THRESHOLD FLEXOELECTRIC EFFECT IN NEMATIC LIQUID CRYSTAL

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THRESHOLD FLEXOELECTRIC EFFECT IN NEMATIC LIQUID CRYSTAL

Y. Bobylev, V. Chigrinov, S. Pikin

To cite this version:

Y. Bobylev, V. Chigrinov, S. Pikin. THRESHOLD FLEXOELECTRIC EFFECT IN NE- MATIC LIQUID CRYSTAL. Journal de Physique Colloques, 1979, 40 (C3), pp.C3-331-C3-333.

�10.1051/jphyscol:1979364�. �jpa-00218760�

JOURNAL DE PHYSIQUE Colloque C3, suppldment au no 4, Tome 40, Avril 1979, page C3-331

THRESHOLD FLEXOELECTRIC EFFECT IN NEMATIC LIQUID CRYSTAL

Y. P. BOBYLEV, V. G. CHIGRINOV and S. A. PIKIN

Institute of Crystallography, Academy of Sciences of the USSR, Moscow 117333, USSR

RBsumB. - Une rksolution complhte du problhme d'instabilitt: flexoklectrique est donnke dans le cas d'un bchantillon d'bpaisseur finie et sans approximations sur les constantes Clastiques.

Abstract. - Total solution of flexoelectric instability problem is given for finite thickness sample and real elastic moduli.

There is a linear coupling between electric pola- rization and orientational deformation in nematic liquid crystal [l]. The appropriate term in the free energy of liquid crystal under the influence of electric field E can be written in accordance with symmetry properties as

where n(r) is director, e, and e2 are flexoelectric moduli. This coupling results in appearance peculiar flexoelectric effect which is a periodic distortion of initial planar orientation. Meier [l] had shown that the infinite liquid crystal must be disturbed and the perturbation must be periodic along the director orientation in the absence of electric field. This defor- mation is not the threshold one and its period is inversely proportional to electric field strength.

It was shown in the paper [2] that another type of flexoelectric instability is possible in nematic liquid crystal layers of finite thickness with a homo- geneous orientation on the walls. New instability arising at certain threshold voltage is characterized by small director deviations from X axis in two planes : on angle 8 in XZ plane and on angle p in XY plane (see Fig. 1). The instability results in arising of the periodic along Y axis domain structure which

is parallel to X axis. In the case of small deformations l p l , 181

+

^{l , n , =} l , n y = p , n , = 0, E, = E, = 0, E, = E and equal elastic moduli K,, = K,, = K the solution corresponding to the boundary condi- tions 0 = p = 0 at z =

+

d/2 takes the form :

8 = 8, cos (qy) cos (nzld)

,

cp ⁼ cpo sin (qy) cos (nzld)

.

⁽²⁾

The instability appears at the threshold voltage U, and the threshold wave vector q, :

where

p = (E, K/4 ne# ') , e# = e1 - e 2 ,

E, is dielectric anisotropy, d is the layer thickness.

Similar flexoelectric instability can exist under the influence of constant as well as alternating electric field. Frequency dependence of the threshold is analysed in reference [2].

Such type of instability seems to be observed for the first time by Vistin 131. Perhaps this instabiIity can be observed in alternating field a t temperatures when the Williams domains do not exist because of negative sign of conductivity anisotropy [4].

Arising of this flexoelectric instability in constant field and the dependence of threshold voltage and wave vector upon dielectric anisotropy were investi- gated in reference [5] in details using n-buty1-n"- methoxyazoxybenzol doped by 2-3-dicyano-4-amy- loxyphenyl ester of amyloxybenzoic acid or nu- cyanphenyl ester of n-heptylbenzoic acid for changing of dielectric anisotropy. The experimental data are in good agreement with the theory and allow one to define K and e # for BMAOB taking

E, < 0 (K = 6.5 ^X 10-' dyne, FIG. I. - Distribution of molecular orientation. e# = 1.7 X 10-4 CGS units) .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979364

C3-332 Y. P. BOBYLEV, V. G. CHIGRINOV AND S. A. PIKIN

The equations (3) were received for equal moduli Kjj so only average values of Kjj and e" were deter- mined in reference [5].

