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Large-scale self-organization of reconfigurable topological defect networks in nematic liquid crystals

Large-scale self-organization of reconfigurable topological defect networks in nematic liquid... ARTICLE Received 25 Apr 2016 | Accepted 13 Sep 2016 | Published 7 Nov 2016 DOI: 10.1038/ncomms13238 OPEN Large-scale self-organization of reconfigurable topological defect networks in nematic liquid crystals 1 2,w 1 1 1 2,w 2 Yuji Sasaki , V.S.R. Jampani , Chiharu Tanaka , Nobutaka Sakurai , Shin Sakane , Khoa V. Le , Fumito Araoka & Hiroshi Orihara Topological defects in nematic liquid crystals are ubiquitous. The defects are important in understanding the fundamental properties of the systems, as well as in practical applications, such as colloidal self-assembly, optical vortex generation and templates for molecular self-assembly. Usually, spatially and temporally stable defects require geometrical frustration imposed by surfaces; otherwise, the system relaxes because of the high cost of the elastic energy. So far, multiple defects are kept in bulk nematic liquid crystals by top-down lithographic techniques. In this work, we stabilize a large number of umbilical defects by doping with an ionic impurity. This method does not require pre-patterned surfaces. We demonstrate that molecular reorientation controlled by an AC voltage induces periodic density modulation of ions accumulated at an electrically insulating polymer interface, resulting in self-organization of a two-dimensional square array of umbilical defects that is reconfigurable and tunable. 1 2 Division of Applied Physics, Faculty of Engineering, Hokkaido University, North 13 West 8, Kita-ku, Sapporo, Hokkaido 060-8628, Japan. RIKEN Center for Emergent Matter Science (CEMS), 2-1 Hirosawa, Wako, Saitama 351-0198, Japan. w Present addresses: Physics & Materials Science Research Unit, 162a Avenue de la Faiencerie, University of Luxembourg, Luxembourg (V.S.R.J); Department of Chemistry, Faculty of Science, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan (K.V.L.). Correspondence and requests for materials should be addressed to V.S.R.J. (email: venkata.jampani@uni.lu) or to F.A. (email: fumito.araoka@riken.jp) or to H.O. (email: orihara@eng.hokudai.ac.jp). NATURE COMMUNICATIONS | 7:13238 | DOI: 10.1038/ncomms13238 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13238 ontrol of the large-scale formation of functional micro on the spontaneous self-organization of NLCs through the and nano-patterns is attracting intense interest in standard reorientation of the director supported by an AC Cthe interdisciplinary field of materials science . Self- voltage, V. For homeotropically aligned NLCs with negative 2,3 organization in soft matter systems, such as colloids , block dielectric anisotropy, the director tends to orient perpendicular to 4,5 6,7 copolymers and liquid crystals , is widely used for designing the applied field above a threshold voltage, VZV . When the th materials with emergent properties and for templating structures director tilt is induced toward the cell (horizontal) plane, with new functionalities. Recently, fabrication of periodic degeneracy remains in the azimuthal angle, j(x,y), which patterns focuses on the integration of top-down (lithographic) results in the formation of a topological defect in the xy-plane, and bottom-up methods because structures prepared solely by and j(x,y) ¼ sf þ c holds. Here, s ¼ 1 and c is a constant self-organization often lack sufficient long-range order, which is value. These topological configurations are often referred 40,41 crucial for practical use. to as umbilics . Contrary to conventional coarsening 42,43 Nematic liquid crystals (NLCs) are anisotropic fluids that behaviour , we show that doping with a small amount of an already possess long-range orientational order of the long axes ionic compound leads to the formation and stabilization of a large of constituent molecules, with the preferred direction called number of umbilics in a square arrangement without the director (n). An advantage of NLCs is that the director can annihilation. The size of the grid is tunable from several be easily controlled by external electric fields owing to its hundreds of micrometres to several micrometres, producing a dielectric anisotropy, as in current LC display applications. high-density of defects. A large single domain is obtained Generally, inhomogeneous director structures, including topolo- spontaneously by the edge effect of the electrodes. Moreover, gical defects, have high elastic energy costs and they appear as optical manipulation enlarges the uniform area to the millimetre 8,9 uncontrollable and unstable features in the bulk unless scale. The arrangement of umbilics can be regarded as a soft 10,11 12,13 geometrically frustrated by a surface as seen in droplets two-dimensional crystal on the micrometre scale, which enables 14–16 and around colloids . Although there are some experimental the direct observation of the moving dislocations in non-uniform systems that exhibit spontaneous periodic patterns under electric arrangements of umbilics. 17–22 23 and magnetic external fields , during heating and in 24,25 submicron thin films , these observations are mostly limited 17,19,23,25,26 to one-dimensional stripes. Two-dimensional patterns Results are relatively rare and are not formed over a large uniform area as State diagram of the micrographic appearance. We begin reconfigurable, tunable patterns. Therefore, the stabilization of by describing the observable textures. Sandwich cells, consisting the complex director fields in NLCs is achieved by top-down of two parallel glass plates coated with indium-tin-oxide 27,28 lithographic approaches such as AFM scratch methods , (ITO), are filled with NLCs (CCN-mn; trans,trans-4,4 -dialkyl- 29 30–34 0 nanoimprint lithography and photo alignment for (1a,1 a-bicyclohexyl-4b-carbonitriles) (Fig. 1a). The ITO glass 29,35 30 multi-stable alignments , light diffraction gratings and is spin-coated with an amorphous fluorinated polymer, 33,36–39 optical vortex applications . However, the templates used CYTOP (CTX-809A, Asahi Glass Co.), dissolved in a fluorinated in these lithographic techniques limit the reconfigurability and solvent (CT-Solv.180, Asahi Glass Co.) to induce the homeotropic 44,45 controllability of the system, and spontaneous self-organization is anchoring . An AC voltage of V ¼ V cos2pft is applied to important for further exploitation of NLCs. the ITO electrodes in order to induce the reorientation of n In this work, we demonstrate stable arrays of defects in NLCs (Fig. 1b,c). The micrographic appearance is studied as a function without preparing a pre-patterned mask. Our approach is based of frequency f and amplitude V . Figure 2a shows four typical CCN-mn C H V<V n 2n+1 th H C 2m+1 m CN F F 2 2 C C F F C C O CF CYTOP 2 n b c z =d V ≥V th 0 0 z =d /2 z =0 x Figure 1 | Schematic illustration of the director configuration in a sample cell. (a) Cross-sectional view of the director configuration below the threshold voltage (homeotropic alignment). Chemical structures shown are the NLC (CCN-mn) and the alignment layer (CYTOP). (b) The director deformation above the threshold voltage. (c) The oblique view of the director field. It is to be noted that the director tilt y is allowed for the arbitrary j. 2 NATURE COMMUNICATIONS | 7:13238 | DOI: 10.1038/ncomms13238 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13238 ARTICLE non-perturbed H alignment and the electric-field-induced G, S or U states, for several sample cells with various conditions. Typical Non-uniform umbilical (U) Grid-like (G) results are summarized in Fig. 3a. Open symbols denote S and G states obtained by adjusting V and f, whereas closed symbols show the U state above V . In our standard experiments, a th solution containing 3 wt% CYTOP is used for spin-coating unless otherwise mentioned. The thickness of the alignment layer l is B120 nm, as estimated by spectroscopic ellipsometry. In this condition, V for the pure CCN-37 (used as received) is almost th Striped (S) Homeotropic (H) constant and small (magenta closed circles in Fig. 3a) due to the normal Fre´edericksz transition. However, the ion-doped samples (open symbols in Fig. 3a) show a significant increase of V at low th frequencies at any cell thickness d, while a good agreement is seen for all the data at high-frequencies. These observations confirm that the sharp increase of V is caused by the ionic contribution. th To obtain more evidence, the thickness dependence of the CYTOP layer is examined, which leads to an important result explaining the V behaviour on l . We dilute the CYTOP th s solution, for example, to 1.0 wt% (blue closed circles) so that the alignment layer becomes thinner (l B17 nm) under the (H) s (S) (U) same spin-coating conditions, which also initially induce the uniform H state. In spite of the same ionic concentration and the same interface, V substantially decreases compared with the th (G) data with open symbols and no further periodic G or S state pattern is formed. Moreover, the data for 0.8 wt% CYTOP solution (l B10 nm, black closed circles in Fig. 3a) agree with those for the pure CCN-37 except for very low-frequency region. These decreases in V indicate that a significant voltage drop th occurs in the CYTOP layers due to the high resistivity which is larger than 10 O  m (ref. 48) (http://www.agc.com/kagaku/ 2 3 10 10 shinsei/cytop/en/data.html), and the effective voltage on Frequency (Hz) the NLC layer is essentially similar to all the cases. To support the finding, we also perform experiments using a surfactant Figure 2 | AC-voltage-dependent states. (a) Four types of textures monolayer of a silane-coupling agent, N,N-dimethyl-N-octadecyl- (umbilical (U), striped (S), grid-like (G) and homeotropic (H) textures) taken 3-aminopropyltrimethoxysilyl chloride (DMOAP, Aldrich) and under crossed-polarizers. P and A denote the polarizer and analyser. All the a polyimide layer (SE-1211, Nissan Chemical), which are images are taken with a constant V ¼ 20 V by varying the frequency. Scale well-known homeotropic surfaces. For SE-1211, effects of the bar, 100mm. (b) The state diagram as a function of frequency and amplitude spin-coating conditions are also examined, as described in of AC voltages. The NLC used here is CCN-37 containing 1 wt% TBABE ions. the Experimental section. The behaviour of these homeotropic The thickness of the NLC sample is 3.0 mmonaverage. surfaces is similar to that of the pure NLC sample (magenta closed circles in Fig. 3a), which shows no substantial change in textures of an NLC, CCN-37, doped with an ionic compound, V . Furthermore, neither the G nor S textures appear. These th 1 wt% tetrabutyl anmonium benzoate (TBABE), taken under results also prove that the increase of V is responsible for the th crossed polarizers. Figure 2b shows the state diagram plotted on pattern formation caused by electrical insulation, that is, the the f-V plane based on the observations of the 3.0-mm-thick cell. field-screening effect of the ionic localization in the vicinity of the The details of the director configuration are addressed later CYTOP layer (Fig. 3b). Thus, the underlying physical origin of (Fig. 4). At high-frequency, the well-known umbilical texture (U), V curves is the Freedericksz transition even though the variation th consisting of randomly located 1 umbilics, is observed. By of V depends on the thickness of the CYTOP layer, l .In th s decreasing the frequency gradually, a striped pattern (S) is formed other words, the bulk reorientation by the dielectric property from the U state. Further decrease of the frequency induces is essential in this phenomenon. We note that polarization orthogonal strips, leading to a formation of a grid texture (G). microscopy also excludes the possibility of the director Finally the dark texture of the homeotropic alignment (H) is reorientation at surfaces due to the surface polarization effect, observed in the low-frequency region. We stress that the that is, there is not observed any typical cloudy pattern that 49,50 emergence of these four states is qualitatively identical in the depends on the surface polarity . The possibility of the other cells with different cell thickness. As shown in Fig. 2b, flexoelectric instability is also excluded for the G or S textures. although the present system is based on the standard Freedericksz Such flexoelectric domains can only be observed in NLCs with transition, all the boundaries of the transitional voltages low dielectric anisotropies and the applied electric field must be separating adjacent states increase sharply as the frequency DC or AC at low frequencies . Besides, the contribution of decreases. Considering that the theoretical threshold voltage of electrohydrodynamics can be ruled out because the tracer pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ¼ p 2K =ejj De (ref. 46), where K is the elastic constant of particles immersed do not exhibit a motion due to the LC flow. 0 3 0 3 the bend deformation, is several volts using typical parameters , In fact, the electrohydrodynamic convection (EHC) pattern the observed V curve in Fig. 2b is markedly different. emerges, overlapping with the G texture for a proper cell th thickness, as described later (Fig. 8b). Moreover, comparing the data denoted by open symbols in Fig. 3a shows that the V curve th Behaviour of threshold voltages. We examine the behaviour shifts to the low-frequency side as cell thickness d increases. of V , which denotes the transitional voltage between the The tendency is also distinct from that caused by the polar surface th NATURE COMMUNICATIONS | 7:13238 | DOI: 10.1038/ncomms13238 | www.nature.com/naturecommunications 3 V (V) 0 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13238 a b d = 4.6 μm, pure CCN37, l =120 nm d = 4.5 μm, 1 wt% TBABE, l =10 nm 40 s d = 4.6 μm, 1 wt% TBABE, l =17 nm V=2V +V s LC d = 11.1 μm, 1 wt% TBABE, l =120 nm d = 4.8 μm, 1 wt% TBABE, l =120 nm Alignment layer d = 2.5 μm, 1 wt% TBABE, l =120 nm d = 2.5 μm, 1 wt% TBABE, l =120 nm s V 48 °C LC l d l s s 1 2 3 4 10 10 10 10 ITO glass Frequency (Hz) Figure 3 | Behaviour of threshold voltage. (a) The threshold voltages for the director reorientation from the homeotropic alignment. The data expressed with open symbols exhibit the grid-like (G) and striped (S) states at lower frequency. On the other hand, the data for the closed circles show only the non- uniform umbilical texture (U). The spin-coating condition is the same for all the data. 3 wt% CYTOP solution is used for spin-coating in all experiments except for the data denoted by closed blue and black circles respectively in which 1.0 wt% and 0.8 wt% CYTOP solutions are used to make a thinner alignment layer. (b) A schematic illustration to explain the voltage drop in the sample cell. The thicknesses of NLC and CYTOP are denoted with d and l , respectively. The solid magenta curve denotes that a substantial voltage drop occurs not only in the NLC layer (V ) but also in the CYTOP alignment layer LC (V ) of the sample cell exhibiting (G) and (S) states. 