The hydrodynamics of surface layers of nematic liquid crystals studied by modulation ellipsometry

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The hydrodynamics of surface layers of nematic liquid crystals studied by modulation ellipsometry

L. Blinov, D. Subachyus, S. Yablonsky

To cite this version:

L. Blinov, D. Subachyus, S. Yablonsky. The hydrodynamics of surface layers of nematic liquid crystals studied by modulation ellipsometry. Journal de Physique II, EDP Sciences, 1991, 1 (4), pp.459-469.

�10.1051/jp2:1991180�. �jpa-00247530�

Classification

Physics

Abstracts 61 30

The hydrodynamics of surface layers of nematic liquid crystals

studied by modulation ellipsometry

L. M

Bhnov,

D B.

Subachyus

and S V

Yablonsky

Institute of

Crystallography,

US S R Academy of Sciences, 117333, Moscow,

Lemnsky

_prosp..

59, U S S-R

(Received14 February

1990, revved 26 November1990,

accepted17

January

1991)

Abstract. An

acoustically

induced Poiseuille flow

changes

the director onentatlon of _a nematic

liquid crystal

in _a

hybnd

cell. Th1n surface

layers

(less than 0 2 _{~L) m} the vlcimty of solid interface

are of

special

interest To

study

them _we used a novel

technique

ofmodulation

ellipsometry

based on

probing

of the interface

layers by

_an evanescent optical wave appeanng m a

liquid crystal

when

light

_is

totally

reflected from the interface between the

liquid crystal

and

heavy glass

The essential details of the

technique

are the

angle

of

light

incidence _is

considerably higher

than the total reflection

angle

and the _a c acoustic excitation _is used with _a consequent ^lock-in detection of the response The

technique

allows _us to measure both the dynamic deviation angle and the static tilt

angle

of the director at the

homeotropically

_onenting interface As _a result, the temperature

dependence of the

anchoring

_energy for that interface _was measured

Introducfion.

The present ^work was initiated to

clarify

_some

featurei

of the thickness and temperature

dependences

of the flexo-electnc effect

acoustically

induced _m

polar

nematic cells with

hybnd

(homeo-planar)

molecular onentation.

Figure

^I

reproduces

the _main result of _{our previous} paper

[I]

_: for thin cells the flexo-electric

voltage

observed at the

first

harmonic of the

applied

acoustic excitation _is

considerably higher

than that for thick _ones and the

corresponding temperature dependence

is

sharper.

At first

sight,

this result

disagrees

with _a

simple theory developed

m

[I],

where the

z-component

^{of the} flexo-electric

polanzation

m a

hybnd

cell

(z

_is

a

layer normal)

is

etermined

by

two (ejj and _e~~), the

cell

_thickness

d and

(«

_and 8) for the _director nentation

at the

lanar

and

^{meotropicoundanes,}

espectively It _should be noted _that the averaged

polarization is independent ^of _the ^concrete

law

^for the _director

distnbution

U(~&j),m~ (~A~,ffi~

1._5

§ CB

~f~

~~~

5 ~'~~

1.0

30 34 36

T~C

Fig I. The temperature

dependence

of the flow induced

voltage

across electrodes

(at

the first

harrnonlc)

for _a cell with the

hybrid

orientation. Cell thickness d

= 7

(1),

17

(2)

and 32

(3)

_~Lm Insert _a scheme of the expenment and definition of the angles.

To make the situation

clearer,

let _{us assume} that the

anchoring

_energy at the

planar

interface is infinite

(W~

= co

)

and the

angle

_«

=

w/2

is

unchanged If,

in

addition,

the static

pretilt angle

at the

homeotropic boundary

_is absent

( 80

= 0

)

and the

dynamic

deviation

angle

is sniall

( 8~

₌ 0 we would have for the

acoustically

^induced

change

m

polarization

_a

simple quadratic

form _:

" =

~

~~~ ~~°

~

~~~ iPo

8

=

o

)

^{en + e~~}

~ ~

~~°~

~(~

~) ~

~~~/j33 t~j

^~~~

The flexo-electnc

voltage Ui

= 3

(P)) /C,

where C is cell

capacity,

should be

quadratic

_{m an}

excitation acoustic field and

independent

of the thickness for

8~ kept

constant. In

fact,

under the above mentioned

assumptions,

the

voltage

must _even decrease with

decreasing thickness, first,

due to a

drop

of the flow

velocity gradient dvJdz

d and the

corresponding

decrease _m

a viscous torque M exerted _on the director in the

vicinifiy

of the

interface, 8~

M

d, and,

second,

due _{to a} correction to the acoustic _pressure caused

by capillary

forces.

