Flow alignment of nematics under oscillatory shear

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Flow alignment of nematics under oscillatory shear

A. Krekhov, L. Kramer, A. Buka, A. Chuvirov

To cite this version:

A. Krekhov, L. Kramer, A. Buka, A. Chuvirov. Flow alignment of nematics under oscillatory shear.

Journal de Physique II, EDP Sciences, 1993, 3 (9), pp.1387-1396. �10.1051/jp2:1993209�. �jpa-

00247914�

J. Phys. II £Fance 3

(1993)

1387-1396 SEPTEMBER1993, PAGE 1387

Classification Physics Abstracts

61.30G 47.20

Flow alignment of nematics under oscillatory shear

A.P. Krekhov

(~),

L. Kramer

(~),

^{A, Buka} (~) ^{and A-N- Chuvirov}

(~) (~)Institute

of Physics, University of Bayreuth, D-95440 Bayreuth, Germany

(~)Physics

Department, ^{Bashkirian Research} Center, Russian Academy of Sciences, 450025 Ufa, Russia

(~) KFKI-Research Institute for Sohd State Physics, Hungarian Academy of Sciences, H-1525 Budapest, P.O.B.49. Hungary

(Received

1

Aprfl1993, accepted

2 June

1993)

Abstract. Using _a simple averaging method _we show that under plane oscillatory shear

a

net torque acts _on the director of _a nematic

liquid

crystal leading to unexpected flow alignment.

Under quite general conditions the usual flow-alignment orientations _are unstable, and the ori-

entation parallel to the shear

(perpendicular

to the

velocity)

becomes _most stable within the shear plane. ^The effect drops out exactly under

(unrealistic)

linear shear unless the director has

a nontrivial _space dependence. An explicit application to plane Poiseuifle flow is

presented.

Pre- liminary experimental data _are shown which demonstrate _a transition under planar oscillatory

shear from _a homeotropically aligned nematic to _a spatially homogeneously deformed state.

1. Introduction.

The

coupling

between flow and director is _one of the basic features of

liquid crystals.

^Thus in _a

steady velocity

field

u(z) along

the _x axis

(plane

shear

flow)

the director of _a nematic will tend to

align

in the _{x -z}

plane

at an

angle

_{off =} + tan~~

(a31a2)

with the _x axis if _a3

la2

> 0 and the shear

8u/8z

is

positive/negative

_{(a3> a2 are} Leslie

viscosities) ill.

The

respective

orientation is the

only

stable

solution,

the _one with the other

sign representing

_an unstable

equilibrium.

The

only

other unstable

equilibrium corresponds

to the orientation

along

the _y axis. A number of

interesting phenomena

result from this

flow-alignment

effect

(see

_{e-g- 11, 2]} and references

therein).

In most nematics

a31a2

is small but

positive,

in _some

materials,

however, _{one can} have

a3/02

< 0

(especially

_near the transition to _a smectic

phase)

and then instead of flow

alignment

_one has _a

tumbling

motion which has also attracted much attention [3].

When the

velocity

field oscillates

periodically

and

symmetrically

around _zero the situation is different. Then for the flow

alignment

_case,

a31a2

_>

0,

and _a linear

ii-e- constant-velocity-

gradient)

shear

flow,

the director oscillates between two

positions

that

approach

the

alignment

angles

+

tan~~(a31a2)

_as the oscillation

amplitude

becomes

large

_[4].

However,

_one should

note

that,

at least for not too

large amplitudes,

_so that instabilities _can be

excluded,

there exists in

principle

_an ^infinite set of

possible (oscillatory) trajectories depending

_on the initial director orientation. Thus there is _no real flow

alignment.

^Of course this _can

apply only

when there is _no other torque

acting

_on the director

(e,g.

surface

effects)

and therefore the statement is difficult to

verify experimentally.

For the non-flow

alignment

_case the situation is rather

similar,

except

possibly

at

large amplitudes

_[4].

Actually

this

degeneracy

is

peculiar.

One has to

postulate

_a

large degree

^of

reversibility

in this

dissipative

system, ^{since the director is}

presumed

to return after each

period exactly

to its

previous position.

