Inhibited pattern formation by asymmetrical high voltage excitation in nematic fluids
aa r X i v : . [ c ond - m a t . s o f t ] S e p Inhibited pattern formation by asymmetrical high voltage excitation in nematic fluids
P´eter Salamon, N´andor ´Eber, Bal´azs Fekete, 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 (Dated: January 1, 2018)In contrast to the predictions of the standard theory of electroconvection (EC), our experimentsshowed that the action of superposed ac and dc voltages rather inhibits pattern formation thanfavours the emergence of instabilities; the patternless region may extend to much higher voltagesthan the individual ac or dc thresholds. The pattern formation induced by such asymmetricalvoltage was explored in a nematic liquid crystal in a wide frequency range. The findings could bequalitatively explained for the conductive EC, but represent a challenging problem for the dielectricEC.
PACS numbers: 61.30.Gd, 47.54.-r, 89.75.Kd
INTRODUCTION
Instabilities in nonlinear dynamical systems can leadto formation of patterns [1]. In fluids, patterns areoften associated with vortex flow induced by variousdriving forces such as temperature gradient (Rayleigh-B´enard convection [2–5]), shear (Taylor-Couette flow [6–9], Kelvin-Helmholtz instability [10, 11]), or electric field(electroconvection (EC) [12]). Anisotropic fluids are es-pecially convenient to study some general features of dy-namical systems, as they prone to show easily observableconvective patterns in applied electric fields because oftheir optical anisotropy.Electroconvection in nematic liquid crystals [12] canbe induced by both direct (dc) and alternating (ac) volt-ages in the same compound. In this paper, we show thatthe application of asymmetrical voltages correspondingto the superposition of a dc and a sinusoidal ac signalcan inhibit the formation of patterns, even if the ac anddc components are an order of magnitude higher than thethreshold voltages of the purely ac or purely dc inducedelectroconvection.Nematic liquid crystals are mostly composed of elon-gated molecules with their long molecular axes fluctuat-ing around an average direction, the director n ( r ) [13].Due to their uniaxial symmetry, nematic materials canbe characterized by two independent dielectric constantsmeasured with electric fields parallel or perpendicular tothe director ( ε || and ε ⊥ , respectively). A positive or neg-ative dielectric anisotropy ε a = ε || − ε ⊥ allows to alignthe director parallel with or perpendicular to the electricfield, respectively [13].Liquid crystals are studied and used mostly in thin(5-20 µ m) films, sandwiched between glass plates withtransparent electrodes providing an electric field alongthe cell normal. A proper treatment can ensure stronglyanchored, homogeneous director alignment at the sur-faces. In a planar cell, the homogeneous director liesin the cell plane. Due to the electric field applied per-pendicular to the initial director, if ε a >
0, an instability occurs at a critical voltage U cF leading to a homogeneousdirector deformation, called the Freedericksz transition.The electric Freedericksz transition can be induced bydc ( U dc ) as well as by sinusoidal ac ( U ac ) voltages offrequency f [14]. In the case of an asymmetrical driv-ing, the applied voltage is described by U = U dc + √ U ac sin (2 πf t ). The onset of the instability can beachieved by different combinations of the two control pa-rameters, U dc and U ac , characterized by a frequency in-dependent threshold curve: a quarter circle in the U ac - U dc plane given by U cF = U dc + U ac . Inside this curvethe system is in its homogeneous basic state; outside theinitial planar state is deformed.If ε a <
0, the electric field exerts a stabilizing torque onthe director, however, an instability can still take placeleading to a periodic director deformation by convection,governed mostly by the Carr-Helfrich mechanism: spa-tial director fluctuations lead to space charge separationdue to the conductivity anisotropy σ a = σ || − σ ⊥ ( σ || and σ ⊥ are the conductivities measured with an electric fieldparallel and perpendicular to the director, respectively);the Coulomb force induces flow forming vortices due tothe constraining surfaces; the flow exerts a destabiliz-ing torque on the director [12]. Above a critical voltage U c , the fluctuations do not decay, but grow to a macro-scopic pattern of convection rolls characterized by a criti-cal wave vector q c . The typical case of electroconvection(standard EC) can be observed in a planar cell filled witha nematic liquid crystal with ε a < σ a >
