Ultrasonic investigation of anisotropic viscosities in a nematic liquid-crystal

HAL Id: jpa-00208574

https://hal.archives-ouvertes.fr/jpa-00208574

Submitted on 1 Jan 1977

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire

HAL, est

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Ultrasonic investigation of anisotropic viscosities in a nematic liquid-crystal

F. Kiry, P. Martinoty

To cite this version:

F. Kiry, P. Martinoty. Ultrasonic investigation of anisotropic viscosities in a nematic liquid-crystal.

Journal de Physique, 1977, 38 (2), pp.153-157. �10.1051/jphys:01977003802015300�. �jpa-00208574�

ULTRASONIC INVESTIGATION OF ANISOTROPIC VISCOSITIES IN A NEMATIC LIQUID-CRYSTAL (*)

F. KIRY and P. MARTINOTY

Laboratoire

d’Acoustique

Moléculaire

(**),

Université Louis

Pasteur, 4,

^rue

Blaise-Pascal,

⁶⁷⁰⁷⁰

Strasbourg Cedex,

^France

(Reçu

^le

16 juillet 1976, accepté

^{le 21 octobre}

1976)

Résumé. ²⁰¹⁴ Nous avons mesuré à 15 MHz la partie ^{réelle R et la} partie imaginaire X ^de l’impédance

de cisaillement du cristal liquide nématique p-n-pentyl

p’-cyanobiphényle

(PCB). ^{Nous avons observé}

que R est égal à X. Ce résultat nous a permis de déduire certains des coefficients de friction de la théorie de Ericksen-Leslie et de vérifier l’égalité ^de Rapini.

Abstract. ²⁰¹⁴ The real part ^{R and the} imaginary part X ^{of the shear}

impedance

for the nematic

liquid crystal p-n-pentyl

p’-cyanobiphenyl

(PCB) ^{were measured at 15 MHz. R and X are found to be} equal in the nematic phase.

Anisotropic

viscosity coefficients are deduced from these measurements and the Rapini equality is verified.

Classification

Physics ^Abstracts

7.130 ^- 7.250

1. Introduction. ^- We present measurements of both real

(R )

^and

imaginary (X)

parts of the ultra- sonic shear

impedance

^{of the}

chemically

^stable

nematic

liquid crystal

^PCB

(p-n-pentyl p’-cyanobi- phenyl).

^{We found R and X to be}

equal

^{within our}

experimental

^errors. From these measurements we

deduced the effective viscosities _{flA’ fiB} and _nc

(see Fig. 1)

^{which are related to} the Leslie coefficients

[1].

We also found that the

viscosity

fiB is

equal

^{to the}

viscosity

nc. From a theoretical

point

^of

view,

^this

equality

^derived

by

^A.

Rapini

^{is based on the}

Onsager

relations

[2]. Thus,

^as

pointed

^out

by

^{de Gennes}

[3],

this

equality provides

^a direct check on the

validity

FIG. 1. ^- Schematic diagram ^{of the} experimental set-up showing

the three orientations in which viscosity coefficients were measured.

The vibration is parallel ^{to the} reflecting ^surface.

(*) ^Work supported by the D.G.R.S.T. under contract 7470 458.

(**) ^{E.R.A. au C.N.R.S.}

of the

Onsager

relations in the

hydrodynamics

^of

liquid crystals.

^The

temperature

variation for the

viscosity

coefficients is discussed in the framework of Imura and Okano’s

theory [4].

2.

Experimental.

^{- In the shear} reflectance tech-

nique,

^the

complex

^shear

impedance

^{Z = R + iX}

.of the

liquid sample

^{is first}

determined,

^{and from}

this the

dynamic viscosity il’

^{and the}

dynamic

^stiff-

ness G’ can be

computed

^{from the}

following

^rela-

tions :

where _p is the

density and f the

^{shear wave}

frequency.