In paper

[I]

the flexo-electric

voltage

_was observed at the first harmonic of the

applied

acoustic excitation. It may be accounted for if

only

the permanent ^tilt

angle 80

is taken into account. One

ought

to

distinguish

between the static

pretilt angle independent

of excitation and the

dynamic

bias tilt

angle 8f

which

might

_occur due to _a

nonlineanty

of the _response,

that _is due _to the

nonequivalence

of the to and fro directions of flow _{m a} monodomam

hybrid

cell. In any case, instead of

(2)

_we have

(for 80, 8~ «1)

3

lPll

=

~~~ ~~~

(80

₊

8m

_sin

wt)2

en + e~~ 2

80

₊

8~ 8(

^~~~

d 4 ^~

~° ~~'~~~

~°~

S

^{~°~ ~~°~}

Thus the flexo-electric _response at the first harmonic

(angular frequency w) depends linearly

on the permanent tilt

angle irrespective

^of its

origin.

Due to the curvature

elasticity 80

can

strongly depend

_on

temperature, anchonng

_energy and cell thickness and therefore

influence the

polar properties

^of a

hybrid

cell _{as a} whole The aim of the present ^work is to

carry ^out direct measurements of the

fro

^and

8~ angles

_m order to understand the

anomalous behavlour of the shear induced

flexo-electricity

and estimate the value for the

anchonng

_energy at the

homeotropic boundary.

A

technique

for modulation

ellipsometry.

It _is

usually accepted

that surface tilt

angles

_may be measured

using

_a total internal

light

reflection

techntque (e.g

see

[2-4]). Unfortunately

for

hybnd

cells _{as our}

experiments

had

shown,

there _{is no}

sharp angular dependence

of the reflected

light intensity

for the

extraordinary

_{ray m} the

vicinity

of the virtual total reflection

angle

because of _a smooth

change

_m the

corresponding

refraction index

along

the

layer

normal and

waveguidmg

_{» a}

light

beam into the bulk of _a

liquid crystal.

For this _{reason we} have

developed

_{a new} version of

ellipsometry

which allows _us to

perform

dynamtc

measurements of

parameters

^{of the}

polarization ellipse

for _a

light

beam reflected from the interface at _an

angle considerably exceeding

the total reflection _one. The choice of such _an

angle provides

_an onentation of the electric

polarization

vector of the _evanescent

wave

perpendtcular

to the interface and _a decrease _in the

depth

of the _wave

penetration

f

wtth increasing incidence

angle [5] According

to _our estimates

f

is of the order of

A

/2

to

A/10, being

_a function of matenal parameters ^and

temperature

The scheme of the expenment ^{is shown} m

figure

2. A monochromatic beam of _a He-Ne

laser

(A

= 0.633

~)

refracted in _a

heavy glass

_{pnsm is} incident upon the

homeotropic

boundary

of _a

hybnd

cell at an

angle

of 80°

(the

total reflection

angles

for _a uniform

homeotropic

onentation are about 61° and 72° for the

ordinary

and

extraordinary

_rays,

respectively).

The

light polanzatton

vector _was at _an

angle

of 45° _to the incidence

plane.

A

AH-plate

onented

#

_{to a}

polarizer

transforms

elliptically polarized

reflected

light

into

linearly polanzed light.

After _an

analyser

the

light

beam _is detected

by

_a

photodiode.

The size of the

light spot

at the interface studied _was about _a few millimeters.

The _a-c- acoustic pressure from _a

loudspeaker (the

_{maxtmum pressure} at the end of the

acoustic

wavegutde

tuned _{in resonance} wtth

frequency

57 Hz _was

l2kPa)

causes the

oscillating

flow of _a

liquid crystal along

the flat

capillary

formed

by

two semicircular

prisms.

The _rear end of the

capillary

_{is open} The flow results in oscillattons of the dtrector

throughout

the cell

including

thtn surface

layers probed by

the evanescent wave. The

corresponding changes

in the refraction index for the

extraordinary

_ray modulate the

shape

of the

polarization ellipse,

that

is,

the

light intensity

behind the

analyser.

The _a-c-

signal

_is

observed both at the first and the second harmonics of the acoustic excitation

frequency

_using

a selective

amplifier.

il

,/

'~

/

,.

i

~~

i

~,

/

^'"

~

, ,,,

l' _,, ,'

j

^"' ,,

j ~~

(

~ 5

~~

2 15 ii /3 Ii

Fig 2

Experimental

set-up

I)

a He-Ne laser,

2)a diaphragm 3)polaroids 4) semlcyhndrical

prisms Wtth

htgh

refraction index,

5)a thermocouple, 6)a liquid crystal (SCB), 7)

an acoustic

waveguide

,

8) _a

loudspeaker, 9)

_a

AH plate

,

10) a silicon

photodiode

,

II _an audio generator, 12) a current

amplifier

,

13) a selective voltmeter

,

14) an

oscilloscope,

15) a

digital

voltmeter

Theory

^of the method.

The

geometry

of the

polarized fight

reflection _is shown _in

figure

3 We need formulas which relate _a

change

_in the director

angles 80

and

8~ averaged

over the

penetration depth

^{of the}

evanescent _wave wtth the

corresponding change

in the azimuthal direction of the

light

polarization.

Some useful expressions may be found,in

[6].

In the _case of _a uniaxial

crystal

with the

optical

_axis

lying

_in the incidence

plane

the

phase

difference between the

extraordinary ~p)

and

ordinary (s)

_{waves is}

~/n ( ni(n ( sm~

q~ ⁿ

( ) N~ ~/N~ ^sin~

_q~

_n$~r)

_{cos q~}

«~

«~ = 2 _arctan

(4)

N

(n~

_njj

^cos~

_q~ +

~/(N~sm~

_q~ n

( )(N~sin~

q~

n$~r))

Here njj, n~ and N

=

1.806 _are the refraction indices of _a

liquid crystal

and

glass TF-10, respectively,

q~ is the incidence

angle

The effective refraction index _is n$~r =

ni _cqs~

8 _{+ n}

( _{sm~ 8,}

where 8

=

8

o +

8~

sin ml

After the second A

/2 plate

the aztmuthal

angle

of the

linearly polarized light

wtth respect to the

polanzation

of the incident

light

_is

[7]

:

230=«~-«~. (5)

For small director deviattons 8 at the interface _we have _a small

change

_m the azimuthal

angle A30. Keeping only

the first term in the

Taylor

_{expansion we} have _:

3(£rp- £r~)

~

