Periodical shear instability of nematics : symmetry considerations

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Periodical shear instability of nematics : symmetry considerations

A.-J. Koch, F. Rothen, J. Sadik, O. Schöri

To cite this version:

A.-J. Koch, F. Rothen, J. Sadik, O. Schöri. Periodical shear instability of nematics : symmetry considerations. Journal de Physique, 1985, 46 (5), pp.699-707. �10.1051/jphys:01985004605069900�.

�jpa-00210011�

Institut de Physique Expérimentale, Université de Lausanne, ¹⁰¹⁵ Lausanne-Dorigny, Switzerland and O. Schöri

Institut de Physique Nucléaire, Université de Lausanne, 1015 Lausanne-Dorigny, Switzerland

(Recu le 10 août 1984, révisé le 17 décembre, accepté ^{le 1 S} janvier 1985 )

Résumé. 2014 Nous montrons qu’à partir ^de considérations de symétrie, ^{il est} possible ^de prévoir ^{la forme} générale

des courbes de seuil séparant ^les différents régimes d’écoulement (uniforme, rouleaux, carrés) ^d’un nématique homéotrope ^cisaillé périodiquement. ^{Le calcul est effectué dans le cas} où la transition se ramène à une brisure de

symétrie caractérisée _par l’apparition de rouleaux de direction 03C8. Nous déterminons également ^{la variation de 03C8}

au seuil lorsque l’on modifie le cisaillement

Abstract. ²⁰¹⁴ We show that from symmetry considerations it is possible ^to predict ^the general shape ^{of the threshold}

curves fixing the boundaries between the different flow regimes (uniform, ^{roll or} square) occurring ^{in a} homeotropic

nematic submitted to a periodical ^shear. Calculations have been made for the case in which the transition is reduced to a break of symmetry characterized by ^the appearance of a roll set in the direction 03C8. The variation of 03C8 at threshold when the shear is modified has also been obtained

1. Introduction

The present paper is the

development

of earlier _expe- rimental and theoretical works on a certain

hydro- dynamical instability

^of nematics under shear

[1-7].

Very general

symmetry considerations ^{about the} system will be used in order to find relations between the parameters

characterizing

the various flow

regimes

at threshold.

The _purpose of our work is to show that the funda- mental

symmetries

of the motion

equations

^{are rich}

enough

^{to lead to not trivial}

scaling

laws. Some of our

results have

already

^{been confirmed}

by experi-

ments

[1-5]

^{or are} consistent with

previous

theoretical calculations made with classical methods [6, 7].

1.1 NEMATIC _LIQUID CRYSTALS. ^- As we do not use

detailed

nematodynamics,

^{we shall}

only

recall three

points [8] :

1)

The nematics are formed

by elongated

^molecules

which have a mean orientation direction described

by

a director n (n ^{and - n are}

equivalent).

This introduces

a

large anisotropy

^{in the} nematics, ^{which are to first}

order

approximation,

characterized

by

^five

indepen-

dent

viscosity

coefficients

(Xi (i

^{= 1, ...,}

5).

2)

^An

alignment elasticity

exists which is described

by

^three

elasticity

^constants

K1, K2, K3.

3)

In addition to the force and

incompressibility equations,

^a

complete description

^of nematics needs a

torque

equation

^{and the condition of}

unitarity

^{of n}

1. 2 EXPERIMENTAL FACTS. ^- We consider a nematic

homeotropically

retained between two

parallel glass plates, separated by

^a distance d. A

periodic

^movement

is

applied

^{to one of them}

(Fig.

1). ^{In the}

Oxyz

frame,

the

displacement

^{of the}

plate

^{reads :}

Experimental

^{data are} available for a few values of x., Xn, Ym, Yn and m, ^n.

They

^{have the}

following

^com-

mon

properties.

When the shear

amplitudes

^are small, ^the

regime

^is

« uniform » : in the nematic

sample,

^the average orien- tation of molecules,

given by

^{the director n, is inde-}

pendent

^{of the}

position.

