Electroconvection in one-dimensional liquid crystal cells

We investigate the alternating current (ac) -driven electroconvection (EC) in one-dimensional cells (1DCs) under the in-plane switching mode. In 1DCs, defect-free EC can be realized. In the presence and absence of external multiplicative noise, the features of traveling waves (TWs), such as their Hopf frequency $f_{\mathrm{H}}$ and velocity, are examined in comparison with those of conventional two-dimensional cells (2DCs) accompanying defects of EC rolls. In particular, we show that the defects significantly contribute to the features of the TWs. Additionally, owing to the defect-free EC in the 1DCs, the effects of the ac and noise fields on the TW are clarified. The ac field linearly increases $f_{\mathrm{H}}$, independent of the ac frequency $f$. The noise increases $f_{\mathrm{H}}$ monotonically, but $f_{\mathrm{H}}$ does not vary below a characteristic noise intensity $V_{\mathrm{N}}^{*}$. In addition, soliton-like waves and unfamiliar oscillation of EC vortices in 1DCs are observed, in contrast to the localized EC (called worms) and the oscillation of EC rolls in 2DCs.

Figure 1

EC in a 1DC prepared by employing the in-plane switching mode. (a) Picture of the cell in the $xy$ plane (with an electrode width of $a=10\mu \mathrm{m}$, a gap of $d=10\mu \mathrm{m}$ between electrodes, and an electrode thickness of $\mathrm{\Delta }z\approx 0.16\mu \mathrm{m}$). The active dimensions for EC were $L_x\times L_y=1\times 1\mathrm{cm}$. (b) Typical EC pattern (at $V=45.2\mathrm{V}$ and $f=200\mathrm{Hz}$), (c) Scheme of EC in the 1DC, (d) 1DC depicted in the $yz$ plane. The slab of an NLC is $b=50\mu \mathrm{m}$. The thick rods and ellipsoids in (c) indicate the director $\mathit{n}(x,y)$ of the NLC and the EC vortices, respectively; ${\mathit{n}}_0$ denotes the initial director of the NLC (${\mathit{n}}_0\parallel \mathit{E}\parallel \widehat{\mathit{y}}$ at the surface). Ac-driven EC is induced between electrodes by applying a voltage above the critical voltage $V_{\mathrm{c}}$.

Figure 2

Phase diagram and various patterns in the 1DC. (a) Well-known DSM. (b) Typical DSM in the 1DC [distinguished from (a)], (c) TWs, (d) EC on disclination patterns. (e) Disclination patterns before EC, (f) SWs. (g) Phase diagram in the ac-frequency and voltage plane. The horizontal pattern in (e) may be caused by the small declination of the director resulting from the electric field [including the effect of small electric-field gradients shown in Fig. 1].

Figure 3

STM and Hopf frequency for TW. (a) Typical STM obtained by successive placement of the one-line image $X$ with an identical time interval $\mathrm{\Delta }t=0.033\mathrm{s}$. A right-TW (RTW with $\theta <0$) is distinguishable from the SW $(\theta =0)$. (b) Transmitted light intensity $I\left(t\right)$ for a TW obtained at an arbitrarily selected point $x_0$ in (a), (c) Calculated power spectra from (b) for an example. The peak in (c) indicates the Hopf frequency $f_{\mathrm{H}}$ (approximately 4 Hz) of the TW.

Figure 4

STM with a steplike increase of the ac voltage $V$ (at a fixed frequency $f=100\mathrm{Hz}<f_{\mathrm{TW}}\approx 120\mathrm{Hz}$). An SW $(\theta =0)$ induced slightly above the EC threshold $V_{\mathrm{c}}(=24.1\mathrm{V})$ evolves into a left-TW (LTW with $\theta >0$) and becomes a more active TW with high $f_{\mathrm{H}}$; for $f<f_{\mathrm{TW}}$, TW arises as a secondary instability. See also Fig. 2. The broken-line arrow indicates the times of the appearance of dislocations resulting from EC vortex-pair creation (or annihilation).

Figure 5

STM with a steplike increase of the ac voltage $V$ (at a fixed frequency $f=200\mathrm{Hz}>f_{\mathrm{TW}}\approx 120\mathrm{Hz}$). A TW (i.e., RTW) arises as a primary instability above the EC threshold $V_{\mathrm{c}}(=38.4\mathrm{V})$ and develops into a more active TW with a high $f_{\mathrm{H}}$; no SW $(\theta =0)$ was observed below $V_{\mathrm{c}}=38.4\mathrm{V}$. See also Fig. 2. The broken-line arrow has the same meaning as those in Fig. 4.

Figure 6

Hopf frequency $f_{\mathrm{H}}$ of TW as a function of the ac voltage $V$. For $f=75$ and 100 Hz $(<f_{\mathrm{TW}}\approx 120\mathrm{Hz})$, an SW ($f_{\mathrm{H}}=0$) starts at $V_{\mathrm{c}}$ and develops into a TW at $V_{\mathrm{TW}}\approx 30\mathrm{V}$, whereas a TW ($f_{\mathrm{H}}\ne 0$) suddenly starts at $V_{\mathrm{c}}$ for $f=200$ and 250 Hz $(>f_{\mathrm{TW}})$. See also Figs. 2, 4, and 5.

