Generalization of Meyer's ferroelectricity to cholesteric phase : electroclinic coupling and prediction of a new ferrocholesteric phase

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Generalization of Meyer’s ferroelectricity to cholesteric phase : electroclinic coupling and prediction of a new

ferrocholesteric phase

J. Marcerou

To cite this version:

J. Marcerou. Generalization of Meyer’s ferroelectricity to cholesteric phase : electroclinic coupling

and prediction of a new ferrocholesteric phase. Journal de Physique II, EDP Sciences, 1994, 4 (5),

pp.751-761. �10.1051/jp2:1994162�. �jpa-00247998�

Classification Physic-s Abstracts

64.70M 77.80

Generalization of Meyer's ferroelectricity to cholesteric phase

_:

electroclinic coupling and prediction of

_{a new}

ferrocholesteric

phase

J. P. Marcerou

Centre de Recherche Paul Pascal, Av. Dr. Schweitzer, F-33600 Pessac, France

(Received 22 March 1993, revised 20 December 1993, accepted ¹⁶

February

1994)

R4sumd. Nous

prdsentons

_une

gdndralisation

de la thdorie de Meyer du couplage trilindaire _entre [es couches smectiques, I'inclinaison de l'axe optique par rapport h la norrnale _aux couches _et la

polarisation

dlectrique

qui est h

l'origine

de l'existence d'une phase smectique C*

ferrodlectrique

ainsi que de I'effet

dlectroclinique.

Celui-ci ddcrit

l'apparition

d'une

polarisation

et d'un

angle

d'inclinaison _en

prdsence

d'un

champ

extdrieur dans [es phases

smectique

C*,

smectique

A et

cholestdrique. Nous prdvoyons une nouvelle phase « ferrocholestdrique _» entre le smectique C* et le

cholestdrique quand

l'ordre local est de type ^C* avec une polarisation

macroscopique

_non nulle.

Nous comparons enfin _nos

prddictions

_avec (es rdsultats expdrimentaux

disponibles

en phase

cholestdrique.

Abstract. We present a generalization of

Meyer's theory

of the trilinear

coupling

between the smectic

layers,

the tilt angle of the

optical

axis with respect to the

layer

normal and the electric polarization ^which accounts for the spontaneous ferroelectricity in smectic C*

liquid

crystals. This

coupling

allows for the description ^of the electroclinic effect, I-e- the appearance of _a

polarization

and _a tilt angle induced by _an extemal electric field in smectic C*, smectic A and N* (cholesteric) phases. ^A new « ferrocholesteric _» phase is

expected

between C* and N*

phases

^when the local order is smectic C*-like with _a small but _non-zero

polarization.

Eventually, a

comparison

is made with experimental data available in N* phase.

1. Introduction.

In 1975,

Meyer [I]

used _a

general

symmetry argument to show that in smectic

phases

made of chiral molecules and

belonging

to local C

~ symmetry group, a spontaneous

polarization

would

develop

as soon as the average molecular direction _n is not

parallel

to the

layer

normal _q. The

direction of this

polarization

^is the so-called C

~

symmetry

^{axis which is the} common normal _to

n and _q. The purpose of this paper is _to show that the _same symmetry argument ^{which is known}

to hold in the SmC*

[I]

and in the SmA

phase

in the presence of _an electric field

(I.e.

electroclinic

coupling) [2]

is also valid in the fluid N *

phase

when _one

applies

_an electric field

normal to the director. One

finds,

within the framework of mean-field

approximation

that

describes

[3-5]

the smecticA to smectic C*

phase transition,

that the inclusion of the

electroclinic

coupling

leads to different

predictions

about the behaviour of the smectic A

phase

which becomes _a low> tilt smectic C*

phase

and the appearance of _a

polarized

induced

fert.ocholesteric phase

in _an electric field. Close to a N*AC*

triple point,

the

possibility

of such _{a new}

ferrocholesteric phase

with local smectic C ^* order is

predicted

without any

applied

field.

~In

the first part we present a mathematical argument ^that we use

throughout

^the paper in order to build up odd _terms in the free energy and

apply

it _to introduce _a

coupling

between the smectic order parameter and the

polarization

in the Landau free energy ; in the _next three pans,

we

develop

our

predictions

about the

polarization susceptibility

_XR we

eventually

_compare

them with

already published experimental

work _on the electrodinic effect in cholesteric

phase.

