Dynamic Motions of Topological Defects in Nematic Liquid Crystals under Spatial Confinement

Topological defects (TDs) have become a sensational topic due to their significant influence on the unusual optical and physiochemical characteristics of the material. To facilitate their application across a wide range of disciplines it is desirable to analyze and gain fundamental understanding of TDs in both equilibrium and nonequilibrium systems. Liquid crystals (LCs) are considered an ideal system for study given the direct visualization of TDs and a straightforward analyzation process. In addition to the equilibrium morphology of LC TDs, it is also of great interest to track and control the formation and annihilation of defects during thermodynamic processes. However, controlling the dynamic behavior of formed defects remains a challenge. Here, nematic LCs confined in a cell exhibiting surfaces with periodic anchoring conditions containing surface topography are relied on. The effects of patterned surface characteristics such as width, periodicity, and degree of curvature on defects dynamic motion, stabilization, and annihilation are explored. The computational experiments recapitulate the TDs transition path and provide free energy‐based predictions of critical distances for defect annihilation. Taken together, this simple approach offers a promising opportunity to control the dynamics of TDs in LCs through chemical patterned surfaces with topography.


Dynamic Motions of Topological Defects in Nematic Liquid Crystals under Spatial Confinement
Tejal Pawale, Justin Swain, Mohammad Reza Hashemi, Giordano Tierra, and Xiao Li* DOI: 10.1002/admi.202300136 encompasses a wide variety of physical systems in which the dominant physical behavior occurs on energy scales comparable to room temperature thermal energy. TDs manifest in such soft matter systems with multiple forms due to the weak molecular interactions and a delicate balance between entropic and enthalpic contributions to the total free energy. Especially, in the liquid crystal (LC) category, its unique molecular anisotropic properties create long-range orientational order, and sensitivity of equilibrium topological structures to external conditions. LC has been considered an ideal system for defect investigation, due to the birefringence phenomena of LCs enabling direct visualization of such TDs and a straightforward analyzation process. [3] In LCs, the long-range molecular axis is defined as a director n. In an equilibrium state, geometric frustration or thermal fluctuations can lead to a transformation from a uniform state to an inhomogeneous structure of the director n, resulting in static TDs with high elastic energy. [4] The defect formation in 1D, 2D, or 3D is represented as point, linear, or planar singularity, respectively. [5] The nematic phase, the simplest LC phase, has been investigated extensively to elucidate the formation and transformation behavior of TDs. The analogous nature of these nematic LC defects makes the mathematics describing the formation and evolution of TDs applicable across different disciplines and length scales. Therefore, stabilizing and manipulating the formed TDs into a desired pattern over large area is essential not only for understanding the underlying physical mechanics, but also to facilitate the application of TDs in templating, molecular self-assembly, [6] photonic devices, [7] diffraction gratings, [8] and vortex generation. [9] Defect topology in an equilibrium state has been extensively studied in both bulk and surface confinements. In bulk, the formation of TDs with non-uniform director field is usually associated with high elastic energy cost, which makes their characteristics very unstable. Previous research work has incorporated techniques such as the introduction of external stimuli, including electric, [10,11] magnetic, [12] and optical field, [9] as well as geometric boundary frustration to create topographic features or confinement [13][14][15] for self-assembly and stabilization of LC defects. The application of external stimuli can control the bulk volume of LC to generate a single domain of defect arrays with high tunability. [16] However, these lattices of Topological defects (TDs) have become a sensational topic due to their significant influence on the unusual optical and physiochemical characteristics of the material. To facilitate their application across a wide range of disciplines it is desirable to analyze and gain fundamental understanding of TDs in both equilibrium and nonequilibrium systems. Liquid crystals (LCs) are considered an ideal system for study given the direct visualization of TDs and a straightforward analyzation process. In addition to the equilibrium morphology of LC TDs, it is also of great interest to track and control the formation and annihilation of defects during thermodynamic processes. However, controlling the dynamic behavior of formed defects remains a challenge. Here, nematic LCs confined in a cell exhibiting surfaces with periodic anchoring conditions containing surface topography are relied on. The effects of patterned surface characteristics such as width, periodicity, and degree of curvature on defects dynamic motion, stabilization, and annihilation are explored. The computational experiments recapitulate the TDs transition path and provide free energy-based predictions of critical distances for defect annihilation. Taken together, this simple approach offers a promising opportunity to control the dynamics of TDs in LCs through chemical patterned surfaces with topography.

