Dislocation dynamics in an anisotropic stripe pattern

The dynamics of dislocations confined to grain boundaries in a striped system are studied using electroconvection in the nematic liquid crystal N4. In electroconvection, a striped pattern of convection rolls forms for sufficiently high driving voltages. We consider the case of a rapid change in the voltage that takes the system from a uniform state to a state consisting of striped domains with two different wave vectors. The domains are separated by domain walls along one axis and a grain boundary of dislocations in the perpendicular direction. The pattern evolves through dislocation motion parallel to the domain walls. We report on features of the dislocation dynamics. The kinetics of the domain motion is quantified using three measures: dislocation density, average domain wall length, and total domain wall length per area. All three quantities exhibit behavior consistent with power-law evolution in time, with the defect density decaying as $t^{-1∕3}$, the average domain wall length growing as $t^{1∕3}$, and the total domain wall length decaying as $t^{-1∕5}$. The two different exponents are indicative of the anisotropic growth of domains in the system.

Figure 1

(a) A schematic drawing illustrating a transition from a zig to a zag and back to a zig region of the system. The black lines represent lines of constant phase between the stripes. There are two horizontal domain walls, one at each transition. Also, one example of each type of step dislocation is shown. The Burgers vector is also illustrated for one of the dislocations. The length of a horizontal domain wall is indicated by $L_i$. (b) A close-up of a section of the system that shows two step dislocations (black squares). The white squares are provided as an aid in counting the zig and zag rolls. One finds that there are two more zig rolls than zag, as expected with two dislocations. The dashed line highlights one of the horizontal domain walls. The scale bar represents $0.15\text{mm}$.

Figure 2

Four images separated by $60\mathrm{s}$ illustrating the collapse of a zig domain. The scale bar represents $0.15\text{mm}$.

Figure 3

Images separated by $150\mathrm{s}$ illustrating the incorporation of an isolated dislocation into a domain wall. The scale bar represents $0.15\text{mm}$.

Figure 4

Eight images separated by $300\mathrm{s}$ illustrating the motion of a single dislocation highlighted by a circle in (a). A dust particle on the surface of the sample is not removed to provide a frame of reference. In (a)–(d), the dislocation oscillates. In (e)–(h), the dislocation is attracted to one of opposite sign and annihilated. The scale bar represents $0.15\text{mm}$.

Figure 5

Images separated by $150\mathrm{s}$ illustrating oppositely charge dislocations moving together, “bouncing,” and ultimately attracting each other. One can also observe the spread of dislocations in a vertical wall that is highlighted in Fig. 6. The scale bar represents $0.15\text{mm}$.

Figure 6

Images separated by $300\mathrm{s}$ illustrating dislocations in a domain wall spreading out in time. The scale bar represents $0.15\text{mm}$.

Figure 7

Images separated by $150\mathrm{s}$ illustrating the nucleation of narrow domain without a dislocation pair. The scale bar represents $0.15\text{mm}$.

Figure 8

(a) Plot of ${\log }_{10}\left(\rho \right)$ versus ${\log }_{10}\left(t\right)$. Symbols are experimental data. The solid line has a slope $-0.32$. For comparison, the dashed line has a slope of $-0.25$. (b) Plot of ${\log }_{10}\left(⟨L⟩\right)$ versus ${\log }_{10}\left(t\right)$. Symbols are experimental data. The solid line has a slope 0.33. For comparison, the dashed line has a slope of 0.25. (c) Plot of ${\log }_{10}\left(L\right)$ versus ${\log }_{10}\left(t\right)$. Symbols are experimental data. The solid straight line has slope of $-0.2$. For comparison, the dashed line in (c) has a slope of $-0.33$.