Nucleation and growth of cholesteric fingers under electric field

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Nucleation and growth of cholesteric fingers under

electric field

P. Ribiere, P. Oswald

To cite this version:

Nucleation and

_growth

of cholesteric

_fingers

under electric field

P. Ribiere and P. Oswald

Ecole Normale

_Supérieure

de

_Lyon,

Laboratoire de

_Physique,

46 Allée d’Italie, 69364

_Lyon

Cedex 07, France

(Received

on December 7, 1989,

accepted

in

_{final form}

in

_February

26,

1990)

Résumé. 2014

Nous avons étudié le comportement d’un cristal

_{liquide cholestérique placé}

entre

deux

plaques

avec un _ancrage

_homéotrope

et en

_présence

d’un

_champ

électrique.

En

_changeant

les deux

_paramètres

de

_contrôle,

à savoir

_{l’épaisseur}

de l’échantillon et

_{l’amplitude}

du

_champ

électrique,

nous avons _pu caractériser différents types de solutions : une

_{phase nématique}

(cholestérique

« déroulé

_»),

des

_doigts

isolés doublement torsadés, un réseau

_périodique

de

_doigts

arrangés

côte à côte et une

_{configuration}

transitoire

_simplement

torsadée et invariante _par translation. En incluant la contribution du

_{champ électrique}

aux calculs de la référence

[3],

nous avons calculé un

_diagramme

de

_{phase théorique qui}

est en bon accord avec les données

exp6rimentales.

Il est _{essentiel, pour}

_expliquer

nos

_{observations,}

de tenir compte de

l’anisotropie

des constantes

_élastiques.

Abstract. 2014 We have studied the behavior of a cholesteric

_{liquid crystal}

sandwiched between two

plates,

with

_{homeotropic anchoring,}

in the presence of an electric field.

_By

_changing

two control

parameters,

namely

the

_sample

thickness and the electric field

_magnitude,

we were able to

characterize different _types of solutions : a nematic

_{phase (unwound cholesteric),}

isolated

_doubly

twisted

fingers,

a

_periodic

_pattern of

_{fingers arranged}

side

_by

side, and a transient

_singly

twisted

translationally

invariant

_{configuration. Including}

the electric field contribution to the calculations of reference

_[3],

we have

computed

a theoretical

_{phase diagram}

that agrees well with the

experimental

data. The consideration of

_{anisotropic elasticity}

in our calculations was essential in

explaining

our observations. Classification

Physics

Abstracts 61.30G -

61.30J

1. Introduction.

A cholesteric

_phase

is a nematic

_liquid

that is twisted in a

_single

direction. This structure is characterized

_by

its

_pitch

_{« p} », i.e. the distance over which the director n

_(the

unit vector

parallel

to the

_molecule)

rotates of 2 ’TT. A convenient way to obtain a

_large-pitch

cholesteric is

to dissolve a chiral molecule in a nematic

_liquid.

For small

concentrations,

the

_pitch

is

inversely proportional

to the concentration.

Cholesteric textures and associated

_topological

defects have

_already

been

_extensively

studied,

particularly

in thick

_{samples [1].}

In this case, the influence of the

_boundary

conditions can be

_disregarded.

The situation is very different in a confined

_geometry.

In

_particular,

when the

_anchoring

on the surfaces

_limiting

the

_sample

is

_{homeotropic (molecules perpendicular}

to the

_surface),

the

resulting

frustration can

_completely

unwind the cholesteric. This nematic-cholesteric

tran-sition has been described

_by

several authors

_{[2]. Recently,}

we have shown that elastic

anisotropy implies

that it is a first order transition

_[3].

Note that the control

parameter

of this

unwinding

transition is not the

_temperature

but rather the dimensionless confinement ratio

where d is the thickness

_sample

and p the natural cholesteric

pitch.

