Confined Electroconvective and Flexoelectric Instabilities Deep in

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Confined Electroconvective and Flexoelectric Instabilities Deep in the Freedericksz State of Nematic CB7CB Kanakapura Seshappa Krishnamurthy, Nani Babu Palakurthy, and Channabasaveshwar V. Yelamaggad J. Phys. Chem. B, Just Accepted Manuscript • Publication Date (Web): 10 May 2017 Downloaded from http://pubs.acs.org on May 10, 2017

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The Journal of Physical Chemistry

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Confined Electroconvective and Flexoelectric Instabilities Deep in the Freedericksz State of Nematic CB7CB Kanakapura S. Krishnamurthy,* Nani Babu Palakurthy, and Channabasaveshwar V. Yelamaggad Centre for Nano and Soft Matter Sciences, P. O. Box 1329, Jalahalli, Bangalore 560013, India

ABSTRACT: We report wormlike flexoelectric structures evolving deep in the Freedericksz state of a nematic layer of the liquid crystal cyanobiphenyl-(CH2)7-cyanobiphenyl. They form in the predominantly splay-bend thin boundary layers and are built up of solitary flexoelectric domains of the Bobylev-Pikin type. Their formation is possibly triggered by the gradient flexoelectric surface instability that remains optically discernible up to unusually high frequencies. The threshold voltage at which the worms form scales as square root of the frequency; in their extended state, worms often appear as labyrinthine structures on a section of loops that separate regions of opposite director deviation. Such asymmetric loops are also derived through pincement-like dissociation of ring-shaped walls. Formation of isolated domains of bulk electroconvection precedes the onset of surface instabilities. In essence, far above the Freedericksz threshold, the twisted nematic layer behaves as a combination of two orthogonally oriented planar half-layers destabilized by localized flexoelectric distortion.

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1. INTRODUCTION

The random evolution of confined structures within an initially homogeneous medium is a remarkable nonlinear phenomenon that occurs in a rich variety of physical, chemical and biological dissipative systems under diverse driving mechanisms.1,2 Some examples of such structures are “pulses” in thermoconvection,3,4 “worms” in electroconvection (EC),5,6 “oscillons” in granular dynamics7 and “spots” in reaction-diffusion systems.8 Localized electroconvection (LEC), which is of particular relevance to this study, was first recognized by Joets and Ribotta in planarly aligned calamitic nematics excited by a. c. fields.9 It occurred at the primary bifurcation, as a system of travelling normal rolls with the wave vector k along the initial alignment axis or director no, confined to nearly elliptical stationary domains embedded in the quiescent fluid. The domains enlarged toward the cell-filling state with increasing control parameter. In subsequent studies5,6 on the nematic liquid crystal I52 in planar geometry, a different class of LEC state was detected. It appeared through a subcritical bifurcation in the form of so-called worms of indefinite length along no and constant narrow width. These dynamical objects consisted of swiftly travelling oblique rolls under slowly drifting envelopes. The LEC instabilities in refs 5, 6 and 9 pertain to nematics with negative dielectric anisotropy [εa= (ε||–ε⊥) < 0, || and ⊥ denoting directions relative to no] and positive conductivity anisotropy [σa= (σ||– σ⊥) > 0]. In such (– +) class of materials, anisotropic electroconvection is described by the standard model10 based on the Carr-Helfrichmechanism.11 Even in (– –) nematics, for which the Carr-Helfrich mechanism does not apply, localized states have been observed in the form of butterfly-like flickering objects at high frequencies and clusters of short oblique rolls at low frequencies.12 In planar (+ +) nematics, the nature of EC depends on the magnitude of εa. For small εa, the Freedericksz state at higher voltages, which is theoretically predicted to be stable against standard EC,13 is


