1. Introduction

Liquid crystals (LC) are an ideal platform for the experimental study of topology [1,2], which has been intensively studied in theoretical physics [3,4], biophysics [5], and material science [6]. A typical example of an LC system with abundant topological deformations is chiral nematic LC (CLC) microspheres, in which various types of topological defect loops are observed owing to an uneven distribution of LC directors [1,7,8]. The director field in the microsphere, which is governed by the surface anchoring and the chiral twisting deformation, is highly distorted and inevitably forms characteristic defects depending on the type of surface anchoring and the helical pitch. In particular, under the homeotropic surface anchoring, the discontinuous helical director field experiences a frustration that results in abundant metastable topological defects [7]. Such complex disclination loops within CLC droplets have been intensively studied via theoretical, experimental, and simulation methods [7–9].

In fluidic LC, it is sometimes difficult to identify the precise topological inner structure, because it is not possible to directly visualize it using either scanning electron microscopy or atomic force microscopy. Polymerization and quick freezing can be used to solidify the liquid phase to prepare a sample for such analyses [10], but the solidification treatment is likely to modify the intrinsic structures. Instead, polarizing optical microscopy (POM) is widely used to identify the topological structures [8]. POM provides an excellent tool to visualize the molecular orientation inside microspheres. The director profile can be estimated by observing the transmittance profile with varying polarizer directions, because the transmittance of light depends on the angle between the director and the polarizers. In particular, the defect lines are usually brighter than surrounding areas due to light scattering near the defect line, where local director ordering is disturbed and approaches isotropic [11,12]. However, POM has very poor vertical (z-axis) resolution and all the topological profiles in horizontal sections at different vertical locations are cumulatively represented in POM images. Understanding the 3D structure of topological deformation is essential in the study of topology in microspheres. However, unlike a LC cell sandwiched by two parallel substrates [13], in which the vertical director profile is usually simple, the topological defect structures in microspheres are complicated, and it is difficult to extract their 3D structure from the images projected on a 2D plane. Thus, the reconstruction of 3D profiles in micro LC droplets has so far been a challenging task [11].

In this study, we combine the fluorescence confocal polarizing microscopic (FCPM) method with POM analysis with the aim of reconstructing 3D topological structures. The confocal stack images provide us the vertical location of disclination defects and allow reconstruction of the full 3D structures. The reconstructed 3D structures can be viewed from different directions, providing a better understanding of the topological structure. Thus, FCPM can provide an excellent tool to study the topological configuration in microspheres.

2. Experiments and results

A CLC mixtures with varying chiral pitch were prepared by doping a right-handed chiral dopant, R-811 (Merck Company, Korea), into a nematic LC mixture, MLC-7026 (Merck Company, Korea) [14]. A fluorescence dye, coumarin-6 (Sigma-Aldrich, Korea), was added by 0.01 wt% into the CLC mixtures. The CLC mixtures were dropped into distilled (DI) water containing 4 wt% of anionic surfactant, sodium dodecyl sulfate (SDS; Sigma-Aldrich). The fraction of CLC was 1 w% in the water-CLC mixture. The CLC mixture did not dissolve in water, and instead, it broke into tiny micro-droplets after shaking the bottle for 3 minutes. The CLC micro-droplets were stably dispersed within the SDS solution, which was subsequently injected into a cell with a cell-gap of approximately 150 μm. The droplet diameters ranged from tens to hundreds of micrometers.

The POM analysis was performed using an optical laboratory microscope (BX41, Olympus, Japan) to characterize the droplets by their textures. A laser scanning type FCPM (K1-Fluo, Nanoscope systems, Korea) was used to obtain horizontal sectional images at different vertical locations, and to reconstruct the 3D configuration of molecular arrangement inside the droplets [15]. The pinhole was set to 1 a.u., and a 405 nm laser beam was used. Objectives with magnitudes of × 20 and × 40 and numerical apertures of 0.5 and 0.75, respectively, were used. All the experiments were performed at 25 °C. A rotatable polarizer was inserted between the objective and the cell. Bright fluorescence signal is detected when the director alignment is parallel to the polarizer axis.

