Temporal response to harmonic driving in electroconvection

The temporal evolution of the spatially periodic electroconvection (EC) patterns has been studied within the period of the driving ac voltage by monitoring the light intensity diffracted from the pattern. Measurements have been carried out on a variety of nematic systems, including those with negative dielectric and positive conductivity anisotropy, exhibiting “standard EC” (s-EC), those with both anisotropies negative exhibiting “nonstandard EC” (ns-EC), as well as those with the two anisotropies positive. Theoretical predictions have been confirmed for stationary s-EC and ns-EC patterns. Transitions with Hopf bifurcation have also been studied. While traveling had no effect on the temporal evolution of dielectric s-EC, traveling conductive s-EC and ns-EC patterns exhibited a substantially altered temporal behavior with a dependence on the Hopf frequency. It has also been shown that in nematics with both anisotropies positive, the pattern develops and decays within an interval much shorter than the period, even at relatively large driving frequencies.

Figure 1

Microscopic snapshots of the EC patterns. (a) Conductive s-EC oblique rolls in a $(-,+)$ nematic (sample thickness $d=12\hspace{0.5em}\mu$m); (b) conductive s-EC normal rolls in a $(-,+)$ nematic ($d=12\hspace{0.5em}\mu$m); (c) dielectric s-EC normal rolls in a $(-,+)$ nematic ($d=11.4\hspace{0.5em}\mu$m); (d) ns-EC pattern in a $(-,-)$ nematic ($d=11\hspace{0.5em}\mu$m); (e) ns-EC roll pattern in a $(+,+)$ nematic ($d=19.5\hspace{0.5em}\mu$m). The double arrow indicates the initial director n for all patterns, while the scale bars denote $100\hspace{0.5em}\mu$m in all subfigures.

Figure 2

Temporal evolution of the first-order fringe intensity just above the onset of instability in samples exhibiting stationary s-EC patterns. The straight line shows the average intensity; the nearly straight, noisy line at the bottom (fluctuating around $I_{\pm 1}=0$) corresponds to the background dark level. The oscillation occurs with twice the driving frequency. $A$ is the amplitude of the oscillations, $B$ the average intensity with respect to the dark level, and $t_0$ the period of the driving frequency. (a) Phase 5A, $d=12\hspace{0.5em}\mu$m, $T=30\hspace{0.5em}{}^{°}$C, $f=600$ Hz, stationary conductive s-EC, $A/B\approx 0.001$, and the inset is the blowup of the intensity oscillations. (b) Phase 5, $d=12\hspace{0.5em}\mu$m, $T=30\hspace{0.5em}{}^{°}$C, $f=1$ kHz, stationary dielectric s-EC, and $A/B=1$.

Figure 3

Same as in Fig. 2, measured in samples exhibiting traveling s-EC patterns. (a) Phase 5, $d=3.4\hspace{0.5em}\mu$m, $T=30\hspace{0.5em}{}^{°}$C, $f=200$ Hz, traveling conductive s-EC, and $A/B\approx 0.57$. (b) Phase 4, $d=3.2\hspace{0.5em}\mu$m, $T=30\hspace{0.5em}{}^{°}$C, $f=400$ Hz, traveling dielectric s-EC, and $A/B=1$.

Figure 4

(a) Frequency dependence of the onset voltage $U_c$ of conductive s-EC rolls determined both with optical microscopy (open circles) and with diffraction (pluses); (b) the ratio $A/B$ (solid squares) of the $2f$-modulated ($A$) and the constant ($B$) part of the diffracted fringe intensity measured in a mixture of Phase 5 and Phase 5A (7:1) at $T=30\hspace{0.5em}{}^{°}$C, as well as the frequency dependence of the Hopf frequency $f_H$ calculated from the WEM (solid line). The frequency ranges of the stationary oblique rolls (OR), stationary normal rolls (NR), and traveling normal rolls (TR) are separated by vertical lines.

Figure 5

Same as in Fig. 2, measured in samples of (a) 10/6, where stationary ns-EC patterns appear at the onset of convection, $d=11\hspace{0.5em}\mu$m, $T^{*}\approx 0.6$, $f=30$ Hz, and $A/B=1$; (b) 8/7 in ($d$, $T^{*}$, $f$) parameter range, where traveling ns-EC patterns appear at the onset of convection, $d=13.2\hspace{0.5em}\mu$m, $T^{*}=0.38$, $f=40$ Hz, and $A/B\approx 0.25$.

Figure 6

Same as in Fig. 2, measured in a $(+,+)$ sample of 5CB below (at $U=20$ V) and above (at $U=50$ V) the onset of the roll pattern [26]. $d=19.5\hspace{0.5em}\mu$m; $T=30\hspace{0.5em}{}^{°}$C; $f=20$ Hz.