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3 found to be destabilized by wormlike structures.14 For large εa, as in pentyl cyanobiphenyl, far above the Freedericksz threshold UF, nonstandard EC is found in the form of extended cellular structures and rolls.15 This study concerns several remarkable localized states that arise, under competing dielectric, flexoelectric (flexo-, for short) and electroconvective modes, in a (+ +) nematic with an underlying heliconical or twist-bend nematic (Ntb) phase. Most striking of these are the wormlike and labyrinthine structures composed of solitary flexodomains. The instabilities discussed herein are mainly from experiments on quarter-turn twisted planar layers that enabled ready separation of bulk and surface effects; however, we also provide several examples of localized states in untwisted planar cells to point out the generality of the results. 2. EXPERIMENTAL DETAILS

The electric field experiments were carried out on planar 90o-twisted and untwisted nematic layers of the dimeric bent mesogen cyanobiphenyl-(CH2)7-cyanobiphenyl (CB7CB), having the phase sequence: Ntb (103.3 oC) N (116.5 oC) isotropic. The sample temperature T was held constant to an accuracy of ±0.1 oC using an Instec HCS402 hot-stage coupled to a STC200 temperature controller. The experiments were performed at 105

o

C, unless

mentioned otherwise. The sample cells (from M/s AWAT, Poland) were sandwich type, constructed of indium tin oxide coated glass plates and having a thickness d in the range 5-20 µm. Spacer rods between the electrodes ensured uniformity of cell gap. The electrodes were coated with polyimide (SE-130 from Nissan Chemical Industries) and buffed unidirectionally to obtain planar anchoring.

Optical textures were examined using a Carl-Zeiss Axio

Imager.M1m polarizing microscope with an attached AxioCam MRc5 digital camera. The voltage source was a Stanford Research Systems function generator (DS345) connected to a FLC Electronics voltage amplifier (model A800). Sine wave field was used for excitation (unless otherwise stated) and the applied voltage U(t)=√2 U sin 2πft was measured with a



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4 Keithley-2002 multimeter. Cited voltages U are the rms values. For measuring the cell capacitance as a function of voltage, an Agilent 4284A precision LCR meter was used. For reference, we take the initial director no as along x in planar untwisted cells; in twist cells, it is along x at the bottom substrate and along y at the top substrate. The applied field E is along the layer normal z. In Figure 1 showing the twisted geometry, L and D are, respectively, the laevo and dextro regions. Figures 1a and 1b show the director distributions for zero and nonzero pretilts, respectively. Region D in Figure 1b involves both splay and twist distortions, with the midplane director orthogonal to z. In what follows, the polarizeranalyser setting will be indicated as P(α)–A(β), where α and β are the angles, in degrees, made by the transmission axes with the x direction.

L

z

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L

D

x

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Figure 1. Sample geometry. (a) Director pattern in the absence of pretilt, with the twist-sense being anticlockwise or laevo in region L and clockwise or dextro in region D. (b) Pre-tilted director pattern (for the cells used). In region D, the twist is coupled to splay, with the midplane director in the xyplane. In region L, the distortion is pure twist. 3. RESULTS AND DISCUSSION

3.1. Anisotropic Electrical Properties. From measurements of cell capacitance and conductance as a function of voltage, we find nematic CB7CB to be positive in both dielectric and conductivity anisotropies. Typical plots of σ(U) and ε(U) obtained at 105 oC are shown in Figure 2. Notably, the material is of high electrical conductivity, with σ of the order of µS/m. The relative permittivity εr of 7.05 below the Freedericksz threshold (≈1.9 V) may be taken as ε⊥; similarly, the extrapolated value of εr=8.82 corresponding to U–1=0, obtained from the εr(U–1) plot, nearly equals ε||. These values are broadly in agreement with


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5 previous measurements.16 The principal conductivity components obtained likewise are

σ⊥=0.93 µS/m and σ||=1.25 µS/m.

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U (V) Figure 2. Electrical conductivity σ of the 90o-twisted planar sample (d=10 µm) as a function of the applied rms voltage U showing the anisotropy σa as positive. 10 kHz. T=105 oC. Inset: Voltage variation of relative permittivity εr indicating a well defined Freedericksz threshold UF (≈1.9 V) and positive dielectric anisotropy εa.

