Crawling and spiraling of cholesteric fingers in electric field

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Crawling and spiraling of cholesteric fingers in electric field

P. Ribière, P. Oswald, S. Pirkl

To cite this version:

P. Ribière, P. Oswald, S. Pirkl. Crawling and spiraling of cholesteric fingers in electric field. Journal

de Physique II, EDP Sciences, 1994, 4 (1), pp.127-143. �10.1051/jp2:1994119�. �jpa-00247944�

Classification Physics Abst;a(.ts

61.30G 61.30J

Crawling and spiraling of cholesteric fingers in electric field

P.

Ribibre,

P. Oswald and S. Pirkl

(*)

Laboratoire de physique, Ecole Normale Supdrieure de Lyon, 46Allde d'ltalie, 69364

Lyon

Cedex 07, France

(Received 7 July J993, receii,ed in final form 29 September J993, accepted J2 Or.tober J993)

Rdsum4. Nous _montrons que deux types de doigts existent dans des dchantillons

hom£otropes

de cristaux liquides cholestdriques

d'anisotropie didlectrique positive

les doigts de premibre espkce dans

lesqueh

le champ de directeurs _est continu, et les doigts de seconde espkce qui sont

topologiquement

singuliers

et de mdme nature que les sphdrulites (aussi

appeldes

_« bulles _»

cholestdriques).

Quand les

premiers

sont soumis h

un champ dlectrique altematif basse frdquence, ils rampent ^{lentement le} long de leurs _axes tandis que les seconds ddrivent perpendiculairement h

leurs _axes et forment des spirales quand une de leurs extrdmitds _est

pidgde

_{sur un} ddfaut. Ce travail complbte les observations de spirales faites rdcemment h Nice par

Kamayd

et Gilli [8] ainsi que par Mitov _et Sixou [9] dans des systbmes similaires.

Abstract. We show that two types ^of

fingers

exist in

homeotropic

samples of cholesteric

liquid

crystals of positive dielectric anisotropy fingers of _a first species in which the director field is continuous, and fingers of _a second species ^which are topologically singular and of the _same nature as spherulites (also called cholesteric bubbles). When the former _are

subjected

to a low

frequency

AC electric field,

they

crawl slowly along their _axes whereas the latter drift

perpendicularly

to their

axes and forrn spirals when _one of their ends is pinned on a defect. This work

supplements

spirals recently observed in Nice by Kamayd and Gilli [8] and by Mitov and Sixou [91 _in similar systems.

1. Introduction.

By confining

a cholesteric of

positive

dielectric

anisotropy

between two

parallel glass plates

which anchor molecules

strongly homeotropically

and/or,

by subjecting

it to an electric field, it is

possible

to unwind it

completely

and to obtain _a

homeotropic

nematic

phase [1-4].

This

phase

transition is

usually

first order and is controlled

by

two parameters the

applied voltage

V and the confinement ratio C

=

d/p

of the thickness _over the

quiescent pitch.

In the parameter

plane (C,

V ), the

homeotropic

nematic and the cholesteric

fingers

coexist _{on a} critical line

j~) Permanent address. University of Chemical Technology, ⁵³²¹⁰ Pardubice Czeck Republic.

V

=

V~(C

). The nematic is stable above this line whereas the

fingers

are stable between this line and the C-axis. In

general,

the director field inside the

fingers

is continuous. These

fingers

will be called cholesteric

fingers of

the

first species (CF-I)

in _contrast with those

having

a

discontinuity

inside, which _we shall call cholesteric

fingers of

^the second

species (CF-2).

^Our

main purpose is _to show

experimentally

the existence of these two kinds of

fingers by

analyzing

their

dynamical properties.

So

far, only

CF-l's have been studied

intensively

because

they

are much easier to

produce

than CF-2's, at least with conventional _non

polymeric

materials. Moreover, ^{there exists} a

topological

model _on the unit

sphere

S~

[5]

which allows _us to

explain

the various

topological properties

of CF-l's _as well _as their main

optical [6]

and

energetic properties [3, 4, 7].

On the critical line, the CF-I's do not

lengthen

because

they

have the _same energy as the

surrounding

nematic

phase.

This observation has been used _to determine

experimentally

the critical

voltage V~.

