Evidence of two new ordered smectic phases in ferroelectric liquid crystals

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Evidence of two new ordered smectic phases in ferroelectric liquid crystals

J. Doucet, P. Keller, A.M. Levelut, P. Porquet

To cite this version:

J. Doucet, P. Keller, A.M. Levelut, P. Porquet. Evidence of two new ordered smec- tic phases in ferroelectric liquid crystals. Journal de Physique, 1978, 39 (5), pp.548-553.

�10.1051/jphys:01978003905054800�. �jpa-00208786�

EVIDENCE OF TWO NEW ORDERED SMECTIC PHASES

IN FERROELECTRIC LIQUID ^CRYSTALS

J.

DOUCET,

^P.

KELLER,

^{A. M.} LEVELUT and P.

PORQUET

Laboratoire de

Physique

des Solides

(*),

Université

Paris-Sud,

^Bâtiment

510,

⁹¹⁴⁰⁵

Orsay,

^France

(Reçu

^{le 21 décembre} 1977,

accepté

^le

1 er février 1978)

Résumé. ²⁰¹⁴ A partir ^de diagrammes de rayons X d’un échantillon orienté ^sous

champ

électrique

de (R-)

chloro-2-propyl-p-hexyloxybenzylid6ne-p’-aminocinnamate

(HOBACPC), nous avons mis

en évidence deux nouvelles phases smectiques ^orientées

qui

apparaissent ^à plus ^basse température que la phase ferroélectrique Sm C*. Ces deux phases ^sont caractérisées par ^un ordre

pseudo-hexagonal

^à

l’intérieur des couches, mais elles diffèrent des phases ^Sm Bc par la direction d’inclinaison des

grands

axes moléculaires qui ^sont inclinés dans un plan

parallèle

^{à un} des côtés de la maille pseudo-hexa- gonale. ^{Une de ces nouvelles} phases présente ^{un ordre} tridimensionnel, alors que l’autre, qui apparait

lorsque

^l’on augmente ^la température, présente ^{seulement un ordre} bidimensionnel, les couches

smectiques étant presque totalement décorrélées. Cette structure est

probablement

^{reliée au} compor- tement

ferroélectrique

^{de cette dernière} phase qui ^est analogue ^{à celui} présenté par la

phase

^{Sm C*.}

Abstract. ²⁰¹⁴ From X-ray experiments performed ^{on oriented}

samples

^of

(R-)-chloro-2-propyl- p-hexyloxybenzylidene-p’-aminocinnamate

(HOBACPC)

by

^{means of an electric} field, ^{we have}

identified two new ordered smectic phases which appear ^at lower temperature than the ferroelectric Sm C* phase. ^{These two} phases ^are characterized

by

^a

pseudo-hexagonal

order within the

layers,

^but

they

differ from the Sm Bc

phases

by the direction of the

long

^{molecular axes which are tilted in a}

plane

parallel ^{to an} edge ^{of the}

pseudo-hexagonal

lattice. One of these new phases ^{shows a three-}

dimensional order, whereas the other _one, which appears when increasing ^the temperature, ^shows only ^a two-dimensional order, the smectic

layers being

almost uncorrelated. This structure is probably

connected with the ferroelectric behaviour of this last

phase,

^{which is} analogous ^to that exhibited

by

^{the Sm C*}

phase.

Classification

Physics ^Abstracts

61.30

The first evidence of the existence

of ferroelectricity

in

liquid crystals

^was

given by

^{R. B.}

Meyer et ^al.

in 1975

[1].

The authors studied a chiral

compound

which exhibits three smectic

phases :

^{two of them were}

identified as a classic smectic A

phase

^{and a smectic C}

with a helicoidal structure

(Sm C*),

^the

third, only produced

^when

cooling

^{the Sm C*}

phase, proved

^to

have a

Debye-Scherrer X-ray

pattern ^{similar to some} of the Sm B patterns, ^that

is,

^{it showed a}

small-angle ring corresponding

^to

Bragg

reflexions from the smectic

layers

^{and one}

sharp ring

^{at 4.5}

A [2]

^which

implies

order within the

layers ;

in the absence of further structural

investigation,

^this

phase

^was

named Sm H*

by

^{R. B.}

Meyer (tilted

^Sm

B).

From

optical observations,

^this ordered smectic

(*) Laboratoire associé au CNRS.

phase

appears ^to be helicoidal

just

^as the Sm C*.

