Surface-polarization effect in the alignment of nematic liquid crystals

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Surface-polarization effect in the alignment of nematic liquid crystals

G. Barbero, A. Chuvyrov, G. Kaniadakis, E. Miraldi, M. Rastello

To cite this version:

G. Barbero, A. Chuvyrov, G. Kaniadakis, E. Miraldi, M. Rastello. Surface-polarization effect in the alignment of nematic liquid crystals. Journal de Physique II, EDP Sciences, 1993, 3 (1), pp.165-173.

�10.1051/jp2:1993118�. �jpa-00247809�

Classification

Physics

Abstracts 6 1.30

Surface-polarization effect in the alignment of nematic liquid crystals

G. Barbero

('),

A. N.

Chuvyrov

(Z), ^{G. Kaniadakis}

(I),

E.

Miraldi(I)

and

M. L. Rastello

(3)

(J) Dip.

Fisica, Politecnico, C. Duca

degli

Abruzz124, l0l29Torino,

Italy

(2) Bashkir State

University,

Ufa, Russia

(3) 1-E-N-G- Ferraris, Str. Delle Cacce91, 10135 Torino, Italy

(Received 3 March 1992, revised 7 August 1992, accepted ²⁸ September 1992)

Abstract. The

change

in the orientation of nematic

liquid crystals

due _to the increase of the

sample

thickness is

experimentally analysed.

The data show that the

phenomenon

is a critical _one,

typical

of _a second order

phase

transition. The

change

in the surface

polarization

induced

by

_a

selective

ion-adsorption

_on the two

boundary

solid substrates is

proposed

as the _cause of the observed facts and discussed. The considered model is correct for

angles

of tilt small with respect

to the initial homeotropic

alignment,

and the theoretical _curves fit both the interferometric and the

voltage-threshold

^{data. A} few comments on the obtained results _are

reported.

1. Introduction.

It is well known that the _mean molecular orientation of _a nematic

liquid crystal,

the so-called

« director _{» n,}

depends

on both the intrinsic symmetry ^{of the}

material,

and the structural

symmetry

^{of the substrate surface.}

The director

profile imposed by

_a solid substrate has been

analysed by

_{many groups} I both

experimentally [2]

and

theoretically [3],

in order _to find _a connection between the observed orientation and the

physico-chemical properties

^{of surfaces} and

liquid crystal.

A

complete understanding

of the

ordering properties

of _a

surface,

due _to the chemical and

physical interactions,

is far from

being achieved,

but

they

_can be described in terms of _an

«

anchoring

_{» energy.}

From _a

phenomenological point

of

view,

the

anchoring

_energy is defined _{as a} function of the director orientation close to the surfaces its minimum coincides with the _mean molecular direction in the absence of bulk torques. ^This energy

depends

_on both the interaction between the

liquid crystal

^and

substrates,

and the interaction among the nematic molecules themselves.

In the absence of

long-range forces,

the surface energy ^is a local

property

^{in the} sense that it does _not

depend

on the

sample

thickness. On the contrary, ^it does

depend

_on the thickness when there _are

long-range forces,

like electrostatic

forces,

due to

ion-adsorption

_on the

surfaces,

_or viscous

forces,

due to the

velocity

field in the nematic volume

[4].

This energy can be

investigated by controlling

the

boundary conditions,

e. g.

by covering

the cell surfaces with suitable

conducting

and/or

insulating

materials and

by rubbing them,

as for the most

widely

used

displays.

166 JOURNAL DE PHYSIQUE II N°

In this paper, the average director

profile

of different nematic

cells,

filled

by capillarity,

has been determined

by measuring,

with standard interferometric

techniques,

their

birefringence

for different discrete thicknesses. In this _manner,

varying

the

sample thickness,

_{one can vary} its base and lateral

surfaces,

and

indirectly

the concentration of the

selectively

adsorbed

ions,

the

sample

volume

being

constant.

