Static and dynamic behavior of a nematic liquid crystal in a magnetic field - Part I : static results

HAL Id: jpa-00207295

https://hal.archives-ouvertes.fr/jpa-00207295

Submitted on 1 Jan 1972

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.

Static and dynamic behavior of a nematic liquid crystal in a magnetic field - Part I : static results

P. Pieranski, F. Brochard, E. Guyon

To cite this version:

P. Pieranski, F. Brochard, E. Guyon. Static and dynamic behavior of a nematic liquid crys- tal in a magnetic field - Part I : static results. Journal de Physique, 1972, 33 (7), pp.681-689.

�10.1051/jphys:01972003307068100�. �jpa-00207295�

STATIC AND DYNAMIC BEHAVIOR

OF A NEMATIC LIQUID ^{CRYSTAL IN A} MAGNETIC FIELD

PART I : STATIC RESULTS

P.

PIERANSKI,

^F. BROCHARD and E. GUYON Laboratoire de

Physique

des Solides

d’Orsay

^{associé au CNRS}

(Reçu

^{le 18}

février 1972)

Résumé. ²⁰¹⁴ Nous discutons théoriquement ^et expérimentalement ^les propriétés statiques ^et dynamiques de la transition de Freedericks dans du MBBA à l’état nématique. Les résultats expéri-

mentaux ont été obtenus, ^{dans la} géométrie planaire ^et homéotrope, ^à partir ^de l’anisotropie ^de

conductivité thermique ^{à travers} le film. Les expériences permettent d’estimer les valeurs de deux constantes élastiques (flexion ^et éventail).

Nos experiences conduisent aussi à la première estimation non ambigue ^de l’anisotropie ^de

conductivité thermique

(k~

²⁰¹⁴

^k)/k

⁼ 0,64 ± 0,04.

Abstract. ²⁰¹⁴ We give ^an extensive theoretical and experimental discussion of the statics and

dynamics of the Freedericks transition of nematic MBBA films in a magnetic field. The experi-

mental data are obtained from measurements of the anisotropy of the thermal conductivity ^across

the film both in the « planar » ^{and «} homeotropic » configurations. Estimated values of the bend and splay ^{elastic constants are obtained.}

Our measurements also give ^{the first} unambiguous value of the anisotropy of the heat conducti-

vity

(k~

^-

^k)/k

^{= 0.64 ± 0.04.}

Classification Physics abstracts :

14-82

1. Introduction. ^- A nematic

liquid crystal (LC)

thin film of thickness d

kept

^{between two}

parallel glass plates

^can be oriented

by

^the solid boundaries.

By rubbing along

^one direction the

glass

^surfaces

[1],

one can obtain

single

^domain

liquid crystals having

the director axis n in the

plane

^{of the film}

(planar case). Using

surfaces which are very clean or chemi-

cally

^treated

by

^an

appropriate

surfactant agent, ⁿ

can be made

perpendicular

^{to the}

plane

^{of the}

plates (homeotropic case).

^{If a}

magnetic

^{field is}

applied

^at

right angle

^{to the}

director,

^{the nematic}

ordering

^is

modified above a critical field

H c’

^the transition

being

second order.

This effect ^was first studied

optically by

^Freede-

ricks

[2]

^for films in the

homeotropic configuration (geometry

^{3 of}

Fig. 1).

^Zocher

[3] interpreted

^these

static

properties using

^the continuum Frank-Oseen

[4], [5] theory.

^The critical field

H,,,

^is

^{given by}

where the index i

(1, 2, 3)

^{refers to the three}

geometries

of

figure

^{1. This relation} expresses the

equilibrium

between the

magnetic

torque ^{due to the}

anisotropic

part ^{of the}

magnetic susceptibility

xa and the

restoring

torque ^{due to} the elastic constant

Ku.

The critical field has been measured for several

geometries

ⁱⁿ

different

liquid crystals,

ⁱⁿ

particular

^{from conosco-}

pic [6]

and dielectric constant

[7]

measurements.

FIG. 1. ^- The three geometries of the Freedericks transition.

The alignment of the molecules close to the glass plate

(z

⁼

d/2)

is « planar » ^{in case 1 and 2 and} « homeotropic » ^{in case 3.}

A large enough magnetic ^field (H > He) ^at right angle ^{with the} initial alignment ^{creates a} « twist » distortion (geometry 2) ^or

a mixture of « bend » and « splay » distortion (geometries ¹

and 3).

In this

article,

^we

analyse

the Freedericks transition from measurements of the thermal

conductivity anisotropy.

