Flexo-electric domains in liquid crystals

phenyl ester of p-pentyloxybenzoic acid (DCEPBA, 8a = - 25 at t = 25 °C) or p-cyanophenyl ester of p-heptylbenzoic acid (CEHBA, 8a = + 29 at t = 25 OC...

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Flexo-electric domains in liquid crystals M. I. Barnik, L.M. Blinov, A.N. Trufanov, B.A. Umanski

To cite this version: M. I. Barnik, L.M. Blinov, A.N. Trufanov, B.A. Umanski. Flexo-electric domains in liquid crystals. Journal de Physique, 1978, 39 (4), pp.417-422. <10.1051/jphys:01978003904041700>.

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Physics Abstracts 61.30



Organic Intermediaries and Dyes Institute, Moscow, K-1, U.S.S.R. (Reçu le 5 octobre 1977, Résumé.

révisé le 9 décembre 1977,

accepté le 5 janvier 1978)

On a étudié les caractéristiques de seuil d’une instabilité dans la forme des domaines flexoélectriques apparaissant dans une fine couche de cristal liquide soumise à un champ électrique continu ; les échantillons sont des phases nématiques obtenues par mélanges de composés esters et azoxybenzènes. On a déterminé les variations de la tension seuil et de la périodicité des domaines en fonction de l’anisotropie diélectrique. On a montré que la distorsion maximum d’une couche homogène correspond à la partie centrale de cette couche. En comparant des données expérimentales avec la théorie de Bobylev et Pikin pour les domaines flexoélectriques à deux dimensions, on a calculé la différence de coefficients flexoélectriques e* e11 2014 e33 pour les cristaux liquides étudiés. 2014


Abstract. The threshold characteristics of an instability in the form of longitudinal flexoelectric domains, arising in thin layers of liquid crystals exposed to a d.c. field were investigated for two nematic mixtures based on azoxy-and ester compounds. The experimental dependences of the threshold voltage and the domain period as functions of dielectric anisotropy were obtained. It was shown that maximum distortion of a homogeneously oriented layer corresponds to the central region of the layer. The difference of the flexo-electric moduli e* e11 - e33 was calculated for the mixtures by comparing the experimental data with the theory for two-dimensional flexo-electric deformation developed by Bobylev and Pikin. 2014


It is well known that in homo1. Introduction. geneously oriented layers of nematic liquid crystals (NLCs) in an external electric field, an instability occurs at a certain threshold voltage. The instability is characterized by a specific spatially-periodic pattern of the molecular distribution which shows up in the form of Williams domains at low frequencies, -

and in the form of chevrones at high frequencies, , Wc (here 6 and e are the electrical conductivity and dielectric constant, respectively) [1]. At the threshold the direction of domains is perpendicular to the director for both domain patterns mentioned above. A mechanism for the formation of the domains have been well investigated both theoretically [2-7] and experimentally [7-9]. It is important that the nature of these domains is electrohydrodynamic. However, there is one more type of instability which is not due to the current flow in an NLC. When the NLC molecules are of the pear- or banana-like form, a specific piezoelectric [10] (or flexoelectric [11]) effect could take place. The one-dimensional theory not accounting for the boundary conditions for homogeneous layers again predicts a field-induced periodic pattern in the form of domains perpendicular to the director [10]. The transverse domains also W >

appear in

homogeneously oriented layers with strong molecular anchorage, exposed to a non-uniform field, because of the gradient flexo-electric effect [12]. In this case a flexo-electric deformation, shows no threshold dependence with external field, and the transition from the aperiodic regime of distortion to the periodic one is of the second order type. Another theory [13] developed for the rigid boundary conditions predicts a novel type of domains which are parallel with the initial direction of the director (longitudinal domains). For the Bobylev-Pikin domains the maximum flexo-electric distortion is located in the middle of the liquid crystalline layer. In addition, there is a sharp threshold at increasing voltage, i.e. this phenomenon is of the first order transition type. It should be noted, that the domain period and threshold voltage are determined not by the sum but by the difference of the splay and bend flexo-electric moduli, e* ei i - e33. We have shown earlier [14-15] that the mechanism of the longitudinal domains, arising in thin layers of nematic liquid crystals (NLCs) of low electric conductivity [16], can, in principle, be understood using the flexo-electric model [13]. In the present paper, the detail experimental investigation of the conditions for the longitudinal domains formation is made and the dependences of the threshold voltage (Ut.;) and the =