3.2. Flexoelectric Instability in Low-Frequency AC Fields. When the frequency f of the applied field is sufficiently large for ionic effects to be negligible and U is gradually raised, the Freedericksz transition in CB7CB becomes discernible through a homogeneous change in birefringence colour at UF ≈1.9 V; more specifically, at 1 k Hz, UF is 1.84 V for region L and 1.94 V for region D, indicating the presence of a slight pretilt. In general, the external bias at which the dielectric reorientation occurs is frequency dependent, showing a pronounced increase with decreasing f in the mHz regime. In particular, below 0.2 Hz, the base state remains unperturbed even at voltages well above the high frequency threshold UF. This suppression of dielectric response is traceable to the large electrical conductivity of the sample. Diffuse counter-ion layers that form next to the electrodes completely screen the applied field E as long as it remains below some critical value Ec, which, for static fields, is Ec=Uc /d =√(2nikBT/ε⊥), where ni is the ion density and kB, the Boltzmann constant.17 In fact, in the mHz frequency regime, the base state gets destabilized first by the gradient



6 flexoelectric distortion, instead of the Freedericksz transition, at a frequency dependent threshold UG>UF. Close to UG, the instability is transient, appearing periodically, with the maximum distortion amplitude corresponding to the peak positive and negative voltages, and disappearing completely in a time less than half the period τ =1/f; in a 90o-twist cell, the domain lines of oppositely twisted regions enclose an angle γ that is acute (obtuse) for positive (negative) voltages. Further, the angular deviation of domain lines, with respect to the diagonal direction, increases continually with frequency (see Figure 3) so that above ~1 Hz the stripes appear either along x or y. By implication, the thin layer above (below) the midplane in which the distortion originates for positive (negative) E, progressively approaches the negative electrode as the sine wave frequency is raised. Clearly, the faster the voltage change at polarity switchings the closer to the cathode would be the plane of maximum deformation. This effect is further confirmed using, for excitation, a low frequency square wave field. Then the flexodomains, which appear transiently soon after every polarity switch, are practically along the easy axis at the cathode. We have previously discussed the spatiotemporal aspect of square wave driven flexoinstability in a bent core nematic.18 Briefly,

Figure 3. Select frames from time lapse recordings of the texture of oppositely twisted neighboring regions L and D; U=4.5 V and f=0.02 Hz (a, b), 0.04 Hz (c, d) and 0.1 Hz (e, f). With increasing f, the plane of maximum distortion shifts progressively away from the z=0 midplane; correspondingly, the pattern wave vector tends toward the horizontal [(a)→(c) →(e)] or vertical [(b) →(d) →(f)]. Patterned state is transient, forming around the peak positive (left) and negative (right) voltages, and prevailing for a few seconds. Scale: 5 µm/div. d=10 µm. P(0)–A(45).


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7 field inhomogeneity due to ionic transport between the electrodes, which is not considered in the Bobylev-Pikin model of bulk distortion, plays a key role in bringing about the polarity dependent surface-like effect. The field asymmetry required to explain polarity dependence is obtained assuming selective adsorption of negative ions and creation of intrinsic double layers.19 Spatial gradients in electric field E, due to ionic movements following each polarity reversal, would lead to the gradient flexoelectric (GF) torque ΓGF=(es–eb) n.∇E contributing to the distortion; es and eb here are, respectively, the splay and bend flexocoefficients that relate the flexopolarization density P with the splay distortion S= n(∇ ∇. n) and bend distortion B= nx(∇ ∇ x n) through P = es S + ebB. Interestingly, in the frequency region 0.2-2 Hz, in addition to the just described gradient flexoeffect, we also observe the steady field volume flexoinstability envisaged in the Bobylev-Pikin model.20 The former effect manifests as narrowly-spaced periodic domains along x or y, while the latter, as relatively widely-spaced domains nearly along the midplane

