Topological Defect Evolutions Guided by Varying the Initial Azimuthal Orientation

Abstract

Topological defects are a key concern in numerous branches of physics. It is meaningful to exploit the topological defect evolutions during the phase transitions of condensed matter. Here, via varying the initial azimuthal orientation of the square alignment lattice in a hybrid cell, the topological defect evolution of liquid crystal during the nematic (N)–smectic A (SmA) phase transition is investigated. The director fields surrounding ±1 point defects are manipulated by predesigning the initial azimuthal orientation. When further cooled to the SmA phase, spiral toric focal conic domain (TFCD) arrays are formed as a result of twisted deformation suppression and unique symmetry breaking after the phase transition. The variation in the azimuthal orientation causes the TFCDs to degenerate from infinite rotational symmetry to quadruple rotational symmetry, thus releasing new textures for the SmA phase. Landau–de Gennes numerical modeling is adopted to reproduce the director distributions in the N phase and reveal the evolution of the topological defects. This work enriches the knowledge on the self-organization of soft matter, enhances the capability for the manipulations of topological defects, and may inspire new intriguing applications.

1. Introduction

Topological defects are discontinuities or nontrivial regions in physical systems that cannot be eliminated through continuous symmetry operations [1]. They are distinguished by topology and play vital roles in various fields of physics, including condensed matter physics [2], cosmology [3], and quantum field theory [4], determining corresponding properties and behaviors. Topological defects inevitably arise during the phase transition of condensed matter because of a spontaneous symmetry breaking process [5]. Thus, it is meaningful to exploit the behaviors and mechanisms of topological defect transformations across the phase transitions. Liquid crystals (LCs) provide an ideal platform for such investigations owing to their unique combination of fluidity, stimuli-responsive optical anisotropy, and abundance of phases and topological defects [6].

The constituent rod-like molecules in the nematic phase (N) of LCs spontaneously orient along a dominant direction, referred to as the director n, where n and −n are physically equivalent [7]. NLCs usually exhibit point defects, line defects, and wall defects. The topological charge s is a conserved quantity characterizing the number of rotations of the director around the defect core, and is considered as a key property of topological defects [7]. The value of s is infinite in theory; however, topological defects with ±1 and ±1/2 are usually observed experimentally. The spatial dimension is another feature of topological defects [8]. For example, point defects can be classified into bulk point defects (three-dimensional) and surface point defects (two-dimensional). The former is usually formed in spherical nematic droplets with perpendicular anchoring surfaces and uniformly aligned NLCs embedded with spherical particles of vertical alignment [9,10,11,12]. The latter has been more extensively investigated due to its ability to freely generate and manipulate under external stimuli such as electric fields [13,14], magnetic fields [15], topographically patterned mechanical scrubbing [16], and preprogrammed photopatterning [17]. Across the N–smectic A (SmA) phase transition, one-dimensional positional order is further introduced in addition to long-range directional order. Accordingly, the LC molecules arrange into parallel layers with nanometer thicknesses. Under hybrid anchoring conditions, these layers are periodically wrapped to generate toric focal conic domains (TFCDs) [18,19]. TFCDs exhibit perfect rotational symmetry and can focus light like a microlens due to their radially gradient refractive index change [20]. Moreover, their specific director distribution enables the transportation of functional nanoparticles to the central singularity [21] and this unique topographic feature facilitates the manufacture of artificial superhydrophobic surfaces [22]. Recently, ordered TFCDs were converted from NLCs with predesigned point defects during the N-S phase transition [23,24]. Up until now, the LC orientation around point defects has been restricted to the radial condition, hindering the behaviors of the corresponding structures. New architectures with unprecedented properties can be released if one can overcome the infinite rotational symmetry limitation.