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01978003904041700


period (Wh) on the NLC parameters (electrical phase of p-n-butoxybenzilidene-p-butylaniline conductivity, anisotropy of the conductivity ulllu, (BBBA) and dielectric anisotropy Ga) cell thickness and temperature was obtained. The voltage dependence of the domain period was also investigated. In addition, a special experiment with twist cells was carried out to find the spatial region of the maximum distortion. having a certain degree of the smectic ordering. The experimental data obtained confirm the flexoThe experimental technique was exactly the same electric nature of the longitudinal domains and allow as reported in [15]. The measurements of Uth and us to calculate the mean elastic modulus and the Wth with d.c. voltage were carried out on thin (up to difference of the flexo-electric coefficients e* for the 2 p) .planar and wedge-shaped cells with rubbed Sn02 NLCs under study. electrodes. It is known that the rubbing technique provides relatively strong molecular anchoring at The investigations were made the Sn02 electrodes. The anchorage energy (W,) 2. Experimental. for p-n-butyl-p-methoxyazoxybenzene (BMAOB) in that case is of the order of 0.1 erg. [18]. -



Consequently, for usual values of the mean elastic modulus K -- 10-6 dyn the characteristic length

nematic interval 25 to 73 °C

b = il W, -- 0. 1 g is always less than the layer thick-



on a

quaternary mixture of the phenylbenzoates



ness and the condition of strong anchorage, which is necessary for the model of Ref. [13], is fulfilled here. Some experiments were also carried out using twist cells of thicknesses of 12 and 20 J.l.

nematic interval - 5 to 62 OC The optical pattern of the longitu3. Results. dinal domains for various applied voltages is shown in figure 1. The crucial condition which must be satisfied for the appearance of the domains depends on a certain relationship between the NLC electrical conductivity and the cell thickness. For example, in pure BMAOB the longitudinal domains arise for the thicknesses up to 20 J.l, the threshold voltage being practically independent of thickness (Fig. 2). Further increase in cell thickness (d) results in an electrohydrodynamic (EHD) mode of an instability in the form of transverse domains or parquet [1 5]. The critical thickness (de) below which the EHD mode of an instability is completely suppressed decreases with increasing conductivity. Under the condition d dc the threshold and period of the longitudinal domains themselves are independent of the electrical conductivity and the anisotropy of the conductivity. The latter was varied in the range 1.8 > ,,/ov > 1.2. Dust particles occasionally located near domain strips do not move with applied d.c. voltage as seen under a microscope. Thus, in contrast to Williams domains, the appearance of the longitudinal domains is not accompanied by the steady-state flow of the liquid. When an a.c. voltage is applied to pure samples of 60 g, only the any thickness in the range 3 d EHD instability with a specific threshold-frequency dependence, Ulh - f 1/[email protected] was observed. We have shown earlier that such an instability can result from the isotropic mechanism [9, 17]. The longitudinal domains can be only seen at infra-low frequencies, f f [15]. Of course, for samples of high conduc- , tivity Williams domains are observed at frequencies -


for each of the components, respectively. At the temperature t 25 °C the starting materials had a d.c. electric conductivity =




frequency dielectric anisotropy

respectively for BMAOB and PB (the indices Il and 1 refer to the direction of the director). When it was necessary, dielectric anisotropy was varied by doping the NLCs with the 2,3-dicyano-4-pentyloxyphenyl ester of p-pentyloxybenzoic acid (DCEPBA, 25 at t 25 °C) or p-cyanophenyl ester of 8a acid + 29 at p-heptylbenzoic (CEHBA, 8a t 25 OC). The conductivity was controlled by doping the NLCs with donor and acceptor impurities [8]. We observed the longitudinal domains also for a quaternary azoxy-compound mixture (a mixture A in [8]) doping with CEHBA, as well as for the nematic =





./i/./c= wc/2 n.


The threshold voltage for longitudinal domains as function of layer thickness (t 25 °C). Dielectric anisotropy

FIG. 2.



: Ea=0 (1), -0.25 (2), -0.35 (3), -0.4 (4) for BMAOB ; Ea= + 0.09 (5) =

for PB.

FIG. 3. (3,4) for

The threshold voltage (1,2) and period at the threshold longitudinal domains vs dielectric anisotropy (d 11.7 J.1, t 25 °C). Materials :1,3-BMAOB, 2,4- PB.