49,50 instability . Furthermore, as the temperature increases, the around þ 1 umbilics is considered to be a radial type rather G state returns to the H state, namely the V curve shifts to the than spiral one . To support this consideration, fluorescent th high-frequency side. These results are also consistent if the ionic confocal polarizing microscopy observations are performed dynamics is considered and a more detailed description of the (Supplementary Fig. 1). This technique enables the director role of CYTOP is provided in the Discussion. mapping by the intensity distribution of the polarized fluorescence from a doped emitter molecule transmitted 52,53 through a polarizer inserted in the optical path . An image Spontaneous formation of a large single domain. The G state is taken at the middle plane of the cell agrees with our microscope usually accompanied by multi-domains (Fig. 2a). Therefore, in observations. These observations lead to a texture consisting of addition to elucidating the director field, realizing a large single two types of umbilics arranged in squares (Fig. 4h). In the present domain is of particular interest because it has huge potential type of electrodes, the corner always has a hyperbolic hedgehog applications. We present a method for creating single domains by defect with  1 strength because of the topological constraint of the director field (Supplementary Fig. 2). The director combining self-organization with a simple top-down approach. Two glass substrates with stripe-patterned ITO electrodes deformation in the z-direction means that lens effects can be typically several hundreds of microns wide are prepared and they observed qualitatively by moving the objective focal plane of are placed so that the ITO stripes cross (Fig. 4a). Then, we apply microscope (Supplementary Fig. 3). an AC voltage to the square regions of the overlap of the ITO stripes. In this experiment, f is gradually increased with a constant V to transform the H state to the G state. First, uniform Direct manipulation of defect arrays by laser irradiation.We arrangements of umbilics are spontaneously formed along the show another important method for actively obtaining a large edges of the square region (Fig. 4b, Supplementary Movie 1). single domain without the edge effect of electrodes. We use an These initial umbilics trigger the epitaxial growth of a optical tweezers technique with an Nd-YAG laser (1,064 nm) to unidirectional single domain of the umbilics, and the G domain manipulate the local structure . Owing to light-induced heating, spreads over the whole square region of the overlapped the director in the laser spot returns to the perpendicular electrodes. About 1,000 defects are packed with a regular spacing alignment, which can be seen as a dark spot (Fig. 5b and in Fig. 4c. To fill the area completely requires several tens to a Supplementary Movie 2). After removing the laser spot, the few hundred seconds, depending on the quality of the sample director finds a more stable configuration while recovering its tilt cells. In our experimental conditions, this approach can be used angle. Thus, scanning the laser spot allows umbilics to rearrange up to the submillimeter scale. to form uniform arrays (Fig. 5c). We can eventually create a The micrographic appearance is changed dramatically by domain on the sub-square centimetre scale (Fig. 5d). A larger rotating the crossed polarizers (Fig. 4c–e). When the direction of scale would be possible by preparing ideal cells under clean room the extended lines connecting the neighbouring umbilics is conditions. parallel to the polarizers, a square lattice pattern is observed. Laser tweezing can be used to create or erase umbilics at A slight rotation of the polarizers (Fig. 4e) allows the topological arbitrary positions in the S state, which appears on the nature of umbilics to be observed. The rotational directions of the high-frequency side of the G state (Fig. 2). We use square four brushes of the adjacent umbilics are opposite, reflecting the electrodes, the edges of which form a regularly ordered S state 1 states. The contrast of the image is inverted by rotating the (Fig. 6a). After preparing a single G domain, we adjust f at a fixed polarizers by 45. Figure 4f,g shows the effects of inserting a V to the conditions where the S state is slightly preferred over the full-wave retardation plate (l-plate) into the setup in Fig. 4c,d, G state. Then, the umbilics are irradiated with a laser to induce respectively. The blue (added retardation) and bright magenta homeotropic alignment. After switching off the laser, the defects (subtracted retardation) regions imply that the director field no longer appear and instead the more stable S state is obtained 4 NATURE COMMUNICATIONS | 7:13238 | DOI: 10.1038/ncomms13238 | www.nature.com/naturecommunications V (V) 0 NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13238 ARTICLE ab Analyser Patterned ITO glass Polarizer Observation area c d A A f g h Figure 4 | Template-assisted self-organization of square arrays of defects. (a) Schematic illustration of sample cell with patterned ITO electrodes for inducing a single domain of the grid-like texture. (b) The initial stage of the formation of the grid-like texture from the edge of electrode. The frequency of the electric field is gradually increased. The contrast and brightness of the image (b) is different from (c–e) for visibility. (c–e) A single domain of defects array spontaneously obtained in an epitaxial way. These images shown are taken under different crossed-polarizers. (f,g) Textures with the insertion of a full-wave plate, denoted by l. The NLC used is CCN-37 and the experimental condition here is V ¼ 17.5 V, f ¼ 110 Hz. The cell thickness is 4.9 mm. Scale bar, 200 mm. (h) The schematic illustration of the director profile in defects array. (Supplementary Movie 3). Once the S state is prepared, we estimated to be B8 mms from observation of the moving can create artificial isolated umbilic arrays surrounded by umbilics. The action of the flow aligns n along the flow direction the S domain (Fig. 6b). The whole area can be transferred and simultaneously distorts it in the orthogonal direction. from the S state to the G state (Supplementary Movie 4). This can be explained by investigating the effect of Poiseuille Importantly, the S state can be obtained again by erasing flow on the umbilics . To apply simple shear, we prepare a setup umbilics, and thus the process is fully repeatable. consisting of a cell with its upper substrate fixed to a motorized translation stage without using spacers. The series of snapshots in Fig. 7 (Supplementary Movie 6) shows that the shear Flow-induced striped texture. Mechanical flow also affects the flow strongly affects the texture. Even this simple experiment director field, n, substantially. When the NLC sample is demonstrates interesting properties. The lines of the grids normal introduced into the cell, the effect of capillary flow on the texture to the shear direction degenerate and disappear, whereas those is visible. Here, we present an observation for a typical cell parallel to the shear directions remain (Fig. 7a). Once the shear is thickness of 4 mm. The flow speed cannot be controlled in our stopped, the G state is spontaneously recovered (Fig. 7b). experimental conditions, because it depends on the cell gap and Comparing the two adjacent lines normal to the shear direction the observation area. Poiseuille flow occurs and the director shows that their behaviour is different. The position of the defects profile can be regarded as two halves of the shear flow region . on the white dashed lines in Fig. 7a,c is almost unchanged under The G structure changes to the S state as the sample flows during shear flow and the motion of the adjacent grid lines is opposite the injection (Supplementary Movie 5). The flow speed is depending on the flow directions. This is because the bottom NATURE COMMUNICATIONS | 7:13238 | DOI: 10.1038/ncomms13238 | www.nature.com/naturecommunications 5 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13238 a bc Before manipulation Homeotropic orientation After manipulation induced by laser λ-plate Figure 5 | Creation of a single domain by an optical manipulation. (a) A multi-domain structure of grid-like pattern which can be obtained by switching on an electric field suddenly. (b) The homeotropic alignment induced by the irradiation of laser light marked with a white dashed circle. (c) A uniform domain created by the optical manipulation. The cell thickness is 3.0 mm. (d) A large single domain of grid-like texture obtained by the help of optical manipulation. The cell gap is 5.9 mm. V ¼ 32 V. The NLCs used here are the 1:1 mixture of CCN-47 and CCN-55. Scale bars, 200 mm. substrate is fixed (Fig. 7d). The blue and yellow strips exchange the number of grids in a single area dynamically (Fig. 8c and positions depending on the shear direction (compare the bottom Supplementary Movie 7). The edge effect of the electrodes figures of Fig. 7a,c). These behaviours are qualitatively consistent is important and helps to maintain the structure. Particularly in with the properties of umbilics. Details of the effect of mechanical narrow electrodes, the high elastic energy cost due to the flow on the G state will be reported in future work. non-uniform deformation quickly relaxes to the single domain. Thus, the edges help produce a reproducible pattern when the field is switched on and off. We also confirm that even numbers Tunable grid spacing. The spacing between umbilics can also of squares (or odd numbers of umbilics) are allowed inside the be controlled. The grid size is measured by varying the cell area because of the unique director configuration at the corner thickness, d. Figure 8a shows that the size is almost proportional (Supplementary Fig. 2). to d, which is tunable from several to hundreds of micrometres The variable grid spacing provides an interesting feature even (Supplementary Fig. 4). This means that a thinner cell can for inhomogeneous arrangements. So far, we are mainly focusing generate high-density umbilics (Supplementary Fig. 5). For thick on obtaining uniform square arrangements of umbilics. However, cells (for example, thicker than 20 mm in our experiments), the imperfect arrangements of umbilics often form dislocations like change in birefringence is clearly observed by polarization atomic crystals. This is a unique feature that is not observed in microscopy as the frequency increases (Supplementary Fig. 6). other analogue systems, such as two-dimensional bubble rafts , This is because the net birefringence is increased by the because in our system liquid crystal umbilics mimic the atoms in increase of the tilt angle, y , in the bulk (Fig. 1b). Further crystals. Dislocations at a grain boundary can be generated increases of the frequency cause electrohydrodynamic convection artificially by using the acute corner of an oblique cross of the (EHC) (Fig. 8b). This is reasonable because EHC occurs for ITO strips (Fig. 9a). The umbilics tend to align along the edge of 57,58 planar and homeotropic anchoring . Even in the EHC region, the electrodes and they prefer to be packed in squares; thus, grain the grid can be maintained with good stability. Because the boundaries appear. Three different cross-polarized conditions director orientation is normal to the hydrodynamic rolls, show three domains: two domains near the electrode edges and a the topological strength of umbilics is easily visualized by the central domain. Although the present system adds neither direction of the rolls. This additional periodic modulation of the tensional nor compressional stress to the defect array in the cell birefringence would offer interesting optical properties. (horizontal) plane, similar to the bubble raft system, frustration The size of the grid also depends on V ; the spacing increases can be induced in the system by changing the grid size. with V . Of course, f must be decreased simultaneously in Figure 9b shows a visualization of moving dislocations that are accordance with Fig. 2. This property can be used to control of trying to reduce the frustration. The electric field