In this work, we investigate the topological defect evolution of LCs across the N-SmA phase transition in a hybrid cell via presetting different azimuthal orientations of a square alignment lattice. The square lattice of the radial alignment is imprinted on the substrate by photoalignment, while a perpendicular alignment is set on the superstrate. When the LC is filled with such a cell, a ±1 point defect array is achieved in the N phase when it is slowly cooled from the isotropic phase. Via varying the initial azimuthal orientation of each alignment unit, the transformations of both the point defects and the surrounding director fields occur correspondingly. The evolution of ordered TFCDs is generated during the N-SmA phase transition due to the inheritance of orientational order. The Landau–de Gennes theory is utilized to model the director distributions of point defects in the N phase, revealing how topological defect profiles change in response to variations in the azimuthal orientation. This work extends the understanding of orientational order control and may release more creative architectures of soft matter.

2. Materials and Methods

The LC device is formed by an LC layer [8CB (NCLCP, Nanjing, China), isotropic—40.5 °C-N–33.5 °C-SmA] sandwiched between a photoalignment agent [SD1 (NCLCP, Nanjing, China), dissolving in dimethylformamide (DMF) at a concentration of 0.3 wt.%.]-covered substrate and a PDMS (Polydimethylsiloxane, Dow Corning, Midland, MI, USA) layer coated with a superstrate [Figure 1a; the molecular structures of SD1, PDMS, and LC 8CB are shown in Figure 1b]. The former gives planar anchoring, while the latter provides homeotropic alignment, and they work together to generate a hybrid alignment. The specific sample fabrication and characterization are presented in Appendix A. A periodic square lattice of +1 and −1 defects is imposed in the SD1 layer using a multistep partly overlapping photoalignment technique [25], with directors varying gradually among adjacent defects (Figure 1), and the detailed processes are given in Appendix B. The director field is depicted as follows [26]:

$$\alpha (x,y)={\displaystyle \sum _i{s}_i\arctan \frac{y-y_{0i}}{x-x_{0i}}}+{\alpha }_0,$$

(1)

where topological charge is set as s_i = ±1, (x_0i, y_0i) are the coordinates of the singularities, and α₀ (the angle between the easy axis and the x-axis) is the initial azimuthal orientation. Here, the cell gap is fixed as h = 8.5 μm, and the lattice constant is set as l = 30 μm. The LC director follows the guidance of the photopatterned substrate, and the undefined director orientation on the alignment singularities leads to the generation of ±1 point defects. α₀ can be freely varied owing to the flexibility of photoalignment. Figure 1a–d show the lattices with α₀ = 0°, 30°, 60°, and 90°, respectively. The director fields surrounding the alignment singularities change according to the variance of α₀.

3. Results and Discussions

When the infilled 8CB is cooled from the isotropic state to the N phase at a rate of −0.3 °C/min, the LC directors follow the local alignment of SD1 on the substrate and gradually turn to a uniform vertical alignment toward the superstrate. Figure 2 reveals the generated N textures observed under polarization optical microscopy (POM). When α₀ = 0°, a square lattice of Maltese crosses is exhibited (Figure 2a). When the polarizers are rotated, the Maltese crosses around +1 point defects rotate in the same direction, whereas those around −1 point defects rotate in the opposite direction. The above phenomena indicate that +1 and −1 point defects are located exactly on the predesigned +1 and −1 alignment singularities. Similar textures are observed in samples with α₀ = 30°, 60°, and 90°, except for obvious rotations of Maltese crosses at the corresponding anchoring angles. The bright field images shown in the insets of Figure 2 further confirm the consistency of point defects on alignment singularities. Cyan and magenta interference colors are simultaneously observed in Maltese crosses around adjacent +1 point defects. These differences are caused by the distinct phase retardations induced by the different tilt angles of the adjacent domains. To more precisely determine the director field around the point defects, a first-order retardation plate (λ = 530 nm) is inserted between the sample and the analyzers. Magenta is observed when n is parallel and vertical to the polarizers, and cyan blue and yellow colors appear when n is parallel and vertical to the slow axis of the λ-plate, respectively [27]. As shown in the bottom images of Figure 2a, cyan blue and magenta emerge twice alternatively in each unit, suggesting n rotates accordingly. The bottom images of Figure 2b–d show the cases with α₀ = 30°, 60°, and 90°, respectively. The square lattices are maintained, while the interference colors change with varying α₀. The interference color variations are consistent with the presetting local alignments α(x,y).

