Light scattering from the free surface near a second order nematic to smectic a phase transition

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Light scattering from the free surface near a second order nematic to smectic a phase transition

D. Langevin

To cite this version:

D. Langevin. Light scattering from the free surface near a second order nematic to smectic a phase transition. Journal de Physique, 1976, 37 (7-8), pp.901-907. �10.1051/jphys:01976003707-8090100�.

�jpa-00208485�

LIGHT SCATTERING FROM THE FREE SURFACE NEAR

A

SECOND ORDER NEMATIC TO SMECTIC

A

PHASE TRANSITION

D. LANGEVIN

Laboratoire de

Spectroscopie

Hertzienne de

l’E.N.S., 24,

^rue

Lhomond,

75231 Paris Cedex

05,

^France

(Reçu

^{le 23}

janvier 1976, accepté

^{le 19 mars}

1976)

Résumé. ²⁰¹⁴ Nous avons étudié le spectre de la lumière diffusée par les ondes de surface excitées

thermiquement à la surface d’un cristal liquide nématique ^{dans un} champ magnétique près ^d’une

transition nématique-smectique ^A du second ordre. Nous avons mesuré la tension superficielle ^et

trois coefficients de viscosité. L’une des viscosités diverge ^{et son} comportement critique ^a été observé

jusqu’à ^{3 mK} de la transition. L’exposant critique ^{mesuré est en} bon accord avec la théorie de

champ

moléculaire pour les deux cristaux

liquides

étudiés : le cyanobenzilidène

octyloxyaniline

(CBOOA)

et l’octyloxycyanobiphényle (M 24).

Abstract. ²⁰¹⁴ We have studied the spectrum ^of

light

^scattered by ^{surfaces waves} thermally ^excited

on the free surface of a nematic liquid crystal ^{in a} magnetic ^{field near a} second order nematic to smectic A phase transition. We have measured the surface tension and three viscosity coefficients.

One of the viscosities diverges and its critical behaviour has been followed to within 3 mK of the transition. The measured critical exponent is consistent with mean field theory ^{for the two studied} liquid crystals : cyanobenzilidene octyloxyaniline (CBOOA) ^and

octyloxycyanobiphenyl

(M 24).

Classification

Physics ^Abstracts

7.130 ^- 7.480 ^- 7.620

1. Introduction. ^- The

study

of second order nematic to smectic A

phase

transitions has

recently

aroused great ^interest. Critical behaviour of several static

[1]

^and

dynamic [2, 3] properties :

Frank elastic constants and Leslie

viscosity coefficients,

^{has been}

predicted theoretically.

^{One can} understand the

origin

of this behaviour with a

simple physical picture : regions

of the nematic

phase

^move

temporarily

^into

smectic

droplets having longitudinal

^dimensions

(parallel

^to the molecular

axis)

^of

order jjj

^{and trans-}

verse dimensions of order

çl.’ Çll

^and

^çl.

^{are the cohe-}

rence

lengths

^{which have been}

proposed

^{to vary as :}

It is

energetically

^very

costly

^{to bend or twist the}

smectic

planes

^{in the}

droplets.

This leads to a

diverging contribution k

to the bend and twist elastic constants

near the

phase

transition : K = K° +

K,

^{where K°}

is the

regular

contribution

and k

is

given by :

Similarly,

viscous forces _appear when the

liquid

^flows

across the smectic

planes

^{of the}

droplets.

These forces

can be characterized

by

^{a new}

viscosity

coefficient _{Y3 ;} Y3 is related to the lifetime of the smectic

droplets

by : T

⁼

Y3/4 A,

^A

being

the first term of the free energy

expansion

in powers of the smectic order

parameter 0 :

This leads to a

diverging

contribution to the twist

viscosity

coefficient ₇₁ and to another of the Leslie

coefficients,

al :

When a mean field type

theory

^is

valid,

^{that is when the} fluctuations of the order

parameter t/J

^{are small}

compared to 1 t/J 12),

^{the critical exponent for}

the coherence

length

^{is v = 0.5.} Within this _appro- ximation _Y3 is a constant and A N

ç - 2.

