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The electrohydrodynamic instability in homeotropic nematic layers
A. Hertrich, W. Decker, W. Pesch, L. Kramer
To cite this version:
A. Hertrich, W. Decker, W. Pesch, L. Kramer. The electrohydrodynamic instability in homeotropic ne- matic layers. Journal de Physique II, EDP Sciences, 1992, 2 (11), pp.1915-1930. �10.1051/jp2:1992243�.
�jpa-00247778�
J. Phys. II France 2 (1992) 1915-1930 NOVEMBER1992, PAGE 1915
tllassification Physics Abstracts
61.30G 47.20
The electrohydrodynamic instability in homeotropic nematic
layers
A. Hertrich, W. Decker, W. Pesch and L. Kramer
Physikafisches Institut der Universitit Bayreuth, D-Bayreuth, Germany (Received 24 February1992, revised 17 July1992, accepted 17 August 1992)
Rdsumd, Nous calculons le seuil d'apparition de convection 41ectrohydrodynamique dans des couches n4matiques orient4es hom40tropiquement poss4dant une anisotropie d141ectrique n4gative aprbs que la transition de Fr4edericksz de type courbure (bend) vers un alignement quas1llorizontal aft eu lieu. Des rouleaux obliques sont observ4s dans un domaine plus grand
que dans le cas de conditions aux limites planes. La pr4diction la plus int4ressante est que dans le cadre d'une analyse faiblement nonlin4aire toutes les solutions de rouleaux sont en fait instables et un passage direct vers
un chaos spatic-temporel a lieu. Dans le
cas de rouleaux obliques
ceci peut Atre accompagn4 d'une rotation permanente locale de l'axe directeur de courbure.
L'application d'un champ magn4tique plan devrait stabiliser [es rouleaux.
Abstract, We calculate the threshold for electrohydrodynamic convection in homeotro-
pically oriented nematic layers with negative dielectric anisotropy that sets in after the bend- Fr4edericksz transition to a quasi-planar alignment has taken place. Oblique rolls are found in
a larger range than in the
case of planar anchoring. The most interesting prediction is that in the weakly nonlinear analysis all roll solutions are in fact unstable
so that
a direct transition to spatic-temporal chaos becomes possible. In the oblique-roll case this may be accompanied by a permanent local rotation of the director bend axis. Application of
a planar magnetic field should stabilize the rolls.
1, Introduction.
Hydrodynamic systems subject to external forcing can undergo transitions to spatially periodic patterns. A well-known example is the Rayleigh-B4nard instability (RB) [Ii, where the driving
force is a temperature gradient applied vertically across a horizontal fluid layer. This system is isotropic in the plane of the layer and is therefore susceptible to reorientation of the pattern
on a large scale with
a slow dynamics. Hence ideal patterns without defects are difficult to observe because restoring forces, which lead to spatially uniform patterns, are small.
This is different in the usual electro-hydrodynamic convection (EHC) in a planarly aligned
nematic liquid crystal layer. The thin layer (thickness d
+~
10 -100 ~tm) is embedded between
1916 JOURNAL DE PHYSIQUE II N°ii
glass plates coated by conducting electrodes and an ac-voltage is applied across [2, 3]. A planar
orientation of the director can be achieved by proper boundary treatment. The pattern-forming
scenario is fairly rich already near the onset of the EHC-instability [2, 4, 5]. Further advantages
of EHC compared with RB-convection are the short characteristic times, large aspect ratios
(= ratio of lateral dimension to thickness), and the possibility of changing very easily external control parameters like the amplitude and frequency of the voltage and applying a magnetic
field. Usually one finds convection rolls with the roll axis perpendicular to the director (normal rolls), but sometimes the roll axis is oblique. Many of the experimental results agree fairly well
with detailed theoretical predictions [5, 4].
The subject of this paper is EHC in the homeotropic configuration where the director is
perpendicular to the confining electrodes. Clearly the configuration is isotropic in the plane of the layer, but with the help of a planar magnetic field it is possible to turn on an anisotropy.
Our goal is the quantitative analysis of the onset of a convective instability in the homeotropic configuration. This should also facilitate optimal designing of experiments.
