Viscosity coefficients in the isotropic phase of a nematic liquid crystal

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Viscosity coefficients in the isotropic phase of a nematic liquid crystal

P. Martinoty, F. Kiry, S. Nagai, S. Candau, F. Debeauvais

To cite this version:

P. Martinoty, F. Kiry, S. Nagai, S. Candau, F. Debeauvais. Viscosity coefficients in the isotropic phase of a nematic liquid crystal. Journal de Physique, 1977, 38 (2), pp.159-162.

�10.1051/jphys:01977003802015900�. �jpa-00208575�

VISCOSITY COEFFICIENTS IN THE ISOTROPIC

PHASE OF A NEMATIC LIQUID ^CRYSTAL

P.

MARTINOTY,

^F.

KIRY,

^{S. NAGAI}

(*),

^{S. CANDAU}

Laboratoire

d’Acoustique

Moléculaire

(**),

Université

Louis-Pasteur, 4,

^rue

Blaise-Pascal,

⁶⁷⁰⁷⁰

Strasbourg Cedex,

^France

and F. DEBEAUVAIS

Centre de Recherches sur les

Macromolécules, Strasbourg,

^France

(Reçu

^le

16 juillet 1976, accepté

^{le 21 octobre}

1976)

Résumé. ²⁰¹⁴ Nous avons mesuré _pour le p-n-pentyl

p’-cyanobiphényle

^les trois coefficients de friction qui apparaissent dans la théorie de de Gennes relative à la

phase

isotrope d’un cristal liquide nématique.

Abstract. ²⁰¹⁴ Using ^various techniques, ^we measured for

p-n-pentyl

p’-cyanobiphenyl ^{the three} viscosity coefficients featured in de Gennes’ phenomenological theory ^{of the} isotropic phase ^{of a}

nematic liquid crystal.

Classification

Physics ^Abstracts

7.130

The

study

of static and

dynamic properties

^{near the}

nematic-isotropic phase

transition in

liquid crystals

has been the

subject

^of considerable recent research

[1].

Although

the transition from the nematic to the iso-

tropic phase

^{is a} first-order

transition,

^the

liquid crystal

^exhibits

pretransitional

effects similar to those in the

vicinity

^{of a} second-order transition. For instance in the

isotropic phase

^the

magnetically-

induced

[2]

and flow-induced

birefringence [3] diverge

as one

approaches

the transition. The

Rayleigh

ratio

[2]

and the relaxation time

[4]

of the order _para- meter show the same behaviour. All these effects

(static

^and

dynamic)

have been discussed

theoretically by

^{de Gennes}

[5]

^{in terms of a} Landau model. Accord-

ing

^{to this treatment}

quantities

^{such as the}

magnetic

and electrical

birefringence diverge

^as

(T - Tc *)y

where

Tc*

^{is a} temperature

slightly

^below

Tc and y

^{is an}

unknown exponent. ^On the other hand the

dynamic

effects are

analyzed

^{in terms of the}

thermodynamics

^of

irreversible processes; the main feature of the

theory

is the _presence of three

viscosity coefficients,

qo, J1 and v.

In this

study

^we report

precise

measurements of the

(*) On leave from National Research Laboratory ^of Metrology, Itabashi, Tokyo, Japan.

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

flow-birefringence

^{in the}

isotropic phase

of the che-

mically

stable material PCB

(p-n-pentyl p’-cyano- biphenyl).

^From these measurements,

together

^with

the

magnetic birefringence

^data

of Filippini et

^al.

[6-7],

we have deduced the

viscosity coefficient It

^{which is}

directly

^{related to the flow}

birefringence.

^{We have also}

measured the

capillary viscosity

ilo and the real and

imaginary

part of the shear

impedance

^{at ultrasonic}

frequencies.

^{From these} measurements we

attempt

^to deduce the

viscosity

coefficient v. All

experiments

^were

made on the same

liquid crystal sample.

^{The transition}

point Tc

^{was 35.2 OC.}

In the flow-induced

birefringence experiments,

under a

velocity gradient

^{G =}

a v/az the ’isotropic phase

^becomes

birefringent

and the difference An in refractive indices is

given by [5, 8]

where A8 is the maximum

anisotropy

^{for a}

perfectly

oriented

liquid crystal, n

^{the mean} refractive index and _Jl a

viscosity

coefficient. The coefficient

A(T)

^is

taken to be

A(T )

⁼

a(T - Tc*)y

where a is an unknown

constant. y and

Tc*

^have

already

been defined. Since

A(T)

^{is small near}

T,,,

^the above formula

explains why LBn/G diverges

^{in the}

vicinity

^{of the} transition.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01977003802015900

160

The

experimental

^{set-up was described in an earlier}

paper

[3].

