Permeative and hydroelastic flow in smectic A liquid crystals

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Permeative and hydroelastic flow in smectic A liquid crystals

R. Bartolino, Geoffroy Durand

To cite this version:

R. Bartolino, Geoffroy Durand. Permeative and hydroelastic flow in smectic A liquid crystals. Journal

de Physique, 1981, 42 (10), pp.1445-1451. �10.1051/jphys:0198100420100144500�. �jpa-00209336�

Permeative and hydroelastic ^{flow in smectic A} liquid crystals

R. Bartolino (*) and G. Durand

Laboratoire de Physique ^des Solides, ^Bât. 510, Université de Paris-Sud, ⁹¹⁴⁰⁵ Orsay, ^France

(Reçu ^{le 7 avril} 1981, accepté ^{le 9} juin 1981)

Résumé.

Nous calculons l’écoulement et la déformation élastique d’une lame de smectique ^{A idéal} comprimée

sinusoidalement entre deux membranes parallèles

couches. A basse fréquence, ^la perméation apparaît ^dans

les deux couches limites prévues précédemment. ^{A haute} fréquence ^la perméation disparaît. L’écoulement est

pure distorsion hydroélastique, où les couches smectiques ^sont gelées ^{dans la matière. La} partie élastique ^de l’impédance mécanique de la lame smectique ^{subit une} relaxation, augmentant ^de 20% par rapport

régime élastique ^{de basse} fréquence. ^{Une tension} de surface finie doit supprimer les couches limites de perméation ^à

basse fréquence.

Abstract.

We compute ^the coupled ^{flow and elastic} distortion of

ideal smectic A liquid crystal normally squeezed ^{between two} oscillating plates parallel ^{to the} layers. ^{At low} frequency, permeation ^{occurs within the} boundary layers ^as already predicted. ^At high frequency, permeation vanishes. The flow is

pure hydroelastic distortion, with the smectic layers

frozen » inside the matter. The elastic part ^{of the} mechanical impedance ^of

the smectic slab undergoes

relaxation, increasing by ²⁰ % ^{above the} purely elastic low frequency regime. ^Surface

tension will suppress the permeation boundary layers ^{at low} frequency.

Classification

Physics ^Abstracts

61.30

62.40

46.30M - 47.55M

Each layer ^{of a smectic A} liquid crystal [1] ^{is a two}

dimensional fluid. In defect-free samples, ^{a flow nor-}

mal to the layers ^is generally associated with layer

motion itself. For fixed layers, ^{a weak} permeative [2]

flow can be induced by ^{a pressure} gradient ^normal

to the layers. ^{A D.C.} squeezing flow has been _pre-

viously ^described [3], ^{for the case of a non ideal smectic}

A containing ^so many defects that the layer ^number

is not a conserved quantity. ^An interesting prediction

of this model is the existence of a permeation boundary layer ^{close to the} boundary plates, ^{across which the}

pressure gradient necessary ^to induce permeation ^can

relax by inducing lateral flow. When the smectic slab is thicker than the boundary layer, the flow should be

permeation ^limited. Experimentally, ^to observe such

a behaviour, ^an oscillating squeezing ^flow [4] ^{is much}

easier to realize. In this paper, ^we have extended the D.C. model of reference [3] ^{to the case of A.C.} squeezing

oscillations. Assuming ^small amplitude oscillations we can keep ^{the ideal} (defect-free) model for the smectic A.

We compute first the flow and the layer distortions

versus frequency. ^{We then} derive the expression ^{of the}

transmitted force through ^{the smectic} layers, ^{i.e. the} quantity of interest for an experimentalist ^{who can} easily ^{measure the} mechanical impedance ^{of a smectic}

slab versus the frequency [5].

(*) On leave from Universita degli ^{Studi di} Calabria, Diparti-

mento di Fisica, Arcavacata di Rende, Cosenza, ^Italia.