Figure 4. (Fi) Patterned states excited in a 90o-twisted layer of nematic CB7CB by sine wave field of frequency 0.58 Hz and voltage 3.3 V. Fi denotes the ith frame of the time series captured with a single polarizer P(45), with the frame rate fr=10f. F1 and F6 correspond to the hybrid volumeflexoelectric and gradient flexoelectric states formed soon after successive polarity changes. F3 and F5 show Bobylev-Pikin states with voltage dependent line density. Notably, Freedericksz reorientation does not occur prior to the onset of periodic instabilities. d= 10 µm; scale: 5 µm/div.



8 director. The two coupled modes appear superimposed soon after each polarity change; but the gradient effect decays fast so that, for better part of the half period before the next polarity switch, only the bulk effect persists with a voltage varying domain density. In frames Fi of Figure 4 exemplifying these features, F1 and F6 show the texture of the hybrid state formed after successive polarity reversals; the gradient flexodomains are nearly along y in F1 and along x in F6. We may note here that, in a time-serial recording with the frame rate fr=2mf, m being an integer, (m–1) frames separate the frames recorded at successive polarity changes. Thus, in frames F2-F5 separating F1 and F6 of Figure 4, only the bulk flexodistortion manifests. On raising f to 2 Hz or more, we find the bulk flexodomains ceasing to form, but, in the Freedericksz state, the gradient flexopattern continues to form transiently at the cathode following each polarity reversal (see Figure 5). The flexopattern, as seen in Figure 5(d-g), exhibits maximum visibility when the incident light vibrates along the domain lines and poor

Figure 5. The periodic gradient flexostate obtained close to the substrates in a 10 µm thick nematic layer. In frames (a-c), m=2, f=2.4 Hz and U=3.7 V. While (a), recorded at one polarity switch, shows vertical stripes, (c), recorded at the next, shows horizontal stripes; and (b) shows the base state, indicating the transient nature of the instability developed soon after a polarity change. In (a-c), the polarizer is at 45o to x. (d-g) Successive frames of a time series recorded with m=1, f=4.8 Hz and U=4.5 V at successive polarity switchings, showing the dependence of pattern-visibility on polarizerorientation. P||x in (d, e) and P||y in (f, g). Scale: 5 µm/ div.

visibility when it vibrates transverse to the lines. This feature, also seen in planar untwisted samples, is understandable since the Mauguin condition of wave guidance is no longer satisfied in the Freedericksz distorted state well above UF. At higher frequencies, the gradient flexobands remain discernible in time lapse recordings even upto about 40 Hz, although the


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9 correlation between their formation and field direction cannot be ascertained. We recall these results later in discussing the formation of flexoelectric worms. 3.3. Localized Electroconvective and Flexoelectric States. Upon increasing U to ~5UF at 1 kHz, and then slowly decreasing f, we find the uniform quasihomeotropic state destabilized first by a pattern of periodic diagonal stripes at f= fc. Observed only in birefringence contrast, the pattern shows different line densities for P(0)–A(90) and P(45)– A(135), as illustrated in Figures 6a and 6b for region L. Previous experiments have shown stripe patterns in the splay Freedericksz state, with the stripes reported to be normal to the initial director no in a (+ +) rodlike nematic,15 and parallel to no in a (+ –) bent-rod nematic.21,22 The director pattern corresponding to the present case, in which the stripes are normal to no, has two possibilities shown schematically in Figures 6c and 6d. The first visualizes a perturbation of the midplane homeotropic state, with the director defined by the

Figure 6. Typical EC pattern observed between crossed polarizers (a) P(0)–A(90) and (b) P(45)– A(135); 45 Hz, 10.9 V. (c, d) Possible director patterns along the midplane easy axis x' (or the wave vector k). Scale: 10 µm/div.