For the

special

^{case where R = X}

(Newtonian fluid),

the stiffness is zero and the

steady-flow viscosity il

is deduced from the relation

In

practice

^the determination of the shear

impe-

dance is obtained from measurements of the reflection coefficient for a shear wave at a

solid-liquid

^inter-

face

[5]

that between ^a fused

quartz

^{bar and a nematic}

liquid crystal.

^The reflection loss r and the

change

ⁱⁿ

phase T

^caused

by

^the

liquid layer

^{are related to the}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01977003802015300

154

TABLE I

Measured and calculated

phase changes produced by application of dibutylphthalate

^to

surface of

ultrasonic unit as

a function of

temperature ^{at 15 MHz}

shear

impedance

^{of the}

liquid by

^the

following

formulae

Zq

is the shear mechanical

impedance

of the fused

quartz

bar and is a constant

(8.29

^x

105 dyn . s . cm- 3),

and 0 is the

angle

of incidence of the ultrasonic wave

to the

reflecting

surface of the bar.

Eq. (3)

shows that R and X

respectively,

^{are deter-}

mined

mainly by

^the

changes

ⁱⁿ

amplitude,

r, and

phase,

Q, and the _accuracy of X is almost

entirely governed by

^the

phase change

T.

Usually T

^is very

small, typically 20,

^{and to} be useful a method must have a fractional

phase

resolution of the order of

one part ⁱⁿ

10’ .

In this

study

^{we have}

compared Rand

^X.

Thus,

^this

comparison

^is

significant only

^{if X is} known with an

accuracy

comparable

^{to that of R. For this reason we}

now

give

^some information about the method used to determine _Q

accurately.

Further details are

given

ⁱⁿ

ref.

[8].

Measurements of such a small

phase change require

^careful attention in order to

distinguish

^{it frofn}

any undesirable

phase

shifts due to another source.

One

possible

^{source of such}

phase change

is any small temperature fluctuation in the fused quartz

bar;

temperature ^must therefore be controlled to within + 0.01 ^{OC. Our}

phase

measurements are obtained

by superposing

the acoustic

pulse

^{on a}

comparison pulse.

By appropriate adjustment

^{of a}

continuously

^variable

delay

^{line and an} attenuator, the ^two

pulses

^{are made}

equal

ⁱⁿ

amplitude

^and

opposite

ⁱⁿ

phase

and therefore cancel each other. After a

liquid

^is

applied

^{to the fused}

quartz

bar,

^this process is

repeated.

^{If AT is the time}

delay required

^{to maintain}

cancellation,

^the

phase

shift _Q is

given by

where n is the number of the reflected echo and

f

^the

wave

frequency.

This method is similar to the

gated

carrier method

[6] (i.e. pulse

modulation of a CW

signal)

except ^{that in} the method we use, the wave

frequency

^{is held constant and AT is read on a conti-}

nuously

variable time

delay

^standard

[7].

Another source of error in the measurement of _(p arises from the feature of this method

requiring

cancellation of the acoustic

pulse

^{with a reference}

pulse. Thus,

^it is essential to obtain an acoustic

pulse

with a very flat top. ^{The acoustic}

pulse

^{is altered if the}

transducer is bonded to the fused quartz bar in such a

way that the shear vibration is not

exactly parallel

^to

the

reflecting ^surface, inducing

^a mode conversion of the shear wave. This effect causes considerable error

in the

phase

^shift measurement and is the

major

limitation of the

experiment.

On the other

hand,

^the

change

ⁱⁿ

amplitude

^is

determined

simply by measuring

^the

drop

^{in the level}

of the acoustic

pulse

^{with a} calibrated attenuator after the

liquid

^is

applied.

^A resolution of 0.01 db is

required,

r

being

^{close to}

unity.

Since

dibutylphthalate displays

^{Newtonian beha-}

viour at ultrasonic

frequencies,

measurements on this

product

^{served as a check on} the calibration of the instrument.