^{This method is also} well suited for

studying

the

dynamics

^of the transition when the characte- ristic times of the transition i, upon

varying

^{the field}

H,

are

large compared

^to the thermal relaxation time zth

[8] expressed by

(k

is the thermal

conductivity,

p the

specific

mass,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01972003307068100

682

c is the heat

capacity).

^The

dynamic behavior,

^which

will be discussed in part ^{II is} characterized

by

^{a time}

constant ^1"

given by [9] :

K is an elastic constant, yi is a

viscosity

coefhcient.

With

typical

^{values of k = 4 x}

^10-4 cal/s

^cm

OK,

c ⁼

0,4 cal/g,

p ⁼ 1

g/cc,

^{K = 10-6}

dynes,

y - 10-1

poise,

^we get

In

chapter II,

^we describe the

sample preparation

and the

general

^{results : an estimate of} the absolute thermal

conductivity k,

^and characteristic times of the transition.

In

chapter III,

^we calculate the static thermal transport

properties

^{in the} presence of a constant

magnetic

^{field in the three}

configurations of figure

^1.

Our

experiments

^were done in the two

complemen-

tary

geometries

^{1 and 3. The} measurements of the effective thermal

conductivity along

^oz

- ke

^{- in the}

presence of distortion

give

^{two sets of values}

K11

and

K33,

^{and two} determinations of

k.Ikil,

^{We call}

kjj

^and

^kl

^{the thermal}

conductivity

when the heat flow is

parallel

^or

perpendicular

^{to the axis of a well}

oriented

liquid crystal.

^The

anisotropy

of the thermal

conductivity

^is

ka

⁼

kil - ^k,.

The second part ^of this article is devoted to the

dynamics

^{of the} Freedericks transition based on the

hydrodynamic

Leslie-Ericksen

[10] theory

^{and on}

thermal as well as

optical

^results.

II.

Expérimental.

^-

A)

MEASUREMENT TECHNI- QUES. ^- The aim of the

experiments

^{was to measure}

small

changes

in thermal

properties

rather than to obtain absolute thermal

conductivity

^{data. No}

special

care was taken to suppress heat leaks

(radiation, convection,

^thermal conduction

through

other heat

paths).

The cell used in the thermal measurements was

designed

^to allow direct

optical microscope

^obser-

vation

during

^the

experiments.

^{It was} surrounded

by

a vertical solenoid and a

pair

^of horizontal axis Helmholtz coils. Fields _up to 600

G,

ⁱⁿ any

direction,

could be obtained.

A schematic

representation

^{of the}

original

set-up is

given

ⁱⁿ

figure

^{2. The LC film}

[1]

^is

squeezed

^bet-

ween two horizontal

(2

^{cm x 3}

cm) glass plates [2],

with a

thin, low-thermal-conductivity ring [3]

^{as a}

spacer. The

sensing

^{elements are} metallic films

deposited

^{on the}

glass plates, using

^standard evapo- ration

techniques.

The heater is a

semi-transparent gold

^film

[4] evaporated

^on the lower face of the

1/10

^mm thin bottom

glass plate.

^The heat conduction

through

^the

glass

^is

large compared

^{to that}

through

the LC. The

temperature

^{of the} upper

glass plate

was

regulated by circulating

^water. The latter

plate

FIG. 2. ^- Schematic description ^{of the thermal} measurements on a L C film (in 1).

was

kept

^thick

(1 mm)

^to avoid effects due to water

0

pressure. A dc heat

input Q

up ^to 5 W can be obtained.

0

An unknown fraction of

Q (of

the order of 60

%)

flows

through

^the

liquid crystal

^{film. In the}

following,

we assume that this fraction does not

depend

^{on the}

amount of heat

input although

^{none of the}

major

conclusions of this work

depend strongly

^{on this}

assumption.

Two identical sets of nickel

[5]-copper [6]

^thermo-

couple films,

^X

shaped,

^and

facing

^each

other,

^were

evaporated

^{on the}

glass plates.

^{The films were 1 000}

^A

thick and not transparent but their width was small

(2 mm)

^and

they

^covered

only

^{a small area of the}

visible field. The two nickel films

[5]

^{were connected}

by

^a thin nickel wire.

The temperature difference between the thermo-

couples

^{- AT - could be} read from the

voltage

difference between two copper wires soldered at the ends of the two copper films

[6].

^The thermoelectric power of these

thermocouples

^{was found to be}

equal

^{to 25}

pV/°C

^{at 25 °C with a}

good reproduci- bility

^between

samples,

^{and without}

aging.

^The

differential

voltage

^{as well as} the absolute

reading

from each

thermocouple,

^{was measured on a}

Keithley

148 nanovoltmeter and recorded on an

X,

^t

plotter.

We also used the temperature

dependence

^{of the}

resistance of the

gold

^{film to cross check our}

results,

with a very

satisfactory

agreement.