The _purpose of this communication is to show that the above

picture

is

only

an

approximation

and that in

general

there does indeed exist _a net torque so that there is in fact real flow

alignment

_even under

oscillatory

shear. The effect is contained in the standard

hydrodynamic description

of nematics [5, 6]. ^{It cancels}

exactly

under linear

shear,

I.e. when au

/8z

is constant

(at

least when the director is assumed _to be _space

independent).

In _a time-

dependent

situation _a

strictly

linear shear is unrealistic.

Actually

_a

time-averaged

net torque is

responsible

for the

pattern-forming

instabilities observed in thin nematic

layers subjected

to

an

elliptic oscillatory

shear

ii,

_8], and _was included in the theoretical treatments of this effect [9]. ^{We mention in}

passing

that the linear _response of nematics _to

small-amplitude oscillatory

shear flow with

rigid

surface

anchoring

of the director _was considered

recently

_[10].

Experiments

under the

plane oscillatory

shear flow considered here show

instabilities,

which up ^to now have remained

essentially unexplained.

A transition to _a

spatially periodic

roll

pattern ^oriented

perpendicular

to the shear with

period

of order of the cell thickness _was found

by

Scudieri in MBBA

ill].

The

frequency

_was 7.5 kHz and the

alignment homeotropic.

Apparently

the roll

instability

was

preceded by

_a

homogeneous

transition

leading

to "biaxial-

type"

behavior. With the _same material and

alignment,

but at much lower

frequency (500 Hz),

_a transition to rolls that _are

parallel

to the shear _was observed at

considerably higher

oscillation

amplitudes by

Guazzelli [8]. ^In this work the

plane

shear _was

just

_a

special

_case of

elliptical

shear. With small deviations from

planarity

the threshold first increased, but then there _was

apparently

_{a crossover} to the usual

elliptic

shear

mechanism,

after which the threshold

decreased. At very low

frequencies (20-100 Hz)

and rather

large amplitudes

with

planar

and

homeotropic alignment

and materials that included non-flow

aligning

_{ones a}

variety

of effects

were observed [4]. ^The

periods

^of patterns were much

longer

^{than the cell thickness} and the orientation

predominantly oblique. Finally,

at ultrasonic

frequencies

_a transition from the

homeotropic prealignment

to a state with _a

planar

component ^{of the}

director,

which _rotates

slowly

and

develops approximately axisymmetric

_wave

phenomena ("autowaves"),

has been

found [12].

In the next section _we present _some

preliminary experimental

observations of _a

homoge-

neous transition under

plane oscillatory

shear which

similarly

to the roll patterns

presumably

appearing

for

higher

shear

amplitudes require

the action of _net torques. Further _{on we} calculate the net bulk torque, ^for

simplicity dealing only

with motion in the _{x z}

plane.

In section 3 _we present our

general averaging

method and

apply

it in section 4 to the

simplest

type ^of

prescribed

nonlinear

shear, namely small-amplitude plane

Poiseuille flow. In section 5

we comment our results and

give

_an outlook for future work. Some detailed

analytic

formulas

are

presented

in the

appendix

^A.

2.

Experiments.

In this section _{we want to} demonstrate

qualitatively

_a transition under

planar oscillatory

shear from _a

homeotropically aligned

nematic to _a

spatially homogeneously

deformed state.

Exper-

N°9 FLOW ALIGNMENT OF NEMATICS UNDER OSCILLATORY SHEAR 1389

~)

3

' ' ' '

', ', ',

' , ',

,', _' '

' , , ,

' ' ,'

Fig-I-

Time resolved measurements of the _zero level of the optical

signal ii),

the acceleration of the _upper glass plate

(2),

and the optical _response signal in arbitrary units

(3).

iments have been carried out with two nematics, EBBA and 7CB. The cell _was

horizontally placed

on a

thermostatting block,

the

height

and orientation of which could be

adjusted

with micrometer _screws. The lower

glass plate

^{of the} cell _was fixed _to the

heating

block while the upper one was

rigidly

^attached to the membrane of _a

loudspeaker.