Nematic droplet without chiral dopant exhibited a hedgehog texture with a single defect point in the center (image not shown here), indicating homeotropic surface anchoring [1,12]. In the CLC droplets with helical structure, complicated topological defects were observed [2,8–10]. The topological defect shapes were diverse, depending on the droplet size and the CLC dopant concentration. Figure 1 shows a typical example of topological defects in a CLC droplet with long helical pitch, in which the helical pitch (~90 μm) was similar to the droplet diameter (~84 μm). Figures 1(a) and 1(b) were obtained under crossed polarizers and without polarizers, respectively, using a POM. As shown in Figs. 1(a) and 1(b), sharp entangled circular defect lines are discernible. The defect lines exhibit clear contrast to the gradual brightness modulation arising from the helical deformation with long pitch. The sharp bright or dark lines along the defect line in Fig. 1(a) indicate that the birefringence changes abruptly across the defect line due to the director distortion. The defect lines appeared dark in the optical microscope image without polarizers [Fig. 1(b)], due to strong light scattering [11–13].

 

Fig. 1 Microscopic analysis for a CLC droplet with a helical pitch of 90 μm and a diameter of 84 μm. a) POM image under crossed polarizers. b) POM image without polarizers. c) Selected FCPM images at different vertical locations from top to bottom of the droplet under a vertical polarizer. d) FCPM images at the same vertical location with varying polarizer directions. The last image in (d) shows the collection of defect lines determined from the other four images. The arrows indicate the direction of polarizer. Scale bar: 30 μm.

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The entangled defect lines observed in the POM images lie within a 2D plane, and the POM images do not provide information regarding the vertical location of the defect lines. Hence, it is not possible to determine whether the defect lines lie on the surface or inside the droplet. To solve this problem, we analyzed the same droplet using a FCPM. FCPM is commonly used for biological samples, in which a target organ in the sample is selectively dyed [16]. However, the fluorescence dye disperses uniformly in the LC medium, and thus the selective dyeing method is not applicable. However, the fluorescence intensity depends on the molecular orientation, and by utilizing a polarizer, the molecular orientation is facilely determined [13,15]. Figure 1(c) shows a stack of FCPM images at varying vertical locations in the droplet, showing textures that are different at the different locations. In addition, the texture shapes depended on the polarizer direction, as shown in Fig. 1(d), in which the arrows indicate the direction of the polarizer. The bright area corresponds to the directors being parallel to the polarizer. The four images in Fig. 1(d) were obtained from the middle layer of the CLC droplet with varying polarizer directions. In the first image in Fig. 1(d), sharp defect lines are discernible on the upper and bottom edges, where the surrounding area appears dark. The whole loop of defect lines at the corresponding vertical section can be determined by rotating polarizers. The last image in Fig. 1(d) is the collection of the defect lines obtained from the four set of FCPM images, and clearly shows a large open circular defect line in the middle section of the droplet. The open circular defect line is expected to extend to different levels of the vertical section.

The defect line was not always represented as a sharp line in FCPM images, but the appearance depended on the angle of inclination of the defect line. Figure 2(a) shows a horizontal defect line with a defect strength of + 1, where the polarization axis is parallel to the defect line. As illustrated in the schematic drawing, all molecules around the defect line are perpendicular to the polarization axis, and hence the surrounding area appears dark. The molecular orientation is disturbed in the defect line, and some portion of molecules are not perpendicular the polarization of laser [10–12]. Hence, the fluorescent light emitted from the molecules close to the defect line is detected as a sharp line. However, as the defect line is increasingly tilted, the portion of molecules that are not perpendicular to the polarizer axis increases [green molecules in the illustration in Fig. 2(b)]. Hence, the thickness of the defect line increases, as indicated in the FCPM image in Fig. 2(b), in which the thin defect line (blue dashed line) becomes thicker (green line in the inset). The defect line appears dark when perpendicular to the polarizer (yellow line in the inset). Such a thick line could be misinterpreted as a helical deformation texture or a defocused defect line at different vertical levels. When the inclination of the line exceeded a certain level close to the vertical, the defect appears to be a point defect with two brushes [FCPM image in Fig. 2(c)], because the molecules on the bright sides are parallel to the polarizer axis as illustrated in Fig. 2(c). Unlike a hedgehog point defect, the vertical defect has a tail line extended to a horizontal defect line (see the orange line and yellow arrows).

 

Fig. 2 The shape of defect lines in the polarized FCPM images (a) for a horizontal defect line, (b) for a tilted defect line, and (c) for a vertical defect line. The top illustrations represent the director orientations around the defect line, and the green and gray directors represent the bright and dark molecular arrangements under the polarizer (blue arrows), respectively. The green lines in the small inset images in (a) and (b) denote the locations of corresponding detect lines in FCPM images. The red circle and green ellipses in the inset image in (c) denote the vertical defect point and the molecular arrangement around it. The extended defect line from the vertical defect is marked by orange lines and yellow arrows in (c).