There are two cases :

- C

>Cc~

1,

and the chiral structure can

_develop

either as isolated

_fingers

or as a

periodic

pattern,

- C

Cc,

and the

_homeotropic

nematic

phase

alone is stable.

It is also well known that another way to unwind a cholesteric structure is to

_apply

an electric field. If the dielectric

_anisotropy

of the molecule is

_positive

it will tend to

_{align along}

the field.

_{By applying}

a

large-enough

electric field

perpendicular

to the helical

_axis,

it is

possible

to unwind the cholesteric structure into a nematic

phase

_[4].

In this

article,

we describe the behavior of a cholesteric

_phase

submitted to both

_plate

confinement and an electric field E. We show that in the

(E, C )

parameter

plane,

there are three different

types

of solutions : nematic and double twisted

_fingers,

which can be either isolated or

_arranged

into a

_periodic

_pattern

_(the

so-called

_{« fingerprint » texture).}

We determine this

_{phase diagram experimentally}

and then extend the theoretical model of reference

_[3]

to the case of an

_applied

electric field.

The

_plan

of the

_body

of the article is as follows : in section 2 we describe the

topology

of a

finite-length finger.

In

_particular,

we show that its two ends are

_{topologically}

different. In

section 3 we describe the

_experimental

set _up. In section 4 we establish the

_{phase diagram.}

Section 5 is a theoretical extension of the model of reference

_[3]

that allowed us to draw up a

phase diagram

in

_good

_agreement

with the

_experimental

data. 2.

_Topology

of a

_finger

of finite

_length.

2.1 DTC MODEL. - The model which we use in this article has been

extensively

described in reference

[3].

A convenient way to visualize the

_topology

of a

finger

is to consider its

_image

onto the unit

_sphere

S2. The

_principle

of this

_mapping

is recalled

_{in figure}

1. At the moment, the

_finger

is assumed to be rectilinear and infinité

_along

the x-axis. In this case the

image

of a

straight

line z =

zo

(parallel

to the

y-axis

i.e. normal to the

_finger)

is a circle

_{passing through}

the north

_pole,

_{parameterized by}

the

_{angle a (zo )}

between its axis and the N.S. axis of the

sphere

and whose the center lies on another circle

_tangent

to the north

_pole

_(whence

a double-twist

_{configuration (D.T.C.}

in the

_{following) energetically}

favorable

[3]) (Fig. 2).

To insure the

_homeotropic

_anchoring

-on the

_{glass plates,}

we take

Fig.

1. -

In this

_model,

the

_angle

_{a o}

_plays

the role of an order

parameter

for the transition :

afro = 0

corresponds

_to the nematic

_phase,

a o # 0 to a cholesteric

_finger.

In

_figure

_3,

we have

represented

the director field for a

_finger

of width A. This molecular

_{configuration}

is continuous and without any

singularity,

in contrast to the model

_{proposed by}

Cladis and Kléman

_[5].

On the other

_hand,

it is

_{topologically equivalent}

to the model

_{proposed by}

Press and Arrott

_[2].

Fig.

2. -

Double twisted

_finger

of width A. On the contour ABCD the director is

_parallel

to the z-axis in order to insure the

_{homeotropic anchoring}

on the

glass plates

and the

matching

with the outer nematic

phase.

The

image

of a line z = _zo

(parallel

to the

y-axis)

onto the

sphere

is _a circle of

angle

a (zo)

and

_{passing through}

the north

_pole.

The

angle

fl is

given by

cos ₈ =

(tan

a /2) / (tan a o/ 2 ).

Fig.

3. - Director-field

representation in a D.T.

_finger.

A nail represents a molecule

_making

an

_{angle K}

with the

_plane

of the

figure ;

its

_length

is

_proportional

to cos K ; the head of the nail is behind the

plane

of the

figure

and the

_tip

is

_pointed

on the observer :

_a)

section

_by

the median

_plane

z =

d/2,

a o = 30° ;

b)

section

by

_a

plane perpendicular

_to the

finger

axis, a o =

30° ;

_c,

d)

idem,

a o = 80°. This

2.2 TOPOLOGY OF A FINGER TIP. - A

_finger

is never infinite. One

_possibility

is that it may end at the

_{sample edge.}

It may also form a

loop

(Fig.