is suddenly 6 NATURE COMMUNICATIONS | 7:13238 | DOI: 10.1038/ncomms13238 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13238 ARTICLE A AA Figure 6 | Creation of a grid-like texture on a striped pattern. (a) Typical microscopic appearance of a single domain filled with striped pattern under different polarization conditions. (b) An isolated domain of grid-like pattern created on the stripe domain with an optical manipulation. CCN-37 is used and the average cell thickness is 3.2 mm. Scale bar, 100 mm. changed to a higher V and a lower f to increase the grid size volume resistivity of 1/s 410 O  m and a small dielectric 0 s (Fig. 8c), and then a lower density of umbilics is required to reach constant of e ¼ 2.0–2.1e (ref. 48). The resistivity is at least s 0 a stable state. Thus, because of the imposed elastic frustration, a around an order of magnitude greater than that of the polyimide 10 13 pair of umbilics annihilates to generate two dislocations, and then surface (10 –10 O  m) (ref. 60), while the dielectric constant these dislocations move apart to relax the structure. The time is still comparable to that of polyimide and LCs, that is, e Be .It s LC course is shown as snapshots in Fig. 9b (Supplementary Movie 8). is a reasonable assumption that the electrical potential, V , LC The dislocation moves in the normal direction of the glide necessary for the reorientation of n is almost the same as the plane (or line). This is natural considering the energetics because Fre´edericksz transition voltage of the pure NLCs shown in Fig. 3a, this motion reduces the density of umbilics. This is a namely the rest of the applied voltage, V  V is used for the LC distinct difference from the atomic crystals where the voltage drop in the alignment layer. We also note that l /doo1 dislocation moves parallel to the direction of the glide line holds because the thickness is l B120 nm in our standard because the atoms never annihilate. spin-coating condition. Then the high-frequency region gives VB{2(e /e )(l /d) þ 1}V BV , because the dielectric LC s s LC LC constant is dominant. However, the conductivity becomes Discussion important at the low-frequency side and V can be approximated The G and S states can be obtained by the combination of dopant as VB{2(s /s )(l /d) þ 1}V , which is greatly affected by LC s s LC ions and a CYTOP layer with a proper thickness. Here, in this the amount of doped ions s . Our preliminary evaluation LC study, we consider the field-screening effect as mentioned above. of the effective conductivity of CCN-37 is of the order of In order to explain the sharp increase of V at low frequencies, th 6  1  1 10 O m at 1 kHz, which is measured without using an we use the configuration shown in Fig. 3b to calculate the voltage alignment layer. Thus, the main reason can be assigned to the applied to the cell. The dielectric constant is e , the LC increased value of s . This supports our experimental data that LC electric conductivity is s , and the thickness of the NLC is d, LC show thicker cells lower the threshold voltage at the same and the corresponding terms for the alignment layer are e , s , s s frequency. Eventually the surface charge density, r , given by and l , respectively. The anisotropy of e and s is not s LC LC r ¼ e {e (s /s )  e }(V /l ) in the low-frequency limit, s 0 LC LC s s s s considered and s includes the ionic contribution. The applied LC becomes high. We can speculate that a high surface charge iot voltage V ¼ Re Ve is written as V ¼ V þ V , where s LC density is achieved on CYTOP after doping with ionic iot iot ~ ~ V ¼ Re V e and V ¼ Re V e are the voltage drop in s s LC LC materials and that it creates a spatial distribution near the the alignment layer and in the NLC layer. Using the complex interface. The expression of r also supports the temperature conductivity, s ~ ¼ s þ ioe and s ~ ¼ s þ ioe , a relation LC LC LC s s s dependence because the ionic localization around the ~ ~ s ~ V =d ¼ s ~ V=2l should hold due to the conservation of LC LC s s surface develops faster owing to the increased mobility, current density. The complex amplitude of the threshold voltage mps ¼ s exp(  W/k T) (ref. 46). LC 0 B ~ ~ applied to the cell is described as V ¼ 2ðÞ l =dðÞ s ~ =s ~ þ 1 V . Since the qualitative behaviour of V is explained by fg s LC s LC th Low and high-frequency limits are evaluated by considering the CYTOP surface, here, we present a detailed s ~ =s ~ ¼ðÞ s þ ioe =ðÞ s þ ioe . Here CYTOP has a high model that can reproduce the transition to the G state from LC s LC LC s s NATURE COMMUNICATIONS | 7:13238 | DOI: 10.1038/ncomms13238 | www.nature.com/naturecommunications 7 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13238 Shearing direction Shearing direction ab c Shearing direction Figure 7 | Effect of a shear flow on the grid-like texture. The snapshots for the effect of the shear flow for the grid-like texture. (a,c) The texture under shear flow. Dashed lines are the guide to the eye. (b) Recovering the grid-like texture. (d) Shematic illustration for the cross section of the director field. The horizontal allows show the direction of shear flow. The NLC used is 1:1 mixture of CCN-47 and CCN-55. Scale bar, 100 mm. the H state prior to achieving the U state. Based on the Fig. 4. The threshold voltage can be regarded as a function of q, observations, the effects of the doped ions (TBABE) and that is, V (q). The real threshold voltage, V , is given by the 0 th the insulating layer (CYTOP) are considered. The details are minimum value of V . A local minimum at q ¼ 0 corresponds to provided in the Method section. We set the origin of the z-axis the transition to the U state, namely, the normal Freedericksz (z ¼ 0) at the center of the nematic slab (Fig. 10a). The transition, and another minimum at qa0 corresponds to the NLC-CYTOP interfaces and the CYTOP-electrode interfaces transition to the G state. Because V depends on the frequency of th ± ± are positioned at z ¼ d/2 and z ¼ h/2, respectively. We focus the applied voltage, permittivities, conductivities, elastic constants on calculating the threshold voltage from the H to G states. In our and thicknesses of NLC and insulating films, the numerical system, the electric potential is given as f ¼0at z¼ h/2 and calculations based on the continuum theory of NLCs are iot f ¼ V cosot at z ¼ h/2, with f ¼ Re½fe . The slightly tilted performed by using the typical material constants for the NLC n is expressed as n ¼ (dn ,dn ,1) and the corresponding potential and CYTOP, and our experimental conditions for d and l. The x y ~ ~ ~ becomes f ¼ f þ df. The translational symmetry in the x-y conductivities s and s , which depend on the concentration of || > plane allows us to write the solution for the grid pattern as dopant ions, are chosen to reproduce the frequency dependence of V in Fig. 2b. The behaviour of V (q) at different typical th 0 dn ðx; y; zÞ¼ yðÞ z cosqxsinqy; ð1Þ frequencies is shown in Fig. 10b. At low frequencies of f ¼ 70 and 200 Hz, the G state is more stable than the U state since the dn ðx; y; zÞ¼ yðÞ z sinqxcosqy; ð2Þ minimum of V is at qa0. However, the two local minima of V 0 0 at q ¼ 0 and qa0 become close at 200 Hz. As the frequency is ~ ~ dfðÞ x; y; z ¼ cðÞ z sinqxsinqy; ð3Þ increased, they become equal (457.5 Hz) and eventually the U where the grid pattern described by equations (1)–(3) should be state becomes more stable (550 Hz). Calculating V (q) by varying rotated by 45 when it is compared with the photos shown in the frequencies allows us to plot V as a function of f (Fig. 10c). th 8 NATURE COMMUNICATIONS | 7:13238 | DOI: 10.1038/ncomms13238 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13238 ARTICLE a b 020 40 Sample thickness (μm) Figure 8 | Tunable grid spacing by sample thickness and applied voltage. (a) Distance of adjacent defects depending on the sample thickness d and electric field strength V . The vertical bars are obtained when V is changed (from V B10 V to 40 V). Larger array size in the bar corresponds to higher V 0 0 0 0 and the frequency is adjusted in each case. The plotted data are the mean value of the minimum and maximum grid size. Closed circles are the data for 1:1 mixture of CCN-47 and CCN-55, and open symbols are for CCN-37. Because of experimental limitations, the range of the thickness used is above 1 mm. We note that there is no significant difference in the grid size between these two NLC samples. (b) A texture accompanying an electro-hydrodynamic convection observed in a relatively thick cell with d ¼ 24 mm. The mixture of CCN-47 and CCN-55 is used. (c) Controllable numbers of arrays in a narrow area. The left, middle, and right micrographs show arrays of 6 6(V ¼ 17.5 V), 4 4(V ¼ 26.2 V), 2 2(V ¼ 39.3 V), respectively. Here CCN-37 is 0 0 0 used and the average cell thickness is 3.7 mm. Scale bars, 100 mm. The magenta lines are the boundaries of the H–G transition evidence, we need to use LC materials that induce homeotropic with qa0 and the black lines are those of the H–U transition with alignment on CYTOP. q ¼ 0. The blue solid circles indicate the intersection of the two In summary, we report unconventional pattern formation in lines. We cannot reproduce the G–S and S–U transitions because NLCs by combining doped ions and a perfluoro polymer our numerical results are limited to the linear stability analysis. (i) alignment layer. The creation of a large single domain of square in Fig. 10c corresponds to Fig. 2, where we use the same material arrays consisting of high-density defects is demonstrated in constant values as in Fig. 10b. Good agreement is obtained, several ways. The system has huge advantages compared with though the numerically obtained position of the boundary previous systems because its self-organization offers highly between the H–G and H–U transitions is located at a little tunable structures that do not require special surface bit higher frequency compared with the experiment. When modifications. The structure can be used directly for diffractive the conductivities s and s are decreased ((ii) in Fig. 10c), the microlens arrays, generation of multiple vortex beams using LC || > 36,61–65 boundary between the H–G and H–U transitions shifts to a lower mesophases as foreseen applications. Because our system frequency, implying that the grid pattern is destabilized by the is self-repairing, it could be applied for sensor applications using decrease in conductivities. In other words, the conductivity director distortion. Polymerizing the structure would provide a stabilizes the G state. However, when the thickness of the soft lithographic template for micro and nanostructures . insulating films is reduced from 0.12 mm(l ¼ 3.24 mm) to 0.05 mm The stabilization of the grids may offer very interesting (l ¼ 3.1 mm) ((iii) in Fig. 10c), similar behaviour is observed, possibility to create freestanding films which provide exposed implying that the insulating film also stabilizes the grid state. If interfaces and are useful, such as for pixelated LC sensor the conductivity is small or the film is thin, the grid state applications in liquid or gaseous environments. Moreover, the disappears ((iv) in Fig. 10c). These results clearly indicate that the system offers a playground for studying the fundamental physics enhanced conductivity of the NLC and the insulating CYTOP of various fields such as microrheology and colloidal science. Our film play a crucial role in forming the G state. experimental results may shed light on creating unconventional To confirm these results, we test typical materials with negative LC textures using ionic effects. Although the data presented here dielectric anisotropy, such as MBBA (Sigma-Aldrich) and an are obtained under standard laboratory conditions, we believe NLC mixture, Phase 5 (Merck). They do not show normal that the quality can be improved substantially under clean and anchoring on the CYTOP surface. For further experimental refined experimental conditions. NATURE COMMUNICATIONS | 7:13238 | DOI: 10.1038/ncomms13238 | www.nature.com/naturecommunications 9 Grid size (μm) ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13238 b t = 15 s t =0 t = 30 s t = 45 s t = 75 s Figure 9 | Dislocations formed by imperfect arrangements of umbilics. (a) Three domains of defect arrays with different orientations observed near a sharp corner of the intersected stripe electrode. Each image is taken by rotating polarizers. (b) The process of annihilation of a pair of defects and the dynamics of dislocations which repel each other. The frustration is generated by increasing the applied voltage from V ¼ 17.5 to 35 V. The cell thickness is 3.0 mm on average and the NLC sample is the 1:1 mixture of CCN-47 and CCN-55. Scale bars, 100 mm. ab c V=V cosωt +h /2 0 (i) (H)–(U) (II) CYTOP (H)–(G) l 70 (Hz) (insulating film) +d /2 0 0.2 0.4 7.2 (ii) d (I) NLC 20 200 (Hz) 0 0.2 0.4 –d /2 6.2 (iii) 457.5 (Hz) (II) CYTOP (iv) 550 (Hz) –h /2 2 3 0 0.2 0.4 0.6 0 0.2 0.4 10 10 V=0 –1 q (μm ) Frequency (Hz) Figure 10 | Theoretical approach for the grid-like state. (a) Schematic of our sample cell and the coordinates used for the calculation. z ¼ d/2 are the CYTOP–NLC interfaces and z ¼ h/2 are the CYTOP-electrode interfaces. The CYTOP thickness l is written as l ¼ (h  d)/2. The electric potential is zero s s at z¼ h/2 and V cos ot at z ¼ h/2. (b) Behaviour of V (q) obtained for four frequencies of 70 Hz (black curve), 200 Hz (blue), 457.5 Hz (magenta) and 0 0 6  1  1  6  1  1 550 Hz (orange). The material constants used are: e ¼ 4, e ¼ 11, s ¼ 2.6  10 O m , s ¼ 2.1  10 O m , K ¼ 4.5 pN, K ¼ 8.5 pN, e ¼ 2, || > || > 1 3 s 15  1  1 s ¼ 10 O m , d ¼ 3 mm and h ¼ 3.24 mm. (c) Dependence of V on