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.
J.
Pliys
II
France
4
(1994)
127-143
1994,
JANUARY
127
PAGE
Classification
Physics
Abst;a(.ts
61.30G
61.30J
Crawling Ribibre,
P.
Laboratoire
Cedex
spiraling
and P. de
07,
(Received
Oswald
and
physique,
of S.
Pirkl
in
electric
field
(*)
Normale
Ecole
fingers
cholesteric
Supdrieure
de
Lyon,
46Allde
d'ltalie,
69364
Lyon
France
7
July
J993,
receii,ed
in
final form
29
September
J993,
accepted
J2
Or.tober
J993)
existent dans des dchantillons hom£otropes montrons types de doigts que deux liquides cholestdriques d'anisotropie didlectrique positive les doigts de premibre espkce dans lesqueh le champ de directeurs continu, seconde espkce qui sont est et les doigts de topologiquement singuliers et de mdme les sphdrulites (aussi appeldes « bulles » nature que cholestdriques). Quand les premiers sont soumis h un champ dlectrique altematif basse frdquence, ils lentement le long de leurs tandis seconds ddrivent perpendiculairement h rampent axes que les leurs forment des spirales quand une de leurs extrdmitds ddfaut. Ce travail et est pidgde sur axes un complbte les observations de spirales faites h Nice par Kamayd et Gilli [8] ainsi que par rdcemment Mitov Sixou [9] dans des systbmes similaires. et Rdsum4.
de
Nous
cristaux
liquid We show that cholesteric types of fingers exist in homeotropic samples of two 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 spherulites (also called bubbles). When the former are subjected to a low frequency cholesteric as AC electric field, they crawl slowly along their perpendicularly to their whereas the latter drift axes This work supplements spirals and forrn spirals when of their ends is pinned on a defect. axes one observed in Nice by Kamayd and Gilli [8] and by Mitov Sixou [91 in similar recently and systems. Abstract.
crystals
of
Introduction.
1.
By confining
of positive dielectric parallel glass plates anisotropy between two strongly homeotropically and/or, by subjecting it to an electric field, it is possible to it completely unwind and to obtain homeotropic nematic phase [1-4]. This a is usually first phase transition order and is controlled by two the applied voltage parameters V and the confinement ratio C d/p of the thickness the quiescent pitch. In the parameter over plane (C, V ), the homeotropic nematic fingers coexist on a critical line and the cholesteric which
anchor
a
cholesteric
molecules
=
j~)
Permanent
address.
University
of
Chemical
Technology,
53210
Pardubice
Czeck
Republic.
128
jOURNAL
DE
PHYSIQUE
N°
II
this stable between is stable above this line whereas the fingers are These fingers general, the director field inside the fingers is continuous. with having a cholesteric fingers of the first species (CF-I) in those will be called contrast cholesteric fingers of the second species (CF-2). Our discontinuity inside, which we shall call of fingers by of these kinds experimentally the existence main is to show two purpose analyzing their dynamical properties. they are much easier to produce studied intensively So far, only CF-l's have been because materials. there exists conventional polymeric Moreover, than CF-2's, with least at a non 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 surrounding critical CF-I's because they have the same the line, the do not lengthen energy as This observation determine experimentally the critical voltage nematic phase. has been used to
V
=
line
V~(C ).
The
and
C-axis.
the
nematic In
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 assumed reason, we in previous calculations [3, 4] that electrohydrodynamic effects negligible. This our were approximation is quite good as long as we are dealing with the « static » properties of fingers (topology, domain of existence and limits of stability in the parameter plane, also absolute called spinodal lines). On the other hand, a careful examination of their dynamical properties (growth in the nematic phase) reveals fine electrohydrodynamical effects exist that up to relatively high frequencies effects, which in the (about lo kHz). These shall examine we lateral drift of line and to the present article, lead to the crawling of the CF- I's near the critical CF-2's which form spirals when of the their ends is pinned to a dust particle. This one difference of dynamical behavior allows distinguish the two kinds of fingers. to us To our knowledge, only spiral formation has been mentioned and studied, first by Kamayd Gilli [8] in a and smectic A phase smectic A~cholesteric phase transition and then, by near a Mitov Sixou in a and cholesteric phase [9]. Although the experiment of Mitov and Sixou is close to these authors do not mention existence of the two types of fingers. By the very ours, Kamay6 and Gilli describe different fingers of contrast, two types of fingers but they compare the smectic A phase form spirals to those of the which cholesteric phase. Thus, it is likely that the fingers of type described different from by Gilli and Kamayd are the fingers of the two species described in this article. second article is organized as The follows. section 2, we briefly recall the In experimental procedure and the phase diagram. We then describe sections 4 in section 3 the crawling of CF- l's and in and 5 the spirals that formed CF-2's. emphasize spontaneously by In particular, shall are we the fundamental topological differences that exist between the fingers of the first and of the species. We shall also see that there exists a strong link between spherulites and fingers second of the species. second =
Experimental
2. The
cholesteric
compound
S811
phase
diagram.