^{Elastic cons-}

tants and viscosities

diverge

^{with the same critical} exponent ^v.

In the critical

region,

when the fluctuations of

ql

become

important,

^{it has been}

predicted, by analogy

to

liquid helium,

^{that v = 0.66.}

Using dynamical scaling

arguments, ^{L ’"}

ç3/2,

^{so that :}

and idem for

al.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01976003707-8090100

902

Experimentally,

^{a wide}

variety

^of exponents have been found. For

K33

^the

study

^of Freedericksz transition

gives

^{values for v between 0.5 and}

1

[4, 5, 6, 7].

^For

K22, light scattering intensity

^studies

gives

^{either v = 0.5 or v = 0.66}

[8, 9].

^For Yb studies in a

rotating magnetic

^field

give

for different

products

exponents ^{between 0.36 and 1.07}

[10].

^{However in}

more recent

experiments : dynamic

of Freedericksz

transition [11, 12],

^NMR

[13],

the authors show that the

regular

part

70,

^{is too}

large

^{to deduce a}

precise

value for the exponent ⁱⁿ the studied temperature range : T -

Tc

> 0.1°. This is

probably

^{also the}

situation with the elastic constants since the above mentioned measurements

(excepted

^ref.

[9])

^{have also}

been made

relatively

^far

from Tc :

^{T -}

Tc ~

^0.1°.

Recently,

measurements

[14], [9] using light scattering

studies of the twist mode have been made much closer to

£ :

^{T -}

Tc Z ^10-3°

with CBOOA and M 24.

The

intensity

measurements

give K22

and the life- times

K22/Y1.

^{The results are} consistent with mean field

theory.

However other ^recent studies with a Poiseuille viscosimeter of the

coefficient 17

⁼

(LX4

+ a3 +

a6)/2 give

^an exponent ^{0.36 for M 24}

[15]. il

^{has a} very small

regular

part ^{and its}

diverging

part ^{is identical to}

71’ ’

In these measurements T -

Tc > ^10-2°.

Despite

^the great ^{number of}

experimental studies,

it is still difficult at the

present

^{time to know what}

theory

should be used to describe the

phase

transition.

This was the motivation for our

experiment.

^We

wanted to

study

^the

divergence

^{of a}

viscosity

^coeffi-

cient in the best

conditions,

^i.e.

provided

^{that :}

- its

regular

part ^is

small,

- the measurements could be made _very close to

Tc.

These two conditions are easy ^to fulfil

by doing

^the

spectrum

analysis

^{of the}

light

^{scattered at the free}

surface of the nematic

liquid.

This method allows the determination of the surface tension and of three

viscosity

coefficients :

The

theory predicts

that the first two are

regular

^at

the transition and that _n3

diverges;

^its

divergent part is :

(since 72

⁼

71). Using (2)

^{one finds :}

We started the measurements with CBOOA for which _{X ray}

experiments

^had

given çll/ç.l 1’-1

⁴

^[16].

We

expected that, n3 being

^much

larger

^than

yl,

^the first condition mentioned above would be fulfilled :

n3 > 3-

^Our

preliminary

measurements confirmed that this condition holds

everywhere

^{in the nema-}

tic

phase [17]. Working

^at temperatures ^{such as}

T -

Tc > 0.1 °,

^{we found a critical} exponent

x ⁼ 0.54 + ^{0.08 in}

good agreement

^{with mean field}

theory.

^{In order to}

improve

^the

precision

^{of this}

determination we

pursued

^the measurements with a

better temperature stabilization

allowing

^{us to work}

at temperatures ^{T -}

Tc > ^10-3°.

^We also studied a new

compound,

^M

24,

^{in order to see if} the critical exponent could be different than for CBOOA as

suggested

in reference

[15].

Thereafter,

^we first describe the temperature ^sta- bilization and the

experimental procedure

in section 2.

We then present ^the

experimental

data and their

analysis

^{in section 2 and we} discuss them in section 3.

2.

Experimental set-up.

^{- 2.1 HOT STAGE. -} The hot stage ^{consists of two}

independent

parts.