2. Mathematical

background.

When molecules _are chiral

(no

mirror

symmetry),

and the local symmetry group is

D~

or

C~

like in N*, SmA _or SmC*

phases,

odd _tensors

T,~,

are allowed

(')

_so that trilinear

couplings

like

T,~, U,

_V~

W, (with

an

implied

summation over

repeated indices)

can enter the free energy. These _{terms can} be written in vectorial notation

AF =TU.

(VXW). (I)

Two of them _are well-known in the

physics

of chiral

liquid crystals

as

they

^lead to the

helicity

of the cholesteric and smectic C*

phases, they

read AF

(N*)

= An

(V

_x

n)

~2)

AF

(C

*

= pc

(V

x

c)

where _n is the director of the cholesteric

phase

^and c the c-director of the smectic C*

phase [6].

The

original

remark of

Meyer [I]

_can be translated _as

leading

to a mixed

product

of the

polarization

P, the director _n and the

layer

normal _q

(see

e. g.

Fig. I)

I-e- to _a term

proportional

to the tilt

angle

0 and to the component

P~

of the

polarization parallel

to the

C~

axis _:

AF (RBM

=

iP (n

_x

q)

₌

tP~

0

(3)

This linear

coupling

has been shown _to describe both the

proportionality

between the

spontaneous ^tilt

angle

_{0~ and}

polarization

_P~ in smectic C*

phase ii, 5]

and between these _same

quantities

induced

by

the electroclinic effect in smectic A and C*

phases [2-5].

The

coupling

coefficient noted _t

[1, 2]

_or C in other papers

[3, 4, 7]

is _a direct _measure of the

chirality

of the

phase

_as it is _zero for _a racemic mixture and

changes sign

from _one enantiomer _to the other.

One of the most

spectacular

results of the

organic chemistry

of ferroelectric

liquid crystals

in

recent years has been _to increase this coefficient

by

a factor of _more than loo from the

historical DOBAMBC

(with

_{P~ a} few

nC/cm~ Ii)

to the record

awarding compounds

of

today

(P~ _~ l 000

nC/cm~ _{[8]). Recently,}

an electroclinic effect has been

reported

in N*

phase [9-12]

with _a

surprisingly large dispersion

of the apparent or

computed

tilt

angles (from

10~ ^~ _to 10~ ^~

radian)

and with _no clear connection _to

Meyer's

argument. ^We now show that this link _can be

easily

established with _a suitable choice of the vectors

entering

the trilinear

coupling

term.

By

a

straightforward generalization,

one finds that _a

Meyer's expression

valid

(ij Namely ^the fully

antisymmetric

^{Levi Civita} tensor such that

T,~==T,,~ =T-,, -T~=,

T,

=

T-,,,

all other elements vanishing identically.

+

~

li

z~+q,

~°

_i

p ⁺ⁱⁱ

o

Fig. I. a) Sketch in real space of the director _n, layer normal _q and

polarization

P which _are coupled

linearly

_so that the

amplitudes

of P and of the tilt

angle

0 between _n and q are

spontaneously proportional

in SmC* phase. The _same coupling leads _to the proportionality of 0 and P _{to an} extemal electric field by

means of electroclinic effect. b)Smectic _{wave vectors} in reciprocal _space, the amplitude of

q,, =

0q,

is proportional to the polarization.

in the three

phases

should include the

polarization

and the

density

modulations in the directions

parallel

and normal _to the local director _:

AF

(RBM

=

©P

(Vii ^P x

V~

P *

with Vii P

= n

(n

VP

) (4)

and

V~

P

= n x

(n

_x VP

).

We consider here that P

(r

is the smectic order parameter

[6]

with _a

preferred

_wave vector q

determined,

in mean-field

approximation, by

minimization of the free energy with respect to its components _qjj

parallel

to the director and q~ normal to it. Under these conditions, Wand Vii P

(

=

iq~

P in the condensed

phase)

both _account for the smectic A

density

modulation

which fluctuates in the N*

phase

while

V~

P

=

iq~ P)

_measures the smecticc type

fluctuations in N* and A

phases.

3.

Consequences

of the electroclinic

coupling

in cholesteric

phase.

