Flashing flexodomains and electroconvection rolls in a nematic liquid crystal
aa r X i v : . [ c ond - m a t . s o f t ] S e p Flashing flexodomains and electroconvection rolls in a nematic liquid crystal
P´eter Salamon , N´andor ´Eber , Alexei Krekhov and ´Agnes Buka Institute for Solid State Physics and Optics, Wigner Research Centre for Physics,Hungarian Academy of Sciences, H-1525 Budapest, P.O.B.49, Hungary and Institute of Physics, University of Bayreuth, D-95440 Bayreuth, Germany (Dated: July 2, 2018)Pattern forming instabilities induced by ultralow frequency sinusoidal voltages were studied in arod-like nematic liquid crystal by microscopic observations and simultaneous electric current mea-surements. Two pattern morphologies, electroconvection (EC) and flexodomains (FD), were distin-guished; both appearing as time separated flashes within each half period of driving. A correlationwas found between the time instants of the EC flashes and that of the nonlinear current response.The voltage dependence of the pattern contrast C ( U ) for EC has a different character than that forthe FD. The flattening of C ( U ) at reducing the frequency was described in terms of an imperfectbifurcation model. Analysing the threshold characteristics of FD the temperature dependence ofthe difference | e − e | of the flexoelectric coefficients were also determined by considering elasticanisotropy. PACS numbers: 61.30.Gd, 47.54.-r
I. INTRODUCTION
Nematic liquid crystals are the simplest paradigm foranisotropic fluids; i.e. liquids with a preferred directionof the orientation of molecules with anisotropic shapewhich is described by the director field n . The anisotropyof their dielectric properties allows controlling the direc-tor by electric fields. The (usually homogeneous) reori-entation of the director by a properly applied voltagechanges the direction of the optical axis and hence thelight transmittance of the sample; this forms the phys-ical background of the liquid crystal displays, [1] usedwidespread in common electronic devices.Applying an electric voltage to a nematic liquid crys-tal layer can, however, often result in the appearanceof spatio-temporal, periodic or disordered structures too.The conditions of their occurrence, the pattern morpholo-gies and their onset characteristics have been extensivelystudied since decades, both experimentally and theoret-ically [1–14].In the mostly studied planar configuration, where thedirector is initially oriented parallel to the confiningplates, one of the electric field induced patterns corre-sponds to spatially periodic, equilibrium director defor-mations (seen as stripes parallel to the director in a po-larizing microscope), occurring due to a flexoelectric freeenergy gain of the deformed state; therefore they havebeen coined flexoelectric domains (FDs) [2]. FDs have sofar been detected in a few nematic compounds only andthey are observable at DC (or very low frequency AC)driving only.A more frequent, but also more complex pattern form-ing phenomenon is the electroconvection (EC) where thedirector distortions are accompanied by space charge sep-aration and hence by material flow; thus having a dissi-pative character. It could be observed in many nemat-ics, some of which possess substantially different materialproperties [3, 4]. EC patterns could be induced in a wide frequency range of the applied voltage (ranging from DCup to several hundreds kHz AC); the resulting convectionrolls are seen in a polarizing microscope as stripes whosedirection may be normal to, oblique or parallel with thedirector. Up to now studies were mostly focussed on theclass of nematics with negative dielectric and positiveconductivity anisotropies and on driving frequencies f within the range of 10 Hz to 10 kHz. In this f range evo-lution of the pattern requires numerous driving periodsafter voltage application. For such conditions the vari-ation of pattern morphologies (conductive and dielectricregimes, oblique and normal rolls) upon the amplitudeand frequency of the applied voltage have been exploredin detail and the mechanism as an electrohydrodynamicinstability has been well understood. A quantitative the-oretical description of the pattern threshold, the criticalwave vector and some secondary transitions (e.g. ab-normal rolls) could be given combining nematodynam-ics with electrodynamics under the simplifying assump-tion of Ohmic conductivity – now called as the standardmodel of EC [5] – or via its extensions by flexoelectricity[6] or by ionic diffusion/recombination [7].Recently interest has arisen to study the behaviourin another, subhertz frequency range, where the patterngrowth/decay times are (much) shorter than the drivingperiod, using compounds which may exhibit both EC andFD patterns. It has been proven experimentally that atsuch ultralow frequencies both for the dielectric [8] andthe conductive [9] EC regimes, as well as for the FD[8, 9] the patterns are flashing, i.e. they exist only in asmall part of the driving period. It has been found thatthere is an f range ( ∼ −
100 mHz) where both EC andFD patterns can exist in each driving half period in theform of successive (time shifted) flashes. Theoretical cal-culations based on the standard model of EC extendedwith flexoelectricity [6] (which is able to describe FDstoo [10]) have justified that flashing patterns are indeedthe solutions of the nemato-electrohydrodynamic equa-tions at ultralow f . The calculated position of the FDflashes within the driving half period showed quantitativematches with the experiments, while for the position ofthe EC flashes the frequency dependence was only quali-tatively reproduced by the calculations, as the EC flashescome earlier within the period than expected [9].In this paper we present further experimental resultson the ultralow f behaviour, however, in a different sys-tem than those reported before. The paper is organizedas follows. Section II introduces our compound and theexperimental method. The new findings are groupedaround three subtopics: Section III A reports on the tem-poral evolution of the patterns within the period; Sec-tion III B deals with the frequency dependence of thethreshold characteristics and Section III C provides dataon the temperature dependence of various material pa-rameters. Finally the paper is concluded in Section IVwith a summary and some closing remarks. II. EXPERIMENTAL
Our measurements have been performed onthe nematic liquid crystal 4-n-octyloxyphenyl 4-n-methyloxybenzoate (1OO8) that shows only a nematicmesophase. The chemical structure of 1OO8 is shown inFig. 1. OO OOCH C H FIG. 1. The chemical structure of the rod-like nematicmolecule 4-n-octyloxyphenyl 4-n-methyloxybenzoate (1OO8).
