Colloid Particles in the Interaction Field of a Disclination Line in a Nematic Phase

On the basis of linear hydrodynamics, we analyze the trajectory of particle-hedgehog systems, attracted by a $-1/2$ disclination (defect line) in a nematic liquid crystal. We show that, as with the interactions between like-particles, the interaction between a particle and a disclination has an electrostatic analogue, the splay replacing the electric field, except for the symmetry properties. The disclination thus attracts the beads along nonradial tracks and in a self-assembling process, or template mechanism, may build a microscopic necklace with them.

Figure 1

(a) Side view of a nematic sample with antagonistic rubbing conditions, parallel to $\mathbf{x}$ in the $y<0$ region, parallel to $\mathbf{y}$ in the $y>0$ one. A wedge disclination line is thus forced along the $\mathbf{z}$ axis. (b) Nematic field in the plane $z=0$ in the case of a $-1/2$ line. (c)–(e) Photographs (width $50\mu \mathrm{m}$) taken before the bead is captured by the disclination line at times $t=-240\mathrm{s}$, $-56\mathrm{s}$, $-2\mathrm{s}$, respectively. The particle is indicated with a grey (red) arrow, and the line by a white (yellow) one. Note that the line is slightly distorted by the attraction exerted by the particle.

Figure 2

Tracks of different particles attracted by a $-1/2$ disclination line. The continuous thin lines indicate the trajectories calculated for $r_0=200$, 60, 45, $19\mu \mathrm{m}$, respectively. Inset: Mean square distance, $⟨\Delta Y^2(\tau )⟩$, after a time $\tau$ along the $\mathbf{Y}$-axis in the average moving frame of the particle (track with same color (blue) open dots in Fig. 2). At short time scales, the motion is diffusive. We deduce the corresponding bead radius $R=0.55\mu \mathrm{m}$.

Figure 3

Log-log plot of $\xi$ as a function of the distance $r$ of the particle to the line. The fit of a power law, performed over almost one decade, yields an exponent $n=1.92$ consistent with the electrostatic model. Inset: The disclination line is able to attract a large number of particles and finally to form a necklace. The particles in the necklace are in solid contact to one another, as its broken shape demonstrates. For a good focusing, the line here is a loop located in the horizontal plane.