In the later

experiments

^the

thermocouples

^were

separated

^{form the LC}

by

^thin

glass plates glued against

^{the initial}

glass plates

^with

optical

epoxy resin. The thermal time constant was

consequently increased,

^{but it was easier to} carry ^{out «} efficient » surface treatments discussed below. The LC thickness could also be varied

continuously by

^a micrometric

displacement

^{of the} upper

plate.

^The accuracy ^on the thickness measurement was rather _poor

(5 %),

due in a

large

part ^{to the} deformation of the

glass plate (effect

^{of the external}

stresses).

^In

particular,

the

temperature

difference

depended

^{on the} pressure of the

cooling

^{water which was}

regulated by

^{a constant}

pressure

supply

^{and a} fine flow-rate

regulator

^{at the} output. Because of the

uncertainty

^on

d,

^{we will} characterize

it,

^whenever

possible, by

the value of

H,,.

B)

^{MATERIAL USED. - We} have studied metho-

xybenzylidène butyl

^{anilin MBBA.} The critical tem-

perature,

Te,

^{for the}

nematic-isotropic

^transition

was around 42,OC. In the initial

experiments [11],

the material used had a

Tc

^{of 36 °C.} The effect of

impurities [6]

may have

explained

^the

significantly

different values of the elastic constants obtained in this report

(see

^III

C).

C)

^THERMAL CONTROL EXPERIMENTS. ^- Several

experiments

^{were carried to check the}

reliability

^of

the

techniques

^used.

1)

^{The heat}

conductivity

^{of MBBA was determined}

from a

comparison

with several materials of _compa- rable known heat

conductivity.

^{If the} measurements are

done in the same conditions

(same applied

power and average

thickness),

^the

changes

^of temperature ^{AT for}

a

given

^{material are}

expected

^to vary

proportionally

to the

changes

^{of thickness Ad :}

(AdIAT)Q,d

^{= oc.k}

where a is

independent

^of the material. This has been checked

experimentally

^{on several}

experiments.

The results of a

typical

^run

using glycerol, amyl

acetate and

planar

^and

homeotropic

^{MBBA are}

given

on

figure

^{3 and}

give :

FiG. 3. ^- Several materials have been studied thermally ⁱⁿ similar conditions. The variation of the temperature drop ^{due to}

a small change ^{of thickness is} proportional to k for known materials and provides ^a measurement of the heat conductivity

of MBBA in the two orientations. (Note straight ^line depen- dence through origin.)

We will see in III that the

anisotropy

^{is in excellent}

agreement ^with measurements in a

magnetic

^field.

The values are of the same order of

magnitude

^as

those for PAA

[12], [13] where kil

^{= 2.8 x}

^{10-4 cal/}

cm . s . deg.

^The

conductivity

^{does not} vary

appreciably

with _{T up} to 40oC. In this paper, ^we restrict our

attention ^to measurements around 20 OC.

2)

^{In a second}

experiment,

^we have studied the thermal time constant -rth

by witching

^on

(off)

^the

power ^{at t = 0} and

watching

^{the rise}

(decay)

^{of AT.}

The results in the

planar

geometry ^are

given

ⁱⁿ

figure

^4.

For

large

^values

^{of t,}

the behavior is

exponential

FIG. 4. ^- A measurement of the thermal diffusivity ^{is obtained}

from the thermal _response when the heat _power is applied. ^In

a planar ^L C, the thermal _response is slower in zero field where the heat conductivity is smaller. The initial slopes ^{are the same}

in the two cases because the saturated t5T value is larger ^{in zero} field. (The anisotropy ^{observed at} saturation is smaller than the maximum anisotropy ^{of k as H was} not very large compared ^to Hc ^{and as this} recorded AT induded also the temperature drop

across the limiting planes.)

with a

single

^{time constant which} reflects the dominant conduction mechanism : _Tlh ^- 3 s. Such

dynamic experiments

^{should in}

principle give

^a measurement of the thermal

diffusivity

^{of the} system

(= K/C,

^{where C}

is the heat

capacitance

and K the thermal conduc-

tance),

^{b is} controlled

by

the slowest heat _process

(the

^heat

diffusivity through

^the

liquid crystal

^is

10 to 50 times smaller than that

through

^the

glass).

We can see

that,

^{when a}

large enough

^{field is}

applied

^{so that} the thermal

conductivity

^{of the LC}

goes

from kl

^to

kll,

the time scale of the curve

AT(t)

is decreased

by

^a

corresponding

^amount.

(The

^satu-

rated AT value is

larger

^{in the}

planar

^{case and the}

initial

slopes

^are

equal.)