frequency f under different conditions: (i) the material constants used are the s th 6  1  1  6  1  1 same with that in b, (ii) same with (i) except for s ¼ 1.5  10 O m , s ¼ 1.2  10 O m , (iii) same with (i) except for h ¼ 3.1 mm and (iv) || > same with (i) except for h ¼ d ¼ 3 mm. Both threshold voltages for the normal Fre´edericksz transition (black lines) and the transition of forming the grid pattern (magenta lines) are plotted. Methods show the nematic phase at room temperature. Experiments are carried out at 25 C Sample preparation and cells. We use the NLC compounds CCN-mn unless otherwise indicated. Moreover, 1 wt% of an ionic compound tetrabutyl 0 0 anmonium benzoate (TBABE, Aldrich) is mixed into NLCs. In preparing the (trans, trans-4, 4 -dialkyl-(1a,1 a-bicyclohexyl-4b-carbonitriles, Nematel GmbH & Co. KG), which possess a negative dielectric anisotropy . Particularly, CCN-37 ion-doped samples, the NLC and ions are diluted with chloroform and mixed by ultrasonic agitation for 1 h. Then, the chloroform is evaporated for 24 h. and the 1:1 mixture of CCN-47 and CCN-55 are used in this work because they 10 NATURE COMMUNICATIONS | 7:13238 | DOI: 10.1038/ncomms13238 | www.nature.com/naturecommunications V (q) (V) V (V) th NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13238 ARTICLE The sample is filled in cells consisting of ITO coated glass substrates with a On the other hand, director n follows the torque balance equation proper distance determined by standard interferometry. Monodisperse spherical @n dF particles are used as a spacer to maintain the cell thickness. The ITO-coated glass g n  ¼ n  ; ð8Þ @t dn substrate is spin-coated by a thin layer of the amorphous perfluoro polymer (CYTOP, Asahi Glass Co.), which induces the perpendicular orientation of n to the where free energy F consists of the Frank elastic energy, f and the electric-field glass surface . In the process of our standard spin-coating, we mix the solute contribution, f as el CTX-809A (a solution containing 9 wt% CYTOP, Asahi Glass Co.) and the solvent CT-Solv.180 (Asahi Glass Co.) with the weight ratio of 1:2. The spin-coating is K K K 1 2 2 2 3 2 f ¼ ðÞ r n þ ðÞ nrðÞ n þ ðÞ nrðÞ n ; ð9Þ made with 800 r.p.m. for 9 s and 3,000 r.p.m. for 15 s. After coating, the substrates 2 2 2 are dried at 70 C for 30 min and baked at 120 C for longer than 30 min. The ratio of the solute and solvent is changed to obtain various thicknesses of the alignment 1 1 f ¼ e E  eðÞ n  E ; ð10Þ layers. The thickness of the CYTOP layer is estimated by spectroscopic el ? a 2 2 ellipsometry (SE-2000, SEMILAB Zrt.), whose results are 120 nm for 3 wt% We neglect the contribution of the NLC flow because there is no observable EHC solution and 17 nm for 1 wt% solution. As alternative alignment layers for homeotropic anchoring, a polyimide surface (SE-1211, Nissan chemical) and a effect, except for in Fig. 8b. e and s are time-independent constants in equations surfactant mono layer (DMOAP, N,N-Dimethyl-N-octadecyl-3- (4) and (5) because the relaxation time of the director fluctuations near the aminopropyltrimethoxysilyl chloride, Aldrich) are tested. In the standard use of threshold voltage is much longer than the period of the AC electric field. Under SE-1211, the stock solution is diluted with the dedicated solvent of equal amount. these conditions, we obtain from equations (4) and (5) Spin-coating condition is the same as the CYTOP. In addition, the stock solution of SE-1211 itself is also coated with 1,000 r.p.m. to have a thicker alignment layer r s ~rf ¼ 0; ð11Þ although a drastic change is not observed. In this condition, the thickness of SE-1211 amounts to 400 nm. For the latter case, glass substrates are taken into a with water solution and 5 vol% DMOAP solution is added to it. After mixing for about s ~ ¼ s þ ioe: ð12Þ 5 min, the excess surfactant is washed with pure water and the substrates are kept at 120 C for 30 min for drying. The effective electrical conductivity is estimated by a Application of Gauss’s theorem to equation (11) at the NLC-CYTOP interface LCR metre (E4980A, Agilent) using an ITO-coated glass cell without having gives the boundary conditions of alignment layers. It must be noted that the anisotropy is not accessible. The 6  1  1 measured value is of the order of 10 O m at 1 kHz, which is used for the ðIÞ ðIIÞ ~ ~ s ~rf ¼ s ~rf at z ¼ d=2; ð13Þ theoretical calculation. z z To prepare patterned electrodes, ITO-coated glasses are fabricated by a in addition to the continuity of potential of standard photo etching method using a positive photoresist (TFR-2950 PM, Tokyo Ohka Kogyo Co., Ltd.). Finally, an AC voltage V ¼ V cos (2pft) is applied between ðÞ I ðÞ II ~ ~ f ¼ f at z ¼ d=2; ð14Þ the ITO-coated glass substrates along the z-direction in order to reorient the director. For shear application, the upper and lower glass substrates are installed on where superscripts (I) and (II) denote the NLC and CYTOP, respectively, and are the motorized stage separately without using spacers. used hereafter. For the homeotropic state, we can easily obtain potential f from the above equations. The corresponding electric field has only the z component E of Polarization light microscope characterization. AC voltage from the function generator is amplified and applied to ITO-coated glasses so that the electric field is s ~ perpendicular to the substrates. The maximum amplitude of the output voltage V ~ 0 E ¼ V in regionðÞ I ; ð15Þ 0 0 s ~ðÞ h  dþ s ~ d is 40 V. Texture observation is usually made by a polarizing microscope (Olympus k s BX51) under different illumination conditions. If necessary, the temperature controller is used. The micrographic appearance is taken by a DSL camera (Sony s ~ E ¼ V in regionðÞ II ; ð16Þ 0 0 ILCE-7R). For our optical manipulation experiments, an Nd-YAG laser (1,064 nm) s ~ðÞ h  dþ s ~ d k s is irradiated to the sample cell on a motorized stage of an inverted microscope (Olympus IX71). The manipulation is made by moving the motorized stage. where s ~ ¼ s þ ioe and s ~ ¼ s þ ioe . Substituting n ¼ (dn ,dn ,1) and k k k s s s x y ~ ~ ~ f ¼ f þ df into equations (8)–(14), and linearizing the results with respect to dn ,dn and df, we obtain x y Fluorescence confocal polarizing microscopy. A confocal laser scanning microscope Leica TCS sp8 is used for elucidating the director configuration of the G 2 2 2 2 2 @dn @ dn @ dn @ dn @ dn @ dn x x y x y x state. The NLC samples are doped with B0.01 wt% of a fluorescent dye, g ¼ K þ þ K  þ K 1 1 2 3 2 2 2 @t @x @x@y @y @x@y @z 7-diethylamino-3,4-benzophenoxazine-2-one (Nile red, Sigma-Aldrich). The Nile "# ð17Þ red molecules orient along the director field due to its shape anisotropy. Thin quartz e 2 @df ~ ~ Re E dn þ E in ðIÞ; 0 x glass plates of thickness 160 mm are used for sample cells. The laser with the 0 2 @x wavelength of 552 nm excites the dye molecule and the emission is detected in the spectral region of 610–660 nm. The polarization of the incoming laser beam is 2 2 2 2 2 adjusted manually with a combination of a quarter wave plate and a linear polarizer @dn @ dn @ dn @ dn @ dn @ dn y x y x y y g ¼ K þ  K  þ K 1 2 3 inserted in the laser path. Simultaneously, polarizing microscopy images by the 2 2 @t @x@y @y @x@y @x@y @z "# ultraviolet transmitted mode are captured during confocal scans of the same sample. ð18Þ e 2 @df ~ ~ Re E dn þ E in ðIIÞ; 0 y 2 @y Theoretical details. The charge density, r, satisfies the Poisson equation r (eE) ¼ r and the charge conservation law r (sE)¼ qr/qt, where E is the electric field, e is the permittivity and s is the conductivity. Here, the ion diffusion dn ¼ dn ¼ 0at z ¼ d=2; ð19Þ x y is neglected for simplicity. By using the electric potential f, these equations are rewritten as ! 2 ~ 2 ~ 2 ~ @ df @ df @ df @dn @dn x y s ~ þ þ s ~  s ~ E þ ¼ 0inðÞ I ; ð20Þ ? a 0 rðÞ erf ¼ r; ð4Þ 2 2 2 @x @y @z @x @y @r Ddf ¼ 0in ðIIÞ; ð21Þ rðsrfÞ¼ : ð5Þ @t ðÞ I ðÞ II In NLCs, e and s are functions of n ~ ~ @df @df ðÞ I ðÞ II ~ ~ s ~ ¼ s ~ ; df ¼ df at z ¼ d=2; ð22Þ k s @z @z e ¼ e d þ e n n ; ð6Þ ab ? ab a a b df ¼ 0at z ¼ h=2; ð23Þ s ¼ s d þ s n n ; ð7Þ ab ? ab a a b where s ~ ¼ s þ ioe and s ~ ¼ s þ ioe . The last terms of equations (17) and ? ? ? a a a with e ¼ e  e and s ¼ s  s , where || and > denote the components of (18) are replaced by the time average over the period of the applied voltage because a || > a || > permittivity (or conductivity) parallel and perpendicular to n, respectively. e and the director fluctuations become slow near the threshold. s are the permittivity and conductivity of the insulating film, which also imposes a The translational symmetry in the x-y plane allows us to write the solution for condition of s oos , s . the grid pattern as equations (1)–(3). Substitution of equations (1)–(3) into s || > NATURE COMMUNICATIONS | 7:13238 | DOI: 10.1038/ncomms13238 | www.nature.com/naturecommunications 11 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13238 equations (17)–(23) yields 16. Wang, X., Miller, D. S., Bukusoglu, E., de Pablo, J. J. & Abbott, N. L. 2 hi Topological defects in liquid crystals as templates for molecular self-assembly. @y e 2 @ y e a a ~ ~ ~ g ¼ 2K q þ E þ K  qRe E c in ðIÞ; ð24Þ 1 0 3 1 0 2 Nat. Mater. 15, 106–112 (2016). @t 2 @z 2 17. Buka, A. & Kramer, L. Pattern Formation in Liquid Crystals (Springer New York, 1996). y ¼ 0at z ¼ d=2; ð25Þ 18. Meyer, R. B. Piezoelectric effects in liquid crystals. Phys. Rev. Lett. 22, 918–921 2 ~ (1969). @ c 2q s ~ c þ s ~  2s ~ E y ¼ 0inðÞ I ð26Þ ? k a 0 2 19. Kuzma, M. R. Nonequilibrium periodic structures induced by rotating and @z static fields in a lyotropic nematic liquid crystal. Phys. Rev. Lett. 57, 349–352 2 ~ (1986). @ c ¼ 0in ðIIÞð27Þ 20. Barnik, M. I., Blinov, L. M., Trufanov, A. N. & Umanski, B. A. Flexo-electric @z domains in liquid crystals. J. Phys. 39, 417–422 (1978). Equations (17) and (18) are reduced to the same equation (24), and the 21. Chigrinov, V. G., Korkishko, T. V., Barnik, M. I. & Trufanov, A. 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Heat convection in liquid crystals 0 00 0 00 c ¼ c þ ic , we have three equations for three quantities y; c and c from heated from above. Phys. Rev. Lett. 30, 736–739 (1973). 0 0 00 00 lz lz lz equations (24)–(27). Substitution of y ¼ y e , c ¼ c e ,and c ¼ c e 0 0 0 24. Lavrentovich, O. D. & Pergamenshchik, V. M. Stripe domain phase of a 0 00 gives a system of homogeneous linear equations for u ¼ y ; c ; c , which can 0 0 thin nematic film and the K13 divergence term. Phys. Rev. Lett. 73, 979–982 @ y only be solved if the determinant vanishes. Because there are two derivatives, 2 (1994). @z 2 ~ @ c 2 25. Lavrentovich, O. D. & Pergamenshchik, V. M. Patterns in thin liquid and , in equations (24)–(27), we have a cubic equation with respect to l . @z crystal films and the divergence (‘surfacelike’) elasticity. Int. J. Mod. Phys. 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This work was supported by JSPS KAKENHI Grant Number JP25103006 and by the 51. Blinov, L. M. Domain instabilities in liquid crystals. J. Phys. Colloq. 40, Foundation for the Promotion of Ion Engineering. C3–247–C3–258 (1979). 52. Smalyukh, I. I., Shiyanovskii, S. V. & Lavrentovich, O. D. Three-dimensional imaging of orientational order by fluorescence confocal polarizing microscopy. Author contributions Chem. Phys. Lett. 336, 88–96 (2001). Y.S., V.S.R.J., C.T., N.S., S.S., K.V.L. and F.A. performed experiments. H.O. performed 53. Smalyukh, I. I., Zribi, O. V., Butler, J. C., Lavrentovich, O. D. & Wong, G. C. L. theoretical analysis. V.S.R.J., F.A. and H.O. designed the project. Y.S., V.S.R.J., K.V.L., Structure and dynamics of liquid crystalline pattern formation in drying F.A. and H.O. wrote the manuscript. All authors have seen and approved the final droplets of DNA. Phys. Rev. Lett. 96, 177801 (2006). manuscript. 54. Jampani, V. S. R., Skarabot, M., Takezoe, H., Musˇevicˇ, I. & Dhara, S. Laser-driven microflow-induced bistable orientation of a nematic liquid crystal in perfluoropolymer-treated unrubbed cells. Opt. Express 21, 724–729 (2013). Additional information 55. Sengupta, A., Herminghaus, S. & Bahr, C. Opto-fluidic velocimetry using liquid Supplementary Information accompanies this paper at http://www.nature.com/ crystal microfluidics. Appl. Phys. Lett. 101, 164101 (2012). naturecommunications 56. Pieranski, P. Generation of umbilics by Poiseuille flows. Eur. Phys. J. E 37, 24 (2014). Competing financial interests: The authors declare no competing financial interests. 57. Nishioka, Y., Kobayashi, F., Sakurai, N., Sasaki, Y. & Orihara, H. Microscopic characterisation of self-assembled colloidal particles in electrohydrodynamic Reprints and permission information is available online at http://npg.nature.com/ convection of a low-birefringence nematic liquid crystal. Liq. Cryst. 43, 427–435 reprintsandpermissions/ (2016). ´ ´ 58. Buka, A., To´th, P., Eber, N. & Kramer, L. Electroconvection in homeotropically How to cite this article: Sasaki, Y. et al. Large-scale self-organization of reconfigurable aligned nematics. Phys. Rep. 337, 157–169 (2000). topological defect networks in nematic liquid crystals. Nat. Commun. 7,13238 59. Bragg, L. & Nye, J. F. A dynamical model of a crystal structure. Proc. R. Soc. A doi: 10.1038/ncomms13238 (2016). Math. Phys. Eng. Sci. 190, 474–481 (1947). 60. Fukuro, H., Sawahata, K., Sato, T. & Endo, H. Optimal alignment materials and This work is licensed under a Creative Commons Attribution 4.0 technologies for various LCDs. SID Symp. Dig. Tech. Pap. 31, 434–437 (2000). International License. The images or other third party material in this 61. Barboza, R. et al. Harnessing optical vortex lattices in nematic liquid crystals. article are included in the article’s Creative Commons license, unless indicated otherwise Phys. Rev. Lett. 111, 093902 (2013). in the credit line; if the material is not included under the Creative Commons license, 62. Yang, B. & Brasselet, E. Arbitrary vortex arrays realized from optical winding of users will need to obtain permission from the license holder to reproduce the material. frustrated chiral liquid crystals. J. Opt. 15, 044021 (2013). 63. Son, B. et al. Optical vortex arrays from smectic liquid crystals. Opt. Express 22, To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/ 4699–4704 (2014). 64. Loussert, C., Kushnir, K. & Brasselet, E. Q-plates micro-arrays for parallel processing r The Author(s) 2016 of the photon orbital angular momentum. Appl. Phys. Lett. 105, 121108 (2014). NATURE COMMUNICATIONS | 7:13238 | DOI: 10.1038/ncomms13238 | www.nature.com/naturecommunications 13 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Nature Communications Springer Journals

