1. Introduction

Liquid crystals (LCs) possess both liquid and solid properties and exhibit a variety of supramolecular structures. The nematic liquid crystal (NLC) phase, which has the greatest symmetry in known LC phases, exhibits a long-range directional order. Hence, the NLC molecules tend to align in one direction, which is the average alignment direction called director n.

Topological defects cannot be eliminated through continuous changes. They are characterized by topological charges k, and the law of conservation of topological charges controls the merging and decay of defects [1]. Topological defects are commonly produced in NLCs because director n is easily deformed for various reasons owing to the soft properties of NLCs. However, controlling the creation of the desired topological defects is difficult. Although methods, such as the dispersion of topological colloids [2–4], change of boundary conditions [5,6], and laser irradiation [7–10], have been used to create topological defects in NLCs, the process remains difficult. Consequently, this difficulty prevents NLCs from emerging as optimal materials for topological defect research.

In this study, we used rubbed bare glass substrates without alignment layers, which allowed DR1 azo dye to be adsorbed [11] onto the glass surface for photoalignment. This provided the initial alignment condition in which NLC molecules were aligned parallel to the glass substrate in the rubbing direction. Because the photoisomerization [12,13] of the DR1 azo dye caused the molecules to be aligned perpendicular to the polarization of light and generated heat [14,15], a laser tweezer was introduced to control the photoisomerization of the laser spot. When the laser spot, which had been in an isotropic state owing to the laser irradiation, became nematic after turning off the laser, the DR1 dye molecules that were adsorbed on the glass surface and aligned in a direction perpendicular to the incident polarization of the laser light caused the LC molecules to be oriented in the same direction along the DR1 molecules. Because the polarization of the irradiated laser light was parallel to the initial director ${{\mathbf n}_0}$, there was a boundary at which symmetry was broken between the outside and inside of the photoaligned domain.

Similar to the early universe, spontaneous symmetry breaking (SSB) occurs during the isotropic-nematic (I-N) phase transition of LCs [16–18] and forms a dense tangle of topological disclinations (topological line defects). However, in the absence of the symmetry-breaking boundary (SBB), disclinations cannot remain because of pair annihilation [19]. In our study, the SBB, which prevented the complete disappearance of the disclinations, was generated by photoisomerization, and the SSB I-N phase transition occurred simultaneously because of the heat generated; this enabled us to create disclinations at the SBB. In the SBB, disclinations [20] with topological charges (winding numbers) of k = +1/2 and k = -1/2 could remain, and the topological charge, which is the topological invariant, is always conserved [21].

Therefore, we were able to rapidly and simply create disclinations at the SBB by the photoisomerization of the azo dye and the occurrence of SSB in the I-N LC phase transition. The method of generating the topological charges presented in this study is significant because it can aid in basic research and future applications.

2. Experimental

In this experiment, 5CB (4′-Pentyl-4-biphenylcarbonitrile, T_c = 35°C) NLCs were doped with 0.15 wt % of Disperse Red 1 (DR1; N-Ethyl-N- (2-hydroxyethyl)-4-(4-nitrophinylazo) aniline, ${\mathrm{\lambda }_{\textrm{max}}}$ = 502 nm) dye. DR1 is a pseudo-stilbene-type azobenzene chromophore. When irradiated, a trans isomer is photoexcited to a cis isomer, which then becomes a trans isomer by absorbing the same wavelength of light or emitting heat. After consecutive trans-cis and cis-trans photoisomerization, the DR1 molecules align perpendicular to the polarization of the irradiating light. In addition, because of the generated heat, the laser spot of the NLCs can move to an isotropic state, and the isotropic domain of the LC undergoes an I-N phase transition after the laser is turned off.

To make NLC cells, the cover (Matsunami) and soda-lime slide (Marienfeld) glass substrates were rubbed with a cloth, and the two glasses were sandwiched, maintaining a gap with the mixture of UV curing epoxy and spacers (1 or 2.5 μm). The samples were then cured under UV light. The cell was filled with the 5CB and DR1 mixture using capillary force in the isotropic phase at 36°C. Except for the experiment corresponding to Fig. 3, cells with a thickness of 2.5 μm were used in all experiments. For obtaining the experimental results presented in Fig. 3, a cell with a thickness of 1 μm was used. The 5CB and DR1 were purchased from Sigma-Aldrich.

