Dielectric studies of Goldstone mode and soft mode in the vicinity of the A-C* transition

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Dielectric studies of Goldstone mode and soft mode in the vicinity of the A-C* transition

S. Khened, S. Krishna Prasad, B. Shivkumar, B. Sadashiva

To cite this version:

S. Khened, S. Krishna Prasad, B. Shivkumar, B. Sadashiva. Dielectric studies of Goldstone mode and

soft mode in the vicinity of the A-C* transition. Journal de Physique II, EDP Sciences, 1991, 1 (2),

pp.171-180. �10.1051/jp2:1991153�. �jpa-00247505�

J

Phys

II1

(199I)

I7I-180 : FtvRIER1991, PAGE 171

Classification

Physics

Abstracts

61 30 77 40 77 80

Dielectric studies of Goldstop< mode and soft mode in the

vicinity of the A4J* traisition

S. M

Khened,

S. Krishna

Prasad,

B. Shivkumar and B K Sadashiva

Raman Rdsearch

Ihstltute, Bangalore

560080, India

(Received

6 August 1990,

accepted

_m

final form

5 November

1990)

Abstract. We report the results of dielectnc studies on three ferroelectnc

liquid

crystals

exhibiting

the A-C* _transition Two relaxation modes _viz

,

the soft mode and the Goldstone mode have been observed _in the

frequency

domain covered Careful measurements have allowed _us to

obsirve the soft mode relaxation _in the C*

phase,

_even

in'the

_{absence ofa bias field,}

over a

larger

temperature range than _in any

previous study

The results _are discussed _in the

light

of the predictions of the

generahsed

mean-field model

1. Inboduction.

The

temperature

^and

frequency dependence

of the

complex

dielectric constant of

ferrielectnc liquid crystals

has been

siudied _quite extensively~[1-4].

The results of these studies may be

briefly

summansed _as follows When the

sample

is cooled

the'loW _frequency

_{static dielectric}

constant _increases

steeply

as the

temperature approaches

the smectic A-smectic C*

(A-C*)

transition and reaches _a broad _{maximum a} few

degrees

below the _transition point

(T~).

The

dispersion regtme"exhibits

tWo'important relaxation

modes, viz.,

the Goldstone mode and the soft~ mode The

former,

appearing

only

_in the C*

phase,

_owes its existence to the _presence of

phase (or azimuthal)

fluctuations of the tilt order

parameter.

The latter _is due to the

amplitude

fluctuations of the tilt

angle

and exists, in the absence of any bias

field, only

very close to the transition. Most of these studies have been done _on

compounds having

low spontaneous

polansation (P~).

In order to

study

the effect of the

magnitude

of

P~

on

dielectric

properties,

We have carried out static and

dispersion

studies _on materials Which _are

structurally similar,

but exhibit Varying

magnitudes

of

P~.

The results are discussed _in the

light

of the

predictions

of the

generalised

Landau model

[4-6].

The effect of bias field _on the soft mode relaxation _is also studied

2.

Experimental.

The structural formulae and

phase

transition

temperatures

of the substances

[7] investigated

for convenience referred to as

I,

II and III _are given below.

The

P~

values measured [8] at

T~-10

^°C are 500

~CC/m2,

720

~CC/m2

^{and 370}

~CC/m2

for

I,

II and III

respectively.

Cl CH3

C~H~~O~hCH=CHCOO§OOC(H(HCH?CH3

^CompoUnd

615 C* 68 5 A 88 0 CH 950

Cl CH3

IOH~~O~hCH=CHCOO~OOC(HCH?~HCH3

CompoUnd Ii

690 C* 78 0 A 92 5

Cl CH3

C~~H?30~hCH=CH

^COO

~

_COOCH2 ^{H H} _{CH2CH3 CompoUnd III}

* *

K 62 5 C* _{68 5 A 88 0}

The

preparation

^{of the}

sample

cell

(typical

thickness

~ 20

~cm)

^{has been descnbed} in an

earlier paper

[8]

The ITO treated

glass plates

_Were coated with _a

polymer

solution and rubbed

unidirectionally

to get

planar

onentation of the molecules The

samples

_Were

aligned by cooling

from the

isotropic phase

at a slow rate

(~

l

°C/h)

and the

alignment

_was checked

by optical microscopy

The

capacitance

^{and the dielectric loss} were measured using an

Impedance Analyser ~HP4192A). During

_any

dispersion

measurement the temperature of the

cell _was maintained to a

cinstancy

of _± 5i1~K The data

acquisition

was handled

by

a

microcomputer which _was also used for storage ^and

analysis

of the data.