In

general, experiments

_are carried _out with _an AC electric field. Its

frequency

is chosen

high enough (typically f

= I

kHz)

in order _to avoid convection. For this _{reason, we} assumed in _our

previous

calculations

[3, 4]

that

electrohydrodynamic

effects _were

negligible.

This

approximation

is

quite good

_as

long

as we are

dealing

with the _« static _»

properties

of

fingers (topology,

domain of existence and limits of absolute

stability

in the parameter

plane,

^also called

spinodal

lines). On the other hand, _a careful examination of their

dynamical properties (growth

in the nematic

phase)

reveals that fine

electrohydrodynamical

effects exist up to

relatively high frequencies

(about lo

kHz).

These

effects,

which _we shall examine in the present article, lead to the

crawling

of the CF- I's _near the critical line and to the lateral drift of

the CF-2's which form

spirals

when _one of their ends is

pinned

to a dust

particle.

This

difference of

dynamical

behavior allows _us to

distinguish

the _two kinds of

fingers.

To our

knowledge, only spiral

formation has been mentioned and studied, first

by Kamayd

and Gilli

[8]

in _a smectic A

phase

near a smectic A~cholesteric

phase

transition and then,

by

Mitov and Sixou in _a cholesteric

phase [9]. Although

the

experiment

of Mitov and Sixou is

very close to ours, these authors do _not mention the existence of the two types ^of

fingers. By

contrast,

Kamay6

and Gilli describe two different types ^of

fingers

but

they

_compare

fingers

of the smectic A

phase

which form

spirals

to those of the cholesteric

phase.

Thus, it is

likely

that the

fingers

of type two described

by

Gilli and

Kamayd

are different from the

fingers

^of the

second

species

described in this article.

The article is

organized

as follows. In section 2, _we

briefly

recall the

experimental procedure

and the

phase diagram.

We then describe in section 3 the

crawling

of CF- l's and in sections 4 and 5 the

spirals

that _are

spontaneously

formed

by

^CF-2's. In

particular,

_we shall

emphasize

the fundamental

topological

differences that exist between the

fingers

of the first and of the second

species.

We shall also _see that there exists _a strong ^{link between}

spherulites

and

fingers

of the second

species.

2.

Experimental phase diagram.

The cholesteric

liquid crystal

_was

prepared by adding

a small amount

(0.46 wtfG)

of the chiral

compound

S811

(from

E.

Merck)

to nematic BCB

(4~n~octyl-4'~cyanobiphenyl

from BDH

Limited).

The

experimental

cell has been described in _a

previous

article [3]. It allows _us to

change continuously

the distance between the two electrodes with _an accuracy of 0. I ~Lm and to

adjust

their

parallelism

to within 10-~ rad. The electrodes have been coated with silane ZLI 3124

(E. Merck)

and all measurements have been made _at 39 _± 0.I °C, I-e- 3 °C below the

cholesteric-isotropic phase

transition. At this temperature,

pi15.5

_~Lm.

The

phase diagram

is

given

in

figure

I. It _was established _as in reference

[3] by observing fingers

of the first

species (sketched

in

Fig. 2)

which form

spontaneously

when the

voltage

is

quickly changed.

Four lines _are visible. Line

Vz (C

is the critical line for coexistence between the two

phases

: on this line, the CF-J's do not

lengthen

because the _two

phases

^have the _same free energy. Lines

Vo

and

V~

are the

spinodal

limits of the nematic

phase

and of the

fingers,

respectively. Finally,

line

V,

separates two

growth

modes of CF-l's _: above it, the

fingers lengthen

from their two

tips,

while below their rounded

tip splits continuously leading

to a

flower-like pattern.

8

~

⁰

V~

V~

V~

# V

b ^~

>

4

> ,,-'

o

o-S I-o 1.5 2.0 2.5 3.o 3.5 4 o

C=d/p

Fig.

I.

Experimental

phase diagram. Lines

I',

(C)

apply

to CF- l's whereas line

VI

(C relates _to CF-

2's. Stable

spirals

are observed in the shaded

region

of the phase diagram corresponding to

i~~ + 0. V

< I'

< V

2 + 0.5 V.