tilt angle in the x'z-plane θ=(π/2)–θo sin (kx'), and the deviation from x' in the xy-plane ϕ =ϕo sin (kx'), where k=2π/λ, λ being the pattern-period. In the second, the tilt is constant, but the azimuthal deviation is periodic, with ϕ =ϕo sin (kx'). Since the wave-guiding effect of the base state would be absent in the reoriented state at U>>UF, the appearance of alternate dark bands D in Figure 6a at identical positions, as successive dark bands, in Figure 6b agrees with



10 the model in Figure 6c. Thus the pattern, which has a period of ~6d, represents what may be called the wide domain mode (WDM), similar to the prewavy mode in bent-core nematics.23 The wide domain structure at frequency fc, under increasing U, disintegrates first into coffee-bean like formations (observed also in planar24 and quasiplanar25 nematics along no) distributed diagonally. While some of the beans disappear altogether, others develop into loops of varied geometry, within which the first of confined states forms. As illustrated in Figure 7, it appears as a system of rolls with the wave vector direction degenerate, indicating the modulation as originating in the bulk isotropic region. The rolls are propagative (see Supporting Information for a movie clip V1.avi showing this feature). If we take the pattern period as the separation between alternate bright bands, it lies between d and 2d, as in the classical Carr-Helfrich (CH) mode. We need to note here that the Freedericksz state in (+ +) materials with a relatively large value of εa/ε⊥, typically exceeding 0.09 in calamitics, remains stable in the CH theory,10 in the absence of contribution to space charges from flexopolarization.

Figure 7. Travelling wave electroconvection localized in domains of varied geometry in 90o-twist cells. Frames (a)-(d) show the wave vector as degenerate in the layer plane. Axially crossed polarizers P(0)–A(90). (a-c) 45 Hz, 11.8 V, d= 10 µm; (d) 80 Hz, 8 V, d= 5 µm. Scale:10 µm/div.

Under increasing dielectric torque, patterned states tend to occur localized away from the midplane. We find this in the new type of worm instability that evolves in the peripheral


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11 region of isolated EC domains such as seen in Figure 7. In Figure 8a, a well developed worm stretched vertically along the boundary of a loop is shown. It exhibits an alternation of birefringence colour from one band to the next, as seen between slightly uncrossed polarizers, indicating periodicity in the deviation φ of the director. Even in natural light, worms are vividly seen. Evidently, they involve large amplitudes of θ and φ. While they may extend in any direction, their wave vector remains essentially normal to either of the substrate easy axes. The basic units of which the worm is constituted are solitary flexodomains, which are analogous to pulses in thermoconvection.3,4 Between axially crossed polarizers P(0)–A(90), each of these units appears as a pair of short parallel birefringent bands, as seen within the

Figure 8. (a, b) Worms extending along either of the substrate alignment directions, at 13 V, 45 Hz. In (b), block arrows indicate the drift directions; dashed arrow points to the edge dislocation where new domains nucleate. (c) Extended state of worms at 13 V, 40 Hz. (d, e) Long flexodomains formed near the two substrates in a planar untwisted sample appearing alternately at successive polarity reversals under square wave excitation; 7.2 V, 4.42 Hz (= fr/2). Double-arrows represent polarizers. Scale: (a, b) 2 µm/div., (c) 4 µm/div., (d, e) 5 µm/div.

encircled regions of Figure 8a. Their appearance along either of easy axes in twist cells shows their evolution as confined to the two predominantly splay-bend regions next to the electrodes. The worm morphology is a result of rapid evolution of such units, one next to the other. Worms are generally metastable and their decay is marked by a step-wise decrease in