^{If we increase the shear}

amplitude,

the flow becomes unstable : above a certain

threshold, convective rolls appear in the

liquid

^cell;

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01985004605069900

700

Fig. ^{1. -} Experimental ^set up. A nematic sample ^{is sand-} wiched between two rectangular plates. The thickness of the cell is d. One imposes ^a displacement ^{of the form} x(t) = x. ^cos (mwt) + x" ^cos (nwt) ^and y(t) ^{= y. sin} (mwt) ⁺

y. sin (scot) ^{on the} upper plate. ^{The lower} plate ^{is fixed.}

their orientation V is defined in

figure

^{3. If we continue}

to increase the

amplitudes,

^{this roll}

regime gives

way,

beyond

^{a second} threshold, ^{to a «} square flow » here

everything

^{is as if two} roll series were

superposed

almost at

right angle [1-4] (Fig.

2). Our purpose is ^to

use symmetry considerations in order to deduce the

shape

^{of the threshold curves for a vast}

family

^of

shears.

2. The symmetry ^method.

2.1 THE MAIN ASSUMPTIONS. - This work uses the

following assumptions

^{based on}

experimental

^data.

Al) ^{We assume} that the shear rate in the bulk of the

sample

is characterized

by

the ratios

Xi/d, yield (i

= m,

n).

^{This is the case} if the border effects due to the strong

homeotropic anchoring

^are

negligible

^and

if the

penetration depth

^{of a shear wave is much}

larger

than the

sample

^thickness.

A2)

^{At threshold the shear rates}

Xi/d, yid(i

⁼ m,

n)

and the inverse P of the Ericksen number

(see § 2. 2)

are

supposed

^to be small. This enables us to retain

only

^{the first terms in a}

Taylor expansion

ⁱⁿ

xdd, yJd and

^{P. The}

experiments [1-5]

^{show that this}

hypo-

thesis has a

large

^{domain of}

validity.

A3)

^{We are} interested

by

^{a threshold}

equation relating

^{the various} parameters

characterizing

^an

experiment,

^i.e. Xi’ yi

(i

⁼ m,

n),

^{OJ and V}

(1).

^As

we do not seek numerical values but rather

scaling

laws, ^the

viscosity

^and

elasticity

will appear

only

^to

ensure the

right

^{dimension and will be labelled}

by

^a

and K

(a

^{and K can} be any combination of

respectively

the five viscosities and the three elasticities,

having

^the

corresponding dimension).

^{We shall not} try ^to find any information on the roll geometry, ^{so we do not intro-} duce a parameter ^{related to this} property.

(1) ^{We assume that P can be fixed} arbitrarily by ^some experimental device. This is still an assumption ^{because no}

such experiment ^has yet ^been performed.

Fig. ^{2. - For a} given sample ^{at fixed} temperature elliptically

sheared at frequency (u/2 ^n, and for xn = y. ⁼ 0, ^the threshold curves are relations between the shear amplitudes

Xm and y.. A gives ^{the border} between uniform and roll

regimes. ^{Curve B delimits the} boundary between roll and square regimes.

Fig. ^{3. - The direction} of the roll series appearing ^{at the} instability ^{threshold is} characterized by ^{the unit vector b. b}

is defined as the normal to the roll axis direction. P is the

angle between b and an axis of the shear figure, arbitrarily

chosen to be the x-axis.

A4)

^At threshold, ^the bifurcation, characterized for instance

by

^the occurrence of a roll lattice, ^is

supposed

^{to be direct; thus we can} linearize the _equa- tions with respect ^to infinitesimal fluctuations.

It must be

emphasized

^{that these}

assumptions

^are

necessary ^to determine the characteristic parameters of the

problem (Al, A3)

and their relative orders of

magnitude

(A2,

A4).

2.2 INTRODUCTION TO THE THRESHOLD DETERMINA- TION. ^- To get ^{threshold curves, the usual}

procedure

described below

(points

^{I to}

IV)

^is inconvenient : it is very

long, requires

^{a lot of}

simplifications

^{and leads}

to poor numerical results

[6, 7].

^{We shall use a similar}

way but ^we

disregard

^all numerical information and

we avoid

solving

^the

nematodynamic equations.

^{Let us}

fundamental

frequency

^{w. This is}

justified by

^the

fact that the excitation has at least the

frequency

^w.

Following

^{this, we can write}

where 6a can be

bvx, bvy, 6v_,, bnx, 6ny, ðnz

^or

6p;

^{b is an}

unitary

^{vector in the}

plane

(xy), ^as defined in

figure

^3.

