The electrohydrodynamic instability in twisted nematic liquid crystals

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The electrohydrodynamic instability in twisted nematic liquid crystals

A. Hertrich, A. Krekhov, O. Scaldin

To cite this version:

A. Hertrich, A. Krekhov, O. Scaldin. The electrohydrodynamic instability in twisted nematic liquid crystals. Journal de Physique II, EDP Sciences, 1994, 4 (2), pp.239-252. �10.1051/jp2:1994126�.

�jpa-00247958�

Clas~ification Phi-sic-s Abstracts

61.30G 47.20

The electrohydrodynamic instability in twisted nematic liquid crystals

A. Hertrich

(I).

A. P. Krekhov

(2)

and O. A. Scaldin

(2)

(') Phy,iLali,che; In,titut der Univer,,tat Bayreuth. 9~440 Bayreuth, Germany

(~) Phy;ic, Department. Ba,hL,riJn Re,earch Center, Ru~,,an Academy of Sc,ence,, 450025 Ufa, Ru~,ia

(Re(.en.e(/ I ,/ii/v /99~i, it-( en.ec/ _iii /iiifil Iii m ?/ O( tit/Jei /99,I, _a(.( epte(/ 9 Noicni/Jei /99,I

Abstract. We present some experiments _on

electrohydrodynamic

convection (EHC) in nematic

liquid

crystals ^with a planar, but twi~ted

configuration

(MBBA). The threshold voltage _is shown to

depend

only weakly _on the twi~t angle. A theoretical analysis ^of the onset behaviour agrees well with the

experiments.

More _sensitive is _a secondary transition to modulation structures, which _are not ea~ily observed in untwisted geometries. A rough estimate of the secondary instability based _on the behaviour of the most rapidly growing modulation mode _{seems to account} well for the observations. Additional theoretical investigations for materials with

positive

dielectric anisotropy conceming ^the

competition

between the Frdedericksz transition and EHC _are included. We predict

the existence of _{a cro~sover} codimension-? point. which could easily be tested by variation of the

twist angle.

1. Introduction.

When _an

alternating voltage

^is

applied

acros~ a thin

layer

of _a nematic

liquid crystal (NLC)

with

planar

uniform (I,e. without

twist)

orientation of the director 6 in the

plane

of the

layer

and the

strength

exceeds _a certain threshold, _an

instability leading

to

electrohydrodynamic

convection (EHC) occurs which leads to

spatially periodic

patterns ^{of convection} rolls, see e-g,

[1, 2].

Besides normal (or Williams) rolls, which _are oriented

perpendicular

to the undistorted

director,

oblique

rolls have been observed

experimentally [3,

4 and

explained theoretically [5~

8J.

Twisted _{structures are}

typically

used for

optical displays

_on the basis of Fr6edericksz transition, but convection instabilities _are

carefully

avoided, Indeed, the

problem

of EHC in

twisted nematics is not _so

widely investigated

_up to now, The _rare

exceptions

_concern

experimental

observations of the different domain _structures in nematics with the

particular

twist

angle

w12

[9, 10],

but

systematic

studies in the influence of the twist

angle

as a natural

parameter of the twisted nematics _on the

possible

domain structures and scenarios of the pattern transitions _are

missing,

The main issue of this paper is _to fill that gap ^and to

investigate

in more detail the

low~frequency

conduction

regime [6]

of EHC in twisted nematics with

different twist

angle

0

~ ~P~ w ~/2,

In section 2 _we first present ^the

experimental

results _on the

primary

threshold of EHC,

depending weakly

on ~P~, The

velocity field,

which is

directly

visualized, has _an axial component, ^{which varies}

approximately linearly

^with ~P~. ^{One observes further} a

secondary

transition to modulated

rolls, strongly

^influenced

by

~P~. ^Stimulated

by

the

experiment

we

performed

_a theoretical

analysis

of the convection

instability.