Usually

lines

V,

are determined

by using

a square wave AC

voltage

of I kHz. To _our accuracy

(usually

2 §G), the

experimental phase diagram

^does not

depend

upon the

frequency

chosen in the range 0.I to loo kHz. Above this limit, dielectric effects

(diminishing

dielectric

anisotropy

_{F~) occur} and the measured

voltages V,

^increase.

3.

Crawling

of

fingers

of the first

species.

It is well-known that each isolated CF-I of finite

length,

which does not contain _a

point

^defect.

has two different

tips

a rounded

tip,

called normal, because the twist inside has the _same

sign everywhere

as in the free cholesteric and _a

sharp tip,

called abnormal, in which the twist is

locally

of

sign opposite

to the free cholesteric

[2, 3].

Between lines V~

(C

) and

V~(C

), the CF~

l's shorten from their ends, whereas between lines

~~~(C

) and

Vj (C

),

they lengthen.

In the

following,

_we shall call v~ (he

velocity

^of the normal

tip

_{and u~ the}

velocity

of the abnormal

tip.

The velocities _are chosen

positive

when the

fingers lengthen.

^Since the molecular

configur-

ations in the two

tips differ,

their velocities _are also

expected

to differ. One _can also introduce the

lengthening velocity

of the

fingers

: vj~~~~

df/dt

_where f _{is the}

length

of the

finger.

This

velocity

is

positive

if the

finger length

^{increases and}

negative

if it decreases. If the

finger

is rectilinear viength ^~ r~ + c~. ^{At V} =

V~,

the

lengthening velocity

of

straight fingers

vanishe.q

by

200~m

al

1111111111111111

1it iii

it

ii it it I

iJll/"/

_{t i}

_{iiii I} illll---~

,

"ii

i

_/ /w~~--~~-~,, _{, ,} ' I

i

I

I'---"'I

_{t, ,,} I

I

II11-"'~i

_I

>

'it

Ii

_i,

_-,"iL

_i

t ii

I

ii ' ,

ii ii

_{I' t}

_ii

I i i , i

it11,

/ /

i

Ii

I , I

i'L%,-'/ I

ii,,

<1,",~-'/

i

i

_{I ~}

, _, '"~--l'/

I

i I

_{I ' '}

~ --v'll/

I

it

_{iii i}

i llllll

f I

Ii it

I Ii I I I J J

Ii

ii ii ii

b)

Fig. 2. a) Cholesteric finger of the first species photographed between crossed polarizers. b) Director field of _a CF-I in _a vertical plane norrnal _to its axis.

~~~~~~~~~~ ~~stmightf<ngerlensth ^"

~~

~ ~~~~ ^~~S _, ~length ~ ~ ~ ~z _~ ~z2 ^W ~.j¹ ~ ~lcngth ~ ~ W~ en

V ~

V~.

In

figure

3 _we have

plotted tip

velocities v~ and v~ as a function of the

applied

RMS

voltage

at

f

=

lkHz and C =1.48. In this

figure,

each

point

represents the average of several

measurements. We have checked that the

tip

velocities _were

independent

of the

finger length

as

long

as the

finger

is _{not too} much curved at its _two ends. The

experimental dispersion

in

velocity

measurements, of the order of _± 5 ~Lm/s, is

probably

caused

by inhomogeneities

of the electrodes due to the surface treatment. One _sees that both velocities v~ ^and v~ vary

linearly

as a

function of the

voltage

in the whole range of accessible

voltages (Vi

<V

<V~).

^The

deviations from the linear law observed _near

Vj

is

probably

caused

by

the

widening

of the

finger tips

which will result in

splitting

below

Vi.

We also note that v~ and u~ do not vanish

simultaneously

_on the critical line when V

=

V~ (at

this

voltage

v([[([~~~~~~~~ =

0).