Our model is the following : ^{we consider a smectic}

slab of thickness 2 d squeezed ^{between two} parallel

membranes of large size L > d (Fig. 1). ^{The two}

membranes vibrate along Oz, ^{normal to} the smectic

layers, ^with amplitudes ^{+ ô} exp(iwt).cos qx ^{x, with}

qx

x/L. ^{We restrict our} analysis ^{to an} incompres-

sible two dimensional flow in the plane (x, z). Using

the same notations of reference [3], ^{we call u the z} displacement ^{of the} layers, Vz and Vx ^the components of the mass velocity, p the specific ^{mass, p the excess}

pressure, B the smectic layer compression ^elastic

constant, 1 ^an averaged viscosity, ^{Â the} permeation

constant. The edges of the smectic drop ^{form a free}

surface. We neglect the surface tension and take p

⁰ (and ^u

0) ^{at the} edges. ^{We are} interested ^{in « thick}

samples, where the thickness d is larger than the boun-

dary layer thickness 1

(mL )1/2 (m ^{is a molecular}

Fig. ^1.

^{The cell} geometry.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198100420100144500

1446

length). ^{In this case we can} neglect ^curvature elasticity compared ^to layer compression elasticity.

The flow in the smectic slab is described by ^the

Navier-Stokes equations :

where _g

B ô2u/ôz2 is the elastic force density ^asso-

ciated with layer compression. ^We describe the viscous

coupling between flow and layer distortion with the

permeation equation :

The boundary conditions on the _upper and lower mem-

branes are :

(sticking condition for the flow) ^and

and

(no permeation through ^the membranes).

1. The simplified ^low frequency ^model. - We, ^first study ^{the non inertial} regime, assuming ^co «n/pd2

(~ 103 ^{to 105} Hz) ^{so that we can} drop the acceleration terms. The incompressibility condition div V

0 results in

which allows the pressure ^to be eliminated. Using (1), (3) ^and (5), and the thickness condition L > d,

one obtains an equation ^{for u}

which on integration ^becomes,

where we call m2 = À YI ^{and ÂB = m2} BI YI

m2/i.

m is a molecular length. Because of the symmetry ^of the problem, /3s

^{0. The} solution of (7) ^{is the} super-

position ^{of the} particular ^{solution u}

oc(x, t) ^{z and}

the general solution of the equation,

which becomes after a Fourier transform :

In order to simplify ^the following, ^we drop the obvious

x and t dependence ^{on u and V. To obtain} (6), ^{we have}

used the low frequency approximation ^wT 1, ^and 1 qz 1 > qx. With B - 10’ and il

^0.1 cgs, ^one finds

’t ~ 10- 8 S so that the low frequency approximation

allows a wide range of frequencies ^to be considered.

We shall discuss later the thickness condition.

For a given frequency ^Wt, solution of the dispersion relationship (8b) gives ^two purely imaginary (diffusion like) ^{roots :}

and the general solution of (8a) is, because of the ± ^z symmetry :

At low frequencies (Wt mqx), ^{the two roots of} (8b)

have the same modulus 1 qz 12 = 1 q’ z 2

qxlm, ^which

is the inverse squared ^{thickness of the} permeation boundary layer [3]. ^At. high frequencies (cor > mqx)’

the two roots are :

qZ is related to the diffusive motion of the smectic

layers, under the elastic restoring ^force proportional

to the curvature qX, ⁱⁿ the absence of permeation.

We shall call this motion the ^« hydroelastic » ^mode.

qz is related to the diffusive motion of the smectic

layers ^{due to} the pure permeation mechanism, ^{in the}

absence of flow. It is the ^« permeation » ^{mode. In}

contrast with the low frequency regime, ^at high ^fre- quencies, ^the permeative ^and hydroelastic ^{modes are} decoupled. ^{For wT~‘ 1, their} spatial ^extension along ^z

is comparable ^{with L} (hydroelastic mode) ^{or m} (per-

meation mode). ^{As we} have assumed _{cvi «} 1, ^the

condition ( qZ I > qx ^{is satisfied even for the} long ^wave- length hydroelastic ^mode.

1.1 CALCULATION OF THE DISTORTION.

We can now compute explicitly the distortion u versus cor,

using ^{the three} boundary conditions on the membra-

nes. (4a) ^{becomes :}

This can be transformed, using ^the dispersion ^rela- tionship (8b) ^{to :}

iwTa+

(4b) ^and (4c) give :

The solution of I

where

So that the distortion u can be written as :

1.2 CALCULATION OF THE VELOCITY FIELD.

V’ z

V z(Z) ^{cos qx x} is obtained directly ^{from the} permeation equation (3)