^The

light wavelength

^{was 6 328}

^A.

^The

temperature

gradient

inside the

experimental

^{cell was}

smaller than 0.004 OC. In

figure

^{1 we} present ^our measured values for

G/An

^{as a} function of temperature.

The results deviate

slightly

^{from a}

straight

^{line because}

of the temperature

dependence of p.

From eq.

(1),

p is

given by :

FIG. 1. ^- Temperature dependence of the ratio of the flow bire-

fringence ^{to the} velocity gradient.

In this

expression, h/A8

^and

A(T)

^{are unknown. The}

coefficient

A(T)

^can be obtained from static

properties

such as the

magnetic birefringence

^{or the Kerr effect.}

Precise measurements of the Kerr effect

[6, 9]

^and

magnetic birefringence [6, 7]

^{in the}

isotropic phase

^of

some nematic substances have been

reported recently.

The difference

Tc - Tc*

^{for a}

given sample

^{has the}

same value when deduced from Kerr measurements as

when deduced from

magnetic birefringence

^measu-

rements. For

PCB, Tc - Tc*

^{= 0.7 ± 0.2°C and}

y ⁼ 1

[6]. Furthermore,

^the

magnetic birefringence

for PCB shows identical behaviour except ^{for a shift} in temperature scale when the transition

point

^varies

slightly

^from

sample

^to

sample (1).

^This

point

^{has also}

been

reported by

Stinson and Litster

[2].

^The

magnetic birefringence

^is

given by [5, 8] :

For

PCB,

^{C = 85 x}

1012 G2/K [7-10].

Thus from _eq.

(3)

^and eq.

(1)

^{one obtains}

or

To

interpret

these data

properly

^{it is} necessary to know

Ax.

^The

macroscopic magnetic

^aniso-

tropy x || -

^{xl in} the nematic

phase

^is

The value

of AX

^{is that in the}

completely

^ordered

phase

and is related with x II

- xl

through

^the

equation Ax = (x

^{|| -}

^{xl)/ s}

where S is the order parameter

[12].

Since the value of S is

normally

^{about 0.5}

(2),

^far

below

Tc, it

is reasonable to assume

There is an

uncertainty

in the absolute value

of it

^due

to that in

Ax. However,

^this

uncertainty

^{does not affect}

the temperature

dependence of u

^{which is}

reported

ⁱⁿ

figure

²

together

^with that of the

capillary viscosity

ilo.

It is apparent ^{that the}

viscosity coefficients p

^and no

obey

^{the usual}

exponential

^law

throughout

^the tempe-

rature range of the

experiment [13].

^{From the}

slope

^of

the curves we deduce the activation _energy

Wno

^{= 8.2}

kcal/mole , Wu

^{= 4.3}

^kcal/mole

which _may be

favorably compared

^{with 7.4 and}

4.7

kcal/mole

^{for MBBA}

[3,15].

A last

point

is the evaluation of the coefficient a.

For

PCB,

_{n ||} ^{= 1.71 and} nl ⁼ 1.53 at 25 °C

[14].

^Then

taking A8/E

^{= 0.35 one obtains}

FIG. 2. - Plot of the viscosity coefficients _{qo, J1.} and v versus

103/T.

(1) Filippini, ^J. C., private communication.

(2) ^From dielectric measurements, Cummins et al. [14] ^esti-

mate S in PCB as 0.46 at 25 °C.

from eq.

(3),

^which agrees

reasonably

^{with our} pre- vious estimate of a ~ 5 x

105

_erg

cm- 3

K-1 from acoustical

absorption

measurements

[16].

We now discuss our ultrasonic shear wave atte- nuation measurements. As shown

by

^{de Gennes}

[5],

the

dynamic viscosity q(cv)

^is

given by

which leads to

where r ⁼

A/v.

Eq. (5)

shows that the coefficient v can be evaluated from the

anomaly

^{in the}

dispersion

^{curve at a) = r.}

We used the shear wave reflectance

technique

described in the

preceding

paper

[17].

^In this method the

complex

^shear

impedance

^{Z = R} + iX of the

liquid

^{is first} determined and from

this,

^the

viscosity

can be

computed using

the formulae

where

f

is the shear wave

frequency.

The measurements of the

complex viscosity

^coeffi-

cient in a wide _range of

frequencies

is difficult to realize for technical reasons.