Introduction
The existence of topological defects (TDs) is ubiquitous in nature due to their occurrence at broad spatiotemporal scale, for instance, in subatomic particles [1] or cosmology [2] and the undeniable role they play in modern condensed matter physics. Soft condensed matter, a subfield of condensed matter, www.advmatinterfaces.de morphology will not attain an equilibrium state in absence of the external stimuli. Whereas the surfaces imposing geometric or chemical frustration provide an alternative way to minimize the overall free energy of the system to reach equilibrium with stabilized defect morphology. Furthermore, through the design of topographical features or chemical contrast on surface pattern, high-precision and complex boundary conditions can be achieved to localize defects at desired positions. Geometric constraints were successfully introduced by generating topographically featured surfaces through mechanical scrubbing, [17,18] photolithography, [19] nanoimprint lithography, [20] light diffraction gratings, [21] and photo alignment. [21,22] These techniques can be used to fine-tune the confinement on LCs down to nanometer scale. In addition, numerous studies have also manipulated TDs in different LC phases by imposing different anchoring conditions to create chemical contrast. [23][24][25] The introduced topographical features or boundary conditions will determine the charge value s of the defect, considering that the elastic energy value will strongly depend on the 2D director distribution that the feature will impose on the LC molecule.
Current research has shown interest beyond controllable TDs at equilibrium, and toward the LC defects dynamics along with the active systems observed either in biological systems [26,27] or active matter. [16,[28][29][30][31] Active colloids where LC as a medium was used to direct particles of all kinds by utilizing the ability to manipulate the anisotropy of the LC director represents a distinct field of active matter. [31] In the area of electro-kinetics the LC director n acts as a navigator to transport the colloids by means of either anisotropic conductivity or permittivity for electrically driven systems. Considering that the activity is performed by the LC medium, the particle characteristics are insignificant. At larger length scales, similar coordinated collective motion is observed in animal groups and has been studied as an active system. [27] Active nematics emerged as a system to elucidate this relation at the meso-scale and gain a fundamental understanding that is applicable universally. The chaotic flow in the nematic phase due to either external or internal energy is called active and it leads to continuous nucleation and annihilation of TDs. [28][29][30][31][32][33][34][35][36][37] The spontaneous dynamics limits the fundamental understanding of defect nucleation, splitting, and annihilation processes. Reported work has relied on composition, [38] light, [34] magnetic field, [28] activity patterning, [33] and confinement [29,30] to control the dynamics of active nematics systems. Similar to the defects in equilibrium, confinement has a profound effect on the collective dynamics of active systems, not only in bulk LCs, but also when confined between capillaries, [32] surfaces, [39] or in shells [30,36] and droplets. [29] The command over transformations of LC defects and their motion will broaden their functional applications.
In the present work, we rely on a 2D patterned surface with combined chemical contrast and topographic features to isolate, track, and direct single topologic defect motion. A nematic LC system is applied on the patterned surface to create defects and thermal fluctuation is used to induce the dynamic motion of defects before reaching their equilibrium. As shown in Figure 1a, a LC cell is created by assembling two substrates faceto-face. One surface contains periodic anchoring conditions and surface topography inducing a height profile at the interface of homeotropic and degenerate planar anchoring stripes, and the other surface has uniform homeotropic anchoring. The www.advmatinterfaces.de system was studied while cooling from the isotropic phase and, especially before reaching equilibrium, the thermal energy fluctuations cause dynamic motion of formed defects. The effect of the patterned surface characteristics on the dynamic motion of defects will be systematically investigated by varying the width of homeotropic/planar stripes along with the periodicity. The evolution of defects from generation, motion, transition, annihilation, and stabilization is studied in terms of time, pattern shape, and dimension. Computational simulations are used to identify the transition pathways of TDs and director field deformation process. By calculating the total free energy of system, the critical distance between defects is determined for annihilations.