We derive Vx

Yx(z) ^sin qx ^x from the incompres- sibility ^condition

1.3 REACTION OF THE SMECTIC SLAB ON THE MEM- BRANES.

The normal stress exerted by ^{a smectic} layer ^{on a lower z} layer ^is

On the lower plate, because of the boundary ^condi-

tion (4a),

To compute p

p(z) cos qx ^{x, we use} the Navier- Stokes equation (2) ^{and the} previously computed Vx(z) ^and Vz(z) (Eqs. ^{(19) and} (17)). ^{We obtain :}

and using (8a), ^the reduced normal stress is simply :

1.4 DISCUSSION.

Contrary ^{to reference} [3], ^we

have assumed no permeation ^on the membranes. The first point ^to consider is why there should be _any

permeation ^{at all close to the} membranes. This follows

directly ^{from the} boundary conditions. On the mem-

branes, ^Ù V- (4c). ^{We also have} avz/az

⁰ (4a),

but êulêz ^{is non} vanishing, ^{because of the mechanical}

reaction of the smectic spring ^{to the} externally ^forced displacement ^{b. This results in}

close to the membranes and permeation ^{must occur.}

In discussing ^the properties ^{of the} squeezed ^smectic

slab the most relevant parameter is the ratio of the

sample ^{thickness to the} boundary layer ^thickness.

Of secondary interest is the reduced frequency ^(DT.

Let us discuss here the case of thick samples

For zero frequency, ^{V and û vanish. u} obeys ^the

standard equilibrium equation

Apart ^from the cos qx x dependence,

^{is the} simple homogeneous distortion :

The transmitted normal stress on the lower plate ^{is :}

The mechanical reduced ^« impedance » ^{of the}

smectic slab is

In principle, ^{it is then} possible, ^{for an ideal} sample, ^to

deduce the unidimensional elastic modulus B in a D.C.

1448

squeezing experiment, by measuring the normal stress

S, ^{versus the} applied ^strain ôld, ^{in the zero} frequency

limit.

It is interesting ^{to look at the} velocity ^{field for} vanishing Wt, ^to make a comparison with the distor- tion profile. Vz and Vx ^are of the order of iw03B4 and

irobldqx’ We have then plotted ^on figures ^{2 and 3 the}

normalized thickness profiles ^of Vz(z)/(irob) ^and Vx(z) dqxlirob ^{for 03C9T}

^{0. One sees} clearly ^{that the} boundary layer region ^{is the} only ^{one where a standard} hydrodynamical ^{flow occurs, with an} exchange ^bet- ween Vz ^and Vx. The maximum on the Vx profile corresponds ^{to the} largest slope ^of VZ, ^{where the} VZ profile joins ^{back to} the linear profile ^of u(z). ^{In bet-}

ween the two boundary layers, ^{we observe a} region

of uniform distortion along ^{both x and z} directions, where Vz - ^{ù and no} permeation ^{occurs. This} region corresponds ^{to that} part of the smectic slab which

undergoes ^a solid-like ^« hydroelastic » distortion.

Fig. ^2.

Normalized velocity profiles ^Re (Vz/iwb) ^for

⁰

and

The linear dotted line gives ^for comparison ^the

frequency normalized distortion u/b. ^{1 is the} permeation boundary layer thickness for

0. Only

half of the profile ⁰

^d

is represented, ^{the other} part ( - d

0) ^is symmetrical.

Fig. ^3.

Normalized velocity profiles ^Re (Vx dqx/irot) ^for

⁰

and

1 ils the permeation boundary layer thickness. The onset of surface tension will force

constant curvature at

frequency ^and suppress the permeation boundary layer.

Increasing ^the frequency, ^{one finds a} regime ^where

the amplitude ^{of the} permeation ^{flow is a maximum.}

2

2

This occurs when ù _ M2 ô2U T TZ2

^{i.e. for} _{(WT)p ~}

mqx. ^qx At this frequency the distortion profile ^is already

Fig. ^4.

Normalized profiles ^{for the}

velocity V ^{and the} layer velocity û, ^for

mqx when permeation ^{is at}

^maximum.

The permeation boundary layer thickness is larger ^{than 1.}

distorted from the elastic profile (25), ^{as shown in} figure ^4. Increasing further the frequency, ^the per- meation amplitude ^decreases roughly ^as mqxlwt, ^which

means that icou and Vz ^{have the same thickness} profile.

The smectic layers ^{are «} frozen » in the smectic mate-

rial, since Ù - Vz. The flow is purely hydroelastic.