The outer part ^{is a} copper box with its

temperature regulated electronically

^to + 0.025° with ^{a contact} thermometer. The inner part ^{is also a} copper

block,

insulated from the outer stage

by

^one centimeter of air. The temperature ^{of the inner} part ^{is about 0.1 °}

higher

than that of the outer part. ^Its temperature ^is stabilized with an A.T.N.E.

(1)

temperature

stabilizer,

and a

platinum

resistance. The

heating

resistance is located around the top off ^the

liquid cell,

^{in order to} avoid condensation of the

product

^{on the}

glass

^window

through

which the laser beam passes. At this level the temperature ^is

regulated

^{to ±} 0.01 °. At the level of the

liquid (7

^cm

lower)

^the temperature is measured with a Hewlett Packard quartz thermometer. The

stability

^at this level is ± ^{0.3 mK over times of one}

hour,

^{and is} achieved about one hour after a tempe-

rature

change.

2.2 SAMPLE PREPARATION. ^- The

products

^used

in the present

experiment

^were

synthetized by

^{J. Dubois}

of Thomson C.S.F. Their transition temperatures

were

(in °C) :

CBOOA crystal ^{l 73’2 Smectic A 82 8 nematic 108-5} isotropic

The

samples

^were sealed under vacuum in the

glass

cells. Their thickness is about 2 to 3 mm and the diameter of the cell is 44 mm. The nematic to smectic A

phase

transition temperature

Tc

has been determined in several _{ways :}

- When the

liquid

^is

smectic,

^{it becomes} transpa-

rent. This arises from the fact that the orientation fluctuations of the

molecules,

^which

give

^{to the nematic}

phase

its turbid aspect, ^are

practically

^{absent in the}

smectic

phase.

^{But as} the transition is second

order,

the

turbidity

^{of the nematic}

phase

decreases close to

Tc

and the transparency

change

^is

clearly

^visible

only

over temperature differences

larger

^{than 0.01 °C.}

- When the

liquid

^is

smectic,

^{one observes} (1) Applications Techniques ^Nouvelles Electronique.

around the reflected beam a characteristic diffraction pattern

[18]

^{which can be seen as close as 1 mK to}

Tc.

- In the nematic

phase,

^{close to}

Tc (T - Tc;5

³

mK),

the

signal

^{to noise ratio of the observed} spectra becomes

suddently

poor,

probably

because of inhomo-

geneities appearing

^{in the}

liquid.

^This indicates that the thermal

gradients

^{in the hot} stage ^{over the 0.5 cm} illuminated

region

^{or the} temperature ^{width of the} transition AT are of the order of 3 mK.

The transition temperature ^{does not} appear ^to drift with time. We observed in the case of CBOOA that the surface tension of the new

samples

^falls

rapidly reaching

^{an almost constant} value after an

equilibra-

tion time of a few

days.

^{After a} few months

period,

the transition temperature ^is still the _same, but the temperature ^{width is}

considerably larger (AT > 0.1°) indicating

^{that the}

product

^has deteriorated. The

biphenyl

^{M 24 seems to be much more} stable and does not appear ^to

change

^{at all.}

2.3 LIGHT SCATTERING MEASUREMENTS. ^- The

experimental

^set up has been described elsewere

[19].

It allows the measurement of the spectrum ^of

light

scattered

by thermally

^{excited surface waves. The}

nematic

liquid

^{is oriented}

by

^{a 3 000 G horizontal}

magnetic

field. When the wave vector q of the fluctua- tions is

perpendicular

^to

H,

the orientation of the molecules is

entirely decoupled

^{from the}

liquid

motion. The spectrum is identical to those of an

ordinary liquid

of surface tension J and

viscosity

112

[20] :

with

p is the

liquid density; r

^{is the square root deter-}

mination

having

^a

positive

^real part.

When q is

parallel

^to

H,

^the spectrum ^{is more}

complex,

^it

depends

^{on three} parameters ^Q, ill and n3

[20] :

with

Close to the

transition,

^the

damping

of the surface

waves in the direction

perpendicular

^{to the}

magnetic

field will not vary very

much,

^{while it}

diverges

^{in the}

direction

parallel

^{to the field. One thus}

eXpect

^to observe a great

anisotropy

of the scattered

light spectrum.