This

unexpected

result shows that the rectilinear

fingers

craw'l in the

sample

w,hile

keeping

a

~~~

'~~~

,

~~$~r~~~~ps

loo

~'

0

C

E

j

'

'~ '~

~'

'

£

", ",

/ §i

"

" "

'

",

',

',

' ', ^"

Vi V2 ',

",

~

'

'

',

" "

l.0 1-1 1.2 1.3 1.4 1.5

V(Volt)

Fig. 3. Velocities u~ and u~ of the norrnal and abnormal tips of CF-l's _{as a} function of the applied ^RMS voltage ~l'= 1000 Hz, C

= 1.48).

constant

length (Fig.

4). This

phenomenon

^is not an artifact due _{to a} thickness

gradient

in the

sample

because both c~ and v~ are

independent

of the orientation of the

fingers

in the

sample plane.

Furthermore,

neighbouring parallel straight fingers

oriented head to foot

always

_move in

opposite

directions. We shall call the abnormal

tip velocity

_{v~ = v~,} the

crawling velocity

~ ~~~~ ~/ ~/ _~~ ~st~aighl^finger

crawl 2 length

The

crawling velocity

is _a

complex

^function of the

sample

^{thickness and of the}

frequency.

In

figure

5 _we

plotted

_v~~~~j versus

frequency

for various values of the confinement ratio C

=

d/p. Surprisingly,

_u~~~~j can

change sign

when the thickness is increased. In contrast, the

cut-off

frequency

^{above which}

crawling disappears (defined

_as the

frequency

at which

v~~~~j is half of its

low-frequency

^{value) is}

roughly independent

^{of the thickness and}

equals

lo kHz. In

figure

6, _we

plotted

_v~~~~,j as a function of the critical

voltage V~

at

f

= I kHz. This

curve confirms the

crawling velocity

^inversion at

V~

_m ^{1.85 V}

corresponding

to C

m 2. We

also note that _v~~~~j vanishes when

V~

_- ^0, as

expected.

Crawling

is _not the

only

manifestation of

electrohydrodynamic

effects.

Indeed,

_{one very} often observes the formation of

rotating spirals

in our

samples

after _a

long period

(several hours in

general).

^These

spirals

involve

singular fingers

_as will be shown in the next section.

4. Characterization of the cholesteric

fingers

of the second

species.

In this section, _we focus _on

fingers

that form

spirals

when the

applied voltage

is held at a value

close to

V~.

These

spirals

have been observed

previously,

first

by Kamay6

and Gilli in _a

smectic A

phase [8],

^{and then}

by

Mitov and Sixou in _a cholesteric

phase [9]. They

are formed from _new

fingers

^which have almost the same

optical

contrast as the classical

fingers

of the first

species.

This _can

explain why

Mitov and Sixou do not mention the _two types ^of

fingers.

Moreover, the nucleation time of

spirals

is much

longer

in _our

experiment

^than in the

experiment

of Mitov and Sixou

(it

varies from half _an hour at

f

=

lo Hz _to several hours _at

f

= I

kHz).

This

probably

comes from the value of the electric field, much smaller in _our

experiment (typically

E

= 600

V/cm)

than in theirs

(E

1

10~

_{V/cm ).}

a

b

c

d

50 _iLm

Fig.

4. -Crawling

finger

of the first

species.

The time interval between _two

photographs

is 2 _mn

(u~~~~j = 20.4 ~m/min, V V~ 3.8 V, f 000 Hz).

-'~ . .

'c 0

E

_"

E ^"

~ ^lo

)

~

(

. V~=

1.58

V,

C=1.72

"

m

V~=

1.85 V,

30 ~

"

M

V~= 4.10

10~

10~

10~ 10~

10~

f

_(Hz)

ig.

5.

C

=

d/p

I 1.5 2 2.5 3

,'~ _i,

, ,

,

,~

,

,

~- , ,

, ,

b ,'

~

,

E

~

- ,

= ,

~ ,

e ',

u ,

>

~',

',

i,,

,,~

20

"~,,

),

30

2 3 4

V~ (V)

Fig. ^6. Crawling velocity _{as a} function of the critical voltage at f

= 1000 Hz.

In the

following,

we focus on the difference between the two types ^of

fingers

and describe how to

distinguish

them.

Figure

7 shows _a

region

of the

sample

in which the two types ^of

fingers

coexist. Their widths

are very close and

they

are almost

indistinguishable

whether between crossed

polarizers (al

_or in

non-polarized light (b).

I

~

~ '~

'

~

~

b

50 pm

Fig. 7. Region of the sample where the two types ^of finger coexist. Between crossed

polarizers

(a) as

well _as without

polarizers

(b), CF-l's and CF-2's _are difficult to distinguish. The small _arrow shows _a CF-I and the large _{one a} CF-2. C

= 3.13, V 3.9 V.

A convenient way ^to differentiate them is _to observe their behavior when the