A similar

electro-optical

^{effect can} be observed in the two

phases :

^under

application

^{of a}

sufficiently-high

electric

field,

^the helicoidal structure is

completely unwound,

^thus

producing

^a

macroscopic polarization

in the

sample.

^The critical field

E,,

^for

unwinding

^the

helix is

related to a torsional elastic constant, ^{to the}

macroscopic polarization

^{and to the full}

pitch

^{of the}

helix. In this

compound,

the critical field varies with the temperature in the Sm C*

phase,

^{from 600}

V/cm

at 94 °C to 6 500

V/cm

^{at 63.5 °C}

[1].

Since

then,

other ferroelectric

liquid crystals

^have

been

synthetized [3] ;

^{some of} them exhibit an

ordered smectic

phase analogous

^to that described

by

R. B.

Meyer.

^We have tried to obtain more infor- mations about the structure of such a type ^{of smectic}

phase by analysing

^the

X-ray

diffraction patterns ^of these

phases

^oriented

by

^means of an electric field.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01978003905054800

549

1.

Sample

^and

expérimental

^{device. - The}

X-ray experiments reported

here have been

mainly performed

on one

compound : (R-)

^chloro-2

propyl-p-hexyloxy- benzylidene-p’-aminocinnamate (HOBACPC).

From D.T.A.

(Fig. 1)

^and

optical observations,

^we established that this

compound

exhibits four smectic

phases :

FIG. 1. ^- Difïerential thermal analyser thermogram of HOBACPC.

The Sm III and Sm C* _appear to be helicoidal with

large optical activity,

^{whereas in the Sm IV}

phase

^this

activity

^decreases

drastically.

^{The last}

phase

^{has a}

large supercooling

range.

We recorded two different kinds

of X-ray patterns : i) polydomain

patterns ^{with a Guinier camera}

using

^a

heating sample

^holder

[4] (Â.

^{= Co}

Kal), ii)

patterns ^of

samples

^oriented

by

^{an electric}

field

(Fig. 2).

^The

sample

container is a

parallelepiped

cell is irradiated with a

point X-ray

beam focused

by

^a double bent

graphite

monochromator. The diffraction

pattern

is recorded on a flat film.

It is

important

^to

point

^out that in this

compound

the critical field E c is

especially

^low

[3],

^{so that the}

experiments

^{are easier.}

2.

X-Ray diagrams

of the smectic

phases.

^{- 2. 1}

POLYDOMAIN ^{PATTERNS. -} In the four

phases,

^we

can observe a

sharp

^small

angle ring corresponding

^to

the

Bragg

reflexion from smectic

layers (Fig. ^3).

^The

diffuse and broad

large angle

band characteristic of Sm A and Sm C

phases,

^{narrows in the Sm III}

phase

and there also _appears a

sharp ring

^{at d - 4.5}

A.

In the Sm IV

phase,

the diffuse band

disappears

^while

three new

sharp rings

^can be observed close ^to the first one.

FIG. 2. ^- Experimental device with electric field : a) sample, b) film, c) sample holder, d) gold electrodes, e) diffracted X-ray

beam, f) ^direct X-ray ^beam.

of 1.5 ^x 1 ^x 5 mm3 with two thin mica

single crystal

windows. A D.C.

voltage

^is

applied

^{between two}

gold

electrodes

separated by

^{1.5 mm. The}

voltage

^can vary from 0 to 2 000 volts. The electric field is

usually perpendicular

^{to the}

X-ray

^beam

(Cu Ka) (both

^{are -}

horizontal). However,

^{we can rotate} the cell around a

vertical axis

by

^{a maximum}

angle

^{of 35°. The cell}

temperature

^{is held constant within}

± 0.5

^{°C. The}

FIG. 3. ^- Powder pattems of the smectic phases of HOBACPC :

a) ^{Sm A} phase, b) ^{Sm C*} phase, c) ^{Sm III} phase, d ) ^{Sm IV} phase.

2.2 INFLUENCE OF THE ELECTRIC FIELD. ^- Because of the thickness

(1 mm)

^{of the}

sample,

^{we cannot obtain}

a

homeotropic

^{texture on} the mica windows. Conse-

quently,

^the

geometry

^under the electric field is rather

complex.

In the Sm A

phase,

the electric field has no

effect,

we then observe a pattern characteristic of a

poly-

domain

(Fig. 4a) :

^a

sharp

^inner

ring

and its second

order, corresponding

^to the reflexions from the smectic

layers (d -

^28.5

Á)

^{and one broad outer}

ring

(d ---

^4.5

Â).