This shows indeed that the thermal conduction

through

^the

liquid crystal

^is

responsible

^{for the}

major

part ^{of the heat flow.}

However,

^{a detailed}

quantitative analysis

^{for this}

multilayer

system ^{would be difficult}

[8].

D)

^SURFACE TREATMENT AND CONTROL. ^-

1)

^Pre-

paration.

^-

a)

^{Planar case. A LC can} be oriented

parallel

^{to a}

glass

^surface

by polishing

^the

glass plate along

^one direction with _paper or abrasive

[1].

It appears

(see 2b)

^{that a rather} strong

polishing

^is

required

^to get ^a

good ordering.

^In the initial _expe-

riments,

^the

thermocouples

^were in direct contact with the LC and

only

^{a weak}

polishing

^was

applied

in order to avoid to scratch the metallic films the

alignment

^remained

imperfect (n

^{in the}

plane

^of symmetry ^of

polishing

but finite tilt

angle

^{at the}

surface).

b) Homeotropic

^{case. On} the clean surfaces of the

glass plates,

^{a thin film of}

hexadecyltrimethylammo-

684

nium bromide was

applied starting

^{with a dilute}

solution in toluene. The thickness of this film corres-

ponded probably

^{to a}

single

^molecular

layer [14].

The LC was then inserted between the

parallel glass plates.

The initial

alignment

^{of the LC takes a}

long

time

(several

^{hours for a}

400 J.1 film) :

^{when the}

liquid

is inserted between the

plates,

the molecules are

strongly

^oriented

by

^{the flow}

parallel

^{to the}

plates.

The

large

^{time relates to} the exclusion of the structural defects

(disclination loops)

^{which is followed}

optically.

2)

^Control

of

orientation. ^-

a)

^Use

of polarized light.

The search for the

principal

axis between crossed

polaroids

^{does not lead to} the value of the tilt

angle

in a poor

planar sample (see la).

It is also insufficient for a

good testing

^{of the}

homeotropic alignment.

We have characterized the orientation more accu-

rately, using conoscopic techniques.

^{In recent expe-}

riments,

^we also used them to

study

the static

[15]

and

dynamic [16]

aspects ^{of the}

magnetic

^transition

quantitatively.

b) Optical images.

^{- The}

study

^{of the textures in}

the _presence of

magnetic

^{fields and} thermal effects has been used to control the orientation but is

beyond

the _{scope of} the present

study.

^The

birefringence

^can

also

give

^a

simple

^{test of}

alignment :

^{If a small dust}

particle

^is located close to the lower face of the

LC,

it

gives

^a

single image only

^{when one of the}

principal

axes is

perpendicular

^{to the LC}

plane.

^{This was}

usually

^{obtained for the} films used in this

study.

When a field

larger

^than

H,,

^is

applied,

^two

images

are seen. The

separation

^goes

through

^{a maximum}

for an intermediate field value. In

large enough fields,

^{where most of the LC is oriented}

along

^the

field,

^{the two}

images

get very close

again.

c) Finally,

the thermal measurements

give

^also

good

^{tests of}

alignment.

^{Let us}

anticipate

^{on next}

chapter

^and

follow,

^on

figure 5,

^the

change

^{of T}

across a

homeotropic

^{film when its thickness is}

suddenly

^decreased

(point A).

^A

long enough

^time

FIG. 5. ^- The thickness of a 500 _p homeotropically aligned ^{L C} is decreased suddenly (point A) ^{to 400} p. Right after this change (point B), ^the temperature drop ^{across the} sample ^increased due to the decrease of the effective heat conductivity in the disor- dered state created by ^{the flow} (the ^same process is reproduced

from 400 to 300 p).

after the

change

^{of thickness}

(point C),

^the tempera-

ture difference across the film is smaller than at the initial

point

^A. However,

right

^{after the}

change of d,

the temperature difference increases : the disordered state

obtained,

^{due to this}

change,

^{has an effective}

heat

conductivity ke

^{smaller than} the ordered state one

kll ^(as kll > ke

>

kl). However,

the oriented state is restored much faster than when the LC was inserted

initially

between the

plates.

^This time becomes even

shorter for smaller thicknesses.

A last thermal test uses the measurement of the

dynamics

^{of the} distortion when a

magnetic

^{field is}

suddenly applied (see

part

II).

^{It turned out to be}

the most sensitive to small

misalignments.

All

samples

^{used in this} report ^were

satisfactorily

tested with the different

alignment

^methods.

III. Static behaviour. ^-

A)

^{THEORY. - We}

study successively

^{the effect of a}

magnetic

^{field on the ther-}

mal

conductivity

^{in the}

geometries

^of

figure

^1.