Let us write down the free energy

density

^of a N*

phase

^close to a smectic _one in the presence of _an

applied

electric field E, it will be the _sum of the elastic energy of the N*

phase F~~,

of the

fluctuating

smectic Landau energy

Fs~

and of electrostatic _terms

involving

the

polarization

and the electric field

F~j.

F~~

=

Kj(div n)~

+

K~(qo

+ n

(V

_x

n))~

₊

_K~(n

x

(V

_x

n))~

2 2 2

Fsm

=

(« (T 7i~) w2

_c~i

(vi P)2

_{+ y}

(T 7ic)(v~

P )2 +

D,~,I vi

w

vii

w

*) (5) F~j

₌

©P

_{(Vii P}

x

V~ P*)

P E ₊

~~

+

~~

ES E,

_E~

2

o~Xi

2

o~Xll

2

JOURNAL DE PHYSIQUE II T 4 N'5, MAY IQ94

The elastic energy contains _as usual _a spontaneous ^twist term that leads

[13]

to _a first order N* to A

phase

transition in order _to compensate ^{for the helix}

unwinding

energy of

K~ q(/2

_per unit volume. The smectic _term shows the _two would-be-transition temperatures

7~~

and

7~c

for the appearance of the SmA _or SmC

phases

while the

negative

Cjj coefficient _accounts for the spontaneous

tendency

to build up

layers

^normal to the nematic director

[14].

The fourth rank

D,~,I

terms which _are

required

for

stability,

_are in

principle

five in _a uniaxial medium but reduce _to three in Fourier space

[13].

Within the electrostatic term, the

polarization

P represents ^the amount of permanent ^molecular

dipoles aligned by

the

applied

electric field and

by

the

Meyer's

electroclinic

coupling.

The

susceptibilities

_e~

=

eo(I

+

x~)

account for electronic _or intramolecular

polarizabilities.

Then, if the direction of the

applied

field is from _{now on} assumed _to be Ox

parallel

to the cholesteric

axis,

_as

long

_as there exists _a local fluctuation of

P,

the appearance of _{a non} trivial

(I,e.

_{# e~ Xi}

E~ polarization P,

normal to the local director _n fl Oz and of _a modulation of

Pin the _y direction will be

energetically

favored _so that

locally

_:

P~

_{= so Xi}

lE,

₊ V~P

V~P

^*

(6)

In fact the situation is _more involved _as this non trivial

polarization

must be derived

by integrating

out the thermal fluctuations of P in the presence of the

applied

field

E~.

^{It is thus} necessary to express the free energy in Fourier space

J

=

l(F

s~ +

F~j)

dV

V =

v

=

( _{I la (T}

TAN + Y

(T

_TAG

q[

₂

_Cqz

q>

P,

₊

Di (q) q))~

q

p2

+ 2

D'(qi q)) q(

₊

D~ q( _P~

P

~

P, E~

+ ^~ F~

E) ₍₇₎

2 _{So Xi} 2

In this

expression,

_z is the local direction of the

optical

axis

(that

_may be held fixed in _an unwound

sample),

_q is summed from 2 «IV ^~/~ to 2 grid where d is _a molecular

length

in the

I

range,

T~~

₌

7~~

+

~~

~~

and

T~c

=

7~c

^{~ ~' ~~} _are the renorrnalized would be transition

« y

temperatures ^{when the}

D,~~i

terms are taken into _account and reduce _to

Djj, D~

(~ 0)

and D'

(D'~

~ Djj

D~

_;

eventually,

_q~

=

2 «la is the

preferred

_wave vector of the smectic

density

modulation where am

301

_{is the}

layer

thickness.

The

integration

of the thermal fluctuations of P

yields

a uniform

polarization P~

=

e~ XR

E~

which

susceptibility

_XR is renormalized and

larger

than the bare _one Xi

,

moreover

the

optimum

wave vector of the

density

modulation that will be retained _at the N* _to smectic

phase

transition is modified. As

usual,

one can assume q~ to be

given by

_{q~ =} 2 «la while the

optimum

_q~

(if

T

~ TAN ~ TAC) ^is non zero in the _y direction in presence of the

applied

field

along

x :

~~

Y

(T

Ac)

