniformly in the bulk and forming defects near the substrates to accommodate the surface anchoring pattern. This argument can well explain the emergence of line defects in the W-state: bulk nematic adopts a uniform director field; for each surface, defects should appear where the difference between the surface-preferred orientation and the bulk nematic orientation angle reaches the maximum value of \(\pi /2\) (Fig. S4C). Therefore, two layers of disclinations will emerge in thick cells, and the separation distance between neighboring line defects for each layer should coincide with the periodicity \(L\) of the anchoring pattern. At medium cell gaps, a delicate competition between the surface patterning and the bulk nematic leads to a complex defect structure, i.e., the C-state defect (Fig. 1D, G, Fig. S4B), the morphology of which is essentially intermediate between the parallel straight lines in the S-state and the web-like lines in the W-state, serving as a transition state between these two defect states (Movie S1). Similar to the W-state (Fig. S4C, Fig. 1C, H, I), the C-state forms a locally uniform director field in the midplane, and the defects are repelled to the two substrates (Fig. S4B).
Our further analysis of the local profiles of these emergent disclinations reveals that they are structurally different from those reported in similar systems50 (Supplementary Material 2.4). The director profiles of these disclinations can be characterized by two angles51,52, namely a twist angle \(\beta \in \left[0,\pi \right]\) and a phase shift angle \(\alpha \in [0,2\pi).\) (“Methods”, Supplementary Information 2.4, Fig. S3). Pure-twist corresponds to \(\beta=\pi /2\); \(+1/2\) wedge winding and \(-1/2\) wedge winding are characterized by \(\beta=0\) and \(\beta=\pi\), respectively. A pure-twist winding with \(\beta=\pi /2\) can be further characterized by the angle \(\alpha\), which can distinguish between tangential-twist (\(\alpha=\pi /2\), \(3\pi /2\)) and radial-twist (\(\alpha=0\), \(\pi\)) windings50. Our calculations show that line defects in the S- and W-states are of the pure-twist type with \(\beta \equiv \pi /2\) (Fig. 1C, E, Fig. S4). Despite the simple geometry of the line defect in the S-state, its local winding periodically varies from radial-twist-I (\(\alpha=0\)), to tangential-twist-I (\(\alpha=\pi /2\)), to radial-twist-II (\(\alpha=\pi\)), then to tangential-twist-II (\(\alpha=3\pi /2\)), and eventually back to tangential-twist-I (\(\alpha=2\pi \equiv 0\)) profiles (Fig. 1H, Fig. S7). The angle \(\alpha\) of the S-state defect changes from \(0\) to \(2\pi\) in a lattice period \(L\): this contrasts with the line defect with uniform \(\alpha\) in similar systems in which the top substrate adopts a uniform anchoring50. However, for the two groups of parallel defect lines in the W-state, the local director field along the curve tangent is nearly constant (Fig. S4C, top plane and bottom plane). The two groups of defects are tangential-twist-II (near top) and tangential-twist-I (near bottom) (Fig. 1H). Within the same group, all the parallel line defects share the same local profile and, therefore, the constant angle \(\alpha\). In the C-state, the twist angle \(\beta\) of the local winding varies along the defect curve (Figs. 1D, G, 2B, Fig. S4B): when the local winding is close to the two surfaces, \(\beta\) approaches \(\pi /2\) (pure-twist); when the curve passes through the midplane of the cell, the local winding is of wedge type, with \(\beta\) approaching \(0\) and \(\pi\) alternatively (wedge-twist). Therefore, the C-state defect belongs to the wedge-twist type. Distinct from 2D liquid crystals, in 3D nematics, wedge disclinations of winding number \(\pm 1/2\) can be transformed through a continuous transformation, with the symmetric twist state found in between. This continuous transformation of the local profiles can also be used to explain the state transitions (between the S-, C- and W-states) as we tune the cell gap.