The Landau–de Gennes theory is further utilized to reproduce the NLC director distribution and investigate the point defect evolution [28]. A traceless and symmetric tensor is adopted to describe the director field; this is expressed as $Q_{ij}=S(n_i{n}_j-{\displaystyle \frac{1}{3}}{\delta }_{ij})$, where S is the uniaxial scalar order parameter. The total free energy F of the NLC is defined as follows:

$$\begin{array}{cc}\hfill F=& {\displaystyle \underset{v}{\int }f}\mathrm{d}v+{\displaystyle \underset{s}{\int }{f}_{\mathrm{surface}}}\mathrm{d}s\hfill \\ \hfill & ={\displaystyle \underset{v}{\int }(f_{\mathrm{bulk}}+f_{\mathrm{elastic}})}\mathrm{d}v+{\displaystyle \underset{s}{\int }{f}_{\mathrm{surface}}}\mathrm{d}s\hfill \end{array}$$

(2)

The bulk energy f_bulk describes the degree of nematic order, which is given as follows:

$$f_{\mathrm{bulk}}={\displaystyle \frac{1}{2}}\mathrm{A}tr{Q}^2-{\displaystyle \frac{1}{3}}\mathrm{B}tr{Q}^3+{\displaystyle \frac{1}{4}}\mathrm{C}{(tr{Q}^2)}^2$$

(3)

Here, A = A₀(T − T*), where T* defines the supercooling temperature of the nematic phase; A₀, B, and C are defined as constants. For Equation (3), the uniaxial scalar order parameter of the bulk equilibrium can be calculated as $S_{\mathrm{eq}}={\displaystyle \frac{B}{4C}}\left(1+\sqrt{1-{\displaystyle \frac{24AC}{B^2}}}\right)$, and it depends on the temperature T. f_elastic is used to describe the contribution of anisotropic elastic free energy, including the all-spatial gradient of tensor Q, which is given as follows:

$$\begin{array}{cc}\hfill f_{\mathrm{elsastic}}=& {\displaystyle \frac{1}{2}}{L}_1{\displaystyle \frac{\partial Q_{ij}}{\partial x_k}}{\displaystyle \frac{\partial Q_{ij}}{\partial x_k}}+{\displaystyle \frac{1}{2}}{L}_2{\displaystyle \frac{\partial Q_{ij}}{\partial x_j}}{\displaystyle \frac{\partial Q_{ik}}{\partial x_k}}+{\displaystyle \frac{1}{2}}{L}_4{Q}_{ij}{\displaystyle \frac{\partial Q_{kl}}{\partial x_i}}{\displaystyle \frac{\partial Q_{kl}}{\partial x_j}}\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{L}_{24}\left({\displaystyle \frac{\partial Q_{jk}}{\partial x_i}}{\displaystyle \frac{\partial Q_{ik}}{\partial x_j}}-{\displaystyle \frac{\partial Q_{ij}}{\partial x_j}}{\displaystyle \frac{\partial Q_{ik}}{\partial x_k}}\right)\hfill \end{array}$$

(4)

where the coefficients L_i (i = 1, 2, 4, 24) represent the splay (k₁₁), twist (k₂₂), bend (k₃₃), and saddle-splay (k₂₄) elastic constants of Frank theory, and the relationships of them can be defined as follows:

$$L_1={\displaystyle \frac{3k_{22}-k_{11}+k_{33}}{6S_{eq}^2}},L_2={\displaystyle \frac{k_{11}-k_{22}}{S_{eq}^2}},L_4={\displaystyle \frac{k_{33}-k_{11}}{2S_{eq}^2}},L_{24}={\displaystyle \frac{k_{24}}{S_{eq}^2}}$$