applied

electric

field is

suddenly changed.

This method is used to measure

accurately

the

~pinodal

limit

V~(C

) of CF-l's _: indeed, above this limit, CF-l's break

spontaneously

in _numerous

places

and

disappear

within _a few tenths of _a second, whereas between

V~

and

V~,

^CF-l's are

metastable and shorten. The _same

experiment

can be

performed

with the CF-2's after

they

have formed

fully developed spirals (Fig.

8). One then observes that CF-2's _are still metastable at

voltages

much

larger

than

V~,

which _means that their

spinodal

limit

VI (C

is much

higher

than that,

V~(C

), ^of CF-l's

(Fig, ii.

Note that

VI

(C ) is _more difficult to measure than

V~(C

because CF-2's shorten very

quickly

at

large voltages.

When the

voltage

is increased from

V~

to

V~,

^{the width of CF-2's decreases}

slowly

while

remaining comparable

to that of CF-

a

b

_e

c

f

soo _pm

Fig. ^8. The _same spiral photographed at increasing voltages (C

= 3.13, circularly polarized light).

From a-f, V 4.2, 4.6, 5. 5.8, 6.4 and 6.8 V ~f

=

000 Hz ).

l's. In contrast, their width decreases _very

quickly

above

V~ (Fig. 9)

the

finger

looks like _a thin thread at

large voltage.

This observation suggests ^that there is _a

singularity

inside the

finger.

30

"

fi

25

. D CF-I

Q

~~ ^Q

_

15

I

~

m

. .

V3 ^{" "} .

~

4.5 5.0 5.5 6.O 6.5 7.O

V

(Volt)

Fig. 9.- Width A of the cholesteric

fingers

^of the two species as a function of the

voltage

(C

= 3.13, f ₌ 000 Hzl.

Another evidence of the difference in

topology

between the _two types ^of

fingers

concerns

their ends when

they

form segments ^{of finite}

length.

Whereas CF-l's have two different

tips,

_a

rounded _one and _a

pointed

one

(Fig. 4,

_we exclude from the discussion

fingers having

a

point

defect inside), CF-2's

always

have _two similar rounded

tips (Fig. lo).

In order to make the segments ^of

CF-2,

_we

subject

_a

spiral

to

voltage VI during

a fraction of _a second and _we then

abruptly

decrease the

voltage

to a value

slightly

above

V~.

Each segment ^{of CF-2 shortens}

symmetrically

from its two ends until a

spherulite

is formed lo, iii

(Fig,

10).

By

contrast, at this

voltage,

any segment ^{of CF-I}

disappears by collapse

of its _two

tips

of

opposite signs.

The transformation

CF-2-spherulite

is irreversible.

Indeed, growing

_a

finger

from _a

spherulite

at small

voltage

never leads _{to a} CF-2 but _{to a} CF-I segment ^with two rounded

tips

and _a

point

defect in the middle of the

finger

(see

Fig.

4c of Ref.

[3]).

It is also

possible

to measure the

voltage VI

for which the

length

of a small segment of CF-2 remains

stationary.

This critical

voltage

is

slightly larger

than

V~

(about ^0, I

V).

This _means

that CF-2's _are

slightly

_more stable than CF-l's. If the

voltage

is decreased below

Vi,

CF-2's undulate _as do CF-l's below

V~.

This

instability

leads to

undulating spirals

(Fig. II). By subjecting

_an

undulating spiral

to a

large voltage,

it is

possible

to break it

regularly

and to nucleate

strings

of

spherulites (Fig.

12).

5.

Spiral dynamics.

We first mention that there is

always

one or several dust

particles

in the center of each

spiral,

on which the end of the

moving

CF-2 is

pinned.

When the dust

particles

are

strongly

anchored

on a

glass plate, they

do not move. If not,

they

_can rotate. This rotation is

rarely

continuous and is

accompanied by

_a chaotic motion of the centre of

gravity

of the

particle.

It is also

important

to mention that

spirals

are much less _numerous than the dust

particles

present ⁱⁿ the

sample they

also,

rarely cling

to the _same

particles

from _one

experiment

to the next. Most of the

spirals

are

single

(as in

Fig. 8)

and _are

equally

left-handed _or

right-handed

(if the

sample

is turned over a left-handed

spiral

transforms into _a

right-handed

one). Some of them are double

it

b

c

~~

d

50 _pm

Fig. lo- Evolution of _a CF-2 at C 3.13, V

= 4.3 V and f 000 Hz. A spherulite forms after the two similar ends meet each other.

(Fig, 13a)

or even

triple (Figs,