At this stage, the usual method consists in the

following operations :

I) The introduction of v ⁼ v. + bv, ^{n = n’} + bn,

p ⁼ p- ⁺

6p

^{in the}

nematodynamic equations.

II)

The linearization with respect ^{to the ð’s.}

III) ^{The choice of a}

degree

^of

precision;

^{the intro-}

duction of a Fourier

development satisfying

^{the border}

conditions with the desired _accuracy and

neglecting

the

remaining

harmonics. This

procedure

^{uses the}

standard method for this type ^of

problems,

i.e. iden- tifications of the

spatial

harmonics and time

averaging

over one

period

^after

multiplication by

^{a suitable}

factor.

IV)

^The

requirement

^{that the} system obtained has

a

non-vanishing

solution. Let ^us notice that this linear system has its second member

identically equal

^{to zero.}

This is obvious for the torque,

incompressibility

^and

unitarity equations.

^For the force

equations,

^the

second member is the time derivative of the

velocity;

as, at

point

^{III, it is} necessary ^to take its time average

over one

period,

^{it is zero.}

To get ^the

scaling

^{laws at} threshold, ^{we shall}

replace

the

procedure

^{I to IV}

by

^the

following.

1)

^{We write a linear}

homogeneous

system ^{of two}

equations

^{for two unknowns}

6a";

^{we choose}

bvkrs

and the

explicit

^{form of}

M(S)

^{follow at}

point

^{2. We}

have to take a system ^{of two}

equations

^{for two un-}

knowns : this is related to the fact that, ^{as we shall see}

in the

following,

^the

symmetries

^{act in a}

plane. 6v/md

and

fI(prol.

d) ^are the non-dimensional combination

occurring naturally

ⁱⁿ the force

equations.

^The

indexes (k, ^r, s) ^{do not}

play

any role in the computa- tion.

2)

^We determine the essential parameters ^{of the}

problem. According

^{to AI-A3,} the dimensionless

expressions

^{that one can form to} characterize the system ^{are the}

following :

a)

The reduced

amplitudes xi/d, Yid

(i ⁼ m,

n)

^of

the shear stress.

b)

A dimensionless number P built with _m, the

frequency appearing

ⁱⁿ

(2) (see ^{[6] (the}

inverse of P is also known as the Ericksen

number)) :

c)

^The

angle

^T between the direction of the vector b and the x-axis

(Fig. 3).

So, ^{the set of} dimensionless parameters characteris- tic of the

instability

^{will be}

3)

^We

develop M(S)

^{in a}

Taylor expansion

^around

xi/d

⁼

ydd

^{= P = 0 and in a Fourier}

expansion

above ’P. This is allowed

by hypothesis

^{A2 and}

by

the

spatial periodicity

of the rolls. So we get :

702

The _upper limits in the sums above a,

P,

y,

ð, Jl

^{and k}

must be chosen from case to case

according

^{to the}

desired amount of information. We show in the next section that it is

possible

^to

simplify

M(S)

by

^{use of}

the

symmetries

^{of the} system.

2. 3 THE SYMMETRIES. - When no shear is

applied,

the system ^is invariant under _any rotation or reflec- tion in the

(xy)-plane.

^{What are the}

symmetries

^when

the nematic is submitted to a

periodical

^{shear ?}

We have to find out among the

changes

of frame allow- ed

by

^the

symmetries

^{of the}

nematodynamic

equa- tions (i.e. rotations, inversions, change ^{of m’s}

sign

and their

combinations)

^{the one which}

gives equiva-

lent border conditions, ^i.e.

equivalent

shear motion.

There are two ways ^to

satisfy

^this

requirement.

2 . 3 .1 The geometrical symmetries ^{in the}

(xy)-plane (this

^{means the}

plane

rotations and

inversions).

We have to retain

only

^those

satisfying :

with

In this

expression,

^{r is a time}

phase

^{and 8 takes} the values ± ¹

according

^{to whether}

R((p, e)

^{includes an inversion}

or not;

x;, y; (i

^{= m,}

n),

cp and r have to be determined

by solving

^this system.

2.3.2 The sign

change of co.

^- Instead of

developing

the shear for the

trigonometric

functions with

positive frequency

to, ^one may do so with

negative frequency -

^{co. This can} be written as :

The left side of this relation has been

expanded

^over

positive

^{and the}

right

part ^over

negative frequencies. By

use of

trigonometric

^function

properties,

^the

sign

of w has been put ^{in front of the}

right

^member.