Results and Discussion
A surface containing topographic features coupled with periodic anchoring conditions was fabricated as the schematics shown in Figure 1a. A polymer brush poly(6-(4-methoxy-azobenzene-4′-oxy) hexyl methacrylate) (PMMAZO) with 0.05 wt% concentration was deposited onto a cleaned silicon wafer to form a 3-4 nm thin film. The grafting density of polymer brush is 2.02 × 10 −2 chains/nm 2 to provide the side-chain azobenzene mesogens perpendicular orientation on the substrate. [40] The anchoring control from the polymer brush can penetrate several micrometers in LC layer on top of it. On top of the uniform PMMAZO brush, an ≈40 nm poly(methyl methacrylate) (PMMA) photoresist layer was deposited for e-beam lithography patterning. The patterned layer with the array of stripe was exposed and then developed such that the opened stripes remain with weak homeotropic anchoring due to PMMAZO brush, [41] while the PMMA photoresist layer provides planar anchoring. [42] The width of homeotropic anchored region is labeled as Ws, while the distance between the two exposed regions of planar anchoring is labeled as D. The periodicity of the pattern can be calculated as P = Ws + D. The surface topography was introduced owing to the height difference between exposed and unexposed region. The atomic force microscopy in Figure 1b demonstrates the height profile of the patterned surface. The topographic difference between the PMMAZO brush coated region and photoresist layer was found to be around 30-40 nm. An LC cell was assembled with the chemically patterned substrate as the bottom surface and octadecyltrichlorosilane (OTS)-modified glass as the top surface without any spacer, while the average thickness of sandwiched cell was measured to be ≈94 ± 30 nm. [23] The cross-sectional SEM images of assembled cells without the use of spacer presented in our previous study ensured the uniformity and consistency of cell thickness in defect motion study. [23] The most commonly used nematic LC 5CB was injected in the LC cell through capillary action at its clearing point and cooled to room temperature naturally. The polarized optical microscopy (POM) images of an array of defects trapped on the pattern region are displayed in Figure 1c,d. When nematic 5CB is anchored perpendicular to the cell surface, it does not show any birefringence effect. Therefore, the black stripes represent the surface coated with PMMAZO brushes, providing homeotropic anchoring. While, the bright stripes between two homeotropic stripes are observed due to the degenerate planar anchoring provided by the photoresist layer. The assembled defect morphology was observed as triangular shape when stable (Figure 1c). When the sample position was rotated corresponding to the crossed polarizers in Figure 1d, the defect was revealed to be a s = +½ defect close to the interface of planar/homeotropic anchoring stripe. The simulated images of defect morphology with director field are shown in Figure 1e,f. The resulting image is acquired from the numerical experiment with D = 0.8 µm, showing a similar equilibrium state to the experimental results.
To analyze the defect-trapping and stabilizing capabilities of patterns in terms of homeotropic anchoring, the width of homeotropic anchoring stripe (Ws) was gradually increased from 500 nm to 7 µm, while keeping the distance between two homeotropic stripes fixed at D = 8 µm. The POM images in Figure 2 were captured once the defects attained an equilibrium state for all the conditions. As clearly seen in Figure 2, ±½ defects with triangular morphology can be observed on the planar anchoring regions with Ws > 1 µm. For Ws values between 500 nm to 1 µm, the pattern appeared to be unable to trap and stabilize the defects. Instead, the patterned region acted as a homogeneous surface with degenerate planar anchoring. Therefore, Ws above 1 µm was set as a critical condition for trapping and stabilizing defects, independent of the D value. When Ws was between 1.6 and 3.2 µm, the ±½ defects with triangular morphology are stabilized at the middle of stripe. The defect morphology observed at narrow Ws (1.6 and 2 µm) stabilized as a string around the mid-length of the pattern. Considering the small spaces that divide the planar stripes, the formed triangular defects are connected to each other, thus creating a chain of defects. And above 2 µm width the individually stabilized defects with distinct structure were observed. As Ws was increased from 3.2 µm, the steady state defects were observed at the middle of stripes and also at both ends of the stripe. The defect position corresponding to the pattern dimensions was plotted for average defect position versus ratio of Ws and D and is included in Figure S1, Supporting Information.
The transition of defects from ±1 to ±½ defect with the triangular morphology is an instant process and was found to be dependent on the width of planar anchoring stripe D.
