Dynamics of cholesteric structures in an electric field
O.~S. Tarasov$^{1,2}$
A.~P. Krekhov$^{1,2}$
and L.~Kramer$^1$
% Motivated by Lehmann-like rotation phenomena in cholesteric drops we study the transverse drift of two types of cholesteric fingers, which form rotating spirals in thin layers of cholesteric liquid crystal in an {\em ac} or {\em dc} electric field. % We show that electrohydrodynamic effects induced by Carr-Helfrich charge separation or flexoelectric charge generation can describe the drift of cholesteric fingers. % We argue that the observed Lehmann-like phenomena can be understood on the same basis. %
fingers.tex
%\documentclass[aps,prl,preprint,groupedaddress]{revtex4}
\documentclass[aps,prl,twocolumn,groupedaddress,showpacs]{revtex4}
%
\usepackage{bm,graphicx}

\begin{document}

%Title of paper
\title{Dynamics of cholesteric structures in an electric field}

%Authors
\author{O.~S. Tarasov$^{1,2}$, A.~P. Krekhov$^{1,2}$, and L.~Kramer$^1$}
\affiliation{
$^1$Physikalisches Institut, Universit\"at Bayreuth,
D-95440 Bayreuth, Germany\\
$^2$Institute of Molecule and Crystal Physics,
Russian Academy of Sciences, 450075 Ufa, Russia}


\date{March 25, 2003}

\begin{abstract}
%
Motivated by Lehmann-like rotation phenomena in cholesteric drops we
study the transverse drift of two types of cholesteric fingers, which 
form rotating spirals in thin layers of cholesteric liquid crystal
in an {\em ac} or {\em dc} electric field.
%
We show that electrohydrodynamic effects induced by Carr-Helfrich charge
separation or flexoelectric charge generation can describe the drift of
cholesteric fingers.
%
We argue that the observed Lehmann-like phenomena can be understood
on the same basis.
%
\end{abstract}

% insert suggested PACS numbers in braces on next line
\pacs{61.30.Gd, 61.30.Jf, 47.20.Ky}


\maketitle



% body of paper here
 

%%%%%% Introduction
%
One of the most interesting manifestations of macroscopic chirality are 
cholesteric liquid crystals.
%
In cholesterics, as in nematic liquid crystals, there is long-range 
orientational order of the elongated molecules along a local axis 
described by the director  $\bm{n}$.
%
Whereas in nematics the elastic forces (or torques) tend to establish a 
uniform orientation of $\bm{n}$, the chiral molecules in cholesterics 
lead in equilibrium to a helical arrangement with $\bm{n}$ 
perpendicular to the helix axis.
%
Choosing the axis along $z$, the structure with pitch $p_0$ is given by
$\bm{n}=(\cos{\varphi(z)},\sin{\varphi(z)},0)$, $\varphi=q_0 z$, 
$q_0=2 \pi/p_0$.
%

%
The helical symmetry leads to interesting dynamical effects.
Particularly intriguing is the Lehmann rotation of the director 
structure in a cholesteric droplet heated from below,
described in 1900 \cite{Leh1900,degennes,chandrabook}.
%
It has not been observed again, which is usually attributed to the 
influence of surface anchoring.
%
However, the electric analog where the temperature gradient is 
replaced by a {\em dc} electric field $\bm{E}$ (``electromechanical 
effect'') has been observed \cite{madpra87}.
%
Traditionally, the explanation is based on phenomenological hydrodynamic
considerations, which by symmetry allow for an additional 
dissipative dynamic coupling between the director and 
electric field (``electromechanical coupling'')
\cite{degennes,chandrabook}.
%
Strictly speaking, here ``hydrodynamic'' means that spatial modulations
must be slow on  the scale of the pitch which is not the case in
\cite{madpra87}.
%
In fact, in the experiments the director structure inside the droplet 
was not a homogeneous helix, but included substantial spay-bend 
distortions and even defects.
%
Another weakness of the approach is that no underlying mechanism has 
been identified that would, at least in principle, allow to determine 
the coefficients involved.
%

%
We have investigate for the first time a driving mechanism for 
chirality-related dynamical phenomena, involving well-established 
electro-hydrodynamic (EHD) effects.
%
Actually, director rotation is not the best choice to study such
phenomena, since special precautions are needed to avoid surface 
anchoring \cite{madpra87}.
%
Another effect is related to the intrinsic length scale defined by
helical symmetry.
%
In a confined geometry this can lead to the spontaneous formation
of spatial structures (``cholesteric fingers'') lacking certain
reflection  symmetries, which under nonequilibrium conditions results
in drift of the structures.
%
This is not affected by surface anchoring.
%
We will show that this drift can be explained by EHD effects and,
at the end, return to the rotation phenomena.
%