Large-scale self-organization of reconfigurable topological defect networks in nematic liquid crystals

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Science, Humanities and Social Sciences, multidisciplinary; Science, Humanities and Social Sciences, multidisciplinary; Science, multidisciplinary
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ARTICLE Received 25 Apr 2016 | Accepted 13 Sep 2016 | Published 7 Nov 2016 DOI: 10.1038/ncomms13238 OPEN Large-scale self-organization of reconfigurable topological defect networks in nematic liquid crystals 1 2,w 1 1 1 2,w 2 Yuji Sasaki , V.S.R. Jampani , Chiharu Tanaka , Nobutaka Sakurai , Shin Sakane , Khoa V. Le , Fumito Araoka & Hiroshi Orihara Topological defects in nematic liquid crystals are ubiquitous. The defects are important in understanding the fundamental properties of the systems, as well as in practical applications, such as colloidal self-assembly, optical vortex generation and templates for molecular self-assembly. Usually, spatially and temporally stable defects require geometrical frustration imposed by surfaces; otherwise, the system relaxes because of the high cost of the elastic energy. So far, multiple defects are kept in bulk nematic liquid crystals by top-down lithographic techniques. In this work, we stabilize a large number of umbilical defects by doping with an ionic impurity. This method does not require pre-patterned surfaces. We demonstrate that molecular reorientation controlled by an AC voltage induces periodic density modulation of ions accumulated at an electrically insulating polymer interface, resulting in self-organization of a two-dimensional square array of umbilical defects that is reconfigurable and tunable. 1 2 Division of Applied Physics, Faculty of Engineering, Hokkaido University, North 13 West 8, Kita-ku, Sapporo, Hokkaido 060-8628, Japan. RIKEN Center for Emergent Matter Science (CEMS), 2-1 Hirosawa, Wako, Saitama 351-0198, Japan. w Present addresses: Physics & Materials Science Research Unit, 162a Avenue de la Faiencerie, University of Luxembourg, Luxembourg (V.S.R.J); Department of Chemistry, Faculty of Science, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan (K.V.L.). Correspondence and requests for materials should be addressed to V.S.R.J. (email: venkata.jampani@uni.lu) or to F.A. (email: fumito.araoka@riken.jp) or to H.O. (email: orihara@eng.hokudai.ac.jp). NATURE COMMUNICATIONS | 7:13238 | DOI: 10.1038/ncomms13238 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13238 ontrol of the large-scale formation of functional micro on the spontaneous self-organization of NLCs through the and nano-patterns is attracting intense interest in standard reorientation of the director supported by an AC Cthe interdisciplinary field of materials science . Self- voltage, V. For homeotropically aligned NLCs with negative 2,3 organization in soft matter systems, such as colloids , block dielectric anisotropy, the director tends to orient perpendicular to 4,5 6,7 copolymers and liquid crystals , is widely used for designing the applied field above a threshold voltage, VZV . When the th materials with emergent properties and for templating structures director tilt is induced toward the cell (horizontal) plane, with new functionalities. Recently, fabrication of periodic degeneracy remains in the azimuthal angle, j(x,y), which patterns focuses on the integration of top-down (lithographic) results in the formation of a topological defect in the xy-plane, and bottom-up methods because structures prepared solely by and j(x,y) ¼ sf þ c holds. Here, s ¼ 1 and c is a constant self-organization often lack sufficient long-range order, which is value. These topological configurations are often referred 40,41 crucial for practical use. to as umbilics . Contrary to conventional coarsening 42,43 Nematic liquid crystals (NLCs) are anisotropic fluids that behaviour , we show that doping with a small amount of an already possess long-range orientational order of the long axes ionic compound leads to the formation and stabilization of a large of constituent molecules, with the preferred direction called number of umbilics in a square arrangement without the director (n). An advantage of NLCs is that the director can annihilation. The size of the grid is tunable from several be easily controlled by external electric fields owing to its hundreds of micrometres to several micrometres, producing a dielectric anisotropy, as in current LC display applications. high-density of defects. A large single domain is obtained Generally, inhomogeneous director structures, including topolo- spontaneously by the edge effect of the electrodes. Moreover, gical defects, have high elastic energy costs and they appear as optical manipulation enlarges the uniform area to the millimetre 8,9 uncontrollable and unstable features in the bulk unless scale. The arrangement of umbilics can be regarded as a soft 10,11 12,13 geometrically frustrated by a surface as seen in droplets two-dimensional crystal on the micrometre scale, which enables 14–16 and around colloids . Although there are some experimental the direct observation of the moving dislocations in non-uniform systems that exhibit spontaneous periodic patterns under electric arrangements of umbilics. 17–22 23 and magnetic external fields , during heating and in 24,25 submicron thin films , these observations are mostly limited 17,19,23,25,26 to one-dimensional stripes. Two-dimensional patterns Results are relatively rare and are not formed over a large uniform area as State diagram of the micrographic appearance. We begin reconfigurable, tunable patterns. Therefore, the stabilization of by describing the observable textures. Sandwich cells, consisting the complex director fields in NLCs is achieved by top-down of two parallel glass plates coated with indium-tin-oxide 27,28 lithographic approaches such as AFM scratch methods , (ITO), are filled with NLCs (CCN-mn; trans,trans-4,4 -dialkyl- 29 30–34 0 nanoimprint lithography and photo alignment for (1a,1 a-bicyclohexyl-4b-carbonitriles) (Fig. 1a). The ITO glass 29,35 30 multi-stable alignments , light diffraction gratings and is spin-coated with an amorphous fluorinated polymer, 33,36–39 optical vortex applications . However, the templates used CYTOP (CTX-809A, Asahi Glass Co.), dissolved in a fluorinated in these lithographic techniques limit the reconfigurability and solvent (CT-Solv.180, Asahi Glass Co.) to induce the homeotropic 44,45 controllability of the system, and spontaneous self-organization is anchoring . An AC voltage of V ¼ V cos2pft is applied to important for further exploitation of NLCs. the ITO electrodes in order to induce the reorientation of n In this work, we demonstrate stable arrays of defects in NLCs (Fig. 1b,c). The micrographic appearance is studied as a function without preparing a pre-patterned mask. Our approach is based of frequency f and amplitude V . Figure 2a shows four typical CCN-mn C H V<V n 2n+1 th H C 2m+1 m CN F F 2 2 C C F F C C O CF CYTOP 2 n b c z =d V ≥V th 0 0 z =d /2 z =0 x Figure 1 | Schematic illustration of the director configuration in a sample cell. (a) Cross-sectional view of the director configuration below the threshold voltage (homeotropic alignment). Chemical structures shown are the NLC (CCN-mn) and the alignment layer (CYTOP). (b) The director deformation above the threshold voltage. (c) The oblique view of the director field. It is to be noted that the director tilt y is allowed for the arbitrary j. 2 NATURE COMMUNICATIONS | 7:13238 | DOI: 10.1038/ncomms13238 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13238 ARTICLE non-perturbed H alignment and the electric-field-induced G, S or U states, for several sample cells with various conditions. Typical Non-uniform umbilical (U) Grid-like (G) results are summarized in Fig. 3a. Open symbols denote S and G states obtained by adjusting V and f, whereas closed symbols show the U state above V . In our standard experiments, a th solution containing 3 wt% CYTOP is used for spin-coating unless otherwise mentioned. The thickness of the alignment layer l is B120 nm, as estimated by spectroscopic ellipsometry. In this condition, V for the pure CCN-37 (used as received) is almost th Striped (S) Homeotropic (H) constant and small (magenta closed circles in Fig. 3a) due to the normal Fre´edericksz transition. However, the ion-doped samples (open symbols in Fig. 3a) show a significant increase of V at low th frequencies at any cell thickness d, while a good agreement is seen for all the data at high-frequencies. These observations confirm that the sharp increase of V is caused by the ionic contribution. th To obtain more evidence, the thickness dependence of the CYTOP layer is examined, which leads to an important result explaining the V behaviour on l . We dilute the CYTOP th s solution, for example, to 1.0 wt% (blue closed circles) so that the alignment layer becomes thinner (l B17 nm) under the (H) s (S) (U) same spin-coating conditions, which also initially induce the uniform H state. In spite of the same ionic concentration and the same interface, V substantially decreases compared with the th (G) data with open symbols and no further periodic G or S state pattern is formed. Moreover, the data for 0.8 wt% CYTOP solution (l B10 nm, black closed circles in Fig. 3a) agree with those for the pure CCN-37 except for very low-frequency region. These decreases in V indicate that a significant voltage drop th occurs in the CYTOP layers due to the high resistivity which is larger than 10 O  m (ref. 48) (http://www.agc.com/kagaku/ 2 3 10 10 shinsei/cytop/en/data.html), and the effective voltage on Frequency (Hz) the NLC layer is essentially similar to all the cases. To support the finding, we also perform experiments using a surfactant Figure 2 | AC-voltage-dependent states. (a) Four types of textures monolayer of a silane-coupling agent, N,N-dimethyl-N-octadecyl- (umbilical (U), striped (S), grid-like (G) and homeotropic (H) textures) taken 3-aminopropyltrimethoxysilyl chloride (DMOAP, Aldrich) and under crossed-polarizers. P and A denote the polarizer and analyser. All the a polyimide layer (SE-1211, Nissan Chemical), which are images are taken with a constant V ¼ 20 V by varying the frequency. Scale well-known homeotropic surfaces. For SE-1211, effects of the bar, 100mm. (b) The state diagram as a function of frequency and amplitude spin-coating conditions are also examined, as described in of AC voltages. The NLC used here is CCN-37 containing 1 wt% TBABE ions. the Experimental section. The behaviour of these homeotropic The thickness of the NLC sample is 3.0 mmonaverage. surfaces is similar to that of the pure NLC sample (magenta closed circles in Fig. 3a), which shows no substantial change in textures of an NLC, CCN-37, doped with an ionic compound, V . Furthermore, neither the G nor S textures appear. These th 1 wt% tetrabutyl anmonium benzoate (TBABE), taken under results also prove that the increase of V is responsible for the th crossed polarizers. Figure 2b shows the state diagram plotted on pattern formation caused by electrical insulation, that is, the the f-V plane based on the observations of the 3.0-mm-thick cell. field-screening effect of the ionic localization in the vicinity of the The details of the director configuration are addressed later CYTOP layer (Fig. 3b). Thus, the underlying physical origin of (Fig. 4). At high-frequency, the well-known umbilical texture (U), V curves is the Freedericksz transition even though the variation th consisting of randomly located 1 umbilics, is observed. By of V depends on the thickness of the CYTOP layer, l .In th s decreasing the frequency gradually, a striped pattern (S) is formed other words, the bulk reorientation by the dielectric property from the U state. Further decrease of the frequency induces is essential in this phenomenon. We note that polarization orthogonal strips, leading to a formation of a grid texture (G). microscopy also excludes the possibility of the director Finally the dark texture of the homeotropic alignment (H) is reorientation at surfaces due to the surface polarization effect, observed in the low-frequency region. We stress that the that is, there is not observed any typical cloudy pattern that 49,50 emergence of these four states is qualitatively identical in the depends on the surface polarity . The possibility of the other cells with different cell thickness. As shown in Fig. 2b, flexoelectric instability is also excluded for the G or S textures. although the present system is based on the standard Freedericksz Such flexoelectric domains can only be observed in NLCs with transition, all the boundaries of the transitional voltages