The purpose of the present paper is the investi- gation of threshold characteristics of flexoeffect using real values of elastic moduli. In the case of small deformations 1cp

1, 1

f3

1 <

1 the free energy of nematic can be written as

Let us take the solution satisfying to boundary conditions 8 = cp = 0 at z =

+

^{d/2 as}

cp = 9 0 sin (qy) exp(ipz) ⁹

8 = 8, cos (qy) exp(ipz)

.

⁽⁶⁾

Inserting (6) into (5) and taking into account the condition of nontriviality of 8, and p, values one can write the following equation for characteristic numbers p :

where s = plq, p = ((E, K1 1/4 ne# ,), a ⁼ K2,/Kl1,

5

⁼ e# E / K l l g.

Equation (7) has 4 roots : f S,, f S,. So one get the complete solution of equation (5) in the form :

L

Minimizing the free energy F with respect to cp and 8 ^(p

-

^{sin (qy)}

^C

(ai cos (qs, z)

+

bi sin (qs, ²⁾⁾ ,

i = l

one can receive the following equations for stationary

orientation of director (8)

L

8 = cos ( q y )

1

[ai(& cos (qsi z ) - yi sin (qs, z ) )

+

i= l

+

bi(pi sin (qs, z)

+

yi cos (qsi z ) ) ]

,

r J I

a2e a2e

^(P

K,,-

+

^K,,-

+

^{e# E -}

+

ay2 dz2 ay boundary conditions. Inserting the solution (8) into

a, and b, are integration constants calculated from

a2s

& , E , boundary conditions one can get the following

+

( K l l - K,,)

m ^{+ e = o .}

condition of nontriviality of ai and bi :

1

^{cos (As,) cos (As,)} 0 0

I

0 0 sin (As,) sin (As,)

p,

cos (As1)

p,

cos (As,) yl COS (As1) y, COS (As,) y, sin (As,) y , sin (As,) -

p ,

sin ( ] . S , ) - f?, sin (As,)

= 0 (9)

where A = (qd/2). Equation ( 9 ) is the dispersion equation connecting field strength E with wave number q. The dependence E = E(q) was calculated numerically for different values of p and K2,/Kl,.

The instability threshold is determined as a mini- ²⁵⁰ mum E = E, = E(qJ on the curve E(q). Critical

potential difference U, = E, d does not depend on ²⁰⁰ layer thickness d.

Dependence of threshold characteristics on para- ¹⁵⁰ meter K2,/Kl1 is different for negative and positive ^(E,.

The values U , and g, increase almost linearly with 400.

increasing of ratio K , , / K , , (Fig. 2) for ^(E, < 0. At

(E, > 0 the threshold voltage increases up to Fredericks ^{50 -~}

transition threshold U, = 4 4 ~ K , , / ( E , ) ~ / ~ and wave

U, $ d / ~

-

5 -

*

4 - 3 -

2 -

K 2 2 / 6

vector decreases to zero (see Fig. 3). At K,, = 0 ^{I 5 I0 45} solution does not exist. Ca~culatiOns were performed _{FIG. 2. -} Dependence of threshold voltage U , (curve 1) and

for the following values of liquid crystal parameters : threshold wave vector q, (curve 2) upon K,,/K,, at E, = - 0.1.

THRESHOLD FLEXOELEmRlC EFFECT IN NLC C3-333

O' - -

-

05 4 45 ; ^{K22 /K41}

FIG. 3. - Dependence of threshold voltage (curve 1) and wave vector (curve 2) upon K,,/K,, at E, =

+

0.1. Dotted line indicates

the threshold voltage of Fredericks transition.

l € E a

K,, ⁼ 0.677 X 10e6 dyne, e' = lOP4 CGS units,

- b 4 -62 . d2 d4

- 0 4 - a 2 ~ O +

&a = - ' a = + The dependence _{FIG. 4.}

- Dependence of threshold voltage (a) and wave vector (b)

characteristics upon ^E, for various e' was calculated upon dielectric anisotropy. Calculations correspond to the following

by using the values K,, = 0.78 ^X 10-6, values of flexoelectric coefficients e* : 1.5 X 10-4 (curve l), 1.8 ^X 10-4 (curve 2), 2 X 10-4 (curve 3) CGS units. Crosses

K,, ⁼ 0.55 X 10-6 dyne [5]. indicate the experimental values.

