From Bubble Domains to Spirals in Cholesteric Liquid
Crystals
S. Pirkl, P. Oswald
To cite this version:
S. Pirkl, P. Oswald. From Bubble Domains to Spirals in Cholesteric Liquid Crystals. Journal de
Physique II, EDP Sciences, 1996, 6 (3), pp.355-373. <10.1051/jp2:1996187>. <jpa-00248302>
HAL Id: jpa-00248302
https://hal.archives-ouvertes.fr/jpa-00248302
Submitted on 1 Jan 1996
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J.
Phys.
II
France
(1996)
6
Bubble
From
Pirkl
S.
(*) and
(Received
1995)
30
Laboratoire
June
PACS.47.20.Ky
Abstract.
a
We
frustrated
Physique (***),
revised
Defects
first
355
(**)
August
24
46
Allde
1995
and
d'Italie,
20
69364
Lyon
November1995,
Orientational
PACS.61.30.Jf
PAGE
Liquid Crystals
Cholesteric
of liquid crystals;
electric
and
Iiquid crystals
Nonlinearity (including bifurcation theory)
PACS.61.30.Gd
in
de
1995,
Spirals in
to
Oswald
P.
1996,
MARCH
Domains
Lyon,
E-N-S-
355-373
Cedex
accepted
07,
France
December
12
magnetic
field
effects
species
from
a
order
on
in
show
how
to
make
liquid crystal
the applied voltage:
either
cholesteric
depending on
metrical
spirals which we refer to as
isolated single spiral and, especially,
kinematical
using a simple
model in
studying the dynamics of the fingers
liquid crystal tends to zero.
twin
the
a
in
finger
an
of the
A-C-
the finger
spirals. We
trajectories
terms
of the
of the
second
second
field.
electric
Two
bubble
scenarios
are
domain
possible,
single spiral, or it forms two symthe long-time
behavior of an
These
of its two tips.
results
discussed
are
with free ends.
motion of
We finish by
curves
forms
then
species
a
describe
when
the
dielectric
anisotropy
of the
Nous
fabriquer un doigt de seconde espbce h partir
d'abord
montrons
comment
sph6rulite dans un cholestdrique frustrA soumis I champ dlectrique
alternatif.
Deux scdpossibles suivant la tension appliqude: soit le doigt forme une spirale simple, soit il
narios
sont
donne
deux spirales jumelles symAtriques
l'une de
l'autre.
ddcrivons
Nous
ensuite le comporteh long
d'une
spirale simple isolde, et plus
particuliArement
les
de ses
trajectoires
ment
terme
Ces
r6sultats
deux
extrdmitAs.
analys6s 1l'aide
modAle
d'un
cin4matique simple en termes
sont
de
de
courbes
extr6mit6s
libres.
Nous
terminons
des doigts
mouvement
aux
par la dynamique
de seconde espbce lorsque la
d141ectrique du cristal liquide tend vets z4ro.
constante
R4sum4.
d'une
Introduction
1.
of spiral
excitable
media is one of the classical problems of nonequiin
waves
pattern-forming systems.
Spirals have been observed in biology [ii, chemistry [2j,
hydrodynamics [3j nonlinear optics [4j and in liquid crystals.
One way to produce spirals in these
systems is to subject a nematic sample to a rotating
magnetic field and a high frequency electric field.
This
experiment was first performed by
Migler and Meyer [5j and was theoretically analyzed by Frisch et al. [6].
Another
way to
produce spirals is to apply an electric field to a frustrated
liquid crystal. In a DC
cholesteric
The
formation
librium
Permanent
address:
University of Pardubice, FES, Department of Physics,
(*
Republic
(**) Author for correspondence (e-mail: [email protected])
(***) Assoc16 au C-N-R-S-, U-R-A. 1325
Q
Les
#ditions
de
Physique
1996
53009
Pardubice,
Czech
JOURNAL
356
PHYSIQUE
DE
N°3
II
field, all the spirals have the same
handedness
and the
of
rotation
[7].
same
sense
from
phenomenon can be qualitatively
understood
symmetry
arguments using a LandauGinzburg model [8]. On the other hand, this model
explain the apparition of both
cannot
right-handed spirals in the presence of a high frequency AC electric field
left-handed
and
which is sufficiently large.
These spirals were first
observed in 1992 by KamayA and Gilli [9j
and Sixou [10] and RibiAre et al. [11] in ordinary
and then by Mitov
in a polymer
mixture
electric
This
cholesterics.