We present ^{in table I a}

comparison

^of Q

(meas)

^and

9

(lit)

^{at several} temperatures for the fused quartz ^bar

we used

(0 ~

⁷⁷⁰

40’). qJ (lit)

^is calculated from the

steady-flow viscosity [9] by

^use

of eqs. (2)

^and

(3).

^We

also

give

^{values of the time}

delay AT,

^the

quantity

which is measured

directly.

^{AT is in the nsec} range.

The

frequency

^{is 15 MHz.}

The values of

R,

^X

and q

^{are listed in table II}

together

with the literature values to make a

compari-

son. The results show that the system described here

can measure R with an error of 100

dyn

^s

^cm-3

^{and X}

within an error of 300

dyn

^s

^{cm- 3.}

^{We also} observe that

our

experimental

set-up

gives

values for X that are

always slightly

lower than for R.

TABLE II

Values

of

^the

in-phase

^and

quadrature

components

of

the shear

impedance of dibutylphthalate

^as

a function of

temperature ^{at 1 S MHz.}

For our

studies,

^{we used}

commercialy

^available

PCB

[10].

^The

nematic-isotropic

transition tempera-

ture was 35.2°C. Orientations A and B

(see Fig. 1)

were obtained

by rubbing

^the

reflecting

surface of the bar and the cover

glass

^with

^Kleenex; homeotropic

orientation

C, by coating

^the

reflecting

surface of the bar and the cover

glass

^{with a}

layer

^of

lecithin,

^{so thin} that it did not

detectably

affect the reflected

pulses.

The three orientations were examined in

polarized light

between crossed

polarizers.

3. Results. ^- Our measurements of R and X are

displayed

ⁱⁿ

figure

^{2. The}

frequency

^{was 15 MHz.}

Results in the

isotropic phase

^are also shown for

comparison.

^{In the}

isotropic phase (1)

^{R and X differ}

significantly indicating

^a relaxation _process associated with order parameter fluctuations as

explained by

de Gennes

[11].

^In contrast, ^one observes in the nema-

tic

phase

^{that R ~ X. In}

fact,

^as

figure

²

shows,

^{X is}

slightly

lower than

R,

^{but we} believe this result to be a

characteristic of the apparatus because the same diffe-

rence was observed for

dibutylphthalate (see

^Table

II).

Since R ~ X in the nematic

phase,

the relaxation

corresponding

^to the fluctuations of the orientational order

parameter S

^{is not}

observed, althought

^the

fluctuations are

roughly symmetric

^about

Tc [15].

^In

fact,

ⁱⁿ the nematic

phase

^the fluctuations of S involve

a

change

in relative orientations of the molecules without

change

ⁱⁿ the director. For this reason one can

expect ^{that the} influence of the fluctuations in S on the

anisotropic

viscosities is weak

(except, perhaps,

^near

the

transition). However,

^{in the}

isotropic phase

^the

situation is

completely

different since the motion of the

principal

^{axis cannot be}

separated

^{from that of S.}

FIG. 2. ^- The real (R) ^and imaginary (X) parts of the shear

impedance in the nematic and isotropic phases ^{of PCB at 15 MHz.}

These results confirm our earlier

study [1]

^{on MBBA}

where we measured

only

^{the real}

part R

of the shear

impedance

and deduced the effective viscosities _l1A and _l1B

assuming

^{that the}

imaginary

part ^{X is}

equal

^to

the real part ^{R. At the same}

time,

^these measurements

on MBBA agree well with measurements done

by capillary viscosimetry

^and

by light scattering

^{from the}

free surface. Near the

transition, however,

^{a difference}

between the ultrasonic and the

light scattering

^results

was observed

[16].

This difference _may be due to the relaxation

corresponding

^to fluctuations in S.

The

non-relaxing

behaviour of PCB

being accepted,

the effective

viscosity

coefficients can be deduced from

our measurements,

using

^formula

(2).

