Influence of External Noise on Various Electrohydrodynamic Instabilities

in a Nematic Liquid Crystal

Jong-Hoon HUH

Department of Mechanical Information Science and Technology,

Faculty of Computer Science and Systems Engineering,

Kyushu Institute of Technology, Fukuoka 820-8502, Japan

(Received June 28, 2012; accepted August 9, 2012; published online September 14, 2012)

We report noise-induced threshold shifts on various electrohydrodynamic instabilities (EHIs) in a nematic liquidcrystal. There exist three characteristic frequencies ( f1, f2, and f3) in the ac-driven EHI system. By controlling thecutoff frequency fc of external multiplicative noise, we examine the dependence of the thresholds on the noise intensityVN for the Williams domain ( f1 < f < f2), defect-free chevron ( f2 < f < f3), low-frequency Williams domain( f < f1), dielectric chevron ( f > f3), and prewavy ( f > f2). Moreover, a dc-driven EHI (isotropic mode) is dealt within the same way. Depending on VN and fc, noise plays an important role in stabilizing or destabilizing EHIs. Theinfluence of noise on the instability mechanisms and their specific structures is discussed on the basis of the presentexperimental results.

KEYWORDS: electrohydrodynamic instability, liquid crystal, multiplicative noise

1. Introduction

Ac-driven electrohydrodynamic instabilities (EHIs) innematic liquid crystals (NLCs) are one of the manyfascinating research topics for understanding pattern forma-tions in spatially extended, nonequilibrium dissipativesystems. These instabilities have been extensively studiedfor the last four decades,1) since a typical EHI was firstobserved by Williams2) and theoretically treated by Carr andHelfrich (CH).3,4) They provide a rich variety of patterns anddynamics that depend on the frequency and amplitude of theac field. On the other hand, noise-induced phenomena haveattracted much attention because of the important (non-trivial) role of noise and fluctuations found in nonlinearsystems.5–8) For instance, stochastic resonance,5,9) multi-plicative stochastic processes,6,10) and noise-induced order ordisorder (or phase transitions)6–8) are well-known, nontrivialnoise phenomena that have been extensively studied in manydifferent fields such as physics, chemistry, biology, andneuroscience.6–10)

In this article, we address threshold problems for variousac- or dc-driven EHIs in the presence of external multi-plicative noises (Figs. 1 and 2). Noise-induced shifts inthresholds for instabilities (or phase transitions) have beenreported in several different systems.6–8,11) Also, the presentnoise-induced threshold problems have been intensivelystudied in NLCs.10,12–16) Unfortunately, most of the previousstudies have treated a typical, well-understood instability[i.e., Williams domain (WD) due to the CH instability]. Incomparison with WD, we present the problems for variousinstabilities described below.

The essential features of EHIs of a planarly aligned NLCsystem [n0 ¼ ð1; 0; 0Þ for the initial director state] can beeasily understood.17,18) When one applies a sinusoidal acfield EðtÞ ¼ E0 sinð2�f tÞ�z across a thin NLC slab betweentwo transparent electrode plates (typical thickness d ¼10{100 �m), one finds a well-ordered pattern above athreshold voltage Vth. Unlike (the usual) isotropic fluids,the director n, which is a unit vector representing the locallyaveraged orientation of rod-like NLC molecules, plays an

important role in pattern-formation processes in anisotropicNLCs.17,18)

In general, there exist qualitatively different EHI regimesdepending on the frequency f of EðtÞ.17,18) In the so-calledconduction regime ( f < f3; Fig. 3), WD occurs by the CHeffect (i.e., spatially periodic charge focusing owing to theelectric anisotropy of NLCs). At low frequencies below acertain characteristic frequency f3, the charge relaxation time�� is much shorter than the period f�1 of EðtÞ (i.e., ��f � 1),whereas the director relaxation time �d is much longer thanf�1 (�d f � 1). Therefore, the periodic director cannotfollow the external field EðtÞ and remains stationary (leadingto order), thereby forming stable WDs. In principle, thedirector field for the WD is identified as a periodicmodulation accompanying vortices in the xz-plane (Fig. 1).For f2 < f < f3 (Fig. 3), however, defect-free chevrons(DFCs) are formed by periodic, zigzag modulation of theWDs, which is due to a superposition of the prewavy (PW)instability (Figs. 3 and 4). Thus, the director field for DFCbecomes more complicated because of an additional,periodic xy-director modulation that is induced by the PWinstability. The details of the DFC were described in aprevious publication.19) Moreover, at much lower frequen-cies ( f < f1 � 1Hz; Fig. 3) a flexoelectric effect-coupledCH instability is induced.20) Such frequency f1 can be foundunder the condition of comparable time scales for WDs (i.e.,f�11 � �d ¼ 1:7 s for our sample cells). Hereafter, we callthis instability low-frequency WD (LFWD).

