Interactions of charged microrods in chiral nematic liquid crystals

We study the pair interaction of charged silica microrods in chiral nematic liquid crystals and show that the microrods with homeotropic surface anchoring form a bound state due to the competing effect of electrostatic (Coulomb) and elastic interactions. The robustness of the bound state is demonstrated by applying external electrical and mechanical forces that perturbs their equilibrium position as well as orientation. In the bound state we have measured the correlated thermal fluctuations of the position, using two-particle cross-correlation spectroscopy that uncovers their hydrodynamic interaction. These findings reveal unexplored aspects of liquid-crystal dispersions which are important for understanding the assembly and dynamics of nano- and microparticles in chiral nematic liquid crystals.

Figure 1

(a) Field emission scanning electron microscope (FESEM) image of a silica microrod. (b) Director twist in $\pi$- and $2\pi$-twisted nematic cells. (c) Percentage of microrods oriented parallel ($\varphi =0^{\circ }$) and perpendicular ($\varphi ={90}^{\circ }$) to the rubbing direction in a $2\pi$-twisted cell. (d) Polarizing optical microscope images of two interacting microrods with elapsed time (Movie S1) [38]. Horizontal arrow indicates the rubbing direction. P, A indicate polarizer and analyzer.

Figure 2

(a) Time dependence of tip-to-tip separation of two interacting microrods in a $2\pi$-twisted cell (Movie S1) [38]. The solid red curve displays the least square fit to the equation $R\left(t\right)={(R_0^5-5{\alpha }_{d}t)}^{1/5}$ for dipolar type interaction. (b) Time dependence of tip-to-tip separation in a planar 5CB cell (without chiral dopant). The solid blue curve displays the least square fit to the equation $R\left(t\right)={(R_0^7-7{\alpha }_{q}t)}^{1/7}$ for quadrupolar interaction. The insets show respective interaction potentials. The dashed blue curve in (a) and dotted red curve in (b) display fits to quadrupolar and dipolar interactions, respectively.

Figure 3

(a) CCD images with elapsed time showing interacting microrods in a side-to-side configuration. (b) Time dependence of the microrods' separation in side-to-side configuration. The solid red curve displays the best fit to equation $R\left(t\right)={(R_0^5-5{\alpha }_{d}t)}^{1/5}$ for dipolar interaction. The inset shows the interaction potential. The dashed blue curve display fits to a quadrupolar interaction. (c) Time coded trajectories (relative coordinates) of microrods approaching from different starting positions in a $2\pi$-twisted cell. By “relative coordinate” we mean relative to the starting point of each trajectory. Grey shaded elliptical regions indicate where the approaching particle is bound with the central particle.

Figure 4

(a) Mean squared displacements (MSDs) of a free microrod (red squares and circles) and a bound microrod (green triangles and stars) in a $2\pi$-twisted cell. The subscripts $F$ and $B$ stand for free and bound microrods, respectively and the superscripts $\left|\right|$ and $\perp$ indicate motions parallel and perpendicular to the rubbing direction. (b) Confining potentials of the bound microrods. Separation along the $y$ direction is normalized.

Figure 5

(a) Diagram of the cell for the electric field experiments. The direction of the electric field between the ITO plates is perpendicular to the director. (b) CCD images with elapsed time showing the effect of ac electric field ($f=1\mathrm{kHz}$) on the bound state in a $2\pi$-twisted cell (Movie S3) [38]. (c) Variation of tip-to-tip separation with increasing (circles) and decreasing (squares) electric field. The vertical arrow indicates the Freedericksz threshold field ($E_{\mathrm{th}}$).

Figure 6

(a) CCD images with elapsed time showing the effect of elastic distortion created by the laser tweezers on the bound state in a $2\pi$-twisted cell (Movie S4) [38]. (b) A linear chain of ten microrods forming successive bound states. They were guided to assemble with the help of the laser tweezers setup.

Figure 7

(a) Longitudinal autocorrelation $\langle R_1\left(t\right)R_1\left(0\right)\rangle$ (blue circles) and cross-correlation $\langle R_1\left(t\right)R_2\left(0\right)\rangle$ (green diamonds and red stars) functions of the microrods in $\pi$- and $2\pi$-twisted cells. Solid lines represent the least square fits to Eqs. (1) and (2) for auto- and cross-correlation functions. (b) Autocorrelation function $\langle {\theta }_1\left(t\right){\theta }_1\left(0\right)\rangle$ (solid circles) and cross-correlation function $\langle {\theta }_1\left(t\right){\theta }_2\left(0\right)\rangle$ of orientational fluctuations (open circles) of two microrods in a $2\pi$-twisted cell. The solid line indicates the least square fit to $\langle {\theta }_1\left(t\right){\theta }_1\left(0\right)\rangle$. Only every third of the experimental data points are shown. The inset shows the histogram of orientational deviation.

Figure 8

(a) Tip-to-tip separation between a pair of microrods in a $2\pi$-twisted cell with time (in semilogarithmic scale). The data are presented here for about 10 minutes but the observation was made for 3 to 4 h to confirm that the bound state is stable. (b) Tip-to-tip separation in $\pi$- (red circles) and $4\pi$-twisted (green squares) cells. The inset shows the tip-to-tip separation with chirality.

Figure 9

Surface-to-surface separation of two spherical DMOAP coated silica microspheres of diameter 3 $\mu \mathrm{m}$ in $\pi$- (red circles) and $2\pi$-twisted (green squares) cells. The ratios of the diameter to pitch are $D/p=1/2$ and 1 for $\pi$- and $2\pi$-twisted cells, respectively. Note that eventually the particles are attracted to join.

Figure 10

Measurement of total charge $Q$ of DMOAP coated microrods in a chiral NLC. (a) Schematic diagram of the cell used in the experiment. (b) Variation of displacement of the particle position with time when a dc field of 10 V is applied between the copper (Cu) tapes, separated by 1.0 mm distance. The solid red line is a linear fit to the data. (c) CCD images showing the motion of a microrod under the field. Note that the microrod moves from the negative to the positive electrode, indicating it is negatively charged.