This
experimental article deals with these spirals and
description of their nucleation from cholesteric
bubbles.
order
In
helical
produce spirals
to
To do
structure.
in
that,
homeotropically,
enough, the helix can
a
cholesteric
confine
can
one
the
sample,
sample
dynamics.
their
it
is
necessary
between
two
includes
It
to partly
glass plates
the sample
a
detailed
unwind
that
its
anchor
surfaces [12]. If
thickness is
to the glass
completely unwind and the
becomes
The
nematic.
structure
be driven by a strong enough electric field when the liquid crystal
transition
dielectric
same
can
anisotropy Ea is positive [13]. By combining these two effects. it is possible to draw up a phase
d/p (the confinement ratio)
diagram in the parameter plane (C, V) [14]. The parameter C
divided by the equilibrium
cholesteric
pitch. A sinusoidal or squarethickness
is the sample
voltage V (measured in Volts, rms) is applied between the two electrodes bounding the
wave
f > 20 Hz typically). In
sample: its frequency f must be large enough to avoid
convection
usual materials, the
nematic-cholesteric
phase transition is first order [14,15] and the cholesteric
fingers coexist with the homeotropic nematic on a critical line V2(C). Note that the nematic
thermodynamic phase but is imposed by the homeotropic anchoring on the glass
is not a ne~v
plates when the sample is too thin.
molecules
small
normal
I-e-
even
=
far, only a few of their features have been
Archimedian
described
know that
their shape is
and that the larger
we
field
frequency,
the
and
nucleation
the
smaller
their angular velocity
their
In practice,
rate
are.
spirals are difficult to observe at j > 10 kHz. Recently, we gave experimental evidence that
fingers of the first species (CF-1 in the following) that
cholesteric
two
types of fingers exist:
cholesteric
fingers of the second species (CF-2) [1ii that
topologically
and
continuous
are
think
be
and
line
defects
along their
Only CF-2s may form
singular
contain
to
axes.
we
spirals.
These fingers have also very
with
singular
bubble
domains [16)
connections
strong
spontaneously form at the isotropic-cholesteric phase transition, or when a looped CF-1
which
collapses in an electric field [17j. In particular, it was shown that every segment of a CF-2
each bubble
transforms
electric field [iii;
domain in a large-enough
into a bubble
moreover,
n%ich
clearly
the
electric
field
defects
along
~vhen
contains
point
its
axis
is switched
two
separate
oil (see Fig. 12 of Ref. [17j). By contrast, the way by which
CF-2s
created
and
into
are
grow
well-developed spirals has not yet been described in detail.
line, spirals
[9-11]. For instance,
article,
this
Iii
critical
the
Near
u.e
observations
ne~v
may
So
occur.
explain how to make spiraling
long-time spiral dynamics
CF-2s
about
and
bubble
from
their
domains.
behavior
for
also give
ive
vanishing
dielectric
anisotropy.
plan of the
The
and
3,
4,
recall
~~e
sho~v
we
Sections
spirals.
the
how
describe
we
5
article
is
follows:
as
phase diagram
the
to
make
how
this
(experimental)
from
and 6
the
bubble
a
In
2,
Section
two-frequency
domain
a
finite
describe
we
cholesteric
segment
the
mixture
of
a
experimental
chosen.
CF-2
and,
set
up
In
Section
in
Section
single spiral or twin spirals.
(theoretical) are devoted to the long-time evolution of single
Section 7 what happens when the field frequency approaches
the liquid crystal
dielectric
anisotropy changes sign.
segment
Finally, ~ve describe in
frequency at ~vhich
crossover
of
destabilizes
to
give
either
a
SPIRAL
N°3
FORMATION
CHOLESTERICS
IN
357
12
o
V~
D
V~
.
V~
a
V~
~
~~
~
8
$
~
6
fi
>
o
0
1.2
1.6
1.4
c
Fig.
diagram
Phase
1.
f
at
(Ea
Hz
100
=
o)
>
2 0
8
d/P
=
and
T
28
=
°C
of
cholesteric
the
mixture
2LI2979+5811.
2.
Experimental
The
electro-optic
Part
cell
used
we
(ITO) are coated
anchoring. The glass plates
electrodes
ential
to
are
which
screws
allow
similar
to
silane
ZLI
3124
two
metallic
fixed
are
to
us
described
is
with
on
that
coupled together by three
holders
electrodes
two
transducer
within
ATA
101
obtain
to
change continuously the sample
adjust the parallelism between the
measured
using an L-V-T-D-type
[14j. The two parallel
strong homeotropic
a
reference
in
(E. Merck)
10~~ rd.
differ-
from 0 to 300 pm
thickness
The
(Schaevitz)
thickness
with
an
and
variations
accuracy
of
performed using a Leitz
Laborlux
12Pol
measurements
~lm.
were
polarizing microscope equipped with an OLympus OM4 photocamera, a Panasonic
CL-700
colour
CCD
Videocamera, a Panasonic
colour monitor, a
AG-6720
Panasonic
videocassette
recorder, and a Sony UP-811 videocopy processor.