Great _{care was} taken _to avoid

ellipticity

of the

oscillatory

motion. No spacers have been

used,

the

glass plates

_were

adjusted parallel

to each other and the distance

(sample thickness)

has been measured. The

sample

_was

placed

into _a

polarizing microscope

combined with _a

photodetector

and the inten-

sity

of the

light passing through

the

sample

_was recorded. An accelerometer _was attached to the _upper

glass plate

in order to _measure the mechanical vibrations.

In

figures

la-c _we show time resolved low

frequency

measurements at three different vibration

amplitudes.

In each

picture

_curve I is the "zero level" of the

optical signal

without

vibration;

curve 2 shows _a

signal proportional

to the

oscillating

acceleration of the upper

plate

and 3 is

the

optical

_response to the vibration. In

figure

la the vibration

amplitude

is

small,

the

optical signal

follows the vibration with double

frequency.

Zero acceleration which

corresponds

to zero

displacement

coincides with _zero

intensity

of the

optical signal.

This shows that the director

(optical

axis of the

system)

reorients

periodically

with the external

frequency, symmetrically

around the initial

homeotropic alignment angle. (The

small asymmetry of the

optical signal

is

probably

not

relevant,

it could arise from _an

imperfect sample adjustment). Figure

16 shows the situation after _a transition to _an

asymmetric

state occurred at

higher amplitude

of the shear. Director reorientations

corresponding

to

positive

and

negative

directions of the

displacement

are not identical _{any more.} The

optical signal

has _now the _same

frequency

_as the external shear and _{moreover a} small

phase

shift _appears between the two

signals. Figure

lc

corresponds

to _a further increase of the vibration

amplitude.

The asymmetry and

phase

shift of the

optical signal

is _{even more}

pronounced.

The threshold field for the onset of the transition _was measured in _a

frequency

_range of10-600 Hz. For both EBBA and 7CB _we found _a linear

frequency dependence

of the critical

velocity,

which _means there is _a

frequency independent

critical

displacement amplitude

_at which the transition _occurs.

Typical birefringence

measurements where the

sample

is

placed

between crossed

polarizers

which make _an

angle

of 45

degrees

with the shear

direction,

have also been carried out. Both substances showed similar behavior. The

birefringence

started at the critical vibration

ampli-

tude and reached _a

frequency independent

saturation value. The saturation value

corresponds

to _a director orientation

angle

which is

substantially

less than 90

degrees.

For EBBA the

experiment

_was also carried out in _an electric field in order to record the "total"

birefringence

curve, I-e- director reorientation from 0 to 90

degrees. Experiments presented

here do not allow _to determine the direction of the symmetry

breaking

orientation.

3. Basic

equations.

We consider _a nematic

layer

of thickness d confined between two infinite

parallel plates.

The z-axis is chosen normal to the

bounding plates,

the

oscillating

flow in the x-direction and the

director fi confined _to the _{x z}

plane.

With this choice _{one can} write

n~ = cos@, _{ny =} 0, _{nz =}

sine,

U~ " U, fly " 0, _uz " 0,

Ii)

where ₌ @(t,

z),

_{u =}

Hit, z)

with ~

< _z <

~ The

hydrodynamic equations

[5,6] reduce to

2 2

~ii@,t

(a2 ^sin~

_a3 ^cos~_@)u

z ⁼

(Kii

cos~ ₊

K33 sin~

_{@)@,zz +}

(K33

Kii sine _cos

@@)~,

(2)

pu,t " -P~ +

8z(-(a2 ^sin~

_a3

^cos~

_{@)@,t +} [M(@) +

N(@)]u,z ), (3)

where P is the _pressure,

2M(@)

_{= a4} +

(05 a2) ^sin~

@,

2N(@)

₌

(a3

+ a6 + 2ai

sin~

@)

cos~

@,

the _{a; are}

viscosity coefficients,

_{~ii = 03 a2}

,

K;; are elastic constants, and _{p is} the

density

of the

liquid crystal.

The notation

f,;

_a

8f/8i

has been used

throughout.