In the dielectric regime, which is at high frequencies( f > f3), �� becomes longer than f�1 (��f � 1), and chargescannot accumulate. The director oscillates in phase withEðtÞ, whereas the charge distribution is stationary. Thus, onefinds a pattern called the dielectric chevron (DCV).17,18) Theactual structure of the DCV becomes much more compli-cated because of the formation of small-scale vortices nearboth electrode plates.

For f > f2 (Fig. 3), furthermore, PW (or inertia) instabil-ity occurs at a threshold voltage VPW before the appearanceof CH instability [i.e., DFC ( f2 < f < f3) or DCV( f > f3)].

17,21,22) PW instability forms wide bands with

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large-scale flows in the xy-plane that can be distinguishedfrom the typical convective rolls (i.e., WD) in the xz-plane.Although the behavior of the threshold VPW and thecharacteristic of the patterns and flows have been investi-gated, the mechanism is not sufficiently understood.However, it is obvious that the other high-frequencyinstabilities (DFC and DCV; Figs. 3 and 4) occur superposedon the PW.19,22)

On the other hand, an isotropic mode (IM) appears underdc fields.17) In contrast to the ac-driven (charge focusing) CHeffect, IM is destabilized from a nonuniform charge densityalong EðtÞ by charge injection through the electrode plates(i.e., the so-called Felici effect). Thus, IM still exists abovethe nematic–isotropic transition temperature TNI.

17) Fromthis point of view, IM is equivalent to thermal convectionsinduced by a temperature gradient T ðzÞ in an isotropic fluid.

In this article, we focus our attention on the influence ofnoise on the various dissipative structures implemented inthe corresponding frequency regions. In particular, noise-induced threshold problems are investigated by controlling

noise intensity VN and cutoff frequency fc. In practice, thespatial property (e.g., characteristic wavelengths �) of eachpattern should be considered because the noise-inducedthresholds depend on � .10,12,14–16) Accordingly, we deal withnoise-controlled WDs (with �WD; Fig. 4) and DFCs (with�1 � �WD and �2 � �PW; Fig. 4) independently, althoughtheir thresholds [VWDð f Þ and VDFCð f Þ] vary continuouslywith no sharp change (Fig. 3). The details of the dependenceof � on VN for WDs and DCVs have been reported in arecent publication,16) and those for other EHIs (DFC, PW,and IM) are currently under study.

This article is organized as follows. The details of thepresent sample cells and the experimental apparatus andtechniques are described in x2. In x3.1, we present thethresholds and patterns of EHIs (WD, DFC, LFWD, DCV,and PW) in the absence of noise (VN ¼ 0). In x3.2 and x3.3,we analyze the influence of noise in the conduction (WD,DFC, and LFWD) and dielectric (DCV) regimes. In x3.4, theinfluence of noise on PW and IM are investigated andcompared with the results of WD. In x4, we summarize theinfluence of noise on various instabilities on the basis ofexperimental results and mention the prospects for futurestudy and applications.

2. Experiments

A sinusoidal ac field EðtÞ was applied across a thin slab ofan NLC [p-methoxybenzylidene-p0-n-butylaniline (MBBA)]sandwiched between two parallel, transparent electrodes(indium tin oxide), to achieve instabilities. A Gaussian-typenoise NðtÞ created by a two-channel wave-generatingsynthesizer (Hioki 7075) were superposed on EðtÞ in acombiner (JFW 50PD), as shown in Fig. 1. The resultantfluctuating sinusoidal field was amplified by a wide-rangeamplifier (0 < f � 500 kHz, FLC Electronics A600). Cutofffrequency fc-dependent colored noise was generated by thelow-pass filters of the synthesizer. The filters allow low-frequency signals to pass through but attenuate signals with

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lds

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Fig. 3. Dependence of threshold voltages on ac frequency f for the

corresponding electrohydrodynamic instabilities (EHIs) in the absence of

noise (VN ¼ 0). Here f3 ¼ 2:45 kHz indicates a critical frequency dividing

the conduction ( f < f3) and dielectric ( f > f3) regimes. See text for the

details of f1 ¼ 1Hz and f2 ¼ 2:15 kHz.