The most

general

symmetry ^{will be a} combination of both

preceding

^{cases. This means :}

il takes the value ± 1,

according

^to the selected

sign

^{of the}

frequency.

^Since

(6)

^{has to} be valid for any time t,

the identification of the coefficients in front of cos

(kmt)

^{and sin}

(kmt) (k

= m,

n)

^{leads to a} system ^for

x,., xm Y’, Y’

^{i and} cp whose solutions

give

^all

possible symmetries

^{for the} considered shear. P’ is

equal

^to tIP

(since

P -

l/ro)

and ’P’ is deduced from the transformation of the vector b

(see Fig. 3)

^{under the rotation}

R(cp, s q) :

Y’ ⁼

B q(T

⁺

’P).

^{In the}

following,

^we shall write S’ for the transformed parameters

After a transformation like

(6),

^the system

(3)

^becomes

Comparing (3)

^and

(7),

^{one finds}

A careful look at the

symmetries existing

^{under the shear stress}

(1)

^shows that, for the most

general

^{choice of}

Applying

^these

symmetries

^to

equations (8,

5)

one gets ^the reduced form of the coefficients Muv

(their

^{detailed form and an}

example

^of calculation is

given

ⁱⁿ

Appendix).

The threshold is obtained

by putting

^det

M S

^{= 0, in order to} get ^a

non-vanishing

solution of the linear

homogeneous

system

(3).

3. Thresholds for some

special

^shears.

Since the

complete

threshold is rather cumbersome,

we shall consider

particular

^cases and show that additional

symmetries

^{lead to very}

simple expressions.

3. I ELLIPTICAL SHEAR. ^- In order to get ^an

elliptic

shear motion, ^we put xn ⁼ y,, ⁼ 0 and m ⁼ 1 :

Resolution of

equation (6)

^{in this case}

gives

the follow-

ing supplementary

symmetry :

It is worth

emphasizing

that the coefficients

A ",k

and

B;:ðll

^are

^dependent

^{on the roll}

wavelength.

So a result obtained for a

particular

^{case cannot be}

generalized

^{to different}

geometries

^without

checking

that no drastic

wavelength

^{variation occurs. A recent}

experiment [5]

has shown that a

complete change

^of

amplitude

^values

(for example going

^{from an}

ellipse

to an

eight

(see

below)) gives

^{rise to rolls of} very

different

shape,

^{so the} coefficients

appearing

ⁱⁿ MUV(S) may

considerably change.

^For

elliptical

shear,

this remark does not introduce an

important

^restric-

tion since the

wavelength

^{variation is, at} this order of

computation, negligible

^{when the}

ellipticity

^varies

[3].

^{Thus we are allowed to use} the additional _sym- metry

occurring when x,

⁼ y 1 in order to reduce

M(S) ; solving (6)

^{for this}

special

^{case, we} get :

If, ^{as was done in}

[7],

^we decide that the result is

good enough

^{when we cut all terms in nO with n} > 2,

we get the threshold

published

in that reference. Let

us recall that this lead also to the fact that P vs.

XI/YI

^{is a} step ^{curve with a}

jump

^of

n/2 when x i

⁼ YI.

JOURNAL DE PHYSIQUE - T. 46, ^mo 5, ^MAI 1985

Thus the

symmetries

^{enable us to} get ^the

complete

form of the threshold

(in ^[7]

^{the symmetry method}

was used to get ^the

higher

order but the lower one was obtained

by

^{a classical}

computation).

^{It also}

allows us to say that, ^as

long

^as it remains

possible

704

to describe the second transition

(i.e.

from rolls to

squares) by

the appearance of ^{a new} roll set, ^the second threshold

equation

will fulfil the same res-

trictions

imposed by

^the

symmetries

^{and so will have}

the same form. It is worth

noting

^{that even a semi-}

quantitative

determination of this second threshold

by

usual methods would involve cumbersome cal- culations.

3.2 EIGHT-SHAPED SHEARS. - Another

interesting

case is obtained

by putting

xn = ym ⁼ 0

(or

x. = yn ⁼

0)

ⁱⁿ

equation (1).

This leads to the

following plate

^motion

(Fig. 4)

(m

odd, n

even).