The comparison of theoretical and experimental results allows to determine the value

e# = e , - e , = 1.7 ^X 10-4 CGS units.

We considered the dependence of threshold cha- racteristics upon surface coupling energy corres- ponding to polar (angle 8, energy density W, [6]) as well as to azimuth (angle cp, energy density W, [7]) director deviations. In the case of symmetric boundary conditions W, = W, = 0 or W,

-

^{W, -+ the}

flexoelectric structure arises at the same threshold conditions. In the case of nonsymmetric boundary conditions W,

+

W, or W, 6 W, the effect can arise only when W, ,> W, > K j j / d or W, ,> W, > K j j / d that is in qualitative agreement with results of refe- rence [7].

These results show that the difference e , - e, can be found with high accuracy from the comparison of theoretical and experimental dependences of flexo- electric threshold upon dielectric anisotropy (see Fig. 4). Observation of this effect in substances having smectic and nematic phases is of particular interest. In this case the threshold voltage must increase proportionally to K,, near the temperature of phase transition to smectic phase if E, < 0. In the same temperature range the flexoeffect for substances with ^E, > 0 must turn to Fredericks effect

which is followed by sharp increasing of the period of modulated structure.

It must be noted that described structure is electri- cally polarized since there is a component of macro- scopic polarization along Z axis 6Pz

-

^{e,,, q8, v,,}

where 8, and S, are the amplitudes of the director deviations. At U > U,- the amplitudes 8, and 43, are proportional to ( U - i.e.

At U ,> U, we have 19, ^N V), I a n d q e,,, E K - l ,

i.e. 6Pz

-

e:,, K -

'

E. Such change of polarization P,(E) under the influence of low frequency electrical field can result in hysteresis phenomena connected with a long time of orientational relaxation z

-

( y / K q 2 ) where y is a viscosity of substance. Thus some hyste- resis loop can be observed at frequencies f

-

^{10 Hz.}

In reference [8] some oscillograms of dielectric hysteresis loop type were observed but perhaps they are caused by nonlinearity of voltampere characte- ristics [10]. Linear dependence q ^.v E at U > U, was observed in reference [8, 91.

Analytic and numerical calculations show that the existence of tilt orientation results in increasing of threshold U,

- ¹

^{cos 0, 1-I where} 8, is the tilt angle, i.e. the effect disappears for homeotropic orientation.

References [l] MEYER, R. B., Phys. Rev. Lett. 22 (1969) 918.

[2] BOBYLEV, Ju. P., PIKIN, S. A., Zh. Eksp. Teor. Fiz. 72 (1977) 369.

[3] VISTIN', L. K., Dokl. Akad. Nauk SSR 194 (1970) 1318.

[4] TICHOMIROVA, N. A., GINZBERG, A. V., KIRSANOV, E. A., BOBYLEV, Ju. P,, PIKIN, S. A., ADOMENAS, P. V., Pis'ma Zh. Eksp. Teor. Fiz. 24 (1976) 301.

[S] BARNJK, M. I., BLINOV, L. M,, TRUFANOV, A. N., UMANSKI, B. A., Zh. Eksp. Teor. Fiz. 73 (1977) 1936.

[6] DE GENNES, P. G., The Physics of Liquid Crystals (Clarendon Press, Oxford) 1974.

[7] PIKIN, S. A., CHIGRINOV, V. G., INDENBOM, V. L., Mol. Cryst.

Liq. Cryst. 37 (1976) 313.

[8] CHISTYAKOV, I. G., VISTIN', L. K., KristallograJiya 19 (1974) 195.

[9] BARNIK, M. I., BLINIV, L. M., TRUFANOV, A. N., UMANSKI, B. A., J. Physique 39 (1978) 417.

[l01 RABINOVICH, A. Z., SONIN, A. S., Zhidkie kristalli (Mezhvuz.

sbornik, Ivanovo) 1977, p. 83.