PCGenerator

Amp

×100

Thermo-

Controller Light

Lens

Lens

Polarizer

Analyzer

CCD Camera

Combinerz

xNLC

Fig. 1. Experimental setup. When an electric field EðtÞ superposed with

an external multiplicative noise NðtÞ is applied across a thin slab of an NLC,a variety of patterns are observed. The bars in closed stream lines indicate

the director nðx; zÞ for a typical electroconvection (WD).

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P( f

) [a

rb. u

nit]

Frequency f [Hz]

fc

Fig. 2. Characteristics of cutoff frequency fc-dependent colored noises

generated from the low-pass filters of the synthesizer. For the three selected

filters ( fc ¼ 1, 10, and 100 kHz), the power Pð f Þ is roughly constant up to

corresponding f ¼ fc and strongly attenuates at f > fc.

J.-H. HUHJ. Phys. Soc. Jpn. 81 (2012) 104602 FULL PAPERS

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frequencies higher than fc. In general, fc is defined as thefrequency at which the power output becomes half thepassband power Pð f Þ ( f < fc). Figure 2 shows the char-acteristics of noise used in this study. For three selectedfilters ( fc ¼ 1, 10, and 100 kHz), Pð f Þ attenuates at the cor-responding frequency f ¼ fc. In this study, the stochasticintensity VN ¼ d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffihN2ðtÞi

p, cutoff frequency fc (inversely

proportional to the correlation time �N) of NðtÞ, thedeterministic intensity V ¼ d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffihE2ðtÞi

p, and frequency f of

EðtÞ were used as control parameters. To investigate theinfluence of noise on a dc-driven EHI (IM), we used a dcfield superposed on the noise.

Three sample cells with planar alignment (d ¼ 50 �m anda lateral size S ¼ 1� 1 cm2) were prepared, which havedifferent f3. f3 was controlled by doping tetra-n-butyl-ammonium bromide (TBAB) into MBBA from 0 to 1.0wt%(see the caption of Fig. 5). In practice, cell 2 (havingintermediate f3) was mainly used except for Figs. 5 and 6.All measurements were performed at a stable temperature(T ¼ 25� 0:2 C) using an electrothermal control system

(Japan Hightech TH-99). At this temperature, electricconductivity � and dielectric constant " for the NLC in oursample cells were measured with an LCZ meter (NF 2341;the measurement frequency was 1 kHz); see the caption ofFig. 5 for the details. Optical patterns for the instabilitieswere observed in the xy-plane parallel to the electrodes usinga charge-coupled-device camera (Sony XC-75) mounted ona polarizing microscope (Meijitech ML9300). To captureand analyze the patterns on a computer, image-processingsoftware (Scion Image Beta 4.0.2) and an image-captureboard (Scion LG-3) were used.

3. Results and Discussion

3.1 EHIs in the absence of noiseTo begin, we examined the characteristics of the thresh-

olds and patterns for various EHIs in the absence of noise(VN ¼ 0). Figure 3 shows typical results for EHI thresholdsin the f–V plane (in cell 2). We define three characteristicfrequencies ( f1, f2, and f3) to differentiate EHIs as describedin x1. In addition to the well-known frequency f3 � 2:45 kHz

(a)

(f)(e)

(d)(c)

(b)

200 m

WD

2

1

PW

μ

λ

λ

λ

λ

Fig. 4. Various patterns induced by ac or dc electric fields. (a) Williams domain (WD) for f1 < f < f2, (b) Low-frequency Williams domain (LFWD) for

f < f1, (c) Defect-free chevron (DFC) for f2 < f < f3, (d) Dielectric chevron (DCV) for f > f3, (e) Prewavy pattern (PW) for f > f2, (f ) Isotropic mode (IM)

in a dc field.

