Surface anchoring as a control parameter for stabilizing torons, skyrmions, twisted walls, fingers, and their hybrids in chiral nematics

Chiral condensed matter systems, such as liquid crystals and magnets, exhibit a host of spatially localized topological structures that emerge from the medium's tendency to twist and its competition with confinement and field coupling effects. We show that the strength of perpendicular surface boundary conditions can be used to control the structure and topology of solitonic and other localized field configurations. By combining numerical modeling and three-dimensional imaging of the director field, we reveal structural stability diagrams and intertransformation of twisted walls and fingers, torons, and skyrmions and their crystalline organizations upon changing boundary conditions. Our findings provide a recipe for controllably realizing skyrmions, torons, and hybrid solitonic structures possessing features of both of them, which will aid in fundamental explorations and technological uses of such topological solitons. Moreover, we discuss how other material parameters can be used to determine soliton stability and how similar principles can be systematically applied to other liquid crystal solitons and solitons in other material systems.

Figure 1

Skyrmions. (a) Skyrmions in ${\mathbb{R}}^2$ (bottom) can be mapped bijectively from field configurations in ${\mathbb{S}}^2$ (top) through stereographic projections ($\mathcal{P}$). The Neel-type (bottom-left) and Bloch-type (bottom-right) skyrmions are related by a rotation ($\mathcal{R}$) of vectors. The vector orientations are shown as arrows colored according to their orientations on target ${\mathbb{S}}^2$, as shown in the inset. (b), (c) A translationally invariant skyrmion along its symmetry axis and a skyrmion terminating at point defects due to boundary conditions, respectively. The detailed field configurations on spheres around the point defects are shown.

Figure 2

LC torons. (a) The director $\mathit{n}\left(\mathit{r}\right)$ along the average orientation of anisotropic mesogens in nematic LCs can be vectorized by the dispersion of ferromagnetic particles with magnetization vectors $\mathit{m}\left(\mathit{r}\right)$ aligning with $\mathit{n}\left(\mathit{r}\right)$. The orientation of vectorized $\mathit{n}\left(\mathit{r}\right)$ is visualized by a 3D arrow. (b) Polarizing optical micrograph of a LC toron in a cell with thickness $d=10\mu \mathrm{m}$. (c) The field configuration of a toron in the vertical midplane cross-section with the field shown as arrows colored according to the inset in Fig. 1. The point defects are marked by red circles. (d) The preimages of a toron of orientations in target ${\mathbb{S}}^2$ shown as cones. A preimage is the region in ${\mathbb{R}}^3$ that maps to a certain orientation on target space ${\mathbb{S}}^2$. (e) Detailed structures of preimages around the top and bottom point defects of the toron in (d).

Figure 3

Structural stability between 2D skyrmion, toron, and uniform state. (a)–(d) Structural stability diagrams of different material parameters dependent on the LC film thickness and surface anchoring, based on Frank-Oseen free-energy minimization. Blue, red, and black squares indicate stable 2D skyrmion, toron, and uniform state. Filled (unfilled) squares indicate the uniform (TIC) state is the lower energy state, and a yellow (cyan) background indicates the solitons have lower (higher) energies than the uniform unwound state. (e)–(g) Midplane cross-sections of a unit cell of a stable toron lattice in a plane perpendicular (e) and horizontal (f), (g) to the far field. The point defects are marked by red circles. (h) Preimages of points on ${\mathbb{S}}^2$ of an individual toron in the lattice indicated by cones in the top-right inset. (i)–(k) Midplane cross-sections of a unit cell of a stable skyrmion lattice in a plane perpendicular (i) and horizontal (j), (k) to the far field. The point defects are marked by red circles. (l) Preimages of points on ${\mathbb{S}}^2$ of an individual skyrmion in the lattice indicated by cones in the top-right inset.

Figure 4

Escape of point defects under weak anchoring conditions. (a)–(d) Snapshots of the point defects of a toron escaping beyond the surfaces and the toron transforming into a skyrmion when the anchoring condition is softened. Midplane cross-sections in a plane perpendicular (parallel) to the far field are shown in the top (middle) panels. The bottom panels show preimages of each structure of points on ${\mathbb{S}}^2$ indicated by cones in the top-right inset.

Figure 5

2D skyrmion. (a), (b) Experimental polarizing optical micrographs of a 2D skyrmion obtained without (a) and with (b) a 530-nm phase retardation plate with its slow axis labeled by the yellow double arrow. (c) Experimental polarizing optical micrograph of an assembly of 2D skyrmions. (d) Midplane cross-sections of a numerically simulated stable skyrmion in a plane perpendicular (top) and parallel (bottom) to the far field. (e) Numerically simulated 3PEF-PM images obtained with circular polarization of a skyrmion in (d) in the midplanes perpendicular (top) and parallel (bottom) to the far field. (f) Experimental 3PEF-PM images obtained with circular polarization of a skyrmion in (a) and (b) in the midplanes perpendicular (top) and parallel (bottom) to the far field. $d=0.8\mu \mathrm{m}$ and $p=1\mu \mathrm{m}$.