The

general

method is described

in section 3. We _use the standard three-dimensional

description

of EHC

neglecting

the

flexoelectric effect. For the numerical calculations _a Galerkin

procedure

was

employed,

yielding

the threshold curve and the

destabilizing

modes for different twist

angles

and

applied frequencies

of the

ac-voltage,

The results of the linear

analysis

and _a

comparison

^{with the}

experiment

are

given

in section 4, At first normal rolls

(Subsect. 4,1)

_are discussed. Then for materials with

positive

dielectric

anisotropy

the

competition

^{of EHC with the}

homogeneous

Fr6edericksz transition is considered

(Subsect. 4.2).

In subsection 4.3 the

stability regime

of

oblique

rolls is

presented

for different twist

angles.

2.

Experimental

results.

The NLC cell consisted of _two

glass

substrates covered

by SnO~

transparent electrodes with the distance between

plates

fixed

by mylar

_spacers of thickness d

=

20 _~Lm. The

liquid crystal

used was MBBA. The substrates _were rubbed

along

_some direction. Then, after

filling

with

NLC and

forming

the uniform

planar alignment

the

plates

were rotated relative to each other

by

the twist

angle

~P~. ^An

altemating

harmonic

ac~voltage

V _at

frequency

20 Hz _was

applied

across the cell.

Electro-optical

measurements have been carried out with _a

polarizing

microscope.

The temperature ^{of the NLC cell} cas stabilized at 25 _± 0.5 °C.

In the untwisted cell _at threshold

usually

^{the well-known Williams rolls} are observed. That scenario carries _over to twisted nematics

(0

~ ~P~ w

~/2),

where _{at a} threshold

V~(~P~)

the

analogous stationary

roll pattem appears

(Fig. la). Optical analysis

shows that these rolls _are oriented

perpendicular

_to the

midplane

director of the undisturbed

layer (the

so-called normal

rolls).

The

experiments

_are in agreement ^{with earlier} observations in MBBA for _a twist

angle

~P~ =

~/2

[9, 10].

From the

experimental

data for different twist

angles

in

figure

2

(circles),

one sees that the threshold

voltage

_V~

depends only weakly

_on the twist

angle

with _a

slight tendency

to decrease. Also the

change

of the

spatial period

is not very

pronounced.

The normal rolls start to

develop

_a

secondary modulatory

structure

along

the roll axis at a

well-defined threshold

voltage V~~ (the

^{so-called I} -dimensional modulation

instability).

^The

deformation is static and has _a sinusoidal

shape

^for twist

angles

0

~ ~P~ w ~/4

(Fig. lb)

and _a

« helical

» one for ~/4

~ ~P~ w ~/2

(Fig.

lc). The modulation stmcture for the small twist

angle (Fig. lb)

looks

quite

similar _to the varicose pattern ^{that have been found in untwisted}

nematics above the normal roll

regime

in _{some cases}

[4, 11].

In contrast to the untwisted case

[3,

4]

the

period

of the observed modulation structure for

large

twist

angles

is much smaller,

sharply decreasing

with

increasing

twist

angle (Fig. lc),

The threshold

V~~

^of modulations

depends weakly

on the twist

angle

ⁱⁿ the range 0

~ ~P~ w ~/4 and increases rather

strongly

with ~P~ ^{in the} range ~/4

~ ~P~ < ar/2

(Fig,

2).

For the cells with twist

angle

_~P~

=

w/2 _a further increase of the

applied voltage

at fixed

frequency

of 20 Hz leads _{to a} two-dimensional domain _structure

(Fig. ld),

with _a threshold of V~~ =

12.5 V. This pattem ^{has been observed} as a

tertiary

bifurcation and looks different from the _« matrix »-domain

[9]

and _« bi-dimensional _{» structures}

[10]

that have been found earlier in

ar/2-twisted

nematics. The

stability

of _our two-dimensional pattern ^is

strongly

sensitive to

the _occurrence of defects that could be inherited from the

previous

state or arise with

increasing

voltage.

^{Growth of the} disturbances _near defects leads

eventually

to the destruction of the

spatial periodicity.

In order to reveal the streamlines of the flow small

impurity particles

(2~4 _~Lm in

diameter)

a) b)

it

cl di

Fig. I.