In the

following

the average

birefringence

will be defined _as

usually

_:

t12 n

(An )

= An

df

=

2Rt (sin~

4l

)

,

(1)

-t12

~

C

~~~~/~~~~~~~~~~~~ ~

_ ^,

fl ^'

/ ^t9'

,1' Wit)

°

j

/

,

,11,,llllll/ till/

(llllll

_Ill

a)

llzllllllllllll (jllllll/ j

( _{$ jj}

j

_~l

t)

Ii

,

_j

(

,

rll/ / II

llllll11(/

II film

_y

z b)

t

«lllllljll/,llllfllllll/, ⁽

f

^~

/

^l

_,/

^,

i ,~ ^~~~~

^~ _4~

_?

Ill II / /, / II Ill Ill IN llllllll Ill Z

C)

Fig.

I. The co-ordinate system ^{used in} text and the schematic director

profile

_: a) in presence of surface

polarization

b) induced

by

a

velocity gradient

c) in the _case of _a Fr£edericks transition.

where R

= I

(n~/n~)~,

^with _n~n~ ^the

ordinary

and

extraordinary

refraction

indices,

and

(sin~ 4l)

is the average of the

squared

sinus of the

angle

between the

optical

axis and the

surface

normal,

and ( the coordinate normal to the

boundary surfaces, posed

at (

=

± t/2 _as

reported

in

figure

I.

The

figure

also shows the

peculiar

differences of the director

profile

in three conditions,

Figure

la refers _to the _case when

only

surface forces _are

present,

so that the

alignment

is

nearly uniformly

tilted from _one surface to the other and 4l

(()

= const.

Figure

16 shows the director

profile

due _{to a} kind of _« memory » of the

filling

_{process : near}

the

boundary surfaces,

at ( ₌ ±

t/2,

the

hydrodynamic

^{forces and the}

anisotropy

of the

liquid crystal viscosity impose

that

4l(()

reaches the maximum absolute value. The

profile

is

antisymmetrical,

that is 4l

((

=

4l

(-

(

),

In

figure

lc the

liquid crystal undergoes

the so-called Fredericks transition in the presence of

an extemal field:

4l(()

reaches the maximum value in the middle of the cell at

(

=

0. The

profile

_now is

symmetrical.

The three _{cases can} be

distinguished experimentally by

interferometric

techniques

and the measurements discussed in this paper refer _to the first _case shown.

Only

the Frdedericks transition with weak

anchoring

conditions

(in

the _sense that 4l

(±

t/2 _can vary under the torque

imposed by

the extemal

field)

_can

give

a result similar _to the

uniformly

distorted

alignment

due to the

only change

in the surface energy.

Indeed for this _reason the _same

phenomena

were

discussed,

_some time ago, ^as

« Frdedericks _» effects due to

spontaneous polarization [5],

but _a critical look at the accuracy of the former

experimental set-up (mainly

the surface treatments and the

samples preparation)

induced _{us to}

repeat

^the

experiments.

With _{a new}

experimental apparatus, designed

to match the necessary

stability conditions,

and with

nearly

strong

anchoring

conditions

imposed by

^the

boundary surfaces,

_as

exposed

in the _next

paragraph,

this

ambiguity

is resolved.

In section

3,

_a model of surfaces

anchoring

_energy that

depends

on the

selectively

adsorbed ions is

proposed.

The

corresponding

surface

polarization

is proper to

explain

the observed

abrupt change

in the director

alignment,

induced

by

the

substrates,

when _a criticaI thickness of the

sample

is reached. The model

predictions

are

compared

with measurements of both _mean

director

profile

and the electric field Fredericks

transition,

_{as a} function of the cell thickness. In the last

paragraph

some comments are

reported

_on the numerical results.

2. The

experimental

results.

The nematic

liquid crystal

used _was

methoxy-benzylidene-butyl-aniline (MBBA),

both pure

and mixed

(3

_:

1)

with

ethyl-hydroxy-benzylidene-butylaniline (EBBA).

Two cells _were

prepared

with unmixed MBBA and the

corresponding

sets of

experimental

data

(points

and squares in

Fig. 2a)

refer to nematic

samples kept

between

glass surfaces, optically

flat within 100 _nm, cleaned in sulfo-cromic acid and covered with

Cr~,

30 _nm thick.

The two sets of data have been obtained in different