J.-H. HUHJ. Phys. Soc. Jpn. 81 (2012) 104602 FULL PAPERS

104602-3 #2012 The Physical Society of Japan

dividing the conduction and dielectric regimes, we find othertwo characteristic frequencies: f1 � 1Hz, which dividesLFWD and WD and f2 � 2:15 kHz, which divides WD andDFC, respectively. The patterns corresponding to theseEHI frequency domains are shown in Fig. 4. For f1 < f < f3,the threshold curve follows VWD,DFC � ð1þ 4�2f 2�2�Þ=½�2 � ð1þ 4�2f 2�2�Þ (for WD and DFC).17,18) Here � denotesthe Helfrich parameter determined by a combination of someviscous and electric constants of the NLC. For f > f3,VDCV � f 1=2 for DCV.17,18) Moreover, VLFWD � f (< f1) forLFWD and VPW � f (> f2) for PW.17,19–22)

DFC [Fig. 4(c)] is often nonstationary (traveling alongthe initial orientation n0 of the director), whereas WD isstationary at the threshold VWD for the wide frequencyregion ( f1 < f < f2). Moreover, DFC should be distin-guished from the defect-meditated chevron (DMC) (thatoften appears at f < f2 in the homeotropic alignment cells)and the DCV ( f > f3) because their pattern-formingmechanisms are completely different from one another.17–19)

From the patterns in Figs. 4(c)–4(e), the wavelength �2 forthe herringbone-like patterns (DFC and DCV) is consistentwith the wide bands (2�PW) for PW [Fig. 4(e)]. PWinstability appears below VDFC and VDCV as shown inFig. 3. However, the wavelengths �1 for the striated rolls inDFC and DCV are quite different from each other [i.e.,�1ðDFCÞ � �WD and �1ðDCVÞ � �WD]. Moreover, thewavelengths for LFWD [Fig. 4(b)] and for IM [Fig. 4(f )]are roughly identical to �WD. These threshold behaviors andpatterns are in qualitative agreement with previous experi-ments17,20–22) and theories.17,18,20) Such characteristics ofpattern structures play important roles in determining theinfluence of noise on the EHIs described below.

3.2 Influence of noise in the conduction regimeWe measured the variation in the threshold of WD

( f1 < f < f2) upon changing fc. To achieve WD, an ac fieldwith f ¼ 30Hz was applied to the sample cells. Figure 5shows the behavior of VWDðVNÞ. We find that the WDthreshold VWD depends linearly on VN. The relationship maybe modeled as V2

WD ¼ V20 þ bV2

N.10,12,14) Here V0 indicates

the threshold for the onset of the WD in the absence of noise(VN ¼ 0) and b indicates the response sensitivity of the WDwith respect to colored noise with fc.

As shown in Fig. 5(a), slope b is positive for all ranges offc (¼ 200Hz to 200 kHz) in cell 1 (having small f3 ¼160Hz); VWD increases with increasing VN (i.e., upwardthreshold shift). In other words, noise plays a role instabilizing WD (i.e., suppression of the onset of WDs). Onthe other hand, for cell 3 (having large f3 ¼ 40 kHz) in

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Fig. 5. Dependence of threshold VWD on noise intensity VN for WD

( f ¼ 30Hz). (a) cell 1 (� ¼ 5:01� 10�8 ��1 m�1, " ¼ 5:42, f3 ¼ 160Hz),

(b) cell 2 (� ¼ 4:30� 10�7 ��1 m�1, " ¼ 5:63, f3 ¼ 2:45 kHz), (c) cell 3

(� ¼ 2:87� 10�6 ��1 m�1, " ¼ 5:59, f3 ¼ 40 kHz). By changing fc, we

checked the theoretical linear relationship V2WD ¼ V2

0 þ bV2N. Here V0

indicates the threshold voltage in the absence of noise (VN ¼ 0), and slope bindicates the sensitivity of the response of WD to noise. See Fig. 6.

-2.0

-1.5

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102 103 104 105 106

cell 1cell 2cell 3

b

fc [Hz]

fc* (cell 2)

Fig. 6. Sensitivity b of the response of WD to noise for each cell

( f ¼ 30Hz). The noise with b ¼ 0 ( fc ¼ f �c ) is neutral to the onset of WDs.

It strongly depends on cells with different f3 (or �).