%
Cholesterics placed between two plane parallel electrodes (separation 
$d$), providing strong homeotropic (perpendicular to the electrodes) 
anchoring of $\bm{n}$, can experience unwinding of the helix due to 
the orienting effect of the electric field and of the boundaries.
%
The unwinding transition, which is typically discontinuous, occurs 
when a combination of the confinement ratio $C=d/p_0$ and the 
electric field strength (or applied voltage $U$) reaches a critical
value \cite{obpPR2000}.
%
Thus, there exists a line in the $(U,C)$ plane where the two phases 
coexist.
%
Near this line one finds the cholesteric fingers (CFs): elongated 
structures that are localized or arranged periodically.
%
At least four types of CFs were observed \cite{bpoPRE1998}.
The director configuration of CF of the first type (CF1) is invariant
with respect to a $\pi$-rotation about the finger axis 
\cite{prearrJP1976}, whereas CF2 has a mirror symmetry with respect to 
the midplane of the cell \cite{gilgilPRL1998,bpoPRE1998}, see 
Figs. \ref{CF2_profiles}, \ref{CF1_profiles} (top).
%
The difference in structure of CF1 and CF2 manifests itself in the 
dynamics.
%
In a {\em dc} field both fingers are observed to drift perpendicular to
their axis  \cite{gilgilLC1994,gilthiJP1997}, whereas in an {\em ac} 
field only CF2 drifts \cite{gilgilLC1994,ropJP1994,gilthiJP1997}.
%
Since CFs also grow along their axis one observes the formation of CF 
spirals.
%
Most measurements of the transverse drift of CFs are based on the 
analysis of spiral dynamics.
%


%
As the motor of the CF1 drift only the electromechanical coupling has 
been proposed \cite{gilthiJP1997}.
%
The magnitude of the drift velocity estimated with the coupling 
coefficient taken from droplet rotation-type experiments  turns out 
to be at least one order higher than observed \cite{gilthiJP1997}.
%
To explain the drift of CF2 in an {\em ac} field, several models were 
proposed \cite{gilgilPRL1998}, including  electromechanical coupling.
%
However, these models fail to describe the recently observed falloff of
the drift velocity $V_{\perp}$ when the frequency $f=\omega/2\pi$ of 
the applied electric field approaches the inverse charge relaxation time
$\tau_q$ \cite{bpoPRE1999}.
%
In the experiments the conductivity, and thereby $\tau_q$, was varied by
using different concentrations of ionic dopant.
%


%%%%%% The equations
%
The mechanism we propose is based on flow induced either by charge
separation through anisotropic conductivity, i.e., the Carr-Helfrich 
effect, (CF2 under {\em ac} driving) or by flexoelectric charge 
generation (CF1 under {\em dc} driving).
%
We use the standard set of nematodynamic equations for the director
$\bm{n}$, the velocity $\bm{v}$ (Navier-Stokes equation) in the Stokes
approximation (neglect of inertial terms), and the electric field 
$\bm{E}$ \cite{degennes,chandrabook}.
%
As usual the electric properties of the material are described by a 
dielectric permittivity tensor
$\epsilon_{i j}=\epsilon_\perp \delta_{i j} + \epsilon_a n_i n_j$,
($\epsilon_\perp$, $\epsilon_\parallel=\epsilon_\perp+\epsilon_a$ 
are the permittivities perpendicular and parallel to the director,
respectively), the analogous conductivity tensor
$\sigma_{i j}=\sigma_\perp \delta_{i j} + \sigma_a n_i n_j$,
and a flexopolarisation
$\bm{P}^{fl}=e_1\bm{n}(\nabla\cdot\bm{n}) + 
e_3 (\bm{n}\cdot\nabla)\bm{n}$.
%
We introduce dimensionless variables
$t = \tau_d \tilde{t}$,
$\bm{r} = d \tilde{\bm{r}}$,
$\bm{E} = (U_0/d) \tilde{\bm{E}}$,
$\bm{P}^{fl} = (\epsilon_0 U_0/d)\tilde{\bm{P}}^{fl}$,
and charge density
${\rho}_{el}=(\epsilon_0 U_0/d^2)\tilde{\rho}_{el}$.
%
Here $\tau_d = \gamma_1 d^2/K_{33}$ is the relevant director relaxation
time ($\gamma_1=$ rotational viscosity, $K_{33}=$ bend elastic
constant), $U_0=\sqrt{K_{33}/\epsilon_0}$ characterizes the typical
voltage ($U_0\approx 1 $V for the materials used). 
%