The various destabilization mechanisms have been discussed before qualitatively [3]. We
here concentrate on the most interesting case of nematics with negative dielectric anisotropy
ea and (ea( not too small, like e-g- the standard material MBBA, where the director prefers an
orientation perpendicular to the electric field. The homeotropic alignment becomes unstable at a critical voltage l§, ~vhere the director starts to reorient and acquires a planar component (Bend-Fr4edericksz-transition). Ideally the director bends homogeneously in the plane of the
layer, I. e. the planar component is pointing uniformly in an arbitrary direction. The isotropy
in the plane is therefore spontaneously broken. Clearly large-scale variations and defects can
spoil the ideal Fr4edericksz deformation and the temporary application of a planar (or slightly tilted) magnetic field might help to achieve the ideal situation. With increasing voltage V a
fully planar situation is gradually approached except in small boundary layers.
It is not surprising then that with further increase of V a secondary instability leading to EHC takes place as a consequence of the conventional Carr-Helfrich-mechanism [6] in analogy
to the case of planar anchoring. ~Te have calculated the onset voltage l~ as a function of the material parameters. As an interesting result we found that oblique rolls, I-e- rolls oriented
obliquely ~vith respect to the bending axis of the director, prevail in most situations. Such oblique rolls are known front the planarly aligned case [2, 4, 5]. Since there are two degenerate oblique directions a superposition leading to a rectangular pattern is also possible.
Going on to the nonlinear problem we find that weakly nonlinear solutions
can be constructed for normal rolls in the usual Way but not for oblique rolls without planar magnetic field. The
reason is that oblique rolls excite a rotation of the planar component of the Frdedericksz
distorted director, which is a soft Inode of the system. Presumably the system goes directly
into a complex spatio-temporal chaotic state similar to the case of planarly aligned nematics with fi.ee boundary conditions [7, 8]. This state of weak turbulence should be suppressible by
a planar magnetic field.
In section 2 ~ve collect the basic hydrodj'namic equations for EHC [4, g-Ill, the equations
for the balance of momentum, angular momentum, and charge conservation. In section 3
we consider the electric Frdedericksz transition and compute the nonlinear Frdedericksz state above l§ from the full equations by a Galerkin procedure. The next step is a linear stability analysis of the Friedericksz state (Sect. 4 which reveals the secondary destabilization to EHC.
We study t-he dependence on various control and material parameters. A discussion of possible scenarios for the dynamic state expected above threshold in section 5 concludes the work. We do not present the explicit calculations because at this time the qualitative aspects appear
most relevant.
N°ii ELECTROCONVECTION IN HOMEOTROPIC NEMATICS 1917
2. Basic equations.
The standard set of electrohydrodynamic equations for nematic liquid crystals is as follows [4].
One has the orientational free energy density
F = (kit (T7 fi)~ + k22(fi (T7 x fi))~ + k33 (i1 x (T7 x i1))~)
2
~(jL0Xa(il H)~ (£06a (fi E)~ (i)
with the elastic constants for 'splay', 'twht'and 'bend'kii,k22,k33 and the anisotropies of the magnetic and electric susceptibilities Xa " Xii Xi and ea = ejj El. The unit vector fi
denotes the director.