low dielectric anisotropies and the applied electric field must be separating adjacent states increase sharply as the frequency DC or AC at low frequencies . Besides, the contribution of decreases. Considering that the theoretical threshold voltage of electrohydrodynamics can be ruled out because the tracer pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ¼ p 2K =ejj De (ref. 46), where K is the elastic constant of particles immersed do not exhibit a motion due to the LC flow. 0 3 0 3 the bend deformation, is several volts using typical parameters , In fact, the electrohydrodynamic convection (EHC) pattern the observed V curve in Fig. 2b is markedly different. emerges, overlapping with the G texture for a proper cell th thickness, as described later (Fig. 8b). Moreover, comparing the data denoted by open symbols in Fig. 3a shows that the V curve th Behaviour of threshold voltages. We examine the behaviour shifts to the low-frequency side as cell thickness d increases. of V , which denotes the transitional voltage between the The tendency is also distinct from that caused by the polar surface th NATURE COMMUNICATIONS | 7:13238 | DOI: 10.1038/ncomms13238 | www.nature.com/naturecommunications 3 V (V) 0 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13238 a b d = 4.6 μm, pure CCN37, l =120 nm d = 4.5 μm, 1 wt% TBABE, l =10 nm 40 s d = 4.6 μm, 1 wt% TBABE, l =17 nm V=2V +V s LC d = 11.1 μm, 1 wt% TBABE, l =120 nm d = 4.8 μm, 1 wt% TBABE, l =120 nm Alignment layer d = 2.5 μm, 1 wt% TBABE, l =120 nm d = 2.5 μm, 1 wt% TBABE, l =120 nm s V 48 °C LC l d l s s 1 2 3 4 10 10 10 10 ITO glass Frequency (Hz) Figure 3 | Behaviour of threshold voltage. (a) The threshold voltages for the director reorientation from the homeotropic alignment. The data expressed with open symbols exhibit the grid-like (G) and striped (S) states at lower frequency. On the other hand, the data for the closed circles show only the non- uniform umbilical texture (U). The spin-coating condition is the same for all the data. 3 wt% CYTOP solution is used for spin-coating in all experiments except for the data denoted by closed blue and black circles respectively in which 1.0 wt% and 0.8 wt% CYTOP solutions are used to make a thinner alignment layer. (b) A schematic illustration to explain the voltage drop in the sample cell. The thicknesses of NLC and CYTOP are denoted with d and l , respectively. The solid magenta curve denotes that a substantial voltage drop occurs not only in the NLC layer (V ) but also in the CYTOP alignment layer LC (V ) of the sample cell exhibiting (G) and (S) states. 49,50 instability . Furthermore, as the temperature increases, the around þ 1 umbilics is considered to be a radial type rather G state returns to the H state, namely the V curve shifts to the than spiral one . To support this consideration, fluorescent th high-frequency side. These results are also consistent if the ionic confocal polarizing microscopy observations are performed dynamics is considered and a more detailed description of the (Supplementary Fig. 1). This technique enables the director role of CYTOP is provided in the Discussion. mapping by the intensity distribution of the polarized fluorescence from a doped emitter molecule transmitted 52,53 through a polarizer inserted in the optical path . An image Spontaneous formation of a large single domain. The G state is taken at the middle plane of the cell agrees with our microscope usually accompanied by multi-domains (Fig. 2a). Therefore, in observations. These observations lead to a texture consisting of addition to elucidating the director field, realizing a large single two types of umbilics arranged in squares (Fig. 4h). In the present domain is of particular interest because it has huge potential type of electrodes, the corner always has a hyperbolic hedgehog applications. We present a method for creating single domains by defect with  1 strength because of the topological constraint of the director field (Supplementary Fig. 2). The director combining self-organization with a simple top-down approach. Two glass substrates with stripe-patterned ITO electrodes deformation in the z-direction means that lens effects can be typically several hundreds of microns wide are prepared and they observed qualitatively by moving the objective focal plane of are placed so that the ITO stripes cross (Fig. 4a). Then, we apply microscope (Supplementary Fig. 3). an AC voltage to the square regions of the overlap of the ITO stripes. In this experiment, f is gradually increased with a constant V to transform the H state to the G state. First, uniform Direct manipulation of defect arrays by laser irradiation.We arrangements of umbilics are spontaneously formed along the show another important method for actively obtaining a large edges of the square region (Fig. 4b, Supplementary Movie 1). single domain without the edge effect of electrodes. We use an These initial umbilics trigger the epitaxial growth of a optical tweezers technique with an Nd-YAG laser (1,064 nm) to unidirectional single domain of the umbilics, and the G domain manipulate the local structure . Owing to light-induced heating, spreads over the whole square region of the overlapped the director in the laser spot returns to the perpendicular electrodes. About 1,000 defects are packed with a regular spacing alignment, which can be seen as a dark spot (Fig. 5b and in Fig. 4c. To fill the area completely requires several tens to a Supplementary Movie 2). After removing the laser spot, the few hundred seconds, depending on the quality of the sample director finds a more stable configuration while recovering its tilt cells. In our experimental conditions, this approach can be used angle. Thus, scanning the laser spot allows umbilics to rearrange up to the submillimeter scale. to form uniform arrays (Fig. 5c). We can eventually create a The micrographic appearance is changed dramatically by domain on the sub-square centimetre scale (Fig. 5d). A larger rotating the crossed polarizers (Fig. 4c–e). When the direction of scale would be possible by preparing ideal cells under clean room the extended lines connecting the neighbouring umbilics is conditions. parallel to the polarizers, a square lattice pattern is observed. Laser tweezing can be used to create or erase umbilics at A slight rotation of the polarizers (Fig. 4e) allows the topological arbitrary positions in the S state, which appears on the nature of umbilics to be observed. The rotational directions of the high-frequency side of the G state (Fig. 2). We use square four brushes of the adjacent umbilics are opposite, reflecting the electrodes, the edges of which form a regularly ordered S state 1 states. The contrast of the image is inverted by rotating the (Fig. 6a). After preparing a single G domain, we adjust f at a fixed polarizers by 45. Figure 4f,g shows the effects of inserting a V to the conditions where the S state is slightly preferred over the full-wave retardation plate (l-plate) into the setup in Fig. 4c,d, G state. Then, the umbilics are irradiated with a laser to induce respectively. The blue (added retardation) and bright magenta homeotropic alignment. After switching off the laser, the defects (subtracted retardation) regions imply that the director field no longer appear and instead the more stable S state is obtained 4 NATURE COMMUNICATIONS | 7:13238 | DOI: 10.1038/ncomms13238 | www.nature.com/naturecommunications V (V) 0 NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13238 ARTICLE ab Analyser Patterned ITO glass Polarizer Observation area c d A A f g h Figure 4 | Template-assisted self-organization of square arrays of defects. (a) Schematic illustration of sample cell with patterned ITO electrodes for inducing a single domain of the grid-like texture. (b) The initial stage of the formation of the grid-like texture from the edge of electrode. The frequency of the electric field is gradually increased. The contrast and brightness of the image (b) is different from (c–e) for visibility. (c–e) A single domain of defects array spontaneously obtained in an epitaxial way. These images shown are taken under different crossed-polarizers. (f,g) Textures with the insertion of a full-wave plate, denoted by l. The NLC used is CCN-37 and the experimental condition here is V ¼ 17.5 V, f ¼ 110 Hz. The cell thickness is 4.9 mm. Scale bar, 200 mm. (h) The schematic illustration of the director profile in defects array. (Supplementary Movie 3). Once the S state is prepared, we estimated to be B8 mms from observation of the moving can create artificial isolated umbilic arrays surrounded by umbilics. The action of the flow aligns n along the flow direction the S domain (Fig. 6b). The whole area can be transferred and simultaneously distorts it in the orthogonal direction. from the S state to the G state (Supplementary Movie 4). This can be explained by investigating the effect of Poiseuille Importantly, the S state can be obtained again by erasing flow on the umbilics . To apply simple shear, we prepare a setup umbilics, and thus the process is fully repeatable. consisting of a cell with its upper substrate fixed to a motorized translation stage without using spacers. The series of snapshots in Fig. 7 (Supplementary Movie 6) shows that the shear Flow-induced striped texture. Mechanical flow also affects the flow strongly affects the texture. Even this simple experiment director field, n, substantially. When the NLC sample is demonstrates interesting properties. The lines of the grids normal introduced into the cell, the effect of capillary flow on the texture to the shear direction degenerate and disappear, whereas those is visible. Here, we present an observation for a typical cell parallel to the shear directions remain (Fig. 7a). Once the shear is thickness of 4 mm. The flow speed cannot be controlled in our stopped, the G state is spontaneously recovered (Fig. 7b). experimental conditions, because it depends on the cell gap and Comparing the two adjacent lines normal to the shear direction the observation area. Poiseuille flow occurs and the director shows that their behaviour is different. The position of the defects profile can be regarded as two halves of the shear flow region . on the white dashed lines in Fig. 7a,c is almost unchanged under The G structure changes to the S state as the sample flows during shear flow and the motion of the adjacent grid lines is opposite the injection (Supplementary Movie 5). The flow speed is depending on the flow directions. This is because the bottom NATURE COMMUNICATIONS | 7:13238 | DOI: 10.1038/ncomms13238 | www.nature.com/naturecommunications 5 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13238 a bc Before manipulation Homeotropic orientation After manipulation induced by laser λ-plate Figure 5 | Creation of a single domain by an optical manipulation. (a) A multi-domain structure of grid-like pattern which can be obtained by switching on an electric field suddenly. (b) The homeotropic alignment induced by the irradiation of laser light marked with a white dashed circle. (c) A uniform domain created by the optical manipulation. The cell thickness is 3.0 mm. (d) A large single domain of grid-like texture obtained by the help of optical manipulation. The cell gap is 5.9 mm. V ¼ 32 V. The NLCs used here are the 1:1 mixture of CCN-47 and CCN-55. Scale bars, 200 mm. substrate is fixed (Fig. 7d). The blue and yellow strips exchange the number of grids in a single area dynamically (Fig. 8c and positions depending on the shear direction (compare the bottom Supplementary Movie 7). The edge effect of the electrodes figures of Fig. 7a,c). These behaviours are qualitatively consistent is important and helps to maintain the structure. Particularly in with the properties of umbilics. Details of the effect of mechanical narrow electrodes, the high elastic energy cost due to the flow on the G state will be reported in future work. non-uniform deformation quickly relaxes to the single domain. Thus, the edges help produce a reproducible pattern when the field is switched on and off. We also confirm that even numbers Tunable grid spacing. The spacing between umbilics can also of squares (or odd numbers of umbilics) are allowed inside the be controlled. The grid size is measured by varying the cell area because of the unique director configuration at the corner thickness, d. Figure 8a shows that the size is almost proportional (Supplementary Fig. 2). to d, which is tunable from several to hundreds of micrometres The variable grid spacing provides an interesting feature even (Supplementary Fig. 4). This means that a thinner cell can for inhomogeneous arrangements. So far, we are mainly focusing generate high-density umbilics (Supplementary Fig. 5). For thick on obtaining uniform square arrangements of umbilics. However, cells (for example, thicker than 20 mm in our experiments), the imperfect arrangements of umbilics often form dislocations like change in birefringence is clearly observed by polarization atomic crystals. This is a unique feature that is not observed in microscopy as the frequency increases (Supplementary Fig. 6). other analogue systems, such as two-dimensional bubble rafts , This is because the net birefringence is increased by the because in our system liquid crystal umbilics mimic the atoms in increase of the tilt angle, y , in the bulk (Fig. 1b). Further crystals. Dislocations at a grain boundary can be generated increases of the frequency cause electrohydrodynamic convection artificially by using the acute corner of an oblique cross of the (EHC) (Fig. 8b). This is reasonable because EHC occurs for ITO strips (Fig. 9a). The umbilics tend to align along the edge of 57,58 planar