Stick-slip motion of surface point defects prompted by magnetically controlled colloidal-particle dynamics in nematic liquid crystals

We explore the dynamics of topological point defects on surfaces of magnetically responsive colloidal microspheres in a uniformly aligned nematic liquid crystal host. We show that pinning of the liquid crystal director to a particle surface with random nanostructured morphology results in unexpected translational dynamics of both particles and topological point defects on their surfaces when subjected to rotating magnetic fields. We characterize and quantify the “stick-slip” motion of defects as a function of field rotation rates as well as temperature, demonstrating the roles played by the competition of elastic forces, surface anchoring, and magnetic torques on the sphere as well as random-surface-mediated pinning of the easy axis of the nematic director on colloidal microspheres. We analyze our findings through their comparison to similar dynamic processes in other branches of science.

Figure 1

Integrated holonomic magnetic and holographic optical manipulation system. (a) Electromagnetic solenoids arranged on a Cartesian aluminum frame mounted on an inverted microscope (not shown). The solenoids are driven by amplified power supplies via computer-controlled DAQ. The HOT is based on a fiber laser and an optical setup with the following optical elements: polarizer (P), lenses of two telescopes (L1, L2, L3, L4), computer-controlled, dynamically addressable liquid crystal based spatial light modulator (SLM), a 100× oil immersion objective (OBJ), half-wave plate (HWP), polarization rotator (PR), and a dichroic mirror (DM). The trapping beam is focused within the sample volume. This manipulation setup is integrated with an optical imaging system capable of both POM and 3PEF-PM imaging. (b) Two solenoids arranged in the $x-y$ plane aligned with the microscope's focal plane. The direction of the net magnetic force due to magnetic field gradients in this configuration is shown by the arrow.

Figure 2

Superparamagnetic bead surface morphology, induced director structure, “walking” motion, and chain-dimer formation. (a) SEM image of a SPMB showing nanoscale surface roughness, which is responsible for a strong surface anchoring of the NLC director orientation at the surface determined to be tangential but with “memory” (nondegenerate); the color of the SEM image is artificial. (b) Polarizing optical micrograph obtained between crossed polarizers and (c) a schematic showing how the SPMB distorts the nematic director field, creating two surface point defects called “boojums,” shown by hemispheres in (c) and seen as dark regions at the particle poles in (b). (d) Rotating an applied in-plane magnetic field H [brown (gray) CCW curved arrows on left] causes the SPMB to “barrel roll” out of the $x-y$ plane [as shown using blue (gray) arrows on the left-side inset to the left from the optical micrograph], causing a “walking” motion of the SPMB along the $x$ axis [green (gray) arrows on the left], while a constant net magnetic gradient force pulls the SPMB in the $y$ direction. The “barrel-roll” motion is accompanied by a precession of one of the boojums around ${\mathbf{n}}_{\mathbf{0}}$, while the other boojum circumscribes a smaller precessional arc. Switching the magnetic field rotation direction from clockwise to counterclockwise and vice versa reverses the direction of velocity component along the $x$ axis. (e) A schematic of the director structure around an assembly of three SPMB particles with six +1 boojums in the director field at the NLC-particle interface [shown by red (dark gray) hemispheres] and side view of the neck regions marked by blue (gray) solid lines. (f) A schematic of a colloidal dimer defining its center of mass $\left({\mathrm{O}}_{\mathrm{c}.\mathrm{m}.}\right)$, center-to-center separation vector between SPMBs (r_cc) and boojum-to-boojum separation vector r_bb for one of the particles of the dimer. [(g)–(i)] Polarizing optical micrographs obtained between crossed polarizers as the dimer is manipulated magnetically and optically to orient r_cc away from the equilibrium orientation at no fields (g) to be, for example, orthogonal (h) or parallel (i) to ${\mathbf{n}}_{\mathbf{0}}$. [(j)–(m)] Brightfield (j) and POM (k)–(m) micrographs showing magnetic rotation of a magnetically responsive microparticle shaped as a trefoil torus knot, with several of 12 induced boojums marked by yellow (gray) arrows. The images were extracted from the Supplemental movie S1 [29], which allows to see the defect-colloidal dynamics more clearly and in detail. The inset in (j) shows a schematic of the side view of a trefoil knot particle oriented with the torus plane FIG. 2. (Continued) orthogonal to ${\mathbf{n}}_{\mathbf{0}}$ and with 12 boojums (not shown), which reside on the diametrically opposite sides of the knotted polymerized tube on the exterior and interior tip points of the knot along ${\mathbf{n}}_{\mathbf{0}}$. Orientations of polarizer (P), rubbing direction (rub, defining orientation of ${\mathbf{n}}_{\mathbf{0}})$, and analyzer (A) are marked by double arrows on the POM micrographs.