However, since p

^and ’10

are known from the measurements

reported

^{here we}

can determine v from a measurement of the real and

imaginary

parts ^{of the}

viscosity

^{at a}

given frequency.

The

frequency

^{used was 15 MHz.} The variation of the

dynamic viscosity 11’ together

with the static vis- cosit6 _qo is shown in

figure

^{3. The}

dynamic

^visco-

sity

^is

systematically

less than the

capillary viscosity

FIG. 3. ^- Variation of the dynamic viscosity q’ ^{as a} function of temperature ^{at 15 MHz. The} capillary viscosity is shown for _compa-

rison.

indicating

the relaxation _process, ^v is estimated

by solving

^the

quadratic equation

^{in v}

(eq. (5b))

with the parameters a, p ^and ilo. The variation of ^v is shown in

figure

^2.

Although

the estimated values

are rather scattered it seems that v has the same tem-

perature

dependence

^as qo. This scatter is due to at least two reasons.

First 11’ and 11"

^{are related to a cer-}

tain

phase change

Q

produced by applying

^the

liquid

on the

reflecting

^surface of the ultrasonic unit and this

phase change

^is very small

[17]. Second,

^the

amplitude

of the

dispersion i.e.

^the

^quantity qo 2 p

^{is weak.}

In

conclusion,

the results

reported

here confirm our

previous investigation

^{on MBBA}

[1].

^In

particular

^the

three

viscosity

coefficients of the

isotropic phase

follow an Arrhenius law.

References and footnotes

[1] ^{For a review see DE} GENNES, ^P. G., ^The Physics of Liquid Crystals (Clarendon Press, Oxford) ^1974.

[2] STINSON, ^{T. W. and} LITSTER, ^J. D., Phys. ^{Rev. Lett. 25} (1970)

503.

[3] MARTINOTY, P., DEBEAUVAIS, F. and CANDAU, S., Phys. ^Rev.

Lett. 27 (1971) ^1123.

[4] LITSTER, ^{J. D. and} STINSON, ^T. W., ^J. Appl. Phys. ⁴¹ (1970)

996.

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

Liq. Cryst. ¹² (1971) ^193.

[6] FILIPPINI, ^J. C., POGGI, Y., ^J. Physique ^{Lett. 37} (1976) ^L-17.

[7] FILIPPINI, ^J. C., POGGI, Y. and MARET, G., Physique ^sous champs magnétiques ^intenses (Grenoble Edition CNRS) 1974, p. 67.

[8] ^{In the} derivation of formulas (1) ^and (3), ^{the order} parameter

Q03B103B2 ^{taken as} dimensionless and normalized is

Consequently ^{the free} energy density may be written as

(to the lowest order)

The coefficient A defined here is related to the coefficient A* of De Gennes’ paper (ref. [5]) by the relation

[9] FILIPPINI, ^J. C., POGGI, Y., J. Physique ^{Lett. 35} (1974) ^L-99

and J. Physique Colloq. ³⁶ (1975) ^Cl-137.

[10] ^{In ref.} [6-7] the Cotton-Mouton coefficient C is given by

0394n ⁼ C03BBH2 where 03BB is the optical wavelength. ^FILIP-

PINI, J. C., private communication.

[11] We ^are grateful ^to GASPAROUX, H., for private communication of measurements of ~~ 2014 ~~.

162

[12] GASPAROUX, H., REGAYA, ^{B. and} PROST, J., ^{C. R. Hebd. Séan.}

Acad. Sci. 272 (1971) ^1168.

[13] The anomalous increase in the viscosity coefficient 03BC reported

for MBBA by STINSON, ^T. W., LITSTER, ^{J. D. and} CLARK, ^{N. A.} (J. Physique Colloq. ³³ (1972) Cl-169)

was due to an error in data analysis (LITSTER, ^J. D., private communication).

[14] CUMMINS, ^P. G., DUNMUR, D. A. and LAIDLER, ^D. A., ^Mol.

Cryst. Liq. Cryst. ³⁰ (1975) ^109.

[15] The absolute values of the coefficient 03BC reported ^{in ref.} [3]

are erroneous by ^{a factor 2} because the formula used to

deduce 03BC was 03BC

= 0394X/3 (0394n/G) (H2/0394n)

^{instead of}

[16] NAGAI, S., MARTINOTY, ^{P. and} CANDAU, S., ^J. Physique ³⁷ (1976) 769.

[17] KIRY, ^{F. and} MARTINOTY, P., Ultrasonic investigation ^of anisotropic viscosities in a nematic liquid-crystal, ^J. Phy- sique ³⁸ (1977) ^153.