Microphotographs

of the domain _structures in twisted MBBA (20 Hz) normal rolls _near V~ ^for ~fiT ~ 90° (a) _; modulation structure near V~~ ^for

4l~

=

20° (b) and ~fi~ =

90° (c) the _two~

dimensional _{structure near} V~~ for ~fi~ = 90° (d). The length of the illustrated area is about 0.27

mm (~~

direction).

were immersed in the NLC. In the untwisted _case ~P~ =

0)

_one observes _a pure rotation with _a

tangential velocity

_v,, For finite twist

(~P~

_#

0)

_an additional small axial component ^{of the}

velocity

_{v~ is} revealed

(Fig. 3),

which

changes sign

between

adjacent

rolls, An

analogous

helical flow has been

reported

for the normal rolls in the _case of ar/2~twisted nematics

[9].

An obvious

physical

mechanism for the appearance of the axial

velocity

_component is the strong

coupling

between director orientation and

hydrodynamic velocity

in nematic

liquid crystals,

A

nonzero transverse component ^{of the director in the rolls leads}

typically

to the helical character

of the flow. This is confirmed also

by

the fact that in untwisted nematics _an axial

velocity

occurs

only,

^{if the director builds} _up a transverse component

(e.g,

ⁱⁿ

oblique

rolls

[3]),

The

continuity

of the helical flow

along

the roll axis is ensured

through physical

boundaries _or domain boundaries. The

dependence

of the

velocity

components v,, v~ and their ratio _on the twist

angle

is

given

in

figure

3, The axial

velocity

_u~ as function of ~P~

increases,

for

large

twist

angles approximately linear,

whereas v, saturates for

large

twist

angles.

3. Theoretical

analysis.

We consider _a nematic slab of thickness d with the z~axis

perpendicular

_to the

layer

and

confining plates

at _z

= ±

d/2,

where,

by

an

appropriate

surface treatment

(see

^Sect.

2),

_a

iz

o V~

~'

~ V~ ~ _,.

lo

t-

2~ _{g ~}

ff

o

~ ₈

~ ~

~ ~

7

o o o o ~

6

5

0 IS 30 45 60 75 90

twist

angle ldegl

Fig.

?. The threshold of the normal rolls V~ and modulation _~tructure ~'~~ as a function of twi~t angle (~0 Hzl, measured in volt. The theoretical _curve~ for <,jj/«~ l.6?4 (~ee Sect. 4) _are dotted,

planar

orientation with _an additional twist

(~P~)

has been

imposed,

which can be

changed continuously.

The undistorted director orientation in the

midplane

(z

0)

is chosen _as the.i~

axis (I,e, the director at the

confining plates

^{i~ twisted}

by

_±

~P~l?).

25ji .ij..,ilii,ji;jiiiiliiiil I

O _V~

~ V ~-- ~~~

20 ^'

~i _Q'

~E

~15

l.

~

j10

°

5~

~ ~~-

__~_-~---%~""' ~~~

0

0 15 30 45 60 75 90

twist

angle ldegl

al

F;g,

3, a) ^The

tangential

_(v~ and axial (r~) ^{velocities for non~al roll; ? % above threshold} as a

f~nct;on of the _twig angle (~0 Hz). b) The ratio v,/r~ from the experimental data and from theory in the

mjdplane

(z 0) of the

layer

(<,jj/<r~ ₌ 1.624),

0.3

~

o exp.

~ ~~~°~'

~

0.2

~ (

~ Q

~

~" ~ _n~

0.J

j

~ ~,

~

r ]

~~(~l',

, , , , , , , ,

0 15 30 45 60 75 90

twist

angle ldegl

b)

Fig- ~ (( r>iifimie</)

We _use the ~tandard set of

electrohydrodynamic equations

for nematic

liquid crystals [5,

6,

12]

in the form

presented

ⁱⁿ an earlier paper

[13].