J.-H. HUHJ. Phys. Soc. Jpn. 81 (2012) 104602 FULL PAPERS

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Fig. 5(c), b is negative for fc ¼ 200Hz to 100 kHz; VWD

decreases with increasing VN (i.e., downward thresholdshift). Namely, noise ( fc < 200 kHz) plays a role indestabilizing WD (i.e., facilitation of the onset of WDs).Notice that even almost white noise [e.g., fc ¼ 100 kHzhaving �N � 8:5 �s (� 30�1 s ¼ f�1 for WD)] could con-tribute to destabilizing WD. Also, b � 0 appears for fc ¼200 kHz. This means the noise makes no contribution to thethreshold shift. For cell 2 (having intermediate f3 ¼ 2:45kHz) in Fig. 5(b), we find that b is positive for fc ¼ 10

to 200 kHz, b � 0 for fc ¼ 5 kHz, and b is negative forfc ¼ 200Hz to 2 kHz.

The stabilization (b > 0) and destabilization (b < 0)effects on the onset of WDs are switched at a characteristiccutoff frequency fc ¼ f �c (b ¼ 0). The colored noise havingfc � f �c no longer functions as normal noise (to suppress theinstability). Accordingly, VWD ¼ 0 exists at V�

N (for cell 2and 3); that is, pure noise-induced WDs appear.23) Forinstance (cell 2), the colored noise with V�

N � 13V andfc ¼ 2 kHz induces WD in the absence of an ac field(V ¼ 0). Voltage V�

N increases monotonically with fc, anddiverges at fc ¼ f �c . At f

�c , noise has no effect; that is, it is

neutral with respect to the onset of WDs (i.e., b ¼ 0).24,25)

Figure 6 shows the response sensitivity b as a function offc. b increases smoothly with increasing fc and saturates atb � 0:5 with the noise approaching white noise ( fc ! 1;�N ! 0). f �c (b ¼ 0) depends strongly on f3.

26) Moreover,b increased smoothly with increasing f (< f2).

16,26) Inparticular, we have found a characteristic relationshipf �c ¼ hf �3 (� ¼ 1:4, h ¼ 0:1), which was described in detailin a recent publication.26) This means that the stabilization( fc > f �c ) or destabilization effects ( fc < f �c ) on WDs aredetermined by the relationship between the external time-scale (1=fc) and intrinsic timescale (1=f3; strongly relates to��). For the crossover at fc ¼ f �c (> f3), the stabilizationeffect of high-frequency components (> f3) and thedestabilization effect of low-frequency components (< f3)in the CH mechanism of WD are canceled out each other.It is independent of f (for f1 < f < f2).

26)

On the other hand, considering the dependence ofstructures on noise,16) we analyzed how the thresholds ofDFC ( f2 < f < f3) and WD ( f1 < f < f2) vary as a functionof fc. Figure 7 shows the behavior of the threshold VDFCðVNÞfor DFC ( f ¼ 2:4 kHz). For relatively higher fc (� 5 kHz),VDFC decreases slightly and then increases smoothly withincreasing VN. This means that the degree of noise-inducedstabilization and destabilization of DFCs changes as afunction of VN (with fixed fc). Moreover, there seems to exista noise intensity V y

N that is most appropriate for achievingDFCs at minimal VDFC (VN ¼ Vy

N); this result may signifythe possibility of a stochastic resonance [compare toVDCVðVNÞ described below]. For example, VDFCð fc ¼5 kHzÞ � 145V (< VDFC ¼ 150V at VN ¼ 0) at VN ¼VyN � 5V. For lower fc (� 2 kHz), VDFC decreases mono-

tonically with increasing VN. In other words, when fc � 2

kHz, noise plays a role in destabilizing DFCs. Therefore, thebehavior of function VDFCðVNÞ cannot be explained by thelinear relationship (for WDs). However, above the small-VN

region (> 10V) where VDFCðVNÞ increases or decreases,VDFCðVNÞ seems to obey the linear relationship. Although thethresholds VWDð f Þ and VDFCð f Þ vary smoothly in Fig. 3, the

influence of noise on these thresholds is quite different fromeach other.

