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The corkscrew instability of a Fr ´eedericksz domain wall in a nematic liquid crystal
The corkscrew instability of a Fr ´eedericksz domain
wall in a nematic liquid crystal
Alberto de L ´
ozar Mu
noz, Thomas Bock, Matthias M ¨
uller,
Wolfgang Sch ¨
opf and Ingo Rehberg
Experimentalphysik V, Universit¨at Bayreuth, D-95440 Bayreuth, Germany
E-mail:
wolfgang.schoepf@uni-bayreuth.de
New Journal of Physics 5 (2003) 63.163.12 (
http://www.njp.org/
)
Received 10 January 2003, in nal form 26 March 2003
Published 6 June 2003
Abstract.
A liquid crystal with slightly positive dielectric anisotropy is
investigated in the planar conguration. This system allows for competition
between electroconvection and the homogeneous Fr´eedericksz transition, leading
to a rather complicated bifurcation scenario. We report measurements of a novel
instability leading to the corkscrew pattern. This state is closely connected
to the Fr´eedericksz state as it manifests itself as a regular modulation along
a Fr´eedericksz domain wall, although its frequency dependence indicates that
electroconvection must play a crucial role. It can be understood in terms of a
pitchfork bifurcation from a straight domain wall. Quantitative characterization is
performed in terms of amplitude, wavelength and relaxation time. Its wavelength
is of the order of the probe thickness, while its ondulation amplitude is an order
of magnitude smaller. The relaxation time is comparable to the one obtained for
electroconvection.
Contents
1.
Introduction
2
2.
Experimental set-up
4
3.
Phase diagram of ZLI-3086
5
4.
Corkscrew instability
7
4.1.
Jump procedure
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
4.2.
Extraction of the domain wall
. . . . . . . . . . . . . . . . . . . . . . . . . . .
7
4.3.
Measurements for increasing amplitudes
. . . . . . . . . . . . . . . . . . . . .
9
5.
Discussion and summary
10
References
11
New Journal of Physics 5 (2003) 63.163.12
PII: S1367-2630(03)58130-7
1367-2630/03/000063+12$30.00
© IOP Publishing Ltd and Deutsche Physikalische Gesellschaft 63.2
1. Introduction
For the study of pattern formation under non-equilibrium conditions, convection in uids is a
classical example due to its similarity to numerous pattern-forming phenomena in nature [
1
].
The most prominent system is RayleighB´enard convection in simple uids, where a horizontal
uid layer is heated from below. If the temperature difference across the layer exceeds a certain
threshold value, the system undergoes a bifurcation from a uniform and motionless state to a
convective state. Convection may occur in the form of regular patterns, e.g. stripes, squares or
spirals, but also in the form of chaotic motion [
2
].
Another system that has been investigated intensively both theoretically and experimentally
in the context of symmetry-breaking instabilities is a nematic liquid crystal under the inuence
of an alternating electric eld [
3
][
5
]. Depending on the frequency and amplitude of the applied
eld in connection with the size and sign of certain material parameters, a huge variety of
convective states may arise [
5
,
6
]. It turns out that the primary instabilities show a high degree
of universality for which a full classication has been achieved [
1
]. A full classication of
the secondary instabilities destabilizing the primary patterns, however, seems a tremendous
task, except for the quasi-one-dimensional case [
7
] where they have been analysed in detail
for RayleighB´enard convection in simple uids [
8
]. Thus a systematic study of the nonlinear
behaviour of nematic liquid crystals is highly desirable.
A nematic liquid crystal is an intrinsically anisotropic uid with uniaxial symmetry which
can be described macroscopically by anisotropic material parameters. The molecules have a
tendency to align parallel to each other, leading to a preferred axis characterized by the director
eld n, and thus to a long-range order in the equilibrium state. Important for the interaction
with an electric eld and with free charges are the dielectric anisotropy,
a
, and the anisotropy of
the electrical conductivity,
a
, respectively. For
a
>
0 (
a
<
0) the director has a tendency to
align parallel (perpendicular) to the applied eld. For
a
>
0 (
a
<
0) charges move preferably
parallel (perpendicular) to the director. In the planar conguration, the liquid crystal forms a
thin uid layer conned between two transparent electrodes, with the director being aligned in
a preferred direction in the layer plane by a suitable treatment of the electrodes. An ac-voltage
U (t )
= 2U
rms
cos(2f t) is applied across the electrodes, where the rms-amplitude U
rms
and
the frequency f serve as the main control parameters of the system.