times,

with different

accuracies,

but with the _same

experimental _apparatus

_: the

complete matching

between the data confirms that the cell thickness is _a

good

control parameter ^{for this}

phenomenon,

The last set of

experimental

data

(Fig, 2b)

refers to _a third nematic

sample

filled with mixed

MBBA and EBBA and

kept

between fused quartz

plates

covered with

ln~O

and

SnO~,

The chosen surface _treatments

give

in any case an

homeotropic alignment,

I-e- the

starting

director

profile

is normal _to the

walls,

with

nearly

_strong

anchoring conditions,

and in that

position

it would continue _to be while the cell thickness is

changing,

unless other _causes, from inside _or outside the

specimen,

_are able _to support a tilted director

alignment.

168 JOURNAL DE PHYSIQUE II N°

. , m m m mm . mm m

, m " "

0

30 ~

jp~j

⁶⁰

a)

0.12

m

~ W

'$ W

~e

C

~

W

m

0

50

jp~j

^loo

b)

Fig.

2.

Average

sinus of the tilt

angle (sin~

~P

)

^~'~ _versus the thickness _t for

samples

filled with pure MBBA (a), _{or a} nematic mixture MBBA ₊ EBBA (b). Points and squares in al refer _{to two sets} of

measurements

performed

in different times but with the _same

experimental facility.

The theoretical

curves are drawn with the value of the parameters discussed in _text.

The parameter ^{under control in} every run has been the cell thickness. This _can be varied

mechanically

with the aid of micrometer _screws, and _a

great

care was delivered in the

design

of

the mechanical device

appropriate

to

modify

the

sample

dimension without

modifying

the

parallelism

of the

boundary

surfaces.

To the _same purposes the metallic materials used for

supporting

the

liquid crystal cells,

and handled _to

change

their

thickness,

were chosen with the smallest thermal

expansion

coefficient and

kept

in _a thermostatic chamber.

The overall accuracy of these micrometric measurements has been of the order of

2 _~Lm for what _concems the absolute

length determination,

but somewhat better if _one looks

at the remake of the

experiment,

as

previously reported.

The

general properties

that the collected data show _are the

following

_ones :

I)

the director orientation is normal _to the

boundary

surface if the

sample

thickness is lower than _a critical

value,

I.e. if t w t~;

it)

for _t

~ t~, ^{the director}

profile

is _a tilted _one, the average

birefringence (An )

# 0 and

depends

on the thickness _;

iii)

for t

~ co,

(An) approachs

_a constant value that

depends

_on the

liquid crystal

used.

One _can summarize in this way the three above-mentioned statements _: the trend of

(sin~ 4l)

~'~, 4l

being

the surface tilt

angle,

is

typical

of _a second-order

phase

^transition near t

= t~ ^{and shows} a limit for _t

~ co.

For what _concems

point

I the _{two sets} of data

referring

to unmixed MBBA

give

the _same

critical

thickness,

_{as can} be _seen in

figure 2a,

with _a numerical value t~ m 32 _~Lm.

On the contrary ^{the critical value measured with the nematic mixture} was t~ m 60 _{~Lm, as can} be _seen in

figure 2b, showing

that also the critical thickness

depends strongly

_on the

liquid crystal properties.

3. The

physical

model.

As

already

mentioned in the introduction, the

change

ⁱⁿ the director

alignment

with the

liquid crystal

cell thickness could be

interpreted,

in

principle,

in two different ways.

According

to the first

point

of

view,

the flow of the

liquid crystal during

the

filling

_process

modifies the surface

anchoring

_energy, and

imposes

an easy

direction, parallel

to the flow direction.

Only

a few papers consider the influence of the flow _on the nematic orientation. Some time ago, Wahl and Fischer

[6]

observed that the nematic average orientation

drastically changed by

a shear

flow,

with velocities _{even so}

negligible

_as

^10~~ mm/day.

Note that this mechanism

implies

the surface tilt