~~

~~~

(0,

+ q~, ⁺

q~)

^and

(0, q~, q~)

represent in the

reciprocal

_space the coordinates of the

maxima of the correlation function

(P~ P_~)

in N*

phase (see

_e-g-

Fig.

I). Third

consequence,

replacing

_q~ and

P~ by

their

optimum

value in

(7)

leads to an increase of the

temperature

^{T' where the} coefficient of

P~ P_~

^vanishes

(I.e.

to a

displacement

of the

transition under

field)

(~0

_{XR ~s}

~

~~

~2

_(~~

~'

" ~AN ~

«

y(T~N TAC)

^'

So

applying

an electric field in _a cholesteric

phase

transforms it _{to a} kind of induced

ferrocholesteric phase

with both _a

polarization (P,

_{= so}

XRE,)

and _a tilt

angle

0

=

q~/q~

(Eq. (8)) proportional

to the electric field and sensitive to the

proximity

of the smectic

phase

via the renormalized

susceptibility

_XR, as

apparently reported

ⁱⁿ recent

experimental

work

[9-12].

In order _to

improve

these

predictions,

_we will then evaluate the

susceptibility

XR in _a model where the

only

consequence of the

chirality

is the onset of the

Meyer

term _we introduced,

neglecting

the

helicity

of the director that leads _to cholesteric and TGB

[13]

phases.

4. Mean field derivation of the

susceptibility

_x~ with smectic fluctuations.

Chen and

Lubensky ii 4]

have

developed

the NAC

phase diagram

as a function of parameters that

correspond

ⁱⁿ _our notations to y

(T T~c

for x-axis and _{+ a}

(T T~~ )

for

y~axis.

This

diagram

is

reproduced

in

figure

2 where the

physical paths

followed when

decreasing

the temperature ^{from the nematic}

phase

_are

straight

lines of

negative slope starting

^from the upper

left _corner

(T»T~~

and

T»T~c).

There _are two kinds of behavior

depending

_on

T~~

and

T~c (the triple point

is reached when

T~c

=

T~~)

_:

I)

if

T~~

_~

T~c,

one encounters a second order nematic _to smectic A

phase

^transition at

T

=

T~~

^with wave vector of the smectic modulation at q~ = q~, q~ =

0,

and _a second order

smectic A to smectic C

phase

transition at T

=

T~c

it)

if

T~~

_~

T~c,

then there is _a direct second order nematic _to smectic C

phase

transition _at T

=

T~c

₌

T~c

₊ 2 3^~ 2 3

(T~c

₊

^3~ T~~)"~

where the _{wave vector at} the transition is

given by

_q

(

= y

(T~c T~c )/2 D~

and

q]

=

q) _yD'(T~c T~c )/2 D~

Djj ^{in these} express- ions

3~

=

«D~ (I D'~/Djj D~ )/y~

=

«D~/y~.

6

4

, ~q

, ,

2 ',

c '

m ',

~ 0

,

, ,

~ ,

~ ,

,

~ A _, C

',

~6

-10 -5 0 5 10

Tac -T

Fig. 2.

Chen-Lubensky

phase diagram in the

vicinity

of the NAC

point.

If T~N _> T~c, one follows the dashed line when

decreasing

the temperature ^{from N} to A and C phases. If TAN < T~c, ^then one goes

directly

from N _to C _at

T~c (T~~

<

T~c

< T~c).

We have then to derive the effective

susceptibility

_XR when

approaching

the N to smectic

phase

^{transition from above. For this} we will define the effective free energy obtained

by integrating

out

P~

fluctuations in

equation (7)

Feff ~~jnTre

1

f

»

t

^~

V

=

P

~

E,

+

~

~~

e~

E)

₊ kT

~~~

~

In G~ ^'

(q, P~) (10)

So Xi

(2 gr)

where

~- ~~,

~,

_« ~T

T~~

_{+ Y}

~T ~~~~ ~~

~

l _)illl'~i

~~ + ~

/~'~~~

_~~~

~~

_~

~~ ~~

minimizing lo)

with

respect

to

P,, yields

the

equation

of _state for the

polarization P,

-E~-I(P~)=0

E0 xi

where I

(P,)