A topological charge was generated in the NLC cell using a laser tweezer. The laser tweezer consisted of a linearly polarized and continuous TEM₀₀-mode wave from a semiconductor laser (Coherent, SF 532, $\mathrm{\lambda }\; $ = 532 nm), an inverted microscope (Nikon Ti-U) equipped with a piezo stage (PI GmbH & Co. KG), and a CCD (Point Grey Research). The laser tweezer played a dual role of facilitating optical microscopy as well as providing the thermal and photoalignment effects caused by the photoisomerization of DR1 molecules. We used Olympus PLN ×20 (NA 0.4) and ×40 (NA 0.65) objective lenses for the laser tweezer and optical microscopy. The ×20 and ×40 objective lenses made laser spots with radii of 3.5 and 6 μm in dye-doped NLC cells, respectively. The ×20 objective lens was used to obtain the results depicted in Figs. 2 and 4, and a ×40 objective lens was used for acquiring the results in Figs. 1 and 3. The intensities were calculated by dividing the incident laser power by the area of the laser spot. The laser light intensity was 0.3 mW/μm² for the experiments corresponding to Figs. 2 and 3, and 0.5 mW/μm² for that corresponding to Fig. 4.

For optical microscopy, polarized optical microscopy (POM) and bright-field microscopy with a polarizer (BFMP) were used to determine the positions and types of topological defects. The POM image with a broadband half-wave plate $(\textrm{HWP};\; \mathrm{\lambda }$ = 400–800 nm), which is inserted after the LC cell between the crossed analyzer (A) and polarizer (P), shows the texture with colors in relation to the alignment direction of the LCs. The colors of the POM with an HWP (POMH) image provide information about the two-dimensional director field n(r). Information about the direction of molecular alignment obtained from POMH images varies depending on the cell thickness because the birefringence of the LC layer changes with its thickness. If the thickness of the cell is 2.5 μm, the green color indicates that the director is along the 2 o'clock direction, which is the direction of the fast axis h of the HWP, whereas the pink color indicates that the director is along the 10 o'clock direction, which is perpendicular to the fast axis h of the HWP. Conversely, if the cell is of 1.0 $\mathrm{\mu}\mathrm{m}$ in thickness, the green and pink colors indicate that the alignment of the director is in the 10 and 2 o’clock directions, respectively.

3. Results and discussion

We fabricated LC cells by rubbing the soda-lime-silica glass substrates, which allowed the LCs to align in a direction parallel to the glass substrate in a low anchoring energy state. Therefore, inside the LC cell, long rod-shaped DR1 molecules were aligned along the LCs in the same direction, and on the glass surface, the -OH end groups of the DR1 molecules were bound to the silanol group on the glass surface for adsorption [11]. Accordingly, the topological defects formed by the planar anchoring condition were line defects [20,22], that is, disclinations formed between two substrates with topological charges k = +1/2 or k = -1/2.

Figure 1(a) shows the I-N phase transition due to the heat generated in photoisomerization, which results in the photoalignment of the adsorbed DR1 molecules on the glass substrates. Laser light, polarized in the direction parallel to ${{\mathbf n}_0},$ incident on the LC cell, forms an isotropic domain P_iso (radius r_iso= 10 μm) with an area larger than that of the laser spot (radius r = 3.5 μm). The area of P_iso is proportional to the intensity of the laser (Fig. 1(c)). On the surface, the adsorbed DR1 molecules align perpendicular to the polarization direction of the laser light. During the I-N phase transition after the laser light is turned off, the LC molecules on the surface of the substrate are anchored by the DR1 molecules that are aligned and adsorbed on the substrates and have easy axis in the same direction along the DR1 molecules. Therefore, the director ${\mathbf n}$ in the laser spot becomes perpendicular to the initial director ${{\mathbf n}_0}$. However, LC molecules outside the laser spot are oriented again in the direction of ${{\mathbf n}_0}$ because they are not aligned by photoisomerization. Thus, the internal director (${\mathbf n}$) and the external director (${{\mathbf n}_0}$) of the laser spot become perpendicular to each other, resulting in the SBB, where the direction of director is changed. In this SBB, the multi-domains created during the SSB I-N phase transition can remain, which enables the formation of topological charges. Therefore, the laser light is turned off at time t = 0 and the tangled disclinations disappear by pair annihilations (from t = 0 to t = 44 ms). However, in the boundary where the symmetry is broken by the photoalignment, the formation of topological defects, as shown in Fig. 1(a) (the POM image at t = 69 ms and the next BFMP image), occurs in this SBB because of the symmetry-breaking I-N phase transition. Figure 1(b) shows the director field n(r) of the topological charge generated in the SBB and is drawn by overlapping the positions of the isotropic domain, laser spot, SBB, so that it aids in understanding the creation geometry of topological charges due to the I-N SSB phase transition. In Fig. 1(c), which depicts the POMH images showing ${\textrm{P}_{\textrm{iso}}}$ according to the laser intensity, the area of ${\textrm{P}_{\textrm{iso}}}$ is observed to increase in proportion to the laser intensity (Fig. S1 of Supplement 1).