0. Theresulting patterns correspond to a spatially periodic sys-tem of convection rolls that appear as dark and brightstripes perpendicular (or oblique) to the initial directorin a microscope.Different modes of EC can be realized at the onset ofthe instability depending on the frequency. Typically athigh f , the dielectric mode is present; then by decreas-ing f , a transition to the conductive mode occurs at thecrossover frequency f c [12]. In cells of typical thickness,this transition is easily observable due to the largely dif-ferent q c of the two modes.A comprehensive theoretical description of the dif-ferent pattern forming modes in nematic liquid crys-tals is provided by the so-called standard model of EC(SM) [15], which has recently been improved by includ-ing flexo-electricity (extended SM)[16]. It combines theequations of nematohydrodynamics with those of electro-dynamics, while assuming ohmic electrical conductivity.The extended SM provides U c ( f ), q c ( f ) and the spatio-temporal dependence of the director n ( r , t ) at onset, inagreement with experiments. For EC, the equations ex-hibit solutions with 3 different time symmetries. One isfound at dc driving, where n ( r ), the flow field v ( r ), andthe charge field ̺ e ( r ) are static. The other two occur atac voltage excitation: in the conductive mode, the direc-tor and the flow is stationary in leading order (if f is muchlarger than the inverse director relaxation time τ − d ), sothe time average of the director tilt over the driving pe-riod is nonzero ( h n z ( t ) i 6 = 0) while space charges oscillatewith f ; in the dielectric mode, ̺ e ( r ) is stationary while n ( r ) and v ( r ) oscillate with f (thus h n z ( t ) i = 0).Pattern formation may also occur at superposing twoelectric signals. The behaviour of EC has been studiedat adding sinusoidal or square wave voltages of two dis-tinct frequencies ( f < f , f being a multiple of f ) anda nontrivial threshold variation and a reentrant patternforming behaviour was found [17, 18]. Here we reporton the mostly unexplored case of superposing ac and dcvoltages. As standard EC may occur at pure dc as wellas at pure ac excitation, one might expect it to arise ata combined, asymmetrical driving too, just like in thecase of the electric Freedericksz transition. We raisedthe questions, how this nonlinear dynamical system re-sponds to the asymmetric excitation and which kind ofpattern morphologies will occur. The consequences ofthe asymmetric driving in standard EC are mostly un-explored experimentally and are challenging also fromthe theoretical point of view; the solution may not beobtained as a simple superposition of the three modesof different time symmetries mentioned above. Recentstudies have shown that the extended SM can well de-scribe the limits of stability in the U ac − U dc plane forthe conductive EC if f is sufficiently low ( f ≪ f c ); thenthe threshold curve is similar to that of the Freedericksztransition [19]. Our present experimental work aimed toprovide a more comprehensive study on a broad range of f . It will be shown that the onset behavior of the systemis qualitatively different depending on the frequency. EXPERIMENTAL
Our experiments were carried out using the nematicmixture Phase 5. It exhibits ε a < σ a > f , bothdielectric and conductive EC can be observed. We used d = 10 . µ m thick cells (E.H.C. Co., Japan) at the tem-perature of T = 30 ± . ◦ C for the measurements. Ap- plying dc voltage, our planar samples showed EC. Thepatterns were observed in a polarizing microscope usingthe shadowgraph technique [21–23]; the EC thresholdswere determined in the U ac − U dc plane at different fre-quencies of the ac signal. The searching for patterns weredone at a fixed ac-voltage varying the dc-component. RESULTS AND DISCUSSION
In Fig. 1, the threshold curves showing the limits ofstability in the U ac − U dc plane are presented for f = 400Hz. They exhibit a strikingly different behavior com-pared to the quarter-circle of the Freedericksz transi-tion. The threshold curves starting from U dc = 0 V and U ac = 0 V form two branches (the ac- and dc-branches,respectively), which do not cross each other in the avail-able voltage range. At U dc = 0 V, U ac &
24 V, regulardielectric EC rolls are seen that lie normal to the initialdirector. Following the threshold curve on the ac-branch,the morphology remains the same, but surprisingly, withincreasing U dc along the curve, U ac also becomes higher;the curve bends away from the origin towards higher U ac , U dc values. If U ac = 0 V, dc EC can be observed as a rollstructure oblique to the initial director. Following thethreshold curve on this dc branch, with increasing U ac EC appears at higher U dc , and the slope remains posi-tive. We note that at U dc >
23 V, besides EC, stripesparallel to the director also appear in patches; they areattributed to flexoelectric domains (FDs [20, 24–27]).
