Annihilation trajectory of defects in smectic-$C$ films

In a two-dimensional liquid crystal, each topological defect has a topological charge and a characteristic orientation and hence can be regarded as an oriented particle. Theories predict that the trajectories of annihilating defects depend on their relative orientation. Recently, these predictions have been tested in experiments on smectic-$C$ films. Those experiments find curved trajectories that are similar to the predictions, but the detailed relationship between the defect orientations and the far-field director is different. To understand this difference, we extend the previous theories by adding the effects of elastic anisotropy and find that it significantly changes the curved trajectories.

Figure 1

Structure of $\pm 1$ defects with different values of ${\theta }_0$. (a) $+1$ defect, ${\theta }_0=0$, $k_S<k_B$. (b) $+1$ defect, ${\theta }_0=\pi /2$, $k_S>k_B$. (c) $-1$ defect, ${\theta }_0=0$, $k_S<k_B$. (d) $-1$ defect, ${\theta }_0=\pi /2$, $k_S>k_B$. The red arrows indicate the initial conditions for the director field, before the defects begin to move. The background colors indicate the local ratio $F_B/(F_B+F_S)$. White areas are mostly splay, while black areas are mostly bend.

Figure 2

Velocity of defects with different topological charges and different ${\theta }_0$ angles as functions of the Frank constants. (a) Varying $k_S$ with fixed $k_B=1$. (b) Varying $k_B$ with fixed $k_S=1$.

Figure 3

Velocity of defects with different topological charges when one Frank constant increases while the other decreases, so that the sum is constant. When $k_S<k_B$, we simulate a $+1$ test defect or boundary defect with pure splay near the core. When $k_S>k_B$, we simulate a $+1$ test defect or boundary defect with pure bend near the core.

Figure 4

Initial conditions for defect annihilation simulations. (a) Optimal relative orientation with $\delta \theta =0$ and $\mathrm{\Theta }=\pi /4$. (b) Misalignment with $\delta \theta =\pi /2$ and $\mathrm{\Theta }=0$. In both cases, the splay area (white) is below the $+1$ defect, and the bend area (black) is above it.

Figure 5

Defect annihilation trajectories for different ratios of Frank constants, with no defect misalignment in the initial conditions.

Figure 6

Defect annihilation trajectories for different ratios of Frank constants, with an initial defect misalignment of $\delta \theta =\pi /2$.