liquid crystal (from
E.
was
prepared by adding
Merck)
to
nematic
BCB
small
a
(0.46 wtfG)
amount
of the
chiral
from
BDH
(4~n~octyl-4'~cyanobiphenyl
Limited). described experimental cell has been The the two continuously the distance between parallelism to within 10-~ rad. The their
(E. Merck)
and
all
cholesteric-isotropic
measurements
phase
transition.
have At
in
previous
a
electrodes
with
electrodes
have
been this
made
at
temperature,
[3].
article an
It
allows
us
to
of 0. I ~Lm and to accuracy with coated silane ZLI been
39
±
0.I
pi15.5
°C, ~Lm.
I-e-
3 °C
change adjust
below
3124 the
CRAWLING
N°
AND
CHOLESTERIC
SPIRALING
FINGERS
129
in reference [3] by observing established diagram is given in figure I. It was as the voltage is when species (sketched in Fig. 2) which form spontaneously critical coexistence between Line Vz (C is the line for lines visible. Four are CF-J's do not lengthen because the two phases have the the two phases : on this line, the same phase and of the fingers, spinodal limits of the nematic and free Lines Vo the V~ energy. are growth modes of CF-l's : above it, the fingers respectively. Finally, line V, separates two continuously leading to a rounded tip splits lengthen from their two tips, while below their
phase
The
fingers of the first quickly changed.
flower-like
pattern.
~
8
0
V~
V~ V~ #
V
b
~
> 4
>
,,-'
o o-S
I-o
2.0
1.5
3.o
2.5
3.5
4 o
C=d/p Fig.
Experimental phase diagram. in spirals are observed
I.
2's.
Stable
i~~
+
0.
V
Usually
the
Crawling
has
that
the
of
the
l's
shorten
The
from we
velocities
their
shall are
kHz.
phase
line
VI (C
diagram
to
CF-
corresponding
to
relates
this
CF-I
tip,
finite
of
AC
does
not
dielectric increase.
and
free
cholesteric
whereas
length,
normal,
called
cholesteric the
limit,
voltages V,
voltage of I kHz. To our depend upon the frequency (diminishing effects dielectric
wave
species.
first
isolated
ends,
Above
measured
rounded
a
everywhere as in the free locally of sign opposite to
following,
loo
to
each
tips
different
experimental
the
and
fingers
of
well-known two
0. I
whereas
l's the
of
using a square phase diagram
by
determined
are
occur
CF-
to
shaded
V.
2 §G),
range
F~)
0.5
+
2
V,
lines
anisotropy
It is
V
-,"iL
i
t
ii
ii I' it11, i'L%,-'/
,
i
,
i
I
,
I
ii,, i I ~ i I I
it Ii
,
'
'
ii
ii
i
I
i
Ii
ii
i I i
/
/
I I f I
~
iii
it
I
ii t
u
', ,
~',
i,,
',
,,~ 20
"~,,
),
30 3
2
4
V~ (V) Fig.
6.
Crawling
velocity
as
a
function
of
the
critical
voltage
at
f
1000 =
Hz.
134
JOURNAL
following, distinguish
the
In how
to
focus
we
are
in
7
shows
II
the
between
N°
of
types
two
fingers
and
coexist.
Their
describe
them.
region close and they are very non-polarized light (b).
Figure
difference
the
on
PHYSIQUE
DE
a
of the almost
sample in which the indistinguishable
of
types
two
whether
~
fingers
between
I~ ~
widths
polarizers (al
crossed
or
'~
'
~
b
50 Fig. well
without
as
CF-I
A field
of the sample polarizers (b), large one a CF-2.