FIG. 1. - The microscope pattern (frame dimensions are 600 x 400 Il) of the longitudinal domain instability (BMAOB, e. 0.25, ul,= 6 x10-13 ohm-’. cm-’, d = 11.7 Il, t 25 OC). Applied voltage 16 V (a), 25 V (b) and 50 V (c). =

The threshold voltage and spatial period of the longitudinal domains depend strongly on dielectric anisotropy. The experimental data for Uth(8a) and Wth(8a) for the BMAOB and FB samples (d 11.7 p) exposed to a d.c. field are shown in figure 3. The threshold voltage increases sharply with increasing 0. For the critical modulus of 8a in the range 8a values Of 8a = - 0.5 (BMAOB) and ga 0.3 (FB) no domains appear at any voltages up to the breakdown value of about 150 V. The period of the domains at the threshold voltage increases with increasing 8a becoming equal to the layer thickness for 8a = 0. =



At fixed e. the period Wth is a linear function of cell thickness (see Fig. 4). The dependence of the domain period on cell thickness for different voltages applied to BMAOB layers is shown in figure 5. The variation of the domain period with voltage in BMAOB and FB for various values of e. is given in figure 6. 4. Discussion. The instability under consideration is a static deformation of the director distribution and not an electrohydrodynamic process because of the absence of flow in the liquid and the independence of the domain threshold voltage of NLC electrical conductivity and its anisotropy. The flexoelectric effect can be responsible for such a deformation. According to the calculations of Ref. [13], an instability in the form of longitudinal domains can result from the flexo-electric effect in a planar layer of finite thickness with strong molecular anchorage at the boundaries. For this instability the maxi-


distorsion is located in the middle of a layer and the domain period decreases with increasing voltage. The formation of the domains looks like a first-order phase transition. The threshold voltage and the domain period at the threshold were derived mum

as :

Here K is the elastic modulus

corresponding to the

K2z, e* is a difference of flexo-electric coefficients for splay and bend deformation (e* = e li - e33) and li e. K14 ne*2. It follows from the equations (1) that the instability can arise only under conditionIl’| 1 or

approximation K





The dependence of the domain period at the threshold FIG. 4. on cell thickness (BMAOB, t 25 OC). Dielectric anisotropy : Ga = - 0.3 (1), - 0.25 (2), + 0.09 (3). -


mentioned above, the condition of strong anchoring was fulfilled in our experiments. The availability of the sharp domain threshold is also in agreement with the theory of Ref. [13]. In order to show that the maximum distortion is in the middle of a cell a special experiment was done with twist cells. In this case we have to see two systems of the longitudinal domains perpendicular. to each other if the maximum distortion takes place near the electrodes. In fact, we observe only one domain system located at an angle about 45° to the rubbing directions. This direction corresponds to the director orientation in the middle of a cell as can be checked by the observation of the chevrone mode with an a.c. voltage. Thus, there is a maximum distortion in the layer center in accordance with the model of Bobylev and Pikin. It should be noted that the exact angle between domain lines and one of the rubbing directions of a twist cell can be changed by field polarity switching from the value slightly less than 450 to the value slightly exceeded 45°. Such a behaviour is not typical of Williams domains and chevrones and demonstrates the linearity of the flexo-electric phenomenon. Let us show now that the experimental dependences of the threshold voltage and domain period on dielectric anisotropy agree with equations (1). These dependences can be presented in the following coordinates : A n(1 + d 2/ [email protected]) VS Uth and B 4 c( W h - d 2)/ (Wh + d2) vs Ba [15]. The values for Ba and Uth are chosen as parameters for the two, respectively. The new coordinates are utilized in accordance with the As

FIG. 5. The dependence of the domain period above the threshold on cell thickness (BMAOB,t = 25 oC, Ba = - 0.25). Applied voltages : U x Uth 14 V (1), 20 V (2), 40 V (3), 60 V (4). -




equations :

FIG. 6. The dependence of the domain period on the inverse of the applied voltage (t 25 °C, d 11.7 g). Dielectric anisotropy 0.25 (1), 0 (2), + 0.15 (3) for BMAOB; 8, + 0.07 (4), e. 0.09 (5) for PB. -









were obtained from set (1). If the equations (1) adequate for the experimental data of Uth(£a) and Wth(Ea), in the new coordinates we have to obtain straight lines with slopese* |IK and K/e*2. Indeed, the dependences A( Uth) and B(ej are linear ;

which are

however, the curves B vs

0. e. have fractures at e. The equations (1) are expected to be more suitable for the range e. 0 since they have been derived for small perturbations of the director distribution. For e. > 0 the latter condition seems to not be fulfilled as, in addition to the flexo-electric torque, the dielectric torque is destabilizing and reinforces the distortion. This consideration is confirmed by the observation of the high diffraction efficiency of the domain structure even at the threshold voltage. For 0 this efficiency is essentially lower. Another e. possible reason for the fracture of the curve B(e.) at 0 can result from the approximation Ba =


can obtain a better fit of the experimental data equations (1) by taking into account the anisotropy of the elastic moduli [19].