4a)

or a _segment of finite

_length

(Fig. 4b,

c).

In the last case two

_{configurations}

are

_possible

_{experimentally :}

either the two

ends of the

_fingers

are identical and there is

always

a defect inside the

_{finger (Fig.}

_4b)

_[10]

or there is no defect and the two ends of the

_finger

are not the same, one end

_{being sharper}

than the other

_(Fig.

4c).

To construct the

_topology

of the

_finger

ends

without

_{introducing singularities}

in the director

field,

it is sufficient to decrease

_continuously

_{a o} to zero at each end over a distance

_roughly

equal

to the

_finger

width. Because the

_finger

is not

_symmetrical

with

_respect

to a

_plane

perpendicular

to its axis

(no

mirror

symmetry

in a

_{cholesteric),}

the ends are not identical. On

Fig.

4. -

the other

hand,

a calculation of n . eurl n shows that at one of the ends the twist has

everywhere

the same

_sign

as the

_spontaneous

twist of the free cholesteric. At the other

_end,

by

contrast, the twist

_{changes sign (Fig.}

_5a).

We shall call the first one « normal

_{tip » (or}

« + »

tip)

and the other « abnormal

_{tip » (or}

« - »

_tip).

When the two ends are

identical,

there is

always

a line defect

_(here

a S = 1

disclination)

joining

the two

glass plates

in the bulk of the

finger.

This is shown in

_figure

5b. It is easy to see

_{experimentally}

that the fracture of a

_finger

(by

applying

an electric

_field,

see

_below)

leads to two ends of

_{opposite signs. Conversely,}

two

ends of

_{opposite signs collapse together (Fig. 6)}

while two ends of the same

_{sign repel}

and remain

_{separated by}

a nematic slice. On the other

_hand,

an abnormal

_tip

can

_collapse

with the side of a

finger

and form a T-like

_{sidebranching (Fig. 7).}

On the

_contrary

a normal

_tip

is

repelled by

the side of a

_{finger :}

the connection is

impossible.

This is

explained

in

_figure

8. This

_property

allows us to characterize

_{experimentally}

the two

_tip

_types.

Thus the abnormal

tips

are

sharper

while the normal ones are rounded.

Fig.

5. -

a)

Director-field

_{representation}

in the two ends of a

_finger

of finite

length

and in the median

plane

z = 0

(a 0

=

30 ).

Calculation of n - eurl n

_along

the x-axis shows _that, in one end, the twist is

everywhere

of the same

_sign

as the _spontaneous twist of the free cholesteric

_{(normal end)}

while in the other, the twist

_{changes sign (abnormal tip). b)}

Director-field

_{representation}

when the two ends are

identical

(normal tips).

A S = 1 disclination appears in the bulk of the

Fig.

6. - Fusion of an abnormal

finger tip

with a normal one

_(C

= 1.92, V = 4.8

V ) : a) just

before

connection ; b)

after connection.

convective

_{instabilities)}

until the cholesteric structure

_completely

_disappears

so that the nematic

_phase

is the

_only

stable

_phase.

Then we decrease

_abruptly

the

_voltage

down to the desired value. If a cholesteric texture _{appears it will be}

_necessarily

_stable,

the nematic

_phase

being

either metastable or unstable.

_By

_{increasing again slowly}

the

_voltage,

it is

_possible

to

determine the

_voltage

at which this solution becomes metastable and then unstable. In this

3.

_Experimental

set _up.

The chosen materials are the eutectic nematic mixture ZLI 2452

_(Merck)

and the chiral molecule ZLI 811

_(Merck).