A full

sylution

^{of the}

^{coupled equations (2)}

^and

⁽³⁾

^with

time-periodic boundary

^conditions

at _{z =} + ^" for Couette flow _or

time-periodic

_pressure

gradient

P~ for Poiseuille flow is

quite

_a

2 ^'

formidable task. For

simplicity

_we will here treat

Hit, z)

_{as a}

prescribed 2ir/w-periodic

function in t with

Hit

+

ir/w, z)

₌

-u(t, z),

_so that the

time-average

<

u(t, z)

> is _zero. This should

not

change

the

qualitative

features and should be _a reasonable

quantitative approximation

for small oscillation

amplitudes

or low

frequencies

such that the viscous penetration

depth fi,

N°9 FLOW ALIGNMENT OF NEMATICS UNDER OSCILLATORY SHEAR 1391

where _a is _a

typical viscosity,

is

larger

^{than the thickness d. In this}

spirit

_{we may even} include

a g-

dependence

in _@, which leads _{to an} additional _term K22@,yy on the

right-hand

side of

equation (2),

and which _we will need for _{a very}

specific

_reason later _on.

With the dimensionless variables

I

= cot, 2 _{= z}

Id,

fi ₌

u/dw, (4)

the

equation

for _can be rewritten _as

@,t

K(@)u,z

₌ e~[P(@)@,zz + ~P'(@)@)~ ⁺ k2@,yy],

(5)

where the tildes have been omitted and

K(@)

_m

(I

^cos~

^sin~

_@)

Ill I), P(@)

=

cos~

_{+ k3}

sin~

@, 1 =

a31a2, k,

=

K;; /Kii,

e~

=

I/(Tdw),

_Td

='tid~/Kii,

_h~(

_f)

_a

_{8h/8f. (6)}

Note that for the

flow-alignment

_case, I _{> o~}

K(@)

^vanishes at ₌ +@fl. Since the

frequency

w is

usually

much

larger

than the inverse director relaxation time _{we may assume} e~ « I,

Neglecting

the _{terms on} the

right-hand

side of

equation (5)

the

remaining

first-order

ordinary

differential

equation

_can be solved and the solution _@o is obtained from

f°° (

= g,

gin, z)

=

f~

^dtu

^{~it, z) ii)}

so that

@o ⁼

90(J/, g(t, z))

is _a

periodic

function in t. For _J/

= +@fl

ii)

cannot be used

(then

Y/ e +@fl).

Actually

the

o-integral

in

ii)

_can be solved

analytically

and @o(11,g) can be

expressed

in terms of

elementary

functions [4]. ^For our purpose this is not needed.

Choosing

the

origin

of t such that

g(t

=

0)

= 0 _one has @o(t =

0)

= Y/. Thus

@o oscillates around

Y/, but in

general

Y/ does not coincide with _{< @o >.} We conclude that

neglecting

the elastic

coupling

in the bulk leads to _a continuous

family

of periodic oscillations of that _can be

parametrised by

^the

"phase"

Y/.

In order _to

investigate

the influence of the elastic

coupling

we use the method of

multiple-

scale

analysis

_[13].

By introducing

_a "slow" time T ₌

e~t,

that modulates

(slowly)

the

periodic

behavior _on the fast time

scale,

_so that

= @(t, y, z,

T)

and

at

_- at +

e~8T,

_{one can} formulate

a

systematic perturbation expansion

of the form

= @o + e@j + e~@2 +..

,

where all the _{@; are}

periodic

in t. Then from equation

(5)

at order e° _one has the solution

ii). However,

the

phase

Y/ is _now allowed to vary on space and slow time T and is undetermined _at this order. At first order in _{e one} has

L91

= 0, ^L _m

at K~(90)g,t, (8)

where L is the linear operator of the

perturbational equation

of

ii)

and _{we can} choose

0j

₌ 0.

Finally,

at order e~ _we find

L82

₌

-80,T

+

P(80)80,zz

+

P'(80)8(,~

+ k280,~~.

