Hydrodynamics of Pair-Annihilating Disclinations in $\mathrm{S}\mathrm{m}C$ Films

The pair annihilation of smectic $c$-director defects with winding numbers $\pm 1$ in a freestanding $\mathrm{S}\mathrm{m}C$ film as a representative of the $XY$ model is studied numerically, considering a full coupling of orientational degrees of freedom and hydrodynamics. A reduction of the annihilation time compared to the nonhydrodynamic treatment is observed. It is demonstrated that the $+1$ disclination moves considerably faster than the $-1$ one primarily due to hydrodynamic flow, weakly assisted also by elastic anisotropy. The stress tensor terms and material parameters relevant for this effect are identified.

Figure 1

Left: radial-hyperbolic pair of annihilating $\pm 1$ disclinations in $\mathbf{c}$ field (the grid has been coarsened for clarity; only the central region of the mesh is shown). Vector heads have been omitted for legibility; yet there is no ambiguity, since in the absence of any external fields the system is invariant under a global transformation $\mathbf{c}\to -\mathbf{c}$. Right: flow velocity field (same region as on the left diagram, coarser still). The $+1$ defect is subject to strong advection, as opposed to the $-1$ one.

Figure 2

Velocity of the defects (relative to $\xi /\tau \approx 27\mathrm{n}\mathrm{m}/\mu \mathrm{s}$) vs interdefect distance, one elastic constant. The $+1$ defect is strongly sped up by the flow, the $-1$ is slightly slowed down (there the flow is opposite to its motion, Fig. 1). Starting with an equilibrium configuration of fixed defects, a nonmonotonic behavior is observed (dashed line).

Figure 3

Position of the defects vs time, measured from the initial middle point between the defects. The cases with one elastic constant: (a) RH (Fig. 1) and (b) TH defect pair, (c) the case without hydrodynamics, (d) RH pair with the ${\gamma }_1$ coefficient doubled. Combined effect of flow and elastic anisotropy, with the average elastic constant $B_0$ fixed: (e) $B_1/B_2=5$ (RH), (f) $B_2/B_1=5$ (TH). Length and time are scaled by $\xi \approx 2.4\mathrm{n}\mathrm{m}$ and $\tau \approx 88\mathrm{n}\mathrm{s}$, respectively.

Figure 4

The effect of elastic anisotropy (without the hydrodynamics). The ratio of elastic constants is (a) 1, (b) 2, (c) 4, (d) 8, and (e) 16; $B_0$ is kept constant. (f) For comparison, the hydrodynamic one elastic constant case with ${\gamma }_1$ doubled is shown.