angle 4l(f)= 4l(- (),

I-e- _an

antisymmetric

director

profile. Furthermore,

it

implies

also _a

unique flow-direction,

whereas in _our

experimental

_set-up it is

only

a noise

effect,

without any well defined direction _: the

flow-effects,

if any, average ^out to zero.

Looking

for _a different way ^to

analyse

the

experimental data, namely

the observed _re-

orientational

transition,

_a

possible explanation

could be the selective ions

adsorption

phenomenon recently

discussed

[7].

According

to this

point

of view the effective

anchoring

_energy

depends

on the adsorbed ions.

If the dielectric

anisotropy

of the

liquid crystal

^is

negative,

and the easy axis due to the surfaces treatment is

homeotropic,

_as

certainly they

are in _{our case,} the

adsorption phenomena give

a

destabilizing

effect _on the surface orientation.

In this framework let _us suppose that the surface

anchoring

_energy is

represented by

_:

f~

= iii

cos~

_4l

+ y

cos~

4l

,

2 4

(2)

where

cos~ _(4l

_{is connected with the surface}

polarization. Actually

the energy ^term associated

to the order-electric

polarization [8]

is

proportional

to

(rj cos~

4l

+r~)~,

where rj and

r~ are the order-electrical coefficients.

Straightforward

calculations show that _a term

proportional

to

cos~

4l should appear in the surface energy, as

reported

in

equation (2),

where ik is the effective

anchoring

_energy,

taking

into _account the _« ions _» effect.

As it is

given by

reference

[9]

_:

iii=w- ~~~~ ^~

.~~, ₍₃₎

2

e~

where _w is the

anchoring

energy

strength,

A is the

Debye screening length,

e is the average dielectric constant, e~ is the dielectric

anisotropy

and

finally

_~r

= ~r

(t)

is the surface

charge

density

due _to the selective ions

adsorption.

_~r

depends

on the

thickness,

_as discussed in

170 JOURNAL DE PHYSIQUE II N°

reference

[9],

and the second _term in

equation (2)

_can be

guessed

_{as a} term

correcting

the usual

Rapini-Papoular' expression

and connected _to the order-electric

polarization [8].

The nematic orientation in the cell _can be obtained

minimizing

the total energy of the

sample,

defined _as the _sum of the bulk and surface contributions. The bulk _term is due _to the elastic deformation of the

nematic,

and it is

proportional

to the square of the

gradient

of the average molecular orientation. In the

hypothesis

of _a uniform orientation of the nematic in the

bulk,

this term vanishes and the orientation _near the surfaces is deduced

by minimizing only

the surface contribution

f~. Simple

calculations

give

an

homeotropic profile

stable if

ik ~ y, that is for _:

«

(t)

_{~ «}

(tc)

₌

l~ ^)~j

^~

⁽⁴⁾

~a Y

Equation (4)

defines the critical thickness t~. ^for t _~ t~, ^{that is if}

~r(t)

_~

~r(t~),

the stable nematic orientation is characterized

by

a tilt

angle given by

_:

sin~

4l

= a

i~r~(t> ~r~(t~)1

,

(5)

where

e~j

^A

"

(2 ye~l'

^~~~

Of _course the tilted orientation is stable

only

if iii

~

0,

that

implies

~r ~ ~r*

=

fi, (7)

~a(

otherwise the stable orientation is the

planar

_one, with 4l

=

ar/2,

not observed in _{our case.}

Note that

according

to this mechanism the

dependence

of

(An )

versus t is due _{to a}

change

of the surface energy. More

precisely

the

phenomenological

parameters

characterizing f~ depend

on t.

Consequently

^{also the} _casy direction deduced with

df~/d4l

₌

0, depends

_on t.

Hence the

equilibrium

condition

stating

that the total torque on the surface has _to be _zero,

gives

_a surface tilt

angle

that

depends

on the thickness.

From

equation (5)

_{one can see,} for _t 5S t~.

sin4lmp /~, ₍₈₎

where

» =

la ^{~i~ ,~)"~ (9)}

Equation (8)

shows that the

homeotropic

~ distorted transition is of the second

order,

_as

experimentally

observed.

According

to the model

presented

in reference

[9]

~r

(t

is

given by

:

"~~~

_"~

2

~~/r

'