The

layer

is characterized

by

the

anisotropy

of the electric

susceptibility

_F~

= Fjj F~_ ^the

anisotropy

of the

conductivity

_«~

=

rrjjla~

I, the elastic constants for

«

splay

_{», «} twist _» and _« bend _» deformations

kj

j, k~~,

k~i,

^{and the}

Leslie

viscosity

coefficients _«j, k

=

1,

,

6, We

neglect

the flexoelectric

effect, presumably justified

for _{not too} thin and clean cell~

[8],

The material parameters used in

explicit

calculations _are listed in the

Appendix,

The

uniformly

twisted nematic

layer

in the undistorted _state below onset is described

by

^the director field

hjj

₌

(fi,n,

n,o, 0 with n,~ = cos ~P~

I ),

_{fi~o =} ^sin ~P~

~ ),

so that the

d d

director in the

midplane

^is

parallel

to the.i~axis. The twist

angle

is allowed to vary in the range 0

~

lfiT

_<

)

_{At the boundaries the director is fixed, h}

±

~ 2 =

(cos (±

4l~/2),

sin (±

4l~/2

), 0),

corre~ponding

to

strong~anchoring

conditions.

The

velocity

field _i~ described

by

two

velocity potential~ f

and _g

[14] (incompres~ibility

condition), such that

v=Vx

(Vxif)+Vxig.

(I)

The

ac~voltage applied

_across the

plate~

is

given by

V (t) ₌ ED^d cos (wt ) _{= ,} 2

V~n

cos

(wt), (2)

The condition V _x E

=

0 is satisfied

by writing

the electric field in the form

E

=

V~P ₊

Eo

cos (wt) I, (3)

where 4l is the induced electrical

potential.

We have introduced dimensionles~ units, I-e-

length~

_are measured in units of ~

and times

ar

in units of the

charge

relaxation time To =

° ~ The main dimensionless control parameter ^is

~«)

j~2

~2

~ ~z2

R ~

=

° °

m

~~ °

(4)

2

kit ^"~ kii ^"~

One should

keep

in mind that for

s~~0

there is _a

competition

between EHC and the

Fr6edericksz transition

leading

to a

hrnneous

distortion of the director. The

corresponding

threshold V~ ^{is fiven}

by

V~ ₌ ar

~"

,

i-e,

R~

₌

sj'

s~ s~

The various

physical quantities

are collected

sysmbolically

in _{a vector u=}

4l, _n,,

n~,

n~,

f,

_g

),

To determine the convection threshold _we linearize around the

uniformly

twisted

layer

characterized

by

4l

= 0, ii~ =

(n~~,

_n~~,

0), f

= g = 0, Because of the

Floquet~

theorem and the translational invariance in

the.<~j'~plane

we can write the modal solutions

&u in the

general

form

~

&u(x, _{y, z,} t ₌ e"~ _e~~~~ +PY~

~j ^&u~(z)

^e~~~~

⁽⁵⁾

mi m

where _q and _{p are} the wavenumbers in the _x~ and

y~direction, respectively.

The

&ulz)

_are

expanded

in _a

complete

set of functions which

satisfy

the

boundary

conditions

(Galerkin method).

For the

velocity potential f

_{we use} Chandrasekhar-functions and for the

remaining quantities

_{a set} of

trigonometric

functions

[15, 13].

After truncation _one obtains _a linear system ^{for the}

expansion

coefficients in

equation (5).

Tests show that

by keeping only

the

leading

terms in

m( [m

_<

I,

_« lowest~order time~fourier

approximation

») and not more

than six z~modes the relative _error of the threshold is of order 10-3 for the

frequencies (mostly 20Hz),

thickness

(d=20~Lm)

and material parameters ^{used here. The condition} Re

(« (q, p))

₌ ⁰

yields

as usual the neutral surface

Ro(q,

_p). The threshold is then

given by

R~

_:= min

Ro(q,

_p)

,

(6)

q, P

which also defines the critical _wave vector

(q~, p~).

Written _out in _more detail _we have to solve _a linear system ^{of the form}

(A

+

RB)

&u

=

«C &u

(7)

If _{one assumes a}

stationary

bifurcation, I-e- Re _«

= Im _«

=

0 _at threshold, the

computation

of

the neutral surface is

simplified.

We then have to solve the

eigenvalue problem

for

« =0

A- ^' B &u &u.