3. Results and discussion.

As mentioned

earlier,

_in the C*

phase

onentational fluctuations of the director _{gtve nse} to two relaxation

modes,

the Goldstone and soft modes Thus the

complex

dielectric constant can be written _as

Ae~

A~S

+ e~

(I)

~~~f)

"

~'~~~~

~

l ₊

of/fG)~~~~~

l ₊

bf/fS)~~~~

where

f~_

and

f~

_are the Goldstone and soft mode relaxation

frequencies, Ae~

and

Ae~

^their respective

strengths,

e' and e" _are the real and imaginary

parts

and e~

represents

^the

sum of the dielectric

strengths

of all other

high frequency modes,

h _is the distnbution

parameter

^{of the mode.}

In the A

phase

the

amplitude

and

phase

fluctuations _are

degenerate

and hence

only

the second and third _{terms on} the

right

hand side of equation

(I)

remain

Before presenting ^{the results of} our

expenments,

it may be relevant to recall _some of the

predictions

Of the

generalised

Landau model

[5]

A

phase

~2 ~2 Ass

~ 2 2

(2)

a

(T- T~)

+

(K~

_ep

)

_qo

fs

= ~

la (T T~)

₊

(K~

_ep 2)

qjj (3)

Is _Ass

= ~

/~

^E~

^{C~ (4)}

N 2 DIELECTRIC STUDIES OF GOLDSTONE MODE AND SOFT MODE 173

C*

phase

neglecting

the

ep2

_term,

p ²

~~~ 2K~ q8

^'~~~

K~

~~

~

2 _aryi ~ ~~~

i p ²

fG AEG

₌

~ _~~

~ (?)

where the

subscripts

_s and G denote soft mode and Goldstone mode

respectively f

and he _are the relaxation

frequency

and dielectnc

strengths

of the

mode, K~

is the elastic constant, _e represents ^the

high

temperature dielectric

susceptibility,

_p and C _are the coefficients of the flexoelectnc and

piezoelectnc

bilinear

coupling,

_{a is} the coefficient of the temperature

dependent

term _in the Landau

expansion, P,

8 and _{q are}

respectively

the

polansation,

tilt

angle

and _inverse

pitch

_in the C*

phase,

and qo = q ^at T~ yi

represents

the rotational viscosity of the modes.

The

experimental

data of

(e*(f), f)

_was fitted to equation

(I)

_using a non-linear least square

technique by

_varying the parameters

Ae~,

_e~,

f~

and h~

(here

_{i represents} different modes

Golflstone

^and soft

mode) [9]

^The distribution parameter

h,

_was found to be small

(=

0,I

Figure

la-c shows Cole-Cole

diagrams,

for the

compound

II

figure

la _in the A

phase, figure16

_in the C*

phase

close _to T~ ^and

figure

lc _in the C*

phase

_away from T~ Plots _in

figure

la and lc exhibit

single

relaxation behaviour and _are due to soft mode and Goldstone modes

respectively

But

figure16

shows the presence of relaxations due _to both the modes

Normally

it is very difficult to resolve the soft mode from the Goldstone mode because _{as seen in} these

figures

the

strength

of the former _is small

compared

_to the latter.