(9)

This

inhomogeneous

linear

equation

for 82 is not

generally

solvable because L has the null

eigenvector

_80,~. The

solvability

condition takes _on the form:

<

V+80,T

> ₌ <

V+[P(80)80,zz

+

jP'(80)8(,~

^{+ k280,~~] >}

⁽¹⁰⁾

JOURN~L DE PHYS]QUE Ii -T 3 N'9 SEPTEMBER 1993

where _we introduced the scalar

product

< V+

f

> ₌

/~~

^dtv+

^{f (11)}

27r o

and V+ is the null

eigenvector

of the

adjoint

operator L+V+

= 0, L+

= -at

K'(go )g,t. (12)

Using

the fact that go

depends

_{on y,} z and T

only through

_i~ and g

equation (10)

_can be cast into the form of _an evolution

equation

for the

phase

_i~:

i~,T = B ₊ Ci~,z ⁺ Di~,zz ⁺ _Ei~)~ ⁺

Fi~,m

+

Gi~§, (13)

where

B, C, D,.

G _are scalar

products involving

the functions _i~, go_(~1,

g),

_g,z and g,zz. The

expressions

_are

given

in

appendix

A. The calculations _are

quite simple

because 80,~ ^and V+

can be calculated

analytically.

Let _us present _more

explicitly

the

expressions

for the _case of small

g(t, z),

which

essentially

amounts to A

Id

< 1 where A is the maximum

displacement

in the

oscillatory

^flow. Then from

equation (7)

for go _up to the second order _{in g one} has:

80 " ~1+

K(~1)g

+

(K(~1)K'(i~)g~

⁺

⁽¹⁴⁾

and for the coefficients in

equation (13)

B =

P'K~

_<

gg,zz ^>

+[PK~l'j

^< g(z ^{>, C =}

2K[PK'

₊

)P'KI'

< gg,z >, D = P +

K[P'KI')

^< g~ >,

E =

)P'

+

(-K'[P'KI'

+

[K[PK'+ jP'Kl'l'))

< g~ >, F

= k2, G

=

k2(KK"l'~

_{< g~ >}

₍₁₅₎

Here P

=

P(i~)

and K

=

K(i~) everywhere.

Equation (13)

^{is the central result of this work.} It describes the slow evolution of the

phase

i~

(the precise

definition of

i~ ^was

given

after

Eq. (7)).

We _see from

(15),

and also from the

general expressions (see Appendix A),

^{that B} is in

general

_nonzero if _{g is}

spatially varying,

I-e- when there is _a shear

gradient (see Eq. (7)).

Nonzero B _means that there is _a net bulk torque

acting

on i~. If

spatial

variations of

i~

play

no role the

stationary

solutions of

(13)

are

simply given by

the _zeros of B

(note

that B

= 0 for _{i~ =}

~8fl).

As

pointed

out

before,

in

general

the

velocity

field _u has to be determined

self-consistently

from

equation (3) together

with

i~ from

equation (13),

from which go ^follows. For that _purpose it would be useful _to also separate the fast and slow time

dependence

in

equation (3).

4.

Oscillatory

Poiseuille flow.

For

simplicity

_we here choose _a geometry where _one has _a shear

gradient

_even at low

frequencies.

Thus _we

apply

the

theory

to

plane

^{Poiseuille flow.} The results _are

expected

to carry over to Couette flow for

frequencies

such that the viscous penetration

depth fi$

_{is of order d.}

N°9 FLOW ALIGNMENT OF NEMATICS UNDER OSCILLATORY SHEAR 1393

For

simple plane oscillatory

Poiseuille flow _one has

u =

a(4z~ 1)

cost, _g = 8az sin t.

(16)

Then

equation (13)

lvith

(15)

_{goes over} into

YJ,T ⁼

a~lllYJ)

+ a~z©lYJ)YJ,z ⁺ lPlYJ) + a~z~l5lYJ)lYJ,zz +

+[~ P'(i~)

₊ a~z~fl(i~)]i~)~ ⁺ k2~1,yy ⁺

a~z~d(i~)i~§, (17)

where the functions

fi

_etc.

are

easily

calculated. In

particular

one has

tj~)

=

16jpj~)K2j~)ji j18)

In the

flow-alignment

_case,

a3/02

_> 0, the function

fi(i~)

has three _zeros

corresponding

to two stable

equilibria

at _{i~ =} 0 and

7r/2 (here

P'

= K'

=

0),

and _an unstable

equilibrium

at the

flow-alignment angle

_{i~ =} 8fl ₌

tan~~(a31a2) (here

K ₌

0).