~~~

3~efr

n~~ nj

~~~~

~ ~~

~~ ~~~

,,

~

_~~

'

~

~ _~

~~

~ '

~

,~$O ~'

,,

'

, i ,

' -

~

, ,

GLASS

^'

[/

_

~l

3

_,

/ ,"

E,

_{s ',"}

§

ZC

I

CC ( £C

z

Fig 3 Onentation of

light polanzation

_vectors before and after

light

reflection from the

boundary

between

glass

_an

liquid crystal (incidence angle

_p

= 80( n-director,

amplitudes (E~(

₌

(E~(

₌

(E(

=

(E(() a)

a side vtew

b)

_{a vtew} to meet the beam 1,

2) light polarization ellipses

for two different director

angles

fJ

I-I')

directions of _a

polanzer

and second

Am plate, 2,2')

variable

position

of _an

analyser, 3-3')

and

4-4')

directions of the

polanzation

of the outgoing beam after

Q4 plate

for the

two 8

angles defintng

the azimuthal

angle

8

with

An~~

=

(ni

n

( 8~/2

_nj and A calculated from

(4).

Thus the

amplitude

of the _ac.

modulation of the azimuthal

angle 3~

₌

A30

_is

ni

n

(

~

nil

n

(

2

8(

₊

8$ 8$

_cos 2w t

~~

~

4 ~ll

~ ~

2 _~

I 4 ^~

~~ ~~

~~~ ~~

4 ~~~

On the other

hand,

the _same

parameter

may be calculated from the effective value of the

a-c

voltage U(w

of the

optical

modulation measured

by

_a selecttve

amplifier.