3.

Experimental

^{results. - We have done a} syste- matic

study

^{of the} spectrum ^{of the scattered}

light

as a function of the wave vector q in the _{range :}

168 q

⁴⁰⁰

cm- l,

and of the temperature. ^For

one

given sample,

^{such a}

study

lasts for about two weeks.

During

^this

period

^we verified that the measure-

ments were

reproducible.

The data have been fitted

by

the theoretical spectra

using

^a

computer

program which minimizes the mean

square distance between

experimental

^and theoretical spectra.

Figure

^{1 shows a}

typical

spectrum ^{taken on M 24}

corresponding

^{to the wave vector} q ⁼ 263

cm-1,

T -

Tc

^{= 70 and to an} horizontal

magnetic

^field

H // q. The solid line is theoretical.

FIG. 1. ^- Spectrum ^of light scattered from the free surface of M 24 in an horizontal magnetic ^{field H} // q, ^for q ⁼ 263 cm-1 and

T - Tc ^{= 70.}

Figure

² shows the same spectrum taken much closer to

Tc :

^{T -}

Tc

^{= 4.7 mK. The}

frequency peak disappeared,

^the

spectrum’is

^now

purely damped

^and

its width _{is very} small

compared

^{to the} spectrum ^of

figure

^1.

^Indeed,

^{close to}

Tc,

’13 becomes

large

^and

904

Pl(v) (eq. (5))

^{can be well}

approximated by

^{a Lorent-}

zian spectrum of half width :

In

figure 2,

the solid line is a Lorentzian curve.

FIG. 2. ^- Spectrum ^of light scattered from the free surface of M 24 in an horizontal magnetic ^{field H} // q, for q ^{= 263 cm-1 and}

For wave vectors q

perpendicular

^to

^H,

^the

experi-

mental

spectra

^are very similar to the one of

figure 1,

and do not

change

very much when T -

Tc

^decreases.

The

anisotropy

^{of the}

spectrum

^{is thus} very

important

close to

7c

The results for _{n2, qi} and Q are

given

^on

figures 3,

⁴

and 5

for CBOOA and M 24. One can conclude from these results that :

- q2 does not

diverge

^{at the}

phase

transition. Its temperature variation is of the usual type ^for

viscosity :

qa

ew ,

^{where W is an} activation _energy.

- 111 cannot be determined closer than about 20 to

Tc

because since _n3 increases the spectrum ^for

q //

^H

becomes

damped;

it is therefore

impossible

^{to extract}

three

parameters (n1 ,

n3,

y)

^{from its}

shape

^{with a}

reasonable _accuracy.

Nevertheless, figures

^{3 and 4}

show that _n1 does not seem to

diverge

^{at the}

phase

FIG. 3. ^- Results of measurements of the viscosities _"’1 and _q2 for CBOOA.

FIG. 4. ^- Results of measurements of the viscosities ₁₇₁ and 112 for M 24.

FIG. 5. ^- Results of measurements of surface tension for CBOOA and M 24.

transition and behaves like a conventional

viscosity

coefficient : _qa

eW kT,

- The surface tension is

approximately

^{the same}

for CBOOA and M 24 and does not vary very much with temperature

(Fig. 5) :

which is of the same order of

magnitude

^{as for}

ordinary liquids.

^In

particular,

the surface tension does not increase

anomalously

^{close to}

T,

^{as was found in}

reference

[15].

^For

CBOOA,

⁶ decreases with time as was mentioned in

paragraph

^{2.2. For fresh}

samples, alp -

^{28 CGS.}

In order to

interpret

^our results for T -

Tc

²⁰

we

supposed

^that q

stayed

^{finite at the}

phase

^transition

as is

predicted by theory

^{and was verified}

by

^ultrasonic

absorption experiments (2).

^We have taken for _ill the values

extrapolated

^{from the}

straight

^{lines of}

figures

³

and 4. The value of a was obtained from the spectra with _{q 1 H. We} then deduced values for ₁₁₃

plotted

ⁱⁿ

figures

^{6 and 7 in}

logarithmic

coordinates.