EC+FD U d c ( V ) U ac (V rms ) no pattern f = 400 Hz dc EC diel. EC f = 200 Hz no pattern dc EC diel. EC U d c ( V ) U ac (V rms ) FIG. 1. (Color online) Morphological phase diagram of aPhase 5 sample at f = 400 Hz, and at f = 200 Hz (insetfrom [19]). The dashed lines, as a guide for the eyes, indicatethe trends of the stability limit curve. Stars indicate those U ac , U dc combinations where the images covering an area of52 µ m × µ m were taken. The initial director lies along thehorizontal direction. The two branches of the stability limit in Fig. 1 cor-respond to patterns with different morphologies and sig-nificantly different wave numbers. In the narrow chan-nel between the two branches no patterns could be de-tected. Surprisingly, the system remains there in thebasic, undistorted state despite of the high voltages ap-plied. For example, at U dc = 32 V and U ac = 55 V,the dc and ac voltage components are more than 6 and 2times larger than the corresponding thresholds for purelydc and purely ac driving, respectively. In the case ofpurely ac or dc driven EC, at voltages so much above thethresholds, the convection would already be in the tur-bulent regime. Our findings thus indicate that using asignal with properly adjusted asymmetry can result notonly in the suppression of undesirable turbulence but alsoin complete inhibition of pattern formation. An unusualsequence of morphologies can be obtained by varying onevoltage component while the other is kept constant. Forexample, at fixed U ac = 40 V, with no dc component, theconvection is turbulent. Increasing U dc , the system be-haves less and less overdriven; it shows regular patternsat U dc ≈
16 V, then if 16 V . U dc .
20 V, there is nopattern at all. Applying higher U dc , electroconvectionsets in again, and becomes turbulent at high values of U dc .The inhibition of pattern formation at combined acand dc driving holds also at higher frequencies. Decreas-ing f , however, leads to a qualitatively different behavior.At f = 200 Hz (inset in Fig. 1), the pattern morpholo-gies in the U ac − U dc plane at onset are similar, but nowthe ac and dc branches cross each other; the pattern-freechannel closes at some voltages, where a morphologicaltransition occurs between the conductive and dielectricroll structures [19].The purely ac driving at f = 80 Hz yields conductiveEC. The ac-branch of the stability limit curve in Fig. 2exhibits a positive slope in the U ac − U dc plane until U dc =2 V, where the conductive roll structure crosses over tothe dielectric one, indicated by a large increase of thewave number. At this morphological transition, the slopeof the curve also changes abruptly: for dc voltages U dc > U ac component until the crossing with the dc-branch (seedotted line in Fig. 2). There an additional morphologicaltransition occurs between the dielectric and the dc ECmodes, shown again by a significant change in the wavenumber.At even lower frequencies, a dc-voltage-induced transi-tion to the dielectric EC does not occur; as a consequencethere is no dramatic change in the critical wave numberalong the stability limit curve in the U ac - U dc plane.Nevertheless, depending on the frequency, the system canshow different characteristics. At f = 20 Hz (see Fig. 3),the ac-branch shows mainly a positive slope that resultsin a larger pattern-free area compared to the expectedquarter-ellipse-shaped threshold curve, found earlier at f = 10 Hz [19] (see inset in Fig. 3).Recently, the critical voltages and wave numbers werecalculated at the onset of EC induced by superposed acand dc voltages [19]. Both analytical and numerical cal- diel. ECcond. ECdc EC U d c ( V ) U ac (V rms ) f = 80 Hz no pattern FIG. 2. (Color online) Morphological phase diagram of aPhase 5 sample at f = 80 Hz. The dashed lines, as a guide forthe eyes, indicate the trends of the stability limit curve. Thedotted line shows