Reentrant prewavy instability in competition between rising and twist modes in ac field-driven electroconvection

We report on the prewavy (PW) instability in ac field-driven electrohydrodynamics that is induced in a nematic liquid crystal (NLC) sandwiched between parallel electrodes. The instability is characterized by a twist mode of the NLC director along the vertical orientation to the electrodes (i.e., the $z$ axis), generating a periodic pattern having a large wavelength (${\lambda }_{\mathrm{PW}}$) in the $xy$ plane. The PW periodic to the preferred director ${\mathit{n}}_0$ of the NLC should be distinguished from well-known electroconvection (EC) such as normal rolls (NRs) and abnormal rolls (ARs) with similar wave vectors. A reentrant PW (PW2) was discovered by employing well-adjusted optical conditions and a dynamic image-process method. The wavelength ${\lambda }_{\mathrm{PW}2}$ of the PW2 accompanying turbulent EC was measured as functions of the applied ac voltage and frequency, which was distinguished from ${\lambda }_{\mathrm{PW}1}$ of the primary PW (PW1) separated from the NR. Moreover, the appearance, disappearance, and reappearance of the PW were investigated for five frequency regions classified in the ac field-driven EC; it was found that the high frequency and high voltage causes competition between the rising mode (θ, tilting angle to the $xy$ plane) and twist mode (ϕ, in-plane angle to the $x$ axis) of the director through electrohydrodynamic coupling between the director field and flows. We discuss how the PW2 can arise by considering another twist mode known as AR instability.

Figure 1

Transmitted optical intensity $I\left(X\right)$ (a) of a prewavy (PW1) (b) obtained in the ${\mathrm{P}}_{//}\mathrm{Q}{\mathrm{A}}_{\perp }$ [i.e., crossed-nicol (CN), upper] and ${\mathrm{P}}_{\perp }\mathrm{Q}{\mathrm{A}}_{\perp }$ [i.e., parallel-nicol (PN), lower] conditions, and flows (c) in the PW1 (cell 3) (Supplemental Material [44]); here, P, Q, and A indicate a polarizer, quarter-wave plate, and analyzer, respectively. Moreover, $I\left(X\right)$ was calculated by the Jones matrix method, in order to compare with the experimental one that was measured at an arbitrarily selected line $\left(X\right)$ in the PW1. The periodic bars in (b,c) indicate the in-plane director $n$_xy in the midplane of the nematic liquid-crystal cell ($z=d$/2).

Figure 2

Phase diagrams in the frequency-voltage plane. (a) Typical thresholds of patterns in cell 1; WD: Williams domain, CV: chevron, GP: grid pattern, PW: prewavy, DSM: dynamic scattering mode. The shaded region for $f_{\mathrm{DL}}<f<f_{\mathrm{cd}}$ indicates coexistence of a PW2 and a DSM2. The dotted thick line between $f_{\mathrm{DL}}$ and $f_{\mathrm{w}}$ denotes a presumable threshold line for the PW1 extrapolated from that for $f>f_{\mathrm{w}}$ [4]. (b) Phase diagrams obtained in cell 2, cell 3, and cell 4; DCV: dielectric chevron; DNR: dielectric normal roll. Note that our results were obtained at a fixed $T=18{}^{\circ }\mathrm{C}$ for cell 1 to cell 3 (MBBA) for which the clearing point was almost the same value ($T_{\mathrm{c}}\sim 41{}^{\circ }\mathrm{C}$) despite different electric properties as shown in Table 1.

Figure 3

Appearance of a primary PW (PW1) and a reentrant PW (PW2) ($f=850\mathrm{Hz}>f_{\mathrm{w}}\approx 800\mathrm{Hz}$) by increasing $V=180$ V (a), 201 V (b), 216 V (c), 275 V (d), and 299 V (e); the optical condition was ${\mathrm{P}}_{//}\mathrm{Q}{\mathrm{A}}_{\perp }$. In particular, (f) is a DIIPM image transformed from (d). The thick bars in (a,b) indicate the in-plane director $n$_xy; accordingly, the director field in the midplane ($z=d$/2) determines the maximal twist angle ${\varphi }_{\mathrm{m}}$ to the initial director ${\mathit{n}}_0(\parallel \widehat{x}$); in other words, $\pm {\varphi }_{\mathrm{m}}$ corresponds to the counterclockwise- and clockwise-twisted directors, respectively; see also Figs. 6 and 11.

Figure 4

Wavelength ${\lambda }_{\mathrm{PW}}$ of the PW1 and PW2 as a function of the ac voltage $V$ ($f=850\mathrm{Hz}>f_{\mathrm{w}}\approx 800$ Hz) (a), and as a function of the ac frequency $f [{\varepsilon }_{\mathrm{PW}}=\left(V^2-V_{\mathrm{PW}1}^2\right)/V_{\mathrm{PW}1}^2\approx 0.1$ for the PW1 and ${\varepsilon }_{\mathrm{PW}2}=\left(V^2-V_{\mathrm{PW}2}^2\right)/V_{\mathrm{PW}2}^2\approx 0.1$ for the PW2] (b). All pictures were obtained in the ${\mathrm{P}}_{//}\mathrm{Q}{\mathrm{A}}_{\perp }$ condition; the PW2 in (a) is a DIIPM image. Note that the dotted line in (a) does not provide the wavelength because of collapse of the PW1; moreover, the two slopes of ${\lambda }_{\mathrm{PW}}\left(V\right)/d$ are different from each other, indicating different decreasing rates of the wavelengths.