=

kT

~~~

~

Cq~

_q~

G(q, P,). ^{(' ')}

(2

_gr

This

equation

can be solved

graphically

_as shown in

figure

3. First in the presence of the

applied

electric

field,

_{one sees} that there

always

^exists an intersection of the

straight

line

(P le~

_Xi

E~)

with the _curve I

(P~).

At

sufficiently

low

field,

there is _a linear

relationship

P~

= e~ XR E~ ^{between the}

position

_{P_~} of the intersection and the value of the field.

Without

field,

the trivial solution that

corresponds

to a

paraelectric phase yields

P~

=0. One may see

graphically

that this will

happen

as

long

as the initial

slope

of I

(P

_{~) for}

P~

=

0 is smaller than

lleo

_Xi On the contrary, ^I-e-

high enough

^initial

slope,

one

predicts

the existence of _a

ferrocholesteric phase

which could be ferroelectric without _a

fully developed

smectic

layering (see

_e-g-

Fig. 4).

1(P,), T<Tc

P, (lh),T>Tc

Pg

COXI

~ ~

coxi

^'

Fig.

3. The

graphical

solution of the equation of _state for the

polarization P,

is

given by

the intersection of the straight line (P,/_{p~ Xi} E~) and the

integral

I (P

~

). One easily checks that the solution P~ # 0

corresponds

to a minimum of the effective energy f.

+

n+

n+

(a) (b)

Fig. ^4. a) First kind of ferrocholesteric phase where the layer normal rotates around the cholesteric axis Ox like in _a TGB phase [13]. The polarization is always parallel to the axis, this configuration should be retained in the induced

ferrocholesteric

when the field is

applied

along Ox _as it is assumed in the _text.

b) Second

possibility

where the

polarization

P (that _may

require

_{a more} involved definition with modulus and phase like the smectic order parameters) rotates together ^with n while the layer ^normal _q processes

making

an

angle

of gr/2 ₀ with the helix axis.

Quantitatively, equation (11)

_may be used to compute ^{the effective}

susceptibility

x~ in the cholesteric

phase.

In the linear

approximation (I.e.

low field without any

bias),

E~ =

P

/Fo

_XR ^{and I}

(P~

_are

proportional

to

P~,

so that _:

1

-

~~

l~

^~

^C

^{~ kT}

^{/2~3 ~l ~l G~ ~~,}

^°

^('2)

So,

the

susceptibility

will

diverge

as soon as the derivative of I

(P,

^is

large enough.

Let _us

now check this

by

_means of a numerical estimate

using

reasonable values for the coefficients involved in the calculation.

5. Numerical simulation of

XR(7l

in the

vicinity

of NAC

point.

Estimating

first the tilt

angle

in a smectic C

phase

at 3 K below the transition

gives

:

~i Y(I~AC

~)

_l

y

l~i

I

₂

_{D~ q)}

₈ ^~

_q2

_loo

~

Using

_now

X-rays scattering

results

[15, 16]

in the

vicinity

of _a nematic _to smectic A

phase

transition, to evaluate G~

'(q)

leads _to

T

T~~

Y'

~ ~ ~ ~

G~

(q,

⁰ a'

~

(l

₊

fj

(q=

qs)

+

ii

_qi

AN

estimating roughly

that _a

(T T~~)

=

a'[(T T~~)/T~~]Y

at T

=

T~~

+ 3 K, and

using y'

w

2 _vii

- 2 v~ m 1.3

if

_q~

- I and

it

_q~

m

0.5

[15]

_{; one may express} all the coefficients

as functions of

D~

^alone :

y/q)

=

D~ /100, a/q)

=

D~/3 000,

_Djj

=

D~

/10 D'

being arbitrarily

taken

equal

to

D~

/200 in the calculation.

Introducing

the dimensionless variables _u

=

q(/q(

and _x

=

q~/q~, equation (12)

becomes

°~~

= l -N udu

x~dxg~(u,x)

OXR ~ ~

with g~ ^' ₌

~

~~~

+ lo

(x~ 1)~

+ loo u~ ₊

u

IT T~c

₊ x~

1] (13)

~

FoXikT©~

and N

=

lo _x

~ ~

4 gr

D~

_q~

eventually,

the ratio

©/D~

_can be extracted from the measurements of the

Meyer

coefficient C and the elastic constant

B~

in smectic

phases [3, 4]

:

Cq) _(T

_TAG

D~ 100B~

so that with eo = 1/36

gr

10~,

_Xi

" 5, kT

= 4 _x

10~~'

_{J, q~}

= 2 w/3 _x

10~~

_m~

~, C

=

3.8 _x

10~

_and

B~/(T T~c)

=

4 _x

10~

; one

finally

estimates the numerical factor N to be

m 23 in

equation (13).