1) Geometry

^{1. - A}

magnetic

^{field is}

applied along

the z axis

perpendicular

^{to the}

liquid crystal

^director

taken

along

^{ox. We call}

O(z)

^the average orientation of the molecule with respect ^to the x axis. The effective heat

conductivity ke

^is

given by :

where

A;[0]

⁼

kl + ka sin’ 0 ; ka

^{is the}

anisotropy

of the heat

conductivity.

The

equilibrium configuration

^{can be obtained}

from the minimization of the Frank free _energy

density (per

^{unit area of the}

wall)

The first term describes the elastic contribution of the

splay

distortion

(K11)

^and the second one that of the

bending (K33).

The distortion is caused

by

^the

anisotropy

^{of the}

magnetic susceptibility (xa).

^The

third term represents ^this

magnetic

energy contri- bution. For small values of the

field,

^{the stable}

configuration corresponds

^to

0(z)

^{= 0. Above a}

critical field H ⁼

H,,,,

^the

liquid crystal

^{becomes dis-}

torted.

HCI

^{is obtained}

by looking

^{for a} minimum of F

keeping

the lowest order term in 0

This limit

equation depends only

^{on one «}

splay »

elastic constant. The distortion is of the form 0 ⁼ a cos

z/çl(H).

The

length

gives

the range of effect of the wall in the presence of H and is called coherence

length (although

^{it is different}

in essence from the coherence

length

ⁱⁿ

superfluids,

for

example,

^{where the coherence}

length

^{is field}

independent).

By applying

^the

boundary condition, 0 [z

⁼ +

d/2]

⁼

0,

one gets :

or

When H is

larger

^than

HC1’

the solution

O(z) depends

on the two parameters

K11

^and

K33.

^{A first}

integral

of the functional derivative of F with respect ^{to 0}

can be

easily

^{obtained :}

where n

⁼

(K33 - K11)/K11

^{and the maximum dis-}

tortion

angle 0.

^{= 0}

(z

⁼

0).

The

integration

^of eq.

(III.S) using

^the

boundary

conditions

gives Om

^{as a} function of I-I.

By using

sin u ⁼ sin

0/ sin 0 m’

^{one finds}

The

integral

^{I is}

larger

^than

n/2

^and one recovers

the fact that

0.

> 0 is obtained

only

^{when H} >

HCI.

Using (111.5),

^the eq.

(III.1)

^{becomes :}

The _eq.

(III.6)

^and

(III.7) give

^an

implicit

expres- sion of the effective

conductivity ke

⁼

f(h),

^where

we write h ⁼

HIH,,.

In the limit where h - 1 is

small,

^these

equations

can be

easily

^solved

The

anisotropy

^{of the thermal}

conductivity ka

^is

obtained

by measuring ke

^for

large

fields where

k,r --> k il .

^{From the initial}

^slope

^{of the}

^ke(h)

^curve,

one gets ^{a value of the ratio}

K11IK33.

The

shape

^{of the}

complete

^curve

k.(h)

^is

easily expressed

^{in the}

limiting

^{case of a} small thermal

anisotropy k.1k, « 1

^{and when the Frank} coefficients

are

nearly equal.

^The

integral

^of

(6)

^and

(7)

^{can be}

expressed

^{then as an}

elliptical integral (we

^{will use}

the notations of reference

[17])

^and

ke

^is

given

^from

the

coupled equations :

The

corresponding

calculated curves are

given

^as

a function

of 11

^on

figure

^{6. One sees}

clearly

^{that the}

value

of 11

^{is rather}

unambiguously

determined from the initial

slope

^alone.

FIG. 6. ^- The decrease of temperature ^{Jf in the} presence of

a field H is calculated in the planar ^case in the limit conditions of formula (III.9) for several values of the anisotropy ^{of the}

elastic constants Ki, 1 and K33. The normalized curve is rather unambiguously characterized by ^{the initial} slope.

2) Geometry

^{2. - In this} case, ^one has a pure twist deformation. The free _energy can be written as :

In the case of small fields

(B small),

the results can

be deduced from

(1) by having tl

^{= 0 and K =}

K22-

The critical field is :

In this geometry, ^the

anisotropy

of k would have

to be measured in the

plane

^of the film. An unknown fraction of the heat flow is

taking place through

^the

end

plates.

^{For this} reason, ^we will not report ^here any

experimental

results in this geometry.

In the limit where

(ka/kl) ^sin’ Bm

^«

1, ke

^is

given

from the

following equations :

686

3) Geometry

^{3. - The free} energy is :

This case is identical to case 1 if one

interchanges Ki l

and

K33, kli

^and

^k1-.