Figure 3

[(a)–(h)] A series of frames extracted from a video and showing rotation of a dimer of SPMBs at an elevated temperature 33.5 °C, right below the nematic-isotropic phase transition temperature of ≈35 °C. Orientations of polarizer (P), rubbing direction (rub) defining orientation of ${\mathbf{n}}_{\mathbf{0}}$, analyzer (A), and the slow axis of a 530-nm wave plate (WP) are marked by double arrows.

Figure 4

SPMB and boojum angular motion with respect to the SPMB dimer center of mass at two different temperatures. A magnetic field is rotated at a slow, constant rate of 0.05 Hz clockwise in the $x-y$ plane, thus rotating a SPMB dimer at the same average rate. Dynamics of SPMBs and boojums is characterized in terms of azimuthal angles θ describing azimuthal orientations of the corresponding separation vectors connecting the dimer's center of mass with these particle centers and boojums, respectively. (a) Colloidal particle and boojum motion at a room temperature. The inset shows a configuration of a SPMB dimer with each SPMB center of mass and each boojum on the polarizing optical micrograph designated as follows for the initial configuration shown in the inset: Left SPMB (S1), right SPMB (S2); each boojum is assigned to its SPMB. For example, S2B2 is the right SPMB, second boojum (lower right boojum). The second POM micrograph without markings shows details of how boojums of each particle position themselves on the microsphere. All colloidal particle and boojum rotational motion versus time are probed with respect to the dimer center of mass as the reference frame; the time dependencies describing rotational motions of defects and particles are labeled at the right side of the plot. Orientations of crossed polarizers (p1 and p2), rubbing direction (rub) defining orientation of ${\mathbf{n}}_{\mathbf{0}}$, and the slow axis of a 530-nm wave plate (wp) are marked by arrows. The thin lines interrupting the dependencies in graphs correspond to the time intervals of rotation when the boojums cannot be localized within the neck regions. (b) Similar colloidal particle and boojum dynamics at an elevated temperature of 33.5 °C, with the arrow indicating slip motion of one of the surface defects.

Figure 5

SPMB dimer rotation and the ensuing “flip-flop” dynamics under the rotating magnetic field. [(a)–(g)] Optical micrographs showing the dimer as viewed in the lateral $x-y$ plane with the rubbing direction along $y$. Shown above each frame is a reference schematics depicting the dimer tilt out of the $x-y$ plane, along with the “flip-flop” indication. The colored (gray) dots distinguish centers of the two SPMBs; the SPMB's split coloring [white and brown (dark gray)] distinguishes the top and bottom hemispheres of the dimer, respectively. (h) Details of the dimer “flip-flop” motion showing a +180° “flip” within its first 180° in-plane rotation, followed by a -180° “flop” during its second 180° in-plane rotation. The coloring of the halves of spheres in the schematics provides insights into the complex rotations of dimers, as seen from different view directions along the axis shown in (a)–(h).

Figure 6

POM micrographs showing details of SPMB dimer rotation [(a) and (b)] with and (c) without “flip-flop” dynamics under magnetic field H rotated in the [(a) and (c)] CW and (b) CCW directions.

