Annihilation dynamics of topological monopoles on a fiber in nematic liquid crystals

We use the laser tweezers to create isolated pairs of topological point defects in a form of radial and hyperbolic hedgehogs, located close and attracted to a thin fiber with perpendicular surface orientation of nematic liquid crystal molecules in a thin planar nematic cell. We study the time evolution of the interaction between the two monopoles by monitoring their movement and reconstructing their trajectories and velocities. We find that there is a crossover in the pair interaction force between the radial and hyperbolic hedgehog. At small separation $d$, the elastic force between the opposite monopoles results in an increase of the attractive force with respect to the far field, and their relative velocity $v$ scales as a $v\left(d\right)\propto d^{-2\pm 0.2}$ power law. At large separations, the two oppositely charged monopoles can either attract or repel with constant interaction force. We explain this strange far-field behavior by the experimental inaccuracy in setting the fiber exactly perpendicular to the cell director.

Figure 1

(a) The microfiber is oriented perpendicularly to the nematic director in the planar cell, and a gigantic Saturn ring is spontaneously created that encircles the fiber along the long axis. Note that the ring is not straight but forms zigzag sections, because it is pinned to the surface impurities at different positions. This ring is cut by the laser tweezers into two separated rings with locally opposite winding numbers. The smooth region in between is called the “topological soliton.” (b) When the ring on the right (assigned the +1/2 winding number) is cut again with the tweezers, a +1/2 loop is created, which shrinks into the +1 point defect. A $-1/2$ winding loop is left on the right side. (c) The same procedure of cutting is performed on the loop on the right side of this +1 defect. This creates a $-1/2$ winding number loop, which shrinks into a $-1$ point defect. [(d) and (e)] The charges of these point defects are tested by small dipolar colloidal particles. They bind to oppositely charged point defects in opposite orientation of their topological dipole.

Figure 2

Sequence of point monopoles with alternating charge. (a) A sequence of point monopoles are created on a fiber by cutting the gigantic Saturn-ring with laser tweezers. (b) The charge of monopoles are demonstrated by dipole.

Figure 3

Annihilation of point monopoles on a fiber from large starting separation. (a) The time sequence of images showing the annihilation of point monopoles. (b) The position of the $+1$ and $-1$ point monopoles as a function of time.

Figure 4

Annihilation dynamics of two point monopoles, released from a large separation of $\approx 100 \mu \mathrm{m}$ in a cell of 20-$\mu \mathrm{m}$ thickness. (a) The optical micrographs of point monopoles at starting and ending position just before annihilating each other. (b) Relative velocity of the monopoles versus their separation. At large separation, the monopoles are approaching each other with a constant velocity of $v_0=2.1$ $\mu \mathrm{m}$/s. This means that the attractive force is constant. At small separations, the velocity and the attracting force show a power-law behavior. The red line shows power-law fit of the velocity as a function of separation $v\left(d\right)=v_0+d^{-\alpha }$, with $\alpha =2$. The inset shows the power-law fit of the velocity as a function of separation, when the background velocity $v_0$ is subtracted, yielding $\alpha =2.0\pm 0.2$.

Figure 5

Schematic representation of the director around the fiber at different angles with respect to the far-field nematic director. (a) The fiber is set perpendicularly to the nematic director. The monopoles are stable in the far-field region, because the force to the left is balanced with the force to the right. (b) The fiber is rotated counterclockwise. The monopoles start to move towards each other with constant velocity to diminish the energetically costly region. (c) The fiber is rotated clockwise. The monopoles are moving away from each to reduce the region with high elastic energy.

Figure 6

Attraction and repulsion of oppositely charged topological monopoles on a fiber, depending on the sign of rotation of the fiber with respect to the cell director. (a) Attraction of the monopoles on the fiber, which was rotated counterclockwise. (b) The two oppositely charged monopoles repel each other when the fiber is rotated clockwise with respect to the cell director. The cell thickness is 20 $\mu \mathrm{m}$. (c) The relative velocity of the monopoles versus their separation in case (a) has a constant value in the far field. It increases at smaller separation of the two monopoles, where it shows a power-law dependence. The red line shows power-law fit of the velocity as a function of separation $v\left(d\right)=v_0+d^{-\alpha }$, with $v_0=10$ $\mu \mathrm{m}$/s and $\alpha =2$. The inset shows the power-law fit of the velocity as a function of separation, when the background velocity $v_0$ is subtracted, yielding $\alpha =2.0\pm 0.2$.

Figure 7

Attraction and repulsion of opposite monopoles on a fiber as a function of the angle of rotation of the fiber with respect to the cell director. Negative velocity denotes repulsion, and positive velocity denotes attraction. The experiments were performed in a cell of 20-$\mu \mathrm{m}$ thickness.

Figure 8

Ratio of the velocities of the $+1$ and $-1$ point monopole during their annihilation on a fiber in a cell of 20-$\mu \mathrm{m}$ thickness. The ratios are plotted for three different samples. The average value of $v_{+}/v_{-}\approx 1.2$.