12 the number of constituting flexounits. With increasing voltage above the worm threshold Uw (or decreasing frequency below fw, fw being the critical frequency at which the instability sets in at a fixed voltage), as the flexodomains grow in number and length, the worm state tends toward the extended state (see Figure 8c). Our identification of worms as of flexoelectric origin rests on two crucial characteristics. First, the wave vector of the worm is found to be normal to the easy axis at the boundary where it forms; in twisted samples, this is evident from the location of focal planes; more definitively, this feature is evident in untwisted samples showing the birefringent domains along no. Second, the evolution of worm structure is polarity dependent as ascertained from time lapse recordings of the domains under low frequency square wave driving, keeping f synchronized with the frame rate. This is exemplified in Figures 8d and 8e showing the textures of a planar sample at successive polarity switchings. Quite often, the flexodomains in a worm exhibit a relative drift along the wave vector. This motion usually takes place outwards of an edge dislocation (Figure 8b) that 5.3

t (s)

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Figure 9. Profiles of transmitted light intensity along x for a rectangle enclosing the worm in Figure 8b as a function of time. Successive profiles are separated by 106 ms. 1 pixel=0.11 µm.

marks the site where new domains evolve in succession and keep drifting away (see Supporting Information for movie clips V2.avi and V3.avi showing this feature for twisted and untwisted cells). The drift speed increases with distance from the dislocation. In Figure 9


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13 illustrating the outward domain drift relative to the central region where new domains nucleate, the speed in the terminal parts is ~3 µm/s. The worm instability threshold Uw is frequency varying (Figure 10). This indicates its probable correlation with some critical value 20

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of electric coherence length ξc=(k /εoεa)1/2(1/Es), where k is the effective elastic constant (practically the same as the splay modulus k11 since the bend modulus k33 is very low) and Es is the frequency varying critical surface electric field. In planar cells, at higher frequencies (>100 Hz), indefinitely long, solitary flexodomains form along x at the worm threshold. Over several minutes, they open out into loops at random intervals along their length. In Figures 11a and 11b showing this unique feature, the rotational symmetry between the textures at the two electrodes may be noted. That the straight segments of the loops are indeed ½-disclinations becomes clear from the texture in Figure 11c. The director fields for the solitary flexodomains and the loops they interconnect are shown in Figure 11d-f. Qualitatively, the unusual structure of loops follows from elastic energy considerations. At the temperature of our study, from ref. 16, the elastic constants are, approximately, k33 (bend) = 0.4 pN, k22 (twist) = 4.5 pN and k11 (splay) = 7.2 pN; thus the elastic energy per unit length for the splay-rich -1/2 line is considerably more than that for the ACS Paragon Plus Environment


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Figure 11. (a, b) Long flexodomains in the opposite boundary regions, with D-shaped loops developed at irregular intervals; 200 Hz, 20 V. (c) Texture obtained by gradually lowering f from 200 Hz to 55 Hz at 12.5 V, showing the loops, each with a disclination L on one side (along y) and worm structure W on the opposite side. Planar cell, d=8.4 µm. (d) The director field corresponding to a solitary flexodomain in the xy-plane; pins indicate out of plane tilts and thick vertical lines, singular lines; (e) The structure in the central xz-plane of the domain. (f) Structure of the loop in the xy-plane; long vertical line is a +1/2 disclination involving low-energy bend-rich distortion; an array of flexodomains form on the opposite side. Scale: 5 µm/div.

bend-rich +1/2 line. The one sided growth of flexodomains in a loop appears thus to minimize the effect of splay and lead to an effective flexoelectric overbalancing of the elastic cost. Existence of the Bobylev-Pikin instability at high frequencies as observed here points, in addition to the presence of strong double layer field, to a high value of effective flexocoefficient e=(es+eb). For CB7CB, a large value of flexoelastic ratio e/(k11+ k33)=3.67 C/(Nm) has been reported.26 Assuming this as temperature independent, we obtain e=27.8


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15 pC/m. This value amply satisfies the Bobylev-Pikin instability condition e2>(k11+ k22)εoεa/2 or e>10.2 pC/m. The director profile visualized in Figure 11f is well supported by the labyrinthine structures such as in Figures 12a and 12b, developed in a planar layer, in the course of transition to the extended flexoelectric state. The bright spots at turning points of the winding domain seen in Figure 12b (see the inset showing enlargement of the indicated rectangular area) correspond to singular regions. Varying contrast and focus between various frames of