To understand the dependence of the width of the planar anchored stripe (D) the width of homeotropic anchoring was kept constant (Ws = 6 µm), while the D was varied. The LC was cooled down to nematic phase from isotropic phase at the rate of 1 °C min −1 and the captured dynamic motion of defects is shown in Figure 3. Similar two-branched triangular defects were trapped and stabilized when the width of the planar anchoring stripes was set as small as 2 µm. However, due to the small size and ambiguity of the development process, it is not possible to distinguish the stabilization process toward ±½ defects while cooling. Also, the stabilization of defects appeared to be slower for D = 2 µm in contrast to other D values. When D is 8 µm, Schillerian structures originated on the planar anchored stripes when cooling from isotropic temperature and reaching the nematic phase temperature. The four branches connected at one point formed a defect with charge of s = ±1. Instant transition to the ±½ defects aligned around the middle of stripe length was observed. In the case of D = 14 µm, these four branched structures became s = ±1 defects and were located at the middle of planar anchored stripe. Meanwhile, the newly formed defects began to move toward the homeotropically anchored stripe on either side of the planar anchored region. After a few seconds, the ±1 defects are aligned in such a way that the defect center is merged into the interface of homeotropic and planar boundary, and the remaining two branches were transformed into a ±½ defects with a triangular morphology. Before reaching room temperature, part of the trapped ±½ defects travelled toward the end of the pattern region and annihilated, whereas others stabilized into a state of equilibrium. In case of D = 14 µm, these deformed triangular defects were located almost above the midlevel of the long side of the stripes.

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The orientation of these triangles was not unidirectional for D values of 8 and 14 µm. When the width D was further increased to 20 µm, although four branched ±1 defects initially formed on planar anchoring stripe, none of the ±1 defects could be transformed into ±½ defects stabilized at the boundary. The formed ±1 defects were mobile as they migrated toward the end of stripe and were annihilated. Therefore, for large value of D, the space between the homeotropic region on either side prevents the connection of ±1 defect branches to the homeotropic-planar interface, as observed in other narrow D cases. The transformation of defects from ±1 to ±½ defect of triangular morphology is an instant process and was found to depend on the width of the planar anchoring stripe D. Therefore, the width of the planar anchoring stripe D was noted to be critical to the development and retention of defects.
Simulations based on the penalized Ginzburg-Landau model provides a way to study the transition path from +1 defect to +1/2 defect while advancing toward the homeotropic-planar boundary and to understand the stabilizing interaction between defect and boundary. Simulations in Figure 4 start with a + 1 defect initialized at the center of the planar domain. The homeotropic anchoring was imposed on the left and right sides to introduce anchoring contrast without any topographic features. According to the thermal process in experiments, the thermal fluctuation-induced flow velocity was responsible for the dynamic motion of the defect and will be directly related to the width, D. In this way, wider domains will generate a stronger flow, thus a constant and uniform velocity field u = (v, 0) is imposed for various values of v corresponding to the size of the domain. Figure 4a,b shows the results considering two values of D and v as the system evolves in time from left to right. By considering a domain with width D = 14 µm (Figure 4a) and reference velocity , for time scale τ, the defect moved to the right side boundary where it interacted with the boundary conditions and became trapped as a + 1/2 defect, exhibiting a triangular shape similar to the D = 14 µm case. Moreover, by further increasing the domain width to 20 µm (Figure 4b) and assuming the induced flow velocity is v = 4v 0 , the defects passed through the boundary and the equilibrium state is similar to Figure 3a for D = 20 µm. It is important to consider a nonzero velocity field inside the region as otherwise it is energetically stable for the defect to remain in the center of the stripe. If we run the same simulation without flow velocity as in Figure 4, the defect tends not to move toward the boundary. In order to simulate the thermal fluctuations experienced by LC materials during cooling from the isotropic phase, it is necessary to introduce significant flow rates into the simulation. The flow velocity will vary according to the volume of material in the planar anchoring region and will influence the behavior of the defect moving to or across the boundary.