The balance of torques acting on the director can be written as follows
r:=&xh=o (2)
The molecular field h [IIi consists of the reversible part
(~~
fi fif~ fiF
(~~~~)~ = -~ fini
(here and in the following n;,j .= 3xjni) and the viscous part
(hvisc); " (~Y2 tX3) Ni (tX2 + tX3l'ljJ~ji (~)
with N
=
~
i1+ ~ilx
(T7 x v) and Aij = )(vij + vjI) The ak denote the Leslie viscosity
dt 2 '
coefficients which satisfy the Parodi relation a2 + a3 " a6 as
In order to reduce equation (2) to two independent components perpendicular to i1we use
local coordinates besides the laboratory system x,y,z (z-axis perpendicular to the layer). In fact we use one of the two orthogonal systems
i1, k x i1, (k x fi) x fi
,
(5)
fi, ixil, (ixfi)xfi, (6)
depending on whether we start with homeotropic configuration (fi = I, ii x fi [= I) or the nearly quasiplanar Frdedericksz state (fi cs k, I x fi [cs I). Here k, I denote unit vectors along
the x- and z-direction. The two components of equation (2) read in the basis (6)
r (I x i1) % n~nzh~ + nynzhy + (n) I) hz = 0
,
(7)
r ((I x i1) x fi) % -rz % -n~hy + nyh~ = 0 (8)
The Navier-Stokes equation (momentum balance) has the form
pm~v=k+T7.(, (9)
with pm the mass density and k the force density = pejE ). Besides the pressure contribution the stress tensor (i) has an elastic (S) and a viscous (T) part
t = -p6;> + s + T
,
(io)
1918 JOURNAL DE PHYSIQUE II N°11
with
So = -$nk~j
,
(ii)
k~i
ljj # al nknmAkmn;nj + £k2ni NJ + £k3njN; + £k4A;j + fY5n;nkAkj + £k6nj nkAki (12)
The incompressibility condition
T7 v = 0 (13)
can be satisfied by introducing two velocity potentials f and g [12] such that
v = T7 x (T7 x If) + T7 x ig =: 6f + eg
,
(14)
~ "
(W'@'~W Ml
' ~ " (&'~&'°) ~~~~
The pressure p is eliminated by operating with 6 and e, respectively, on equation (9) leaving
us with two independent equations.
Finally we have the Maxwell equations in the quasistatic approximation,
T7 x E
= 0
,
(16)
T7 (eiE + ea (i1 E)i1)
= pep
,
(Ii)
60
together with charge conservation
~
~ ~ i~
~ ~~~~
and Ohm's law
j = al E + aa (fi E)fi + pejv (19)
Equations (Ii) to (19) lead to
jT7.iaiE+aa(fi.E)fiji+ ((+v.T7)jT7.ieiE+ea(fi.E)fiji=0. (20)
In the following we will consider the case of a harmonic ac-voltage V applied perpendicularly
to the plates, I. e. V is given by
v(i) = Eo d cos(wi)
= v5 j~cos(wi) (21)
Equation (16) is satisfied by writing the electric field in the form E = -T74l + ED cos(wt) I
,
(22)
where 4l is the induced electrical potential.
We assume infinite extension in the plane of the layer (x-y-plane) with the confining plates
at z = + ~. At the boundaries the director is fixed homeotropically fi
= (0,0,1), the velocity
2
field obeys v = 0 (consequently 3~v~
= 0, 3yvy = 0, and 3z~z
= 0), and the potential 4~
vanishes.
We have introduced dimensionless units which are listed in Appendix B together with the material parameters used for explicit calculations. Thus we measure lengths in units of ~,
~
N°li ELECTROCONVECTION IN HOMEOTROPIC NEMATICS 1919
times in units of the director relaxation time rd " /~~ ~, and introduce El = ~~,al
=
3~ '~
El al
An important dimensionless material parameter (the ratio between charge relaxation time To "
~°~~ and director relaxation time rd) is al
Q = ~~~~~~°~~ With 71 " £k3 £k2 (23)
71d~'i
The main dimensionless control parameter h
~2 d2 v2 fi
R = $ e @
,
l§ = ~ ~ (24)
2~ ( 60 lea
(l§ = Frdedericksz threshold). The magnetic field H is scaled by the Frdedericksz field H
=
h Hi, Hi = d/~~ ~~~ Finally it should be mentioned that we have neglected the flexoelectric effect, which is presumably justified for not too thin and clean cells [5].
3, The Frdedericksz transition,
For small applied voltages and h < I the whole system is in the basic state corresponding to
a homeotropic configuration, I-e- i1= (0,0,1), f
= g = 0. The aim is to characterize the first bifurcation which for ea < 0 (and lea not too small, see below) is the Frdedericksz transition.
We collect the various physical quantities symbolically in a vector u = (4l,n~,ny, f,g).