Another important frequency is that where the z

extension of the hydroelastic ^{mode is} comparable ^with

the thickness. From (11a), ^{this occurs for} (wT)H ~ qx ^{d 2.}

Of _course, (Wt)H ^is larger ^than (wt)p, ^{because we are} considering ^{the thick} sample ^case (qxlm) ^{d 2} > 1.

Above (wt)H’ ^{where the} viscous forces dominate the elastic forces, ^{the flow must be the same as in a} simple

viscous fluid. One can derive the velocity profile ^from

the limit of (17) ^and (19) ^at large ^{wT. It is} simpler ^to

recalculate it from the high frequency limit of the

equations of motion. For high frequency, ^ù

Vz,

from (3). (6) gives 84ul8z4

a4 vZ/az4 - ^{0. From} (23)

we see that _p is independent ^{of z,} which leads to a

standard Poiseuille flow for Yx. The solution is simply :

and

This high frequency Poiseuille profile is shown in

figures ^{2 and 3.}

1. S MECHANICAL IMPEDANCE OF THE SMECTIC SLAB.

-

For our practical problem, ^{it is} interesting ^{to consi-}

der the frequency dependence ^of the normalized impe-

dance Z of the smectic slab, previously defined in

equations (27) ^and (23). ^{For Wt} well below (Wt)p, ^one

obtains the following expansion :

where J1 ^is the dimensionless thickness

We are now considering ^{the case of} large J1 ^{so that} (30)

reduces simply ^{to :}

In addition to the elastic contribution 1, ^{we should} observe a viscous contribution proportional ^{to the}

membrane velocity ^{icvô. In the case of a} simple ^viscous fluid, ^one should have (see later) :

(31) shows that in _presence of permeation, ^{the smec-}

tic slab behaves mechanically ^{as a} spring in series with

a simple viscous fluid squeezed between the same two

membranes, ^{the thickness} of which is (ld)1/2, the square root of the product ^{of the} sample thickness d by ^the permeation boundary layer thickness l= (mlqx)1/2.

For wt larger ^than (wt)H’ the normalized impedance

is as follows,

Using ^the high frequency ^limit (11a) ^for qz, ^one obtains the expression :

the imaginary part ^{of Z} diverges ; ^{since for} large ^fre-

quency the viscous forces dominate the elastic forces,

this imaginary part ^{must be the} normalized impe-

dance of a simple viscous fluid slab, ^{in the same} geome-

try. ^{This can} be checked directly by solving ^{the con-}

ventional ^« thrust bearing » problem ^{in our} geometry.

It is interesting ^to understand the 20 % increase in the elastic part ^{of Z. This} high frequency strengthening

is simply ^{related to} the fact that the smectic layers

are « frozen » in the material. The high frequency

distortion profile u(z) ^{does not} minimize the elastic

Fig. ^5.

Normalized impedance of the smectic slab

the

frequency. Above the maximum permeation frequency (Wt)p ’" ^mqx,

the friction decreases and the apparent elastic modulus increases

by 20 % (see text).

free energy anymore. The apparent reduced elastic constant can be estimated in the following way,

using ^{for u} equation (28), ^one readily ^{finds Z} = 5

^1.2.

The frequency dependence ^{of Z} is shown in figure ^5,

which demonstrates the relaxation of the apparent elastic constant above (wT)H’ ^{from 1 to 1.2.}

2. Discussion.

2.1 THE INERTIAL REGIME.

In the previous calculation, ^{we have} neglected ^inertia

terms. The results represent ^{the wT}

0 limit of the

more general ^{solution. It is useful to} understand what

happens when inertia is taken into account. As usual, dealing with small amplitude oscillations, ^{we can}

linearize the Navier-Stokes equations (1) ^and (2), writing :

We now introduce the maximum second sound

velocity ^c by

In addition to the low frequency ^and thickness condi- tions wT 1 and qz > qx, ^we introduce the additional condition :

which means that, ^for the molecular frequency T - 1,

the second sound wavelength ^{is much} larger ^{than the}

molecular length ^{m. We} finally obtain the new dis-

persion relationship :

The second sound velocity, ^{for a} distortion of wave

number q(qx, qz), ^{is c} qx qz ^{With our} assumption ^{of a}

q2

thin sample, ^{we must} have q - qz. The frequency

ws

cqx is just ^the frequency ^{of the} second sound ^wave associated with the layer distortion. As long ^{as úJ} ws, the previous discussion remains valid. Close ^{to w}

_ws, the membrane vibrations will excite, at resonance, second sound waves. In practice, úJs is of the order of 104 Hz. The resonant excitation of second sound

waves at úJs has not yet ^been observed, although ^it

could correspond ^{to the} unexplained ^low frequency

mode found in Rayleigh scattering [6].