Figure 5

(a) A transient CV changing into defect chaos with the elapse of time [top: 1 s after applying $V=201$ V ($f=850\mathrm{Hz}<f_{\mathrm{w}}\approx 800$ Hz) > $V_{\mathrm{c}}=182$ V; middle: 5 s; and bottom: 20 s]. (b) Appearance and disappearance of localized EC (i.e., NR) in the PW1 with the elapse of time (top: 1 s after applying $V\approx V_{\mathrm{c}}$; middle: 3 s; and bottom: 5 s).

Figure 6

Pattern evolution ($f_{\mathrm{DL}}<f=600\mathrm{Hz}<f_{\mathrm{w}}$) by increasing $V=80$ V (a), 86 V (b), 122 V (c), 124 V (d), 129 V (e), and 146 V (f). All pictures are DIIPM images obtained in the ${\mathrm{P}}_{//}\mathrm{Q}{\mathrm{A}}_{\perp }$ condition. A defect-lattice (DL) pattern is well visualized in (a,b). A PW1 hidden in the (usual) ${\mathrm{P}}_{//}$ condition is revealed in (d), which is more clearly visualized in (e,f) as its wavelength decreases; see also Figs. 3 and 11.

Figure 7

(a) A DL pattern in the ${\mathrm{P}}_{//}$ condition (left) and ${\mathrm{P}}_{//}\mathrm{Q}{\mathrm{A}}_{\perp }$ condition (right) in cell 3 ($V=65$ V and $f=1200\mathrm{Hz}<f_{\mathrm{w}}\approx 1300$ Hz). (b) An abnormal roll (AR) pattern at $V=19$ V (left) and an immature DL at $V=21$ V (right) in the same ${\mathrm{P}}_{//}$ condition in cell 2 ($f=150\mathrm{Hz}<f_{\mathrm{w}}\approx 245$ Hz).

Figure 8

Pattern evolution (a) for the normal roll (NR) region ($f=450\mathrm{Hz}>f_{\mathrm{L}}\approx 300$ Hz) by increasing $V$ from 15 to 112 V (from upper to lower), and (b) for the oblique roll (OR) region ($f=100\mathrm{Hz}<f_{\mathrm{L}}$) by increasing $V$ from 7.1 to 68 V (from upper to lower). The DSM1 (left) and DSM2 (right) coexist in the turbulent patterns in the lowest row, which are optically distinguished from each other. There is no evidence of the PW. All pictures were observed in the ${\mathrm{P}}_{//}$ condition.

Figure 9

Pattern evolution ($f=300\mathrm{Hz}>f_{\mathrm{cd}}\approx 245$ Hz in cell 2) by increasing $V=24$ V (a,b), 41 V (c), 48 V (d), 53 V (e), and 60 V (f). (a) was obtained in the ${\mathrm{P}}_{//}$ condition, and (b–f) in the ${\mathrm{P}}_{//}\mathrm{Q}{\mathrm{A}}_{\perp }$ condition. A PW1 appears as a background pattern of a DCV near $V_{\mathrm{c}}$, and the PW1 and DCV are perceivable in (a). The PW1 (b) develops into a complicated pattern as its wavelength decreases (c) and collapses by turbulent EC invading the periodic PW bands (i.e., clockwise- and counterclockwise-twisted director fields) (d); for higher $V$, turbulent EC showing the DSM1 (gray) and DSM2 (black) is generated (e) and becomes complicated (f).

Figure 10

(a) A DNR-developed DCV (DCV1) observed at $f=250\mathrm{Hz}$ (> $f_{\mathrm{cd}}\approx 90$ Hz in cell 4) and (b) A PW-originated DCV (DCV2) observed at $f=2000\mathrm{Hz}$ (> $f_{\mathrm{cd}}\approx 1400$ Hz in a cell); see cell 4 (MBBA:$\mathrm{EBBA}=50$:50 in wt %) in Ref. [38]. Both DCVs were observed in the ${\mathrm{P}}_{//}\mathrm{Q}{\mathrm{A}}_{\perp }$ conditions. Note that the wide striped domains in both DCVs are indistinguishable from each other although they have completely different origins; those of DCV1 originate from twisting modulation of the director via generation of defects, whereas those of DCV2 are induced by the PW instability providing the twisting modulation.

Figure 11

A possible mechanism for the appearance of the PW1 and PW2. (a) The director $n$ in EC having a twist mode (ϕ) as well as a rising mode (θ); (b) normalized (transmitted optical) intensity $I$ of the PW1, $I\propto {\varphi }_{\mathrm{m}}=c_{\varphi }\sqrt{{\varepsilon }_{\mathrm{PW}}}$; (c) appearance of the PW2 in the case of $f_{\mathrm{DL}}<f<f_{\mathrm{w}}$; and (d) appearance of the PW1 and PW2 in the case of $f_{\mathrm{w}}<f<f_{\mathrm{cd}}$. The PW1 cannot be observed because of the suppression by a DL on it for $f_{\mathrm{DL}}<f<f_{\mathrm{w}}$ (c), whereas it can occur with the PW2 because of $V_{\mathrm{PW}1}<V_{\mathrm{c}}$ for $f_{\mathrm{w}}<f<f_{\mathrm{cd}}$ (d). See also Figs. 3 and 6.