the boundary between the patternless basicstate and the region of dielectric electroconvection. Stars in-dicate those U ac , U dc combinations where the images coveringan area of 52 µ m × µ m were taken. The initial directorlies along the horizontal direction. U d c ( V ) U ac (V rms ) dc EC no pattern cond. EC f = 20 Hz no pattern f = 10 Hz U ac (V rms ) U d c ( V ) FIG. 3. (Color online) Morphological phase diagram of aPhase 5 sample at f = 20 Hz, and at f = 10 Hz (inset from[19]). The dashed lines, as a guide for the eyes, indicate thetrends of the stability limit curve. The dotted line shows theboundary between the patternless basic state and the regionof conductive electroconvection. Stars indicate those U ac , U dc combinations where the images covering an area of 52 µ m × µ m were taken. The initial director lies along the horizon-tal direction. culations predicted that the dc-branch has positive slopein the U ac - U dc plane at higher frequencies and negativeslope at lower frequencies. This is in good agreementwith the experimental data in Figs. 1-3. The theoreticalwork also pointed out that the ac-branch should exhibitnegative slope if U dc > f = 20 Hz or f = 80 Hz), in contrast to the theoreticalexpectation, the slope of the ac-branch was experimen-tally found positive; i.e. in the presence of a dc bias theEC instability sets in at higher U ac values. The standardmodel of EC, which assumes constant ohmic conductiv-ity, cannot account for this finding. One should note,however, that assuming a voltage independent (ohmic)conductivity in the case of a weak electrolyte, such as aliquid crystal, is not always realistic. If the applied acvoltage is not symmetric, i. e. a nonzero dc-componentis present, the number of effective charge carriers maydecrease, because a fraction of ions is immobilized at theelectrodes coated with insulating (polyimide) surfaces.Therefore the bulk conductivity of the liquid crystal isexpected to decrease with increasing U dc ; this was actu-ally verified by simultaneous conductivity measurements.Consequently, during the experiments shown in Figs. 1-3the conductivity changes from point to point in the U ac - U dc plane, in contrast to the constant σ value assumedin the theoretical calculations [19].In the case of the purely ac driven conductive EC, thethreshold voltage at a fixed frequency increases if theconductivity is reduced [12, 28]. This increment is largerat higher frequencies, being closer to the conductive-dielectric crossover. This behavior offers a qualitativeexplanation to the shape mismatch between the expectedand measured stability limit curves in the conductiveregime. Application of U dc results in lowering the con-ductivity, which leads to a higher onset U ac of EC thanexpected for a constant σ . The higher the frequency,the more probable that this threshold increment flips theslope of the ac branch from negative to positive, as foundin Figs. 2 and 3. Lowering the conductivity by the dc biasreduces the crossover frequency f c as well. If due to thisreduction f c becomes lower than the driving frequency, adc-voltage-induced transition from the conductive to thedielectric mode occurs, as was actually found at f = 80Hz (Fig. 2).Understanding the characteristics of the dielectric ECat an asymmetric voltage driving is more challenging. Onthe one hand, calculations have shown that the ac- anddc-branches of the stability limit curve do not connectsmoothly, as there is a sharp change in the critical wavenumber [19]; these features were confirmed by the exper-iments at lower frequencies in the dielectric regime (insetin Fig. 1). On the other hand, a discrepancy exists, sincein contrast to the theory, the experimental slope of theac-branch is positive. The dc-voltage-induced σ reduc-tion (which was a clue for the conductive EC) does nothelp here; in the case of dielectric EC the theory pre-dicts