Motion of a colloidal particle in a nonuniform director field of a nematic liquid crystal

We investigate the dynamics of a single spherical particle immersed in a nematic liquid crystal. A nonuniform director field is imposed on the substrate by a stripe alignment pattern with splay deformation. The particle of homeotropic anchoring at the surface is accompanied by hyperbolic hedgehog or Saturn-ring defects. The particle motion is dependent on the defect structure. We study the two types of motions theoretically and confirm the obtained results experimentally. The particle accompanied by a hyperbolic hedgehog defect is pulled to a deformed region to relax the elastic deformation energy. The motion occurs in the direction heading the hyperbolic hedgehog defect of a particle in a twist region. The position exhibits a weak S-shaped change as a function of time. The particle accompanied by a Saturn-ring defect shows insignificant motion due to its relatively small deformation energy.

Figure 1

(a) Schematic diagram of alignment in the LC cell. The upper substrate $(z=0)$ possesses an alignment pattern, which is alternately repeated in the $x$ and $y$ directions. The lower substrate exhibits uniform alignment ($x$ direction). The pattern of the upper substrate is repeated every $2L_1$ and $2(L_2-L_1)$ in the $y$ direction. The LC director (n) rotates in the x-y plane and is a function of $y$ and $z$. The rotation angle $\varphi (y,z)$ is defined as the angle with respect to the $x$ axis. $d$ is the cell gap. (b) Cell texture. The red line corresponds to the director direction on the surface of the upper substrate, $L_1=60\mu \mathrm{m}$, $L_2=100\mu \mathrm{m}$. The bright region exhibits a ${90}^{\circ }$ twist alignment, while the dark region displays parallel alignment. The cell gap equals $70\mu \mathrm{m}$. (c) and (d) are images of the defects around particles acquired with the polarizer positioned in the $y$ direction, and the analyzer positioned at ${45}^{\circ }$ to the $x$ direction. Both particles displayed in the images are located in the twisted region.

Figure 2

(a) Images of a DC particle moving along the $y$ direction acquired at different time intervals. (b) Graph of the particle position as a function of time. The position at $y=0$ corresponds to the center of the twist region, and that at time $t=0\mathrm{s}$ corresponds to the onset of observation. The bright lines in (a) represent the boundary between the twist and parallel alignment regions. The radius of the particle equals $11.9\mu \mathrm{m}$.

Figure 3

Positional variation of a QC particle as a function of time. (a) Images of a QC particle for different times. (b) Experimental results showing the positional variation of different particles (indicated by different symbols and colors). “1” indicates the particles in (a). “2” and “3” images show the distorted QC particles. The white lines in two small particle images guide the Saturn ring of each particle. The others have clean QC with an expected symmetric Saturn-ring. $y=0$ corresponds to the center of the twist region.

Figure 4

(a) Free energy of interaction $\left({\mathcal{F}}_P\right)$ calculated for a DC particle in different positions in the twist region. The dipole of the DC particle is assumed to be oriented along the $-y$ direction, in line with the experimental result shown in Fig. 2. (b) Force ${\vec{\mathit{F}}}_P\left(y,z\right)$ acting on the DC particle in the twist region. The black arrows indicate the strength and direction of the force. Red solid arrow lines represent the possible paths of a particle in a quasiequilibrium state for different initial positions.

Figure 5

(a) Side views of a particle in the LC cell obtained with a confocal microscope. The left image corresponds to the middle of the twist region, while the right image was acquired near the boundary. The $z$ position of the particle is the farthest from the upper substrate in the middle of the twist region, getting closer upon approaching the boundary. (b) Force acting on the particle (black arrows) and trajectories of particles (red arrow lines) starting from different positions inside the LC cell.

Figure 6

Experimental and calculated particle trajectories. Different experimental trajectories were fitted by adjusting the initial conditions. Time $t=0\mathrm{s}$ corresponds to the onset of observation.