In heating it melts to nematic from the crystallinephase at 63.5 ◦ C, while the clearing point ( T NI ) equals to76.7 ◦ C. The nematic phase can be supercooled down to53 ◦ C. The material parameters of 1OO8, such as the di-electric anisotropy ( ε a = ε k − ε ⊥ ), the optical anisotropy( n a = n k − n ⊥ ), the anisotropy of the diamagnetic sus-ceptibility ( χ a = χ k − χ ⊥ ), and the bulk elastic constants( K , K , K ) were determined as the function of tem-perature using a method based on magnetic and electricFreedericksz-transitions [15]. Here ε and n denote thedielectric permittivity and the refractive index, respec-tively; the subscripts k and ⊥ correspond to measure-ment directions parallel with and perpendicular to thedirector.The compound was investigated in commercial sand-wich cells (E.H.C. Co.) with ITO electrodes coated with Two abbreviation styles are known in the literature for the mem-bers of the 4-n-alkyloxyphenyl 4-n-alkyloxybenzoate homologousseries. Here we have adopted the one used by Nair et al. [16]According to the alternative style by Kochowska et al.[13] thesame compound could also be abbreviated as 1/8. rubbed polyimide layers for planar alignment. The elec-trode area was 1 cm . The thickness of the empty cells( d = 10 . − . µ m) was measured by an Ocean Op-tics spectrophotometer. During the measurements thetemperature of the sample was kept constant within 0.01 ◦ C in an Instec HSi heat stage controlled with an mK-1board. The sample was driven by a sinusoidal voltage˜ U ( t ) of an Agilent 33120A function generator via a high-voltage amplifier: ˜ U ( t ) = √ U sin(2 πf t ).The electric field induced patterns were observed bya Leica DM RX polarizing microscope in transmissionmode with white light illumination using the shadow-graph technique [17] (the polarizer was removed, whilethe analyser was set to be parallel with the rubbing direc-tion). The imaging system was equipped with an EoSensMC1362 high speed camera interfaced by an Inspecta-5frame grabber. After waiting one or two periods of thedriving signal following the application of the voltage tothe sample (or waiting 5 seconds at frequencies higherthan 0.2 Hz), a sequence of 1000 images was recorded.The acquisition of the first image was triggered by thezero crossing (from negative to positive) of the appliedvoltage.In addition to the optical observations the electriccurrent through the cell was monitored by measuringthe voltage drop on a relatively small, known resistanceconnected in series with the sample. Simultaneouslythe driving waveform was also recorded by a TiePieHandyscope HS3 oscilloscope. The data acquisition andprocessing system was fully automated. III. RESULTS AND DISCUSSIONA. Flashing contrast and current
Applying a low frequency (e.g. f = 50 mHz) sinusoidalvoltage to the cell, patterns appear above a thresholdvoltage in a narrow time window in each half period ofdriving. Two distinct pattern morphologies were foundwith different thresholds, similarly to previous observa-tions on other nematics [9]. Representative snapshots ofthe patterns and their 2-dimensional (2-d) Fourier trans-forms (the spectral intensities) are presented in Fig. 2.The two morphologies can be attributed to oblique con-ductive EC rolls (a zig-zag pattern, Fig. 2a) and to flex-odomains (Fig. 2b); the latter appear as stripes parallelto the initial director alignment.For a quantitative analysis of the pattern evolution itis necessary to provide a proper definition for the pat-tern contrast, which has a minimum (ideally zero) in thehomogeneous state and increases as the pattern emerges.A common procedure is to perform a 2-d Fourier trans-formation of the images in order to find the critical wavevector q c = ( q x , q y ) of the pattern (where the Fourier am-plitudes have maxima) and to define the contrast C q asthe sum of the spectral intensities in a region around q c .It is clear from Fig. 2 that the two pattern types observed (a) q x (units of π /d) q y ( un i t s o f π / d ) −7 −5 −3 −1 1 3 5 7−7−5−3−11357 (b) q x (units of π /d) q y ( un i t s o f π / d ) −7 −5 −3 −1 1 3 5 7−7−5−3−11357 FIG. 2. Snapshot images and their 2-d Fourier transforms (a)for electroconvection and (b) for flexodomains at f = 50 mHzand U = 19 V. The images cover 200 µ m x 200 µ m area. Theinitial director orientation lies horizontally. in 1OO8 (EC and FD) are characterized by different q c vectors, i.e. they are well separated in the Fourier space.Therefore this contrast definition allows distinguishingthem not only from the initial homogeneous state, butalso between each other.Alternatively, a mean square deviation of the imageintensities, C s = h (Φ − h Φ i ) i , may also serve as a mea-sure of the contrast. Here Φ is the intensity of an in-dividual pixel and hi denotes averaging over the wholeimage. This definition is simpler, though