B)

EXPERIMENTS. ^-

1) Homeotropic

^{case. - In}

figure

7, ^we present ^a

typical

^set of recorded thermal

FiG. 7. ^- Homeotropic ^{film : A} magnetic ^{field is} applied ^{in the} plane of the film (point A) ^{and the} temperature drop ^across

the sample increases. When the field is decreased by steps, ^the decrease of ô T is exponential.

results

AT(t).

^The

400 Jl

^{thick LC film was oriented}

perpendicularly

^{to the}

glass plates.

^{In the}

experiment,

we first

applied,

^{in zero}

field,

^a

given

^{power to} pro- duce a temperature difference

fllll

^of the order of one

degree.

The nanovoltmeter is zeroed so that this AT is taken as the base line. Additional

changes,

ôT ⁼ OT -

AT,,,

^are read with ^a

larger sensitivity.

At the initial time t ⁼ 0

(point A),

^{a dc}

magnetic

field is

applied perpendicularly

^{to the LC axis and}

AT is found to increase. At

points B,

^{the field is}

suddenly

^decreased

by steps. Finally,

^{for zero}

field,

AT returns to its

original value,

^with

usually only

small drift effects. The

original

^{value is}

already

^reco-

vered as soon as the field is below a certain threshold which

corresponds

^to the Freedericks critical field.

Above this

limit,

^the

sign

^{of the}

temperature change

in

field,

^{ô T} >

0, corresponds

^{to a decrease of the}

thermal

conductivity.

The time constant, which will be studied in part

II,

^are

large compared

^to the thermal ones,

especially

^{in thick films. A}

study

^{of the}

dynamics

^{of the} transition is essential to get ^the

equilibrium

^value

ô T(H),

^whenthe temperature

changes

are small

(H N jHc).

a)

^Critical

fields.

^{- In low}

field,

^H ;:

HC3’

^{we find}

that ô T varies

linearly

^with H, ⁱⁿ

agreement

with III. A.

We get

Hr ,,,

^{with a}

^{5 %}

^{accuracy from a linear extra-}

polation

^{of ôT to zero. In}

figure 8,

^we

give

^{the varia-}

tion of

IIH,,

^{versus d for three film} thicknesses. From

FIG. 8. ^- The critical field varies as the inverse of the thickness,

in the two geometries studied. The slopes ^{can be used to} get values of Kl and K33.

the

slope

of the linear

variation,

^{we can deduce a} value of the bend elastic constant

K33

around 260 :

b)

^Thermal

equilibrium

^behavior

for

^H >

Hr.

^-

Our results for

samples

^of different thicknesses are

given

^{in the} normalized units :

FIG. 9. ^- The relative increase of the temperature drop ^across homeotropic L C films of different thicknesses is plotted ^{as a}

function of the normalized field.

They

should define a

unique

^curve, function of the

splay

^{elastic constant}

Kl i

^{as well as on}

K33.

In par-

ticular,

^{when h is close to}

1,

the initial

slope

^{is obtained}

from :

This

equation

^can be deduced

easily

^from eq. 111.8 for the

planar

^case.

- The observed

shape

^{of the curve} obtained from the different data

points

^of

figure

^{9 is similar to}

that of the limit calculation

leading

^to

figure

^6.

However,

^{there is a rather}

large

^and

systematic smearing

of the results

leading

^{to a}

larger anisotropy

for thinner films. This was observed

reproducibly

^for

different

samples

^{of the same thickness and was}

found to be

relatively

insensitive to the thermal

gradient

^{across the}

sample. (In ^fact, H,,,

^and

K11

should

only

^vary

slowly

^with temperature ^{close to}

Tc [6]).

- Close to

Hc,

^{the extreme}

slopes

^{1 and 2}

give

^a

value of the ratio

K331K, 1 :

- In

large

^fields

(h » 1), ke

^{is close from}

kl.

Only

the molecules close to the

glass ^surface,

^within

a coherence

distance (

^oc

IIH,

contribute to

kll.

This

simplified analysis

^{as well as the use of the}

limiting expression

^111.9

predict

^an

asymptotic

^behavior

ke(h)

^{= 1 -}

alh.

^This form contradicts the

prediction ke(h)

^{= 1 -}

P/h2

^{based on the « swarm » model in}

the thermal

experiments

^{of reference}

[18].

^The present

experiments

^{do not extend to fields}

large enough

to separate ^{between these two forms. In}

independent optical experiments

^{in much}

larger

^fields

(up

^to

h ⁼

20) [15],

^we have shown that the

optical

^bire-

fringence

^does vary

linearly

^with

Ilh.

-

Finally,

^our results indicate a value of the

anisotropy

^{of the heat}

conductivity :

between

2)

^{Planar case. - The}

experimental

^{results are}

similar in this case

(geometry

^{1 of}

Fig. 1)

^{to those}

in the

homeotropic

geometry. ^{When a field h} > 1 is

applied,

^the temperature difference is found to decrease. This is consistent with the result

ka

> 0

already

^obtained.

a)

^Critical

fields.