Figure 7

SPMB and boojum angular motion with respect to the SPMB dimer's center of mass at room temperature characterized in terms of azimuthal angles θ describing azimuthal orientations of the corresponding separation vectors connecting the dimer's center of mass with these particle centers and boojums, respectively. A magnetic field is rotated at a constant rate of 0.02 Hz clockwise in the $x-y$ plane, thus rotating the SPMB dimer. The inset shows a configuration of the SPMB dimer with each SPMB center of mass and each boojum designated as follows: Left SPMB center (S1), right SPMB center (S2), and each boojum assigned to its SPMB. For example, S2B1 is the right SPMB, second boojum (lower right boojum). The figure shows all particle and boojum rotational motions versus time using the dimer center of mass as the reference frame; the experimental curves are labeled on the right side of the plot. Sharp jumps in the angle versus time curves indicate the “boojum slipping.” The thin lines interrupting the dependencies in graphs correspond to the time intervals of rotation when the boojums cannot be localized within the neck regions.

Figure 8

SPMB and boojum angular motions with respect to the dimer's center of mass at an elevated temperature of 33.5 °C. A magnetic field is rotated at a constant rate of 0.02 Hz clockwise in the $x-y$ plane, thus rotating the SPMB dimer, causing the boojum motion. The inset shows a configuration of the dimer with each SPMB center of mass and each boojum designated as follows: Left SPMB (S1), right SPMB (S2), and each boojum assigned to its SPMB. For example, S2B1 is the top right boojum on the surface of the right SPMB. The figure shows rotation of separation vectors connecting the SPMB dimer's center-of-mass reference frame and the particle centers of mass and boojums described using angle versus time (labeled on the right side of the plot). Boojum slipping is apparent. Sharp angular jumps are “boojum slips.” The thin lines interrupting the dependencies in graphs correspond to the time intervals of rotation when the boojums cannot be localized within the neck regions.

Figure 9

Boojum angular motion with respect to the corresponding SPMB's center of mass at room temperature. Inset image in (a)–(d) indicates the SPMBs boojum that is tracked for a total of four plots. Each line indicates a different angular frequency of magnetic field and dimer rotations: 0.01 Hz (black), 0.02 Hz [red (dark gray)], and 0.05 Hz [green (light gray)]. Note that boojum slipping frequency and amplitude are both dependent on the rotation rate. A representative instance of boojum stick-slip is called out with a black arrow in (a). The thin lines interrupting the dependencies in graphs correspond to the time intervals of rotation when the boojums cannot be localized within the neck regions.

Figure 10

Boojum angular motion with respect to its SPMB's center of mass at an elevated temperature of 33.5 °C. The inset image in (a)–(d) indicates the tracked boojums on a colloidal dimer for a total of four plots. Each line corresponds to a different angular frequency of dimer rotation: 0.01 Hz (black), 0.02 Hz [red (dark gray)], and 0.05 Hz [green (light gray)]. The comparison of these data shows that boojum slipping period and amplitude are dependent on the rotation rate and temperature. A representative instance of boojum stick-slip is called out with a black arrow in (a).

Figure 11

Detailed analysis of a representative single boojum “slip” event using a high-speed camera operating at 500 fps. (a) Boojum's angular position change θ versus time due slipping motion with respect to the dimer's center of mass, with a sigmoidal fit [blue (gray) line] yielding the time constant ${\tau }_s\sim 0.0036\mathrm{s}$ (fitting with $R^2=0.995)$. (b) Boojum angular slipping motion ${\theta }_n$ with respect to the colloidal dimer's center of mass after the constant change of angle with respect to time subtracted and exhibiting a sigmoidal [red (dark gray) line] fit of ${\tau }_s\sim 0.0034\mathrm{s}$ (fitting with $R^2=0.987)$ and exponential [green (light gray) line] ${\tau }_e\sim 0.0012\mathrm{s}$ (fitting with $R^2=0.946)$.

Figure 12

Displacement of boojums $l$ along the SPMB surface as a function of the angle $\varphi$ between ${\mathbf{r}}_{\mathbf{cc}}$ and ${\mathbf{n}}_{\mathbf{0}}$ at (a) 0.01 Hz and (b) 0.02 Hz rotation rates of magnetic field. The top-right inset in (a) shows labeling of boojums on the two spheres of a colloidal dimer. The bottom insets depict changes of positions of encircled boojums as a result of slip events indicated by arrows.