diminishing ac threshold voltages for a lower con-ductivity. Consequently, when increasing U dc , the U ac component at the onset of the instability is expected tobe even smaller than without considering the change in the conductivity, while experiments show U ac increasingwith U dc . Moreover, this increment becomes larger athigher frequencies, leading finally to the inhibition of pat-tern formation, i.e. the extension of the stability limit tosuch U ac , U dc voltage components that are several timeshigher than the threshold voltages at purely ac or purelydc voltage excitation (see the channel in Fig. 1).The (extended) standard model of EC is not able toaccount for the inhibited pattern formation and for thedramatic effect of the frequency on the onset characteris-tics of the dielectric EC at asymmetrical voltage driving.Whatever unexpected the inhibition of electroconvec-tive pattern formation shown above is, it is not fully un-precedented. A much less pronounced inhibition has al-ready been reported earlier [17, 18] for samples driven bya superposition of harmonic or square waves, with fre-quency ratios of a small integer number, i.e. by signalswith the same time symmetries. In that case the pattern-free region could be extended only by a few percent ofthe higher f ac voltage, in contrast to the huge incrementshown in Fig. 1.We anticipate that ionic effects originating in the elec-trolytic nature of liquid crystals, such as voltage depen-dent conductivity (electro-purification), internal voltageattenuation in the cell, and Debye layers at the bound-aries may play an important role. Some earlier reportsindicated that the dielectric EC rolls are rather localizedat the cell surfaces than in the bulk [29, 30]. Taking intoaccount that significant electric field gradients may existon the length scale of the dielectric rolls in the vicinity ofthe electrodes, this might account for why the behaviorof dielectric EC is more anomalous than the conductiveEC.In order to include the above mentioned effects andtherefore to give a more complete explanation of our find-ings, the basic assumption of the (extended) SM on theohmic conductivity should be given up. A weak elec-trolyte model (WEM), accounting for ionic dissociation-recombination processes, has been developed more thana decade ago [31–34]. The WEM introduced additionalvariables (the ionic concentrations) with rate equationswith two additional time scales (for the recombinationand migration times of ions), and with the relevant(mostly unknown) material parameters (e.g. ionic mobil-ities). The resulting set of partial differential equations,which is even more complex than that of the extendedSM, has only partially been analyzed to prove the ionicorigin of the Hopf bifurcation (travelling waves) in AC-driven EC. We expect that the WEM, generalized withinclusion of flexoelectric phenomena, would be a propertheoretical tool to describe the dc voltage induced phe-nomena, including the inhibition of the pattern forma-tion. Such an analysis is, however, a huge theoreticalchallenge for the future. CONCLUSIONS
In summary, we reported convective pattern forma-tion in a nematic fluid induced by asymmetric voltagesignals, exhibiting a rich variety of morphological tran-sitions. The experiments showed, in contrast to our in-tuition and the predictions of the (extended) standardmodel of EC, that the joint action of ac and dc voltagesrather inhibits pattern formation than favors the emer-gence of instabilities. While for the conductive EC aqualitative explanation based on the change of conduc-tivity could be given, the question of why the patternformation is largely inhibited in the dielectric mode athigh frequencies still needs to be precisely answered inthe future. The unexpected suppression of pattern for-mation at high applied voltages can open new horizons in studies of (sub)criticality or director fluctuations in elec-tric fields in voltage ranges where investigations were be-lieved to be impossible due to the occurrence of patternsor turbulent flow of the material. Our finding also risesthe question whether analogous effects can be found inother dynamical systems, such as isotropic EC, or shearinduced turbulent convection combined by electric fields. ACKNOWLEDGEMENTS
