Dynamics of topological solitons, knotted streamlines, and transport of cargo in liquid crystals

Active colloids and liquid crystals are capable of locally converting the macroscopically supplied energy into directional motion and promise a host of new applications, ranging from drug delivery to cargo transport at the mesoscale. Here we uncover how topological solitons in liquid crystals can locally transform electric energy to translational motion and allow for the transport of cargo along directions dependent on frequency of the applied electric field. By combining polarized optical video microscopy and numerical modeling that reproduces both the equilibrium structures of solitons and their temporal evolution in applied fields, we uncover the physical underpinnings behind this reconfigurable motion and study how it depends on the structure and topology of solitons. We show that, unexpectedly, the directional motion of solitons with and without the cargo arises mainly from the asymmetry in rotational dynamics of molecular ordering in liquid crystal rather than from the asymmetry of fluid flows, as in conventional active soft matter systems.

Figure 1

RBF method for defining nodes. (a) Random points scattered in two dimensions. (b) A Gaussian basis set illustrated by centered Gaussian surfaces at the location of each node in (a). Colors represent the relative height of the surfaces. (c) A surface that passes through the points constructed from a unique linear combination of the Gaussians from (b). (d) Periodic square lattice, (e) face-centered cubic lattice, (f) tetrahedral mesh, and (g) Halton-type scattered node distributions in three dimensions. Computational volumes are shown in dimensionless units along the axes solely to illustrate the differences in the spatial distribution of nodes within the computational grids.

Figure 2

Structure of a toron with the topology of an elementary skyrmion and knotted streamlines. A skyrmion embedded in a confined chiral nematic LC with (a)–(d), (g)–(k) no applied field, $U=0\mathrm{V}$, with uniform far-field director ${\mathbf{n}}_0=\left\{0,0,1\right\}$ and (e) and (f) applied field, $U=3.1\mathrm{V}$, with uniform far-field director ${\mathbf{n}}_0=\left\{0,1,0\right\}$. The direct comparison shows agreement of RBF-FD simulated 2D cross-sectional midplanes in (a) x-y and (b) x-z of $\mathbf{n}\left(\mathbf{r}\right)$ with the (c) traditional FD simulated 2D x-y cross-sectional midplane of $\mathbf{n}\left(\mathbf{r}\right)$ using arrows colored according to the corresponding points on ${\mathbb{S}}^2$ (inset). The orientation of ${\mathbf{n}}_0$ on ${\mathbb{S}}^2$ is denoted using cones. (d) Corresponding 3D representation of $\mathbf{n}\left(\mathbf{r}\right)$ described in part (c) by means of isosurfaces representing the $z$ component of the director, $n_z=0.95$ (blue, outer), $n_z=-0.2$ (red, inner), regions around singularities (gray, top/bottom) and an x-z cross section of the director through both point defects visualized using black line segments. (e) 2D x-y cross-sectional midplane of a vectorized skyrmion with applied field. (f) Corresponding 3D representation of $\mathbf{n}\left(\mathbf{r}\right)$ described in part (e) by means of isosurfaces described in (c). The structure of TIC in which the skyrmion is embedded is labeled in the green rectangle. A zoomed-in view at the x-y cross section through point defects of skyrmions, confined in a chiral nematic LC with no applied field, (g) above and (h) below the skyrmion tube and x-z cross section through point defects (i) above and (j) below the skyrmion. Additional arrows in (i), (j) denote locations of (g), (h) cross sections. (k) Streamlines tangent to director within the twisted region of a toron (magenta surface, outer) that trace a Hopf link (black), pentafoil knot (cyan, dark gray in black and white images), and quatrefoil knot (yellow, light gray in black and white images). All numerical results displayed in this figure are based on the physical parameters of a chiral mixture of MLC-6609 and ZLI-811.