For
the
used
pictures
measurements
on
we
Apple
Macintosh
Centris
660AV
Temperature
chosen
°C
30
and
regulated
is
is
computer.
an
observations
All
~0.05
and
~0.05°.
tO
liquid crystal was prepared by adding 0.615% by weight of the chiral molecule
the two-frequency
nematic
mixture
ZLI 2979 (E. Merck). Its equilibrium
measured
by the
conventional
Cano wedge method
and is equal to 16 ~lm.
The
cholesteric
The
5811
(E. Merck)
pitch
p
was
sample
critical
to
thickness
homeotropic anchoring
cholesteric-nematic
the
The
phase
d~
16.6
=
was
front
and
diagram of the
below
~lm
determined
in
CF-1s
was
Hz) and high frequency (30 kHz).
low-frequency limit If
100 Hz) and
Figure I. In this figure, V2 is the critical
100
=
nematic,
spinodal limit
boundary
Vo is the
shows
Finally, l~
worm-like
isolated
splits continuously
performed near C
the
fingers
leading
=
1.7.
grow
to
a
the tip of
of the
their
flower-like
cholesteric
completely
helix
the
distance
unwinds
in
between
low frequency if
corresponding to the
positive dielectric
anisotropy (Ea rzS +2) is shown in
line on which the CF- is coexist with the homeotropic
and % is the spinodal limit of the
nematic
CF-1s.
given
The
between
from
the
which
wedge sample by measuring
the wedge.
reference
in
of this
part
the
tips,
growth
two
while
growth
at
both
modes
of
the
Vi, the rounded
[14]. Most of our
below
patterns
[18]
=
phase diagram
above Vi,
tip of each finger
experiments
were
CF-1s:
JOURNAL
358
Forming
3.
in
Process
section, we show that
give precise data on
this
Fingers
Cholesteric
of
PHYSIQUE
DE
CF-2s
of
the
nucleate
may
II
bT°3
Second
Species
bubble
domains.
from
For
concreteness,
voltages (in Volts, r-m-s-) and velocities (in ~lm Is) at
shall
~ve
of frequency j
We recall that
C
An
A-Cvoltage
100 Hz was
used.
1.7.
wave
square
(typically
voltage
obtained
by
applying
large
V) to looped
bubble
domains
be
easily
10
a
can
domains
stable.
CF- is: at this voltage, only bubble
are
isolated
domains is
The experimental
procedure to obtain
segments of a CF-2 from bubble
the following:
measured
=
=
first,
.
switch
we
destabilize
T~vo
stages
2a
2c)
to
bubble
bubble
domain
that
is
CF-
to
faster
invade
is
CF-
are
rise
of
is
CF-1
a
the
CF-2
V
of the
core
of the
spiral
is
of
segment
a
polarizers
the
constant,
difficult
two
because
the
CF-
two
At
this
after
the
is.
Immediately
we
11'
and
core
crossed
held
is
a
segment.
disappeared,
6.3
Vn m
the
of
eliminate
the
the
~ve
to
V~"
than
completely
have
voltage between
spiral emerge from
switch
above
the
between
an
reason,
while
~lm),
separated by
defects
formation
of
tens
il.
6.65
"
tips (Figs.
rounded
lengthen,
domains
l§
than
with
optical contrast
(Fig. 3). if the voltage
faster
much
a
bubble
~.hich
smaller
CF-1
and
point
two
to
has
voltage slightly
shorten
of
long enough (a few
CF-2, and the
sample. For this
~vhole
slightly
domain
the
than
the
the
increase
the
thii~d,
.
starts
segments
t~vo
they
split, giving
When
intact.
lengthen
second,
voltage,
.
bubble
of the
2d to 2f). This segment
different
from the two others
CF-1s
the
sides
both
6.15 V
is
segments
two
below
m
voltage
This
First,
(
value
critical
a
(Fig. 2).
considered.
be
on
below
out
(Figs.
CF-2
a
stretch
must
grow
remains
voltage
the
and
rzS
6.72
Only
V.
this
in
of
range
Indeed, below Vn, two
CF-1s
a
the
from
from
the
ends
of
of
they
do
bubble
dumain
segment
to
start
grow
a
as
a
shortens, leading again to a bubble domain.
belon~ il.
Above V~", the segment of a CF-2
CF-2
but it drifts
perpendicularly,
At I/p, the length of the
remains
constant,
segment
voltages
its
to
of
segments
1[~) by
In
definition.
the
bubble
Further,
the
of
thei~e is
CF-2s.
bending
spirals.
~Te
Recall
~~oltage ii
a
segment.