Let the d-c-

light _intensity

after _an

analyser obey

the Malus law

(experimental

_curve I _in

Fig 4).

The

angular

behaviour of the _a-c- modulation

signal (curve 2) corresponds

to the first denvative _to the intensity curve It _means that the acoustic excitation

merely

moves the whole _curve I

along

the _~oaxis. Thus there _{is a} direct

correspondence

between the

amplitude

of the

change

in the azimuthal

angle 3~

at

frequency

_w and the effective value -of the _a-c

voltage U(w

at the

photodetector output

3~=~~"~a/.

Here _a and AU _{are static} values of the

changes

_w the

angle

of the orientation of the

analyser

installed

manually

for calibration and the

corresponding change

of _a

photovoltage

from the

photodiode (as

_a rule _{a was} about

5)

lj(uJ)

_yV

II,mv

45

30

15

0

-45° o° 90°

Fig 4 d-c

hght'intensity

after _an

analyser (I)

and the _a-c

signal

of the

acousto-opttcal

effect (2) as

functions of

angle

_X between a

polarizer

and an

analyser

From

equation (7)

_we have the

following expressions

to calculate

80

and

8~.

2

/ ^~~"

a ₌ A ~'~

~~

80 8~ (8)

AU ni

2

/ ^~~~

^"

a = A

~~ ~~

8$ (9)

AU 4 ni

Experhnental

results and dhcussion.

All the measurements _were carried out wtth a nematic

hqutd crystal p-pentyl-p'-cyanobi- phenyl (SCB) having

the

cleanng point

at 34.5 °C. For the

planar

onentation _we used _a

rubbing

of thin

polyvinylketal layers.

The

homeotropic

onentation was obtained using thin

layers

of chromium

distearyl

cltloride

(CDC).

All

temperatwe dependent _parameters

required

for the calculations

(nj,

_n~, ^and elastic moduh

Ki,, K~~)

were taken from

[8, 9].

The

oscillating

flow of SCB _was induced

by

_an acoustic _wave from _a

loudspeaker

at

frequency

57 Hz

optimal

from the

point

^of _vtew of the _maximum acoustic power ^at the end of the

waveguidmg

tube whose

length (about

1.5

m)

_{was m} acoustic _resonance with tints

frequency.

The

optical

_{response was} ^{detected both} at 57 and l14 Hz.

Since the

signal always

contains _a

component

at the fundamental

frequency,

tints

implies

that there is _{a non-zero} tilt

angle 80

It may be either of static _or of

dynamic

nature. If the

permanent angle

is

dynamic

its

magnitude

^would

depend

_on the acoustic _{power. our}

expenment

^{shows that}

U(w ) depends linearly

on the acoustic power thus

80

is _a static

pretilt angle.

Figures

5 and 6 show the temperature

dependences

of the modulation

signals

at the fundamental

U(w)

and double

U(2 w) frequencies

of the

applied

acoustic pressure for _a rather thick

(32 ~)

and thin

(7 ~) cells, respectively

The

signal U(w )

is

nearly

6 times

htgher

m a thinner cell _m

spite

of the fact that

U(2

_w

)

is 3 times decreased It _means that the

pretilt angle 80

_is

considerably

wider for thin

hybnd

cells.

U,pv U(oJj,v U(zKJJ,rV

2 .

.