As described in x1, DFCs form on the background of PW[see �2 of the DFC in Fig. 4(c), which is similar to the width(2�PW) of bands of PW in Fig. 4(e)]. Because noise plays arole in destabilizing PW instability, threshold VPW decreasessmoothly with increasing VN (see x3.4). Therefore, wespeculate that the influence of noise on the PW instabilitydominates for small VN (< 10V) but the influence of noiseon WD dominates for large VN (> 10V). This may explainwhy the threshold VDFC shows such a peculiar behavior inFig. 7.

Furthermore, we measured the variation in the thresholdof LFWDs ( f ¼ 0:8Hz < f1) upon changing fc. For thismuch lower ac frequency f , the LFWD pattern blinks on andoff at f ¼ 0:8Hz; that is, the electroconvection-accompanieddirector structure blinks at f ¼ 0:8Hz. Owing to the blink ofthe pattern, the rolls of LFWDs [Fig. 4(b)] are viewed lessregular than those of stationary WDs [Fig. 4(a)]. Althoughthe flexoelectric effect is added to the CH generationmechanism for LFWDs, dominant convection structureshold (similar to WDs). Thus, the influence of noise onLFWDs seems to be similar to that of WDs, as shown inFig. 8. The linear relationship [for WDs in Fig. 5] can bealso used for LFWDs, except when fc ¼ 1 kHz; suchexceptional case sometimes appears also in WDs.16) Thefunction VLFWDðVNÞ is qualitatively the same as VWDðVNÞ inFig. 5. Moreover, pure noise-induced LFWDs appear (e.g.,VLFWD ¼ 0 at VN ¼ V�

N � 15V and fc ¼ 1 kHz). The flexo-electric effect does not appear in the influence of noise onWDs (at present experimental limitations), whereas it givesrise to unignorable deviation from the CH mechanism[especially, for low frequencies comparable with 1=�d; seealso the sharp change of the threshold function Vthð f Þ aroundf1 in Fig. 3].20)

3.3 Influence of noise in the dielectric regimeIn the dielectric regime ( f ¼ 2:8 kHz > f3), the threshold

variation of DCVs was measured in the same way. Figure 9

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VD

FC[V

]

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[V]

4035302515105 20

Fig. 7. Dependence of threshold VDFC on VN for DFC ( f ¼ 2:4 kHz) in

the change of fc.

J.-H. HUHJ. Phys. Soc. Jpn. 81 (2012) 104602 FULL PAPERS

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shows the threshold function VDCVðVNÞ for different fc.Clearly, the influence of noise on VDCV is markedly differentfrom VWDðVNÞ, VDFCðVNÞ, and VLFWDðVNÞ found in theconduction regime ( f < f3). Noise gives rise to almost nothreshold-shift until a considerable VN (> 60V) is applied(compare with the situation for WDs, DFCs, and LFWDsin Figs. 5–8). As VN increases above 60V, VDCV decreasessmoothly and then increases rather steeply after reaching aminimal value at a characteristic intensity Vy

N. Voltage V yN

is almost identical (VyN � 140V) for fc ¼ 50, 100, and

200 kHz, and the minimal threshold VDCVðVN ¼ VyNÞ de-

creases with increasing fc. This means that DCVs can betriggered with minimal ac-electric force by selecting anappropriate Vy

N. This VN-dependent behavior of VDCV isreminiscent of stochastic resonance in spatially extendedsystems.27) Although we did not measure VDCVðVNÞ for large

VN for lower fc (� 10 kHz) because of experimentallimitations, we expect the results to be similar to those ofwhite-like noise with higher fc (� 50 kHz). Similar to thecase of DFCs, such unique behavior of VDCVðVNÞ may bestrongly affected by PW, which serves as a backgroundpattern for DFCs (see below).

In fact, a primary (normal-roll) instability that appearsbelow VDCV should be studied to understand the thresholdproblem in the (pure) dielectric regime;17) it can be separatedfrom the PW. The present NLC cells are not available forthis. The normal-roll instability could be observed inMBBA with much lower f3

28) (or other NLCs such asMerck Phase 529)).