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The vertical defect line in Fig. 2(c) exhibited a texture with two brushes, and the bright brushes rotated when rotating the polarizer [inset image in Fig. 2(c)]. Considering that LC molecules in the bright brushes align parallel to the polarizer, the defect line is undoubtedly + 1 defect, which does not accord with those predicted by Sec et al [7]. Sec et al. simulated the complicated defect lines within a CLC droplet, in which the strength of defect lines were −1/2 with threefold brushes. The defect type may be diverse depending on the elastic properties of LCs and anchoring type, but in our sample, the defect lines were all + 1 defects.

In this way, we analyzed the defect lines at each horizontal section with rotating polarizers for the droplet shown in Fig. 1, and connected the open circular defect rings in each stacking section [Figs. 3(a)-(d)]. Then, we could construct the whole defect line within the droplet, as illustrated in Fig. 3(e). By rotating the reconstructed 3D defect rings, we were able to reproduce the top view of the defect image [the last image in Fig. 3(e)]. One can notice that the top view of the defect loop coincides with the inset POM image.

 

Fig. 3 (a)-(d) The defect loops for the CLC droplet in Fig. 1 roughly at z = 1/5, 2/5, 3/5, and 4/5 of the droplet from the top, respectively. The last image in each row represents the combined defect loops in each vertical section. (e) Combined defect loops, POM image, and the reconstructed 3D defect loops based on the sectional images. (f)-(j) 3D reconstruction of the defect loops for another CLC droplet with a vertical defect line, via the same procedure. Droplet size: 120 μm. Helical pitch: 136 μm. Arrows: polarizer axes. Scale bar: 30 μm.

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Using the same method, we reconstructed the 3D defect structures for another CLC droplet with a vertical defect line [Figs. 3(f)-3(j)]. By analyzing the stacked FCPM images, we reconstructed the entire 3D defect loops within the droplet as shown in Fig. 3(j). The FCPM images in Fig. 3(f) show a point defect with two dark brushes, indicating the vertical defect line in the middle part. Owing to the blurred observation for the highly slanted defect line, the bottom tail of the vertical defect was not clearly discernible in the FCPM images [Figs. 3(h) and 3(i)]. However, by combing the FCPM images with the POM image, a reasonable reconstruction of the full defect loops was obtained. The purple dashed line in Fig. 3(j) was inferred from the defect line in POM image.

For a droplet with a short pitch helix, it is rather difficult to identify the defect loops from the helical structures in both the POM and FCPM images. Figure 4(a) shows selected FCPM images from a droplet with short helical pitch. The FCPM images clearly show the finger print-like helical periodic structure on the top side of the droplet. On the other hand, the bottom side image was rather smooth with sparse defect lines. The 3D construction of FCPM images in Fig. 4(b) also shows onionskin-like structures on the top side, but the other side is rather smooth. The shape looks like a stacked pile of different sized bowls.

 

Fig. 4 (a) FCPM images for a droplet with short helical pitch for different vertical levels (top, middle, and bottom levels), and (b) the reconstructed 3D images using the stacked FCPM images. The yellow arrows denote the top view direction. Droplet size: 130 μm. Helical pitch: 26 μm. Scale bar: 30 μm.

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3. Conclusion

To date, a FCPM analysis has been adopted to study a bulk LC sample [13,17,18] but not for a CLC droplet in a spherical confinement. We have demonstrated that FCPM is an excellent tool to reconstruct the 3D defect structures inside CLC microspheres with long helical pitches, which has been previously challenging. The defect lines within one horizontal section can be reconstructed by combining FCPM images with varying polarizer directions at a fixed vertical position. Then, by piling the sectional images, we could reconstruct the entire 3D defect structure, including vertically running defect lines. In particular, we found that the defect lines in our sample had a defect strength of + 1, as opposed to those predicted by Sec et al [7]. In CLC droplets with short helical pitches, FCPM can visualize the rough 3D configuration of helical deformation within the droplet.

The FCPM method still has limitations in identifying the topological structures within a CLC microsphere; in particular, the birefringent property of LC distorts the polarization of emitted light, which partially disturbs the identification of molecular ordering [13,18]. However, defect lines in microspheres are clearly discernible as demonstrated in this study, and it shows potentiality of FCPM method in the study of topology and defects in LC microspheres.