Figure 7

Hopf frequency $f_{\mathrm{H}}$ of a TW as a function of the noise intensity $V_{\mathrm{N}}$ (at a fixed reduced voltage $\varepsilon =0.15$) for $f=100$ and 150 Hz. A TW with an almost constant low $f_{\mathrm{H}}$ remains until a characteristic intensity of $V_{\mathrm{N}}^{*}\approx 10\mathrm{V}$ and then becomes faster with increasing $V_{\mathrm{N}}$. The solid lines are for visual guidance.

Figure 8

STM with a steplike increase of the noise intensity $V_{\mathrm{N}}$ (at fixed values of $V=46.0\mathrm{V}$ and $f=200\mathrm{Hz}$). No noticeable variation of the STM was observed from $V_{\mathrm{N}}=0$ to 21.5 V; however, the STM shows a dramatic change of EC from $V_{\mathrm{N}}=23.1$ to 27.9 V. When $V_{\mathrm{N}}$ is increased, the pattern evolution ($\mathrm{RTW}\to \mathrm{SW}\to \mathrm{LTW}$) is observed; i.e., the traveling direction of TW is reversed after the appearance of an SW $(v=0)$. The transmitted light intensity $I\left(t\right)$ measured at a fixed point $x_0$ shows the evolution. See Fig. 12 for $\mathrm{S}{\mathrm{W}}^{*}$.

Figure 9

Velocity $v$ and Hopf frequency $f_{\mathrm{H}}$ of a TW with respect to the noise intensity $V_{\mathrm{N}}$ (at fixed values of $V=46.0\mathrm{V}$ and $f=200\mathrm{Hz}$). At two fixed points $(x_0,x_1), v$ and $f_{\mathrm{H}}$ were measured by increasing $V_{\mathrm{N}}$. Around $V_{\mathrm{N}}=25\mathrm{V}$, a complicated EC (i.e., $\mathrm{S}{\mathrm{W}}^{*}$) is observed with the typical SW ($v=0, f_{\mathrm{H}}=0$); see also Figs. 8 and 12. SW and $\mathrm{S}{\mathrm{W}}^{*}$ are found at $x_0$ and $x_1$, respectively. The solid lines are for visual guidance.

Figure 10

Hopf frequency $f_{\mathrm{H}}$ of a TW as a function of the temperature $T$ (at a fixed reduced voltage $\varepsilon =0.15$ and $f=100\mathrm{Hz}$). At $T=40{}^{\circ }\mathrm{C}$ (near the nematic-isotropic point $T_{\mathrm{NI}}\approx 43{}^{\circ }\mathrm{C}$), an unexpected $f_{\mathrm{H}}$ was measured, which originated from the atypical TW displayed in the upper part. To validate the prediction of the WEM, the electrical conductivity σ was measured as a function of $T$ using a 2DC containing the same NLC [11], as shown in the inset. See also Fig. 11. The solid lines are for visual guidance.

Figure 11

STM at different temperatures $T$: (a) $18{}^{\circ }\mathrm{C}$, (b) $24{}^{\circ }\mathrm{C}$, (c) $38{}^{\circ }\mathrm{C}$, and (d) $40{}^{\circ }\mathrm{C}$. The STMs (a)–(c) show that the Hopf frequency of the TWs (i.e., LTW) decreased monotonically with increasing $T(<40{}^{\circ }\mathrm{C})$; however, an unexpected TW (d) appears at $T=40{}^{\circ }\mathrm{C}$ (near the nematic-isotropic transition temperature $T_{\mathrm{NI}}\approx 43{}^{\circ }\mathrm{C}$) after experiencing an SW-like wave (c) at $T=38{}^{\circ }\mathrm{C}$. See also Fig. 10.

Figure 12

Coexistence of a TW and an SW appearing in alternate pattern shifts along the $y$ direction at fixed values of $V=35.5\mathrm{V}, f=100\mathrm{Hz}$, and $V_{\mathrm{N}}=29.5\mathrm{V}$ (a), as well as the STMs of the TW (b) and SW (c). The TW (i.e., LTW) is typical, whereas the SW is atypical; SW and $\mathrm{S}{\mathrm{W}}^{*}$ appear simultaneously in a single EC lane. Oscillating waves (i.e., $\mathrm{S}{\mathrm{W}}^{*}$) are observed between typical SWs ($\theta =0, f_{\mathrm{H}}=0$); e.g., see the $\mathrm{S}{\mathrm{W}}^{*}$ showing modulating EC in the middle of $X$. The two kinds of arrow symbols in (a) indicate TWs and SWs, respectively. See also Fig. 8 for $\mathrm{S}{\mathrm{W}}^{*}$.

Figure 13

Localized EC (i.e., SLWs) at $f=250\mathrm{Hz}$ and $V=56.8\mathrm{V}$ (a), 58.8 V (b), and 62.1 V (c). SLWs (a) with different lengths travel on the EC lanes; when $V$ is increased, they become a typical TW (c) by combining with each other (b). See also Fig. 14.

Figure 14

STM for localized EC (i.e., SLW) at fixed values of $V=56.8\mathrm{V}$ and $f=250\mathrm{Hz}$, constructed using a single lane of Fig. 13. The dark regions indicate non-EC for time and space. Careful observation of these dynamics indicates that the SLWs travel towards the right-hand side with constant velocity. An SLW appears, travels (i.e., the gray region), and disappears, and then another appears at random time intervals (i.e., the width $\mathrm{\Delta }t$ of the black region: $0.2\mathrm{s}<\mathrm{\Delta }t<3$ s in this study). See also Fig. 13.