The concentrations are

respectively

99.396 % and 0.604 % ww

which

_corresponds

to a cholesteric

_pitch

close to 16.7 um at room

_temperature

_{(20 °C) (from}

Merck

_data).

The mixture is

_placed

between two

_{conducting glass plates.}

The

_plates

are coated with the silane ZLI 3124

_(Merck)

to orient the molecules

_{perpendicular}

to the

_plates

(homeotropic anchoring).

The slides are fixed to two metallic holders

_{coupled together}

via

three differential screws. The thickness of the

_sample

can be

_{changed continuously}

from zero

to 500 um with an _accuray of 0.1 um. The thickness variations are measured

using

a

Fig.

7. - Junction of

an abnormal

_{finger tip}

with the side of an other

finger (C

= 1.92, V = 4.8

V ) :

a) just

before connection ;

b)

after connection. A T-like

_{sidebranching}

is formed.

the

_parallelism

between the two

_plates

to about ±

10-4

rd. Observations are made

_through

a

polarizing microscope

Leitz between crossed

polarizers.

4.

_{Experimental phase diagram.}

Fig.

8. - Director-field in the median

plane (z

=

0) (a 0

= 30

J. a)

T-like

sidebranching

formed after

connection of an abnormal

tip

with the side of a

_{finger ; b)}

because the twist is of the same

_sign

in a

normal

_tip

and in the side of a

_finger,

the connection is

_impossible

and it remains a nematic slice between

them.

way, it is

possible

to determine an

_{unambiguous phase diagram.}

Five

_regions

can be delimited

according

to the values of C and V.

_{They correspond}

to three fundamental solutions :

- the first

one is the

_homeotropic

nematic

_{phase (N)}

which is

_uniformly

dark between crossed

_{polarizers ;}

- the second

one

_corresponds

to the isolated

_{fingers (IF).}

Their width is well defined and

they

grow

only

from their two ends

_{(Fig. 9) ;}

- the third

one is a

_periodic

_pattern

of cholesteric

_{fingers (PP).}

For certain values of the

parameters,

the isolated

_fingers

are unstable with

_respect

to

_{tip-splitting instability. Only}

a

periodic

pattern

can

_develop

which fills _{up the space}

_{(Fig. 10)}

and leads to the so-called

«

_{fingerprint »}

texture. Note that

_only

the normal

_tips

can

split.

It is

_important

to note that these two solutions

(IF

and

_PP)

are not transient and do not

change

after

_growth.

Thus

_{they correspond}

to a minimum of the free energy.

Let us also mention a

_{translationally}

invariant

_{configuration (TIC)}

which is twisted

_only

in the direction normal to the

_{glass plates.}

This solution is unstable with

respect

to the

_periodic

pattern

of

_fingers

and can be observed

_{only transiently (Fig. 11 ) [12].}

Going

back to the

_{phase diagram}

of

_figure

_12,

we observe that :

- in

region

1

_{(at high voltage}

or _very small

_thickness),

the nematic is stable and the cholesteric is

_{unstable ;}

- in

region

II,

the nematic is still stable while the isolated

_fingers

are metastable :

_they

Fig.

9. - Growth of

an isolated

_finger.

a, b,

c)

Near the critïcal line

_(line

2 in the

_{phase diagram),}

the

finger

grows very

slowly

and remains rectilinear : C = 1.92 and V = 4.95 V.

d)

When _we decrease the

voltage,

the

_finger

undulates

_{spontaneously (explanation given}

in

_{Appendix II) :}

C = 1.92 and

V = 4.30 V.

unstable :

_they

_{break up}

_{spontaneously}

in numerous

_places

before

_disappearing

in a few tenths of a

_{second ;}

- in

region

III,

the nematic is metastable. The isolated

_fingers

are stable and

_lengthen

from their ends.