CF-2,
also
that
V for
6.57
rzS
of
CF-2.
a
that
suppose
that
Initial
an
general
the
CF-
is
while
at 1~2
axes
Formation:
now
describe
we
Recall
along their long
<
V
vector
t
the
of the
the
of the
finger
rounded
ends
of the
used
make
small
lengthening
route
to
(resp. vt) changes sign
vtn
which
corresponding
and of the
the
vtn
sketch
first
never
keeping
stage of the
form
vtn
velocities
a
~[~.
<
rotates
i,oltage.
In
=
i,t.
it (resp.
Precise
and vt
vt,,
at
are
values
given
in
about
180°.
iii
an
A-C-
of
a
spiral from
electric
field
a
small
but
may
and
that
length [11].
constant
Stage
small
isolated
its
formation
spirals
segment
of
initial
T~vo
different
behaviors
CF-2
a
length is three to
easily the CF-2 deformation, we introduce
describe
more
and originating in the finger tip. During the initial stage
it,
and
longitudinal velocity
the
domains.
voltages V~'. ~j,
follon~ing section,
Spiral
~Ve
from
to
formation
of the
ends
CF-2
I.
segment
craiv.I
added
schematically
of the
vt,
CF-2
a
different
of the
Table
plot
~ve
that,
and
is
drift
This
responsible for the
is
Figure 5,
In
CF-
(Fig. 4).
axis
~j'
below
4.
n~ill
are
has
been
formed
larger than its width.
To
t parallel to the finger axis
vector
a
investigate in this paragraph, the
we
observed
depending on the value of
ten
times
FORMATION
SPIRAL
N°3
1.68,
=
d
b
e
axis.
the
two
finger
At
faster,
6.2
=
voltages,
By
=
=
it
of the
finger
linearly
the
I-e-
distance
length of
the
V is close
when
lengthens
almost
large voltages,
and
I-e-
contrast,
ends
varies
the
pictures showing the nucleation of
V, f
loo Hz. a) t
o; b) t
of
V
f
so pm
CF-2
a
19 s;
=
from
c)
t
=
a
bubble
23 s;
d)
domain
t
=
32 s;
(unpolarized
e)
t
=
58 s;
s.
small
At
its
case,
Sequence
359
a
C
Fig. 2.
light). C
f) t =102
CHOLESTERICS
IN
to
in
time
V is
when
between
finger
no
the
its
while
ends
two
approach
tend
to
very slowly perpendicularly
strongly bending (Fig. 6) so that
During this stage, the length of the
Vn, the finger drifts
to
from
each
other.
(Fig. 8).
close
to
two
ends
longer
Vi,
increases
finger behaves differently. it now drifts
change significantly (Fig. 7). In this
linearly as a function of time (Fig. 8).
the
does
not
JOURNAL
360
PHYSIQUE
DE
II
N°3
sopm
Segment
Fig.
3.
1'
6.8
=
V, f
of
CF-2
a
imbedded
in
CF-1
a
observed
between
crossed
polar12ers.
C
=
1.75.
Hz.
100
=
Table 1.
C
j
i~
1.7,
=
100 Hz
=
Values
6.18
6.3
6.55
6.63
vtn
1.3
0.95
0.25
0
vt
0.75
0.62
0.25
0.14
6.72
0
At the end of this initial stage, the
CF-2
bifurcates
towards
either a single spiral, or twin
spirals (Fig. 9). This
evolution
depends on both the voltage and the shape symmetry of
the spiraling finger. For
instance, at large voltages, twin spirals almost always form because
each end independently of the other.
the finger winds
around
The two spirals then strongly
and deform in the middle (Fig. 9A). By contrast, single spirals are very often
interact
obtained
if there is a small
This
scenario
at small voltages.
asymmetry when the two tips collide,
occurs
inside (Fig. 98). If the shape is symmetric.
that one tip goes outside while the other
enters
so
simultaneously go inside and twin spirals develop. In Figure 10. ~~e show
the two tips can
well-developed single and twin spirals.
In the following section, we focus our
single spirals.
attention
on
Long-Time
5.
In
this
to
a
move
Evolution
section,
freely.
velocity.
C
particle
dust
As
This
=
or
of
a
1.7
and
j
to
any
other
noted
behavior
above,
is
the
Single
Hz.
100
=
defects
Also,
as
for
spirals
Figure 1la, we show such a spiral and
lines). The applied voltage is close to Vn IV
a
circle
of
center
O
and
of
radius
r
rzS
31
Archimedian
are
In
~lm
only consider spirals that are
gla8s plates. In this case, their
we
the
on
spira18
these
same
Spiral
the
=
that
and
anchored
are
trajectories
6.5
V).
whereas
rotate
We
the
see
a
on
of its
that
external
two
the
tip
at
anchored
two
ends
constant
can
angular
particle [9-11].
dust
tips
not
I
and
internal
traces
a
E (dashed
tip I traces
curved
open
FORMATION
SPIRAL
N°3
IN
CHOLESTERICS
361
a
b
c
d
so pm
Fig.
f
=
Tranverse
4.