Also~

_as will be discussed

later,

the

strength

of the soft mode

rapidly

decreases _as T~

T(

_is increased However owing ^to the

large polarization

in

compound

I _we have been able to resolve the two _{over a}

temperature

range of 0.8 °C _in the C*

phase

For the _same

reason the soft mode could be studied _up to 3 °C above the transition. In

fact,

this point is

quite

clear when _we note that for the three materials these ranges are

0.8,

3 0

for1,

0

5,

5 for II and 0

07,

0 5 for III

respectively Temperature

variation of the soft mode dielectric

strength ind

_relaxation

_frequency

are

plotted

m

figures

2-4 The salient features _{seen are}

(i) f~

decreases

linearly

with reduced temperature

(T- T~),

the value of

f~

at

T~~fo)

is

considerably

different for different materials

~fo)~~ (fo)jj~-(fo)nj. (ii)

the _inverse dielectric

strength llie~,

unlike

f~,

_is non-linear in the A

phase,

indeed it appears ^to have _a

power-law behavlour,

but has _a linear temperature

dependence

_in the C*

phase

Most of the earlier experiments ^{have also shown the linear behaviour of}

f~,

^but

only

_a few observations of

non-linear variation of

I/Ae~

exist From equation

(2j

_{we see} that

I/Ae~

will

linearly

decrease with

temperature only

if

temperature dependence

of the

piezoelectnc

coefficient

C,

elastic constant

K~

and the flexoelectnc term

p~

_are

_negligible Experimentally

it _is known that the value of the

piezoelectnc

coefficient C _remains constant in the A

phase

in materials

possessing

low _as well _as

high polarization [10]

Thus _one

possible explanation

for _a non-linear vanation of

I/Ae~

could be that the twist to piezo energy ^ratio may be

changing

with

temperature

~'Afe will _see later that this _is true _in the C*

phase )

But if the _ratio of elastic _to viscosity terms is

weakly temperature dependent ill ],

then

only

the term

(a (T T~)/2

_{ar y}

i

)

will influence the thermal variation of

f~.

This could be the _reason for the linear behavlour of

f~. Multiplying

equations

(2)

and

(3)

to el1mlnate the elastic term, we get

equation (4).

Based _on the arguments

presented

above _{we can} expect the

product AeJ~

to be non-linear

only

_in the immediate prox1mlty of the transition Thls _is indeed the

expenmental

behaviour _{as seen in}

20

T-Tc

₌ -0.12°C

(b)

~

(a)

T-Tc

₌ ° °8° C

~~ o.ikHz

~ii

~°

i0kHz

~11 °

~_~ __~~~o~ 45 90

15 ^C~

~0

_{15 30}

£'

_o ^~"~

0 30 60

~l

~,

lfi0

T-Tc

=-0 75°C

(c)

£"

~

° ~ ~~~

0

~0

_{120 240}

~i

Fig I -Refiresentative Cole-Cole

diagrams

for compound II _in

a)

A phase, b)C*

phase

near

T~, c) ^C* phase _away from T~

02 ^'

i

0 _.

~ O°

~o

-

-

0 - ..

~

~ ' ,

-

. .

. ,o

. ^~ o

. ₂₀

T-Tc(°C)

Fig

trength (

he j

~) for _compound 1.

~

N 2 DIELECTRIC STUDIES OF GOLDSTONE MODE AND SOFT MODE 175

0 lfi 200

~.

_ ~

.

~.

.

. ' ^°

~ ^{. i O}

~ ~ ~ , ^O -

~

^{° °~}

o

~

~

~

_~O - "

. ~ O j

0 04 ~.

o

°°

50

~S

O

ooO~~p~

-05 o 05 io 15

T-Tc

(°C)

Fig

3 -Thermal variation Of

f~

and Aej ^' for

compound

II

24

03 _. .

~

18

0

~ o o

i iw

w ol

.

_.