FIG. 4. ^- Patterns of HOBACPC oriented by ^{an electric field} perpendicular ^{to the} X-ray ^{beam. The} sharp spots ^at large angles ^{come from} the mica windows : a) ^{Sm A} phase, b) ^{Sm C*} phase, c) ^{Sm III} phase, d) ^{Sm IV} phase.

In the Sm C*

phase,

when the electric field is

applied,

^the

rings split

^into two crescents

(Fig. 4b).

In

fact,

^{we have a texture in} which the smectic

planes

are

parallel

^to the electric field and where the molecules

are tilted with respect ^to the normal to the smectic

layers ;

^their

long

^{axes lie in a}

plane perpendicular

^to

the electric field

(as predictee by

^the symmetry pro-

perties

of the ferroelectric Sm C*

phase [1]).

^The

directions normal to the smectic

layers

^{are located on}

the

plane perpendicular

^to the electric

field,

^as repre- sented in

figure

^5.

Thus,

the maxima of the

Bragg

reflexions from the smectic

layers

^are

aligned

^{in a}

direction

perpendicular

^to the electric

field,

^whereas

the maxima of the outer

ring

^are

aligned parallel

^{to the}

electric field since in that

direction,

^{the outer}

ring

^is

always

ⁱⁿ the reflexion condition. When the electric field is switched

off,

^{we obtain a}

polydomain

pattern

as in

figure

^4a.

In the Sm III

phase (Fig. 4c),

^{the inner}

ring

presents the same aspect ^{as in the Sm C*}

phase,

whereas the

FIG. 5. ^- Texture of the ferroelectric smectic phases ^{of HOBACPC}

oriented by ^an electric field. The electric field (E) ^is perpendicular ^to

the X-ray ^beam (X).

outer

ring splits

^{into six} spots

equally spaced.

^{Two of}

them, lying

^{on an axis}

parallel

^to the electric

field,

^are

somewhat broader and more intense than the four

551

other spots. ^{This can be}

explained by

the fact that we

find the same texture effect as in the Sm C*

phase.

Moreover,

^we also have a fluctuation effect of the orientation of the axes

parallel

^to the smectic

layers

since the six spots ^{are not}

point-like

^{but crescent-}

shaped.

^{As a matter of}

fact,

^{we can}

produce

^{a better}

alignment

of the domains if we wait a

long

^time

(15 hours)

ⁱⁿ the electric field : the inner

ring disap-

pears and the six outer spots

sharpen.

^The spots ^also

sharpen

^{under a}

higher

electric field until it reaches a

saturation value.

Nevertheless,

^{we never obtain}

point-

like reflexions as in the diffraction

patterns

^of

single

domains of a Sm B

phase [2].

^{Let us} remark that we

only

^have

qualitative

information about the influence of the electric field on the

alignment

of the molecules in the smectic C and smectic III

phases,

ⁱⁿ

particular

we cannot measure any value of the critical field.

Finally,

^{when the}

sample

^is

supercooled

^{in the}

smectic IV

phase

under the electric

field,

^{the six}

reflexions seen in the smectic III

phase

^{remain at the}

same

place;

ⁱⁿ

addition,

^there appear four weak reflexions

corresponding

^to smaller reticular

distances, doubling.

^{the four}

spots lying

^out of the electric field direction

(Fig. 4d).

^{In this} case, the pattern ^remains the same when we switch offthe D.C.

voltage.

3. Structures of smectic fi and IV

phases.

^{- If we}

refer to the

powder

and oriented

sample

patterns ^of both

phases,

^{we can} establish that

they

^{are formed}

by

^a

succession of

layers

^{in which the centres of mass of the}

molecules are on a two-dimensional network.

Thus,’

the two

phases

^are very similar.

We

^{have to determine}

for each

phase :

1)

^The two-dimensional lattice within each

layer, 2)

The direction of the molecular axes with respect

to the normal to the

layer

^{and to the} two-dimensional unit

cell,

3)

^{The nature} of correlations between the succes-

sive

layers.

3.1 DETERMINATION OF THE TWO-DIMENSIONAL LATTICE WITHIN EACH LAYER. ^- The pattern ^{of the} oriented

sample

of the Sm III

phase

^{is formed of}

six

equidistant Bragg spots,

^the symmetry ^{of which}

seems to be

hexagonal.