2

~

22

'

Fig

5

Fig. 6.

flm

,Jgj

3

a a

a ~

a ~

d'

a

~

. oO

~

i ^OO

~

0

22 26 30 34

Fig 7

Temperature

behaviour of the

amplitude

of the

oscillating angle

for the director _at the

homeotropic boundary

for _vanous cell thtcknesses d and onentant thtcknesses d' _curve

I)

d

=

7 _~m, d'

=

0.06 _{~m, curve} 2) d

=

32 _~m, d'

=

0.06 _{~m, curve}

3)

d

=

32 _~m, d'

=

0 6 _~m

deviation

angle

_is wider for the thtcker cell for the two reasons mentioned above

(due

the effective _pressure enhanced

by capillary

forces and

higher

_viscous

torque acting

on the director at the

interface). [or

_{thicker CDC}

_layers

m the _same cell

(d

= 32 _~ the calculated

(apparent) amplitude 8~

_is smaller

presumably

due to the screening of _a part ^{of the}

evanescent wave

by

_{a passive} CDC

layer

wtth similar refraction indices. It should be noted that all the

attempts

to observe the modulation of

light totally

reflected from the

planar

interfaces of the _same

hybrid

cells _were unsuccessful. This result confirms the assumption of the strong

(infinite energy) anchonng

at the

planar

interface.

The

temp<rature

^{behaviour of} the static

pretilt angles 80

_is shown in

figure

8 for different cell thicknesses In accordance with _our earlier results

[10]

this

angle

_increases

dramatically

with

decreasing

thickness due to _{an increase m} the elastic torque. ^The

80

temperature

dependence

allows _us to calculate the

temperature dependence

of the

anchoring

_energy

ll~

for SCB

homeotropically

onented

by

_a CDC

layer. According

to

[11, 12]

W~

80

=

?~

K~~

(11)

3z _&=

&~

Where

~~

~ '

~~'~ ^{~~ ~~~~~} '~~~~

_~~

'

~ ^~~ ~~~ll/~33

l

)

COS~ _{X +}

@@

_Jeg

o ^°

I

o

O o

O

3 /

,

2

0

o

~~

~

(

~C

Fig

8 Temperature

dependence

of the

pretilt angle 80

curve I) d ₌ 7 _~m, d'

=

0 06 _{~m curve} 2)

d

=

32 _~m, d'

= 06 _~m, curve

3)

d

=

32 ~m, d'

=

0 06 ~m

erflcm'~

10

a O

§ O O

0

~~ T

,°c

Fig 9. The anchoring _energy of SCB at the CDC treated glass surface _{as a} function of temperature

and d _is the cell thickness. The energy W~ calculated from

equation (I I)

_is shown _m

figure

9.

It _{is m} accord with recent

experimental

data for MBBA

[13]

(we have _no data for

SCB).

Figure

10 shows the

product 80 8~

^{calculated from the modulation}

elhpsometry

data. A

sharp growth

of this

product

with

&creasing

thickness and increasing

temperature

near the

phase

transition accounts for the

analogous

« anomalies _m the linear flexo-electric _response

[I]

which is determined

by

the _same factor

1< s~ e~. e~,Je3z

3

1

~ z

Z

e

o

zo zz z~

T °c

Fig

10 The temperature

dependence

of the

product 808~ responsible

for the temperature

dependence

of the

flexo-voltage

at the first harmonic of excitatton (Fig I). Curve

I)d=7~,

d'

=

0 06 _{~, curve} 2) d

=

32 _~, d'

= 0 6 _~, curve

3)

d

= 32 _~, d'

= 0.06 _~.

Conclusions.

For the

dynamic investigations

of thin interfacial

liquid crystal layers

_a novel

technique

of modulation

ellipsometry

has been

developed theorettcally

and

expenmentally.

The _evan- escent _wave allows

probing

interfacial

layers

at a

depth

of the order of _a tenth of _{a mtcrometer}

even m the _case of _a nonuniform dtrector distribution

along

the

layer

normal. Both the static

pretilt angle

and the

dynamic

devtation of the director _at the

homeotropic

interface _were studied. The esttmate shows that _{one can measure}

pretilt angles

as small _as 0 II

(The sensitivity

of _our set-up is limited

by parasitic

modulation

signals

caused

by

the _acousto-

optical

effect due to the

compressibility

of the

liquid)

The temperature behaviour of

anchoring

energy W~ was also studied wtth the _same

techntque

The data obtained from the

elhpsometric

measurements allowed _a

qualitative explanation

of the anomalous thickness and

temperature dependences

of the

acoustically

induced

changes

_m the flexoelectnc

polarization

of

hybrid

cells observed earlier _m

[1].

Acknowledgements.

The authors _are

grateful

to Dr V N Reshetov for many fruitful discussions

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