We can try ^to estimate the influence of inertia on Z,

when increasing ^the frequency, well below _ws Resum-

1450

ing the calculation of the pressure (Eq. (22)), ^{we find}

for the reduced normal stress :

instead of a (Eq. (23)). ^{The detailed} expression of Sz

is clumsy. ^Note only that inertia introduces W2 terms in the mechanical impedance of the slab.

2.2 THE ASYMMETRIC CELL.

To measure

the mechanical impedance ^Z, experimentally ^{it would}

be more convenient to excite the vibrations of only

one (the upper) ^{membrane, and to deduce Z from the}

transmitted force exerted on the other static (the lower)

membrane. One can find the solution of this non

symmetric problem by superposition ^{of the} previous

« symmetric » solution with that of the « antisym-

metric » problem, where the two membranes are

assumed to vibrate in phase ^{as :}

The superposition ^{of the two solutions} gives ^the boundary conditions of a vibrating upper plate, ^with amplitude ² 03B4, ^{and of a} static lower plate ^{with zero} amplitude.

So long ^{as we} remain in the non inertial approxi- mation, ^the

antisymmetric » solution is the uniform distortion u(z)

u(d), Vz

iwô, V x

^{0. The trans-}

mitted force across the smectic slab remains the same.

In the non inertial regime, ^{we must} also take into account the inertial force associated with the oscil- lation of the centre of mass. This results in an addi- tional _pressure term :

i.e. to an additional term in the mechanical reduced

impedance :

For the second sound resonance ws, this remains a

weak correction - q’ d 2 ^1, because the sample

is thin.

2. 3 SURFACE EFFECTS.

In our model, ^{we have} neglected the surface tension at the free edges ^{of the}

smectic slab. This can be justified physically ^{if we use a} surfactant, ^for example, ^or if the free edges ^{break into} steps ^to allow the smectic layers ^to glide ^{across the}

side surface of the sample. ^{It is} interesting ^{to estimate}

what happens ^{in the} presence of a smooth curved edge

surface with surface tension.

Assume as previously that the free edges ^{are vertical}

in the absence of membrane deformation. At low

frequencies, ^the velocity profile Vx is shown in figure ^3.

The lateral displacement ^{of the} edge ^{is :}

The Laplace pressure due to surface tension is

since we restrict our model to two dimensions.

V, ^{is the} superposition ^{of the} hydroelastic part

i03C903B4 Z/d ^{and of a} Poiseuille-like flow within the _per- d

meation boundary layer ^{of thickness}

This Poiseuille flow has an amplitude v., - ^p

iwô 1 ^d

Writing ^the incompressibility ^{condition across the} boundary layer, ^{one finds a} contribution to the lateral

displacement of the order : _{y -} Lô/d ^{and a} Laplace

~

y ô

pressure P1 ~y. ^{Note now that, because m is a}

p m d

molecular dimension, y/m is of the order of B, ^the smectic layers compression modulus, ^{so that :}

The physical meaning ^of equation (46) ^is important.

Let us recall an argument from de Gennes [1]. ^The permeation boundary layer appears because the pressure gradient ^{normal to the} layers ^{can be relaxed} by ^a lateral flow at distance 1 from the membranes.

The lateral drop ^of pressure is primarily ^{due to the}

viscous drag. ^The Laplace pressure (46) is of the order of the normal elastic stress across the smectic

layers. ^From equation (31), ^we know that in the low

frequency ^{limit the} pressure gradient - ^B au g ^{~z is much}

weaker than the elastic ^stress. The permeation ^boun- dary layer ^{thickness must increase.}

To understand qualitatively ^{how this occurs let us}

consider the Navier-Stokes equation (2). Including

the Laplace surface term, the pressure is now :

Using ^the permeation ^condition equation Vz ~ À P/1

and the incompressibility, ^{we obtain :}

to be compared ^{with Lm} in the absence of surface tension. For zero frequency, 1 becomes infinite. This is obvious because hydrostatic equilibrium ^{in the}

presence of Laplace ^pressure implies ^{a constant cur-}

vature for the edge ^{of the} squeezed ^smectic slab, ^so

that the profile ^of figure ^{3 is not an} equilibrium profile.