100
Hz).
trajectory.
with
an
directions
at
motion
further
We
of
interval
arbitrary
plotted
fixed
CF-2
a
between
the
11d.
in
that
Its
the
a
varies
following
is
at
evolution
time
(Figs. 11b,c).
direction
evolution
segment
micrographs
two
linearly
angular velocity. Finally,
and
constant
Figure
explained
in
Time
is 6
of two
We
=
1.68, V
see
angles
a
that
the
and
~
that
can
be fit to
a
Oi
two tips
the point I
that
time, which
means
of this spiral
plotted
the
length
we
and
Vi
=
and
V
6.72
"
mn.
in
nonlinear
section.
voltage Vi (C
quadratic
law.
as
and
rotate
in
traces
out
a
This
function
OE
make
opposite
a
of
circle
time
dependence
is
JOURNAL
362
PHYSIQUE
DE
N°3
II
~
I
,
~''
>
~
V~
V~
Fig.
ically
ends
as
of the
function
a
region).
gray
representation
Schematic
5.
'-normal"
below
;alues
~[,
CF-2s
and
CF-2
(see
undulate
Kinematical
6.
Vn, the
voltages
of the
of
Ref.
spirals
are
iii).
Above V~*, the
Single
for
ends
unstable
with
velocies
of the
only
are
ends
Tangential
spiral.
a
identical
two
Stable
V.
make
to
way
corresponding
of the
Model
the
and of the
applied voltage
of
Below
(utn)
CF-1s
CF-2s
in
of the
rounded
plotted
I/n and it
schemat-
are
between
the
collapse
given
are
to
velocities
jut
CF-2s
observed
respect
of
formation
into
Table
Voltage
v~*
V~
V~
bubble
CF-
(shaded
is.
while
Precise
domains.
1.
Spirals
~Ve
point of view the evolution of a CF-2.
propagation and from the gron,th of its
ends.
with its length, we shall
describe it by a
Because
the finger is thin in comparison
two
single curve with free ends. A similar description has been used for describing the 801id-liquid
interface in
dendritic
crystal growth jig] and the propagation of wave fronts in weakly, excitable
two-dimensional
media [20].
In
section,
this
show
that
Let tin be the
and I the
By convention,
Experimentally,
local
we
radius
shall
of
vt
vn
from
o
(resp.
depends
curvature),
vt
on
<
whereas
free end
0)
when
vo
Note
the
constant
vt
the
tangential velocity of
its
free
ends.
single spiral).
(resp.
shortens)
(Fig. 12).
lengthens
curve
k(I) of the curve (k
1/R where R is the
within
experimental
In the following,
errors.
internal
tip
in
the
case
of
a
=
take:
=
i>o(v, j)
vt
where
(the
curvature
is
vt
ii~
j.
curve,
one
local
the
normal
its
oriented
the
from
measured
>
theoretical
a
results
velocity of
normal
length
arc
from
examine
we
shape simply
its
IV, f), D(V, j) and
that
D has
the
same
vt
(V, j)
are
dimensions
=
vtll/, f)
functions
as
a
Djv, j)k
12)
of the
diffusion
(i)
applied voltage V and of its frequency,
Experimentally, stable spirals
coefficient.
CHOLESTERICS
IN
FORMATION
SPIRAL
N°3
363
%
50~Lm
A)
~
c
a
d
b
00 pm
B
100 Hz):
1.84, f
of a spiral at voltage V
7.3 V
Vn (C
formation
near
finger
tips in
indicate
position
of
and
Dots
time
106
CF-2s
0
growing
at
two
s.
s
and a bending of the left tip inside.
of 18 s. Note a velocity decreasing at tips collision
intervals
time
50 s; d) t
80 s.
20 s; c) t
0; b) t
B) Micrographs (crossed-polarizers illumination): a) t
Fig.
A)
Initial
6.
Superposition
stage of
=
=
develop when these three quantities
below
decreased
the voltage is
some
(see Figs. 11 and 12 of Ref. [11] ).
or
twin
by
spirals.
shown
Mikhailov
Let
k
=
et al.
k(I, t)
are
positive.
critical
[20j, equations
be
the
natural
value
(I)
If D
becomes
[11]), the
and
equation
(2)
can
of the
=
=
=
=
As
=
of
negative (this happens
CF-2s
destabilize
describe
curve.
the
It
can
and
formation
be
shown
when
undulate
single
[19, 20j
of
JOURNAL
364
PHYSIQUE
DE
II
N°3
~~~~
A)
c
a
d
b
~~ ~~
B)
Fig.