%
We choose the $z$ axis perpendicular to the bounding electrodes and 
allow for a drift of the structure with velocity $V_\perp$ in the $x$ 
direction, transverse to the finger's long axis  by replacing
$\partial_t \rightarrow \partial_t - V_\perp \partial_x$.
%
Then the equations can be written as (tildes are omitted, 
$\bm{V}_\perp=V_\perp {\hat x}$)
%
\begin{eqnarray}
%
&&[\partial_t + (\bm{v}-\bm{V}_\perp)\cdot\nabla
+\gamma'_2 \underline{\underline{\delta}}^\perp \underline{\underline{A}}
-\bm{\Omega}\times] \bm{n}  =
-\underline{\underline{\delta}}^\perp \bm{h}^r ,
\label{tb_eqn}\\
%
&&p_{, i} - T^v_{j i, j} + h^v_k n_{k, i} =
\rho_{el} E_i - V_{\perp} S_{j i, j} ,
\label{ns_eqn}\\
%
&&\frac{\tau_q}{\tau_d}
[\partial_t + (\bm{v}-\bm{V}_\perp)\cdot\nabla] \rho_{el} + \rho_{el}
=
\nonumber \\
&& \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\nabla\cdot[ - \epsilon_\perp \xi_H(\bm{n}\cdot\bm{E})\bm{n} 
+ \bm{P}^{fl} ] .
%
\label{cc_eqn}
%
\end{eqnarray}
%
supplemented by the incompressibility condition $\nabla\cdot\bm{v}=0$,
the Poisson equation $\rho_{el}=\nabla\cdot[ \epsilon_\perp \bm{E}
+\epsilon_a (\bm{n}\cdot\bm{E})\bm{n}+\bm{P}^{fl}]$, which was
already used in the charge conservation equation (\ref{cc_eqn}),
the electrostatic condition $\nabla\times\bm{E}=0$,
and the director normalization $\bm{n}^2=1$.
%
The notation $f_{, i}=\partial f/\partial x_i$ is used throughout.
%
The generation of space charges is characterized by the
Helfrich parameter
$\xi_H=\sigma_a/\sigma_\perp - \epsilon_a/\epsilon_\perp$
in Eq.  (\ref{cc_eqn}).
%


%
The director equation (\ref{tb_eqn}) couples to the flow field through 
the local fluid rotation
$\bm{\Omega}=(\nabla\times\bm{v})/2$ 
and the hydrodynamic strain tensor 
$A_{i j}=(v_{i, j} + v_{j, i})/2$ 
with 
$\gamma'_2=\gamma_2/\gamma_1$ 
and the projection tensor
$\delta_{i j}^\perp=\delta_{i j} - n_i n_j$.
%
Coupling to the elastic and electric torques is through
$h_i^r = \delta F/\delta n_i$
with the free energy density
$F = \frac{1}{2} k_1 (\nabla\cdot\bm{n})^2 +
\frac{1}{2} k_2 (2\pi C + \bm{n}\cdot(\nabla\times\bm{n}))^2
+ \frac{1}{2} k_3 (\bm{n}\times(\nabla\times\bm{n}))^2 -
\frac{1}{2} \epsilon_a (\bm{n}\cdot\bm{E})^2 -
\bm{P}^{fl}\cdot\bm{E}$.
%
Here $k_i=K_{ii}/K_{33}$ with the elastic constants $K_{ii}$.
%
 

%
In Eq. (\ref{ns_eqn}) the Stokes approximation is justified since the 
processes of interest are controlled by $\tau_d \sim 1$ s and the
charge relaxation time 
$\tau_q=\epsilon_0\epsilon_\perp/\sigma_\perp \sim 10^{-3}$ s,
which are much larger than the viscous relaxation time
$\tau_v=\rho_m d^2/\gamma_1 \sim 10^{-6}$ s ($\rho_m=$ mass density).
%
The elastic part of the stress tensor (Ericksen tensor) has been 
eliminated and the pressure redefined:
$p=p_0+F$ \cite{Leslie_ALC4_1_1979}.
%
The  viscous  stress tensor is
$T_{i j}^v = \alpha'_1 n_i n_j n_k n_m A_{k m} +
          \alpha'_2 n_i N_j +
          \alpha'_3 n_j N_i
      + \alpha'_4 A_{i j} +
          \alpha'_5 n_i n_k A_{k j} +
          \alpha'_6 n_j n_k A_{k i}$,
where $\alpha'_i=\alpha_i/\gamma_1$ with the Leslie viscosity 
coefficients $\alpha_i$.
%
Moreover 
$\bm{N}=(\partial_t+\bm{v}\cdot\nabla)\bm{n} - \bm{\Omega}\times\bm{n}$
and 
$h_k^v=N_k +\gamma'_2 n_j A_{j k}$.
%
The bulk force $\rho_{el}\bm{E}$ is the Coulomb force and 
$S_{ij}=\alpha'_2 n_i \partial_x n_j+\alpha'_3 n_j \partial_x n_i$.
%