For frequencies larger ^than ylr¡L (_ 102-103 Hz)

i.e. for

the model without surface tension remains valid.

There should be a critical frequency (COr)d ^smaller

than (wt)p ^{for which 1 - d, i.e. for thick samples,}

The permeation boundary layer should exist for

frequencies ^between (Wt)d ^and but ^must disap-

pear ^at low frequencies. Remember that from our

previous analysis, ^it disappears also above (W’t)H’ ^Of

course, ^a correct description ^{of what} happens ^between (Wt)d ^and (cor), ^{must take into account the} Vz ^flow

induced by ^{the non uniform} Laplace pressure. This results in capillary ^{surface wave emission. The} pro-

pagation of capillary ^{surface waves on a smectic A has} already ^{been discussed for an} unconfined geometry

by ^D. Langevin [7]. ^The problem ^{of their} propagation

and resonances in the confined geometry of the smectic slab _appears to be more complicated. ^{As we do not}

know much [8] about the real state of smectic free edges,

we shall not try ^to discuss this question ^here.

The conclusion of this section is that, ^{in the} presence of surface tension, ^the permeation boundary layers

must disappear ^{at low} frequencies. ^{For thick} samples they could be observable around the low frequency y/ ilL (i.e. ^{close to} the maximum permeation frequency),

and lead to the excitation of capillary ^{surface waves.}

This result demonstrates the important effect of surface

properties ^on the three dimensional permeative ^flow

inside a smectic A liquid crystal.

2.4 CURVATURE ELASTICITY.

In the simplified model, ^{we have} neglected nematic-like curvature

elasticity. ^{Is this} assumption ^{correct ? In} the absence of surface tension the largest ^curvature appears ^at the

edges ^{of the} permeation boundary layers. ^The ampli-

tude of the layer displacement ^u which is curved is

u - bl/d, ^so that the maximum curvature energy

density is of the order of :

We have used the relationship ^{K - Bm2 for the}

Frank curvature elastic constant K. The nematic curvature energy is always negligible. ^{The same result} obviously applies ^{in the} presence of surface tension,

since the edge ^{curvature decreases.}

3. Conclusion.

We have computed the flow and distortion profiles ^{inside a «} thick » ideal smectic A

slab, ^{submitted to a small} periodic layer compression.

In the absence of surface tension, permeation ^occurs

at low frequencies ^{in the two} boundary layers ^as already predicted. ^At high frequencies, the smectic

layers ^are frozen into the material. The flow is Poi- seuille-like. The elastic part ^{of the} mechanical impe-

dance undergoes ^a relaxation, increasing by ²⁰ %.

The inertial regime ^at higher frequencies should lead to a resonant second sound excitation. In the _presence of surface tension, ^the permeation boundary layers ^are suppressed ^{at low} frequencies. At intermediate fre-

quencies, they should lead to excitation of surface

capillary ^waves. Systematic observations of the free surface and measurements of the surface tension of a

smectic A are required ^to check which model should

apply.

Acknowledgments.

^We have benefited from dis- cussions with ^M. Kléman, ^Ph. Martinot-Lagarde ^and

J. Friedel.

References

[1] See for instance :

GENNES, ^P. G., ^The Physics of Liquid Crystals (Clarendon, Oxford) ^1975.

[2] HELFRICH, W., Phys. ^Rev. Lett. 23 (1969) ^372.

[3] ORSAY GROUP

LIQUID CRYSTALS, J. Phys. ^{C 1} (1975) ^305.

[4] BARTOLINO, R., DURAND, G., Phys. ^{Rev. Lett. 39} (1977) ^1346.

[5] DURAND, G., BARTOLINO, R., CAGNON, M., ⁱⁿ Liquid Crystals of

^{and two} Dimensional order (Springer Verlag, Berlin) ^1980.

[6] RICARD, L., PROST, J., ^J. Physique Colloq. ⁴⁰ (1979) ^C3-83.

[7] LANGEVIN, D., ^J. Physique ³⁷ (1976) ^737.

[8] LANGEVIN, D., ^J. Physique ³⁷ (1976) ^755.