A
Initial
I.
Superposition
tips in
time
stage
of
intervals
of
three
of 50
of
formation
gro,ving
s.
Note
a
CF-2s
a
(crossed-polarizers illumination): a)
t
spiral
at
bending of
0; b) t
=
at
time
voltage
0 s,
both
=130
V
=
106 s, and
tips inside
s;
c)
t
=
7.6
V
400
Dots
before
250 s;
Vi (C
near
s.
collision.
their
d)
t
=
indicate
=
400
s.
1.84, f
100 Hz):
position of finger
=
B) J>Iicrographs
FORMATION
SPIRAL
N°3
CHOLESTERICS
IN
365
4oo
300
~
E
3
~
b
200
ioo
-A-
V=7.3V
+
V=7.6V
o
3
2
0
Fig.
Length
voltages,
8.
small
At
that
k is
of
function
of
linear,
whereas
as
law is
of the
solution
a
a
almost
CF-2
a
the
boundary
which
to
stage (C
during the initial
large voltages
time
6
it is
at
+
tl~~
/~ k(I)vn(I)dl
+
the
at
On
case.
°I
(which depend
(3)
physical problem)
considered
the
on
curve
[21j.
the
other
spiral rotates
frame, 8
with
fl +
=
angular
li
velocity
~
that, far from
the
Argsh(8)
origin (8
rd),
>
limit, k
this
solution
of the
-
0
and
vn
~
8~+2
p
(82 +1)3/2
steadily
/~
.
equation ii) is
the tips.
not
calculation
gives (in the
sufficient
to
then
very
the
towards
determine
easy
value
an
p
(5a)
(5b)
a
steadily rotating
Archimediaii
~ ~~
~~
whatever
rotating spiral tends
any
p(fl, t)
~~~
equation of
natural
'~'
It is
vo.
general equation (3)
.
=
8$@
~
+
2gr
the
reads:
~~~~
In
p
"~~
straightforward
A
-w.
~
so
~~ ~
equation
wt):
=
spiral
be
must
only be solved numerically
of stationary
the
existence
experimentally.
observe
we
~~~'~~
rotating
Hz).
I
=
This
100
=
nonlinear.
o
conditions
of the
ends
general
the
1.84, f
~)
-k~vn
=
Unfortunately, this equation can
hand, it can be used to show
spirals that
k(I) which are close to the
Archimedian
k
solutions
of the system.
attractors
These steadily rotating spirals are dynamical
spiral
with polar
Archimedian
consider an
To show this result, let us
(Fig. 11b):
added
in
=
strongly
following integro-differential equation:
~~
0t
S
4
(mn)
Time
to
of p.
~~~
This
Archimedian
which
that
check
is fixed
means
this
function
k(I)
is
a
that:
spiral;
by
the
boundary
conditions
at
JOURNAL
366
PHYSIQUE
DE
II
N°3
a
b
c
d
~
loo
~
~m
illumination)
of micrographs at C
1.84 and f
100 Hz (crossed-polarizers
T,vo
9.
sequences
A) twin spirals (V
bifurcation
sho,ving the
to~vards:
I.6 V): a) t
o; b) t
20 s: c) t
80 s;
Ii'= 7.3 v): a) t
d) t
20 s; c) t
o; b) t
So s: d) t
140 s.
170 s. B) single spiral
Fig.
=
=
=
=
=
=
This
has
checked
been
particle [11]. In this
and on its ability, to
By
simply
contrast.
as
a
the
case,
=
=
=
=
=
experimentally
the
of the spiral is
anchored
dust
when the
center
on
a
pitch of the spiral crucially, depends on the size of the particle
rotate.
pitch
function
of
seems
model
to
be
well
parameters
defined
vu,
when
ut
and
the
D
ends
from
are
our
free.
It
can
experiment.
be
calculated
Indeed,
n,e
FORMATION
SPIRAL
N°3
a
CHOLESTERICS
IN
200
367
b
pm
Well-developed spirals (crossed-polarizers illumination) at
10.
Hz): a) Single spiral; b) Twin spirals. Micrographs are taken 24
Fig.
100
from
small
CF-2s
V
mn
=
6.8
V
(C
spirals
after
1.76, f
=
start
to
=
grow
segments.
in Figure 11a that the spiral is tangent to the circle of radius r that point I traces
out.
