Electric-field-induced shape transition of nematic tactoids

The occurrence of new textures of liquid crystals is an important factor in tuning their optical and photonics properties. Here, we show, both experimentally and by numerical computation, that under an electric field chitin tactoids (i.e., nematic droplets) can stretch to aspect ratios of more than 15, leading to a transition from a spindlelike to a cigarlike shape. We argue that the large extensions occur because the elastic contribution to the free energy is dominated by the anchoring. We demonstrate that the elongation involves hydrodynamic flow and is reversible: the tactoids return to their original shapes upon removing the field.

Figure 1

Aspect ratio of the tactoids as a function of the tactoid volume before applying the electric field. Left inset: The observation under crossed polarizers (white double-headed arrows) reveals the bipolar structure of the tactoids. The two tactoids look different because the axis of the tactoid on the right is exactly parallel to the analyzer direction whereas that of the tactoid on the left lies at an angle of $\sim 3^{\circ }$. Right inset: Schematic drawing of a tactoid, with dashed lines representing the director field.

Figure 2

Time evolution of the tactoids under a field $E_{\text{rms}}=160{\mathrm{V}\mathrm{mm}}^{-1}$, $f=300$ kHz, and relaxation after switching off the field. Experiments: (a)–(e) The field is applied respectively for time ${\tau }_{\text{on}}=0$, 120, 210, 640, and 1080 s. (f)–(i) The sample has relaxed for ${\tau }_{\text{off}}=230$, 380, 530, and 820 s after the field removal. The double-headed arrows in (a) show the orientations of the polarizer P and the analyzer A. In (b) the field direction $\mathrm{E}$ (white arrow) and the shadows cast by the out-of-focus electrodes (black dotted lines) are shown. Note that in (e) the contrast varies and is inverted in some regions, due to introducing a Berek compensator. Simulation: (j)–(o) Time evolution of simulated tactoids under an electric field. (j) Equilibrium tactoid shape. The director field is bipolar and the aspect ratio is $\sim 2$. Because we employ a diffuse interface model the tips may appear slightly rounded. (k)–(o) The tactoid 200, 400, 800, 1600, and 30 000 simulation time steps after applying the electric field. The droplet stretches to an aspect ratio of $\sim 16$ and becomes cylindrical with conical tips.

Figure 3

Axial ratio of the tactoids as a function of their volume and the time under field $E_{\text{rms}}=160{\mathrm{V}\mathrm{mm}}^{-1}$ and $f=300$ kHz. The data corresponding to the same time under field are plotted with the same symbol. Blue circles are taken after 120 s, red triangles after 210 s, green squares after 410 s, black diamonds after 640 s, purple inverted triangles after 850 s, and orange triangles after 1080 s. The data measured for the same tactoid lie almost on a vertical line, as the volume of the tactoid is approximately independent of the time under the field. This is demonstrated by the inset, which shows the volume of the tactoids at different times as a function of the final volume after 1080 s under field.

Figure 4

Time dependence of length $L$ (red markers) and width $D$ (black markers) of tactoids of different sizes. $L$ and $D$ are normalized by their initial values in the absence of an electric field. (a) Experiments: Three chitin tactoids with $L{D}^2=2.0\times {10}^4\mu {\mathrm{m}}^3$ (triangles), $1.5\times {10}^5\mu {\mathrm{m}}^3$ (dots), and $3.7\times {10}^5\mu {\mathrm{m}}^3$ (squares), subject to a 500 kHz ac field $E_{\text{rms}}=154{\mathrm{V}\mathrm{mm}}^{-1}$. (b) Simulations: Three tactoids with $LD=\pi 8^2$ (triangles), $\pi {14}^2$ (dots), and $\pi {16}^2$ (squares).

Figure 5

Length $L$ (red markers) and width $D$ (black markers) of the tactoids as a function of the tactoid volume $L{D}^2$ before the application of the field (open circles), after 61 000 s under $E_{\text{rms}}=154{\mathrm{V}\mathrm{mm}}^{-1}$ (triangles) and after 3600 s under $E_{\text{rms}}=309$ ${\mathrm{V}\mathrm{mm}}^{-1}$ (solid symbols). The no-field data are fitted with a power law with coefficients ${\beta }_L=0.32$ and ${\beta }_D=0.34$, very close to the $V^{1/3}$ dependence expected for constant aspect ratio $L/D$. The 154 ${\mathrm{V}\mathrm{mm}}^{-1}$ data are piecewise fitted with power laws with respectively ${\beta }_L=0.41$ and 0.60, and ${\beta }_D=0.30$ and 0.20, with the crossover at $L{D}^2\sim 2\times {10}^4\mu {\mathrm{m}}^3$. For $E_{\text{rms}}=309{\mathrm{V}\mathrm{mm}}^{-1}$, the coefficients are respectively ${\beta }_L=0.44$ and 0.64, and ${\beta }_D=0.28$ and 0.19, with the crossover at $L{D}^2\sim 5\times {10}^3\mu {\mathrm{m}}^3$.

Figure 6

Simulation results showing how the final aspect ratio $L/D$ of a tactoid depends on its anchoring strength $L_0$ and the electric field strength. Blue diamonds represent an electric field strength of 0.06, red triangles 0.10, green circles 0.15, and black squares 0.20.