it has the dis-advantage of not being able to distinguish various patternmorphologies. Actually C s would coincide with C q if thesummation of the spectral intensities were extended overthe whole Fourier space.Figure 3 exhibits and compares the time dependence ofcontrast within a driving period for both definitions givenabove, measured in a d = 10 . µ m thick cell at T − T NI =21 . ◦ C driven by an f = 22 mHz, U = 18 V voltage.Figure 3a shows C q obtained by the Fourier method forthe EC (solid line) and the FD (dashed line) patterns.Both curves exhibit a single peak in each half period,but at different time intervals; hence these two patterntypes are well separated not only in the Fourier space,but in time as well. In Fig. 3b the contrast C s calculatedby the square deviation is plotted. This curve has two,well separated peaks per half period (looks similar to thesuperposition of the two curves in Fig. 3a); thus can alsobe used to detect the appearance of both pattern types.Therefore, for simplicity, in the following we will use C s as the measure of the contrast of the patterns.Figure 3c depicts the time dependence of the electrical EC FD (a)
Fourier ( a r b . un it s )( a r b . un it s ) C q I ( A ) (b) C s Current (c)Standard deviationt /
FIG. 3. (Color online) The time dependence within a drivingperiod (a) for the contrast C q obtained by Fourier technique;(b) for the contrast C s calculated from the square deviation;and (c) for the electrical current I through the liquid crystal. t = 0 corresponds to the zero crossing (from positive to neg-ative) of the applied voltage. The dashed-dotted lines showthat the peaks of EC and of the current coincide. current which was measured simultaneously with imageacquisition. At this f and U the current is highly nonlin-ear; it can be characterized by sharp peaks rather than bya harmonic response. It can be deduced from the figurethat, surprisingly, the location of the maxima of the cur-rent peaks coincide precisely with the contrast peaks cor-responding to the EC flashes (see the dash-dotted verticallines in Fig. 3). Numerous different voltages, frequenciesand temperatures were tested. Though at various con-ditions the time instant of the EC flash may change [9],it still equals to that of the current peak; thus we canconclude that this is not an accidental coincidence. Wesuggest that the current spikes trigger the emergence ofthe EC pattern. Therefore it appears earlier within thehalf period (a phase-locking behaviour) than expectedotherwise.We note that the spiky behaviour of the current isnot a consequence of the appearance of the EC pattern.Current spikes have been detected at low voltages (muchbelow any pattern threshold) where no patterns are ob-servable and also in the isotropic phase. We think thatthe nonlinear current behaviour is due to ionic effectsand to the presence of insulating polyimide orienting lay-ers on the electrode surfaces of the cell. The presence of(relatively low) concentration of ionic impurities in thenematic makes it to behave as a weak electrolyte. Inthe studied ultralow frequency range the current due tothe linear impedance of the cell (i.e. the capacitive andthe ohmic components) is at least an order of magnitudesmaller than the transient currents due to building or de-stroying the Debye screening layers near the electrodes;the latter occurs at each polarity reversal of the voltage.In order to describe the behaviour of weak electrolytesin electric fields several models were developed, differ-ing in their sets of assumptions [18–26]; i.e. they takeinto consideration different subsets of the possible ef-fects listed below: generation and recombination of ions;different mobilities, diffusion coefficients and charges ofionic species; surface adsorption; charge injection; chem-ical reactions; voltage attenuation due to the orientinglayers; etc. Due to the complexity of the models theymostly focused on the linear response and calculated thelow frequency complex impedance which could be com-pared to low f dielectric spectroscopy data.Recently theoretical calculations of the nonlinear cur-rent characteristics in response to a low frequency sinu-soidal voltage driving were also reported [18, 19], yieldingcurves similar to those shown in Fig. 3c, however, withoutcomparison with experiments. This gives the hope thatafter measurements or intelligent guesses of the unknownmaterial parameters of the model the measured currentresponse can be reproduced; it is remaining a task for thefuture.The nematic being a weak electrolyte has consequenceson the pattern formation processes. It was shown thatthe weak electrolyte model (WEM) of EC [7], which con-siders ionic dissociation and recombination, can accountfor the travelling of EC roll patterns found occasionallyat frequencies above a few tens Hz. This model has notyet been analysed for low driving frequencies; due to itshigh complexity it remains a challenge for the future todecide whether it is able to describe the phase locking ofEC flashes to current spikes. B. Threshold characteristics