Consequently, the normal propagation velocity vanishes at point [21j. In addition, the radius
of
Ro of the spiral at point I is very close to r (within experimental errors). It yields:
curvature
see
r
rzS
~
Ro
(7)
SQS
vo
Because
-w,
the
shape
observed
also
we
stationary
is
and
spiral
the
at
constant
angular velocity
~°~~
~~
w
=
=
while, far from the
(8)
D
r
center:
equations (8) and (9)
From
rotates
have:
calculate:
we
p
2gr~
(10)
=
vt
This
the
relation
To
shows
conclude
spiral. It
is
this
pitch of the spiral crucially depends
particle.
not pinned on a dust
the
that
tip when it
internal
is
section,
let
calcule
us
the
evolution
time
=
t
first
while
at
the
short
the
the
total
tangential velocity
of
length L(t) of the
given by the general equation [20j:
j /~ kvndt
The
of
on
term
in
second
the
one
time, L(t)
r-h-s-
must
the
expresses
corresponds
be
linear
to
its
2vt
due
to
from
the
(11)
the
two
normal
tips.
propagation
This
formula
of the
shows
curve
that
time
in
L(t)
lengtening
lengthenig
+
=
2vtt
when
t
<
~~
v~
(12a)
JOURNAL
368
Sense of
of the
PHYSIQUE
DE
N°3
II
rotaI>on
spiral
~
.
j
~
/
~~~~
a)
b)
~~
~~
6
~
5
~
D
tip
(Q)
200
@
°
c
§
~
4
400
fi
d
~
~
°
C
6Qo
i
~
~
I
100o
o
20
lo
Fig.
tips.
al Single spiral
11.
this
In
example,
the
spiral; c) Angles cv and ~5 as
c) Total length as a function
By
contrast,
Indeed,
vn
=
in
vo.
L(t)
this
Using
C
at
and
i-I
of
time.
of
time.
The
solid
nonlinear
at
limit, one
assume
can
equations (6) and (11),
L(t)
~
=
p
These
time
dependences
are
V
observed
=
line
solid
is
long time,
that
we
v(t~
the
the
on
The
trajectories of its two
representation of the
line is the
least-squares fit to a linear law;
least-squares fit to a parabolic law.
The
G-S V.
revolutions
4
function
a
becomes
=
tip did
inner
5
o
30
~
the
dots
~vhen
the
spiral
the
indicate
circle; b)
Schematic
growth
from
everywhere
is
the
tips is negligible.
Archimedian
and
that
obtain:
when
t
»
~~
(12b)
v~
experimentally (see Figs.
8 and
I
id).
SPIRAL
N°3
FORMATION
CHOLESTERICS
IN
369
R<o
~~ ~
~
E o=L(t)1
R>0
o=oi
v~>o
Fig.
Oriented
12.
normal
the
to
curve
tangential
the
curvature
(k
let
us
Finally,
spiral
velocity
is R
free
it, n)
=
the
free
that
the
of
ends.
+7r/2.
ends (vt
The
The
unit
vectors
and
t
propagation
normal
is
positive
when
trajectory
of the
external
the
curve
n
are
parallel
respectively
velocity is vn
lengthens).
while
van
#
The
local
and
vt
radius
is
of
1/R).
=
note
polar
with
with
curve
with
tip E tends
towards
logarithmic
a
equation:
)°
p
=
lo)j
(fl
pu exp
(13)
t
and
Experimentally,
j
100 Hz.
Spiral
We
know
Behavior
that
~lm
Is,
vt
the
Vanishing
at
SQS
0.38
~lm
Is
and
D
2.8 ~lm~
rzS
Is
C
at
=
1.7, V
=
6.5 V
the
shown
in
and
is
Figure
close
13.
A
to
kHz
sinusoidal
our
cholesteric
frequency j~
crossover
5
Anisotropy
Dielectric
anisotropy of
dielectric
temperature,
room
temperature
as
0.09
"
=
7.
At
vo
at
30
°C.
voltage
At
was
mixture
close
is
the
to
crossover
used
to
2
is very
kHz.
sensitive
It
frequency, l§
performed
this
frequency.
to
with
increases
and
Vo
experiment
the
diverge
[22]. We
frequency approaches j~. In particular, we
(Fig. 14). We see
measured
their drift velocity at Vi, where
length
their
remains
constant
function
that vd~ift
linearly
of
Vi
when
the
frequency
ive know from
increases
increases.
a
as
drift
of
the
CF-2s
does
depend
[1ii
that
the
velocity
previous
measurements
not
on frequency
f
of
kHz.
Thus,
the
of
mainly
due
the
the voltage.
increase
increase
to
< 2 -3
as long as
vdiiit is
dielectric
Note
that the
contribution
does
increase
this
experiment
since Vi
in
not
to
energy
know
be
im~ersely
proportional
the
is proportional to Vo (see Fig. 13), which
to
to
we
square
dielectric
f~. This clearly means
that
Thus EaE~ remains
root of Ea.
constant
at Vi when f
effects
alone explain the
observed
phenomena.