 

Fig. 1. (a) POM images showing how the SSB I-N phase transition occurs over time after the incident polarized laser is turned off (t = 0 ms) and the BFMP image showing the 1/2 topological charge pair formed when the isotropic domain completely transformed to the nematic phase. Pair of 1/2 topological charges, which can be seen in the POM and BFMP images (t = 69 ms), indicate the completion of the phase transition. Values k = +1/2 and k = -1/2 refer to the topological charge values of each defect. ${{\mathbf n}_0}$ is the initial direction of the director formed by the rubbed substrates. Laser spot (whose radius is approximately 3.5 μm) is indicated by the green dotted line in the POM image (t = 0 ms). In addition, the isotropic domain (P_iso) formed by the generated heat is indicated by a purple arrow. Laser was turned off after 30 s of exposure (${\textrm{t}_{0}}$ = 30 s) and the radius of the isotropic domain (r_iso) was 10 μm. (b) Schematic depicting the locations of the laser spot, isotropic domain, and SBB. Black line indicates director field n(r). (c) POMH images showing the isotropic domain formed when the power of the incident laser is changed. Laser power ${\textrm{P}_{\textrm{in}}}$ and ${\textrm{r}_{\textrm{iso}}}$ are given, which shows that the area of the isotropic domain is proportional to the power of the laser. (d) POM (left), POMH (middle), and BFMP (right) images show the photoaligned areas and created topological charge pairs when the laser was turned off after ${\textrm{t}_{0}}$ = 150 s in (c). Power of the laser is given in the POM image, and the thickness of the NLC cell used was 2.5 μm. Visualization 1 and Supplement 1 are available.

Download Full Size | PDF

Figure 1(d) consists of microscopy images that show the phase transition that results from suddenly turning off the laser light after photoalignment, as shown in Fig. 1(c). The angle $\phi $ between the director ${{\mathbf n}_0}$ and the NLC molecules in the laser spot changes from 90° to 0° as it moves from the center to the periphery because of the Gaussian profile of the laser intensity. Further, $\phi $ becomes 90° more quickly regardless of the location in the laser spot for higher intensities. Thus, $\phi $ becomes 90° at the center of the laser spot, and eventually the molecules inside the spot are aligned perpendicular to ${{\mathbf n}_0}$. In addition, Fig. 1(d) shows that topological charges are generated only when the intensity of the incident laser light is high, possibly because lower intensities produce poor photoalignment but also because of the size of P_iso. Topological defects cannot be created if the I-N phase transition does not occur in the SBB (2-20 mW of Fig. 1(d)). In the BFMP image in Fig. 1(d), one can clearly observe the topological charge pairs that are formed, and the corresponding topological charge values are indicated. Because the topological charge k must be preserved when the topological defects are formed, the k = +1/2 and k = -1/2 disclinations are formed in the same number. Hence, dipolar, quadrupolar, and multipolar configurations (Figs. 2(a) and (c)–(e)) are possible.

In Fig. 2(a), we present a series of microscopy images that show the generated topological charges. Figures 2(b)–(e) show the schematic director fields (i) and microscopy images (ii)–(iv) of the topological charges created in the SBB. There may not be topological charges, as shown in Fig. 2(b), but when a topological charge is formed, it is formed only in the SBB and obeys topological charge conservation.

 

Fig. 2. (a) POMH (left), POM (middle), and BFMP (right) images show topological charges generated by the SSB I-N phase transition after an average irradiation of 90 s with a constant laser intensity (I = 0.3 mW/μm²) followed by suddenly turning off the laser. Radius of the laser spot was 6 μm; the polarization direction of the incident laser light was indicated by Lp, and the radius of isotropic domain P_iso was r_iso= 10 μm. (b)–(e) Schematic director fields n(r) (i), POMH (ii), POM (iii), and BFMP (iv) images showing topological defects and photoaligned regions formed after the I-N phase transition. (b) Domain where the molecular alignment direction changes from ${{\mathbf n}_0}$ to ${\mathbf n}$ by photoalignment. (c) and (d) k = +1/2 and k = -1/2 topological charge pair generated in the SBB; (e) 1/2 topological charges in the quadrupolar configuration (Lehmann cluster) [23]. Thickness of the LC cell was 2.5 μm.