When increasing the voltage U
rms
beyond a certain critical value U
c
(f )
, the homogeneous
and motionless basic state, with the director oriented parallel to the boundaries, becomes unstable.
For materials with
a
<
0 and
a
>
0, the phenomenon of electroconvection is encountered at
threshold [
9
]. The most prominent primary pattern consists of normal rolls, where the roll axis is
perpendicular to the director eld. In addition, oblique rolls can be observed in liquid crystals with
sufciently low
a
for small driving frequencies [
6
,
10
]. For a material with
a
>
0, on the other
hand, the homogeneous Fr´eedericksz transition occurs for a critical voltage U
f
= k
11
/
0 a
which is independent of the driving frequency [
3
]. Here, k
11
is the splay elastic constant of the
liquid crystal.
When increasing the driving voltage in the case of electroconvection, the primary patterns
destabilize via secondary instabilities, such as, for example, the long-wave zigzag or skewed-
varicose patterns [
11
,
12
]. For certain frequencies, a homogeneous instability can also be
observed, which is caused by a twist in the director eld leading to abnormal rolls [
13
,
14
].
These secondary instabilities typically give rise to the formation of domains in the liquid crystal,
so that the two possible and equivalent states are separated by domain walls.
New Journal of Physics 5 (2003) 63.163.12 (
http://www.njp.org/
) 63.3
Figure 1.
Magnication of the corkscrew pattern. The shadowgraph technique
gives the impression of a chiral object.
This paper is concerned with the case that the Fr´eedericksz transition and a convective
instability can be observed in the same system, which is possible for nematics with weakly
positive
a
and positive
a
. The competition between the two instability mechanisms promises
an interesting bifurcation scenario. For certain frequencies, for example, we nd a convective roll
pattern as the primary threshold, which upon increasing the voltage develops into a Fr´eedericksz
state characterized by the presence of domain walls. Although the birefringence method indicates
a normal Fr´eedericksz state, it is modulated by convection, which must be present near the
domain walls. We speculate that this is the reason for the stability of so many domain walls,
which otherwise would be expected to decay until only one wall is left. Typical for this state
is the appearance of a zigzag instability along the domain walls, similar to the one observed
in a liquid crystal experiment using homeotropic alignment and a combination of electric and
magnetic elds [
15
]. A further increase of the voltage destabilizes the domain walls into a novel
state that, due to its peculiar appearance, we have termed the corkscrew pattern. Figure
1
shows a strong magnication of such a state. The caustic line produced by the shadowgraph
method gives the impression of a chiral object, although this does not necessarily mean that the
underlying director eld is chiral as well. Indeed, we do not have a model for the director eld
yet.
An example sequence for increasing voltage is shown from left to right in gure
2
. The
preferred director orientation is in the horizontal direction, i.e. roughly perpendicular to the lines.
The zigzag instability of the domain walls can be seen in the middle part of gure
2
, while it is
obvious from the right part, that the corkscrew pattern is a new state different from the zigzag
lines. It rather appears as a further instability on top of the zigzag pattern, but may also occur
along straight domain walls, as will be discussed later. Depending on the driving voltage and
frequency, the corkscrew state (right part of gure
2
) may be rather dynamic, as can be seen
from the accompanying MPEG movie.
Our goal is to characterize the corkscrew pattern in terms of its range of existence in phase
space and to measure its critical voltage and wavelength as functions of frequency, so as to foster
its theoretical description. It turns out that the occurrence of the pattern depends strongly on the
New Journal of Physics 5 (2003) 63.163.12 (
http://www.njp.org/
) 63.4
Figure 2.
A convective roll state at f
= 20 Hz near the primar