%%%%%% Solving
%
We solve Eqs.(\ref{tb_eqn})-(\ref{cc_eqn}) in a perturbative way:
$\bm{n} = \bm{n}_0 + \bm{n}_1 + \dots$,
$\bm{E} = \bm{E}_0 + \bm{E}_1 + \dots$,
$\bm{v} =  \bm{v}_1 + \dots$,  $V_\perp=V_{\perp 1}+\dots$,
$\rho_{el} =  \rho_{el 1} + \dots$.
%
At lowest order the electric charge is neglected.
Therefore no bulk force arises in Eq.(\ref{ns_eqn}) and thus
$\bm{v}_0\equiv 0$.
%
$\bm{n}_0$ and $\bm{E}_0$ are obtained  from
%
\begin{eqnarray}
\label{n0_eqn}
\underline{\underline{\delta_0}}^\perp \bm{h}_0^r = 0 , \;\;
%
\nabla\cdot[
\epsilon_\perp \bm{E}_0 +
\epsilon_a (\bm{n}_0\cdot\bm{E}_0)\bm{n}_0] = 0 .
\label{E0_eqn}
\end{eqnarray}
%
We write
$\bm{E}_0=E_0(\bm{\hat z} - \nabla\phi_0)\cos(\omega\tau_d t)$
with $E_0$ the applied electric field and $\phi_0$ the induced 
potential.
%
Previous results show that the director configurations obtained from 
Eqs.(\ref{n0_eqn}) and the stability diagrams are in good agreement
with the experiments \cite{obpPR2000,bpoPRE1998,bpoPRE1999}.
%
We have solved Eqs.(\ref{n0_eqn}) in two dimensions (infinite extension
of the CF in the $y$ direction) by a relaxation method.
%
 

%
At first order Eq.(\ref{cc_eqn}) gives
%
\begin{eqnarray}
%
\label{rho_el1}
\rho_{el 1} &=&
-\frac{\cos(\omega\tau_d t) + \omega\tau_q\sin(\omega\tau_d t)}
      {1 + \omega^2\tau_q^2}
E_0 \epsilon_\perp \xi_H
\nonumber\\
&&\times
\nabla\cdot\{ [\bm{n}_0\cdot(\bm{\hat z}-\nabla\phi_0)]\bm{n}_0 \}
\nonumber\\
&&+e_{fl}
\nabla\cdot\{ \bm{n}_0(\nabla\cdot\bm{n}_0)
+ \frac{e_3}{e_1}(\bm{n}_0\cdot\nabla)\bm{n}_0 \} ,
%
\end{eqnarray}
%
where $e_{fl}=e_{1} d/(\epsilon_0 U_0)$.
The {\em dc} case is covered by setting $\omega=0$.
%
Next one solves the Navier-Stokes Eq.(\ref{ns_eqn}) (together with the 
incompressibility condition) at first order.
%
Taking the $\bm{curl}$ eliminates the pressure and leads to an 
inhomogeneous ODE for $\bm{v}_1$ with the coefficients depending on
$\bm{n}_0$.
%
The form of the inhomogeneities [right-hand side of (\ref{ns_eqn})] 
give solution of the form
%
\begin{equation}
%
\bm{v}_1 = 
\frac{\epsilon_\perp \xi_H E_{eff}^2}{1+\omega^2\tau_q^2} \bm{f}_1
 + e_{fl} E_0 \bm{f}_2 
 + V_{\perp 1} \bm{f}_3 ,
\label{v1}
%
\end{equation}
%
where $E_{eff}=E_0/\sqrt{2}$ ($E_0$) for {\em ac} ({\em dc}) driving and
the second term appears only in the {\em dc} case (then also 
$\omega=0$).
%
For the {\em ac} case a time  average of Eq.(\ref{ns_eqn}) was taken.
%
The functions $\bm{f}_i$ depend only on $\bm{n}_0$ and the $\alpha'_i$
($\bm{f}_2$ depends also on the ratio of the flexocoefficients 
$e_3/e_1$).
%
We have determined the functions $\bm{f}_i$ [from the linearized 
Eq.(\ref{ns_eqn})] by a Galerkin expansion with trigonometric
functions (for CF1) and Hermite polynomials (for CF2) as trial
functions.
% 

%
\begin{figure}[ht]
\includegraphics[width=7cm,height=3cm]{fig1a.eps}\\
\includegraphics[width=7cm,height=3cm]{fig1b.eps}
\caption{Director profile (top) and induced
velocity profile (stream lines) (bottom) of CF2. $U=1.9$V, $\omega
\tau_q=1$, 5CB material parameters \protect\cite{5cb}.}
\label{CF2_profiles}
\end{figure}
%