In the

following,

^{we shall}

design

^this type ^of

figure by

^«

eight-shaped

^{shear ».}

Experiments

^{have shown}

that the

wavelength

^{is not} very sensitive to the varia- tion of shear

amplitudes [5],

so we can use the limit

cases xm ⁼ 0

and yn

^{= 0 to} get ^more restriction on

M" ; x. ⁼ 0

(y.

⁼

0)

introduced in

(6) produces

^a

new symmetry :

Fig. ^{4. -} Examples ^of eight-shaped ^{shear motions. We} give

here the figures ^one obtains for (m, n) = (1, 2) ^and (m, n) = (1, 4).

This leads to the

following

reduced matrix

M(S)

Let us notice that in the case of an

eight-shaped

shear, ^{P is}

proportional

^to xm yn ;

by calculating

det

M(S)

^{= 0, we} get

(Fig. 5)

(For

the sake of

simplicity

^we have included the

T-dependence

^{in the} coefficients a’ and

b’.)

For a

given frequency,

^the

instability begins

^at

the lowest shear

satisfying (9).

^{1’his leads to the}

minimization of

x’y’ld’

^with respect ^{to T when} this

angle

^{is no}

longer imposed by

^{some external}

device but left free.

Doing

this, ^{we obtain an}

implicit equation

^{for P,}

independent

^{of the shear}

amplitude

and of the

frequency.

^The

experiment [5]

^{shows that}

P remains constant over a

large

^{domain of shear}

amplitude

^{and that our threshold}

equation

^well

reproduces

^{the observed} data. The

shape

^{of the}

threshold curve in the case of

« eight

shear » is

pictured

ⁱⁿ

figure

^5.

Fig. ^{5. - Threshold curve under} eight-shaped shear motion.

A delimits the uniform and the roll regimes. ^{B is the roll-}

square threshold.

sm

and sn

^can take the values ± 1; xm

and xn

^are

chosen so as to be of the ^same

sign.

The resolution of

equation (6)

with these shear

amplitudes

^{leads to the}

following

additional sym- metries :

Fig. ^{6. -} Flower-shaped shears. At given frequency co/2 n

and for fixed (m, n), ^{the threshold curve} (11) depends ^{on the}

ratio r ⁼ (sn xn)/(sm xm). ^We give ^{here some shear} figures

one gets ^{for m = 1 and n = 4} by varying ^r.

It is

interesting

^to observe that when the

product

Sm Sn

changes

^its

sign,

^the symmetry ^{of the shear}

figure changes.

^{In fact, a}

figure with Sm Sn

^{= + 1}

has ^a rotation axis of order

(m - n) ;

^if sm sn = - 1, the order of rotation axis is

(m + n). Figure

⁶ illustrates this

change

^of symmetry ^{for a}

gradual

variation of the ratio ^{r =}

(Sn xn)/(sm xm).

If we assume

again

that, ^at this order of calculation,

the roll

wavelength

^is

nearly

^{constant, we can use}

the limit case of the

ellipse

^to

simplify

^{the matrix}

M(S).

^{Let us consider the case} xn ⁼ 0; this leads to a certain threshold which has to be the

elliptical

one for a motion

frequency

^{of w’ = mm.}

Applying

the same argument ^for xm ⁼ 0, ^we

finally

get

Before

writing

^{this we have}

replaced

^{the sums above k}

by

the introduction of a

T-dependence

^{in the coeffi-}

cients. One can observe that the threshold is invariant under the whole symmetry group

previously

defined,

and in

particular

under the rotations of order (m ±

n).

Anew derivation with respect ^{to IF will lead to an}

implicit equation

^for P,

independent

^{of the}

magnitude

of the shears and of the

frequencies.

The

shape

of the threshold surface

(11)

ⁱⁿ

phase

space

OXm

^{P is}

given

ⁱⁿ

figure

^{7. The}

deep valley

shown is due to the annulation of the term

(sm mx;

+ sn

nx;), leading

^{to a} sixth order threshold in _{x;, y;.} This can be

interpreted

^{as a}

stability

^increase

of the sheared nematic

liquid

^{when the}

amplitudes

are

approximately

^related

by xm

⁼

(n/m) xn.

^{In this}

case, the threshold, up ^to the sixth order, ^reads

706

Fig. ^{7. -} Shape ^{of threshold curve under} flower-shaped

shear.