Region
7.
and
the
convenient
way
suddenly
is
V~(C ) of CF-l's disappear and metastable
and
=
differentiate
to
This
indeed,
above
within shorten.
a
few
types of finger CF-2's difficult are
two
and
C
changed.
the
tenths
3.13,
them
is to is
limit, of
same
a
their
observe used
to
CF-l's
second,
experiment spirals (Fig. 8). One
The
to
Between
distinguish.
polarizers (a)
crossed
The
small
as
shows
arrow
a
3.9 V.
V
method
this
coexist.
can
behavior
when
the
accurately measure break spontaneously
the
whereas
be
performed
between
with
V~ the
in and
CF-2's
applied electric ~pinodal limit places
numerous
V~, after
that CF-2's still then observes developed are VI spinodal their limit (C is that larger than V~, which means difficult CF-l's (Fig, ii. Note that VI (C ) is more that, V~(C ), of than to When the voltage V~(C because CF-2's shorten very quickly at large voltages. slowly while remaining comparable decreases from V~ to V~, the width of CF-2's
formed
voltages
fully
:
where
CF-l's
pm
CF-l's
are
they
have
metastable much
much
measure
is to
at
higher than
increased that
of CF-
N°
CRAWLING
I
CHOLESTERIC
SPIRALING
AND
135
FINGERS
a
b
e
f
c
soo Fig.
8.
l's.
The
a-f,
From
In
thin
finger.
contrast,
thread
spiral
same
4.2,
V
at
4.6,
their
large
5.
photographed 5.8,
width
voltage.
6.4
and
decreases This
at
6.8
pm
increasing V ~f
voltages 000
=
very
quickly
observation
above
suggests
(C
3.13, =
circularly
polarized
light).
Hz ).
V~ (Fig. 9) that
there
is
finger singularity
the a
looks inside
like
a
the
136
PHYSIQUE
DE
JOURNAL
N°
II
30
fi
"
25
D
.
Q
CF-I
Q
~~
I
_
15 ~ m .
. "
V3
4.5
Fig. (C
9.=
3.13,
Width
A
f
000
of
=
the
5.5
5.0
cholesteric
fingers
"
6.O
V
(Volt)
of
the
two
.
~
6.5
species
7.O
as
a
function
of
the
voltage
Hzl.
of the difference in topology between the two types of fingers concerns they form segments of finite length. Whereas CF-l's different tips, a have two point having fingers rounded and a pointed (Fig. 4, exclude from the discussion a one one we CF-2's always have two similar rounded defect inside), 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 slightly above V~. Each of CF-2 shortens the voltage to a value abruptly decrease segment symmetrically from its two ends until a spherulite is formed lo, iii (Fig, 10). By contrast, at this voltage, any disappears by collapse of its two tips of opposite signs. The segment of CF-I CF-2-spherulite is irreversible. Indeed, growing a finger from a spherulite at transformation CF-2 CF-I rounded tips and a point small voltage leads but segment with two to a to a never middle of the finger (see Fig. 4c of Ref. [3]). defect in the It is also possible to the voltage VI for which the length of a small segment of CF-2 measure remains stationary. This critical voltage is slightly larger than V~ (about 0, I V). This means CF-l's. If the decreased below CF-2's slightly stable than voltage is that are more leads to undulating spirals Vi, CF-2's CF-l's undulate do below instability V~. This as (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). evidence
Another
their
5.
ends
Spiral first
We
which
on
when
dynamics. mention
that
the
of the
end
always one or several dust particles in the center of each spiral, anchored moving CF-2 is pinned. When the dust particles are strongly is rarely continuous This rotation and If they rotate. not not, move. can
there
is
glass plate, they do of gravity of the particle. It is also important accompanied motion of the is by a chaotic centre particles present in the sample than the dust spirals much less mention that numerous to are from experiment Most of the spirals particles cling the to the next. they also, rarely to one same (if left-handed right-handed the sample is turned equally Fig. 8) and single (as in or are are of them right-handed one). Some double into spiral transforms left-handed are a over a on
a
N°
CRAWLING
I
AND
SPIRALING
CHOLESTERIC
FINGERS
137
it
b
c
d
~~
50 Fig. lotwo
similar
Evolution ends
meet
of
a
each
CF-2
at
C
3.13,
V
4.3 =
V
and
f
pm 000
Hz.
A
spherulite
forms
after
the
other.
(Fig, 13a) or even triple (Figs, 13b, cl. We have also single ~pirals twice as thick as observed usual (Fig. 13d). By observing disappearance at large voltage, we have seen that they their composed of two CF-2's placed side by side. The multiple spirals are rare, so we shall were in a focus in the following single ones. Stationary spirals are observed the on common more small voltage range, usually between Vi and VI + 0.5 V (hatched region in the phase diagram of Fig, I). At larger CF-2's voltages, easily unpin from dust panicles and spirals quickly disappear.
138
DE
JOURNAL
~
500 Fig.
Fig. ii. develops spiral is Fig,
12.
V~~ the
7
=
the
N°
II
~
pm
500 Fig.
ii.