Figure 3

Torus knots and electric switching of director streamlines. (a)–(d) Streamlines tangent to the director field lines at various applied voltages. The streamlines originate from the cell midplane and propagate along the director field until they terminate on a substrate, close on themselves to form knots or become longer than $200\mu \mathrm{m}$. The color bar indicates lengths of continuous streamlines, with blue (dark gray in black and white images) representing the short streamlines that crash to the substrates quickly $(\sim 10\mu \mathrm{m})$ and red (lighter gray in black and white images) streamlines loop around the torus many times and often form a {$p$, $q$} torus knots $(\ge 200\mu \mathrm{m})$. (e) Contour length, $S$, over the winding number, $q$, of some of the torus knots that are found for $U<1\mathrm{V}$, plotted as a function of applied voltage. The top left inset shows examples of the tracked torus knots and unknots at $U=0.8\mathrm{V}$. The top purple curve (circles) is the circumference of the $\varphi =\pi /2{\mathrm{C}}_{\infty }$ axis. Crosses, asterisks, squares, and triangles mark length of the Hopf unknot, pentafoil knot, quatrefoil knot, and trefoil knot, respectively. The torus knots tracked each have insets representing their topology and $\{p,q\}$ winding numbers, with the corresponding voltage dependencies of the contour lengths indicated. (f) Representative torus surfaces between $\varphi =\pi /4$ and $\varphi =\pi /2$ near to where the respective torus knots and unknots are found including the $\varphi =\pi /2{\mathrm{C}}_{\infty }$ axis. (g) Rectangles schematically representing the unwrapped torus surfaces shown in (f) with the same colors, where each torus is shown $p·q$ times to indicate how many times and how the director streamline slides on the torus surface before looping on itself. The thin black lines indicate the director streamlines that loop around the two axes of the torus to form various torus knots. In this figure the numerical results displayed are based on the physical parameters of a mixture of MLC-6609 and ZLI-811.

Figure 4

Structure of a laterally stretched skyrmion at no fields. (a) 3D and (b)–(j) 2D representations of the stretched skyrmion structure embedded within the uniform far-field director ${\mathbf{n}}_0=\left\{0,0,1\right\}$. (a) The 3D structure is visualized using the Pontryagin-Thom construction colored by the azimuthal orientation of $\mathbf{n}\left(\mathbf{r}\right)$ (as displayed in the upper right), small circular gray isosurfaces representing regions around singularities, and an x-z cross section of the director through the skyrmion visualized using black line segments. 2D x-y cross-sectional planes through the (b) top, (c) midplane, and (d) bottom of vectorized $\mathbf{n}\left(\mathbf{r}\right)$ using arrows colored according to the corresponding points on ${\mathbb{S}}^2$ (inset) and the orientation of ${\mathbf{n}}_0$ on ${\mathbb{S}}^2$ denoted using a cone. The black stars represent the point singularities (corresponding to the light gray cross-sectional surfaces in part a). (e) A y-z cross section along the length of the structure. (f) An x-z cross section through the midplane of the structure. Additionally, the x-y cross sections through point defects (denoted with black stars) of the structure (g) near the top substrate and (h) near the bottom substrate and x-z cross section through point defects (i) near the top and (j) near the bottom of the structure. All the numerical results displayed in this figure are based on the physical parameters of a mixture of MLC-6609 and ZLI-811.

Figure 5

Structure of a laterally stretched skyrmion in an applied field. (a) 3D and (b)–(f) 2D representations of a stretched skyrmion structure at $U=4\mathrm{V}$. (a) Isosurfaces representing the $z$ component of the director $n_z=-0.2$ (red, gray) and regions around singularities (small circular light gray surface) and x-z cross section of the director through the skyrmion visualized using black line segments. 2D x-y cross-sectional planes passing through the (b) top, (c) midplane, and (d) bottom of vectorized $\mathbf{n}\left(\mathbf{r}\right)$-configurations using arrows colored according to the corresponding points on ${\mathbb{S}}^2$ (inset) and the orientation of ${\mathbf{n}}_0$ on ${\mathbb{S}}^2$ denoted using a cone. The black stars represent the point singularities (corresponding to the gray surfaces in part a). 2D x-z cross sections of the extended structure through (e) the skyrmion and (f) one of the hyperbolic point defects near the top substrate. Additional arrows in (f) denote locations of the (b)–(d) cross sections. In this figure, all numerical results are based on the physical parameters of a mixture of MLC-6609 and ZLI-811.