^{- From the linear relation}

between

Hc!

^and

1/d (Fig. 8),

^one gets :

b)

The normalized thermal results ^are

given

^on

figure

¹⁰ for different film thicknesses.

The extreme values of the initial

slope give

^{a value}

of the

anisotropy

^{of the elastic} constants :

From the behavior in

large fields,

^{we obtain a new}

FIG. 10. ^- The relative decrease of the temperature drop

across planar L C films in field (same ^{situation as} Fig. 7).

value of the heat

conductivity anisotropy (between

h ⁼ 0 and

3) :

C)

DISCUSSION. ^- A

comparison

^{between the}

planar

^and

homeotropic

^results

give

correlative infor- mations on the

anisotropy

of the heat

conductivity

and on the elastic constants

Kl 1

^and

K33-

1) Anisotropy of

^{the thermal}

conductivity.

^{- The}

magnetic

measurements gave ^two values for the

partial anisotropy

^{between h = 0 and h = 3 :}

The direct measurements of _{II gave} a value of the total

anisotropy k.1k,

^{= 0.64. If one}

estimates,

^from the calculated results of

figure ^6,

^{that the}

partial anisotropy

^{for h = 3 is 15 to 25}

%

smaller than the saturated _one, we can see that the agreement ^between these results is excellent and that the last value can

be retained with an accuracy of + ⁵

%.

This consis- tency ^{is a}

good

confirmation of the

reliability

^{of our}

measurements.

Although

^{there has not been any}

previously publish-

ed thermal

conductivity

^{data on MBBA some results}

have been

given

^{in other nematic LC.} Fisher and Frederickson

[12]

^{have measured an increase}

of ke

across a

sample

^{of PAA at}

right angle

^{with a shear}

flow.

They implied

^a

negative anisotropy.

^{A value}

of the same

sign

^{was found from heat}

diffusivity

measurements in PAA oriented in an electric field

[19].

This

sign

^of

ka

^{is not} consistent with ^{recent measure-} ments on PAA

aligned

^{with a}

magnetic

^field

[13].

Recent heat

diffusivity

measurements in another LC

- DBA - would also

give

^a

positive sign of ka [18].

However,

^the

alignment

^{close to the boundaries}

was never defined or controlled in _any of these _expe- riments. In this

respect,

^our

experiments provide

^the

first

unambiguous investigation

of the heat conduc-

tivity anisotropy.

^The

experiments

of references

[18]

and

[19]

^were

interpreted

^{in terms of a}

field-independent,

oriented

boundary layer having

^{a heat}

conductivity

688

different from the bulk. We have found no evidence of such a

layer

^{in our results.}

2)

^Elastic constants. ^- Our determinations of the elastic constants from

Hc

^{and from} the behavior above

Hc

^are rather scattered. Before

analysing

these

results,

^{let us} consider the

reported

^{values of}

elastic constant

K11

^and

K33

^{for MBBA. Williams}

and Cladis

[6]

^obtained

K331X.

^{= 6.8 ± 0.7 from the}

conoscopic

measurement of

HC3.

^{Robert and Labru-}

nie

[20]

^measured

K331X,, =

^{7.3 ± 0.4 x 10-’ and}

Kll/K33 ’"

^{0.7 from}

birefringence

measurements in

homeotropic

^{MBBA distorted}

by

^an electric field.

Haller

[21]

^{indicated a value}

of K33 -

^{5.1 x 10-’}

at 220 and

K, 1 IK3 3 =

^{0.9. Rondelez and Hulin}

[7]

obtained K3 3 = 7.4 + 1 X 10-7 and K, 1 = 5.3 + 0.5 x 10-7

from their dielectric measurements. The

published

results

already

^{indicate a}

fairly large scattering.

^In

addition, they

^{were obtained on} films thinner than

ours. In order to

study

^a

possible

^thickness

effect,

we have

developped,

^{to a}

larger

accuracy, the ^cono-

scopic technique

^used

by

^{Williams and}

Cladis,

^and obtained

optically,

^the thickness and the value of

HC3

for

homeotropic

films of various thicknesses. For

a

175 g

^thick

film,

^{we have obtained}

K33/Xa

^{= 7.7:t 0.5}

and for two

350 Jl

^{thick ones}

K331X.

^{= 7 ± 0.5 and}

8.3 + ^0.5.

If we come back to our

results,

^{we see that the} values of

K33

^and

K11

^{deduced from the critical}

field measured

thermally

^{are too} small. This disa- greement ^{should not be taken too}

seriously

^{as the}

thermal effect is

only

^an indication of

H,,

^{and is an} order of

magnitude

^{less accurate}

than,

^for

example,

the

optical

measurements we have used

[15].