One to

The coefficients K and e* were determined from the data for e. 0. At t 25 °C they are 6.5 x 10-’ dyn and 1.7 x 10-4 CGS units for BMAOB, 9.3 x 10-’ dyn and 1.8 x 10-4 CGS units for PB. The order of the values for K agrees well with the results of direct measurements of the moduli K 1 l, K22. It should be mentioned that the difference of the flexo-electric moduli was measured for the first time, other experiments usually give their



[20-22]. Substituting the experimental values for K and e* into the inequality (2), we conclude that the instability 0.56 and in can appear in BMAOB only for 1 Ba1 0.44. These calculated values PB only for lca1 agree well with the experimental ones for the range 0, where the threshold voltages diverge at some Ba critical values of Ba, see figure 3. According to equation (1), the domain period Wth is proportional to cell thickness. This dependence is observed, see figure 4, and from the value Of £a and the slope of the curve, which is equal to sum

the ratio



be determined. The


Kle*’ 2.5 for BMAOB at e. 0 agrees well .with the result obtained from figure 3. The linear relationship between the domain period value of


and cell thickness also holds for voltages above the threshold, see figure 5. Such a behaviour is expected from the analysis of equation (1) of Ref. [13]. Besides, the theory predicts the proportionality W - U which was confirmed experimentally for U » Ulh (Fig. 6). In principle, the ratio K/e* can again be determined independently from the function W(d) at U » Uth. However, there is a discrepancy between these data and the results of the calculation of K/e* from the threshold characteristics of the instability. The discrepancy seems to result from the fact that the condition of small deformation of the molecular distribution [13] is not fulfilled at the voltages well above threshold. The threshold voltage of longitudinal domain formation is almost independent of temperature. This was checked experimentally for doped BMAOB with Ea N 0. Thus, the strong temperature dependence of the modulus K is compensated by the same dependence of the flexo-electric coefficient e*. This is in qualitative agreement with a microscopic theory for the flexo-electric effect [23, 24]. All the results obtained for BMAOB and PB are also reproduced for a quaternary mixture of azoxycompounds (a mixture A). For BBBA the flexoelectric domains can be observed only after doping the substance by CEHBA so as to obtain a very small value of dielectric anisotropy (e. z 0). Therefore, the comparison of our experimental data with the theory of Ref. [13] leads to the conclusion that the longitudinal domains arising in thin homogeneously oriented nematic layers of low electrical conductivity exposed to a d.c. electric field can be explained by the flexo-electric model. Additional theoretical and experimental studies seem to be necessary to investigate the role of the boundary conditions and the elastic moduli anisotropy as well as to determine the distortion geometry. _

Acknowledgment. We are grateful to Dr. P. V. Adomenas and Dr. B. M. Bolotin for the samples of DCEPBA and PB, Dr. S. A. Pikin and Dr. A. G. Petrov for valuable discussions. We would like also to thank Mrs. N. I. Mashirina for the measurements of the dielectric constants of NLCs. -



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[15] [16]

[17] BARNIK,




Zh. Eksp. Teor. Fiz. 72 (1977) 756. [18] RYSCHENKOW, G., KLÉMAN, M., J. Chem. Phys. 64 (1976) 404. [19] The authors are very grateful to a referee of J. Physique for supplying the results of corresponding estimations. [20] SCHMIDT, D., SCHADT, M., HELFRICH, W., Z. Naturforsch. 27A (1972) 277. [21]PROST, J., PERSHAN, P. S., J. Appl. Phys. 47 (1976) 2298. [22] DERZHANSKI, A., PETROV, A. G., MITOV, M. D. (to be published in J. Physique). [23] HELFRICH, W., Z. Naturforsch. 26a(1971) 833. [24] DERZHANSKI, A. I., PETROV, A. G., C. R. Acad. Bulg. Sci. 25 (1972) 167.