This _means that the

integral

in this

equation

has _to reach _a value of 0.044 in order _to describe _a cholesteric _to ferrocholesteric

phase

transition.

This

integral

is evaluated in _two steps, ^{first the}

integration

_{over u} variable _can be done

analytically

then the second

integration

_{over x} is made

numerically using

a

Romberg

method.

The

T~~

transition temperature ^{has been held}

arbitrarily

at 310 K while the

integral

was

computed

^for

T~c ranging

^{from 300} to 330K and for T

ranging

^from

T~~ (resp.

T~c)

₊ 0.001 _{to +} lo K. The result of the most

divergent integral (that corresponds

to the NAC

point

where

T~~

=

T~c)

is sketched in

figure

5. One _{can see} that the maximum value is

larger

than 0.4 _so that the 0.044 limit for _a transition _to ferrocholesteric

phase

^is reached about 0. I K before

T~~c.

One _{sees more}

precisely

in

figure

6 the

plot

of the

integral

as a function of

T~c

_{on one} axis

(309.4

_~

T~c

~

310.5)

and of the distance AT to the nematic-smectic

phase

0.6

0.4

ii

11 0.2

0

lE-4 0.001 0.01 o-I I lo

T-Tc

Fig.

5. Plot of the most

diverging

integral (Eq. (13)) at the NAC point (T~~

T~c).

1

0,45

0,3 '..

o,15

., ..

0 '..

0,001 0,002

0,008 0,022

0,064 0,181

0,512

AT _{31o 3 31o} 309,7 309,4 TAC

Fig.

6. Plot of the integrals

(Eq.

(13)) in the

vicinity

of the NAC point, ^the figure 5 corresponds to a

section _at T~c ₌ 3 lo K.

transition _on the second axis. Then

,

the cholesteric _to ferrocholesteric

phase boundary

is

given by

the

plane

with _an elevation of I/N

(0.044

in _our

estimation),

the _contours 0.15 and 0.3 _are

represented

ⁱⁿ

figure

6

showing

that the extent of the ferrocholesteric

phase

would be _a small

closed domain around the NAC

point.

Using

this

simulation,

_{one can}

predict

the behavior of the

susceptibility

_XR from

equation (13)

in two cases :

I)

if _one reaches the ferrocholesteric

phase,

_x~ will

diverge

as shown in

figure

7 _;

it)

if not, _XR will increase _at the N-Smectic transition from _{Xi to a} finite value.

These

predictions

_are useless if _not

compared

to

experiments,

so we will _now

give

a short discussion about

already reported

and

possible

future

experiments.

6. Discussion.

Let _us first recall that the calculation has been done under the

assumption [17]

that the initial NAC

phase diagram

is the

Chen-Lubensky

_one

(without chirality) ii 4].

We have _not taken into

account the further refinements of Renn and

Lubensky ii 3],

who first showed the

displacement

of NA and AC

phase

boundaries due _to

chirality,

neither _we did for the AC shift due _to

Meyer's

term

[2, 5]

_nor for the

possible

_appearance of TGB

phases ii 3]

in the _{same area} of the

phase diagram.

These results have to be

joined together

in order _to

place

the ferrocholesteric

phase

in _{a more} realistic

phase diagram,

with

hopefully

_{a greater} existence domain.

iooo

ioo _o

~ .

m _.

i~

^{lo ".}

m

i

o-1

0.001 0.01 o-I I lo

T.Tc

Fig. 7. _{XR Xi as a} function of T T~ (T~ ^{is the} temperature ^{where the} integral in equation (13) reaches the value I/N). There is _no

simple

prediction for the value of the apparent ^critical exponent.

Nevertheless, the _structures sketched in

figure

4 may be revealed

by

_some accurate

X-rays scattering

_or

polarization

sensitive

experiments.

Retuming

to the _x~

calculation,

_an immediate check of its

validity

should _use dielectric spectroscopy

techniques.

A series of

experiments

done in pure

compounds

close _{to a} NAC

point

has been

reported by Legrand (Fig.