and homeotropic anchoring . Even in the EHC region, the electrodes and they prefer to be packed in squares; thus, grain the grid can be maintained with good stability. Because the boundaries appear. Three different cross-polarized conditions director orientation is normal to the hydrodynamic rolls, show three domains: two domains near the electrode edges and a the topological strength of umbilics is easily visualized by the central domain. Although the present system adds neither direction of the rolls. This additional periodic modulation of the tensional nor compressional stress to the defect array in the cell birefringence would offer interesting optical properties. (horizontal) plane, similar to the bubble raft system, frustration The size of the grid also depends on V ; the spacing increases can be induced in the system by changing the grid size. with V . Of course, f must be decreased simultaneously in Figure 9b shows a visualization of moving dislocations that are accordance with Fig. 2. This property can be used to control of trying to reduce the frustration. The electric field is suddenly 6 NATURE COMMUNICATIONS | 7:13238 | DOI: 10.1038/ncomms13238 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13238 ARTICLE A AA Figure 6 | Creation of a grid-like texture on a striped pattern. (a) Typical microscopic appearance of a single domain filled with striped pattern under different polarization conditions. (b) An isolated domain of grid-like pattern created on the stripe domain with an optical manipulation. CCN-37 is used and the average cell thickness is 3.2 mm. Scale bar, 100 mm. changed to a higher V and a lower f to increase the grid size volume resistivity of 1/s 410 O  m and a small dielectric 0 s (Fig. 8c), and then a lower density of umbilics is required to reach constant of e ¼ 2.0–2.1e (ref. 48). The resistivity is at least s 0 a stable state. Thus, because of the imposed elastic frustration, a around an order of magnitude greater than that of the polyimide 10 13 pair of umbilics annihilates to generate two dislocations, and then surface (10 –10 O  m) (ref. 60), while the dielectric constant these dislocations move apart to relax the structure. The time is still comparable to that of polyimide and LCs, that is, e Be .It s LC course is shown as snapshots in Fig. 9b (Supplementary Movie 8). is a reasonable assumption that the electrical potential, V , LC The dislocation moves in the normal direction of the glide necessary for the reorientation of n is almost the same as the plane (or line). This is natural considering the energetics because Fre´edericksz transition voltage of the pure NLCs shown in Fig. 3a, this motion reduces the density of umbilics. This is a namely the rest of the applied voltage, V  V is used for the LC distinct difference from the atomic crystals where the voltage drop in the alignment layer. We also note that l /doo1 dislocation moves parallel to the direction of the glide line holds because the thickness is l B120 nm in our standard because the atoms never annihilate. spin-coating condition. Then the high-frequency region gives VB{2(e /e )(l /d) þ 1}V BV , because the dielectric LC s s LC LC constant is dominant. However, the conductivity becomes Discussion important at the low-frequency side and V can be approximated The G and S states can be obtained by the combination of dopant as VB{2(s /s )(l /d) þ 1}V , which is greatly affected by LC s s LC ions and a CYTOP layer with a proper thickness. Here, in this the amount of doped ions s . Our preliminary evaluation LC study, we consider the field-screening effect as mentioned above. of the effective conductivity of CCN-37 is of the order of In order to explain the sharp increase of V at low frequencies, th 6  1  1 10 O m at 1 kHz, which is measured without using an we use the configuration shown in Fig. 3b to calculate the voltage alignment layer. Thus, the main reason can be assigned to the applied to the cell. The dielectric constant is e , the LC increased value of s . This supports our experimental data that LC electric conductivity is s , and the thickness of the NLC is d, LC show thicker cells lower the threshold voltage at the same and the corresponding terms for the alignment layer are e , s , s s frequency. Eventually the surface charge density, r , given by and l , respectively. The anisotropy of e and s is not s LC LC r ¼ e {e (s /s )  e }(V /l ) in the low-frequency limit, s 0 LC LC s s s s considered and s includes the ionic contribution. The applied LC becomes high. We can speculate that a high surface charge iot voltage V ¼ Re Ve is written as V ¼ V þ V , where s LC density is achieved on CYTOP after doping with ionic iot iot ~ ~ V ¼ Re V e and V ¼ Re V e are the voltage drop in s s LC LC materials and that it creates a spatial distribution near the the alignment layer and in the NLC layer. Using the complex interface. The expression of r also supports the temperature conductivity, s ~ ¼ s þ ioe and s ~ ¼ s þ ioe , a relation LC LC LC s s s dependence because the ionic localization around the ~ ~ s ~ V =d ¼ s ~ V=2l should hold due to the conservation of LC LC s s surface develops faster owing to the increased mobility, current density. The complex amplitude of the threshold voltage mps ¼ s exp(  W/k T) (ref. 46). LC 0 B ~ ~ applied to the cell is described as V ¼ 2ðÞ l =dðÞ s ~ =s ~ þ 1 V . Since the qualitative behaviour of V is explained by fg s LC s LC th Low and high-frequency limits are evaluated by considering the CYTOP surface, here, we present a detailed s ~ =s ~ ¼ðÞ s þ ioe =ðÞ s þ ioe . Here CYTOP has a high model that can reproduce the transition to the G state from LC s LC LC s s NATURE COMMUNICATIONS | 7:13238 | DOI: 10.1038/ncomms13238 | www.nature.com/naturecommunications 7 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13238 Shearing direction Shearing direction ab c Shearing direction Figure 7 | Effect of a shear flow on the grid-like texture. The snapshots for the effect of the shear flow for the grid-like texture. (a,c) The texture under shear flow. Dashed lines are the guide to the eye. (b) Recovering the grid-like texture. (d) Shematic illustration for the cross section of the director field. The horizontal allows show the direction of shear flow. The NLC used is 1:1 mixture of CCN-47 and CCN-55. Scale bar, 100 mm. the H state prior to achieving the U state. Based on the Fig. 4. The threshold voltage can be regarded as a function of q, observations, the effects of the doped ions (TBABE) and that is, V (q). The real threshold voltage, V , is given by the 0 th the insulating layer (CYTOP) are considered. The details are minimum value of V . A local minimum at q ¼ 0 corresponds to provided in the Method section. We set the origin of the z-axis the transition to the U state, namely, the normal Freedericksz (z ¼ 0) at the center of the nematic slab (Fig. 10a). The transition, and another minimum at qa0 corresponds to the NLC-CYTOP interfaces and the CYTOP-electrode interfaces transition to the G state. Because V depends on the frequency of th ± ± are positioned at z ¼ d/2 and z ¼ h/2, respectively. We focus the applied voltage, permittivities, conductivities, elastic constants on calculating the threshold voltage from the H to G states. In our and thicknesses of NLC and insulating films, the numerical system, the electric potential is given as f ¼0at z¼ h/2 and calculations based on the continuum theory of NLCs are iot f ¼ V cosot at z ¼ h/2, with f ¼ Re½fe . The slightly tilted performed by using the typical material constants for the NLC n is expressed as n ¼ (dn ,dn ,1) and the corresponding potential and CYTOP, and our experimental conditions for d and l. The x y ~ ~ ~ becomes f ¼ f þ df. The translational symmetry in the x-y conductivities s and s , which depend on the concentration of || > plane allows us to write the solution for the grid pattern as dopant ions, are chosen to reproduce the frequency dependence of V in Fig. 2b. The behaviour of V (q) at different typical th 0 dn ðx; y; zÞ¼ yðÞ z cosqxsinqy; ð1Þ frequencies is shown in Fig. 10b. At low frequencies of f ¼ 70 and 200 Hz, the G state is more stable than the U state since the dn ðx; y; zÞ¼ yðÞ z sinqxcosqy; ð2Þ minimum of V is at qa0. However, the two local minima of V 0 0 at q ¼ 0 and qa0 become close at 200 Hz. As the frequency is ~ ~ dfðÞ x; y; z ¼ cðÞ z sinqxsinqy; ð3Þ increased, they become equal (457.5 Hz) and eventually the U where the grid pattern described by equations (1)–(3) should be state becomes more stable (550 Hz). Calculating V (q) by varying rotated by 45 when it is compared with the photos shown in the frequencies allows us to plot V as a function of f (Fig. 10c). th 8 NATURE COMMUNICATIONS | 7:13238 | DOI: 10.1038/ncomms13238 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13238 ARTICLE a b 020 40 Sample thickness (μm) Figure 8 | Tunable grid spacing by sample thickness and applied voltage. (a) Distance of adjacent defects depending on the sample thickness d and electric field strength V . The vertical bars are obtained when V is changed (from V B10 V to 40 V). Larger array size in the bar corresponds to higher V 0 0 0 0 and the frequency is adjusted in each case. The plotted data are the mean value of the minimum and maximum grid size. Closed circles are the data for 1:1 mixture of CCN-47 and CCN-55, and open symbols are for CCN-37. Because of experimental limitations, the range of the thickness used is above 1 mm. We note that there is no significant difference in the grid size between these two NLC samples. (b) A texture accompanying an electro-hydrodynamic convection observed in a relatively thick cell with d ¼ 24 mm. The mixture of CCN-47 and CCN-55 is used. (c) Controllable numbers of arrays in a narrow area. The left, middle, and right micrographs show arrays of 6 6(V ¼ 17.5 V), 4 4(V ¼ 26.2 V), 2 2(V ¼ 39.3 V), respectively. Here CCN-37 is 0 0 0 used and the average cell thickness is 3.7 mm. Scale bars, 100 mm. The magenta lines are the boundaries of the H–G transition evidence, we need to use LC materials that induce homeotropic with qa0 and the black lines are those of the H–U transition with alignment on CYTOP. q ¼ 0. The blue solid circles indicate the intersection of the two In summary, we report unconventional pattern formation in lines. We cannot reproduce the G–S and S–U transitions because NLCs by combining doped ions and a perfluoro polymer our numerical results are limited to the linear stability analysis. (i) alignment layer. The creation of a large single domain of square in Fig. 10c corresponds to Fig. 2, where we use the same material arrays consisting of high-density defects is demonstrated in constant values as in Fig. 10b. Good agreement is obtained, several ways. The system has huge advantages compared with though the numerically obtained position of the boundary previous systems because its self-organization offers highly between the H–G and H–U transitions is located at a little tunable structures that do not require special surface bit higher frequency compared with the experiment. When modifications. The structure can be used directly for diffractive the conductivities s and s are decreased ((ii) in Fig. 10c), the microlens arrays, generation of multiple vortex beams using LC || > 36,61–65 boundary between the H–G and H–U transitions shifts to a lower mesophases as foreseen applications. Because our system frequency, implying that the grid pattern is destabilized by the is self-repairing, it could be applied for sensor applications using decrease in conductivities. In other words, the conductivity director distortion. Polymerizing the structure would provide a stabilizes the G state. However, when the thickness of the soft lithographic template for micro and nanostructures . insulating films is reduced from 0.12 mm(l ¼ 3.24 mm) to 0.05 mm The stabilization of the grids may offer very interesting (l ¼ 3.1 mm) ((iii) in Fig. 10c), similar behaviour is observed, possibility to create freestanding films which provide exposed implying that the insulating film also stabilizes the grid state. If interfaces and are useful, such as for pixelated LC sensor the conductivity is small or the film is thin, the grid state applications in liquid or gaseous environments. Moreover, the disappears ((iv) in Fig. 10c). These results clearly indicate that the system offers a playground for studying the fundamental physics enhanced conductivity of the NLC and the insulating CYTOP of various fields such as microrheology and colloidal science. Our film play a crucial role in forming the G state. experimental results may shed light on creating unconventional To confirm these results, we test typical materials with negative LC textures using ionic effects. Although the data presented here dielectric anisotropy, such as MBBA (Sigma-Aldrich) and an are obtained under standard laboratory conditions, we believe NLC mixture, Phase 5 (Merck). They do not show normal that the quality can be improved substantially under clean and anchoring on the CYTOP surface. For further experimental refined experimental conditions. NATURE COMMUNICATIONS | 7:13238 | DOI: 10.1038/ncomms13238 | www.nature.com/naturecommunications 9 Grid size (μm) ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13238 b t = 15 s t =0 t = 30 s t = 45 s t = 75 s Figure 9 | Dislocations formed by imperfect arrangements of umbilics. (a) Three domains of defect arrays with different orientations observed near a sharp corner of the intersected stripe electrode. Each image is taken by rotating polarizers. (b) The process of annihilation of a pair of defects and the dynamics of dislocations which repel each other. The frustration is generated by increasing the applied