13b, cl. We have also observed

single ~pirals

twice as thick _as usual

(Fig. 13d). By observing

their

disappearance

at

large voltage,

we have _seen that

they

were

composed

of two CF-2's

placed

side

by

^{side. The}

multiple spirals

are rare, so we shall focus in the

following

_on the more common

single

ones.

Stationary spirals

_are observed in _a small

voltage

_range,

usually

between

Vi

and

VI

+ 0.5 V

(hatched region

in the

phase diagram

of

Fig,

I). At

larger voltages,

^CF-2's

easily unpin

from dust

panicles

and

spirals quickly

disappear.

~ 500 pm ~ 500 _pm

Fig. ii. Fig. 12.

Fig. ii. al

Undulating spiral

(C

= 3,13, V

= 4 V and f

= 000 Hz). This undulation

spontaneously

develops when the voltage _is decreased below

Vi.

hi The _same spiral at V

= 4,I V. At this voltage, the

spiral

is stable.

Fig, ^12. By

subjecting

_an undulating spiral (photograph (a), ^I' ₌ ^3.8 VI to a large voltage (close to V~~ = 7 VI during _a fraction of _a second, it is possible to break it into small, regularly spaced pieces. If the voltage is decreased before all of these pieces have disappeared, ^{it is} possible to obtain strings of

spherulites (photograph (b), V

=

4.2 V). Each

spherulite

comes from the collapse of _a piece of CF-?.

C

= 3.13.

We have first

analyzed

the

shape

of

single spirals. They

can

always

be fitted to

good

accuracy

by

an Archimedian

spiral

whose

polar equation

is _p

(H)

=

3L(H wt).

The

pitch

of the

spiral

is .S

=

2 arJ~ and _w is its

angular velocity.

The transverse

velocity

of the

finger

at

infinity

is _v~,

=

3Lw. In

figure14a (resp,14b),

we

plotted

_w

(resp,

i~~~) as a function of

voltage

^{i'for different}

spirals

observed in the _same

sample (C

=

3,13).

While _w

(and consequently

:f and 3L) vary from _one

spiral

to another (these

quantities probably depend

on the size and _on the

mobility

of the dust

particle

which

pins

each

spiral),

_v~~ is

independent

of the

spiral

^chosen (this feature _was also underlined

by Kamayd

and Gilli

[8]).

More

surprisingly,

v~,is also

independent

of the

voltage

in the small _range which _can be

investigated.

On the other

hand,

_v~~

depends

_on C and

f

as shown in

figure

15. In

particular,

_v~, decreases

strongly

above _a

cut-off

frequency

^{of about 3 kHz and vanishes above loo kHz. This cut-off}

frequency

is

r

a

b

_c

soo pm

d

Fig. 13. a) Double spiral (C 3,13, I' 4? V) b) triple

spiral

(C

= 3.13, V 4.2 V) cl the

same triple spiral at V 4.5 V. The three branches have split off from the _center where _no dust particle ^is visible _note that the ends of the three branches _are identical d) thick simple

spiral

formed by two CF- 2's

placed

side by side (C 3.13, V

=

4 VI.

A

j

O. 8 D Spiral2

h Spiral3

a A

c

~ ^~

+ a

(a)

# ^°

8 ^~

a

+ A

~ A

+

+ a

+ ~

A +

~ A

+

I

~

4.O 4.1 4.2 4.3 4.4 4.5 4.6

V

(Volt)

25

20 h $ $ ^{~ ~ "} k

~ ~

- +

C 15

E

( (b)

+

~

D Spira12 h Spira13

4.O 4.1

V _(Volt)

Fig. 14.-a) Angular velocity _w as a function of voltage V for different spirals (C

= 3.13, f =1000 Hz). b) Transverse drift velocity _{v~~ as a} function of voltage V for the _same spirals (C

= 3.13, f

=

000 Hz).

comparable

to that _we found

previously

for the

crawling

of the CF-l's.

Finally,

_we note that

v~~ decreases

strongly

with C and _seems to vanish at C

~ l.3. This result, obtained

by extrapolating

_v~~(CI to zero, suggests that

spirals

should

disappear

^{when C} _~ ^l.3.

6. Tentative

topological

model for CF-2's.

Our observations show

clearly

that CF-2's _are different from CF-l's in

spite

of their

resemblance

through

^the

microscope.

The

major

observation is that the

collapse

of any ^CF-2 segment

gives

birth _{to a}

spherulite (also

called cholesteric

bubble).

In reference

[I ii,

_we have

suggested

that the

singularity along

the

spherulite

axis is

actually

two

point

defects of

opposite

strength.

In the present ^article, we propose that CF-2's have the same

topology

as

spherulite~

in the

plane perpendicular

to the

finger

^axis. The two

point

^defects now are

replaced by

two

25

-j

~