_{They usually}

nucleate from dust

_particles

which are

always

_present

in the

samples, showing

it is indeed a first order transition. Note that on the line

_{dividing regions}

II and

III,

the two

_phases

have the same free energy so that

_{finger tips}

are motionless. As in the

preceding region,

it is

_possible

to measure the

_finger

width as a function of the electric field

(Fig. 13).

Note a

_tendancy

of the

fingers

to

_{spontaneously}

_undulate,

_especially

near the

_region

IV discussed

below ;

- in

region

IV,

the nematic is still metastable. The isolated

_finger

solution is unstable with respect to

_{tip-splitting instability}

and

_{replaced by}

the

_periodic

solution which

_thereby

becomes the

_only

stable

solution ;

- in

region

V,

the nematic is unstable. «

_{Quenching »}

the

_sample

from

_region

1 into this

region

leads to the

_spontaneous

formation of the transient TIC

_{configuration.}

This solution is unstable

_(perhaps

_metastable)

and

_slowly

relaxes towards a double twist

_periodic

structure

Fig.

10.

- Growih

of a

_periodic

pattern

of fingers ;

C = 1.23 and = 1.18 V. Successive

photographs

are at intervals of 4 s. Because of the circular geometry of the domain,

splitting

of the

fingers

is

necessary to insure a constant mean

_wavelength.

Note that this process leads to numerous

_edge

dislocations in the

finger

pattern. Note also that normal

_{tips split}

alone, and that the

original

abnormal

tip

remains

_{unchanged during}

the

_growth

and well visible on the

pictures (the

arrow indicates the

Fig.

11. -

Spontaneous

and

_{homogeneous development}

of a T.I.C.

_{configuration.}

C = 1.92 and

V = 0.

a) t

= 1 s

_{(after switching}

off the

_{voltage) ; b) t}

= 3 _{s ;}

c) t

= 2 min.

Fig.

12. -

Experimental phase diagram. I)

nematic

_{phase ; II)}

metastable isolated

_{fingers : they}

Fig.

13.

_{- Finger}

width

_(in

units

_{of p)}

versus the

_{voltage (in volt)}

for various values of the confinement

ratio C. The width measurement is not

_{precise (+ 15 %)}

because it

depends

upon the

focusing

and the

optical

conditions. Our measurements have been made between crossed

_polarizers

and

_{by focusing}

in the middle of the

_sample.

We conclude this section

_{by noting}

that line 2 in this

_{phase diagram}

is the critical line

( C

=

c c ( V))

for coexistence between the _two

phases.

Line 3 is the

spinodal line

above which

the cholesteric isolated

_fingers

are

_unstable,

and line 0 the

_{spinodal line}

of the nematic below which the TIC

_{configuration spontaneously develops. Finally,}

line 1 is unusual : above

_it,

isolated

_fingers

grow, while

below,

only

a

_periodic

_pattern

_develops.

5. Theoretical model.

In this

section,

we

_adopt

the calculations of reference

_[3a]

to include electric field effects. The basic

geometrical

model is the DTC model described in section 2. As in reference

[3a],

we

assume that the

_trajectories

of the director on the

_sphere

are circular and we minimize the

energy with

respect

to the width A of the

_finger

and its

_angle

_{a o.} We recall that the free energy

per unit volume of a cholesteric

_phase

is

_(in

S.I.

_{units) [6] :}

where the

_{Ki ’s ( i}

=

1,

2,

3 )

_are the elastic _constants

for,

respectively, splay,

twist,

and bend

deformations

_{and q}

= 2 TT

/p.

The last term is the electric-field contribution. In the

_following,

we shall define the energy of the

_homeotropic

nematic state to be zero. We also assume, for

the sake of

_simplicity,

that E is constant and the

_{applied voltage}

is

_{given by V}

= Ed.

