Traveling waves and worms in ac-driven electroconvection under external multiplicative noise

In the presence of external multiplicative noise, ac-driven electroconvection (EC) in a nematic liquid crystal is investigated. Noise-induced traveling waves (TWs) including localized ones (worms) are found with a typical, stationary wave. Three kinds of TWs are classified by their dynamic features (e.g., noise-intensity-dependent Hopf frequency and space-time map). Moreover, ac frequency-dependent threshold voltages of EC are examined in high noise intensities causing abnormal charge redistribution of the EC cell, and the roles of ac and noise fields with respect to TWs are elucidated in successive pattern evolutions. The mechanism of TWs is discussed in terms of a locally perturbed dynamic conductivity due to the noise field additionally applied to the EC; such a conductivity can be related to a weak-electrolyte model for a Hopf bifurcation to a TW.

Figure 1

Typical ac-driven electroconvection patterns in the absence of noise. (a) Stationary wave (SW) at a driving ac frequency $f=50\mathrm{Hz}$ and a reduced ac voltage $\varepsilon =\left(V^2-V_{\mathrm{c}}^2\right)/V_{\mathrm{c}}^2=0.1$, (b) Traveling wave (TW) at $f=200\mathrm{Hz}$ and $\varepsilon =0.1$. To show their dynamics, a one-dimensional light-intensity profile arbitrarily selected in each pattern is presented in the order of time (from $t=0$ to $t=4\mathrm{s}$). Each frame is $x\times y=0.421\times 0.316\mathrm{m}{\mathrm{m}}^2$.

Figure 2

Typical space-time maps for SW (a) and TW (b), and the power spectrum $S(k,\omega )$ (c) and $P\left(f\right)$ (d) for TW (after binary FIG. 2. (Continued) image processing). The maps were obtained by placing an arbitrarily selected one-dimensional image (Fig. 1) in time series; each frame is $x\times t=640\mathrm{pixels}\times 600\mathrm{pixels}$ ($=0.421\mathrm{mm}\times 60\mathrm{s}$). To measure the Hopf frequency $f_{\mathrm{H}}$ ($={\omega }_{\mathrm{c}}/2\pi$) of TW, we define an angle $\theta$ ($tan\theta ={\omega }_{\mathrm{c}}/k_{\mathrm{c}}$), as shown in (c); the velocity $v$ ($=tan\theta$) of TW was calculated by using $2\pi f_{\mathrm{H}}=k_{\mathrm{c}}tan\theta$. Also, $f_{\mathrm{H}}$ was estimated from the average $P\left(f\right)$ of the map (b) for TW.

Figure 3

Threshold voltage $V_{\mathrm{c}}$, Hopf frequency $f_{\mathrm{H}}$, and wave vector $k_{\mathrm{c}}$ of Williams domain (WD) as a function of ac frequency $f$. WD $(f<f_{\mathrm{cd}})$ are divided by $f_{\mathrm{TW}}$ ($\sim 160\mathrm{Hz}$): SW $(f_{\mathrm{H}}=0)$ for $f<{\mathrm{f}}_{\mathrm{TW}}$, and TW $(f_{\mathrm{H}}\ne 0)$ for $f>{\mathrm{f}}_{\mathrm{TW}}$. Moreover, a typical pattern (chevrons: CV) appears in the dielectric regime $(f>f_{\mathrm{cd}})$, which is discriminated from SW and TW in the conduction regime $(f<f_{\mathrm{cd}})$ [2].

Figure 4

Dependence of Hopf frequency $f_{\mathrm{H}}$ on the noise intensity $V_{\mathrm{N}}$. $f_{\mathrm{H}}$ was measured at a fixed $\varepsilon =0.01$ (near onset). TW1–$3(f_{\mathrm{H}}\ne 0)$ appear for $f=170$, 200 Hz $(>f_{\mathrm{TW}})$, whereas $\mathrm{SW}(f_{\mathrm{H}}=0)$ and TW1–3 appear for $f=50\mathrm{Hz} (<f_{\mathrm{TW}})$. TW3 is defined as a localized TW (or worms) [see Fig. 8]. The solid lines are a guide for the eye. See also Fig. 5 for SW and TW1–3.

Figure 5

Typical space-time maps (at a fixed $\varepsilon =0.05$) for (a) SW ($f=50\mathrm{Hz}, V_{\mathrm{N}}=5\mathrm{V}$); (b) TW1 ($f=200\mathrm{Hz}, V_{\mathrm{N}}=10\mathrm{V}$); (c) TW2 ($f=200\mathrm{Hz}, V_{\mathrm{N}}=20\mathrm{V}$); (d) TW3 ($f=200\mathrm{Hz}, V_{\mathrm{N}}=30\mathrm{V}$). Moreover, some noticeable TWs were observed: (e) Beating TW2 ($f=200\mathrm{Hz}, V_{\mathrm{N}}=25\mathrm{V}$) and (f) pair-waves creation and annihilation of TW3 ($f=170\mathrm{Hz}, V_{\mathrm{N}}=28\mathrm{V}$). The arrows in (d) indicate the traveling direction. Each frame is $x\times t=640\mathrm{pixels}\times 600\mathrm{pixels}$ ($=0.421\mathrm{mm}\times 60\mathrm{s}$).