%
Finally, the director equation (\ref{tb_eqn}) at first order gives
%
\begin{eqnarray}
%
\label{n1_eqn}
\underline{\underline{\delta_0}}^\perp \bm{h}_1^r +
\underline{\underline{\delta_1}}^\perp \bm{h}_0^r &=&
(\bm{v}_1\cdot\nabla - V_{\perp 1} \partial_x)\bm{n}_0
\nonumber\\
&&+ \gamma'_2 \underline{\underline{\delta_0}}^\perp
\underline{\underline{A_1}} \bm{n}_0 
- \bm{\Omega}_1\times\bm{n}_0 ,
%
\end{eqnarray}
%
where $\bm{h}_1^r$, $\underline{\underline{\delta_1}}^\perp$ are
linear in $\bm{n}_1$.
%
The homogeneous problem
$\underline{\underline{\delta_0}}^\perp \bm{h}_1^r +
\underline{\underline{\delta_1}}^\perp \bm{h}_0^r = 0$
is solved by the translation mode $\partial \bm{n}_0/\partial x$.
%
The solvability condition for the inhomogeneous problem [with $\bm{v}_1$
substituted from (\ref{v1})] fixes the drift velocity $V_{\perp 1}$.
%
In physical units we obtain
%
\begin{equation}
%
V_{\perp 1}= \frac{d}{\gamma_1}
\frac{\epsilon_0 \epsilon_\perp \xi_H}
     {1+\omega^2\tau_q^2} E_{eff}^2 \frac{I_1}{I_0+I_3}
     + \frac{e_1 E_0}{\gamma_1} \frac{I_2}{I_0+I_3} ,
\label{vd}
%
\end{equation}
where 
$I_0 = \langle \bm{n}_{0, x}\cdot\bm{n}_{0, x} \rangle$,
$I_1 = \langle \bm{n}_{0, x}\cdot\bm{g}_1 \rangle$,
$I_2 = \langle \bm{n}_{0, x}\cdot\bm{g}_2 \rangle$,
$I_3 = \langle \bm{n}_{0, x}\cdot\bm{g}_3 \rangle$.
%
The last term appears only in the {\em dc} case.
%
The functions $\bm{g}_1$, $\bm{g}_2$, $\bm{g}_3$ are easily expressed 
in terms of the $\bm{f}_1$, $\bm{f}_2$, $\bm{f}_3$, respectively.
%
The scalar product is defined by
$\langle \bm{a}\cdot\bm{b} \rangle = \int\int(\bm{a}\cdot\bm{b})dx dz$.
%



%%%%%% Results
%%% CF2 results
%
We first discuss our results on CF2 drift in an {\em ac} field.
%
Since in the experiments the arms of CF2 spirals are well separated 
\cite{bpoPRE1999}, we have in the computations chosen a box width
large compared to the width of the director structure (isolated
finger).
%
The director profile  $\bm{n}_0$ together with the stream lines of the
flow from (\ref{v1}) as computed for the  parameters of the material 5CB
(4'-n-pentyl-4-cyanobiphenyl) \cite{5cb} used in the experiments are
shown in Fig. \ref{CF2_profiles}.
%
In Fig. \ref{cf2comp} our results for $V_{\perp 1}$, which is of order 
of $v_1$, versus reduced frequency $\omega \tau_q$ are shown, together
with the experimental results (the different symbols relate to different
impurity concentrations between $2 \cdot 10^{-5}$ and $0.05$ wt\%)
\cite{bpoPRE1999}.
%
The authors of \cite{bpoPRE1999} have scaled the frequency down by a 
factor of about $1.75$ in order to account for the fact that the
charge relaxation time $\tau_q$ was measured in the isotropic phase at
temperature $T=40$ $^\circ$C (the experiments were done at
$T=30$ $^\circ$C).
% 
The three curves correspond to $\sigma_\parallel/\sigma_\perp=1.3$,
$1.4$ and $1.5$, which agrees well with measurements in pure 5CB where
$\sigma_\parallel/\sigma_\perp=1.44$ \cite{jadz}.
%
$\sigma_\parallel/\sigma_\perp$ is the only parameter not given in
\cite{bpoPRE1999}.
%

%
\begin{figure}[ht]
\includegraphics[scale=0.45]{fig2.eps}
\vspace*{-0.5cm}
\caption{
Drift velocity of CF2 versus reduced frequency: points
are experimental data from \protect\cite{bpoPRE1999},
lines are calculations from (\protect\ref{vd}):
for $\sigma_\parallel/\sigma_\perp=1.3$ (solid),
$\sigma_\parallel/\sigma_\perp=1.4$ (dashed),
and $\sigma_\parallel/\sigma_\perp=1.5$ (dash-dotted).
$C=1.77$, $d=31\,\mu$m, $U_0=1.9$ V, 5CB material parameters
\protect\cite{5cb}.}
\label{cf2comp}
\end{figure}
%

%
Another experimental result born out by the model is the
approximate linear dependence of the drift velocity of CF2
as a function of applied field for samples with different
thicknesses but fixed confinement ratio $C$ \cite{bpoPRE1999}.
%
The CF2 appear only around an electric field such that
$(E_0 d)^2 = 
\pi^2/(\epsilon_0 \epsilon_a)
(4C^2K_{22}^2- K_{33}^2)/K_{33}$
\cite{obpPR2000}.
%
Using this to eliminate $d$ from (\ref{vd})
one obtains the linear dependence of the drift velocity on $E_0$.
%