Figure 12. (a,b) Successive frames of a time series showing labyrinthine directed growth (along x) of flexodomains in a planar untwisted sample; 3.96 Hz, 8 V. (c-f) Asymmetric labyrinthine growth of disclination loops in a 90o-twisted nematic layer; loops are localized near the top substrate in (c,d) and bottom substrate in (e,f); (c,e) 60 Hz, 8.5 V, (d,f) 28 Hz, 12.5 V. Scale: (a,b) 10 µm/div, (c-f) 5 µm/div. P(0) in (a,b), P(45)-A(135) in (c-f).

the time series is a consequence of polarity dependent build up of flexodistortion in one halfcycle and its partial decay in the next. Labyrinthine structure very similar to that in Figure 12a is also found for the twisted sample, except that the directed growth near the two substrates now occurs along orthogonal directions (x, y), as exemplified in Figure 12c-f. This is to be expected since, in twisted samples in the quasihomeotropic state (U>>UF), the



16 azimuthal orientation would be nearly saturated in the boundary layers so that the distortion therein is mostly splay-bend, as in planar samples. The dynamics associated with the winding flexopattern is rather complex (see Supporting information for a movie clip V4.avi showing this feature), involving space charges generated by flexopolarization and also influenced by the underlying bulk electroconvection. Broadly, under ionic influence, flexodomains develop, along their length, a regular modulation (see Figures 12c and 12e) that is propagative, just as found in the so-called corkscrew instability27 within the Brochard-Leger walls.28,29 The results from low frequency experiments (see Figure 5) suggest a possible correlation between the generation of flexoelectric worms and the gradient flexoelectric (GF) instability. The latter, as earlier noted (Figure 5), appears at the cathode after each polarity switch as very closely spaced domains extending along the alignment direction therein. This instability is cell filling in the absence of other superimposing patterned states, but becomes localized when competing modes exist. In Figure 13, we present textures of coexisting GF and worm instabilities. While, in Figures 13a and 13b, the extended worms formed near the two

Figure 13. Frames from timelapse recordings showing narrow gradient flexoelectric domains formed transiently (at the cathode, as inferable from experiments in the mHz region) after each polarity


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17 reversal, and relatively stable worms, in a twist cell. (a, b)10 Hz, 7.1 V, P(45). (c) 20 Hz, 9 V, P(45)– A(135). d=10 µm. Scale: 5 µm/div.

electrodes, due to their relatively longer decay time, appear to change marginally between polarity reversals, the GF domains switch between the alignment directions at the substrates. It appears that the surface-like GF distortions trigger the evolution of worms in substantially thick layers contiguous to the two substrates. The GF domains formed along y in Figure 13c are confined to diagonal bands of the coexisting prewavy instability. We also see here localized EC (e.g., in the region demarcated by the ellipse) and emphatic worm-distortion. In general, depending on the structure of coexisting instabilities, GF domains are found confined to diagonal stripes, circular patches or regions surrounding electroconvecting domains.

Figure 14. (a) A ring-shaped wall in a 10 µm thick sample exhibiting a banded birefringent structure; 30 Hz, 13.0 V. (b) Ring walls of various sizes drifting diagonally in the arrow-direction; 28 Hz, 12.3 V. (c) A pair of loops from the dissociation of a ring wall.