Flexodomains and electroconvection both are thresh-old phenomena; i.e. the patterns with a criticalwavenumber q c = | q c | occur above a threshold voltage U c . Determination of U c and q c is therefore the primarytask at pattern characterization. At high frequencies( f >
10 Hz) for
U > U c patterns usually develop withinseconds; therefore thresholds can easily be estimated byincreasing U as the voltage at which the pattern becomesperceptible by eyes in the microscope. This simple tech-nique practically does not work at our ultralow frequencydriving, since the driving period is quite long and in ad-dition the patterns appear as flashes, which means theycan be observed only in a short time window.In order to determine U c precisely one has to followquantitatively the emergence of patterns from the ho-mogenous state; i.e. to record and then analyse thecontrast-voltage curves. As the contrast varies withinthe driving period (as shown in Fig. 3b), the maximum C m of the contrast C s in the FD (or EC) peak can beregarded as a measure to what extent the FD (or EC)pattern has been developed at a given applied voltage.In an ideal case (perfect bifurcation) the contrast C m should be zero at voltages below the threshold. Exper- imentally a nonzero background contrast C b is alwaysfound even in the homogeneous state at no applied volt-age ( C b = C s ( U = 0)). This background contrast comesfrom various sources: the electronic noise of the camera,the thermal fluctuation of the director in a planar ne-matic, imperfections of the orientation or inhomogeneityof the illumination. This background was automaticallysubtracted from each data point; thus it will not be in-dicated in the forthcoming figures.As the voltage is increased above U c , the initial pla-nar director orientation n = (1 , ,
0) becomes unsta-ble and a spatially periodic director distortion δ n = n lin A exp[ i ( q x x + q y y )] appears. Here n lin = (0 , n y , n z ) isa linear eigenvector, A ∝ p U − U c characterize the am-plitude of the distortion, and q c = ( q x , q y ) is the wavevec-tor of the pattern. The spatially periodic director dis-tortion results in a shadowgraph image whose intensitymodulation I s depends on the amplitude of the verticaldistortion An z . For small distortion amplitudes (not toofar from threshold) the intensity modulation in the lead-ing order is given[27] by I s = c a A + c p A with the firstorder amplitude term and the second order phase term.For EC patterns (normal rolls with q c = ( q x , I s ∝ A . In case of FD, where q c = (0 , q y ), the relevant contribution to the shadow-graph intensity is of the second order: [28] I s ∝ A . Thecontrast of the shadowgraph image defined as the meansquare deviation of the image intensities is then C s ∝ I s .Thus the maximum of the contrast within the drivingperiod is expected to be C mEC ∝ ( U − U cEC ) for an ECpattern and C mF D ∝ ( U − U cF D ) for the FD [8]. Inthe vicinity of the threshold ( U − U cF D ) ≈ U c ( U − U c );therefore C mEC as well as √ C mF D should grow linearlywith the voltage.Figure 4 shows the measured √ C mF D ( U ) curves for afew frequencies. It is seen that the linear relation near thethreshold is obeyed quite well; though the transition issmeared a little (due to imperfections and/or the occur-rence of subcritical fluctuations). Therefore the thresholdvoltage U cF D is actually determined by a linear extrap-olation, as the intersection of the horizontal axis withthe line fitted onto the linear section of the C m ( U ) curveslightly above the suspected threshold. This procedureis going to be referred as method A.The voltage dependence of C mEC for EC is shown inFig. 5 for several driving frequencies. It is clearly seenthat the frequency affects not only the threshold volt-ages, but also the character (the shape) of the C mEC ( U )curves. Evidently the linear relation holds only at highfrequencies; there the thresholds U cECA can be deter-mined by extrapolation (method A).Below 1 Hz, however, there is no sharp increase of thecontrast; the C m ( U ) curves show rather a slow gradualincrease, while the contrast levels and thus the visibilityof the patterns vary in the same range as at high fre-quencies. The determination of thresholds is then notso straightforward. In lack of a well defined linear partof the contrast curve, method A becomes unreliable; the C / m F D ( a r b . un it s ) U (V)
10 mHz 50 mHz 365 mHz
FIG. 4. (Color online) The voltage (rms) dependence of thesquare root of the FD contrast peaks for different frequencies(symbols). The dashed lines indicate the linear extrapolation.