Clearly u = 0 in the basic state and the Frdedericksz transition is determined by the existence of exponentially growing solutions of equations (1) (22) linearized around u
= 0
US) = it@)exP(«1) (25)
Because of the applied ac-field
,
the function d(i) is w-periodic (Floquet-theorem) and can be expanded in a (truncated) Fourier series. The growth rate a(q; R,. .) turns out to be real in the relevant cases. Because of translational invariance in the z-y-plane the modal solutions
fi(i) are harmonic in z and y. Without a (planar) magnetic field isotropy ensures that one
can choose the wavevector in the z-direction so that we can then classify the modes by a
wavenumber q.
From the symmetries
V - V, (@, nz, ny, f,g) - +(-*, n~, ny, f, g), (26)
z - -z, (*, n~, n~, /,gj
- +(a, n~, n~, /, -g), (2i)
of the linearized equations one deduces that solutions fi(i) of the form T lx,v, z,i + ))
= -Tlz,v, z,t)
,
W (x, v, z, t + ))
= w(x, v, z, t)
,
W = G, %, f,
exist which are called the conductive modes. The opposite time symmetry (4 even and the other quantities odd) refers to dielectric modes [5], which are relevant only at high frequencies
and which we do not consider in this paper.
1920 JOURNAL DE PHYSIQUE II N°11
A simple analytic approximation is obtained by retaining only the lowest nonvanishing time- Fourier contributions and employing a harmonic z-dependence, which is strictly valid only with
free-slip boundaries for the velocity. This leads to
4
= cos(z) cos(qx)(4~asin(wt) + 4lb cos(wt))
,
G = N~ cos(z) sin(qz), q = Ny cos(z) sin(qx), (28) f = F cos(z) cos(qz), y = G sin(z) cos(qz).
One finds a bend-twist mode (bt) with n~
= f = g = 0 and a bend-splay mode (bs) with ny = g = 0 and this remains unchanged in the presence of a magnetic field in z- or y- direction.
From the condition a(q; R,. = 0 one obtains the corresponding neutral curves Rbt = I h( + k[~q~
,
(29) h( + kiiq~
Rbs (30)
~ '
where
C " I+q~cl>
~l ~2 ~
2~/2~2 ~~'b
~ ~l~~~~~~b ~ ~i~~ ~~~ ~ ~~~~~ ~~~
'
b b ~
ab " (q~ + 1) +'~ 6b
" (q~ + 1) + 6~
~l ~~~ ~ ~~ ~ ~~~~~ ~ ~~~ ~ ~~ ~~~
+ (2£k1 £k2 + £k3 + 2£k4 + as + tk6)q~
2
82 " £k3q~ tX2
C reduces to I at q = 0 so that the two modes then become equivalent and correspond to the planar bend-Frdedericksz distortion.
For realistic materials with ea < 0 one easily sees that Cl < 0 at q = 0. Therefore both, Rbt and Rbs, obtain their minimum (= I) at q = 0. In addition one finds a local minimum of the
bend-splay threshold Rbs at large values of q, where 82 has become negative (a3 < 0!). The instability corresponds to an electrohydrodynamic mode with a nonvanishing velocity field [3]. For MBBA-like materials a3 (<( a2 this minimum is considerably higher than I, so the homogeneous Frdedericksz transition corresponds to the first destabilization. The order
reverses only for values of ea extremely near to zero (ea > -0.0003) and the minimum at q # 0
persists into the range of positive ea.
The rigorous solution for q # 0 can be found numerically. We use a Galerkin method
expanding the velocity potential f in Chandrasekhar functions Cn(z) [13, 14] (which vanish
together with their first derivatives at the boundaries) and the other variables in trigonometric
functions with the appropriate symmetry and boundary conditions. In figure I the neutral
curves computed by the Galerkin method are plotted. If no other frequency is mentioned we
used w = 127 in our units, that is about 0.15 in units of the inverse charge relaxation time To
((ro2~)~l cs 34 lh). Note that for h > I the layer is in the Frdedericksz state already at zero
voltage.