FIG. 6. ^- Results of measurements of the viscosity n3 for CBOOA.

FIG. 7. ^- Results of measurements of the viscosity n3 for M 24.

The full line curve is relative to the _power law 113 ⁼ C(T - Tc) - x

and the dashed line curve to the law 113 ⁼ C(T - Tc) - x ^{+ D.}

(2) Martinoty, P., private communication.

For

CBOOA, assuming

^that ’13 varies with tempera-

ture

according

^to the power law :

and

doing

^{a least} squares fit with these three _para-

meters one finds :

The value of

Tc

^{is in}

good

agreement with the value determined as

explained

ⁱⁿ

paragraph

^{2.2 :}

For fixed C and

Tc values,

^{the mean} square ^error is twice as

large

^{for x = 0.57 and x = 0.47. This}

gives

an order of

magnitude

^{for the}

uncertainty

^{on x, and}

we will say that :

We have not taken into account the contribution of the

regular

part

ilo - ^{ew .}

^{One can see in}

figures

⁶

and 8 that if it

exists,

^{it is} very small

(3).

^{Indeed :}

a[

^is

generally small, a2

^is

negative

^{for the known}

nematics and

afl

^is

positive

^{for the} nematics pre-

senting

^{a smectic A}

phase (4).

^Then

y?

>

yz

^and

FIG. 8. ^- Results of measurements of A ⁼ ?13/tI, ^{for CBOOA}

and M 24.

(3) ^{Note that for T -} T,, -> 2°, ^the 173 values are not very ^accu- rate. This comes from the fact that they ^are determined by ^{a three} parameters ^{fit of} experimental spectra ^with eq. (5), ^{with unknown} q 1

and a values.

(4) Pieranski, P., private communication.

906

qg 7?

^{0.5 P}

[10, 11].

^{This is} of the order of our

uncertainty

^on n1.

For M

24,

^{the same three} parameters ^{fit to} eq.

(6) using points

^{with T -}

Tc

^0.6°

(Fig. 7) gives :

The

7c

value is also in

good

agreement ^{with the} value determined as

explained

^{in 2. 2}

For fixed C and

Tc,

^{the mean} square ^error is twice

as

large

^{for x = 0.54 and x = 0.46. Therefore}

For T -

Tc

>

0.6°,

^the

experimental . points

^no

longer

^{fit the} power law

[6].

^{We are lead to attribute}

this fact to the contribution of the

regular

part

tjo

which should be

negative.

^We indeed have measured

negative

values of _n3 for T -

Tr

> 5°

(see Fig. 8).

As 11

^is

always positive [21],

^{we must} admit that for M

24, a°

^is

large

^and

negative.

^{It would be}

interesting

if an

independent

measurement

(with

^{a Poiseuille}

viscosimeter for

example)

^{could confirm our} assump- tion.

n3 should then _vary with temperature

according

^{to a}

more

complicated

^law

where W is an activation energy. Our ^measure- ments accuracy is not sufficient to

perform

^{a five} parameters ^least square fit. For T -

Tc

^small

enough (T - Tc ;$ 2°)

^the

regular

^{term D}

^e’/kT

^{will not} vary very much with temperature. ^{We then drawed in}

figure

^{7 a} dashed line

corresponding

^{to the law}

where

C, Tc

and x have the above values and

One can see that the agreement ^{with the}

experimen-

tal

points

^is

reasonably good.

Remark. ^- The curves were

analysed doing

^the

usual three parameters ^least square fit with _eq.

(6) ;

^it

provides

^a

Tc

value which falls within the

experimen-

tal error bar.

Nevertheless,

^{if one} considers the

experimental T,

^{value itself}

(see § 2 . 2)

^the

experimental points

^no

longer

^{fit the same} power law in all the temperature range either for CBOOA or for M 24.

Far from

Tr

^the

points

^{are not affected}

by

^the

Tc . change

^{and the} exponent ^{is 0.5. Close to}

T,

^the

exponent

is lower : x - 0.33 in

agreement

^{with the} Helium

analogy.

