Flexo-electric domains in liquid crystals

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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�. �jpa-00208775�

FLEXO-ELECTRIC DOMAINS IN LIQUID ^CRYSTALS

M. I.

BARNIK,

^{L. M.}

BLINOV,

^{A. N. TRUFANOV} and B. A. UMANSKI

Organic

Intermediaries and

Dyes Institute, Moscow, K-1,

^U.S.S.R.

(Reçu

^{le 5 octobre}

1977,

^{révisé le 9 décembre}

1977, accepté

^{le 5}

janvier 1978)

Résumé. ²⁰¹⁴ 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 homo-

gène

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.

Abstract. ²⁰¹⁴ The threshold characteristics of an

instability

in the form of

longitudinal

^flexo-

electric 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.}

Classification

Physics ^Abstracts

61.30

1. Introduction. ^- It is well known that in homo-

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, ,

W > 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

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

^condi-

tions

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 conduc-

tivity [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

418

period (Wh)

^{on the NLC} parameters

(electrical conductivity, anisotropy

^{of the}

conductivity ulllu,

and dielectric

anisotropy Ga)

cell thickness and tempe-

rature 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.

The

experimental

data obtained confirm the flexo- electric nature of the

longitudinal

domains and allow

us to calculate the mean elastic modulus and the difference of the flexo-electric coefficients e* for the NLCs under

study.

2.

Experimental.

^{- The}

investigations

^{were made}

for

p-n-butyl-p-methoxyazoxybenzene (BMAOB)

with a nematic interval 25 to 73 °C

and on a quaternary mixture of the

phenylbenzoates (PB)

^{with a} nematic interval - 5 to 62 OC

where

for each of the components,

respectively.

At the

temperature t

^{= 25 °C the}

starting

^materials

had a d.c. electric

conductivity

and a low

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-pentyloxy- 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).

^The

conductivity

^was controlled

by dop- ing

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

phase

^of

p-n-butoxybenzilidene-p-butylaniline (BBBA)

having

^{a certain}

degree

of the smectic

ordering.

The

experimental technique

^was

exactly

^{the same}

as

reported

ⁱⁿ

[15].

^The measurements of

Uth

^and

Wth

^{with d.c.}

voltage

^{were carried out on thin}

(up

^to

2

p) .planar

^and

wedge-shaped

cells with rubbed

Sn02

electrodes. It is known that the

rubbing technique provides relatively

^{strong molecular}

anchoring

^at

the

Sn02

electrodes. The

anchorage

energy

(W,)

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

b = il W, -- 0. 1 g is ^always

less than the

layer

^thick-

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.

3. Results. ^- The

optical

pattern ^{of the}

longitu-

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

any thickness in the _{range 3} d 60 _g,

only

^the

EHD

instability

^{with a}

specific threshold-frequency dependence, Ulh - f 1/2@

^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

./i/./c= wc/2

^n.

FIG. 2. ^- The threshold voltage ^for longitudinal ^{domains as a}

function of layer ^thickness (t ^{= 25} °C). Dielectric anisotropy : Ea=0 (1), -0.25 (2), -0.35 (3), -0.4 (4) for BMAOB ; Ea= + 0.09 (5)

for PB.

FIG. 3. ^- The threshold voltage (1,2) ^and period ^at the threshold

(3,4) ^for 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

modulus

of 8a

^{in the} range 8a 0. For the critical values

Of 8a = -

^0.5

(BMAOB)

^and ga ^{= -} 0.3

(FB)

no domains _appear at any

voltages

up ^to the break- down 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 conside-}

ration is a static deformation of the director distri- bution 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 flexo-}

electric effect can be

responsible

^{for such a defor-}

mation.

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 ancho- rage ^at the boundaries. For this

instability

^{the maxi-}

420

FIG. 4. ^- The dependence of the domain period ^at the threshold

on cell thickness (BMAOB, ^{t =} 25 OC). Dielectric anisotropy :

Ga = - 0.3 (1), - ^0.25 (2), ^{+ 0.09} (3).

FIG. 5. ^- The dependence of the domain period above the thres- hold 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).

FIG. 6. ^- The dependence of the domain period ^on the inverse of the applied voltage (t ⁼ 25 °C, d ⁼ 11.7 g). Dielectric anisotropy

e. ^{= -} 0.25 (1), ⁰ (2), ^{+ 0.15} (3) ^for BMAOB; 8, ⁼ + 0.07 (4),

- 0.09 (5) ^{for PB.}

mum 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

as :

Here K is the elastic modulus

corresponding

^{to the}

approximation K

⁼

KI

1 ⁼

K2z,

^{e* is a} difference of flexo-electric coefficients for

splay

and bend defor- mation

(e*

= e l i -

e33) and li

⁼ e.

K14 ^ne*2.

^It

follows from the

equations (1)

^{that the}

instability

^can

arise

only

^under

condition Il’ |

^{1 or}

As was 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 elec-}

trodes. 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 obser-}

vation 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

depen-

dences can be

presented

^{in the}

following

coordinates : A ⁼

n(1

⁺

d 2/ W2@h) 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

equations :

which were obtained from set

(1).

^{If the}

equations (1)

are

adequate

^{for the}

experimental

^{data of}

Uth(£a)

and

Wth(Ea),

^{in the new} coordinates we have to obtain

straight

lines with

slopes e* |IK

^and

K/e*2.

Indeed,

^the

dependences A( Uth)

^and

B(ej

^are

linear ; however,

^{the curves B vs} e. have fractures

at e.

^{= 0.}

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 ful- filled _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

e. 0 this

efficiency

^is

essentially

lower. Another

possible

^reason for the fracture of the curve

B(e.)

^at

Ba ⁼ 0 can result from the

approximation

One can obtain a better fit of the

experimental

^data

to

equations (1) by taking

^{into account the}

anisotropy

of the elastic moduli

[19].

The coefficients K and e* were determined from the

experimental

^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

sum

[20-22].

Substituting

^the

experimental

values for K and e*

into the

inequality (2),

^we conclude that the

instability

can appear in BMAOB

only for 1 Ba 1

^{0.56 and in}

PB

only for lca 1

^{0.44. These} calculated values agree well with the

experimental

^{ones for the} range Ba

0,

^where the threshold

voltages diverge

^{at some}

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

the ratio

Kle*’

^can be determined. The

experimental

value of

Kle*’

⁼ 2.5 for BMAOB

at e.

0 agrees well .with the result obtained from

figure

^3.

The linear

relationship

between the domain

period

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}

depen-

dence 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 _azoxy-

compounds (a

^mixture

A).

^For BBBA the flexo- electric 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 homo-

geneously

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

sûpplying

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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Phys.-Usp.

17(1975)658.

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

[12] DERZHANSKI, A., Plenary ^{Lecture at} the Conversatorium on

Liquid Crystals (RZESZOW, Poland), 1975.

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

369.

[14] BARNIK, ^M. I., BLINOV, ^L. M., TRUFANOV, ^A. N., ^UMAN- SKI, B. A., 2nd Liquid Crystal Conf. of ^Soc. Countries, Sunny Beach, Bulgaria (1977) Abstracts, p. 35.

[15] BARNIK, ^M. I., BLINOV, ^L. M., TRUFANOV, ^A. N., ^UMAN- SKI, B. A., Zh. Eksp. Teor. Fiz. 73 (1977) ^1936.

[16] VISTIN’, ^L. K., Dokl. Akad. Nauk SSR 194 (1970) ^1318.

[17] BARNIK, M. I., BLINOV, L. M., PIKIN, S. A., TRUFANOV, A. N.,

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.