~~~~

with r =

~r~/p~,

^where _{p~ is} ^the ions concentration in the

bulk,

_{~r~ is} the

asymptotic

value of the surface

charge density,

and both concentrations refer to an infinite

sample.

By substituting equation (10)

into

(4)

_one obtains _:

sin~

_4l

=

awl ^~~~~~~

~

~~~~~~

~

l. ^(ll)

(

₊ t/r

) (

₊

t~/r)

It is

possible

to estimate the order of

magnitude

of the parameters

appearing

in the model.

For what _concems the surface parameters

[8, 10]

_one has

wm10~~erg/cm~

_{and y}

m

10~ ^~ ₌

10~~ erg/cm~

for the electrical

[9,

^it ones one has _:

e~(

_m

0.5,

_{e m}

5,

A _m 0. I

=

I _~Lm, ~r~ m 10~ ^~

C/cm~,

_p~

m 10~ ^~

= 10~ ^~

C/cm~.

Consequently

_{: r}

=

~r~/p~

^m10 ÷

10~

_~Lm,

a =

e~(

^Al

(2 ye~)

m

10~

=

10~ (cm~/C)~, awl

m

10~

=

10~.

By using equations (4)

and

(10)

_{one can} calculate the critical thickness t~.

tc

12 ^e~(w

^y

⁾

"~

t~ + 2

r e~ y

From this

equation

and

using

the values of the

physical

_parameters listed

above,

_one has t~ m

(IN)

_r, that is of the correct order of

magnitude.

In

figures

^2a and b two theoretical _curves,

computed

with

equation (11),

and the

corresponding

sets of

experimental data, (sin~ 4l)"~

_{versus t, are}

_reported.

It is

straightforward

to see that the agreement ^{of the} theoretical _curves with the

experimental points

is

fairly good only

_near the critical thickness. This is due to the very naive model

proposed

for the ion

adsorption,

described

by equation (10).

As is well

known,

this

expression

is similar _to the

Langmuir isotherm,

and

consequently

it _{can run} well

only

in the limit of small

~r values. A _more

general expression

_can be obtained

only by taking

into account the

electrostatic

repulsion

between the adsorbed

ions, neglected

in this

analysis.

The model used leads in fact to a

polynomial expression

that contains

only

three

parameters, namely

_one is the critical thickness and the other two are related _to the ion concentrations in the bulk and _on the surfaces.

By

_means of the above-mentioned

model,

nevertheless, _one can deduce the critical

voltage

U for the Fr£edericks transition _as a function of the thickness of the

sample.

To this aim

4

0

lo t

jp~j

³²

Fig.

3. The theoretical _curve and the

experimental

data collected for the threshold voltage in the Frdedericks transition U _versus the

sample

thickness _t. The

samples

used _were the _{same as} in

figure

2a.

172 jOURNAL DE PHYSIQUE II N°

measurements of the threshold

voltage

U for the Frdedericks transition have been

performed,

and the uniform

homeotropic alignment

of the director for

samples

with _t

~ t~ was confirmed.

In

samples

filled with pure MBBA U _was

falling

with

increasing

thickness of the

crystal,

till

a

complete vanishing

_was reached for _{t m} t~,

supporting

the existence of _a permanent ^{tilt of the}

director,

as in the

optical

measurements.

By minimizing

the total free energy of the

sample,

obtained

by summing

_up the bulk _term

(including

the elastic term and the extemal field

interaction)

and the surface _{one, as} shown in reference

[12],

_one obtains that the critical

voltage depends

on the elastic

(k)

and electric

(e~) properties

of the

nematic,

the effective surface energy

(iP)

and the

sample

thickness

(t).

Its

analytical expression

is

~t= ~tg (i~ _{), (12)}

" k UC 2 UC

where U~ ₌ ar

kleo

_e~ ^is the critical

voltage

ⁱⁿ the strong

anchoring

_case, and U is the _one in the weak

anchoring

_case.

By using

also

equations (3)

and

(10),

it is

possible

to calculate

U/U~

_versus t.

The

experimental

results

conceming

the critical

voltage

and the theoretical _curve derived from the

physical

models _are

reported

in

figure

3. The agreement ^between

points

^and curve is

obtained with the _same values of the

parameters

used above.

4. Conclusions.

The influence of the thickness of _a

sample

on the average orientation of _a nematic

crystal

^has

been considered. The

experimental

data show that the

reorienting phenomenon

connected with the thickness of the

sample

is _a critical _one. More

precisely

it is

experimentally proved

that for t _~ t~, ^where t~ ^is a critical

thickness,

the

samples

are in _a

homeotropic

orientation. For t m t~ ^the average orientation

changes according

to

(t t~)~'~, typical

of _a second order

phase

transition.

The

experimental

data are

analyzed by using

a

simple model,

in which the surface _energy

depends strongly

on the adsorbed ions.

If _{one assumes} in the

expression

of the surface energy a term

proportional

to

cos~ 4l,

connected with the order-electric

polarization,

a transition from the

homeotropic

to the tilted director

configuration

must be

expected

if _one suggests ^{that the selective ions}

adsorption

process follows the

Langmuir

law for the

perfect

_gases.

In the _same frame of the

Langmuir-adsorption

law the threshold

voltage

^for the Frdedericks transition has been

analyzed

_: the theoretical

predictions

_are in

good

_agreement with the

experimental

data for _t

< t~.

In contrast, the theoretical

(sin~ 4l)~'~

_curves and the

experimental

data _are in

good

agreement

only

for _t 5St~. ^{This fact} can be

interpreted

in the _sense that the

Langmuir

law must be considered

only

_{as a very} crude

approximation,

valid

just

in the

beginning

of the

transition,

for _t

m t~. ^{Work is in} progress ^to

improve

the theoretical model.

References

[1] JEROME B.,

Rep. Frog.

Phys. 54 (1991) 391.

[2] See for instance COGNARD J., Mol. Cryst. Liq.

Cryst. Supp.

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