(8)

The determination of R

using (8)

is

technically

_more convenient than

using (7).

We have

always

^{checked the}

consistency

of the results of

(8)

with the results of

(7), Actually

our

computation

was obtained

by modifying

the code used

previously

to calculate the onset of EHC _{as a}

secondary

bifurcation of the Fr6edericksz distorted _state in

homeotropically

oriented

nematic

layers [13].

^The present case is easier to deal with because the director

configuration

of the undisturbed state is

simpler

than the Fr6edericksz _state,

4. Theoretical results.

4.I NORMAL ROLLS. The case of normal rolls is obtained

by setting

_p

= 0 in the

Floquet

ansatz

(5).

In

figure

4 the neutral _curve

Ro(q)

is shown for MBBA

(material

_{parameters see}

Appendix)

^{for different twist}

angles

^{and fixed}

frequency

_wTo

=

0.6. One sees that the

threshold _curves remain

virtually unchanged

for different twist

angles (4l~

₌ 0, _~b~

=

0.3 _ar,

4l~

₌ 0,5 _{ar, see}

figure caption),

13

lz

'~

~ 4, = 0.3n

4~ ⁼ 0.5n

c- it

if

w

f

₁₀

~

9

8

7

1.0 1.5 z.0 z.5 3.0

wave number q

Fig. 4. Neutral _curve (in Volt) for normal rolls ~p =0) at

wT~=0,6

for the twist angles

~fi~ 0, ~fi~ =

0.3 _~ and ~fi~ =

0.5 _n, We have used standard parameters of MBBA

(Appendix)

for all theoretical calculations, unless

explicitly

remarked,

In

figure

5 the

frequency dependence

of the threshold R~ ^{and the critical wavenumber} q~ is shown for different

4l~,

For low

frequencies

the threshold increases

only

_very

slightly

with the twist

angle

whereas the rise of the critical wavenumber is _more

pronounced.

This _can be

possibly interpreted

in _terms of the Carr-Helfrich mechanism

[16, 17]

from which _we expect that the threshold

voltage

in the lowest-order time~fourier

approximation

is

independent

of the thickness d and the wavenumber q is of order I/d for uniform

planar alignment.

A small

but _nonzero twist

angle 4l~

reduces the effective thickness

resulting

in the increase of the

wavenumber _q without

altering

the threshold

voltage,

One finds for _nonzero twist

angle

that the deviations n, and _fi~ from the basic twisted

configuration ho

~ (n~o, n,,o, 0) _are maximal in the

midplane (where ho

^is

parallel

to the _x-

direction),

The

out-of-plane

deviation n~ is maximal _at the roll center, whereas the

in-plane

deviation n~ is maximal between the rolls. Furthermore the critical mode solution contains _a

non-vanishing velocity potential

_g, This leads _{to an} axial

velocity

component v~, ^{which is in} agreement ^{with the}

experiments (see

Sect.

2).

The

velocity

field v~ is maximal in the

midplane

between the rolls like v~. One should

keep

in mind that all

quantities

_are

periodic

in the horizontal

plane,

where the

period

contains _two rolls. In

figure

3b _we have

plotted

the ratio between the axial (v~ ) ^{and the}

tangential

(v~) component ^{in the}

midplane

t~).

Clearly

this ratio

go

t

~~

~

p ⁴⁰

w

if

ii

^~~

zo

lo

o

o-o o.5 1-o 1.5 z-o z.5

frequency

_uj

a)

4.5

4.0

~

j

,/

3.5

Jn /

E

,/

I

3.0 _,"

g

_~ ^/

~ [

~ Z.5

[

z-o

1.5

o-o o.5 1-o 1.5 z-o z.5

frequency

_ui~

b)

j

Fig. 5. (a) Thre~hold voltage _V~ Iv _versus frequency _wTj~ for ~fi~ 0 (solid), 0.3 _n (dashed) and 0.5 n (dotted), (b) Wavenumber _{c/~ i'eisiis}

frequency

_wTjj for ~fi~ 0, 0.3 _n and 0.5 _n,

I, well de,cribed

by

a linear function of the twi;t

angle.