3.4 Influence of noise on other instabilitiesFigure 10 shows the threshold function VPWðVNÞ for

PW ( f ¼ 2:4 kHz > f2). As VN increases, VPWðVNÞ de-creases smoothly after being nearly constant at small VN

(< 10{40V). In other words, adequately large noiseintensity destabilizes PWs. However, above the much largerintensity VN � 85V, regular PWs [in Fig. 4(e)] disintegrateupon the appearance of noise-initiated turbulence.16) More-over, for large VN > 10V, VPWðVNÞ depends on fc. Thesedestabilization effects may explain the behavior of VDFCðVNÞand VDCVðVNÞ because PW serves the background patternfor DFCs and DCVs. The decrease of VDFC for small VN

(< 10V, Fig. 7) and the almost null variation of VDCV forsmall VN (< 60V, Fig. 9) may be affected by the super-position of PWs.

As reported in previous studies,17,21,22) PWs have aperiodic director structure [n ¼ nðx; yÞ] in the xy-plane(i.e., the electrode plane), which can be distinguished fromthe other patterns [n ¼ nðx; zÞ for WDs; n ¼ nðx; y; zÞ forDFCs and CVs]. The in-plane director rotation for PWseems to respond simply to the noise field. Although themechanism of PW instabilities is not sufficiently understood,the periodic director modulation in PWs is facilitated by theappropriate VN; namely, noise plays a role in destabilizingPWs.

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VD

CV [

V]

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Fig. 9. Dependence of threshold VDCV on VN for DCV ( f ¼ 2:8 kHz) in

the change of fc. A minimal threshold exists at the most relevant noise

intensity for each fc (e.g., VDCV ¼ 138V at VyN ¼ 140V for fc ¼ 200 kHz).

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VL

FWD

2 [V

2 ]

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*2

700600500400300200100

Fig. 8. Dependence of threshold VLFWD on VN for LFWD ( f ¼ 0:8Hz) inthe change of fc. Similar to the case of WD in Fig. 5, the relationship

V2LFWD ¼ V2

0 þ bV2N roughly fits except for fc ¼ 1 kHz. Here V0 indicates the

threshold voltage for LFWD in the absence of noise (VN ¼ 0).

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]

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Fig. 10. Dependence of threshold VPW on VN for PW ( f ¼ 2:4 kHz) in the

change of fc.

J.-H. HUHJ. Phys. Soc. Jpn. 81 (2012) 104602 FULL PAPERS

104602-6 #2012 The Physical Society of Japan

In addition, we investigated the influence of noise on dc-driven IMs, as shown in Fig. 11. Depending on fc, noisecontributes to stabilizing ( fc � 500Hz) or destabilizing( fc � 200Hz) IMs. The function VIMðVNÞ agrees qualita-tively with VWDðVNÞ and VLFWDðVNÞ, although the linearrelationship does not seem to hold definitely (see also theinset of Fig. 11). Although the pattern of IM is somewhatdifferent from those of WD and LFWD (Fig. 4), similarconvection structures (without additional structure-modula-tions such as PW) may result in the qualitatively similarresponse to noise.

4. Summary

We investigated the influence of noise on the thresholdsfor various EHIs depending on f . Noise intensity VN andcutoff frequency fc play a crucial role in determining theinfluence with respect to the corresponding EHI structures(mechanisms).

For the typical WD in the conduction regime ( f1 <f < f3), the expanded CH theory,12) in which quasi-whitenoise is considered, explains the upward shift in thethreshold VWDðVNÞ, which can be intuitively understood.Thus, the noise with a higher fc (> f �c ) contributes tostabilizing WDs because the onset of WD is suppressed bythe obstruction of periodic charge injections and the randomrecombination of space charge (due to the superposition ofnoise). The downward shift in VWDðVNÞ that occurs for noisewith a lower fc (< f �c ) is an interesting noise-inducedphenomenon. Namely, sufficiently colored noise contributesto destabilizing WDs. This may be understood by regardingnoise ( fc < f �c ) as Fourier modes satisfying the CHconditions (��f � 1 and �d f � 1). In particular, one canfind pure noise-induced WDs (VWD ¼ 0 at VN ¼ V�

N).Moreover, these explanations are applicable to LFWD( f < f1) because the flexoelectric effect is negligible to theinfluence of noise on LFWD. Also, dc-driven IMs showqualitatively similar to its influence on WDs because all ofthem are, in principle, (similar) convective instabilitieswithout additional structure-modulations such as PW. In the

case of WD, LFWD, and IM, which have similar EHIstructures, the flow field seems to be a dominant factor forthe noise-induced threshold shifts because the director fieldremains stationary.