60 Hz 10 Hz 1.4 Hz 0.19 Hz 0.05 Hz 0.033 Hz C m E C ( a r b . un it s ) U (V)
FIG. 5. (Color online) The voltage (rms) dependence of thecontrast peaks C m of EC for different frequencies (symbols).Solid lines are fits with the imperfect bifurcation model, thedashed lines indicate the linear extrapolation. choice of points used for the extrapolation (the dashedlines in Fig. 5) is to some extent arbitrary.An alternative way (method B) is to select (arbitrar-ily) a critical contrast value C (the dash-dotted line inFig. 5) where the EC pattern is visible by eye. The volt-age U cECB , where C mEC ( U cECB ) = C , can be regardedas another estimate of the threshold. In case of forwardbifurcations, which the standard EC pattern formation isan example for, the contrast increases continuously fromzero. Therefore U cECB slightly overestimates the thresh-old.The change in the shape of the C mEC ( U ) curves maybe interpreted so that the nearly perfect bifurcation (athigh f ) becomes imperfect at lower f . For an imperfectbifurcation the amplitude of the director distortion A satisfies the equation εA − gA + δ = 0 . (1) Here ε = U /U cEC − U is the rms applied voltage, U cEC is the threshold voltage, g > δ ≥ δ = 0 corresponds to the perfectforward bifurcation). For g > δ > ε > − A = ( δ g ) / F (˜ ε ) ,F (˜ ε ) = ( ˜ ε ˆ f (˜ ε ) + ˆ f (˜ ε )) for ˜ ε ≤ ,F (˜ ε ) = 2 √ ˜ ε cos ( 13 arctan ( p ˜ ε − ε > , ˜ ε = 23 ε (2 gδ ) / , ˆ f (˜ ε ) = (1 + p − ˜ ε ) / . (2)As mentioned above, the maximum contrast C mEC ofthe EC patterns observed using the shadowgraph tech-nique is proportional to A . In Fig. 6 the dependence of A on the applied voltage U is shown for different val-ues of the imperfection parameter δ at fixed values of U cEC and g . It demonstrates that the shape of the curvechanges substantially if the imperfection ( δ ) increases. A ( a r b . un it s ) U (V) FIG. 6. (Color online) Square of the pattern amplitude A as a function of the applied voltage U for δ = 0 .
01 (solidline), δ = 0 . δ = 0 . U cEC = 5, g = 0 . For a precise quantitative analysis we can use the samebackground subtraction here, just as was done with theexperimental data; therefore the contrast depicted inFig. 5 will be related to the amplitude as: C mEC = C max − C b = α [ A ( U ) − A ( U = 0)] , (3)where C max is the maximum contrast of the pattern, C b is the background contrast at U = 0, and α > C mEC ( U ) curves by this phenomenologicalmodel for imperfect bifurcation using four parameters: α , g , δ and U cEC (method C).The actual value of the scaling parameter α is deter-mined by the optical set-up and the optical properties.As q c of the EC pattern depends weakly on f , we canassume that α is frequency independent. Its value couldbe obtained from the fit at f = 60 Hz, leaving only threefree parameters for the fits at lower frequencies. f (Hz) FIG. 7. Frequency dependence of the imperfection parameter δ . The results of the fit procedure are shown by solidlines in Fig. 5. The match with the experimental dataare quite convincing. The frequency dependence of theimperfection parameter δ is plotted in Fig. 7. It clearlyshows – what we have already expected from the exper-imental data in Fig. 5 – that the imperfection grows atlower frequencies. Several reasons could be responsiblefor the increase of the apparent imperfection.In planar samples aligned by rubbed polyimide layers asmall director pretilt at the confining plates is practicallyunavoidable. Such pretilt is known to yield imperfect bi-furcation (i.e. lack of a sharp threshold) in the case ofsplay Freedericksz-transition. The effect of a tilted align-ment on the EC characteristics has theoretically beenstudied only for high frequencies [29]; the pretilt modi-fied U c , but did not affect the sharpness of the threshold,which is in agreement with our observations (Fig. 5) athigh f .Decreasing the frequency of the applied ac voltage wellbelow the inverse director relaxation time may, however,alter the situation as one enters the regime of quasistaticdirector response. Here a small pretilt may enhance thedirector deformations and correspondingly the contrastof the pattern can develop already at lower voltage am-plitudes compared to the high frequency case. Unfortu-nately a detailed theoretical analysis of this regime in thepresence of pretilt is not yet available.The nonlinear electric current characteristics presentedin Sec. III A may provide another reason for the apparentsoftening of the ultra-low f EC thresholds. The coinci-dence of the electric current peaks and the EC flashesclearly shows the strong correlation between pattern for-mation and ionic phenomena: the massive ionic flowhelps the electro-hydrodynamical instability to emerge. The spatial distribution of the current is not necessarilyuniform, mainly due to surface inhomogeneities (whichmay originate e. g. from crystallization of the com-pound) or small variations in the cell thickness and/orpretilt. The current inhomogeneities may locally reducethe threshold of EC. In fact this effect has been observed:the EC pattern first appears in germs and extends gradu-ally to larger area by increasing the voltage. The locationof these germs can be identified even in the well devel-oped pattern as small spots/patches of higher contrast(a few such spots can be seen in Fig. 2a). The contrast C mEC ( U ) of the pattern plotted in Fig. 5 is calculatedover the whole image; thus a continuous increase of thearea filled with pattern leads to a continuous increase of C mEC ( U ). Consequently a locally sharp transition yieldsa softened, gradual contrast variation. While ionic effectsare mostly negligible at high frequencies (linear currentresponse), they become crucial at ultralow frequencies(spiky current response), which may explain the increaseof the imperfection parameter for f → U c ( V ) f (Hz) FD (method A) EC (method A) EC (method B) EC (method C)
FIG. 8. (Color online) The frequency dependence of thresholdvoltages (rms) of EC and FD determined by various methods.