A

_{straightforward}

calculation

_gives

_{the energy} F of a

finger

_{per unit}

length :

The terms

_{I, J, L, M,}

N are sums of

_{integrals Ii depending}

_{upon a o}

_{(see Appendix 1) :}

with

_K32

=

K3/K2

and

K12

=

KI/ K2- It

can be

easily

checked that :

Let us first consider an isolated

_{finger (a}

case that was not considered in Ref.

_[3]).

Minimization with

_respect

to k

_{(or À)}

leads to the

_finger

width :

The

_{corresponding}

energy is after substitution into

equation (4) :

Because of the

_inequalities

noted in

_{equation (6),}

_{L + M / C + f (E)}

may vanish and

become

_negative

for certain values of the

_{parameters E,}

C _{and a o : in such} cases, the isolated

finger

is not a solution

(its

width becomes

infinite).

This is in

agreement

with our

_experiment,

which shows that isolated

_fingers

do not

_always

exist. We shall see later that for these values of the

_parameters

there still exists a

_periodic

solution. To

_push

further our theoretical

predictions,

we need to minimize F with

_respect

to _{a 0} which

_requires

numerical calculations. In

_figure

14 we

_{plot F (in}

units

_{of K2)}

for two

_particular

values of C =

d /p

and various values

of the

voltage V

(in volt).

Calculations have been made with

_K12

=

1.78,

K32

= 2.38 and 7r

KI/sa 80

= 1.91 V

(see below).

We now discuss these two

_examples

in

_greater

detail.

In

_figure

_14a,

C = 1.10. The isolated

finger

exists for all electric field values. For

V

_{:> V 3 =}

1.0

_V,

the

_only

minimum is _{a o} = 0 : the nematic is

stable,

the cholesteric unstable. For

_{V3:> V}

:>

V 2 =

0.62

V,

there are two minima in _ao = 0 and

ao

= 0

near to

_{ir /4:}

the nematic is stable and the cholesteric isolated

_finger

is metastable.

_{If V V 2,}

there are still two

minima,

but the cholesteric isolated

_finger

is now stable while the nematic becomes metastable.

In

_figure

_14b,

C = 1.16. The discussion remains

unchanged

as

_along

as V > V* = 0.45. For this value of the field and

below,

a

_divergence

of the

_finger

width appears : the located

finger

solution vanishes to the benefit of the

_periodic

solution that we discuss

_briefly

below

(see

Ref.

_[3a]

for more

details).

Up

to now, we have been interested

_only

in localized solutions. It is also

_possible

to

consider

_periodic

_pattern

of

_{fingers arranged}

side

_by

side. In this case the

_quantity

to minimize is

_{F (À,}

_{a o)/A}

_{i.e. the energy per} unit area of the

_pattern.

A

_simple

calculation

_{gives (in}

units of

_{K2 q) :}

for an

_{optimal wavelength :}

One sees

_immediately

that A is

_always

defined and

_{depends through}

_{a o upon the electric}

Fig.

14. - Free _{energy per unit}

_{length (in}

units

_{of K2)}

of an isolated

_finger

as a function of the

_voltage.

a)

C = 1.1 and the

voltage

varies from 0 to 1.5 V

by

increment of 0.1 V ;

b)

C = 1.16 and the

voltage

varies from 0 to 1.7 V

_by

increment of 0. 1 V ; when V is lower than V * .,: 0.45 V, the energy is no

_longer

defined for all values of ao and the

finger-width diverges.

Thus a

_periodic

structure is

_{expected (and}

indeed observed

_{experimentally)}

for all values of the

_voltage

lower than

_{V * (C).}

As a matter of fact we have never observed the

_divergence

of the width of an isolated

_finger

but

_only

an increase in width

_(see

_{Fig. 13)}

and line 1 in the

experimental phase diagram

does not

_exactly

coincide with

_{V * (C ).}

To determine

_analytically

the value of the field

_{V1 (C)}

below which the

_periodic

solution

_{develops (with necessarily}

V1(C)

>

V*(C)),

we must

_analyze

the

_{tip-stability}

of a

_growing

isolated

_finger.