In Fig. 1M, the focus of the microscope is adjusted in the experiments to provide a clear view of the top and bottom parts separately. In the particular case of the defect network state (the W-state), colloids are found to assemble in both groups of disclinations (top layer and bottom layer), giving rise to a 3D configuration (Fig. 1P, Movie S5). This is potentially helpful for reversible reprogramming of colloids as building blocks to achieve multiple functions.
We also show that the geometric details of the emerging disclinations, including neighboring defect spacing \(T\), defect orientation angle \(\omega\), helical periodicity \({T}^{*}\), and projected helix diameter \({A}_{{xy}}\) (C-state), are fundamentally dictated by the geometric moiré pattern regardless of defect states (Fig. 2A, B, Supplementary Material 2.4). As \(\Psi\) increases, the disclinations become closer to each other and their orientations further rotate (Fig. 2B, Movies S6 and S7). Larger \(\Psi\) gives rise to more frequent variations in the anchoring angle mismatch \(\Delta \theta\) in space, the disclinations therefore appear denser in the system. Both T and ω can be understood by the geometric moiré pattern28. The theoretical values \(T=L/\sqrt{2(1-\cos \Psi )}\) and \(\omega=\Psi /2\) for the first-order \((\pm 1,\mp 1)\) mode in the geometric moiré pattern (Fig. S8) are quantitatively matched by those measured from all three defect states in both simulations and experiments (Fig. 2D, E). This programmable variation in disclination lines can be directly utilized to template colloidal self-assembly. For instance, when colloidal particles are added to the nematic moiré in the S-state, they tend to self-assemble into linear chains, the position and orientation of which follow the geometric features of the nematic moiré (Fig. 2C, Movie S8). Interestingly, these self-assembled colloidal chains can also impact straight-line disclinations (with a local twist profile) by inducing entangled structures (Fig. 2C insets). In an earlier work, the disclinations around particles were found to be sensitive to the cell gap40. Different from the closed entanglement loops around particles34,41,42, we show the interaction between existing defect curves and particles, as particles are attracted to the straight-line disclinations (the S-state). In our case, the defect curve (S-state) in Fig. 2C has a constant pure-twist profile akin to the assembly structures in ref. 53. The projected helix diameter \({A}_{{xy}}\) in the C-state increases as \(\Psi\) decreases (Fig. 2F, Movie S6) or \(H/L\) increases (Fig. 2G, Movie S1). Upon the transition from the C-state to the W-state, \({A}_{{xy}}\) approaches \(T\) as the neighboring wavy-like defects intersect and start to form web-like defects (Fig. 2F, G, Movie S1).
Different length scales that emerge from the defect structures stem from moiré modes of different frequencies, the high-frequency ones of which are usually difficult to see in isotropic moiré patterns (Supplementary Material 2.4). For the C-state, there are two emerging geometric parameters associated with the helical-like defects, namely the period of the helix, \({T}^{*}\), and its helical diameter projected onto the \({xy}\) plane, \({A}_{{xy}}\) (Fig. 2B). For the W-state, \({T}^{*}\) is defined as the spacing between adjacent defect crossings on the same pseudo line. It turns out that \({T}^{*}\) in both states can be understood by the periodicity \({T}^{{\prime} }\) for the \((\pm 1,\pm 1)\) mode in the geometric moiré pattern (Figs. S8, S9), as \(\frac{{T}^{*}}{{T}^{{\prime} }}=2\) in both states (Fig. 2F, G). Therefore, our nematic moirés provide a simple method to tune the topology and geometry (shapes, periodicities, orientations, etc.) of the disclinations using the geometry of moiré patterns, which can further guide the colloidal assembly.