The next step is the computation of the Frdedericksz state above threshold [9, 10] with the full nonlinear equations. For nonzero planar magnetic field h~ (chosen along the z-axis without loss of generality) the director bends in the z-z-plane. For zero magnetic field one has
N°11 ELECTROCONVECTION IN HOMEOTROPIC NEMATICS 1921
60
l~
50 ~
,
40
30
20 3
lo
0
0.00 1.25 2.50 3.75 5.00 6.25 7.50 8.75
~l
Fig-I- Neutral curves for the primary instabilities. Curve I shows the threshold for the bend-twist mode (+ Fr4edericksz transition at q = 0) and curve 3 the threshold for the bend-splay mode with a
convective part. Curves 2 and 4 belong to the other z-symmetry with higher thresholds.
1.oo
~x 1.85
o-So
o-So
o-w
o-lo
o.oo
-z.oo -i.oo o.oo i.oo z.oo
-n /2 +n /2
Fig.2. Fr4edericksz-bending of the director in x-direction for different values of R across the layer (z = + ~ corresponds to the upper and lower confining plate in our units).
2
a continuous degeneracy with respect to rotation of the bend plane around the z-axis and the Frdedericksz transition spontaneously singles out an axis from the z-y-plane. We again choose the z-z-plane as the bend plane.
In the Frdedericksz state there is no flow and only the electric potential 4li and the director hi (Al = I) have to be determined. In principle the equations could be written in an integral
form with the induced field to be determined selfconsistently. We found it easier to solve the
original set of equations by a Galerkin method, which afterwards facilitates the investigation of the secondary bifurcation. As a representative example we have plotted n~ for zero magnetic
field in figure 2. As expected one observes that the planar profile in the center region becomes
1922 JOURNAL DE PIIYSIQUE II N°11
more pronounced with increasing voltage, I-e- R. Note that fi is constant as a function of time in the lowest-order time-Fourier approximation. This approximation is excellent
as long as w
is not too small.
4, The second destabilization Electrohydrodynamic convection,
Since above the Frdedericksz transition the director field aligns more and more quasi-planarly
one expects a second threshold Rc for a convective instability in analogy to planar EHC.
To calculate Rc a linear stability analysis of the Frdedericksz solution uf
= (4l~,n[,0,0,0)
computed in section 3 is performed. One then has to solve a linear problem for a perturbation
li1 around uf similar as before. Because of the Floquet-theorem the general form of the solution is given by
li1(z, y,z,t) = e~~iii(z, y, z,t) (31)
where iii has the periodicity of the external electric ac-field. Because of translational invariance in the z-y-plane the modal solutions have the general form
cc~
iii(z, y, z, t)
=
e'(§~+PV) £ film(z)e~~~~
,
(32)
m=-cc~
where q and p are the wave numbers in z- and y-direction, respectively. In our calculation we have confined ourselves again to the lowest-order time-Fourier approximation in most cases. As in the calculation of the Frdedericksz state we expand with respect to z in systems of functions
which satisfy the boundary conditions (Galerkin method). For the velocity potential f we have
again Chandrasekhar-functions and for the remaining quantities a set of trigonometric functions
as in section 3. After truncation one obtains a linear system for the expansion coefficients in
equation (32). The condition Ile(a(q, p)) = 0 yields as usual the neutral surface Ro(q,p). The threshold is then given by
Rc ~"
ljljllRo(q, p) (33)
which also defines the critical wave vector (qc,pc).
Written out in more detail we have to solve a linear system of the form
(A + Rfl li1
= aCiiJ
,
(34)
where the matrices A, B, C depend on the Fr6edericksz solution, which is itself a function of
R (A " A("f(R))> .I'
If one assumes a stationary bifurcation, I-e- Rea
= Ima
= 0 at threshold, the computation
of the neutral surface simplifies. We then have to solve the eigenvalue problem for a
= 0
A~~B li1
=
-( li1 (35)
One starts with a value R, determines the Frdedericksz solution uf and the matrices A, B.
Then we have to solve (35) until the R used in the matrices A,B (with uf) coincides with the inverse of the eigenvalue. In figure 3a the neutral curve at p = 0 is shown for MBBA for the lowest and the second lowest mode.
Alternatively one solves the generalized eigenvalue problem (34) for given R. The neutral surface is determined from the condition that the largest real part of the a's changes sign. In this way we verified that the threshold indeed occurs with Ima = 0. Figure 3b shows the