voltage from V ¼ 17.5 to 35 V. The cell thickness is 3.0 mm on average and the NLC sample is the 1:1 mixture of CCN-47 and CCN-55. Scale bars, 100 mm. ab c V=V cosωt +h /2 0 (i) (H)–(U) (II) CYTOP (H)–(G) l 70 (Hz) (insulating film) +d /2 0 0.2 0.4 7.2 (ii) d (I) NLC 20 200 (Hz) 0 0.2 0.4 –d /2 6.2 (iii) 457.5 (Hz) (II) CYTOP (iv) 550 (Hz) –h /2 2 3 0 0.2 0.4 0.6 0 0.2 0.4 10 10 V=0 –1 q (μm ) Frequency (Hz) Figure 10 | Theoretical approach for the grid-like state. (a) Schematic of our sample cell and the coordinates used for the calculation. z ¼ d/2 are the CYTOP–NLC interfaces and z ¼ h/2 are the CYTOP-electrode interfaces. The CYTOP thickness l is written as l ¼ (h  d)/2. The electric potential is zero s s at z¼ h/2 and V cos ot at z ¼ h/2. (b) Behaviour of V (q) obtained for four frequencies of 70 Hz (black curve), 200 Hz (blue), 457.5 Hz (magenta) and 0 0 6  1  1  6  1  1 550 Hz (orange). The material constants used are: e ¼ 4, e ¼ 11, s ¼ 2.6  10 O m , s ¼ 2.1  10 O m , K ¼ 4.5 pN, K ¼ 8.5 pN, e ¼ 2, || > || > 1 3 s 15  1  1 s ¼ 10 O m , d ¼ 3 mm and h ¼ 3.24 mm. (c) Dependence of V on frequency f under different conditions: (i) the material constants used are the s th 6  1  1  6  1  1 same with that in b, (ii) same with (i) except for s ¼ 1.5  10 O m , s ¼ 1.2  10 O m , (iii) same with (i) except for h ¼ 3.1 mm and (iv) || > same with (i) except for h ¼ d ¼ 3 mm. Both threshold voltages for the normal Fre´edericksz transition (black lines) and the transition of forming the grid pattern (magenta lines) are plotted. Methods show the nematic phase at room temperature. Experiments are carried out at 25 C Sample preparation and cells. We use the NLC compounds CCN-mn unless otherwise indicated. Moreover, 1 wt% of an ionic compound tetrabutyl 0 0 anmonium benzoate (TBABE, Aldrich) is mixed into NLCs. In preparing the (trans, trans-4, 4 -dialkyl-(1a,1 a-bicyclohexyl-4b-carbonitriles, Nematel GmbH & Co. KG), which possess a negative dielectric anisotropy . Particularly, CCN-37 ion-doped samples, the NLC and ions are diluted with chloroform and mixed by ultrasonic agitation for 1 h. Then, the chloroform is evaporated for 24 h. and the 1:1 mixture of CCN-47 and CCN-55 are used in this work because they 10 NATURE COMMUNICATIONS | 7:13238 | DOI: 10.1038/ncomms13238 | www.nature.com/naturecommunications V (q) (V) V (V) th NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13238 ARTICLE The sample is filled in cells consisting of ITO coated glass substrates with a On the other hand, director n follows the torque balance equation proper distance determined by standard interferometry. Monodisperse spherical @n dF particles are used as a spacer to maintain the cell thickness. The ITO-coated glass g n  ¼ n  ; ð8Þ @t dn substrate is spin-coated by a thin layer of the amorphous perfluoro polymer (CYTOP, Asahi Glass Co.), which induces the perpendicular orientation of n to the where free energy F consists of the Frank elastic energy, f and the electric-field glass surface . In the process of our standard spin-coating, we mix the solute contribution, f as el CTX-809A (a solution containing 9 wt% CYTOP, Asahi Glass Co.) and the solvent CT-Solv.180 (Asahi Glass Co.) with the weight ratio of 1:2. The spin-coating is K K K 1 2 2 2 3 2 f ¼ ðÞ r n þ ðÞ nrðÞ n þ ðÞ nrðÞ n ; ð9Þ made with 800 r.p.m. for 9 s and 3,000 r.p.m. for 15 s. After coating, the substrates 2 2 2 are dried at 70 C for 30 min and baked at 120 C for longer than 30 min. The ratio of the solute and solvent is changed to obtain various thicknesses of the alignment 1 1 f ¼ e E  eðÞ n  E ; ð10Þ layers. The thickness of the CYTOP layer is estimated by spectroscopic el ? a 2 2 ellipsometry (SE-2000, SEMILAB Zrt.), whose results are 120 nm for 3 wt% We neglect the contribution of the NLC flow because there is no observable EHC solution and 17 nm for 1 wt% solution. As alternative alignment layers for homeotropic anchoring, a polyimide surface (SE-1211, Nissan chemical) and a effect, except for in Fig. 8b. e and s are time-independent constants in equations surfactant mono layer (DMOAP, N,N-Dimethyl-N-octadecyl-3- (4) and (5) because the relaxation time of the director fluctuations near the aminopropyltrimethoxysilyl chloride, Aldrich) are tested. In the standard use of threshold voltage is much longer than the period of the AC electric field. Under SE-1211, the stock solution is diluted with the dedicated solvent of equal amount. these conditions, we obtain from equations (4) and (5) Spin-coating condition is the same as the CYTOP. In addition, the stock solution of SE-1211 itself is also coated with 1,000 r.p.m. to have a thicker alignment layer r s ~rf ¼ 0; ð11Þ although a drastic change is not observed. In this condition, the thickness of SE-1211 amounts to 400 nm. For the latter case, glass substrates are taken into a with water solution and 5 vol% DMOAP solution is added to it. After mixing for about s ~ ¼ s þ ioe: ð12Þ 5 min, the excess surfactant is washed with pure water and the substrates are kept at 120 C for 30 min for drying. The effective electrical conductivity is estimated by a Application of Gauss’s theorem to equation (11) at the NLC-CYTOP interface LCR metre (E4980A, Agilent) using an ITO-coated glass cell without having gives the boundary conditions of alignment layers. It must be noted that the anisotropy is not accessible. The 6  1  1 measured value is of the order of 10 O m at 1 kHz, which is used for the ðIÞ ðIIÞ ~ ~ s ~rf ¼ s ~rf at z ¼ d=2; ð13Þ theoretical calculation. z z To prepare patterned electrodes, ITO-coated glasses are fabricated by a in addition to the continuity of potential of standard photo etching method using a positive photoresist (TFR-2950 PM, Tokyo Ohka Kogyo Co., Ltd.). Finally, an AC voltage V ¼ V cos (2pft) is applied between ðÞ I ðÞ II ~ ~ f ¼ f at z ¼ d=2; ð14Þ the ITO-coated glass substrates along the z-direction in order to reorient the director. For shear application, the upper and lower glass substrates are installed on where superscripts (I) and (II) denote the NLC and CYTOP, respectively, and are the motorized stage separately without using spacers. used hereafter. For the homeotropic state, we can easily obtain potential f from the above equations. The corresponding electric field has only the z component E of Polarization light microscope characterization. AC voltage from the function generator is amplified and applied to ITO-coated glasses so that the electric field is s ~ perpendicular to the substrates. The maximum amplitude of the output voltage V ~ 0 E ¼ V in regionðÞ I ; ð15Þ 0 0 s ~ðÞ h  dþ s ~ d is 40 V. Texture observation is usually made by a polarizing microscope (Olympus k s BX51) under different illumination conditions. If necessary, the temperature controller is used. The micrographic appearance is taken by a DSL camera (Sony s ~ E ¼ V in regionðÞ II ; ð16Þ 0 0 ILCE-7R). For our optical manipulation experiments, an Nd-YAG laser (1,064 nm) s ~ðÞ h  dþ s ~ d k s is irradiated to the sample cell on a motorized stage of an inverted microscope (Olympus IX71). The manipulation is made by moving the motorized stage. where s ~ ¼ s þ ioe and s ~ ¼ s þ ioe . Substituting n ¼ (dn ,dn ,1) and k k k s s s x y ~ ~ ~ f ¼ f þ df into equations (8)–(14), and linearizing the results with respect to dn ,dn and df, we obtain x y Fluorescence confocal polarizing microscopy. A confocal laser scanning microscope Leica TCS sp8 is used for elucidating the director configuration of the G 2 2 2 2 2 @dn @ dn @ dn @ dn @ dn @ dn x x y x y x state. The NLC samples are doped with B0.01 wt% of a fluorescent dye, g ¼ K þ þ K  þ K 1 1 2 3 2 2 2 @t @x @x@y @y @x@y @z 7-diethylamino-3,4-benzophenoxazine-2-one (Nile red, Sigma-Aldrich). The Nile "# ð17Þ red molecules orient along the director field due to its shape anisotropy. Thin quartz e 2 @df ~ ~ Re E dn þ E in ðIÞ; 0 x glass plates of thickness 160 mm are used for sample cells. The laser with the 0 2 @x wavelength of 552 nm excites the dye molecule and the emission is detected in the spectral region of 610–660 nm. The polarization of the incoming laser beam is 2 2 2 2 2 adjusted manually with a combination of a quarter wave plate and a linear polarizer @dn @ dn @ dn @ dn @ dn @ dn y x y x y y g ¼ K þ  K  þ K 1 2 3 inserted in the laser path. Simultaneously, polarizing microscopy images by the 2 2 @t @x@y @y @x@y @x@y @z "# ultraviolet transmitted mode are captured during confocal scans of the same sample. ð18Þ e 2 @df ~ ~ Re E dn þ E in ðIIÞ; 0 y 2 @y Theoretical details. The charge density, r, satisfies the Poisson equation r (eE) ¼ r and the charge conservation law r (sE)¼ qr/qt, where E is the electric field, e is the permittivity and s is the conductivity. Here, the ion diffusion dn ¼ dn ¼ 0at z ¼ d=2; ð19Þ x y is neglected for simplicity. By using the electric potential f, these equations are rewritten as ! 2 ~ 2 ~ 2 ~ @ df @ df @ df @dn @dn x y s ~ þ þ s ~  s ~ E þ ¼ 0inðÞ I ; ð20Þ ? a 0 rðÞ erf ¼ r; ð4Þ 2 2 2 @x @y @z @x @y @r Ddf ¼ 0in ðIIÞ; ð21Þ rðsrfÞ¼ : ð5Þ @t ðÞ I ðÞ II In NLCs, e and s are functions of n ~ ~ @df @df ðÞ I ðÞ II ~ ~ s ~ ¼ s ~ ; df ¼ df at z ¼ d=2; ð22Þ k s @z @z e ¼ e d þ e n n ; ð6Þ ab ? ab a a b df ¼ 0at z ¼ h=2; ð23Þ s ¼ s d þ s n n ; ð7Þ ab ? ab a a b where s ~ ¼ s þ ioe and s ~ ¼ s þ ioe . The last terms of equations (17) and ? ? ? a a a with e ¼ e  e and s ¼ s  s , where || and > denote the components of (18) are replaced by the time average over the period of the applied voltage because a || > a || > permittivity (or conductivity) parallel and perpendicular to n, respectively. e and the director fluctuations become slow near the threshold. s are the permittivity and conductivity of the insulating film, which also imposes a The translational symmetry in the x-y plane allows us to write the solution for condition of s oos , s . the grid pattern as equations (1)–(3). Substitution of equations (1)–(3) into s || > NATURE COMMUNICATIONS | 7:13238 | DOI: 10.1038/ncomms13238 | www.nature.com/naturecommunications 11 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13238 equations (17)–(23) yields 16. Wang, X., Miller, D. S., Bukusoglu, E., de Pablo, J. J. & Abbott, N. L. 2 hi Topological defects in liquid crystals as templates for molecular self-assembly. @y e 2 @ y e a a ~ ~ ~ g ¼ 2K q þ E þ K  qRe E c in ðIÞ; ð24Þ 1 0 3 1 0 2 Nat. Mater. 15, 106–112 (2016). @t 2 @z 2 17. Buka, A. & Kramer, L. Pattern Formation in Liquid Crystals (Springer New York, 1996). y ¼ 0at z ¼ d=2; ð25Þ 18. Meyer, R. B. Piezoelectric effects in liquid crystals. Phys. Rev. Lett. 22, 918–921 2 ~ (1969). @ c 2q s ~ c þ s ~  2s ~ E y ¼ 0inðÞ I ð26Þ ? k a 0 2 19. Kuzma, M. R. Nonequilibrium periodic structures induced by rotating and @z static fields in a lyotropic nematic liquid crystal. Phys. Rev. Lett. 57, 349–352 2 ~ (1986). @ c ¼ 0in ðIIÞð27Þ 20. Barnik, M. I., Blinov, L. M., Trufanov, A. N. & Umanski, B. A. Flexo-electric @z domains in liquid crystals. J. Phys. 39, 417–422 (1978). Equations (17) and (18) are reduced to the same equation (24), and the 21. Chigrinov, V. G., Korkishko, T. V., Barnik, M. I. & Trufanov, A. N. ~ ~ ~ boundary conditions for c are obtained by replacing df with c in equations Freedericksz transition in electric fields near the dielectric sign reversal (22) and (23). At threshold voltage V , the relaxation time diverges so that the time th frequency. Mol. Cryst. Liq. Cryst. 129, 285–300 (1985). derivative term on the left-hand side of equation (24) vanishes. Then, equation (24) 22. Lonberg, F., Fraden, S., Hurd, A. J. & Meyer, R. E. Field-induced transient @y with ¼ 0, together with the other equations (25)–(27) gives V . The above th @t periodic structures in nematic liquid crystals: the twist-Fre´edericksz transition. equations (24)–(27) cannot be solved analytically for general cases had, except for Phys. Rev. Lett. 52, 1903–1906 (1984). h ¼ d. Then, we briefly explain how to calculate V numerically. Expressing th 23. Pieranski, P., Dubois-Violette, E. & Guyon, E. Heat convection in liquid crystals 0 00 0 00 c ¼ c þ ic , we have three equations for three quantities y; c and c from heated from above. Phys. Rev. Lett. 30, 736–739 (1973). 0 0 00 00 lz lz lz equations (24)–(27). Substitution of y ¼ y e , c ¼ c e ,and c ¼ c e 0 0 0 24. Lavrentovich, O. D. & Pergamenshchik, V. M. Stripe domain phase of a 0 00 gives a system of homogeneous linear equations for u ¼ y ; c ; c , which can 0 0 thin nematic film and the K13 divergence term. Phys. Rev. Lett. 73, 979–982 @ y only be solved if the determinant vanishes. Because there are two derivatives, 2 (1994). @z 2 ~ @ c 2 25. Lavrentovich, O. D. & Pergamenshchik, V. M. Patterns in thin liquid and , in equations (24)–(27), we have a cubic equation with respect to l . @z crystal films and the divergence (‘surfacelike’) elasticity. Int. J. Mod. Phys. 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Rev. Lett. 111, 093902 (2013). in the credit line; if the material is not included under the Creative Commons license, 62. Yang, B. & Brasselet, E. Arbitrary vortex arrays realized from optical winding of users will need to obtain permission from the license holder to reproduce the material. frustrated chiral liquid crystals. J. Opt. 15, 044021 (2013). 63. Son, B. et al. Optical vortex arrays from smectic liquid crystals. Opt. Express 22, To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/ 4699–4704 (2014). 64. Loussert, C., Kushnir, K. & Brasselet, E. Q-plates micro-arrays for parallel processing r The Author(s) 2016 of the photon orbital angular momentum. Appl. Phys. Lett. 105, 121108 (2014). NATURE COMMUNICATIONS | 7:13238 | DOI: 10.1038/ncomms13238 | www.nature.com/naturecommunications 13

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