A 1D sinusoidal splay-bend pattern generates a 2D lattice of ring defects
In the previous system, a 1D cusp-like pattern can generate 1D periodic defect structures. Here, we demonstrate that an alternative 1D pattern can generate a 2D defect lattice. Specifically, we consider a 1D sinusoidal splay-bend pattern for the two surfaces. Let \(\theta \left(x,y\right)=\frac{\pi }{2}\sin (2\pi x/L)\) for both surfaces prior to rotation (Fig. 3A, B, Fig. S1). We first use the same theory to plot the predicted disclinations and find a 2D lattice of loops of alternating sizes (Fig. 3D, Supplementary Material 3.1, Figs. S10, S11). This predicted configuration agrees very well with the thin-cell simulation (\(H/L=0.1\)) and experiment (Fig. 3H, Fig. S11). As the cell gap increases, both simulation and experiment show that smaller defect loops (Loop-I) shrink and eventually self-annihilate, and larger loops survive (Fig. 3E, G, Figs. S12, S13, Movie S9). Simulations also find that the surviving defect loops gradually switch from a more round-like shape (Loop-II, Fig. 3E, F, Fig. S13B) to a rhombohedral shape (Loop-III, \(H/L\in \left[0.2,0.6\right]\), Fig. 3E, G, Fig. S13C) in the planar view. This transition with an out-of-plane buckling is akin to the transition from the S-state to the C-state in the 1D cusp-like pattern (Figs. S12, S13). There is an excellent agreement between the simulated and experimental POM images (Fig. 3H, I), again underscoring the fidelity of the simulation method. These novel optical images consist of a lattice of dark grains and rings, in sharp contrast to that formed from the cusp-like pattern and the corresponding isotropic moiré pattern (Figs. 2B, 3C). For even thicker cells (\(H/L \, > \, 0.6\)), a web-like defect is formed (Fig. 3E), akin to the thick-cell results of the cusp-like pattern (Figs. S4C, S12F). As we increase the angle \(\Psi\) from \({0}^{\circ}\), the defects first enter the simulation window around \(\Psi={6}^{\circ }\), as shown in Fig. S14A. Then we observe one array of closed defect loops at \(\Psi={12}^{\circ}\) (Fig. S14B) and multiple arrays of defect loops with a distance of \(T\) after \(\Psi \ge {24}^{\circ }\). As \(\Psi\) increases, the aspect ratio of the loops decreases, and the loops become rounder, as seen in the simulation and theory (Supplementary Information 3.3, Fig. S14I).
A Schematic of the system with the dashed thick rods indicating the geometry of the pattern. The inset is the sinusoidal pattern in a period \(L\). B Superposed surface-preferred director field of the two anchoring patterns and C its mapped geometric pattern with identical period \(L\) and rotation angle \(\Psi\). D Theoretically predicted defect prediction for \(\Psi={3}^{\circ}\), showing defect period \(T\) and tilting angle \(\omega\). E From left to right: the simulation of \(H/L=0.1\) has two groups of defect loops, \(H/L=0.3\) only has surviving larger loops, and \(H/L=1.0\) gives rise to web-like defects (\(\Psi={30}^{\circ}\)). F, G A more detailed comparison of Loop-I, Loop-II, and Loop-III in the simulations and the classical wedge-twist loop. POM images (H) and the corresponding simulated images (I) for thin cells (left) and medium-gap cells (right) at \(\Psi={30}^{\circ}\). J Moiré period \(T/L\) in theory, simulations, and experiments. K Moiré tilting angle \(\omega\) in theory, simulations, and experiments. Scale bar: \(50{{{{{\rm{\mu }}}}}}{{{{{\rm{m}}}}}}\). Source data are provided as a Source Data file.