^{In the Sm IV}

phase,

the intense

Bragg

spots ^{remain at the same}

place ;

^{we then} suppose that the structures of the Sm III and Sm IV

phases

ⁱⁿ

each

layer

^{are almost}

identical ;

^{the weak extra}

Bragg

spots

give

information about the mode of

stacking

successive

layers.

^{In both}

phases,

^the distance between nearest

neighbour

^{molecules is about 5}

A ;

this is the distance

usually

observed in the Sm B

phases.

3 . 2 DEFINITION OF THE DIRECTION OF THE MOLE- CULES. ^- The situation seems clearer for the Sm III

phase

since the ferroelectric

properties

^are similar for the Sm C* and Sm III

phases.

^{This means that the}

molecules are tilted with

respect

^to the normal to the

layer

and that the molecular electric

dipoles

^{lie in the}

same direction in the Sm C* as in the Sm III

phases.

Under an electric

field,

the molecular axes lie in a

plane perpendicular

^to this electric field whereas the real

hexagonal

network in each

layer

is oriented in such a

way that the electric field is

perpendicular

^{to one}

edge

of the unit cell. The structure of each

layer

differs from the structure of a

Sm Bc layer (Sm B

with tilted

molecules) by

the relative orientation of the

tilting plane

of the molecules with respect ^{to the}

hexagonal

lattice of each

layer :

^,

- in the Sm III

phase,

^a

plane

^{normal to the}

layer plane

^and

parallel

^to the molecular axes, is

parallel to

one

edge

^{of the}

hexagonal

two-dimensional lattice.

We then

suggest naming

^{such a structure}

pseudo- hexagonal

type

II layer

^structure

(Fig. 6b) ;

1

FIG. 6. ^- Tilt direction of the long ^{molecular axes with} respect ^to the lattice. a) type ^{1 : Smectic} Bc, b) type II : Smectic IV and Smectic III of HOBACPC (the ^arrows indicate the dipole direction).

- on the contrary, ^{in the Sm}

Bc phases,

^the

plane

normal to the

layer planes

^and

parallel

^to the molecular

axes is

perpendicular

^{to one}

edge

^{of the}

hexagonal

two-dimensional lattice

(pseudo-hexagonal

type

Ilayer structure) (Fig. 6a) ;

- the case of the Sm IV structure may be derived from the Sm III

phase. Thus,

^{it seems} reasonable to assume that the

layer

^{structure is the same for the two}

phases,

^the differences

coming

^{from the}

stacking

^of

successive

layers.

^{We shall see} below how these

assumptions

^are consistent with the

interpretation

^of

the

Debye-Scherrer

patterns.

3.3 NATURE OF CORRELATION BETWEEN SUCCESSIVE LAYERS. ^- In the absence of

patterns

of oriented

samples

^{in various}

positions,

^the

Debye-Scherrer

patterns

give

^more information about this

point.

The

powder

pattern ^{of the Sm IV}

phase (Fig. 3d)

^is

similar to a pattern ^{of a Sm}

Bc phase

^{which is charac-}

teristic of a three-dimensional order with fluctuations of

large amplitude

around the mean

positions

^{of the}

molecules. We can index the

pattern

with a C-face centred monoclinic lattice where c is

parallel

^{to the}

molecular _axes, and a and b are located in the

layer .plane (b being

the twofold

axis).

^{Two sets of lattice}

constants fit with the observed

Bragg

reflexions. The first one

corresponds

^{to a}

pseudo-hexagonal

type ^I

layer

^{structure with a}

(10.37 Â)

> b

(5.02 Á), f3

^{= 124.8°}

and c ⁼ 33.7

A;

^{but the c}

length

doesn’t fit the molecular

length

^{which cannot} exceed 30

Á.

The second set

corresponds

^{to the}

pseudo-hexagonal

type ^II

structure with a

(5.33 À)

^b

(8.52 A), fi

^{= 109.6°}

and where c ⁼ 29.4

A

is consistent with the molecular

length ;

the indices of the observed

rings

^{are :}

(001), (002), (111), (110), (020), (111) (the (021) ring

^{is not}

visible).

Therefore,

^the

assumption

^{of a}

pseudo-hexagonal

type ^{II structure} is consistent with both the

powder

and oriented

sample

patterns ^{of the Sm IV}

phase.

On the pattern ^{of the Sm III}

phase (Fig. 3c),

^the

(hkl)

reflexions are blurred out into a broad band where there

only

^{remains one more or less}

sharp ring.