Note also that only twin spirals are
cannot
observed at large voltages (close to f~) because of the tendency for CF-2s to strongly bend at
their ends (Fig. is). Finally, we have
that the higher the frequency, the faster the
observed
CF-1s
crawl along their long axis at l§ (for instance, v~~awi @ 132 ~lm/mn at f
4900 Hz
~lm/mn
V).
and V2
j
Hz
and
53.8 V whereas
100
6.58
3
at
V2
v~rawi @
also
observed
the
behavior
of
the
CF-2s
as
the
~
=
"
=
"
JOURNAL
3m
PHYSIQUE
DE
N°3
II
ho
a
m
v~
v~
so
40
i~
~
~3
I
30
O
20
io
o
1000
3000
2lt©
4000
Frequency (Hz)
Fig.
13.
Voltages
and V2
Vo
frequency
us.
30
at
°C and C
i-I-
=
70
6o
~50
C
E
~
~
~~
~
~
20
lo
10
0
20
30
50
40
V2 (V)
*
Fig.
8.
We
14.
Drift
velocity
of
a
small
segment
of
a
CF-2
function
a
as
of
Vi
at
increasing frequency,.
Conclusion
have
shown
how
a
CF-2
forms
from
a
bubble
specific to the two-frequency mixture chosen, but was
octyl cyanobiphenyl) and 5811 (0.5% by weight). Two
finger of the second species lengthens, depending on
domain.
also
scenarios
its
nucleation
This
observed
degree
are
of
in
a
process
mixture
possible
when
symmetry
and
of BCB
a
on
is
not
(4-n-
cholesteric
the
value
SPIRAL
N°3
FORMATION
IN
CHOLESTERICS
371
a
b
e
f
c
200
Fig.
a)
t
Initial
15.
=
o; b)
t
stage of
45 s;
=
c)
t
of twin
formation
=105
d)
s;
t
=195
~m
spirals
s;
e)
at
t
=
large voltage (C
225 s; f) t
285
=
1.7, V
=
=
48
V, f
=
4700
Hz).
s.
applied voltage: either the finger leads to a single spiral, whose inner tip describes a
long time, or it gives twin spirals. We emphasize that only twin spirals are observed
j~ and the dielectric
anisotropy vanishes.
at very large voltages ~vhen j
theoretical
From a
point of view, single or twin spirals result from both the lengthening
and the normal
Their
spatio-temporal
ei<lution
be predicted
propagation of the CF-2s.
can
provided that vn and vt are known as a function of the control
parameters (V, j) and of the
k(I) of the finger. In particular, we have shown how to determine the model
local
curvature
observation
of single steadily rotating spirals.
parameters vu vt and D experimentally from the
On the other hand,
conditions
boundary
be specified at the free tips in order to select
must
solution
all the possible
solutions
of equation (3). A priori,
conditions
these
not
are
a
among
of the
circle
at
~
:
372
JOURNAL
the
same
CF-2s
II
media, but further investigations
understand
from
important to
now
very
Only such a model will give the suitable
excitable
in
as
particular,
In
PHYSIQUE
DE
it is
propagate.
N°3
are
a
determine
to
necessary
them.
microscopic point of view why
boundary
conditions
at the free
ends.
From
experimental point
an
voltage
to
induce
properties
medium's
would be interesting to slowly
spirals by analogy with what happens in
spatially or temporarily changed [20].
of view, it
drift of the
a
are
modulate
the
excitable
media
applied
when
Acknowledgments
This
Poste
supported by C-N-E-SRouge provided by the CNRS.
work
was
N°
contract
0253.
Slavomir
Pirkl
is
grateful
for
the
References
[ii
[2j
[3]
[4j
[5]
[6]
[7]
[8j
[9j
[10]
[iii
[12j
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[14j a)
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I.
FORMATION
SPIRAL
N°3
IN
CHOLESTERICS
373
M., Kogure O., Kato Y., Jpn J. Appt. Phys. 13 (1974) 1457; b) Haas W-E-L-,
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media, it is usually assumed that boundary
In
excitable
conditions
at the free ends are of
~~
~~~
~
the form:
With this choice,
show
numerics
-vt
vt
[16j a)
Kawachi
Adams
[17]
[18j
[19]
[20]
[21j
~°
t
=
°
,
i=o
dt
"
°i
i=L(t)
rotating spiral traces out a circle. On
a streadily
spiral cuts this circle at a right angle at point I, in disagreement with
cholesterics.
These boundary
conditions
certainly not the correct
are
that
[22] We
on
the
found
the
tip
inner
that
shape
the
of the
of
voltages l§, l§', etc in Volts rms do
applied voltage (sinusoidal or square-wave).
values
of
not
the
our
ones
hand, the
other
observations
in
depend
our
in
system.
2%
within