The local topological structures of these defect loops are different from the pure-twist and wedge-twist defect loops (Fig. 3G, Fig. S13) reported in driven and active nematic systems51,54. Along a standard wedge-twist loop (Fig. 3G), its \(\beta\) profile continuously varies from pure-twist (\(\beta=\pi /2\)) to \(+1/2\) wedge (\(\beta=0\)), then back to pure-twist (\(\beta=\pi /2\)), to \(-1/2\) wedge (\(\beta=\pi\)), and finally returns to pure-twist (\(\beta=\pi /2\)) (Fig. S3A). For thin cells and medium-gap cells, the three kinds of wedge-twist defect loops characterized by the profile of angle \(\beta\) are generated in both simulations and experiments (Fig. 3D–I). Both Loop-I and Loop-II are of elliptical shape and exhibit pure-twist profiles (\(\beta=\pi /2\)), except for the two end regions along their long axes (Fig. 3F). Loop-I and Loop-II are also different: Loop-I and Loop-II (blue loops in Fig. 3D) exhibit negative and positive anchoring angle differences \(\Delta \theta\), respectively (Fig. 3D), which implies that \(\Delta \theta\) changes oppositely from the inside to the outside region of the loop (Fig. S10). Moreover, the two end regions along the major axis in a Loop-I have a \(+1/2\) wedge-like profile (\(\beta \, < \, \pi /2\)), whereas the end regions along the long axis of a Loop-II have a \(-1/2\) wedge-like profile (\(\beta \, > \, \pi /2\), Fig. 3F, Figs. S12, S13). As the cell transits from a thin gap to a medium gap, a Loop-II can evolve into Loop-III, a nontrivial, distorted, 3D loop (Movie S9). When the tangent vector rotates along the Loop-III for one revolution, it goes through both the \(+1/2\) wedge profile and \(-1/2\) wedge profile 2 times and passes the pure-twist profile 4 times, which is also distinct from the wedge-twist loop (Fig. 3G). Note that our system uses a patterning technique and quasi-static rotation operation to manipulate topological structures, and defect loops are stable except when they are smaller than ~\(1200\ {{{{{\mu }}}}{{{{{\rm{m}}}}}}}^{2}\) in the experiments. For energy reasons, the small loop gradually shrinks and eventually annihilates. The topological charges of the three kinds of loops are all zero (Supplementary Information 3.2) and they can self-annihilate if not prevented by laden particles. Since there is no more than a \(\pi\) rotation from the top to the bottom patterned substrate, knot formation is topologically forbidden.
Although the spatial variation details of the two 1D splay-bend patterns (cusp-like and sinusoidal) are different, both can produce the moiré period \(T\) and the tilting angle \(\omega\) in terms of defect geometries and optical patterns (Fig. 3H, I, Fig. S14, Movie S10). Note that \(\Psi={90}^{\circ}\) is an interesting case in which theory predicts a square lattice of round defect loops. This is indeed confirmed in experiments and simulations (Fig. S15), which also identify a metastable state in which rectangle-like disclination loops with their sides along the \(x\)- or \(y\)-direction appear (Fig. S15). These defects can emerge when neighboring defect loops coalesce into large ones.
When colloidal particles are incorporated into the sinusoidal pattern system, we observe various loop-shaped colloidal assemblies in the experiment. The expansion of the disclination loops during a twist operation can be used to collect particles, which leads to loop-shaped colloidal self-assembly structures (Fig. 4A). Importantly, when a particle-laden defect loop undergoes shrinkage in the system, colloidal particles can prevent the self-annihilation of the defect through jamming (Fig. 4B). By contrast, a particle-free defect loop will disappear during the same shrinking process (Fig. 4B). We therefore expect that the interplay of dynamic disclinations and colloids in nematic moiré cells will give rise to more intriguing phenomena.
A The nucleation and expansion of the disclination loops can collect nearby particles. \(\Psi={90}^{\circ}\). B The colloidal assembly undergoes jamming as defect loops shrink and prevent their self-annihilation. C In a similar scenario without colloids, the defect loop is eventually self-annihilated. \(\Psi={30}^{\circ}\). t is time. Scale bar: \(50\ {{{{\mu }}}}{{{{{\rm{m}}}}}}\).