The frequency dependence of the threshold voltagesof both patterns can be seen in Fig. 8. It depicts the U cEC values determined by all three methods introducedabove. The data by methods A (extrapolation) and B(comparison) almost coincide, while the thresholds ob-tained from fitting to the imperfect bifurcation modelare significantly larger at lower frequencies. This is notsurprising since methods A and B intrinsically assumethat no deformation exists below a threshold, while animperfect bifurcation actually means a thresholdless de-formation with U c being a parameter only.Otherwise the U cEC ( f ) curve exhibits the expected be-haviour. The reduction of the threshold at lowering f inthe 0 . < f <
10 Hz range corresponds to the theoreticalpredictions and matches the behaviour of other nematics[30]. The increase of U cEC toward ultralow frequenciesis attributed to the internal attenuation due to the insu-lating polyimide alignment layers on the electrodes [9].The frequency dependence of U cF D seems to be signif-icantly weaker than that of U cEC in the same f range.Taking into account the internal attenuation, the actualFD threshold voltage (on the liquid crystal layer) growsmuch stronger with f than the apparent threshold plot-ted in the figure (the voltage applied to the cell), whichis in agreement with the theoretical predictions [10].Figure 8 clearly shows that the two distinct patterns,EC and FDs, coexist in a relatively wide (0.02 Hz < f < f > . U cEC is much lower than U cF D . Thus the EC contrastspikes become much broader and the EC pattern doesnot decay fully before FD should emerge. Under suchcondition the FD pattern (which has a lower contrastthan EC) cannot be recognized any more.As the frequency is reduced, at around 0 . − .
07 Hzthere is an intersection of the two threshold curves ( U cF D and U cEC ). At f below this intersection the threshold ofFDs is lower than that of EC; thus upon increasing thevoltage FD is the first instability, EC sets on at a highervoltage. This is in accordance with the finding that whenapplying a pure DC voltage, no EC pattern, only FDs canbe detected. ob li qu e d i e l ec t r i c FD EC q c ( un it s o f / d ) f (Hz) ob li qu e c ondu c ti v e FIG. 9. (Color online) The frequency dependence of thethreshold wave numbers for EC and FD.
Characterization of the threshold behaviour is incom-plete without addressing the frequency dependence of thecritical wave number q c = | q c | . Figure 9 exhibits the rele-vant curves both for EC ( q EC ) and FD ( q F D ). The valueswere determined using the 2-d fast Fourier transforma-tion (FFT) of images taken slightly above the thresh- old, at U = 1 . U c , in order to have sufficient contrastfor the evaluation. Note that for the oblique EC rolls q EC = q q x + q y , while FDs are parallel to the initial di-rector, so q F D ≈ q y . The wave numbers increase for bothpatterns with the frequency. In the case of FD there isa moderate f dependence even at ultralow frequencies.For EC, the change of q EC seems to be very small until5 Hz. Between 5 and 10 Hz, however, the wave numberincreases suddenly, which is attributed to the transitionbetween oblique conductive and oblique dielectric EC. Toour knowledge no such transition was reported before inthe literature. We note that the obliqueness angle de-creases with the frequency, the Lifshitz-point is reachedin the dielectric regime at f L ≈
80 Hz.
C. Temperature dependence of the flexoelectriccoefficients
Though several experimental methods have been pro-posed to measure the flexoelectric coefficients, measure-ments usually cannot be done without serious compro-mises [14]. Analysis of the threshold parameters ( q cF D , U cF D ) of the flexoelectric instability is one of the possiblemethods. Its drawback is that only a few compounds ex-hibit this effect, because: 1) the material needs to have aquite low dielectric anisotropy ( | ε a | ≪ K = K = K ):˜ U cF D = 2 πK | e − e | (1 + µ ) , (4)˜ q cF D = πd r − µ µ , (5)where e and e are the splay and bend flexoelectric co-efficients, respectively, and µ = ( ε ε a K ) / | e − e | . (6)According to Eq. (5) the flexodomains can only existfor the material parameter combination | µ | <