Download Full Size | PDF

In Fig. 3, we present microscopy images (a)–(e) to show that the topological charge pairs were created using the laser tweezer in the π-domain wall formed by the existing k = +1/2 and -1/2 charges. The corresponding schematic director field n(r) (Fig. 3(f)) is also shown. The polarization of the incident laser light (Lp) is parallel to the LC molecules in the center of the $\pi $-domain wall (Figs. 3(a) and (f)). Therefore, Fig. 3(a) shows that when the laser light is incident, an isotropic domain is formed due to the thermal effect caused by the photoisomerization of the DR1 molecules. Subsequently, as shown in Fig. 3(b), a 1/2 topological charge pair is formed by the SSB phase transition and photoisomerization by the DR1 molecules adsorbed on the substrates. After repeating this process several times, a series of topological charge pairs can be generated, as shown in Figs. 3(c)–(e). The generated topological charge pairs can be clearly observed inside the white box in Fig. 3(e). Figure 3(f) shows the director field n(r) of the newly created k = +1/2 and -1/2 topological charge pairs. The alignment direction of the $\pi $-domain walls between the newly formed topological charge pairs is perpendicular to that of the existing $\pi $-domain wall.

 

Fig. 3. Result of new topological dipoles formed in the existing domain wall between the k = +1/2 and k = -1/2 topological charges. (a) POMH image showing the isotropic domain generated by the irradiated laser light (Lp) on the π-domain wall. Direction of Lp is the polarization direction of the incident laser light and is parallel to the LC molecular aligned direction of the π-domain wall. (b) POMH image showing the process of forming topological dipoles. (c)–(e) POMH (c), POM (d), and BFMP (e) images showing the result of the formation of topological dipoles made by changing the position of the laser beam. (e) Configuration of the created topological charges. Existing +1/2 and -1/2 topological charges are indicated by the + and - inside the circles, and the newly generated topological charges by the SSB phase transition are indicated by the + and - inside the white box. (f) Schematic director field n(r) corresponding to (c)–(e). Because the thickness of the used LC cell is 1 μm, the direction information in accordance with the color of the POMH texture is different from that of the 2.5 μm cell, and therefore, pink and green colored regions in the POMH image represent the direction of molecular alignments in the 2 o’clock direction (parallel to h of HWP) and the 10 o’clock direction (perpendicular to h of HWP), respectively. Visualization 2 is available.

Download Full Size | PDF

The distortion in an NLC is described as an elastic field corresponding to an electric field of the electrostatic force [24,25]. Therefore, the defect that creates distortion becomes a topological charge, and an elastic field is formed around it. An elastic field in the direction of the +1/2 charge to the -1/2 charge is formed between the existing k = +1/2 and k = -1/2 topological charges, in which the newly generated 1/2 topological dipoles are aligned within the elastic field. In other words, the generated topological dipoles are aligned in one direction because of the strong external elastic field (distortion by the existing π-wall), as if the electric dipoles are aligned within the strong electric field.

 

Fig. 4. (a) Schematic director field n(r) of the photoaligned domain generated by slowly moving the laser light (Lp), which was linearly polarized, parallel to the initial director ${{\mathbf n}_0}$. To create topological defects, the linearly polarized laser light Lp ($\theta$ = 50°) parallel to the molecular alignment direction of the SBB was irradiated on the SBB. (b) Schematic director field n(r) of topological charges generated by irradiating Lp. Experimental results are shown in (c)–(f). (c) and (d) POM images showing the experimental results corresponding to (a). Isotropic domain (P_iso) was formed by irradiating laser light inside (c) and outside (d), the photoaligned area. Director field of the photoaligned area can be confirmed from the locations of a pair of boojums on the surface of the isotropic domain. Topological dipoles were created at the domain boundary, which was partially photoaligned by the laser light. (e) POMH image depicting the experimental result corresponding to (b). (f) BFMP image showing the charge configuration of the created topological dipoles. Intensity of the laser light was 0.5 mW/μm², and it took approximately 30 minutes to generate the photoaligned domain (40 × 40 μm²$)$. Thickness of the used NLC cell was approximately 2.5 μm. Visualization 3 is available.