A 2D defect lattice pattern generates hybrid disclinations
To further demonstrate the versatility of the proposed nematic moiré, we turn to 2D anchoring patterns. Specifically, we consider a surface pattern of a 2D lattice of defects consisting of two interweaving square lattices of bend-type \(+1\) and \(-1\) defects (Fig. 5A). This 2D pattern is similar to the dot screen pattern in isotropic moirés28 (Fig. 5B, C). Here, we focus on a thin-film scenario with \(H/L=0.3\) (Fig. S16), in which the system forms disclinations connecting two imprinted defects from opposite surfaces at equilibrium. Before rotation, i.e., \(\Psi={0}^{{{{{{\rm{o}}}}}}}\), each pair of same-charge surface defects at the same \((x,{y})\) coordinate will be connected by two disclination lines of half-integer charge instead of one disclination line of integer charge due to elastic reason55; these two disclinations repel each other elastically, and they appear as loop-like defects in 3D20 (Fig. S17A), and as shown in Fig. S17C–E, a nematic between loops and patterned surface defect core positions has depressed order and a quasiloop that connects two \(+1\) defect cores and two \(-1\) defect cores has \(\beta \in [0,\frac{\pi }{2}]\) and \(\beta \in [-\frac{\pi }{2},0]\), respectively (Supplementary Information 4.2).
A Schematic of the system with the patterned top substrate in pink and the bottom substrate in dark green. The inset is a schematic of the 2D pattern. B The map** from the dot-screen pattern to the \(\pm 1\) defect lattice pattern (left) and the superposed 2D geometric square gratings formed at \(\Psi={5}^{\circ}\). C Superposed 2D geometric moiré pattern at \(\Psi\, \approx \,{36.8}^{\circ}\). D Simulated defect structure in the nematic moiré at \(\Psi={5}^{\circ}\). E, F Experimental bright field snapshots corresponding to two boxed regions in the dashed box in (B, D). G Superposed period \(T/L\) from geometric moiré pattern theory, simulation, and experiment. H Superposed tilting angle \(\omega\) from geometric moiré pattern theory, simulation, and experiment. Yellow is for the \(\Psi \le {21}^{\circ }\) regime with \(({{{{\mathrm{1,0}}}}},-{{{{\mathrm{1,0}}}}})\) as the dominating moiré, and blue is for the \(\Psi \in [{34}^{\circ },{40}^{\circ }]\) regime with dominating moiré \(\left({{{{\mathrm{1,2}}}}},-2,-1\right)\), with a singular point (infinitely large period) at \(\Psi={\tan }^{-1}(3/4).\) For the other angle ranges, \(\Psi \in [{21}^{\circ },{34}^{\circ }]\) and \(\Psi\, > \,{34}^{\circ }\), no dominating moiré is observed (using green). I Nematic moiré in simulation for \(\Psi \, \approx \,{36.8}^{\circ}\). Its period and tilting angle are \(\widetilde{T}\) and \(\widetilde{\omega }.\) J Experimental coloured POM snapshots (top) and bright field image (bottom) of a periodic defect structure at \(\Psi\, \approx \,{36.8}^{\circ}\) corresponding to the boxed region in (C). K Fast rotation simulation snapshots at \(\Psi={15}^{\circ}\) (left) and \(\Psi={180}^{\circ}\) (right) with the simulation time ratio being \({\tau }_{{{{{{\rm{s}}}}}}}\)/\({\tau }_{{{{{{\rm{cell}}}}}}}=35.4\) (“Methods”). Scale bar: \(50{{{{{\rm{\mu }}}}}}{{{{{\rm{m}}}}}}\). Source data are provided as a Source Data file.