Finally, we turn to the confined instability manifesting as ring-shaped walls with embedded fluctuating domains. It is observed, without a definite threshold, above the onset voltage Uw(f) (or below the frequency fw(U)) of the linear worm instability, at defect sites like spacer rods, where the fluid convects strongly. In its primitive state, the ring structure is a ACS Paragon Plus Environment


18 few microns across, appearing as a collection of a few jiggling birefringent spots. Coalescence of such units produces perceptibly circular objects (Figure 14b). Repetitive birefringent bands disposed radially within a ring, as in Figure 14a, indicate a periodic structure involving strong θ and φ deviations of the director. These walls exhibit complex dynamics arising from their general diagonal drift coupled to phase propagation of the distortion wave; the latter is caused by random evolution of new domains, as in linear worms. While the walls at constant U and f vary generally in size, a slight change in U or f considerably affects their size (Figure 15) (see Supporting Information for the movie clip

Figure 15. Radius r of two ring walls as a function of t upon changing f from (a) 30 Hz to 28 Hz at 12.8 V and (b) 27 Hz to 30 Hz at 12.6 V. Continuous curves are quadratic fits. Inset: Two frames separated by 23.6 s, from the time series from which the data in (a) are derived. White arrow indicates the direction of drift of the walls. Black arrows indicate the applicable axes. Scale: 5 µm/div.

V5.avi showing the ring walls shrinking under increased f). The identification of ring structures as walls is evidenced by their metastability and dissociation into two loops, just as Brochard-Leger walls28,29 undergoing pincement into two half-strength disclinations of opposite topological charge. Figure 14c shows the two loops derived from a ring wall; one of the loops with its two ends attached to a spacer rod, shows vertical bands in the bottom segment; the other is a loop with its winding section to the left. Obviously, the two loops are


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19 formed close to the electrodes, on opposite sides of the midplane. The asymmetric finger-like structure of loops is a characteristic feature found, as a rule, in the disintegration of any ring wall. In time image series with very low exposure time, we find the two loops generally differing in visibility, with the maximum contrast alternating between them, indicating their polarity sensitivity. Clearly, these loops are of flexoelectric origin and ring walls are generated through some form of coupling between the loop instabilities at the two substrates.

3. CONCLUSIONS

This study reveals a variety of patterned states evolving far above the Freedericksz threshold in a highly conducting (+ +) nematic liquid crystal with an underlying heliconical nematic phase. Initially, as the constraint is increased, the extended prewavy-like structure is destabilized by a confined travelling roll instability. This is followed by the evolution of open worms near the two substrates; their formation is seemingly influenced by the polaritydependent gradient flexoelectric surface instability. The worms here are, very significantly, polarity sensitive, consisting of Bobylev-Pikin type flexodomains; they differ from the subcritically appearing electroconvection worms in (– +) nematics. Labyrinthine pattern of extended worms forming a section of hybrid loops is the second unprecedented finding. A related observation is of circular bulk walls disintegrating into asymmetric loops, somewhat as in pincement of Brochard-Leger walls. Our interpretation of surface instabilities as basically flexoelectric phenomena seems well founded. The structure of flexodomains as visualized may need refinement taking into account pretilt and azimuthal deviations (expected even in planar samples under low coherence lengths30). It is also interesting to consider if molecular conformational changes, such as found in CB7CB subjected to high magnetic fields,31 are possible at the boundaries under the strong double-layer fields. A



20 theoretical description of the localized states resulting from the coupling between different instability modes in the highly nonlinear regime remains a challenging problem.

■ ASSOCIATED CONTENT *S Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: ……………………… Five video clips, V1.avi to V5.avi, are provided for a better appreciation of the dynamical phenomena. The details pertaining to these clips are given in SI.pdf.

■ AUTHOR INFORMATION Corresponding Author *Phone: +91 80 23084236. Fax: +91 80 28382044. E-mail: [email protected], [email protected] Notes The authors declare no competing financial interest.

■ ACKNOWLEDGMENT We are thankful to Prof. N. V. Madhusudana for many useful discussions and to Prof. G. U. Kulkarni for the experimental facilities. One of us (CVY) acknowledges the financial support received from SERB, DST, Govt. of India, under Project No. SR/S1/OC-04/2012.

■ REFERENCES (1) Cross, M. C.; Hohenberg, P. C. Pattern Formation Outside of Equilibrium. Rev. Mod. Phys. 1993, 65, 851-1112.


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Confined Electroconvective and Flexoelectric Instabilities Deep in

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