After comprehensive analysis involving the impact of anchoring strength on defect stabilization it was found that planar anchoring with width of D = 8 µm provides better control over the position of the defect morphology along the stripe. Whereas, homeotropic anchoring width around Ws = 7 µm promotes formation of individual and distinct triangular defect morphology. Thus, the dynamic motion of defect morphology upon cooling from isotropic to nematic phase for pattern dimension of D = 8 µm and Ws = 7 µm was recorded as shown in Figure 5. As the LC cell cooled, the generated defects became trapped in the pattern area, and in particular a set of defects formed into a line in the middle of stripes. In addition, the  Figure S4a, Supporting Information. www.advmatinterfaces.de development of ±1 defects to ±½ defect can also be observed at the two ends of the planar stripes. Once the phase transition temperature is crossed at t = 1 s, as shown in Figure 5a, the ±1 defects formed at both sides of the stripe end, and then the core of ±1 defects moved toward either edge of the planar stripe. After 2 s, most of ±1 defects were split into ±½ defects with the center of defect resting on either side of the stripe. The slower orientation propagation was noted for the middle area of the patterned stripes compared to the two ends. Once the ±1 defects were formed, the transition of ±1 defects to ±½ defects was similar to the other defects until they stabilized as a triangular morphology. Before reaching room temperature, while the system still has residual energy from heating, some defects at the end of patterned stripes either moved out of the pattern and disappeared, or moved toward another ±½ defect (t = 9 s) to annihilate both, if they are in close vicinity to each other. Such dynamic motion and defects annihilation happened within a few seconds from t = 5 s to t = 9 s. By further cooling the system, the mobility of defects became less active considering reduced flow velocity, so it took a longer time (from t = 1 min to t = 7 min) for two defects to approach and annihilate. Most of the defects formed at the middle of the stripes are stabilized and remain in a steady state after 7 min. To gain an overview of the change in defect location over time, a graph of average defect location after cooling from isotropic phase to a steady state is plotted (Figure 5b). Using the bottom of the strip www.advmatinterfaces.de as the initial point and the top as the final point, the average defect location (µm) along the length of the strip was measured. Each point on the graph represents position of the defect as a function of time, where 0 s represents the Iso-N phase transition. When temperature is right below isotropic-nematic transition, the defects formed, moved rapidly, and annihilated within the first 10 s, and then the defects were stabilized on the pattern surface and remained constant after 7 min, becoming an equilibrium configuration.
Accordingly, two ±1/2 defects of opposite topological charge were simulated on the homeotropic-planar boundary separated by a distance α to observe the tendency for the defects to annihilate for several values of α. As shown in Figure 5c, the energy curve, computed using the free energy functional given in "Numerical Methods," for ten different values of α shows the evolution of the total free energy over time. When α ≤ 20 µm, two defects appeared close to each other, they moved together to annihilate as shown in Figure 6a. When α ≥ 80 µm, two defects appeared far away from each other, they moved away from each other and escaped out of the boundary. Figure 6b shows the case of a defect moving out of the planar region, and the opposite charge defect (not shown) moves out of the other side. These dynamics are shown by the steep decrease in energy at times monotonically increasingly in time as α moves closer to 50 µm. For intermediate distances between two defects, 20 µm < α < 80 µm, the defects do not move, or move relatively slow, such that the energy curves of these simulations quickly plateau and remain nearly constant for the duration of the numerical experiment. As shown in Figure 5c these energy curves converge to the same nonzero value, indicating the system has achieved an energetically stable state. The curves suggest a critical value of α c ≈ 20 µm such that for α ≤ α c or α ≥ 100 − α c it is energetically favorable for the system to annihilate the defects, either by moving together or by moving out of the region, to reduce the total energy to zero. Our numerical results are consistent with the theoretical fact that establishes that the distance between two annihilating defects first decreases linearly with respect to time and then it decreases to zero in proportion as the square root of time, [43][44][45] this fact is illustrated in Figure S5, Supporting Information.