The theoretical

analysis

^{of the}

experiment

^is

reported

^{in our}

previous

paper

[1] ]

and shows that the nematic fluid behaves

exactly

^{like an}

ordinary

^{fluid of}

viscosity

lIA, lIB and _qc,

depending

^{on the}

geometrical

conditions. In terms of the Ericksen-Leslie

theory,

the calculation leads to the

following

^relations

[1, 3] :

with

where _a; are the Leslie coefficients.

If the Parodi’s relation

(a2

+ a3 = a6 -

a5)

^is

used,

^{one obtains} flB ⁼ tlc. This

equality

^{is a conse-}

quence of the

Onsager reciprocal

relations. The real parts

RB

^and

Rc

^are

displayed

ⁱⁿ

figure

^{3. It is} apparent that

RB

⁼

Rc in

all the nematic _range.

Consequently

we

have IIB

⁼ fle. Measurements of X

(at

^21.32

OC,

25.60 OC and 33.60

OC)

^{show that}

XB

⁼

Xc

^{and lead to}

the same conclusion.

FIG. 3. ^- The real (R) part of the shear impedance ^{as a function}

of temperature ^{at 15 MHz} for orientations B and C.

The

viscosity

coefficients _11A and _qc as a function of

1000/ T

^{are shown in}

figure

⁴

together

^{with the}

capil- lary viscosity.

^We observe that the viscosities _r¡A (1) ^The results in the isotropic phase together with flow-induced

birefringence ^{results are} discussed in the next paper [12].

156

and _Nc follow the usual

exponential

^{law in}

practically

all our temperature range of

investigation

^{and do not}

present ^the characteristic curve near the transition temperature

Tc

where the order

parameter S

goes ^to

zero. This can be understood if we look at the curve

of S

against

^{T where S}

(2)

^decreases

strongly only

ⁱⁿ

the last

degree

^before

Tc.

On the other

hand,

^for temperatures

sufficiently

^{far below}

T,

^{we observed no}

marked difference

between

^{and the}

capillary viscosity.

This result indicates that the molecules

align

in the

capillary nearly parallel

^to the direction of flow

(Y2lYl ~ 1) [1-3].

FIG. 4. ^- Plot of the viscosity coefficients versus 103 IT : . Capil- lary viscosity; 0 nB ^or tlc; ⁺ r¡ A. The dashed line is the extrapolation

of the viscosity data of the isotropic phase into the nematic phase.

From the

slope

^{of the curves we deduce} the activa- tion _energy. Within the limit of

experimental

error, the activation

energies

^for ’Niso, ’nA and _’1c are

approxima- tely

^{the same :}

Imura and Okano related the Leslie coefficients t ()

the orientational order parameter

[4]. They

^deduces

the

following

relations which show the difference in behaviour between the various coefficients _{on :}

a4 does not involve the

alignment properties

^{and is}

independent

of S in the lowest order.

In terms of S the

viscosity

coefficients can be written

as

where the coefficients a,

B1

^and

C1

^{are constants which}

are

expected

^to

depend

very little on

temperature.

In

figure 5, nA -niS

^and

nc - nis

^are

plotted against

temperature. ^The

quantity . NA - S ^nis

is small and

approximatively

constant,

indicating

^{that a is small}

and

positive,

^{but for}

nc - nis

^{s we observe an exponen-}

tial behaviour

indicating

^that

B1 /2 C 1

^is

slightly temperature-dependent. Exponential

behaviour has also been observed in other

nematics,

^where

however,

the behaviour

of y1/S

⁼

C1

^was

strongly

temperature-

dependent [14].

I iG. 5. ^- Temperature dependence ^of S, (l1¡so ^- llA)/S ^and (niso ^- llc)/ S. The data for S are deduced from ref. [13].

4. Conclusion. ^- In summary,

using

^{a shear wave}

reflectance

technique

^we have measured at 15 MHz the real and

imaginary

parts of the shear

impedance

of PCB and found them

equal

^{within our}

experimental

errors. This result enables us to deduce the effective

viscosity

coefficients _nA, 1JB and nc, and to check the

Rapini equality (nB

⁼

1Jc).