9 of Ref.

ii 2])

and shows that the dielectric

strength

Pm I +XR increases when

approaching

the N to A

phase

transition _as

expected.

The

interpretation

of

optical experiments reported by

most other authors

[9-1Ii

is _more involved.

First _we do _not compute a tilt

angle

but _a dielectric

susceptibility,

the link between these

quantities

has to be

expressed

like in

equation (8)

which

gives

the _most

probable angle

between the

fluctuating

smectic modulation and the

optical

axis at a

given

temperature ^{in N}

phase.

Then,

assuming

that the

fluctuating layer

normal is fixed in the direction of the

optical

axis at rest, one may think that the

light intensity

modulation is

directly proportional

to XR and that

optical experiments

are

again

_a

proof

of

Meyer's ferroelectricity

in cholesteric

phases.

7.

Summary.

We have

generalized

the symmetry argument ^due to R. B.

Meyer

which is _at the

origin

of the invention of ferroelectric

liquid crystals.

We have shown

that,

_as

long

as three elements _meet

together, namely

^smectic

layering,

^{tilt of} the

optical

axis and electric

polarization,

one can

derive, in mean-field

approximation,

a

theory

of electroclinic

coupling

that _covers

cholesteric,

smectic A and smectic C*

phases

and is consistent with

experimental

results. We have then shown that _a

spontaneously

ferroelectric

phase

is

likely

to appear close _{to a} NAC

point

_as

long

as the

Meyer's coupling

is

strong enough.

Acknowledgments.

We would like _to thank T. C.

Lubensky

and J. Prost for their

encouraging

reactions to this work and most of all, _we thank the anonymous referee whose

help

was invaluable.

References

[1] Meyer R. B., Liebert L., Strzelecki L., Keller P., J. Phys. Lent 30 (1975) 69.

[2] ^Garoff S.,

Meyer

R. B.,

Phys.

Rev. Left. 38 (1977) 848 Phys. Rev. A19 (1979) 338.

[3] Dupont L.,

Glogarovh

M., Marcerou J. P., Nguyen H. T., Destradec., Lej6ekL., J. Phys. II France I 1991) 83

[4] Dupont L., thesis 476, Universitd de Bordeaux (1990).

[5] Glogarovh M., ^Destrade C., Marcerou J. P., ^Bonvent J. J., Nguyen H. T., Ferroelectrics121 (1991) 285.

[6] de Gennes P. G., The

Physics

of

Liquid Crystals

(Clarendon Oxford Press, Oxford, 1974).

[7] Glogarovh M., Pavel J., Liq. Cryst. 6 (1989) 325.

[8]

Kobayashi

S., Ishibashi S., Mol.

Cryst.

Liq.

Ciyst.

220 (1992) 1.

[9] Zili L., Petschek R. G., Rosenblatt C., Phys. Rev. Leit. 62 (1989) 796

Zili L., Di Lisi G. A., Petschek R. G., Rosenblatt C., Phys. Rev. A 41(1990) 1997.

[10] Komitov L., Lagerwall S. T., Stebler B., Andersson G., Flatischler K., Ferroelectrics l14 (1991) 167.

ill] Etxebarria J., Zubia J., Phys. Rev. A 44 (1991) ^6626.

[12]

Legrand

C., Isaert N., Hmine J., Buisine J. M., Pameix J. P., Nguyen ^H. T., Destrade C., J.

Phys. II Franc-e 2 (1992) 1545.

[13] Renn S. R., Lubensky T. C., Phys. Rev. A 38 (1988) 2132 Lubensky T. C., Renn S. R., Phys. Rev A 41 (1990) 4392 _; Renn S. R.,

Phys.

Rev. A 45 (1992) 953.

[14] Chen J., Lubensky T. C., Phys. Ret,. A14 (1976) 1202.

[15] Bouwmann W. G., de Jeu W. H., Phys. Ret>. Lett. 68 (1992) 800.

[16] Nounesis G., Blumm K. I., Young M. J., Garland C. W., Birgeneau R. J., Phys. Rev. E 47 (1993) 1910.

[17] One _must remark however, that _most of the optical measurements have been done with commercial mixtures designed to have _a

compensated pitch

(I.e.

diverging

for

alignment purpose) just

above the N _to A

phase

transition. So the N _to A phase boundary ^is not too different from the Chen-Lubensky _one.