1. Thisleads to the requirement | ε a | < | e − e | / ( ε K ) thatshould be valid for materials showing FDs. CombiningEqs. (5) and (6) yields: | e − e | = s ε ε a K q cF D − q cF D . (7)For 1OO8 both q cF D and U cF D were measured as thefunction of temperature using 10 mHz ac sine voltage.We assumed that 10 mHz is low frequency enough to beconsidered as a quasistatic case, hence we have fitted theresults with a static model. Therefore U cF D here is pre-sented in voltage amplitude values instead of rms, sinceFD appears when the driving voltage reaches its max-ima. Therefore U cF D in Fig. 10 is presented in voltageamplitudes instead of rms values. -24 -22 -20 -18 -16 -14 -12 -10 -820304050602.02.53.03.54.0 (b) U cFD experimental U cFD calculated using K= (K +K )/2 U cFD calculated using K and K U c F D ( V ) q c F D ( un it s o f / d ) T - T NI ((cid:176)C)(a) FIG. 10. (Color online) The temperature dependence of (a)the wave number q cF D and (b) the voltage U cF D (amplitude)at the onset of flexodomains. Both q cF D and U cF D increase strongly toward highertemperatures. Above T − T NI = − ◦ C we could notdetect flexodomains up to the voltage of 135 V.In order to determine | e − e | we have measured somematerial parameters of 1OO8 using methods based onelectric and magnetic Freedericksz-transitions. The tem-perature dependence of ε a and of the diamagnetic suscep-tibility anisotropy ( χ a ) is shown in Fig. 11a. ε a is nega-tive and relatively small, as it was expected. Thereforein our planar sandwich cell geometry the dielectric in-teraction stabilizes the planar structure; no electric fieldinduced Freedericksz-transition occurs. The values andthe thermal behaviour of χ a are in the regular range ofthose in rod-like nematics. This also holds for the elas-tic constants K , K , and K , which are plotted inFig. 11b. We note that K is shown only for the sake ofcompleteness; we do not use it further on.The temperature dependence of | e − e | , presentedin Fig. 12, was calculated from the measured databy two different techniques. The first method (squaresymbols) was based on the analytical formula, Eq. (7),of the one-elastic-constant approximation, taking K =( K + K ) /
2. The second technique (triangle sym-bols) utilized the recent theory [10] of flexoelectric do-mains that takes into account the anisotropic elasticity( K = K ), calculating | e − e | numerically. As seenin Fig. 12, the second method provided values about 7% -24 -20 -16 -12 -8 -424681012-0.5-0.4-0.3-0.2 K K K (b) a E l a s ti c c on s t a n t ( p N ) T - T NI ((cid:176)C)(a) a ( un it s o f - ) a a FIG. 11. (Color online) The temperature dependence of (a)the dielectric ( ε a ) and the diamagnetic ( χ a ) anisotropies, (b)the three elastic moduli. -24 -22 -20 -18 -16 -14 -12 -10 -83.54.04.55.05.56.06.57.0 T - T NI ((cid:176)C) | e - e | ( p C / m ) using K = (K +K )/2 using K and K FIG. 12. (Color online) The temperature dependence of thecombination | e − e | of the flexoelectric coefficients. higher than those by the first one; both values of | e − e | fall in the regular range of that of rod-like nematics.In order to check the consistency of our models and theobtained data, we have calculated U cF D using the | e − e | values determined from q cF D . The results, depicted inFig. 10b, show that the first model gave about 2% lower,while the second one about 11% higher values for U thF D than the experiments.Knowing the temperature dependence of | e − e | gaveus an opportunity to compare our results with the pre-dictions of the molecular theory of flexoelectricity. It isexpected [31–33] that the difference of flexoelectric coef-ficients should be proportional to the square of the orderparameter S ( T ): | e − e | = ˆ eS ( T ) , (8)where the proportionality constant is denoted by ˆ e .In Fig. 12 | e − e | is decreasing with the tempera-ture, which is consistent with the similar tendency ofthe order parameter. For a more quantitative compari-son, the knowledge of S ( T ) would be essential. S ( T ) canonly be accessed via measuring physical quantities thatare directly coupled to it. The diamagnetic susceptibil-ity, which is already determined from the Freedericksz-transition measurements (Fig. 11a) is a good candidate,since it should be proportional to S [35]: χ a ( T ) = ˆ χS ( T ) , (9)where ˆ χ is a constant. In order to determine ˆ χ , and S ( T ),the generalized form of the empirical Haller-extrapolation[34, 35] method is applied, via fitting the experimentaldata of χ a ( T ) with: χ a ( T ) = ˆ χ (cid:18) − β TT NI (cid:19) γ , (10)where β , γ are constants, and the temperature data ( T , T NI ) is measured in the Kelvin-scale. The result of thefit can be seen in Fig. 11a (solid line). The parameters ofthe best fit correspond to: ˆ χ = 1 . × − , β = 1, and γ = 0 .