(5)

The second integral in Equation (2) is the anchoring energy, which includes the homeotropic anchoring energy of the superstrate covered with a PDMS layer and the planar anchoring contribution of the substrate covered with an SD1 layer. The calculation of homeotropic anchoring energy is defined by the Rapini-Papoular expression: $f_{\mathrm{surface}}={\displaystyle \frac{1}{2}}{W}_{\mathrm{h}}tr{(Q-Q_{\mathrm{s}})}^2$. W_h represents the anchoring strength and Q_s is the priority tensor of the surface. The planar anchoring energy is calculated using the following equation: $f_{\mathrm{surface}}={\displaystyle \frac{1}{2}}{W}_{\mathrm{p}}({\tilde{Q}}_{ij}-{\tilde{Q}}_{ij}^{\perp })({\tilde{Q}}_{ij}-{\tilde{Q}}_{ij}^{\perp })$, where ${\tilde{Q}}_{ij}=Q_{ij}+{\displaystyle \frac{1}{2}}{S}_{\mathrm{eq}}{\delta }_{ij}$, using the projection operator: ${{\tilde{Q}}_{ij}}^{\perp }=P_{ik}{\tilde{Q}}_{kl}{P}_{lj}$, $P_{ij}={\delta }_{ij}-{\nu }_i{\nu }_j$. Here, the tensor P_ij will subtract the normal of the Q tensor, and v represents the normal of the substrate. The total free energy of the LCs is further minimized based on the Euler–Lagrange equation, and is solved by a finite difference method. Here, the parameters in the simulations are selected as A = −0.172 × 10⁶ J/m³, B = 2.12 × 10⁶ J/m³, and C = 1.73 × 10⁶ J/m³; thus, the scalar order parameter is S_eq = 0.533 [28]. The homeotropic anchoring strength of the PDMS superstrate is W_h = 6 × 10⁻⁶ J/m² and the planar anchoring strength of the SD1 substrate is W_p = 1 × 10⁻⁴ J/m² [29]. For the elastic constants, we choose k₁₁ = k₃₃ = 2k₂₂ = 9.5 × 10⁻¹² N and k₂₄ = k₂₂. Finally, our simulation is conducted on a cubic lattice with periodic boundary conditions along two lateral directions.

Figure 3 reveals the simulated LC directors of samples with different α₀. The director n distributions of three typical cross sections are further selected to study the defect evolution along with the variation in α₀. They are the middle layers in the x-y cross section (top images in Figure 3), and the x-z cross section passing through −1 (y = 0) and +1 (y = l/2) point defects with y = 0 (middle images in Figure 3) and y = l/2 (bottom images in Figure 3), respectively. Here, φ is defined as the azimuthal angle between the director n and the x-axis. Figure 3a shows the φ distributions and director fields in the three cross sections for the sample with α₀ = 0°. The top image reveals that the director n is radially distributed within each unit due to the guidance of the preset alignment. The middle image in Figure 3a reveals the variation of directors in the z dimension. According to the simulation, a +1 point defect is classified into two types: (1) +1 converging type (+1_c), in which the director field will converge toward the z axis; (2) +1 diverging type (+1_d), in which the director field will diverge from the z axis. Moreover, for the −1 defect point, the converging and diverging director fields coexist in orthogonal planes, resulting in only one type. The cross section passing through the center of the +1 point defects (bottom image in Figure 3a) proves that there is no azimuth rotation for the +1 point defects, differing from the case of the −1 point defects; the calculated director distribution is consistent with the interference color in the POM image (Figure 2a). The director fields and φ distributions of samples with α₀ = 30° (Figure 3b), α₀ = 60° (Figure 3c), and α₀ = 90° (Figure 3d) are also simulated, indicating that the directors rotate with increasing α₀, which is consistent with the experimental interference colors. In the x-z cross sections with y = 0 (middle images), the variation in the director field astride the −1 point defect indicates a transformation from the convergent type to the divergent type. The bottom images indicate that the type of +1 point defect remains unchanged, whereas slight deformation occurs due to the increase in the azimuthal component of the surrounding directors as α₀ increases. Thus, the variation of α₀ leads to an azimuthal rotation of the director fields.