%%% CF1 results
%
We now turn to CF1 in a {\em dc} field.
%
From the symmetry of the director profile $\bm{n}_0$ one now has 
$I_1=0$, so one is left with the last term in Eq.(\ref{vd}) relating to
flexoelectric charge generation.
%
Unfortunately one now has to cope with various uncertainties.
%
First, the approximation of isolated fingers is not valid, since in the
experimental spirals neighboring fingers are packed closely
\cite{gilthiJP1997,hinkukMCLC84}.
%
Second, in the {\em dc} case one has to expect screening of the electric
field by Debye layers, as evidenced in \cite{hinkukMCLC84} and also
suggested by the voltage offset $\sim 2$V in the current-voltage curve
presented in \cite{gilthiJP1997} [both using the material MBBA 
(4-methoxy-benzylidene-4'-n-butylaniline)].
%
%
\begin{figure}[ht]
\includegraphics[width=7cm,height=3cm]{fig3a.eps}\\
\includegraphics[width=7cm,height=3cm]{fig3b.eps}
\caption{
Director profile (top) and induced
velocity profile (stream lines) (bottom) of CF1. $U=0.2$V,
$L=L_F=1.95$, MBBA material parameters \protect\cite{mbba}.}
\label{CF1_profiles}
\end{figure}
%
%
Thus we have solved Eqs.(\ref{n0_eqn}) on a width $L$ in the $x$ 
direction  with periodic boundary conditions.
%
$V_{\perp 1}$ turns out to be sensitive to $L$: it decreases with 
decreasing $L$ and even changes sign at $L\approx 1.6$.
In the absence of experimental data on the finger width we have -- on 
one hand -- minimized the free energy density with respect to $L$ for
given values of the electric field leading to the ``optimal'' box
width $L_F(U)$.
%
Typical values of $L_F$ (in units of $d$) are $2 \div 2.5$ for a voltage
$0.2 \div 3$V.
%
In Fig. \ref{CF1_profiles} an example of the calculated director profile
and the stream lines of the flow from (\ref{v1}) are shown.
%
%
\begin{figure}[ht]
\includegraphics[scale=0.45]{fig4.eps}
\vspace*{-0.5cm}
\caption{
Drift velocity of CF1 versus voltage: points
are experimental data from \protect\cite{gilthiJP1997} shifted by $2$V;
lines are calculations from (\protect\ref{vd}).
Solid line: $L=L_F(U)$; dashed: $L=1.9$; dot-dashed: $L=2.1$.
Flexocoefficients $e_{1}=-1.05\cdot 10^{-11}$ C/m,
$e_{3}=-1.25\cdot 10^{-11}$ C/m.
$C=1.14$, $d=12\,\mu$m, MBBA material parameters
\protect\cite{mbba}.}
\label{cf1comp}
\end{figure}
%
%
We have also made calculations for some fixed values of $L$.
%
Our results for $V_{\perp 1}$ as a function of $U$, corrected by a 
screening of $2$V, are given in Fig. \ref{cf1comp}, together with the
experimental data \cite{gilthiJP1997}.
%
Typical values of the flexocoefficients for pure MBBA were chosen
\cite{dmdJPL82}.
%


%%%%%% Conclusion
%
In conclusion, we have developed an EHD model for the drift of CF1 and
CF2.
%
For CF1 in a {\em dc} electric field, flow induced by flexoelectric 
charge generation can describe the drift, but no quantitative conclusion
could be drawn.
%
For CF2 a Carr-Helfrich-like mechanism describes {\em quantitatively}
the drift in an {\em ac} electric field.
%
Moreover, our preliminary studies show that the EHD model with 
flexoelectric charge generation (as in CF1) can describe the rotation 
of cholesteric droplets in a {\em dc} electric field \cite{madpra87}, 
which has hitherto been interpreted in terms of phenomenological
electromechanical coupling.
%
Furthermore, it seems very likely that analogous thermo-hydrodynamic 
effects, which lead to very efficient convection phenomena in liquid
crystals \cite{thermo}, can also describe the original Lehmann
rotation \cite{Leh1900}.
%
This then suggests that so far there is no clear experimental 
manifestation of the (unspecified) phenomenological electro- or
thermo-mechanical coupling.
%
The EHD model provides a very general mechanism for forces and motion
that can be applied to other director structures, like  other types of 
cholesteric fingers (CF3 and CF4) and defects in an electric field.
%


%%%%%%
%
We thank J. Baudry, and P. Oswald for very useful discussions on their
experiments and help with the computations of the director structures. 
%
Financial support by DFG Grant Kr690/14-1, RFBR Grant 02-02-17435 and
the European graduate school ``Nonequilibrium phenomena and phase 
transitions in complex systems'' funded by DFG are gratefully 
acknowledged.