Undulating spiral (C
al when
PHYSIQUE
voltage
=
3,13, V
decreased
is
below
4 V
and
Vi. hi
The
=
f
=
same
000 Hz). spiral at
This
12.
spontaneously
undulation 4, I V.
V =
pm
At
this
voltage,
the
stable.
By subjecting an during a fraction
VI
voltage spherulites
decreased
before
(photograph
(b), V
is
(photograph (a), I' 3.8 VI to a large voltage (close to possible to break it into small, regularly spaced pieces. If these pieces have disappeared, it is possible to obtain strings of from the collapse of a piece of CF-?. V). Each spherulite comes
undulating of
second,
a
all
of
4.2 =
spiral
=
it is
3.13.
C =
shape of single spirals. They can always be fitted to good 3L(H wt). The pitch of spiral whose polar equation is p (H) velocity of the finger at 2 arJ~ and w is its angular is .S velocity. The transverse of plotted w (resp, i~~~) as a function infinity is v~, 3Lw. In figure14a (resp,14b), we (C 3,13). While (and i'for different spirals in the sample observed voltage same w consequently :f and 3L) vary from one spiral to another (these quantities probably depend on the which pins each spiral), v~~ is independent of the size and on the mobility of the dust particle Kamayd and Gilli [8]). More surprisingly, underlined by feature also (this spiral chosen was which be investigated. On the other small in the of the voltage independent range can v~, is also particular, decreases strongly above 15. In in figure f shown C and depends hand, v~~ a v~, as on This cut-off frequency is loo kHz. vanishes above cut-off frequency of about 3 kHz and We
have
accuracy the spiral
by
first
analyzed
the
Archimedian
an
=
=
=
=
N°
CRAWLING
I
SPIRALING
AND
CHOLESTERIC
FINGERS
139
r
a
b
c
soo
pm
d 13.
same
visible 2's
spiral (C
a) Double triple spiral at V
Fig.
note
placed
that side
by
the
4.5
ends side
(C
V.
The
of the
3,13,
I'
three
branches branches
three
3.13,
V
=
4 VI.
4?
are
V)
b) triple
have
split
identical
off
spiral (C from
d) thick
the
3.13,
V
4.2
=
center
where
simple spiral
no
cl the V) particle is by two CF-
dust
formed
JOURNAL
140
PHYSIQUE
DE
II
N°
j
A
Spiral Spiral
D
O. 8
h a
2 3
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
(Volt)
V
25
20
~
~
$
$
h
"
k ~
~
+
-
15
C
E
(
(b)
~
+ Spira12 Spira13 4.1
D h 4.O
V
Fig. f (C
Angular
14.-a)
=
3.13,
f
comparable decreases
v~~
extrapolating 6.
b)
Hz).
=1000
Tentative
velocity
Transverse
as
w
function
a
velocity
drift
as
v~~
of a
voltage function
V
for
of
voltage
spirals
different for
V
the
(C
=
3.13,
spirals
same
Hz).
000 =
to
(Volt)
that
v~~(C I
previously
found
we
strongly
with to
topological
C
zero,
model
and
suggests for
for
seems
that
the to
crawling vanish
spirals
at
should
of
C
~
Finally,
CF-l's.
the l.3.
disappear
This when
result, C
~
we
note
obtained
that
by
l.3.
CF-2's.
CF-l's in spite of their CF-2's different from clearly that are is that the collapse of any CF-2 observation microscope. The major In reference [I ii, have cholesteric bubble). called spherulite (also birth gives we to a segment 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 replaced by two point defects The two now are the plane perpendicular to the finger axis.
Our
observations
resemblance
show
through
the
N°
CRAWLING
I
SPIRALING
AND
CHOLESTERIC
FINGERS
141
25
-j ~
~~~
~
o 2
3
4
c=d/p
D
D
o
D
a
O
-
T
°
Cl
a
E
ii
~~~ ~
r
.. A 40 (1989) 3974. [8] al Gilli J. M. and Kamayd M., Liq. Cryst. ii (1992) 791; b) Liq. Ciyst 12 (1992) 545. [9] Mitov M. and Sixou P., J. Phj's. II Frant.e 2 (19921 1659 Mol. Cryst. Liq. Crysi. 231 ii 993) 11.
[10]
Kawachi
a)
b)
Haas
W.
M., E. L.
Kogure and
O.
Adams
and J.
Y.,
Kato
E., Appl
Appl. Phys. 13 (1974) 1457 25 (1974) 263 c) Appl. PhW.
Jpn Phys.
Lett.
Cijst
13
J.
535.
[I ii
Pirkl
S.,
Ribibre
P.
and
Oswald
P.,
Liq.
(1993)
413.
Lett.
25
(1974)