Having demonstrated the control over defect creation, transition, and their dynamic motion toward stabilization, we next tested the extent to which the degree of curvature confinement can direct the defects. For simulations we consider a planar stripe region of 100 µm length and sine curved domains with amplitudes of 0, 10, and 20 µm. First a +½ defect was initialized at mid-length of the pattern and the motion of a defect escaping through either end of the planar stripe was captured (Figure 7a-c). Depending on the degree of curvature the dynamic motion of defect is found to be faster for larger degree of curvature, that is, for amplitude of 20 µm (Figure 7a) in comparison to 10 µm (Figure 7b). Whereas for the stripe with 0 amplitude (Figure 7c) the defect remained stationary since any movement to either side will break the symmetry of the system, resulting in a small increase of the energy, and though the system will have a smaller elastic energy if the defect disappears, it will require the system to gain energy to reach the state with less energy. On the other hand, for amplitude 10 and 20 µm due to the fact that the center of the right boundary is the point with higher curvature, any movement to the side will decrease the energy because moving to the side produces a slightly better relation between the director vector and the planar boundary condition. Similar defect behavior was recapitulated in experimental observation for a curved stripe with identical topographic features used previously. A 90 µm long, 7 µm (Ws) wide homeotropic stripe with 40 degrees of curvature and 8 µm (D) planar spacing was used to confine nematic LC. The POM images during cooling from Iso-Nematic phase are as shown in Figure 7d. Around the stripe length median a ±½ defect is formed at the homeotropic/planar interface and with time the defect can be seen to move toward the end of the stripe where it is equilibrated. To test the impact of defect position on defect dynamics the initial defect position in simulation was shifted 10 µm from the stripe median for each case (Figure 7e-h). As a direct consequence, the dynamic motion of the defects toward the stripe end was noted to be present even for stripe with 0 degree of curvature (Figure 7g). At the same time for each degree of curvature confinement the defect motion turned out to be swifter than the previous case. This is due to the symmetric orientation of the defect, in the rectangular region its horizontal component perfectly aligns with the planar boundary condition of the right boundary, such that the decrease of the elastic energy as the defect moves along the boundary is not impacted by the boundary condition, that is, it is only impacted by the configuration in the rest of the domain. While for curvature amplitudes of 10 µm (Figure 7e) and 20 µm (Figure 7f), the defects are never at rest because the orientation of the defect cannot align with the planar boundary condition of the right boundary, so that the only way of drastically reducing the elastic energy is by eliminating the regions  Figure S3, Supporting Information, and close-up images given in Figure S4b-e, Supporting Information, as well as simulated optical images.

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where the orientation of the molecules does not align with the planar boundary requirement. In fact, the higher the curvature of the domain, the higher the discrepancy between the planar boundary condition with the orientation of the defect, resulting in a stronger need of eliminating those discrepancies, that is, the higher the curvature the faster the defects have to be annihilated. The recorded POM images in Figure 7h were consistent with simulation results for the defect positioned at a distance from the mid-length of the stripe. In addition, the time required for the ±½ defect en route to the end of the stripe was revealed to be significantly lower. Thus, confirming the position-dependent nature of defects dynamic motion.
The surface anisotropy plays a key role in manipulating the orientation of LCs, and TD formation, transformation, annihilation, as well as stabilization. Here, both the surface topography and chemical contrast provided through lithography process and LC brush layer/photoresist layer lead to different interactions of LC molecules on the respective surfaces. On the surface with uniformly degenerate planar anchoring, the defects are generated with random distribution and order parameter, which could be either +1 or −1 defect ( Figure S2, Supporting Information). By creating the surface anisotropy, specifically the stripe pattern studied in this work, the type of defects is restricted, the location of defect is controlled, the

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dynamic motion of defect is directed, and the annihilation of defect is manipulated. Thermal expansion or contraction causes volume changes in nematic LC, which can induce flow velocity [46] to rearrange LC molecules in the system. Therefore, thermal fluctuations induced upon cooling down from isotropic phase trigger the defect motion before the system reaches equilibrium. While the relatively weak homeotropic anchoring strength of PMMAZO brush provides a good balance between the flexibility of molecular rotation to accommodate orientation changes and the energy that is conducive to stabilize defects. Therefore, as in Figure 3, the formed +1 defect between two homeotropic stripes is able to move toward the interface and transform into a +½ defect within a few seconds, [3,47,48] as well as when two defects are close enough, they can move toward each other and become annihilated. The geometry design of the long stripe pattern with different D and Ws values has demonstrated that transformation and stabilization of triangle like +½ defects in the middle of stripe pattern is dimension-dependent.

Conclusion
In this study, we fabricated a patterned surface comprising of surface topography and periodic anchoring conditions to manipulate the dynamic motion of formed LC defects and stabilize them. Parameters such as time, width of homeotropic and planar anchoring regions, and periodicity of defect formation, motion, and transition to equilibrium morphology are analyzed. The transition of defect from ±1 to ±1/2 at the interface of planar and homeotropic anchoring represents decomposition in order to release the strain energy of the system is observed. The equilibrium state of this defect can only be reached when the width of the homeotropic anchoring area is larger than 1.0 µm and the pattern period is 9.0 µm. The width of planar anchored region D was observed to have influence on the position of defect morphology along the long edge of the pattern. The annihilation of the defects is observed when two defects are present in close vicinity. From the simulations, the critical distance between defects leading to annihilation was calculated to be 20 µm. Furthermore, impact of the degree of curvature confinement and defect position on the dynamic motion of defect was studied. The pattern-assisted dynamic motion of defects in LC system can further the potential applications that require autonomous motion of defects.