%%%%%%
%
\begin{thebibliography}{99}
%
%
\bibitem{Leh1900}
O.~Lehmann,
Annalen Phys.
\textbf{2},
649
(1900);
C.~W. Oseen,
Trans.\ Faraday Soc.
\textbf{29},
883
(1933).
%
\bibitem{degennes}
P.~G. de Gennes and J.~Prost,
\textit{The Physics of Liquid Crystals}
(Clarendon Press, Oxford, 1993).
%
\bibitem{chandrabook}
S.~Chandrasekhar,
\textit{Liquid Crystals}
(Cambride University Press, 1992).
%
\bibitem{madpra87}
N.~V. Madhusudana and R.~Pratibha,
Mol.\ Cryst.\ Liq.\ Cryst.\ Lett.
\textbf{5},
43
(1987);
Liq.\ Cryst.
\textbf{5},
1827
(1989).
%
\bibitem{obpPR2000}    
P.~Oswald, J.~Baudry, and S.~Pirkl,
Phys.\ Rep.
\textbf{337},
67
(2000).
%
\bibitem{bpoPRE1998}    
J.~Baudry, S.~Pirkl, and P.~Oswald,
Phys.\ Rev.\ E
\textbf{57},
3038
(1998).
%
\bibitem{prearrJP1976}    
M.~J. Press and  A.~S. Arrot,
J.\ Phys.\ (France)
\textbf{37},
387
(1976).
%
\bibitem{gilgilPRL1998}    
L.~Gil and  J.~M. Gilli,
Phys.\ Rev.\ Lett.
\textbf{80},
5742
(1998).
%
\bibitem{gilgilLC1994}
J.~M. Gilli and L.~Gil,
Liq.\ Cryst.
\textbf{17},
1
(1994).
%
\bibitem{gilthiJP1997}
L.~Gil and S.~Thiberge,
J.\ Phys.\ II (France)
\textbf{7},
1499
(1997).
%
\bibitem{ropJP1994}   
P.~Ribiere, P.~Oswald, and S.~Pirkl,
J.\ Phys.\ II (France)
\textbf{4},
127
(1994).
%
\bibitem{bpoPRE1999}   
J.~Baudry, S.~Pirkl, and P.~Oswald,
Phys.\ Rev.\ E
\textbf{59},
5562
(1999);
J.~Baudry, S.~Pirkl, and P.~Oswald,
Phys.\ Rev.\ E
\textbf{60},
2990
(1999).
%
\bibitem{Leslie_ALC4_1_1979}    
F.~M. Leslie,
Adv.\ Liq.\ Cryst.
\textbf{4},
1
(1979).
%
\bibitem{5cb}			
5CB material parameters used (at 30 $^\circ$C): 
viscosities ($\cdot 10^{-3}$ N$\cdot$s/m$^2$)
$\alpha_1=-6.6$, 
$\alpha_2=-77.0$, 
$\alpha_3=-4.2$, 
$\alpha_4=63.4$, 
$\alpha_5=62.4$, 
$\alpha_6=-18.4$, 
elastic moduli ($\cdot 10^{-12}$ N) 
$K_{11}=3.5$, 
$K_{22}=1.6$, 
$K_{33}=3.5$ and $\epsilon_\perp=7.3$, $\epsilon_a=8.9$.
%
\bibitem{jadz}			
J.~Jadzyn and P.~Kedziora,
Mol.\ Cryst.\ Liq.\ Cryst.
\textbf{145},
17
(1987).
%
\bibitem{hinkukMCLC84}		
H.~P. Hinov and E.~Kukleva,
Mol.\ Cryst.\ Liq.\ Cryst.
\textbf{109},
203
(1984).
%
\bibitem{mbba}			
MBBA material parameters used (at 25 $^\circ$C):  
viscosities ($\cdot 10^{-3}$ N$\cdot$s/m$^2$)
$\alpha_1=-18.1$, 
$\alpha_2=-110.4$, 
$\alpha_3=-1.1$, 
$\alpha_4=82.6$, 
$\alpha_5=77.9$, 
$\alpha_6=-33.6$, 
elastic moduli ($\cdot 10^{-12}$ N) 
$K_{11}=6.66$, 
$K_{22}=4.2$, 
$K_{33}=8.61$ and $\epsilon_\perp=5.25$, $\epsilon_a=-0.53$.
%
\bibitem{dmdJPL82}
I.~Dozov, Ph.~Martinot-Lagarde, and G.~Durand,
J.\ Phys.\ Lett. (France)
\textbf{43},
L-365
(1982);
N.~V. Madhusudana and G.~Durand,
J.\ Phys.\ Lett. (France)
\textbf{46},
L-195
(1985).
%
\bibitem{thermo}
see, e.g., L.~Kramer and W.~Pesch,
Annu.\ Rev.\ Fluid Mech.
\textbf{27},
515
(1995).
%
%
\end{thebibliography}
%