At a small rotation angle (\(\Psi={5}^{{{{{{\rm{o}}}}}}}\)), the corresponding dot screen moiré exhibits a tilt super lattice consisting of periodically overlap** lattice points with a moiré period \(T\) and orientation angle \(\omega\) (Fig. 5B). The emerging defect structure in the nematic moiré is highly correlated with this super lattice, with different regimes in the unit cell engendering defects of different types and shapes (Fig. 5D, Figs. S18, S19). The loop size is small near the unit cell center and gradually increases with the distance from the center, while their windings approach to the pure-twist profile (Fig. S18). Therefore, there is a correlation between the size and the twist angle of the loops (Fig. S19). Experiments have also confirmed this finding (Fig. 5E, F). The periodicity of these defect structures and their orientation angles from both experiments and simulations match well with the geometric moiré theory (Fig. 5G, H, Fig. S20, Movie S11). At large rotation angles, \(\Psi \ge {21}^{{{{{{\rm{o}}}}}}}\), the disclinations in the nematic moiré pattern appear to be aperiodic (Movie S11, Supplementary Material 4.4, Fig. S21).
Interestingly, a periodic defect structure is observed around a special angle \(\Psi={\tan }^{-1}3/4 \, \approx \, {36.8}^{{{{{{\rm{o}}}}}}}\) in the simulation (Fig. 5I), with a lattice constant \(\widetilde{T}\) (different from the moiré period \(T\)) and orientation angle \(\widetilde{\omega }\) (different from the moiré tilting angle \(\omega\)) (Table 1). The rotation angle \(\Psi={\tan }^{-1}3/4\) represents a Pythagorean triple, leading to a super lattice containing overlap** points from opposite layers (Table 1, Fig. 5I). This intriguing defect configuration was also observed in the experiment (Fig. 5J). Note that \({\tan }^{-1}(3/4)\) is a singular point in the moiré theory (divergent black curve in the blue region in Fig. 5G)28 with an infinitely large moiré period \(T\) (see the calculations in Supplementary Information 4.4). In the simulation and experiment, the defect structures are found aperiodic for \(\Psi \in [{34}^{\circ },{40}^{\circ }]\) except \(\Psi \, \approx \, {36}^{\circ}\) (Fig. S21). The emerging defect period \(\widetilde{T}\) and angle \(\widetilde{\omega }\) of the superlattice around the rotation angle \(\Psi \, \approx \, {36}^{\circ}\) agree between the experiments and simulations and can be explained by the Pythagorean triple theory (Fig. 5I, Table 1, Fig. S22) rather than the moiré theory.
In conventional moiré patterns, the translation of one layer can lead to the translation of the emerging pattern in the orthogonal direction. This feature is also found in our 2D nematic moiré (Movie S12). We further confirm that the rotation center will not alter the defect period and tilting angle for small \(\Psi\) (Fig. S24, Supplementary Material 4.6). For the special angle \(\Psi={\tan }^{-1}3/4\), however, the emergence of the periodic structures comes from the overlap** of defects and therefore, the defect configuration is sensitive to the choice of the rotation center (Fig. S23). Note that the defect structure is also sensitive to the rotation speed. When the rotation is in the quasi-static limit, the system shows periodic disclinations during the rotation and can return to the initial state after a π-turn (Fig. S17). However, if the rotation speed is fast (Materials and Methods, Supplementary Material 6.2.2), the system will enter a disordered defect state and be stuck in a state comprised of space-filling curve defects after a π-turn, akin to the Truchet pattern (Movie S13). The hysteresis of the defect structure is rooted in the complex free energy landscape of the system. During substrate rotation, there is a competition between the surface-preferred director field and bulk elasticity. Slow rotation will render the system remaining in the free energy minimum constrained by the anchoring condition; for fast rotation, the bulk director has no time to reorient to satisfy the anchoring condition, thereby leading to a different, path-dependent director field. This also explains the rotation rate dependence of the defect structure in the 2D pattern. When the ratio is lower, there are more bulk defects than near-surface defects and the total volume of defects is lower (Fig. S25).