~ o _-

. o

~ ~o

o -01

~_~ Oc) <

Fig 4 Thermal _variation of

f~

^and Aej for compound III

0 8

lo

o

_

~o°

'u

~o°

~o

~ ~o

'~

₀

_~oo°

%

,

°°

w~

0

'oooooo~

%~'

0

-lo 0 lo 20 30

T-Tc (°C)

Fig 5 -Plot of the product

AeJ~

_vs temperature for compound1

oo

-

0 75

~

°

i O

u O

I ~O°

w

~

~O~'

~ ~fl~

~

~~~£°

~ ooO

O

~

-0 5 0 0 5 lo 15

T-Tc

(°C) Fig 6 -Temperature

dependence

of

AeJ~

for

c6mpound

II

o

f

0 04 _{o o}

w o °

I

~

°

~

₀₀₂

~

°

~j

o o

°

0 o

-01 0 01 02 03 04 05

T-Tc

(°C) Fig 7 Plot of

AeJ~

_vs temperature ^for compound III

figures

5-7 It must be

pointed

out that all these effects _{are seen over a} very small range for

compound

III

,

the range itself

being

much

bigger

for materials I and

II, understandably

due to increased

piezoelectnc coupling Following

Gouda et al

[12],

_we have tried to describe the

temperature variation of the

product AeJ~

with _a

power-law AeJ~oz (T T~)fl

,

fl

values obtained for

I,

II and III _are 0

45,

0 37 and 019

respectively

The

figures

8 _to 10 show the _variation of

Ae~

and

f~

as a function of temperature ^The

qualitative

behaviour _{is in} very

good

_agreement with the theoretical

prediction [6].

In

particular,

it is _seen that the _{minimum in}

f~

_occurs not at the transition, but

slightly

below it.

Using

equation

(7)

^and

knowing

the parameters

f~, Ae~,

P and ₈ and the relation y~ = yj

sin~

_8,

y~ can be calculated Such _a calculation done at T~ 3

(P

and 8 values _are taken from Ref

[8])

for the

compound

II gtves y~ =

10 mPas This may be

compared

with the value of 25 mPas for the _same

compound

obtained

[8] by employing

_a field-reversal

method The

discrepancy

_may

probably

be attributed to the fact that the

parameters

involved have been obtained

by

different

techniques According

to

Legraqd

and Parneix

[13]

the

relative twist to piezo-energy parameter

Q~

can be defined _as

Q~

"

(Kq ~/xw C~) (where

K is

the renormahzed twist elastic constant, _{xw "}

(e~

I

))

These authors also _{give a more} 4 ar

practical

definition for

Q~:

~

2arxw

Q

_"

fi

G

N 2 DIELECTRIC STUDIES OF GOLDSTONE MODE AND SOFT MODE 177

400

~QQ ^{O O O O}

O O O O

M

.

X

.

d$

.

$

~i

₂₀₀ .

~ ^. _ioo

~6

Tc(°c)

Fig 8 -Temperature

dependence

of Goldstone mode

frequency ~f~)

and dielectnc

strength (Ae~)

for compound1 ^'

~~~

o o o o _

° 0

o o o o o

~~~ ° °° _°o ^-

ooo ^o

~

. . . * . . .. ..

°~~~ 0 5 df

~ .. ^~D

~i

^{~~~ ~ .}

~

^~

160

$

0

'~

o

_, _~ ~

T-Tc (°C)

Fig. 9 Thermal variation of

f~

and Ae~ for

compound

II

240 14

O

_ o

180 -

o o o o o o o o owooo _oo

#

O o 07 ~

~D

~ .

. ^~

$

^12°

. .

_

.

~ 60

~

Fig

10 Thermal variation Of

f~

and

Ae~

for

compound

III

Using

this

relation,

_we have calculated

Q~

for the

compound

II at two different

temperatures

,

the

values

are 0 004 at T

= T~ ^{6 °C and 0 09 at T} = T~ ^{0 02 °C These numbers} may be

compared

with the value

Q~

= 0.2 for DOBAMBC

[I] (P~

= 30 ~CC/m~) ^and

Q~

= 0 003 for _a matenal

[13]

with

P~

=

500 ~CC/m~ ^This appears ^to suggest

thi

_{the value of}

_Q~

is

inversely proportional

to the

magnitude

of

P~ (and

hence the

piezoelectnc coefficient).

But _more

systematic

measurements need to be carried out in order to establish this

relationship.

As mentioned earlier _in the C*

phase

the dielectric

strength

of the Goldstone mode drowns

that due to the soft mode _-even

slightly

away from T~ One well known

technique

to suppress

the Goldstone

mode,

_so that the soft mode effect _can be

clearly observed,

_is to

apply

_a DC bias field which _is

high enough

to unwind the helical structure. We have used this method to

study

the soft mode relaxation _in

one,material,~vlz., compound

II

Figures

I I and 12 _are

plots

of I

/Ae

and

f~

as a function of temperature obtained _{using a} bias value of 20

kV/cm

for the sake of _comparison the _zero bias values _are also

plotted.

While the _zero bias data show

sharp

qlinima ^{at the} trans1tlon, the _{minimum is} broad _in the presence of _a bias

vo[tage

As

expectqd

the values obtained _at the _{transition are} also different _in the _{two cases}

(Ae~)~~ ok.

43 5_;

(Ae~)~

~o ⁼ 5 8

,

~f~)~

o ⁼ 3.8

kHz, ~f~)~

~o ⁼ 53 7

kHz,

here E denotes the value of the bias field

applied.

x E=20

~

~

~ ( ^{O E=0 ~}

~x

x x x x x

x x

~

jG3

x

~ x

W o 2