Compared to WDs, the threshold VDFC for DFCs ( f2 <f < f3) shows quite a different behavior. At higher fc, VDFC

decreases smoothly and then increases with increasing VN.For lower fc, however, it decreases monotonically withincreasing VN. In particular, the stabilization and destabiliza-tion effects are altered by VN; therefore, the influence of noiseon DFCs is more complicated. The difference betweenVWDðVNÞ and VDFCðVNÞ should be explained by their pattern-formation mechanisms (i.e., director structures and flows).PW instability, a background pattern, is crucial in determin-ing the influence of noise on DFCs. Threshold VPW decreasessmoothly with increasing VN, independent of fc [after novariation (in small VN region)]. This behavior seems to berelated to the peculiar behavior of VDFCðVNÞ [and VDCVðVNÞ].

Moreover, PW also serves as a background pattern forDCVs in the dielectric regime ( f > f3). For DCVs, theinfluence of noise on VDCV is more noticeable and is notexpected from conventional theories.10,12,14) This resultsuggests that an optimal noise condition with an appropriateVN and fc may facilitate the onset of DCVs at the lowestdriving force. In other words, external noise that matches theintrinsic characteristics of the system is most effective intriggering instability (i.e., the oscillation of the director forDCVs), similar to stochastic resonance.5,9) This result forVDCVðVNÞ may also be understood on the basis of thedifference in the pattern-formation mechanisms for bothregimes. Contrary to WDs, the director oscillation (forDCVs) requires a stationary charge distribution, as describedin x1. For rather small VN, such a charge distribution mayhold for small modulations with sufficiently large V . Forlarge VN, however, the noise breaks the stationary chargedistribution. Nevertheless, to obtain the director oscillation,a stationary charge distribution must be constructed byincreasing V . This phenomenon leads to an increase in VDCV

via a minimal VDCV. In practice, the influence of noiseon DCVs may become complicated owing to the largedissipative energy of small-scale convections and thebackground-pattern PWs, as well as the director oscillation.The influence of noise may be changed by the dominantinstabilities in the system. Moreover, the details of the noise-induced pattern changes and the threshold shifts (for WDsand DCVs) were reported in our previous publications.16)

Unfortunately, there is no theoretical explanation for suchinfluences of noise in the dielectric regime ( f > f3),although the linear stability analysis has been intensivelystudied in the conduction regime (only for WD withf1 < f < f2).

10,12,14)

The stabilization effect from high frequency componentsof noise and the destabilization effects from low frequencyones always compete with each other, depending on internaltimescales of EHIs such as 1=f3 (or ��). In the case of WD,LFWD, and IM, such competition results in relatively simplechange in threshold shifts (i.e., a linear change). Thecouplings of the noise field to the director and flow fieldsof EHIs determine the response to the multiplicative noise.The details of the couplings could help us to understand theunique threshold problems of various EHIs. These are

5

10

15

20

25

30

35

0

200kHz10kHz1kHz500Hz200Hz100Hz

VIM

_dc [

V]

VN [V]

8

9

10

11

0

8070605040302010108642

Fig. 11. Dependence of threshold VIM on VN for dc-driven IM in the

change of fc.

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currently under study, in association with noise-controlledEHI structures.

Furthermore, compared with the influence of noise on theFredericks instability (FI) in NLCs,16) we also find theimportance of the underlying structures for EHIs. FI withoutfinite characteristic length (i.e., a spatially homogeneousstate) shows a simple threshold shift. Independent of f andfc, the threshold VF for FI decreases monotonically withincreasing VN. Thus, FI can be induced only by the pure(effective) intensity of the applied fields, independent of thetype of electric fields (ac, dc, or noise only or mixed). Seethe details of VFðVNÞ in our previous reports.16)

The present noise influence on nonlinear systems is veryimportant for understanding nonlinear dissipative systems innoisy environments. In particular, the nontrivial phenomenasuch as noise-induced subthreshold instability (or purenoise-induced instability) and stochastic resonance (in DFCsand DCVs) give us useful hints for applications to realbiosystems and neuroscience.30)

Acknowledgements

This study was partly supported by a Grant-in-Aid forScientific Research from the Ministry of Education, Culture,Sports, Science and Technology of Japan and the JapanSociety for the Promotion of Science (No. 22540394).

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