^This

pattern

^is very similar to that of the

L/3’ phases

ⁱⁿ

lyotropic systems [5].

^{In such}

phases,

^{we have no}

correlations between the two-dimensional lattices of successive

layers;

^{we can then}

reasonably

^assume

that this also accounts for the case of the Sm III

phase.

Indeed,

^{if there are no} correlations between the

layers,

the

intensity

distribution in the

reciprocal

space consists of

(001) points corresponding

^to

Bragg

reflexions from the

layer planes

^and

by

^continuous

(hk0)

^rows

parallel

^to

c*,

^limited

by

the molecular structure factor

[6] ;

^{as a} consequence, the

(hk0)

and

(hkl) Bragg

spots ^are

replaced by

fractions of rows with

length c*,

^{centred on the}

(hk0)

reflexions corres-

ponding

^{to the} three-dimensional ordered

phase.

^If

we assume that the structure within the

layers

^{is similar}

in both Sm IV and Sm III

phases,

^{and if we take into}

consideration all the

scattering

^vectors

having

^their

extremities on the

portion

^{of rows with}

length c*,

^we

can build a theoretical

qualitative Debye-Scherrer pattern

of the Sm III

phase concurring

rather well with the observed one

(Fig. 7).

^The

sharp ring

^corres-

ponds

^{to the}

(020)

reflexions and the broad band to the

(110)

^reflexion

(as

^{in the Sm IV}

phase,

the reticular

spacings

^{are such that}

d(110)

>

d(020).

Therefore,

^{the structure of the Sm IV and Sm III}

phases

within each

layer

^{seems to be} very much alike.

The two

phases

differ in the

stacking

of the smectic

layers

^{over one} another : in the absence of an electric field or any other external

forces,

^two successive

layers

in the Sm IV

phase

^are correlated

(three-dimensional order)

^whereas

they

^{are not in the Sm III}

phase.

^This

structure of the Sm III

phase

^{is in} agreement ^{with the}

optical

observations

(high rotating power)

^which

probably indicates,

^{as in the Sm C*}

phase,

^{a helicoidal}

stacking

of the smectic

layers.

^{When an} electric field is

applied,

^{there occurs an}

alignment

^{of the}

hexagonal

lattices of successive

layers,

^{and if we wait a}

long

time

(15 h)

^the

long

^{molecular axes} also have a

tendency

^to

align,

^thus

forming

^a monodomain.

FIG. 7. ^- Effects of decorrelation of smectic layers ^{on the} powder diagrams. a) Reciprocal space ^- Each reflexion is replaced by ^a portion ^{of row} parallel ^to (00/). The reflexions are located in the

plane (P). Those which are also located on the Po plane (perpendi-

cular to 00l), ^{that is to} say the (0 ^± 20) reflexions, give ^{no broaden-} ing ^effect going through the Ewald’s sphere. b) Ring profiles.

(11C) rings ^arc broadened whereas (020) ^are not, and the ^sum of the two rings intensities give ^the expérimental profile (in ^full line).

We may wonder whether these two new

liquid crystal

structures are related to the chiral and ferro- electric

properties

^{of the}

compound.

^We

have, therefore, performed

^other

X-ray experiments

^on

various ferroelectric

compounds.

4. Other ferroelectric

compounds.

^- 4. 1 OTHER

FERROELECTRIC PURE COMPOUNDS. - A few other ferroelectric

compounds

present ^{one ordered smec-} tic

phase.

^{We measured}

powder diagrams

^{on seve-}

ral of them : the

compounds n

^{= 5 and n = 6 of the}

S

(-) p-alkoxybenzylidene-p’-amino-2-methyl-butyl-

cinnamate :

and the :

p-decyloxybenzylidene-p’-amino-6-methyl-octyl-cinnamate

553

The

Debye-Scherrer patterns always

show small

angle rings corresponding

^to successive order of reflexions from the

layer plane,

^{and a}

sharp ring

at 4.5

A superimposed

^{on a} broad band. This

diagram

is characteristic of a

stacking

^{of non} correlated

layers

formed of tilted molecules with a

pseudo-hexagonal type

II array, like the Sm III

phase

of HOBACPC.

The

méan _position

of the

(110)

band and the

sharp (020)

^line

gives

^{the value}

d(110)

^and

d(020)

^for

the local lattice.