~~~xxxXx _xx~

o

~

~ o

~

°°%~

0

-15 0 15 30

T-Tc

°C

, .,

Fig

II Plot of

Ae/

_versus temperature ^with no bias

(o)

and 20

kV/cm

bias field

(x)

for

compound

II

300

x E=20

° E=0

' ~' <x

~

x

- x

M _x ^~

X x

~ Ox

-

~ o'

W x ~

~ x

~ ~$

O~O~~

-1 5 0 5 3 0

T-Tc

^(°C)

Fig 12 Plot of

f~

_versus temperature ^with no bias (o) and 20< kV/cm bias field (x) for

cbmpound

II

N 2 DIELECTRIC STUDIES OF GOLDSTONE MODE AND SOFT MODE 179

Static dielectnc constant measurements were also made _on these matenals. We present ^the results for _a

representative

_{case, viz.,}

compound

II.

Figures13

and 14 show static

measurements at different

frequencies

and

applied

bias

respectively

The features _seen here

are s1mllar _to the _ones observed

by

other

authors,

viz,

(I)

at

higher frequencies

_or

high

bias

voltages,

the Goldstone mode vanishes and

only

the soft mode _remains,

(2)

there _{is a}

maximum in the dielectric constant at

thi

_{transition when measured either at}

_high frequencies

or with

high

bias

voltages

300

0 3 kHz

240

,,

180 _4kHz '.,

£ ... "."" ",, ..

"'

120 ,

o 5kHz

60

" ~,

lokHz ^{~ ~'}

-3 -2 -1 0 2

T-TC(°cJ

; ,

Fig

13 Influence of the

frequency

of measurement on the dielectnc constant of

compound

II

,

40

E=O

30

",

: ~

~'

~_~

)~_o

loo ^'

£ _;

~~

E ₌ 8

°~

j

T

c (°~)

..

~

',

lo

~,,__

-2 -1 0 2

T-Tc ^(°C)

Fig ¹⁴ Effect of bias field _on the static dielectric constant of

compound

II

JOURNAL DE PHYSIQUE II -T I, M 2, FiVRIER 1991

4. Conclusions.

We

llave

_{studied the effect of the}

_magnitude

_of

_P~

on the soft mode and Goldstone mode relaxations Because of the

large P~

value in one of the three matenals

used,

_we could

study

the soft mode relaxation _in the C*

phase

_{over a} temperature range as

high

_as 0 8 °C from T~. The results obtained compare well with the

predictions

of the

generahsed

mean-field

model

Acknowledgments.

We _are

highly

indebted to Prof S

Chandrasekhar,

who initiated _our studies _on Ferroelectric

liquid crystals,

for his _very keen interest _in this work and for many valuable discussions

References

ill

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LEVSTIK I

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BLiNC R and FILIPiC C., J

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Vaughan,

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