The

experimental

valuer (O) tend to be somewhat lower except at very ;mall ~P~, ^which

correipond

to a very low

tangential velocity.

In

figure

6 _we have

plotted

_V~ and

cj~ ^{as a} function of the twist

angle

for _wTjj

= 0.6 and

different values of _«,,

=

cTjj/cr~

I, The threshold

typically

increa;es

slightly

with the twist

angle

is _{one uses} the standard parameter ;et for MBBA [6(. However, with

increasing ajj/«~

the effect becomes ~maller and at

«jj/«~

₌ ^2.0 we ob;erve

finally

a

~light

decrease

(~o,I WI of the threshold

voltage

with _an increa;e of the twi~t

angle

as ob~erved

experimentally,

For _a,

=

0.6?4

(cTjjla~

₌ ^1.624, ;ee the dotted line in

Fig.

?), I,e. _a value

7.75

u,=0.5

7.50

j

~ 7.25 ^~

~

w _7.00

7i

~

g~s

u,=0.6

g-so

6.z5 g =0,7

6,00

0.0 0,1 0.Z 0.3 0A 0,5

twist

angle

4,

In

al

1,675

1.650

°.~°.5

~l.625 ~

)

~

f

_t,600

~ __,,--."

~$ l,575 o=0.6 ~,-~--"~' ^"~'

i

^~'~~~~~'

i.550

1.525 _o =0.7

1.500

0.0 0,1 0.Z 0.3 0A 0.5

twist

angle

_4,

In

b) Fig. 6. a) Thre,hold voltage I

~

(vi _{i<.i iui} twi,t angle ~fij [or ir,~ (,jj/1,~ ₌ 0.5, 0.6 and () 7.

b) Wavenumber _(/~ i<,i.ifii twi;t angle (fij for (r, 0.5, (),6 and 0.7.

slightly changed

in

comparison

to the standard value 0.5, the theoretical threshold coincides with the

experimental

one at Ah

=

0. Note that the wavenumber q~

always

increases with twist

angle.

An alternative

possibility

to get a threshold

decreasing

with twist

angle

^is to

modify

the

dielectric

anisotropy

_F~ to

positive

or

only slightly negative

values, like in

figure

8.

Although

the

secondary

destabilization is outside the scope of the linear

analysis

we can try

an estimate of the threshold

V~~.

In the

regime

bounded

by

the neutral _{curve one can} determine

a line which separates ^the

regimes

of maximal

growth

rates with _nonzero and zero

wavenumber p,

respectively.

In

figure

7, in addition _to the neutral _curves for

ajjla~

=

1,624.

iz

I j

1

?

,

^j

I j i i j

,

, , >

)

. > >

i

~f

G£

9

_,'

~

8

7

6

.5

wave nwnber q

Fig, 7. -

«11 /«~ theoretical _{threshold oincides} with the

experimental at ~fi~

=

0.

Also _plotted are _the

linesthat _separate the regions where the most rapidly growing

modes have p

= 0 and _p ~

(for all ~fi~ ^for

which

we have experimental

points in

Fig. 2).

those lines _are

plotted

for

4l~

=

0 and ~P~ = 0.5 _{~, as} well _as for those intermediate

angles,

for which _we have

experimental

data

points

from

figure

2, In

particular

_{we can now} consider the

voltage V~

^which separates the p = 0 _case from the p # 0 _case at q ₌ q~

(see Fig. 7).

In

figure

2 the

separation voltage V~(~P~)

has been included

(upper

dotted

curve).

We _note that

V~(~P~

)

gives

a

fairly good

estimate for

V~~,

^{The fact that the linear}

growth

rate criterion _at q~ could

give

a reasonable estimate for the transition to modulated rolls is in agreement ^{with the}

experience

that

already

in the _case ~P~ ₌ ^{0 the behaviour of} the fastest

growing

modes

[6]

is

correlated to the

zigzag instability (which

_may

actually

lead to modulated

rolls).