When the sample is further cooled to the SmA phase, the LC molecules locally arrange into parallel layers and form TFCDs. In each domain, the LC layers are wrapped around a defect pair (red) of a straight line and a circle (the inset in Figure 4a). As reported previously [24], when α₀ = 0°, TFCDs are generated only at +1_c point defects due to the conforming converging director fields. Moreover, +1_d point defects change to domains with positive Gaussian curvatures and are alternatively embedded into the TFCDs, resulting in a square TFCD array with a lattice constant of $\sqrt{2}l$ (top image in Figure 4a). In this case, the TFCD exhibits infinite rotational symmetry, which is further verified by the circular contour in the bright field image of each TFCD (bottom image in Figure 4a). For the case of α₀ = 30°, TFCDs are still formed, but are rotationally distorted. The distortion is attributed to the azimuthal twist of the LC director fields guided by the anchoring (top image in Figure 4b). Due to the inheritance of orientational order across the N-SmA phase transition, distortion induces the deformation of the LC layers. As a result, TFCDs are split into four partial lobes, whose boundaries are clearly observed as dark curves in bright field images (bottom image in Figure 4b). In this case, spiral TFCDs exhibiting chirality are formed due to symmetry breaking. Accordingly, the TFCD units degenerate from infinite rotational symmetry to quadruple rotational symmetry. Notably, the core of the spiral TFCD is enlarged compared with that of the standard TFCD, while the diameters remain $\sqrt{2}l$. We find that the integrity of the spiral TFCD remains until α₀ > 45°. When α₀ = 60°, the spiral unit is divided into four individual parts and forms a windmill-like texture (Figure 4c). When α₀ = 90°, a new texture is formed. Thus, via changing α₀, the infinite rotational symmetry of the TFCD is broken and the shape of the repeatable unit is varied, which enriches the self-assembly of SmA hierarchical architectures.

Via reasonably predesigning the initial azimuthal orientation of the alignment lattices, the mechanisms and behaviors of topological defect evolutions during the N-SmA phase transition are investigated. In previous research, the direction fields of the TFCD transformed from +1 point defects were restricted to a radial distribution. Here, through programming the photoalignment patterns, the infinite rotational symmetry restriction is broken, and new SmA textures are generated. Here, topological defect evolutions occur at the millisecond scale to the second scale. This approach significantly enhanced the direct observation of these evolutions, which is unachievable in quantum physics and cosmology. New properties and applications can be reasonably expected as new ordered LC textures are fabricated. It also brings about new possibilities for existing diffraction elements, particle assembly, and asymmetric imaging.

4. Conclusions

The topological defect evolutions of LCs are investigated via varying the initial azimuthal orientation of the square alignment lattice in a hybrid cell. As α₀ increases from 0° to 90°, the twist deformations of the director n lead to significant director field variations surrounding +1 and −1 point defects. The Landau–de Gennes theory is adopted to analyze the director distributions and defect transformations. The simulations indicate that the directors rotate with increasing α₀, and there is no azimuth rotation for the +1 point defect, while the −1 point defects exist. Further theoretical analysis reveals that the −1 point defect changes from a convergent type to a divergent type, while the type of +1 point defect remains unchanged. Although twisted deformation is suppressed in the SmA phase, discontinuous LC layers are formed due to the inheritance of orientation order of the above textures. This results in various spiral TFCDs when α₀ ≤ 45°. For larger α₀, the unit is divided into four individual lobes. TFCDs degenerate from infinite rotational symmetry to quadruple rotational symmetry, and the shape of the repeatable unit drastically changes with the variance of α₀. This work enriches the fundamental understanding on the self-organization of soft matter systems and may open the door for advanced applications of topological defects.