However,

^{in another} paper

[17]

^{on a nematic which}

exhibits a second-order nematic ^to smectic A

phase

transition we report ^a relaxational behaviour for _qc.

This effect is

probably

^{related to}

the non-hydro- dynamic

relaxation of the director.

Acknowledgments.

^{- We thank Y.} Thiriet for his

help

^{with the}

experimental

^device.

(2) S is deduced from the formula AX ⁼ aS where a is an average molecular susceptibility. ^For PAA, a ⁼ 2.42 x 10-’. Values of AX are taken from ref. [13].

References and footnotes

[1] MARTINOTY, ^{P. and} CANDAU, S., ^Mol. Cryst. Liq. Cryst.

14 (1971) ^243.

[2] Using the Ericksen-Leslie theory, ^{see ref.} [1]. ^{In the FLMSP} theory ^this equality ^is obvious, ^see FORSTER, D., LUBENSKY, T., MARTIN, P., SWIFT, ^{J. and} PERSHAN, P., Phys. Rev. Lett. 26 (1971) ^1016.

Note added in proof : ^The equality ^{of these two} viscosities follows from the basic hydrodynamic considerations.

The fact that the stress tensor is symmetric ^{would have}

to be true even if Onsager’s general ^{relations were not}

true. PERSHAN, ^P. S., private communication.

[3] ^DE GENNES, ^P. G., ^The Physics of Liquid Crystals (Clarendon Press, Oxford) 1974, p. 169.

[4] ^{IMURA and} OKANO, Japan ^J. Appl. Phys. 11 (1972) ^440.

[5] MASON, ^W. P., BAKER, ^W. O., ^Mc SKIMIN, ^{H. J. and} HEISS, ^J. H., Phys. ^{Rev. 75} (1949) ^976.

[6] MooRE, ^R. S. and Mc SKIMIN, ^H. J., ⁱⁿ Physical ^Acoustics (Academic Press), ^volume 6, 1970. Devices measuring phase ^{shift are} described in this paper.

See also BARLOW, A. J. and SUBRAMANIAN, S., ^{Br. J.} Appl.

Phys. ¹⁷ (1966) ^1201.

[7] Ad-Yu Model 20 B1.

[8] THIRIET, Y., Thesis, Strasbourg, Université Louis Pasteur, in preparation.

[9] LAMB, J., in Molecular Motions in Liquids (D. ^Reidel Publishing company), p. 53.

[10] ^{PCB was} purchased from B.D.H. Chemicals.

[11] ^DE GENNES, ^P. G., Phys. ^{Lett. A 30} (1969) 454; Mol. Cryst.

Liq. Cryst. 12 (1971) ^193.

[12] Viscosity coefficients in the isotropic phase ^{of a nematic} liquid-crystal. MARTINOTY, P., KIRY, F., NAGAI, S., CANDAU, S., DEBEAUVAIS, F., ^J. Physique ³⁸ (1977) ^159.

[13] ^{We are} grateful ^to GASPAROUX, ^{H. for} private communication of measurement of X~-X~.

[14] PROST, J., Thesis, ^{Bordeaux I} (1973).

GASPAROUX, H., HARDOUIN, F., ACHARD, ^{M. F. and} SIGAUD, G., ^J. Phys. Colloq. ³⁶ (1975) ^Cl-107.

JAHNIG, F., Inter. Conf. on Liq. Cryst., Bangalore ^{3-8 dec.}

(1973), ^Pramana Supplement.

[15] RAO, ^{K. V.} S., HWANG, ^{J. S. and} FREED, J. H., Phys. ^Rev.

Lett. 37 (1976) ^515.

[16] LANGEVIN, ^{D. and} BOUCHIAT, M. A., ^J. Physique Colloq.

33 (1972) ^Cl-77.

[17] KIRY, F., MARTINOTY, P., ^{to be} published ^and MARTINOTY, P., oral communication at the First European ^Conference

on Thermotropic ^{Smectics and their} Applications, ^Les

Arcs 15-18 décembre 1975.