^{With this}

hypothesis

^we should have to consider the

possibility

^of a crossover between the critical and the mean field type

regime

^{in the}

region

T -

Tc ~ ^10-2°.

4. Discussion of the results. ^- We have found that the

divergent

part

q3

^{of the}

viscosity

coefficient :

follows a power law with a critical exponent ⁱⁿ

good

agreement ^{with mean field}

theory.

^The

regular

part

q3o

is

negligible

^{for CBOOA in} all the nematic _range and for M 24 it becomes

important only

^{for T -}

Tc

> 0. 6°.

From our results and the y, measurements

by

^other

authors we can deduce the ratio of the two correlation

lengths jjj /ji ^using

^eq.

(3). Taking the yl

^{1 values of}

references

[10]

^and

[11],

^and

interpreting

^{them with}

x

= 0.50,

^we obtain for CBOOA at T -

Tr

⁼

^{1 °,} yi -

0.22 P. We

measured 3 N

^{2 P}

(with

p -

1)

^at

the same temperature. ^{Then we deduce :}

in

good

agreement with the value obtained

by

X rays measurements

[16].

^{For M 24 and T -}

Tc

⁼

1°, Yi -

^{0.17 P}

[15].

^{We measured}

q3 -

^{7 P}

(again

^with

p -

1)

^{at the same} temperature. ^{We then deduce :}

Our results for the critical exponent x ^{of M} 24 agree with those of reference

[14]

^but

disagree

^{with refe-}

rence

[15]

where it is found x ⁼ 0.36. We do not

presently

understand the

origin

of the difference.

De Gennes has

suggested that, despite

^{the simi-}

larities between smectics A and

superfluids,

^the

critical exponents may not be the same in the two

cases

[1].

^{In a smectic}

A,

^the fluctuations of the order parameter

phase

^are

divergent;

this effect has no

counterpart ⁱⁿ

superconductors. Moreover,

^{the fluc-} tuations of the orientation of the molecular axis in smectics A are

anisotropic :

^their range is of the order

of j

ⁱⁿ the direction normal to the

layers

^{and of}

//-in

the

plane

^{of the}

layers [2].

It may be

possible for ç"

11

and ji

^to have different critical

exponents.

^{This was}

already

found with X ray

experiments

^{on CBOOA}

[16] :

ç" 1’-1 ^{(T -} Tc) - 0.75 and (T - Tc) - 0.6.

^From

eq.

(3)

one can see that in these conditions the critical exponent ^for

n3

^{would be}

higher

^{than the one}

of f, -

It has also been

predicted recently

that the nematic

to smectic A

phase

transition should

always

^{be first}

order

[24].

^In

practice,

the theories mentioned above would still be

valid, provided

that the effective transi- tion temperature

Tc

^is

replaced by

^a temperature

T*,

lower than

Tc,

which is related to a hinted second order transition.

From our

results,

^{as we} have observed a critical behaviour _very close to

Tc,

^{we can}

place

^an upper limit of 40 mK on T* -

Tc

^{for CBOOA and 3 mK for M 24.}

Thus if the

phase

transition is not second

order,

^{it is}

very

weakly

first order.

5. Conclusion. ^- We have measured three visco-

sity

coefficients and the surface tension in the nematic

phase

of CBOOA and M 24. One of the viscosities exhibits a critical behaviour close to the nematic to smectic A

phase

transition. The measured critical exponents ^{are in}

good

agreement ^{with mean field}

predictions.

This contradicts several

previous experi-

mental studies. However our measurements have been done two decades in temperature ^{closer to the}

phase

transition than most of them

[4, 13]

^{and are not}

sensitive to

background

subtraction

problems. This

may be the entire source of

discrepancy

between these works and ours. Recent measurements

performed

closer to the

phase

transition are in agreement ^with

ours

[9,14] excepted

those of reference

[15].

^{We do not}

presently

understand the

meaning

of this difference.

It is

possible

^{that the} theories have been over

simplified

and that it is _necessary to introduce new critical exponents. ^{It will be}

important

^to

investigate

^this

point

^{with new}

experiments

^and

by repeating

^or

reanalysing

many of the

previous

^ones.

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