
Electrically driven convection in liquid crystals
Jana Heuer, Thomas John, Dirk Pietschmann, Ralf
Stannarius

Introduction

Electrohydrodynamic convection (EHC) in anisotropic
fluids is a
standard system for dissipative pattern formation. A large variety
of electroconvection patterns in an external electric ac field has
been reported. Different combinations of the anisotropies of
conductive and dielectric material constants result in distinct
types of EHC structures. In the classical EHC studies, liquid
crystals with positive conductivity anisotropy and negative or
weakly positive dielectric anisotropy have been used. Both the
spatial (normal rolls, grid patterns) and temporal properties
(stationary, harmonic, subharmonic, travelling patterns) of these
systems are multifaceted and have been investigated thoroughly in
the course of the last 30 years. Beside the rich experimental
results, an extensive model based on the CarrHelfrich mechanism has
been developed that yields not only a linear stability analysis but
also a weakly nonlinear description. It explains the temporal
behaviour of the electroconvection system.
The
experimental observation is performed with a µm thin
transparent
glass cell wherein the nematic liquid crystal is sandwiched. Applying
an electric AC field normal to the cell will lead to an accelaration of
the charges that exist due to impurity. In the inhomogenous director
field these carges will separate into charge clouds. The flow
field
driven by this ionic flow couples to the director deflection. Thus,
convection rolls arise that are visible by means of the shadowgraph
method in the polarazing microscope.
The pattern appears above a certain threshold voltage with different
wave numbers that depend on the excitation frequency.

Sketch of the
convection and director patterns in a planar cell
Literature:
e. g. L. Kramer, W. Pesch: Electrohydrodynamic instabilities
in
nematic liquid crystals in: Pattern Formation in Liquid
Crystals, p. 221, editors A. Buka, L. Kramer (Springer, NY, 1996).

(a): oblique conduction rolls
(b): normal conduction rolls
(c): normal dielectric rolls
(e): higher instabilites
(g): ground state, no convection rolls

EHC patterns near the threshold voltage:
At higher voltages the pattern becomes more complex:

Unusual
electroconvection of bentcore nematics

In the materials studied in our
group, the signs
of the conductivity and dielectric anisotropy are different from the
standard EHC materials. The observed patterns differ
significantly from the classical types. Partially these patterns can
be described within the CarrHelfrich mechanism, but in some cases
novel or adapted models have to be discussed.
We have proposed a novel
mechanism that leads to patterns that are
qualitatively different from those of the conventional EHC. In this
context we investigated a bentcore (bananashaped) liquid crystal with
a positive
dielectric and a negative conductivity anisotropy [Tamba2007]. The
orientation
and optical behaviour of the observed patterns are no longer
describable within the classic
model. In contrast to the common EHC, the pattern evolves from a
distorted ground state above the socalled splay
Fréedericksz
transition. Moreover, the convection rolls are oriented along the
initial director easy axis. The usual stripe patterns are perpendicualr
to this axis or slightly tilted. Because of these qualitative
differences, we derived a new basic mechanism that
adapts the CarrHelfrich theory and is based on a twist instability
causing a modulation in the cell plane after a splay
Fréedericksz transition. Our model predicts qualitatively
correct the threshold behaviour and optical characteristics of the
observed electroconvection patterns [Stannarius2007,Heuer2008].




M.G. Tamba, W. Weissflog, A. Eremin, J. Heuer, R. Stannarius:
Electrooptic characterization of a nematic phase formed by
bent core mesogens, Eur. Phys. J. E 22,
85 (2007).
R. Stannarius, J. Heuer: Electroconvection in nematics above
the splay
Fréedericksz transition, Eur. Phys. J. E 24 27,
(2007).
J. Heuer, R. Stannarius, M.G. Tamba, and W. Weissflog. Longitudinal
and normal electroconvection rolls in a nematic liquid crystal with
positive dielectric and negative conductivity anisotropy.
Phys. Rev. E, 77
056206, (2008).
Subharmonic
patterns