When _{a3 goes} to _zero two

equilibria

coalesce and for _{a3 >} 0 there remains _an unstable

equilibrium

at i~ = 0

(no changes

at i~ =

7r/2).

Since for

typical

nematics 8fl is

small,

_{one expects} the solution

i~ ⁼ 0 to be

only

very

weakly

^stable

(small

domain of

attraction) compared

to _{i~ =}

7r/2.

This is also borne out

by

^the fact that the

quantity

U(il)

₌

-16P(i~)K~(i~), (19)

which

plays

^{the role of} _a

potential (up

to a constant

factor)

for the motion of _{i~, I-e-}

fi(i~)

=

-U'(i~),

is much lower _{at i~}

=

7r/2

than at _{i~ =} 0.

Our conclusion about the relative

stability

of the two minima of U _can be substantiated further

by considering

^{the effect of} fluctuations and the direction of motion of _a domain wall in the _y direction

connecting

the _{two states}

(here

_we need the _y

dependence). Clearly

for a~ _« the last _term in

equation (17)

_can be

neglected compared

to the _one before it.

Discarting

_z

dependence

_as

before, equation (17)

represents _a

simple Ginzburg-Landau equation.

It is well known that in this

theory

domain walls

always

move in the direction that lowers the

potential

and localized fluctuations and

perturbations

_can

only

lead from the

higher

to the lower state.

When

oscillatory

Poiseuille flow is induced in _a thin slab in the usual _way

by application

of _a pressure difference _{one must} be _aware of the fact that _{as a} result of

compressibility

the _pressure

decays along

^{the flow} direction _{x on a}

length

^b

~

d/pc~ /(3wo))

where _c is the sound

velocity

and _{a a}

typical

^shear

viscosity

[14]. ^{Since b} » d for all reasonable

frequencies, experiments

should nevertheless be

possible. Moreover,

the

decay

is avoided when Poiseuille flow is induced

by oscillating

in

parallel

both

plates

of _an

open-ended

slab.

5. Discussion.

Our

theory

shows that under

oscillatory

Poiseuille flow there is _a net bulk torque ^which renders the

time-averaged homeotropic

position

(<

8 _>

=

7r/2) globally

stable within the shear

plane.

Actually

the

interpretation

of the effect is

fairly straightforward.

Let _us consider _a much

simpler

system,

namely

that of _a

parametrically land oscillatory)

driven

highly-damped particle

described

by

_an

equation

of the form

8,t

"

K(8)au(t)

where

u(t)

is

27r/w-periodic

and _a is

some factor. This system ^does not loose the _memory of its initial

conditions,

I-e- _one has _a continuum of solutions

including

the

(neutrally stable) equilibrium position.

The situation is

changed immediately

when _an identical

particle

is

coupled

to the first _one via discrete diffusion

and the second

particle

is not driven with the _same

amplitude (but

with the _same

frequency),

I.e. when _one has

81,t

=

K(81)aiu(t) D(81 82),

82,t "

K(82)a2u(t) D(82 81) (20)

with _al

#

_a2. It _can then be shown

quite generally

that the

equilibrium position, K(8)

= 0,

becomes

unstable,

which is the

analog

of the flow

alignment angle becoming

unstable _{as a} result of

spatial coupling,

and that the

long-time

solutions _are attractors

(in fact,

it is

easily

seen that the

time-averaged divergence

of the flow in

phase

_space defined

by equation (20)

is

-2D,

I-e-

negative).

The additional

dissipation

_comes from the fact that the

inhomogeneous driving always

activates the diffusive

coupling,

which attempts to restore

homogeneity.

Our

treatment takes

properly

_care of the

singular

nature of the

perturbation.

Our result appears ^to hold for _more

general plane

shear flow. A

simple

estimate of the

magnitude

of the effect shows

that,

in order for the flow-induced torque to overcome strong

(planar)

surface

anchoring,

the

displacement amplitude

of the fluid has _to be of order of the thickness d.

Unfortunately

then the

small-amplitude approximation

invoked in section 3 breaks down. A _more accurate treatment within the framework of the

theory

is in progress.