Figure 6

Formation of skyrmions and synchronization of south-pole to north-pole preimage vectors. (a) POM images showing gradual self-alignment of preimage vectors within the TIC. Trajectories of south-pole (red, light gray in black and white images) and north-pole (blue, darker gray in black and white images) preimages after applying voltage are overlaid on the last frame. The elapsed time is indicated on the POM video frames. (b) An angle characterizing the azimuthal orientations of the preimage dipoles versus time for the five solitons labeled in (a). (c)–(e) POM images showing interaction of the north-pole and south-pole preimages after they were pulled apart by laser tweezers. Crossed polarizers in POM images are marked with white double arrows. North-pole and south-pole preimage separation (f) for many applied voltages, represented by a surface colored according to the magnitude of separation, and (g) for the minimum and maximum voltages applied. The points corresponding to the images in (c)–(e) are marked by red dots on the $U=4.1\mathrm{V}$ curve. The experimental results presented in this figure are based on a chiral mixture of MLC-6609 and ZLI-811 where thickness and pitch are about $10\mu \mathrm{m}$.

Figure 7

Dynamics of skyrmion's preimages. (a) North-pole and south-pole preimage velocities during skyrmion formation, where the surface is colored according to the velocity magnitude. (b) Position and (c) velocity of the preimages during attraction for the two extreme voltages. (d), (e) Logarithmic representations of the preimage velocities and the velocity speed anisotropy during attraction at (d) $U=3.5\mathrm{V}$ and (e) $U=4.1\mathrm{V}$. In this figure, the experimental results are all based on a mixture of MLC-6609 and ZLI-811 where cell thickness and pitch are about $10\mu \mathrm{m}$.

Figure 8

Anisotropy of north-pole and south-pole preimage velocities. (a) Speed anisotropies of the preimages, where the surface is colored according to magnitude of speed anisotropy. (b), (c) Voltage dependence of (b) the maximum speed anisotropy and (c) the north-pole and south-pole preimage separation measured at the maximum speed anisotropy. All experimental results presented in this figure are based on a chiral mixture of MLC-6609 and ZLI-811 where cell thickness and pitch are about $10\mu \mathrm{m}$.

Figure 9

Translational motion of a stretched skyrmion structure. (a) Voltage profile of the square waveform used for modulation, where $f_{\mathrm{m}}=0.5\mathrm{Hz}$. The black arrows indicate (b) corresponding POM images of a stretched skyrmion structure in the sample (top). Translational motion of the structure is shown over many modulation cycles (bottom). White double arrows indicate the polarizer orientation. (c), (d) Intensity profiles of the extended structure within one period of voltage modulation, extracted from POM images shown in (b). (e) Periodic change in width, $w$, as defined in (b), of the structure, where voltage modulation starts at 0 s. Green (dotted) and red (dashed) lines indicate where $U=2\mathrm{V}$ was turned on and off. (f) Lateral displacement of the structure over time. (g) Intensity profiles corresponding to the initial (blue, dark gray) and final (orange, lighter gray in black and white images) positions of the structure extracted from the corresponding POM images. All experimental results presented in this figure are based on a chiral mixture of E7 and CB-15 where thickness and pitch are about $30\mu \mathrm{m}$.

Figure 10

Translational motion of a stretched skyrmion structure. The thickness and pitch are close to $10\mu \mathrm{m}$. The applied voltage is always $U=4.2\mathrm{V}$, which is modulated at $f_{\mathrm{m}}=2\mathrm{Hz}$ where noted in top left corner and not modulated otherwise. The polarizer orientations are marked with white double arrows and elapsed time is noted in the bottom right corner corners of images. In this figure, the experimental results are based on a chiral mixture of MLC-6609 and ZLI-811 where thickness and pitch are about $10\mu \mathrm{m}$.