In the

following

subsections _we

give

_some additional theoretical results

pertaining

to

experimental

parameters not

explored

so far

by

the

experiments,

4.2 THE FRfEDERICKSz TRANSITION. For

s~~0

the neutral _curve

Vo(q) diverges

for

q - 0, For _s~

~ 0

Vo(q

tends _{to a} finite value for _q

- 0 and

Vn(0)

= V~

corresponds

_to the

threshold of the

spatially

uniform Fr6edericksz transition, which is

given by

_~

,fi

_in

the untwisted cell.

Changing

_s~ (for fixed

w)

_{we can} define _a critical value s~~, where the Fr6edericksz transition _crosses the threshold of EHC for ~P~ = 0,

Slightly

above s~~

crossing

of the thresholds _can be achieved

by increasing

the twist

angle,

This feature is demonstrated

by

two threshold _curves in

figure

8a, A situation where the _two thresholds coincide

(codimension-

2

point)

can be obtained

by varying

_~P~ and is of

special

interest. In that _{context we} remind also

the _occurrence of _a codimension-2

point

_e~~

(w~

in untwisted nematics obtained

by varying

the

frequency

_w

[6].

In _{our case} with the additional parameter ~P~ this

point

evolves into _a codimension-2 line

e~~(w~,

~P~~)

(Fig. 8b).

4.3

OBLIQUE

ROLLS. A necessary ^{condition for the normal rolls to have the lowest threshold} is @~R~(q,

p)/@p~

_~ ^{0 for the}

points

_q

= q~, p =

0 _on the neutral _curve

R~(q, p).

^Otherwise there exists _a lower threshold at p~ #

0,

which

pertains

to

oblique

rolls. For the standard

7.00

6.75

(

~ °

4, = 0.5R 6.50

c-

f~

gz~ e, = 0.26

3 7i

~ 6.00

5.75

5.50

"~"',,,~~

5.z5

""'~""'

o-o o.5 1-o 1.5 z-o

wave number _q a)

0.270

o,z65

g

GJ

o,z60

o.z55

0.250

0.00 0,10 0.20 0.30 0.40 0.50

twist

angle 4, In

b)

Fig.

8, a) ^Neutral _curve (in Volt) for e~= 0.26 and

4l~=0

(solid),

4l~=0.5

_« (dotted)

(W_To 0.6 ). b) Codimension~2 line _p~~(w~, 4l~ for _wT~ 0.6. Here the thresholds for the Frdedericksz transition and the EHC transition become equal.

MBBA parameters there exist _no

oblique

rolls _{at onset} in the whole range 0 _w ~P~ < ar/2 in agreement with _our

experiment.

But if _we

change

the material parameters _F~ or «jj

/«~ slightly,

then _one has

oblique

rolls at threshold for

frequencies

_w below _some value w~

(Lifshitz point).

As _a

typical example

_we

used in _our calculations _a

slightly higher

value for

«jj/«~

than 1.5

(standard),

which is

possibly

realized in _{not too} clean materials. The

dependence

of the roll

obliqueness p~/q~

_on

frequency

for different twist

angles

for

«jj/«~

=

2.0 is shown in

figure

9.

By

0.40 j

0.00 different

0.10 twist angles 4,/K

0 30

~

020 '~

",

i

cl ',, ', j

)

~'

~ 0.20

0.26

'

',

~ ~~

0.28

'~

~

~ ~~

~

'

,

)

., ^{' '>}

'"

'i

0.00

0.0 0.I 0.3 0A 0.5

frequency

_uj

Fig. 9. Roll obliquene~~ p~/cj~ _ve1311s frequency _wTj, for different _twist angles ~fi~/« and

«jj/«~

=

2.0. For _~fiT

~ 0.~ _« there _{are no} oblique roll;.

increasing

the

frequency

_we reach the Lifshitz

point

_{w =} w ~ where the

oblique

rolls

disappear.

Clearly

twist reduces the

oblique

roll

angle

and decreases the Lifshitz

point.

In

figure

10 _we show the

dependence

of _{w~ on} the twist

angle

^{for different values of} _«~. ^In

experiments

described in section 2 the

frequency

20 Hz is

larger

^than

w~/2

_ar even for _an untwisted cell.