\end{document}
%

cond-mat0303549.extracted_bib
\begin{thebibliography}{99}
%
%
\bibitem{Leh1900}
O.~Lehmann,
Annalen Phys.
\textbf{2},
649
(1900);
C.~W. Oseen,
Trans.\ Faraday Soc.
\textbf{29},
883
(1933).
%
\bibitem{degennes}
P.~G. de Gennes and J.~Prost,
\textit{The Physics of Liquid Crystals}
(Clarendon Press, Oxford, 1993).
%
\bibitem{chandrabook}
S.~Chandrasekhar,
\textit{Liquid Crystals}
(Cambride University Press, 1992).
%
\bibitem{madpra87}
N.~V. Madhusudana and R.~Pratibha,
Mol.\ Cryst.\ Liq.\ Cryst.\ Lett.
\textbf{5},
43
(1987);
Liq.\ Cryst.
\textbf{5},
1827
(1989).
%
\bibitem{obpPR2000}    
P.~Oswald, J.~Baudry, and S.~Pirkl,
Phys.\ Rep.
\textbf{337},
67
(2000).
%
\bibitem{bpoPRE1998}    
J.~Baudry, S.~Pirkl, and P.~Oswald,
Phys.\ Rev.\ E
\textbf{57},
3038
(1998).
%
\bibitem{prearrJP1976}    
M.~J. Press and  A.~S. Arrot,
J.\ Phys.\ (France)
\textbf{37},
387
(1976).
%
\bibitem{gilgilPRL1998}    
L.~Gil and  J.~M. Gilli,
Phys.\ Rev.\ Lett.
\textbf{80},
5742
(1998).
%
\bibitem{gilgilLC1994}
J.~M. Gilli and L.~Gil,
Liq.\ Cryst.
\textbf{17},
1
(1994).
%
\bibitem{gilthiJP1997}
L.~Gil and S.~Thiberge,
J.\ Phys.\ II (France)
\textbf{7},
1499
(1997).
%
\bibitem{ropJP1994}   
P.~Ribiere, P.~Oswald, and S.~Pirkl,
J.\ Phys.\ II (France)
\textbf{4},
127
(1994).
%
\bibitem{bpoPRE1999}   
J.~Baudry, S.~Pirkl, and P.~Oswald,
Phys.\ Rev.\ E
\textbf{59},
5562
(1999);
J.~Baudry, S.~Pirkl, and P.~Oswald,
Phys.\ Rev.\ E
\textbf{60},
2990
(1999).
%
\bibitem{Leslie_ALC4_1_1979}    
F.~M. Leslie,
Adv.\ Liq.\ Cryst.
\textbf{4},
1
(1979).
%
\bibitem{5cb}			
5CB material parameters used (at 30 $^\circ$C): 
viscosities ($\cdot 10^{-3}$ N$\cdot$s/m$^2$)
$\alpha_1=-6.6$, 
$\alpha_2=-77.0$, 
$\alpha_3=-4.2$, 
$\alpha_4=63.4$, 
$\alpha_5=62.4$, 
$\alpha_6=-18.4$, 
elastic moduli ($\cdot 10^{-12}$ N) 
$K_{11}=3.5$, 
$K_{22}=1.6$, 
$K_{33}=3.5$ and $\epsilon_\perp=7.3$, $\epsilon_a=8.9$.
%
\bibitem{jadz}			
J.~Jadzyn and P.~Kedziora,
Mol.\ Cryst.\ Liq.\ Cryst.
\textbf{145},
17
(1987).
%
\bibitem{hinkukMCLC84}		
H.~P. Hinov and E.~Kukleva,
Mol.\ Cryst.\ Liq.\ Cryst.
\textbf{109},
203
(1984).
%
\bibitem{mbba}			
MBBA material parameters used (at 25 $^\circ$C):  
viscosities ($\cdot 10^{-3}$ N$\cdot$s/m$^2$)
$\alpha_1=-18.1$, 
$\alpha_2=-110.4$, 
$\alpha_3=-1.1$, 
$\alpha_4=82.6$, 
$\alpha_5=77.9$, 
$\alpha_6=-33.6$, 
elastic moduli ($\cdot 10^{-12}$ N) 
$K_{11}=6.66$, 
$K_{22}=4.2$, 
$K_{33}=8.61$ and $\epsilon_\perp=5.25$, $\epsilon_a=-0.53$.
%
\bibitem{dmdJPL82}
I.~Dozov, Ph.~Martinot-Lagarde, and G.~Durand,
J.\ Phys.\ Lett. (France)
\textbf{43},
L-365
(1982);
N.~V. Madhusudana and G.~Durand,
J.\ Phys.\ Lett. (France)
\textbf{46},
L-195
(1985).
%
\bibitem{thermo}
see, e.g., L.~Kramer and W.~Pesch,
Annu.\ Rev.\ Fluid Mech.
\textbf{27},
515
(1995).
%
%
\end{thebibliography}