In EHC experiments different regimes
can be observed that are
distinguishable clearly due to their spatiotemporal behaviour. Their
occurance depends on the excitation wave form and the excitation
frequency. The classic EHC studies, using sinus and square waves,
described two dynamic pattern regimes: Conduction rolls have stationary
director fields whereas in dielectric patterns the director deflections
change with the periodicity of the
excitation. We found a new regime with a dynamics that leads to a
system response with twice the period of the excitating voltage
[John2004]. Necessary for the occurrence of these novel patterns are
particular asymmetries of the applied wave form
[John2005,Stannarius2005,Heuer2006]. Antisymmetry, time
reversal symmetry and dichotomy of the excitation wave forms suppress
the
subharmonic regime. A new rigorous method to compute
the optical
properties of electroconvection patterns has been implemented on the
basis of FDTD (Finite Difference Time Domain) approaches [Bohley2005] 
e.g. sawtooth excitation: 
Spacetime plots of the three pattern regimes
(pseudocolor presentation of the optical transmission intensity) 
T. John and R. Stannarius. Preparation of subharmonic
patterns in nematic electroconvection. Phys. Rev. E, 70
025202(R), (2004).
T. John, J. Heuer, and R. Stannarius. Influence of excitation
wave forms and frequencies on the fundamental time symmetry
of the system dynamics, studied in nematic electroconvection,
Phys. Rev. E 71, 056307 (2005).
R. Stannarius, J. Heuer, and T. John. Fundamental relations
between the symmetry of excitation and the existence of spatiotemporal
subharmonic structures in a patternforming dynamic system,
Phys. Rev. E 72, 066218 (2005).
C. Bohley, J. Heuer, R. Stannarius. Optical properties of
electrohydrodynamic
convection patterns: rigorous
and approximate methods, J. Opt. Soc. Am. A 22,
2818 (2005).
J. Heuer, T. John, and R. Stannarius. Reentrant EHC Patterns
Under Superimposed Square
Wave Excitation, Mol. Cryst. Liq. Cryst. 449,
11 (2006).
Instability thresholds under
timereversed excitation wave forms

Standard model (CarrHelfrich)
Dynamic patterns that are described by differential
equations
of two
or more dynamic variables in general exhibit different trajectories of
these variables when the excitation wave form is changed. This leads in
general to different thresholds for the instability of the ground
state, even in a linear stability analysis (Floquet analysis)
[Stannarius2009]. It has been demonstrated by a linear stability
analysis of a simple twovariable model for EHC, and confirmed in
experiments [Heuer2008], that this dissipative pattern forming system
belongs to a special class of systems that is insensititve to such a
time reversal. The reason lies in the special structure of the
underlying differential equation system, which contains a matrix that
has the same temporal dependence in all offdiagonal elements
[Heuer2008].
The test of the mathematical modeling was performed in
experiments with superimposed square wave functions. The trajectories
of the two dynamic variables (charge density amplitude q and director
deflection amplitude
φ) differ for time mirrored functions (see figures below). The
thresholds and onset wavelengths are identical, within experimental
uncertainty, for both excitations.

"Forward" excitation wave form. The wave form is
asymmetric with respect to time reversal, its time mirror
(right) differs from the forward
wave form when the phase shift θ is not zero or multiple
integers of
90°.

Thresholds and pattern wavelengths of the
electroconvection pattern for forward (open symbols) and backward
(solid symbols) excitation with square waves in a 1:4 frequency ratio
[Heuer2008]

Trajectories of the system variables (charge q and director
deflection φ) for different phase shifts θ
between the high and
low frequency components. The trajectories for 0° coincide,
since the wave form only changes sign under time reversal. The
90° trajectories also coincide because the wave form is
symmetric, the 45° trajectories differ noticeably, while the
thresholds remain unchanged [Heuer2008]. 
Beyond the standard model
The experiments were later extended
to superimposed harmonic
waves, and it was found that in the vicinity of the cutoff frequency
separating conduction and dielectric regimes, the thresholds as well as
the critical wavelengths differ [Pietschmann2010]. This was observed
earlier, and a socalled weak electrolyte model was developed by
Treiber and Kramer to account for charge dissociation and recombination
processes. The breakdown of the classical EHC model is reflected in our
experiments in the different cutoff frequencies, pattern thresholds
and critical wavelengths in that region. Similar experiments with time
reversed excitation functions have been performed in Faraday wave
experiments [Pietschmann2013], where the thresholds were also found to
be independent of the timedirection of the excitation. 
Superimposed sine waves with frequency ratios 1:2 and phase
shifts corresponding to a forward/backward pair. The threshold voltages
and pattern wave lengths show a clear mismatch in the vicinity of the
socalled cutoff frequency [Pietschmann2010]. In that region, the
standard EHC model is obviously incorrect for the description of the
pattern onset (see pattern stability diagram on the right hand
side). 

R. Stannarius. Timereversal
of parametrical driving and the asymptotic stability of the
parametrically excited
pendulum.
Am. J. Phys., 77
164, (2009).
J. Heuer, T. John, and R. Stannarius.
Time reversal of the excitation wave form in a dissipative
pattern forming system. Phys. Rev. E, 78
036218, (2008).
D. Pietschmann, Th. John, and R. Stannarius. Nematic electroconvection under
timereversed excitation.
Phys. Rev. E, 82
046215, (2010).
D. Pietschmann, R. Stannarius, C. Wagner, and Th. John. Faraday waves under
timereversed excitation.
Phys. Rev. Lett, in press,
(2013).
Last modified: Feb
23, 2013,
ralf.stannarius@ovgu.de