Figure 11

Directional motion of torons. (a) Experimental and (b) RBF-FD simulated skyrmion position when voltage modulation is turned on (positive curve) and off (negative curve). The dotted line in (a) represents where the magnitudes of these opposite displacements are equal. The inset shows a soliton with both the south-pole to north-pole vector and the direction of motion along the $y$ axis. (c), (d) Experimental skyrmion velocity (c) with nonzero fill ratio, $U_1/U_2$, and (d) with $U_1/U_2=0$. The corresponding square waveform amplitude modulation schemes are shown in the inset. The chiral mixture is MLC-6609 and ZLI-811 where thickness and pitch are about $10\mu \mathrm{m}$.

Figure 12

Analysis of presence of flows in LC cells at amplitude-modulated applied fields. (a) Surface Plasmon Resonance (SPR) spectrum of the GNR nanoparticles used as tracers to detect possible presence of flow and TEM image (inset). (b) Depth-resolved backflow displacement amplitude corresponding to Fourier component at 1 Hz in a $10\mu \mathrm{m}$ thick cell, $U=5\mathrm{V}$, $f_{\mathrm{c}}=1\mathrm{kHz}$, $f_{\mathrm{m}}=1\mathrm{Hz}$, and $50\%$ duty cycle and characteristic dark-field image of 7 GNRs at a depth of 4 μm (inset). (c) Fourier analysis of GNR motion for $U=0-8\mathrm{V}$, $f_{\mathrm{c}}=1\mathrm{kHz}$, $f_{\mathrm{m}}=1\mathrm{Hz}$, and $50\%$ duty cycle. Insets show displacement trajectories obtained for 15 cycles of amplitude modulation and the point cloud of nanoparticle positions for each trajectory offset in the $y$ direction for clarity. These experimental results presented in this figure are based on a mixture of MLC-6609 and ZLI-811 with thickness and pitch of about $60\mu \mathrm{m}$.

Figure 13

Analysis of skyrmion translation past tracer particles. (a) Fluorescence image of nanocubes in the midplane cross section of a skyrmion's north-pole preimage in a 60-μm-thick cell. (b) POM image of the north-pole preimage and its vicinity. (c) Time-coded trajectories of nanocubes during amplitude modulation for $U=0-8\mathrm{V}$, $f_{\mathrm{c}}=1\mathrm{kHz}$, $f_{\mathrm{m}}=2\mathrm{Hz}$, and $50\%$ duty cycle where the north pole traversed through nanocubes in the bulk LC (black, bottom right to top left). (d) Fluorescence images of nanocubes in the midplane cross section of the north- and south-pole preimages of a skyrmion in a 60-μm thick cell. (e), (f) POM images of the soliton (e) before and (f) after applying voltage. (g) Time-coded trajectories of nanocubes as the skyrmion traverses through (black line, top left to bottom right). Elapsed time is given in seconds. The experimental results presented in this figure are based on a chiral mixture of MLC-6609 and ZLI-811 with thickness and pitch of about $60\mu \mathrm{m}$.

Figure 14

Translational motion of soliton with and without entrapped melanin-resin microparticle cargo. (a) Bright-field microscopy of a skyrmion (top) and a skyrmion with cargo (bottom). (b) POM images of a skyrmion (left) and a skyrmion with cargo (right) during voltage modulation at $f_{\mathrm{m}}=2\mathrm{Hz}$ and $75\%$ duty cycle. (c) Corresponding displacement of the skyrmion (red, gray in black and white images), skyrmion plus trapped cargo (black), and an untrapped GNR (blue, darker gray in black and white images). (d) POM images of a skyrmion transporting cargo during voltage modulation at $f_{\mathrm{m}}=8\mathrm{Hz}$ (top row) and $f_{\mathrm{m}}=2\mathrm{Hz}$ (bottom row). (e) Corresponding displacement of the skyrmion along $y$. Polarizer orientations are marked with white double arrows on the first frame and the elapsed time stamp is given at bottom-right of each POM frame. In this figure, experimental results are based on a mixture of MLC-6609 and ZLI-811 where thickness and pitch are about $10\mu \mathrm{m}$.