Download Full Size | PDF

Figure 4 shows the result of topological charges that are formed at the SBB by photoalignment with the polarization parallel to the boundary direction. The linearly polarized laser light (Lp) in parallel with ${{\mathbf n}_0}$ is slowly moved to form a domain that is vertically aligned with ${{\mathbf n}_0}$. Figure 4(a) shows the director field n(r) created in this manner. To create a topological charge, the polarization direction of the laser light incident on the symmetry-breaking domain boundary was Lp ($\theta$ = 50°), which was parallel to the molecular alignment direction of the domain boundary. As shown in Fig. 4(a), the photoalignment (within 100 s) in the isotopic state was performed by the linearly polarized laser light (Lp) located in the SBB. If the laser is turned off after the photoalignment, the phase transition occurs followed by the creation of topological charges. This is shown in the schematic director field n(r) in Fig. 4(b), and the experimental results are presented in Figs. 4(c)–(f). Figures 4(c) and (d) are POM images depicting the experimental results that correspond to Fig. 4(b). In Figs. 4(c) and (d), we observe that the isotropic domain with planar anchoring is induced by strong laser light, and at the boundary of the isotropic domain, a pair of antipodal surface topological defects called boojums [26] are formed in the direction of the director in a nematic. The position of a pair of boojums varies when the laser light is located inside and outside the photoaligned domain. Inside the photoaligned domain (Fig. 4(c)), a pair of boojums is induced on the top and bottom of the isotropic domain, whereas outside the photoaligned domain (Fig. 4(d)), it is induced on the left and right of the isotropic domain. Hence, the initial orientation ${{\mathbf n}_0}$ and the orientation ${\mathbf n}$ inside the domain are perpendicular to each other, and it is confirmed that the intended photoaligned domain is formed. In addition, topological dipoles are formed at the edge of the domain. Figure 4(e) is a POMH image showing the experimental results that correspond to Fig. 4(b), and the orientation direction information of LC molecules related to the color of the domain is reflected in Fig. 4(b). Figure 4(f) is a BFMP image showing the charge values of the topological defects formed by photoalignment and I-N phase transition. In other words, the topological charge was formed because the orientation direction was partially changed in the symmetry-breaking region by the photoalignment.

In summary, we were able to easily create 1/2 topological charges in NLCs using a laser tweezer. DR1 azo dye-doped 5CB nematic LCs were used, and an LC cell was fabricated using soda-lime-silica glass substrate to utilize the photoalignment effect of DR1 adsorption on the substrate. Using unidirectionally rubbed glass substrates, LCs were oriented in the rubbing direction parallel to the substrate, and DR1 molecules aligned parallel to the LCs were adsorbed on the glass substrates. When the laser light that was polarized in the direction parallel to ${{\mathbf n}_0}$ was incident on the LC cell, DR1 molecules absorbed the laser light, and then photoisomerization occurred. An isotropic domain was formed in the laser spot owing to heat generation, and the DR1 molecules adsorbed on the substrates were simultaneously aligned perpendicular to the polarization of the laser light. Thereafter, when the phase transition occurred after the laser was turned off, the DR1 molecules that were adsorbed on the surface and aligned perpendicular to the polarization direction anchored the LC molecules in the same direction. Consequently, they possessed easy axes perpendicular to the laser polarization when the nematic phase was formed. Therefore, when the I-N phase transition occurred after irradiation for a certain period, the director of the LC changed in the easy axis direction, and this photoalignment was more effective at higher intensities of the irradiating laser. Meanwhile, the boundary of the photoaligned domain, the symmetry-breaking region, can be formed of multiple domains by symmetry breaking that occurs during the phase transition, allowing 1/2 disclinations to remain. Furthermore, topological charges can be created by the partial photoalignment of the symmetry-breaking region. The topological charge must be preserved such that equal numbers of k = +1/2 and k = -1/2 topological charges are created.

4. Conclusions

For decades, topological defects in LCs have been considered disadvantageous. However, their formation goes beyond academic research and has become a very important and interesting research topic in diverse areas. Despite many efforts to create desirable topological defects, it is difficult to control the director field to achieve the same. However, we were able to generate 1/2 topological charges owing to the SSB phase transitions and photoisomerization in NLCs doped with azo dyes. The elastic field created by the topological charge in the NLCs corresponds to the electric field created by the electric charge.

In addition, because the mechanism of charge creation and formation of an elastic field are associated with a wide range of disciplines, from the SSB phase transitions in the early universe to quantum chromodynamics (QCD) strings, research on topological charges is of great importance. Therefore, our proposed methods can further expand the understanding of topological charges and can possibly be extended to nematic colloids [27], where various types of feedback can be observed by controlling the defects induced by light and colloidal defects.