^A

possible

additional cause of

disagreement

^{would be the effect}

of the temperature

gradient

^{across the}

sample, although

we have found ^no

appreciable change

^{in the}

Hc

value when

changing

^the power ^across the

sample by

^a factor of three.

If we consider the results ^on the thinner films

(d

^{= 300}

Il),

the different determinations of the ratio

K11IK33

^{are found to be}

compatible

^and

give

^{a value}

of

Kl lI K33 ^-’

^{0.7 ± 0.15.}

Remark. - In a

preliminary report

^{we had obtained} values

K11/Xa

^{= 1.4 + 0.3 and}

Kll/K33

^{= 0.25 + 0.05}

in

striking disagreement

with these latest results. It may be instructive to

analyse

^{the causes. A}

possible

reason is related to the _poor

quality

of the material used

(Tc = 36 OC).

Williams and Cladis

[6]

^found

that, by

the addition of a surfactant agent,

K33

^decreas-

ed

by 14 %

per percent ^of

impurity.

^{Another reason}

was the _poor

alignment

^{of the LC}

(see

the discussion in part

II).

In the _presence of ^a finite

tilt,

9, of the director at the

glass surface,

the critical field vanishes.

However,

the distortion

changes rapidly

^{around a value of}

field which decreases _very

rapidly

^as lfJ becomes

larger. Typically

^for qJ ⁼

10°,

^the apparent ^critical field would decrease

by

³⁰

% (and

^the

K33

^determi-

nation

by

⁶⁰

%).

IV. Conclusion. ^- Our results

provide

^{the first}

complete

^and

unambiguous study

^{of the}

anisotropy

of the thermal

conductivity

^{of a nematic LC.}

Yun and Frederickson

[29]

^has

reported

^some

effects of heat

generation

^{in LC}

subjected

^to

magnetic

fields.

However,

^as

pointed

^out

by

^{Kessler and}

Long- ley-Cook [13],

^the

persistent changes

ⁱⁿ temperature

are more

probably

^{due to the}

change

^{of the thermal}

impedance

between the LC and the

exterior,

^{due to} the

anisotropy

^{of k in a field. Our results} support this last

explanation.

^{It would be of interest to}

study

now the

coupling

effects between thermal

gradients

and other transport

properties,

^{via this}

anisotropy

term in the nematic state.

The static

study

^of the Freedericks transition has confirmed the results of the Frank-Oseen

description.

However the thermal method is too insensitive to get

accurate determination of the elastic constants.

Optical experiments

appear to be more

appropriate

but a very fine control of the surface conditions is

required.

Acknowledgments.

^{- We are}

grateful

^{to P. G. de}

Gennes for several

illuminating

discussions

along

this work. We thank Mrs. Williams and Mrs. Cladis for

helpful

^{critics based on their}

optical study

^{of the} Freedericks transition. We have benefited from the

experience

^{of the}

Groupe

d’Etude des Cristaux

Liquides d’Orsay

^{and from a} critical review of this work

by

^{C. Mitescu.}

One of us

(E. G.)

^{wants to}

acknowledge

^the

Physics Department

^{of the}

University

^of

California,

^Los

Angeles,

^for

welcoming

^him

during

^{a time where} part ^{of the work was done. He had}

stimulating

^dis-

cussions with F. J. Kahn and L. B. Kovalenko.

Note added in

proof :

^M.

Longley-Cook (Thesis, University of Arizona, 1971)

^{has done}

independently

similar measurements on PAA. His results are dis- cussed in terms of our initial report

[11]

and agree with those

presented

^here.

References

[1] ^CHATELAIN (P.), Bull. Soc. Franç. Mineral, 1943, 66,

105.

[2] FREEDERICKSZ (V.), ^ZOLINA (V.), ^Trans. Faraday Soc., 1933, 29, ^919.

[3] ^ZOCHER (H.), ^Trans. Faraday Soc., 1933, 29, ^945.

[4] ^FRANK (F. C.), ^Disc. Faraday Soc., 1958, 25, 1.

OSEEN (C. W.), ^Trans. Faraday Soc., 1933, 29, ^883.

[5] ^RAPINI (A.), ^PAPOULAR (M.), ^PINCUS

(P.), C.

^{R. Acad.}

Sci., Paris, 1968, 267, ^120B.

RAPINI (A.), ^PAPOULAR (M.), ^J. Physique, 1969, ^{C 430.}

[6] ^WILLIAMS (C.), ^CLADIS (P. E.), ^{to be} published ^{in Solid}

State Communications, 1972, 10, 357.