This is a

difficult calculation that we have not done.

_{Nevertheless,}

the

_experiment

shows that the

tip-splitting instability

occurs as soon as the width

_a,

p. of an isolated

finger

is

roughly equal

to 1.4

App

where

_App

is the

_wavelength

of the

_{corresponding periodic}

solution

_[9].

This

experimental

crude

criterion,

which results from a

dynamical

_{process, allows} to estimate a theoretical value for

_{V1 ( C) (line}

1 in the

_{phase diagram).}

Finally

we calculate the

spinodal

line

corresponding

to the nematic

_{phase (line}

0 in the

phase diagram).

It is the curve below which the T.I.C.

configuration develops spontaneously

[7].

In reference

_[3],

we showed that this occurs at zero electric field when C >

K32/2.

This calculation can be

easily

extended to the case of an

_applied

electric field. It

yields :

In

_figure

15,

we

_{plot Vo (experimental)}

as a function of

V (C -

K32/2 )

and find a

_good

linear

_dependance.

From these

data,

we obtain

_{K = 2.38}

and

’TT / J KI/ £a £0= 1.91 V

knowing

that

_KI2:::::::

1.78

_(from

Merck

_data).

In

_conclusion,

we have all the theoretical tools we need to

_predict

the

_{phase diagram,}

represented

in

_figure

16. In

_spite

of a few shifts of the calculated curves with

_respect

to the

experimental

ones

_(certainly

due to the crude

_assumption

that director

_trajectories

are

_always

Fig.

15. -

Vo (experimental)

versus

V C - K32/2

with

_K32/2

= 1.19.

Fig.

16. - Theoretical

phase diagram.

we show in

_figure

17 theoretical

_predictions

for the width A of an isolated

_finger

as a function of the electric field. Here

_again,

a

_{good qualitative}

_agreement

is found with the

_experimental

data

_{(Fig. 13).}

All these results

_fully

confirm the

validity

of our

_{geometrical approach}

of this

unwinding

transition. Once

_again,

note that these calculations took into account

_anisotropic

elasticity (whence

our

_{approximations) ;}

_otherwise,

it would be

_impossible

to

_explain

the first-order character of this transition and most of our observations.

_Finally

we

_give

in

_appendix

II a

_rough

calculation which accounts

_{qualitatively}

for the undulation

_instability

observed in isolated

_fingers.

In the

_future,

it would be

_interesting

to extend this work to the

_dynamics

of the nematic-cholesteric interfaces that we

_previously

observed _{in the presence of} a

_temperature

gradient

v

Fig.

17. -

Finger-width

versus the

_{applied voltage (in volt)}

for various values of the confinement ratio

C : theoretical

prediction

based on

_{equation (7).}

-Acknowledgments.

We thank F.

Lequeux

and J. Bechhoefer for useful discussions. This work was

_{supported by}

D.R.E.T. contract No

_88/1365.

with and

Appendix

II.

Undulation

_instability

of an isolated

_finger.

The

_experiment

shows

_clearly

that an isolated

finger

tends to undulate

_{spontaneously}

with a

wavelength

that decreases with the electric field. This undulation results from a balance between the energy decrease due to the

_{finger’s larger}

area and the elastic _{energy increase due}

to its local

_bending.

To derive this

_result,

let a and A be the

_amplitude

and

_wavelength

of the undulation :

Calculation of the

_length

_{increase per unit}

_{finger, length (along}

the

_{x-axis) yields}

the free

energy

gained :

where

_FI.F.

is

_{given by equation (8).}

By

analogy

with rod

_elasticity,

we assume that the bend _energy of a

_finger

is

_proportional

to

the square of its curvature

_{1 /R.}

Thus we write the curvature _{energy in the form :}

K is a Frank

_{modulus, X =}

À d the area of a vertical section of the

_finger

and C a numerical

constant.