~~~

~

o

2 3 4

c=d/p

D D o a D

- O

T ^°

Cl _a

E

ii

~

~~~

<r

5 _a

a

io~

_iO~

io~ io~ io~

f

(Hz)

Fig.

15. al Transverse drift velocity _v~~ as a function of C _at voltage V

=

VI

~f =1000 Hz).

b) Transver~e drift velocity _{v~~ as a} function of frequency f (C ₌ 3.13, V ₌ 4.15 VI.

singular

lines _or disclinations

(of strengths

S

= + I and S

=

Ii joining

the _two ends of the

fingers. Figure

16a shows the director field in _a vertical

plane perpendicular

to the

finger

axis while

figure

16b shows the director field in the median

plane parallel

to the

glass plates.

^One

sees in

figure

16b that the two ends of the

finger

are identical, in _contrast with CF-l's which

have two different

tips.

In

spite

of this fundamental difference, it

clearly

_appears,

by

comparing figures

2b and 16a, that director fields inside both

finger

types ^{resemble each} other, which

explains why they

are so difficult _to

distinguish through

^the

microscope. Moreover,

both

director fields

identically

match the

homeotropic

nematic

phase.

This

explains why

the

abnormal

tip

of _a CF-I _can merge into the side of _a CF-2,

just

_as it would have with _a CF-I, and form _a T-like sidebranch (see

Fig.

2a)

[3].

7. Conclusion.

The main conclu~ion of this article is that there exist

experimentally

_two

finger types

:

fingers

of the first

species

^which _are

topologically

continuous and

fingers

of the second

species

which

I I I I I I I

I / / i I I I I I I

I / /

J i I

I I

I i I

I I I

I I

_J

, ~

4

~, , i I

I I

I J

i _, i

I I

I ~ i

I I

I I

_{J ,} I

I

J J ,

i I

I I

_/ -,

I I _i

_{J -,}

I I

I

i _i i

I I

i _,

I

I

I > 7 I

I I

_{I >} 7 I

I

~,

-, /

l

~,

~'/ l

I ~%~--ll1 ~~--~ll I

I ~'~~---~>--~---ll 7 I

I ~"" ~ / llllf I

I ~"'i I I till / I

I I I I I I I I I I I I I I

a)

b)

Fig, 16. Director field in _a cholesteric finger of the second species a) ^section by a plane perpendicular

to the

finger

axis b) section by a median plane parallel to the glass plates. The two ends of the

finger

_are

identical in _contrast with cholesteric

fingers

of the first species.

are

singular.

^{The former have}

differently shaped

ends and crawl

along

^their _axes ⁱⁿ an AC electric field. The latter have similar

ends, collapse by leaving spherulites,

and _move

perpendicularly

to their _axes. This _transverse motion _may lead to

spirals.

The

question

now is _to

explain

the

origin

of these

electrohydrodynamic phenomena.

Acknowledgments.

We thank L. S. Tuckerman for fruitful discussions. This work _was

supported by

DRET

Contract No. 92/1313/DS/SR.

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Stegemeyer

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