Annihilation of edge dislocations in smectic-$A$ liquid crystals

This paper presents a theoretical study of the annihilation of edge dislocations in the same smectic plane in a bulk smectic-$A$ phase. We use a time-dependent Landau-Ginzburg approach where the smectic ordering is described by the complex order parameter $\psi (\vec{r},t)=\eta e^{i\varphi }$. This quantity allows both the degree of layering and the position of the layers to be monitored. We are able to follow both precollision and postcollision regimes, and distinguish different early and late behaviors within these regimes. The early precollision regime is driven by changes in the $\varphi \left(\vec{r}\right)$ configuration. The relative velocity of the defects is approximately inversely proportional to the interdefect separation distance. In the late precollision regime the symmetry changes within the cores of defects also become influential. Following the defect collision, in the early postcollision stage, bulk layer order is approached exponentially in time. At very late times, however, there seems to be a long-time power-law tail in the order parameter fluctuation relaxation.

Figure 1

The geometry of the problem. The edge dislocations are initially separated by a distance $L_0$. The distances $L_x$ and $L_z$ describe the simulation cell volume. Bold lines indicate the region over which calculations are carried out. Thin lines determine the smectic layers. The two dots mark the centers of defects.

Figure 2

The order parameter profile of an isolated dislocation located at $z=0$. Left panel: smectic layer profile with contour plot of $\eta (x,z)$ superimposed. Contours correspond to $\eta ∕{\eta }_b=0.95$ (dotted line), 0.8 (dash-dot-dotted line), 0.6 (dash-dotted line), 0.4 (short dashed line), 0.2 (solid line), and 0.05 (solid line). Right panel: spatial variation of the order parameter along the $x$ axis at $z=0,\xi$ (bottom line), $2\xi ,3\xi ,4\xi ,5\xi ,6\xi$ (top line).

Figure 3

Order parameter profile at different stages during the annihilation process. Left panel: smectic layer profile with contour plot of $\eta (x,z)$ superimposed. Contours correspond to $\eta ∕{\eta }_b=0.95$, 0.8, 0.6, 0.4, 0.2, 0.05. Right panel: spatial variation of the order parameter along the $x$ axis is shown at $z=0,\xi ,2\xi ,3\xi ,4\xi ,5\xi ,6\xi$. (a) The late precollision regime, $L=5\xi$; (b) exactly at the time of collision, $L=0$; (c) the early postcollision regime, $t=\tau$ and $\eta (0,0)∕{\eta }_b=0.23$.

Figure 4

The dynamics of the precollision regime. (a) The $L\left(t\right)$ dependence. Neglecting the initial anomalous dependence we roughly obtain $L\propto \sqrt{(t_c-t)}$. Care should be taken because the observed $L\left(t\right)$ dependence on $L_0$ is due to memory effects. The results are obtained at a temperature where $\xi \sim d_0$. (b) The dependence $v\left(L\right)$ on a log-log plot. The law $v\propto 1∕L$ would result in a straight line with slope $-1$.

Figure 5

Precollision regime: the time evolution of the characteristic core size width in the direction along the line connecting the defects and perpendicular to it. The diameters ${\xi }_x$ and ${\xi }_z$, characterizing the core shape, were measured in the $x$ and $z$ directions, respectively, between opposite points at which $\eta ={\eta }_b∕2$.

Figure 6

$\eta (x=0,z=0)$ time evolution in the late precollision and postcollision regimes.

Figure 7

Influence of elastic anisotropy, nematic distortions, and hydrodynamics on the $L\left(t\right)$ dependence. In all cases $L_0=60\xi$. (a) $C_{\Vert }∕C_{\perp }=1,{\kappa }_n=\infty$, the solid line; (b) $C_{\Vert }∕C_{\perp }=5,{\kappa }_n=\infty$, the dashed line; (c) $C_{\Vert }∕C_{\perp }=1,{\kappa }_n=0$, the dotted line; (d) $C_{\Vert }∕C_{\perp }=1,{\kappa }_n=\infty$, addition of advection terms, the dash-dotted line. The ratio $a_{\Vert }∕a_{\perp }=4$, typical for nematogenic molecules, is taken. Note that in cases (c) and (d) the curves nearly overlap.