Figure 6

Light-intensity profiles $I\left(t\right)$ obtained (in arb. unit) from an arbitrarily selected temporal line $t\left(x_0\right)$ in each space-time map (Fig. 5): $I_{\mathrm{B}}\left(t\right)$ for a beating TW from Fig. 5 and $I_{\mathrm{T}}\left(t\right)$ for a typical TW from Fig. 5.

Figure 7

Dependence of Hopf frequency $f_{\mathrm{H}}$ on the conductivity ${\sigma }_{\perp }$ (at a fixed $\varepsilon =0.01$ and $f=200\mathrm{Hz}$). The relation $f_{\mathrm{H}}\propto {\sigma }_{\perp }^{-1/2}{d}^{-3}$ predicted in the WEM is confirmed for $V_{\mathrm{N}}=0,10\mathrm{V}$ (TW1), whereas it is inapplicable to $V_{\mathrm{N}}=25\mathrm{V}$ (TW2). The frequency $f_{\mathrm{H}}$ (and velocity $v$) of TWs decreases (i.e., to become slower TW) and then becomes zero [i.e., SW($f_{\mathrm{H}}=0$)] with increasing ${\sigma }_{\perp }$ (i.e., with increasing temperature $T$); ${\sigma }_{\perp }={\sigma }_{\parallel }\sim 5.66\times {10}^{-8}{\mathrm{\Omega }}^{-1}{\mathrm{m}}^{-1}$ at the nematic-isotropic transition temperature $T_{\mathrm{c}}\sim 40{}^{\circ }\mathrm{C}$ for the NLC.

Figure 8

Different threshold functions $V_{\mathrm{c}}\left(f\right)$ for EC in the two regions (in the presence of noise $V_{\mathrm{N}}=30\mathrm{V}$). (a) TW3 (worms) and two selected regions (dotted rectangular places); (b) coexistence of chevrons (CV) and TW3 $(f=180\mathrm{Hz})$; (c) threshold voltages $V_{\mathrm{c}}\left(f\right)$ for the two regions. The patterns (a,b) were observed at the points symbolized as $*$ and × in (c), respectively. Each frame for (a) and (b) is $x\times y=0.421\times 0.316\mathrm{m}{\mathrm{m}}^2$.

Figure 9

Space-time maps $(f=30\mathrm{Hz}<f_{\mathrm{TW}})$ with increasing the ac voltage $V$ [(a–d) at $V_{\mathrm{N}}=0$], and with increasing the noise intensity $V_{\mathrm{N}}$ [(e–h) at $\varepsilon =0.01$]: $V=6.35\mathrm{V}$ (a), 7.44 V (b), 8.68 V (c), 9.92 (d), and $V_{\mathrm{N}}=12.2\mathrm{V}$ (e), 19.5 V (f), 21.8 V (g), 27.0 V (h). Increasing $V$ gives rise to defect chaos (b), whereas increasing $V_{\mathrm{N}}$ induces TW (f) which develops into the faster TW $(\mathrm{TW}1\to \mathrm{TW}2)$. Each frame is $x\times t=640\mathrm{pixels}\times 600\mathrm{pixels}$ ($=0.421\mathrm{mm}\times 60\mathrm{s}$).

Figure 10

Space-time maps $(f=200\mathrm{Hz}>f_{\mathrm{TW}})$ with increasing the ac voltage $V$ (i.e., $\varepsilon$) (at a fixed $V_{\mathrm{N}}=25\mathrm{V})$: $\varepsilon =0.10$ (a), 0.45 (b), 0.50 (c), 0.90 (d). When increasing $V$, TW2 (a) degrades into TW1 (b), and then changes into defect chaos (c,d). No TW is found in high ac voltages. Each frame is $x\times t=640\mathrm{pixels}\times 600\mathrm{pixels}(=0.421\mathrm{mm}\times 60\mathrm{s})$.

Figure 11

Dependence of Hopf frequency $f_{\mathrm{H}}$ on a reduced ac voltage $\varepsilon (f=200\mathrm{Hz})$ in the absence $(V_{\mathrm{N}}=0)$ and presence $(V_{\mathrm{N}}=10,25\mathrm{V})$ of noise. TW2 $(V_{\mathrm{N}}=25\mathrm{V})$ and TW1 ($V_{\mathrm{N}}=0$, 10 V) change into completely different EC not traveling $(f_{\mathrm{H}}=0)$ with increasing $\varepsilon$. The solid lines are a guide for the eye. See also Fig. 10.