Adding

the two _terms, we

_get

the

_free-energy

variation :

This

_quantity

is minimized when

We conclude that an isolated

finger

undulates

_{spontaneously}

as soon as

_FLF.

is

_negative,

that is in

_region

III of the

phase diagram.

As one

_approaches

the critical line

2,

FLF.

goes to zero and ll becomes infinité : close to this line the

_fingers

must be almost rectilinear while _{undulations appear}

_{progressively}

as the electric field is decreased

( | FI.F. [

increases as E

_decreases).

These theoretical

_predictions

are _{in very}

_{good qualitative}

accord with our observations.

_{Quantitative comparison}

with the

_experiment

_requires

a

calculation of the numerical constant C. This is a rather tedious calculation that we have not

done.

References

[1]

BOULIGAND _Y., J.

_Phys.

France 34

₍₁₉₇₃₎

603 ; 35

₍₁₉₇₄₎

959.

[2]

For the

_experimental

evidence of the

_unwinding

transition induced

_by

the

_plates

confinement see :

a)

BREHM, FINKELMANN H. and STEGEMEYER H., Ber.

_{Bunsenges. Phys.}

Chem. 78

₍₁₉₇₄₎

883.

b)

HARVEY T., Mol.

_{Cryst. Liq. Cryst.}

34

₍₁₉₇₈₎

224.

For a theoretical calculation in

isotropic elasticity

and observations under

magnetic

field see :

c)

PRESS M. J. and ARROTT A. S., J.

_Phys.

France 37

₍₁₉₇₆₎

387.

d)

PRESS M. J. and ARROTT A. S., Mol.

_{Cryst. Liq. Cryst.}

37

₍₁₉₇₆₎

81.

[3a]

LEQUEUX F., OSWALD P. and BECHHOEFER J.,

Phys.

Rev. A 40

₍₁₉₈₉₎

3974.

[3b]

LEQUEUX F., J.

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France 49 (1988)

967.

[4]

DE GENNES P. G., Solid State Commun. 6

₍₁₉₆₈₎

163 ;

MEYER R. _B.,

_{Appl. Phys.}

Lett. 14

₍₁₉₆₉₎

208.

[5]

CLADIS P. E. and KLÉMAN M., Mol.

_{Cryst. Liq. Cryst.}

16

₍₁₉₇₂₎

1.

[6]

DE GENNES P. G., The

Physics

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_{Liquids Crystals (Oxford University}

Press,

Oxford)

1974.

[7]

We showed in reference

_[3]

that both the

_periodic

solution and the T.I.C.

_{configuration}

become unstable at the same critical _parameters values. Thus, the selection between these two

solutions is

_{purely dynamical,}

the T.I.C.

_{configuration having}

a _greater

growth

rate than the

periodic

solution.

[8]

OSWALD P. et al.,

Phys.

Rev. A 36

₍₁₉₈₇₎

5832.

[9]

It is

possible

to show that the critical line for both solutions,

namely

the isolated

_fingers

and the

periodic solution,

is the same

(line

2 in the

phase diagram).

The reason for which we did not

observe the

_periodic

solution in

_region

III, but

_only

the isolated

_fingers,

is

dynamical

and related to the nucleation and

growth

processes. On the other hand, our static calculation

shows that the isolated

_fingers

can

_only

be observed in a well defined

region

of the

phase

diagram, namely

between the lines

_V2(C)

and

V*(C).

[10]

Such a defect has been

previously

observed

by

several authors. See for instance :

a)

KAWACHI _M., KOGURE O., KATO _{Y., Japan} J.

_{Appl. Phys.}

13

₍₁₉₇₄₎

1457 ;

b)

BHIDE V. G., CHANDRA S., JAIN S. C., MEDHEKAR R. K., J.

_{Appl. Phys.}

47

₍₁₉₇₆₎

120 ;