Thus we can state that the ordered

phases

^{of ferro-}

electric

compounds

^{have a}

pseudo-hexagonal layer type

^{II structure} with tilted molecules and no corre-

lations between

layers,

except for the Sm IV

phase

of HOBACPC.

4.2 RACEMIC MIXTURE OF HOBACPC. - The race-

mic mixture of HOBACPC presents ^{the same}

phase

succession as the chiral

compound (the phase

^tran-

sitions were detected

by

D.T.A. and

optical

^obser-

vations).

Although

^the

X-ray powder

patterns of the Sm III

phase

^are identical for the chiral

compound

^{and its}

racemic

mixture,

^{we could not} orientate the

sample

with an electric field _up to 13 000

V/cm.

We don’t observe _any

change

^{in the}

X-ray

patterns from thé Sm III to the Sm IV

phase

of the racemic mixture. This

point

^{has to} be clarified in a more

complete study.

^One

possible explanation

^{is that we}

have a demixtion of the two

optical antipodes

^into very small domains in the Sm IV

phase, especially

^{in a}

direction

perpendicular

^{to the}

layer planes :

^{in such a}

case, the

rings

^{would be} very broad and

might

^not

even be visible at all.

5. Conclusion. ^- In the ferroelectric

compound HOBACPC,

^we have identified two novel ordered smectic

phases appearing

^at

temperature

^{lower than} the Sm C* range. In both

phases,

the molecules are

packed

^{in a}

hexagonal

array within each

layer ;

^the

long

^{molecular axes are} tilted in such a way that

they

lie in a

plane parallel

^{to one}

edge

of the two-dimen- sional

hexagonal lattice. ^Thus,

^these

phases

differs from the Sm

Bc phase (Sm

^B

tilted)

in which this

plane

^is

perpendicular

^{to one}

edge

of the two-dimensional

hexagonal

lattice. The difference between the struc- tures of the two Sm IV and Sm III

phases

^{is due to the}

stacking

mode of the

layers :

in the Sm III

phase,

successive

layers

^are almost uncorrelated whereas

they

^are correlated in the Sm IV

phase,

^thus

implying

a three-dimensional

order ;

^the very low associated

enthalpy

^{and the}

optical

observations are consistent with these features.

Other ordered smectic

phases analogous

^{to the}

Sm III

phase

^were observed in a few ferroelectric

compounds,

^{but no}

phase of type

^{Sm IV was found.}

In

HOBACPC,

^{the structure} within each

layer

^can

be understood from a steric

point

of view : the main contribution to the

dipole

component ^{comes from} the C-Cl

bond,

and this bond

mostly

remains in a

direction

perpendicular

^{to the}

long

^axes of the mole- cules. The orientation of the

hexagonal

^two-dimen-

sional lattice is in this case such that the Cl atoms

enjoy

^more space than in a Sm

Bc

structure. The same

argument

should also account for the other ferro- electric

compounds

which have an

asymmetric

^carbon

tied to a transverse

dipole.

^{Let us}

point

^out that in the

case

of HOBACPC,

the formation of the Sm IV

phase

is

probably

^favoured

by

^the

high dipolar

interaction.

Nevertheless,

^further

experiments performed

^on

oriented

samples

^are

required

for a better understand-

ing

^{of the} structures. The

question

^{of the nomen-}

clature of these two smectic

phases

^remains open because the

subscript

H* is somewhat

confusing

^and

certainly

not accurate.

Acknowledgments.

^{- We wish to thank Mr. L. Des-}

champs

for his technical assistance and Mrs M. C. Comes for

revising

^the

English manuscript.

References

[1] MEYER, ^R. B., LIEBERT, L., STRZELECKI, ^{L. and} KELLER, P., J. de Physique ^{Lett. 36} (1975) ^{L. 69.}

[2] DOUCET, J., LEVELUT, A. M. and LAMBERT, M., ^Mol. Cryst.

Liq. Cryst. ²⁴ (1974) ^317.

[3] KELLER, P., Thèse de 3e Cycle, Université Paris-Sud (1977).

[4] BIGARÉ, M., Rev. Mater. Constr. Trav. Publics 32 (1965) ^598-599.

[5] TARDIEU, A., LUZZATI, V., REMAN, F. C., ^J. Mol. Biol. 75 (1973) 777.

[6] GUINIER, A., X-ray diffraction (W. ^H. Freeman & Company,

San Francisco, London, England) 1963, Chapitre ^7.