2. Besides the dimensionless SI quantity of ˆ χ , itsmolar version is often used: ˆ χ M = ˆ χM m /ρ , where M m ,and ρ are the molar weight, and the density, respectively.Using M m = 356 . gmol , and ρ = 1 gcm one gets ˆ χ M =585 × − mol , that value fits well in the range of earlierresults [35, 36] obtained for different compounds with twoaromatic rings.Combining Eq. (8) and Eq. (9) yields: χ a = a p | e − e | , (11)with a = ˆ χ/ √ ˆ e . a ( un it s o f - ) |e - e | ((pC/m) ) measurement linear fit through the origin FIG. 13. (Color online) The relation between χ a and p | e − e | . Figure 13 provides a test of this relation, as it plotsthe measured χ a values against p | e − e | calculated forthe same temperatures (determined from the model withanisotropic elasticity). The fit corresponding to Eq. (11),represented by the dashed line, seems to be quite goodin spite of the fact that there was only one fit parame-ter. The best fit results a = 0 . C/m ) − . , that with ˆ χ determined above, yields ˆ e = 18 . IV. CONCLUSIONS
We have investigated the pattern forming phenomenainduced by ultralow frequency sinusoidal voltages appliedonto the calamitic nematic liquid crystal 1OO8. It wasfound that the behaviour in this low frequency range ischaracteristically different from that typical for high fre-quencies: here patterns appear as flashes in a short timeinterval within each half period of driving. Two kinds ofpattern morphologies were detected: electroconvectionrolls and flexodomains. The types of patterns differ intheir wave vector (EC rolls are oblique to, whiles FDs areparallel with the initial director); moreover their flashesoccur subsequently with a time separation, though in thesame (and each) half period of driving. These scenariosare similar to those reported recently [8, 9] for the ne-matic mixtures Phase 5 and Phase 4.Electric current measurements carried out simultane-ously to pattern recording indicated strongly nonlinearcurrent responses: the time dependence of the currentshowed sharp peaks after each polarity reversal of theapplied voltage. The current nonlinearity in 1OO8 wasmuch more pronounced than in Phase 5. This behaviouris attributed to the ionic conductivity of the liquid crys-tal. The transient current may be due to the motion ofions during building up a Debye screening layer at theelectrodes, while the (insulating) polyimide coating en-suring the planar alignment blocks the charge transferthrough the electrodes.We found that, interestingly, the time instant of theflashing EC patterns (the time of the EC contrast peak)and that of the electric current peak coincide. This coin-cidence holds for all voltages, frequencies and tempera-tures that we have tested. The shape of the current signalis not affected by the occurrence of EC significantly, in-dicating that it originates from the more robust ionic ef-fects described above. This is also supported by the fact,that the current peaks could be observed below as well asabove the EC threshold, and even in the isotropic phase.We think, that the current peak has a significant effect onthe formation of EC, but not vice versa; the appearanceof the EC flashes is synchronized to the current peaks.Recently we reported a comparison [9] between the mea-sured and the theoretically calculated time instant of theEC flashes for Phase 5. It indicated that in the exper-iment at ultralow f EC occurred earlier within the halfperiod than expected from the extended standard modelof EC [6, 10]. We suggest that the phase locking of EC tothe ionic current peaks might be the reason for this mis-match (the extended standard model does not considerionic effects). We guess that an adequate extension ofthe theory to weak electrolytes could reveal this problemand additionally explain the role of the robust currentpeaks in the pattern formation; proving that, however,represents a great theoretical challenge for the future.Studying the threshold characteristics of the patternswe found that the behaviour of EC and FD are essentiallydifferent. Flexodomains have a sharp threshold, i.e. the0pattern contrast increases suddenly for
U > U c . For ECthis holds only at high f ; reducing the frequency theEC threshold becomes gradually less sharp (the contrastchanges smoothly with the voltage). On the one handit hinders the precise determination of the EC threshold.On the other hand, we showed that this tendency canbe followed quantitatively using an imperfect bifurcationmodel. In this approach the amount of imperfection in-creases as the frequency is lowered.EC and FD have different frequency dependence oftheir thresholds. At high f the EC threshold is lower,while at DC driving flexodomains are seen. Therefore itis not surprising that there is a crossover between EC andFD at around 60 mHz, where their thresholds becomeequal. Such a scenario was already anticipated from mea-surements on Phase 5, but could first be demonstratedexplicitly now on 1OO8.Interestingly, the two kinds of patterns can appear inthe same half period in some frequency range on bothsides of the crossover point, including frequencies wherethe two thresholds are quite different. This is made pos-sible by the narrow time interval and time separation ofthe flashes.The q cEC ( f ) curve of 1OO8 shows a discontinuity at f c ≈ f c indicates a fairly low electrical conductivity which also helps distinguish-ing between EC and FD patterns by increasing their timeseparation and may also be responsible for the enhancednonlinearity of the current.Measuring the critical wave number of the flexoelec-tric domains offers a way to calculate the combination | e − e | of the flexoelectric coefficients using theoreticalmodels based either on the one-elastic-constant approxi-mation or on a rigourous handling of anisotropic elastic-ity. It has turned out that the values determined by thetwo methods differ only by about 7%. The reason for thissmall difference is that the relevant material parameters( K , K and ε a ) of 1OO8 fall into that range, where q cF D is only slightly sensitive to the elastic anisotropy.The threshold voltages of FDs, calculated from the the-oretical models using the above values of | e − e | , showa satisfactory agreement with the measured data; thisproves the consistency of the models.In cooling 1OO8 has a nematic temperature rangeof about 25 degrees. The temperature dependenceof the elastic moduli, the dielectric and the magneticanisotropies was determined for the whole nematic range.For | e − e | data could be obtained only for the lowertemperature part of the nematic phase as flexodomainsdid not exist for T − T NI > − ◦ C. The temperaturedependence of | e − e | was compared with that of χ a ;the latter being proportional to S ( T ). It was found that | e − e | ∝ S is satisfied, as it is expected from themolecular theory of dipolar flexoelectricity, and also theproportionality constant was determined. ACKNOWLEDGEMENTS
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