Even if there _{are no}

oblique

rolls _at threshold,

they play

an

important

roll at

higher voltages

in the nonlinear

regime

of EHC

[18,

3.

4]

and

analogously

for

Rayleigh~Bdnard

convection

(19]. They

can lead to

zigzag

structures, where the

oblique

roll

regimes

become unstable with respect to the

equivalent degenerate configuration.

Such _an cross~roll

instability

_can lead _{to a}

bimodal,

rectangular

structure and

pictures

like

figure

ld seem

possible.

o.6z5

tT,=i.Z

~ 0.500

$

_"..

§ "',

0.375

"~

$

_'

~

~

,~

0.250

,

3

~

fT,r0.85

I

0.lZ5

j

', _j

r )

0.000

0.0 0.1 0.2 0.3 0A

twist

angle

_4,

In

Fig. lo. Lifshitz

frequency

_w~

T~j ^i'eisiis twist angle ~fi~ for _«~ <rjj/tr~ -1

=

0.85, 1.0 and 1.~_

5. Conclusion.

We have examined _a twisted

planar

nematic

configuration experimentally

as well _as

theoretically by

linear

analysis

of the standard

hydrodynamic equations.

Reasonable agreement is obtained

by adjusting

the

conventionally

used value of _«jj

/«~ l.5)

to a

higher

value, which

appears not unreasonable for not too clean substances.

Depending

_on

_«jj/«~

_{one can} either have _a

slight

increase _or decrease of the threshold with

increasing

twist

angle.

In

general

there

are

only

small

changes

of threshold and critical _wave vector with

increasing

twist

angle. Very typical

is the appearance of _an axial

velocity

with twist. A rather drastic

change

is

predicted

for

the _occurrence of

oblique

rolls and the Lifshitz

point,

which _can

presumably only

be observed

with materials with

slightly changed

parameters, ^More

spectacular

results _are related _to the

nonlinear

regime.

The

experiments

show _a

secondary

bifurcation of the modulation type

leading

to a stable distortion

perpendicular

to the roll _axes. The modulation

instability

_{seems to}

be

generic

for EHC in nematics

[20].

In

experiments

one

typically

observes in the untwisted

case

pronounced zigzag

structures, whereas stable undulated _{structures seem to} represent more an

exception [4]. Theoretically

_on the other hand undulated _structures have been identified _as

one of the attractors

[20~23].

Thus twisted nematic

configurations

seem suitable for _a detailed

investigation

of

interesting

modulation scenarios. The

rigorous

nonlinear

analysis

has to be done in that _case, but it appears that the

tendency

of the

growth

rates of the linear modes

give

a

useful hint how the modulation

instability

varies with twist

angle,

^because the

simple

estimate of the second bifurcation based on linear

growth

rates is in

good

agreement ^with

experiment.

Acknowledgements.

We wish _to thank

especially

W, Pesch and L, Kramer, as well _as W. Decker for useful

discussions,

reading

the

manuscript

and

help

^{in the}

programming,

Financial support

by

the

Deutsche

Forschungsgemeinschaft (Sonderforschungsbereich

213,

Bayreuth)

is

gratefully

acknowledged.

One of _us

(A.

K,) ^wishes to thank the

University

of

Bayreuth

for its

hospitality,

Appendix.

Material parameters.

The

computations

_are carried out for the

following

MBBA material parameters at 25 °C

[24, 25].

conductivities

[10

^~

(am )~

« l.5

«~ l.0

dielectric _constants

F 4.72

e~ 5,25

viscosity

coefficients 10~ ^~

~(

m

«~ -18,1

«~ l10.4

«~ l-1

«~ 82.6

«~ 77.9

«~ 33.6

JUURN~L OE PHYSIQUE II -T 4 N 2 FEBRLARY1~~4

elasticity

coefficients

[10~

^~~ N

kit

6.66

k~~ 4.2

k~~ 8.61

In

general

_we used the

layer

thickness 20 _~Lm and the

frequency

20 Hz (w_To

=

0.6)

as in the

experiment.

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