Figure 6

Toron. (a), (b) Experimental polarizing optical micrographs of a toron obtained without (a) and with (b) a 530-nm phase retardation plate with its slow axis labeled by the yellow double arrow. (c) Experimental polarizing optical micrograph of an assembly of multiple torons. (d) Midplane cross-sections of a numerically simulated stable toron in a plane perpendicular (top) and parallel (bottom) to the far field. (e) Numerically simulated 3PEF-PM images obtained with circular polarization of a toron in (d) in the midplanes perpendicular (top) and parallel (bottom) to the far field. (f) Experimental 3PEF-PM images obtained with circular polarization of a toron in (a) and (b) in the midplanes perpendicular (top) and parallel (bottom) to the far field. $d=1.05\mu \mathrm{m}$ and $p=1\mu \mathrm{m}$ .

Figure 7

Skyrmion/toron hybrid. (a), (b) Experimental polarizing optical micrographs of a skyrmion/toron hybrid structure obtained without (a) and with (b) a 530-nm phase retardation plate with its slow axis labeled by the yellow double arrow. (c) Midplane cross-sections of a numerically simulated stable skyrmion/toron hybrid in a plane perpendicular (top) and parallel (bottom) to the far field. (d) Numerically simulated 3PEF-PM images obtained with circular polarization of a skyrmion/toron hybrid in (c) in the midplanes perpendicular (top) and parallel (bottom) to the far field. (e) Experimental 3PEF-PM images obtained with circular polarization of a skyrmion/toron hybrid in (a) and (b) in the midplanes perpendicular (top) and parallel (bottom) to the far field. $d=1\mu \mathrm{m}$ and $p=1\mu \mathrm{m}$.

Figure 8

Twisted wall and CF-3. (a) A LC twisted wall. (b) Schematic of a twisted wall of directors with head-tail symmetry going through a full $\pi$ rotation. (c) The director of the twisted wall in (b) winds around its order-parameter space ${\mathbb{S}}^1/{\mathbb{Z}}_2$ exactly once and ${\mathbb{S}}^1/{\mathbb{Z}}_2\cong {\mathbb{S}}^1$. (d) The twisted wall in (a) with the field vectorized and visualized by arrows. (e) A translationally invariant CF-3 along the direction perpendicular to the cross-section with strong boundary conditions with (top) the 2D director orientations confined in the yz-plane and (bottom) 3D director orientations. The two half integer defect lines near the substrates are visualized by reduction of $\mathrm{S}$ at the defect cores. (f) Detailed field configurations visualized along circular loops of radius of 20 nm around the cores of the top and bottom singular disclinations with (left) 2D director orientations confined in the yz-plane and (right) unconfined 3D director orientations. (g) Structural stability diagram of twisted wall, CF-3, and uniform state dependent on the LC film thickness over pitch ratio and effective anchoring strength. $L$ is the elastic constant related to the average Frank elastic constant as $L=2K/\phantom{2K9S_{\mathrm{eq}}^2}9S_{\mathrm{eq}}^2$. (h) Experimental polarizing optical micrograph of twisted walls in a thin cell with $d=0.8\mu \mathrm{m}$ and $p=1\mu \mathrm{m}$. (i), (j) Experimental 3PEF-PM images obtained with circular polarization of twisted walls in (h) in the midplane perpendicular to the far field (i) and in the plane parallel to the far field (j) through the yellow dashed line in (i). (k) Experimental 3PEF-PM image obtained with circular polarization of a CF-3 in the vertical plane between the arrows in (l). (l) Experimental polarizing optical micrograph of a CF-3. In (k) and (l), $d=10\mu \mathrm{m}$ and $p=12.5\mu \mathrm{m}$.

Figure 9

2D and 3D simulations of soliton structures. (a), (b) 2D simulations of an elementary skyrmion and a high charge skyrmion bag ($|N_{\mathrm{sk}}|=2$) visualized by colors based on the orientation of the vectorized field [Fig. 1, inset]. (c), (d) 3D simulations of an elementary skyrmion and a high charge skyrmion bag, respectively, shown by preimages of points on ${\mathbb{S}}^2$ indicated by cones in the top-right inset and the midplane cross-sections. (e), (f) The midplane cross-sections in (c) and (d) of an elementary skyrmion and a skyrmion bag, respectively.