P is of third order

in xi (i

= n4 n), ^{as in case of}

eight- shaped

shear. Shear motions which

give

way ^to such thresholds (P -

xt)

^{seem to} be related

by

^some

common characteristic. Let us define the ^area of a

figure

^as

being positive

if the borders are followed clockwise and

negative

^otherwise

(see Fig. 8). Using

this definition,

figures

^with

x. - (n/m) x’

^{have a}

zero area since the total area of the ^«

petals

^{» is}

almost

equal

^{to the} centre area, but with an

opposite sign.

4. Conclusions.

In this _paper we have shown that symmetry ^conside- rations lead to some

interesting predictions

^{about the}

instability

^threshold

appearing

ⁱⁿ

periodically

^sheared

nematics : ^we have studied both the

shape

^{of the}

threshold curve and the orientation of the

instability

induced structure. In all the considered cases the structure orientation at threshold is

independent

of the variation of the shear

figure

except ^{for some}

jumps.

Moreover, ^{we do not} need any detailed infor- mation on the

regime

which becomes unstable.

In the case of a transition from uniform to roll

regime

followed

by

^a transition from roll to square flow,

this remark shows that both threshold curves have

Fig. ^{8. - The «} petals » ^{of the shear} figure ^{are followed} clockwise, ^{so their area is} positive; ^{the centre of the} figure

is followed counterclockwise and thus is negative. ^{The total}

area; ^when

x’ - (n/m) x;,

^{is almost zero} (here (m, n) = (1, 4)).

the same

shape (only

the numerical constants can be

different).

We want to

emphasize

that the method can be

applied

^{to other}

problems.

^It may be

particularly

useful to get

qualitative

^{ideas of threshold curves in a}

lot of involved instabilities. The

only required

^condi-

tion is that there is

enough

information to determine the

right

^{set of} parameters

entering

^the

problem,

their order of

magnitude

^{and the} symmetry group

acting

^{in each case.}

Finally

^we propose this method

as a useful tool to

qualitatively

^{solve a wide} range of symmetry

breaking problems.

Acknowledgments.

We are

grateful

^to F. Reust for his

help

^{and his}

assistance in

developing

^the symmetry considerations.

We thank E. Dubois-Violette, ^E. Guazzelli, ^E.

Guyon,

P. Pieranski for fruitful discussions and for their

hospitality

in Paris and

Orsay.

^We

acknowledge

^the

referees for their useful comments.

During

^{the matu-}

ration of this paper, ^we have had

stimulating

^talks

with A.

Bay

and M. Jorand. We are indebted to S. Herranz, N. Leiser for technical assistance and to S. Moch for a careful

reading

^{of the}

manuscript

Let us notice that we have retained

only

^terms up ^to third order in _{xi, yi}

(i

^{= m,}

n).

^{P is at} least of second order in _{x;, yi} as it can be verified

by inspection

^of

M(S).

^{We shall} show, ^{with an}

example,

^{how it is}

possible

^{to obtain}

this reduced form; ^for instance, ^{a term like}

must vanish. Let us first evaluate this term after a rotation of _n, with the

help

^{of S’}

(see

^{the table} in §

2. 3) :

We shall now evaluate the left member of

equation (8) by using

the table of

symmetries given

in § ^{2. 3 :}

So

equation (8) implies M(S’)

⁼

M(S) ;

this leads to

References

[1] PIERA0144SKI, ^{P. and} GUYON, E., Phys. Rev. Lett. 39 (1977)

1280.

[2] DREYFUS, ^{J.-M. and} GUYON, E., ^J. Physique ⁴² (1981)

283.

[3] GUAZZELLI, E., Thesis, Orsay (1981).

[4] GUAZZELLI, ^{E. and} GUYON, E., ^J. Physique ⁴³ (1982)

985.

[5] GUAZZELLI, ^{E. and} KOCH, A.-J., ^J. Physique ⁴⁶ (1985)

673.

[6] DUBOIS-VIOLETTE, ^{E. and} ROTHEN, F., ^J. Physique ³⁹ (1978) ^1039.

[7] SADIK, J., ROTHEN, F., BESTGEN, W., DUBOIS-VIO- LETTE, E., ^J. Physique ⁴² (1981) ^915.

[8] ^DE GENNES, ^P. G., ^The Physics of Liquid Crystals (Cla-

rendon

Press)

^1974.