Financial support by the Hungarian Research Fundgrants OTKA K81250 and NN110672 are gratefully ac-knowledged. The authors are indebted to Werner Peschand Alexei Krekhov for fruitful discussions. [1] M. C. Cross and P. C. Hohenberg
Rev. Mod. Phys. ,851 (1993).[2] G. Ahlers, S. Grossmann, and D. Lohse, Rev. Mod. Phys. , 503 (2009).[3] G. Ahlers, E. Bodenschatz, D. Funfschilling, S. Gross-mann, X. He, D. Lohse, R. J. A. M. Stevens, and R.Verzicco Phys. Rev. Lett. , 114501 (2012).[4] K. Petschel, S. Stellmach, M. Wilczek, J. L¨ulff, and U.Hansen
Phys. Rev. Lett. , 114502 (2013).[5] R. du Puits, L. Li, C. Resagk, A. Thess, and C. Willert
Phys. Rev. Lett. , 124301 (2014).[6] S. G. Huisman, D. P. M. van Gils, S. Grossmann, C. Sun,and D. Lohse
Phys. Rev. Lett. , 024501 (2012).[7] Y. Duguet and P. Schlatter
Phys. Rev. Lett. , 034502(2013).[8] S. G. Huisman, S. Scharnowski, C. Cierpka, C. J. K¨ahler,D. Lohse, and C. Sun
Phys. Rev. Lett. , 264501(2013).[9] K. Deguchi, A. Meseguer, and F. Mellibovsky
Phys. Rev.Lett. , 184502 (2014).[10] Y. Kuramitsu, Y. Sakawa, S. Dono, C. D. Gregory, S. A.Pikuz, B. Loupias, M. Koenig, J. N. Waugh, N. Woolsey,T. Morita, T. Moritaka, T. Sano, Y. Matsumoto, A.Mizuta, N. Ohnishi, and H. Takabe
Phys. Rev. Lett. ,195004 (2012).[11] O. A. Hurricane, V. A. Smalyuk, K. Raman, O. Schilling,J. F. Hansen, G. Langstaff, D. Martinez, H.-S. Park,B. A. Remington, H. F. Robey, J. A. Greenough, R.Wallace, C. A. Di Stefano, R. P. Drake, D. Marion, C.M. Krauland, and C. C. Kuranz
Phys. Rev. Lett. ,155004 (2012).[12] L. Kramer and W. Pesch, In ´A. Buka and L. Kramer (ed-itors),
Pattern Formation in Liquid Crystals (Springer-Verlag, New York, 1996) pp. 221255.[13] P.G. de Gennes and J. Prost,
The Physics of Liquid Crys-tals (Oxford Science Publications, 2001).[14] I. Stewart,
The Static and Dynamic Continuum Theoryof Liquid Crystals (Taylor & Francis, New York, 2004).[15] E. Bodenschatz, W. Zimmermann, and L. Kramer,
J.Phys. (France) , 1875 (1988). [16] A. Krekhov, W. Pesch, N. ´Eber, T. T´oth-Katona, and ´A.Buka, Phys. Rev. E , 021705 (2008).[17] J. Heuer, T. John, and R. Stannarius, Mol. Cryst. Liq.Cryst. , 11 (2006).[18] D. Pietschmann, T. John, and R. Stannarius,
Phys. Rev.E , 046215 (2010).[19] A. Krekhov, W. Decker, W. Pesch, N. ´Eber, P. Salamon,B. Fekete, and ´A. Buka, Phys. Rev. E , 052507 (2014).[20] N. ´Eber, L. O. Palomares, P. Salamon, A. Krekhov and´A. Buka, Phys. Rev. E , 021702 (2012).[21] S. Rasenat, G. Hartung, B. L. Winkler, and I. Rehberg, Exp. Fluids , 412 (1989).[22] S. P. Trainoff, and D. S. Cannell, Phys. Fluids , 1340(2002).[23] W. Pesch and A. Krekhov, Phys. Rev. E , 052504(2013).[24] Y. P. Bobylev and S. Pikin, Zh. Eksp. Teor. Fiz. , 369(1977).[25] A. Krekhov, W. Pesch, and ´A. Buka, Phys. Rev. E ,051706 (2011).[26] P. Salamon, N. ´Eber, A. Krekhov and ´A. Buka, Phys.Rev. E , 032505 (2013).[27] ´A. Buka and N. ´Eber, Flexoelectricity in Liquid Crystals.Theory, Experiments and Applications , (Imperial CollegePress, London, 2012).[28] W. Pesch, A. Krekhov, private communication .[29] N. Gheorghiu, I. I. Smalyukh, O. D. Lavrentovich, andJ. T. Gleeson,
Phys. Rev. E , 041702 (2006).[30] H. Bohatsch and R. Stannarius, Phys. Rev. E , 5591(1999).[31] M. Treiber and L. Kramer, Mol. Cryst. Liq. Cryst. ,311 (1995).[32] M. Dennin, M. Treiber, L. Kramer, G. Ahlers, and D. S.Cannell,
Phys. Rev. Lett. , 319 (1996).[33] M. Treiber, N. ´Eber, ´A. Buka, and L. Kramer, J. Phys.II (France) , 649 (1997).[34] M. Treiber and L. Kramer, Phys. Rev. E58