Applications of nematic moiré patterns
Beyond the versatility of the proposed simple twist method, we proceed to demonstrate the applications of nematic moiré cells. For defect-based applications, existing research efforts have been devoted to the forward design problem: which topological structures with what properties can be formed from a given geometry or a given pattern. Moreover, for a given anchoring pattern, all the possible defect structures are often quite limited to a few metastable states. This greatly limits the applications of nematic disclinations. Here, we propose using the moiré effect to address the inverse problem of designing arbitrary pixelated defect regions.
To this end, we are inspired by the application of moiré methods in the protection of documents and products56. Based on the moiré intensity profiles, new patterns are generated between two specially designed periodic dot screens, one of which is a microstructured image that is located on the document itself (bottom, green layer, Fig. 6A), while the other (top, pink layer, Fig. 6A) plays the role of a revealer. In the design of anti-counterfeiting materials, the superposed new pattern of a highly visible repetitive moiré pattern of a predefined intensity profile shape and color gives the authenticated properties, documents, or products. Here we use simulations to demonstrate that a judiciously chosen nematic moiré pattern can be used to engender arbitrary shapes represented by defect regions. Specifically, we design a circular pattern for the top substrate and a Latin-letter pattern (U, S, or T) for the bottom substrate (Fig. 6B). Combining the understanding that disclination lines appear where the twist angle of the pattern director has a \({90}^{\circ}\) difference in a thin cell44,45, we propose an inverse design possibility and map the “document” pattern (bottom) and “revealer” pattern (top) to LC patterns. At a small twist angle \(\Psi={4}^{{{{{{\rm{o}}}}}}}\), an amplified letter region of topological defects much larger than the periodicity of the pattern emerges, which consists of many defect loops (Fig. 6C). Therefore, we anticipate that nematic moiré cells can generate various programmed shapes and can potentially be used for inverse design, printing, self-assembly, photonics, and anti-counterfeiting materials57.
Inspired by geometric moiré, using alternate surface patterns, various Latin letter-shaped defects following the moiré period and tilting angle can be generated. A By putting a “revealer” pattern (circular) on top of a “document” pattern (T-shape), an array of amplified T shapes can be revealed in the geometric moiré. One of the generated T shapes is noted by a dashed blue frame. The inset is the circular LC pattern mapped from the circular geometric pattern. B Map the “document pattern” at the bottom to the LC pattern (in the shape of U, S, and T). C Results of three simulations for \(\Psi=4^\circ\): Overlap** the circular LC pattern on the top and the three Latin letter LC patterns on the bottom sequentially, pixelated regions in the shape of U, S, and T comprising defect loops are generated in the three simulations.
Lastly, we study how nematic moirés respond to external electric or magnetic fields. A uniform nematic cell can undergo the so-called Frederiks transition if an electric field applied normal to the cell surface is above a threshold value1. Twisting can lower the threshold voltage (Supplementary Material 5.1). For comparison purposes, we choose \({\Psi }_{0}=\Psi={12}^{{{{{{\rm{o}}}}}}}\) for both twisted nematic cells (TNCs) and nematic moiré cells. To characterize the transition in nematic moiré cells, we introduce a tilting angle \(\phi\) to represent the angle between n and the \({xy}\) plane. The measured \(\bar{\phi }\) from both uniform cells and TNCs in the simulations agree well with the above theory (Supplementary Material 5.2). We consider the 1D cusp-like pattern and compare the average tilting angle \(\bar{\phi }\) in the midplane of the cell as a function of the applied voltage \(V\) for the three defect states with the same cell gap \(H\) in the simulation (Fig. S26). Similar to the effect of twisting in planar cells, spatial distortions of the nematic imposed by the surface patterning can also facilitate the onset of the Frederiks transition (Fig. S26B, Movie S14, Supplementary Material 5.2, 5.3). Indeed, the nematic moiré pattern exhibits a threshold voltage approximately \(10\%\) lower than that of TNCs, showing the promise of nematic moiré patterns in applications such as displays and responsive materials.