Experimental Section
Numerical Methods: The numerical approach to simulating nematic LCs was based on the penalized Ginzburg-Landau model, [49] which had proved useful to represent point defects. This model uses the director vector, n, to describe the local average orientation of the molecules. The energy functional was defined as n n n ∫ ε ( ) where ε, K > 0 were parameters that balance the competition between the elastic energy and the contribution from the potential function, F(n), which weakly imposed a unit constraint on the length of the director vector and was given by The dynamics of the system were described by the corresponding convective L 2 -gradient flow, n u n δ δ ( ) where u was a velocity field, and δE/δn denoted the variational derivative of the energy with respect to n. The choice of K and ε depended on the scale of the domain and in all simulations K/ε = 10 3 was used.
and 2D sine-curve transformations of the rectangular domain with amplitudes 10 and 20 µm were considered. In all simulations, H = 100 µm and D ≤ 20 µm. The numerical method that was used to compute the solutions was based on combining a linear finite element approximation in space using triangular elements with semiimplicit time stepping algorithms in time. [50] Materials: The thermotropic 5CB (4-pentyl-4-biphenylcarbonitrile) was purchased from Sigma-Aldrich. 5CB revealed nematic phase at room temperature (20 °C) and the temperature of a LC-isotropic phase transition at 35 °C. Heptane, isopropyl alcohol (IPA), dichloromethane (DCM), OTS, and chlorobenzene were purchased from Sigma-Aldrich. PMMAZO was discussed by Stewart and Imrie. [41] Silicon wafers were purchased from Wafer Pro. Glass slides were purchased from Fisher Scientific Surface Modification: Silicon wafers and glass slides were deposited in piranha solution (7:3 of H 2 SO 4 and H 2 O 2 , respectively) at 130 °C for 1 h in order to cleanse surfaces. Silicon wafers and glass slides for OTS modification were anchored by immersing in OTS solution (13.8 µL OTS/120 mL heptane) for 1 h, washed with DCM several times, and quickly dried under a nitrogen flow. For PMMAZO-modified silicon wafers, polymer brush thin films were spin-coated on silicon wafers by an appropriate amount of PMMAZO solution (0.05 wt% PMMAZO/ chlorobenzene) at 4000 rpm for 1 min and then the wafers were annealed at 250 °C in nitrogen environment for 5 min. During annealing, the hydroxyl groups in PMMAZO reacted with the silanol groups of the native oxide, forming a brush layer on the substrate. Afterward, wafers were sonicated in chlorobenzene for 5 min and repeated three times, in order to remove non-grafted PMMAZO. All surfaces were quickly dried by nitrogen after completing the modification processes. A PMMA photoresist was deposited onto PMMAZO-modified wafer to form a 30 nm film and baked at 160 °C temperature for 5 min. The surface was exposed to light beam to form a pattern with arrays of stripes by photolithography. The pattern was developed in the n-amyl acetate for 15 s followed by rinse with IPA.
Cell Preparation: LC cells were prepared with OTS-modified glass slides and patterned wafers. The assembly process was demonstrated by the schematic (Figure 1). LCs were heated to 45 °C on the heating stage, and 3.5 µL of LCs was injected from one side of the cell.
Characterization: Images of CLC cells were obtained by polarization microscopy (BX53 Olympus). Periods of striped pattern and sizes of fingerprint domains were measured by the image processing program (ImageJ). PMMAZO film thicknesses were measured by alpha-SE ellipsometer. Contact angles of the modified wafers were measured by Dataphysics measuring device. Compensator angles were measured by U-CTB Berek compensator (Olympus).

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author. www.advmatinterfaces.de and X.L. also acknowledge support of startup funding from University of North Texas. G.T. and J.S acknowledge support of startup funding from University of North Texas. The authors acknowledge the use of the facility resources provided by the Center for Nanoscale Materials, a U. S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under Contract No. DE-AC02-06CH11357, and University of North Texas' Materials Research Facility. X.L. thanks Rui Zhang for helpful discussions.