We point out that _a net torque can also arise from the term

proportional

to i~~,z ⁱⁿ

equation (13).

To activate this term, an asymmetry across the

sample

with respect to the _{z =} 0

plane

has to be present which could be induced

externally by asymmetric boundary

conditions. This effect _can destabilize the

homeotropic alignment

and would lead to _a state where _i~

changes monotonically

_across the cell.

Let _{us now} comment _on the

possibility

of

out-of-plane

motion under

planar oscillatory

shear.

Neglecting

the elastic

coupling

_one

again

has the situation that the system ^does not relax, and _{one now} has _a two-parameter

family

of

trajectories. Clearly

_none of the out-of

plane trajectories

_crosses the shear

plane.

From the theoretical treatments of the

elliptic

shear

instability

it is known that the threshold

diverges

in the

planar

limit [9]. ^{This must} mean

that, taking

into account elastic

coupling,

the

in-plane

motion of

flow-aligning

nematics under

planar

shear is indeed

linearly

stable for all

amplitudes.

Therefore transitions to

out-of-plane

motion _can

presumably only

be

explained by assuming

the existence of stable

out-of-plane

orbits that coexist with the

in-plane

orbits but _never bifurcate. Therefore the transitions should be

hysteretic.

A

study

of the out-of-

plane

motion

using

the method

presented

here is in progress. Whereas in

low-frequency experiments

the critical

amplitudes

of the oscillations

were found to be of order of the thickness d _{[8] or}

larger

_[4],

they

_were much smaller

(below

0.1

~m)

for the

high-frequency

measurements

[11,

12]. ^{The latter situation} seems difficult to

explain by

_our

theory,

except

by assuming

that the effective

(averaged)

surface

anchoring

becomes weak at

high frequencies.

At _very

high frequencies

it may be _necessary to include other effects like

compressibility

[15].

The

theory presented

here shows that in

plane oscillatory

shear flow _a net

hydrodynamic

torque acts in

general

_on the

director, opening

the

possibility

for reorientational transitions of the kind

presented

in section 2. More

detailed, quantitative experimental

studies of the

phenomenon

_are in progress.

Acknowledgments.

We _are

grateful

to W. Pesch for _a critical

reading

of the

manuscript.

Two of _us

(A.B.

and

A-K-)

wish to thank the

University

of

Bayreuth

for its

hospitality.

Financial support ^from Deutsche

Forschungsgemeinschaft (SFB

213,

Bayreuth)

_as well _as the

Hungarian Academy

of Sciences

(OTKA 2976)

_are

gratefully acknowledged.

N°9 FLOW ALIGNMENT OF NEMATICS UNDER OSCILLATORY SHEAR 1395

Appendix

A.

From

equation (7)

_we have go ₌ go(i1,

g)

,

where _{i~ =}

il(T,

_y,

z),

_g

=

g(t, z)

and

80,T

= i1,T80,~, 80,z = i1,z80,~ ⁺

g,z80,g,

80,y = i1,Y80,~,

80,zz = i1,zz80,~ ⁺ _il~z80,~~ ⁺ i1,zg,z280,~g ⁺

g,zz80,g

+ g~z80,gg,

80,yy = ~l,vv80,~ + ~l(y80,~~.

II)

We _can calculate

80,~, 80,g, 80,m,

_80,gg, and 80,~g ^from our solution

(Eq.(7)):

a~ /°° ]I=

o, j2)

~

which

gives

_80,~

"

fl@. Analogously,

_{80,g =}

K(80), _80,m

=

fl@~'°#(l'~,

_80,gg

=

K(80)K'(80)

and 80,~g =

@K'(80).

The null

eigenvector

^V+ of the

adjoint

_operator

(12)

can be

easily

found _as V+

=

@

_and

we choose

C(il)

=

K(il)

_so that

V+80,~

=1. Then for the coefficients in

equation (13)

_we will have:

B =

K(il)(< P(80)g,zz

> + <

[P(80)K'(80)

₊

)P'(80)K(80))g)z ^>),

C = 2 _<

[P(80)K'(80)

+

)P'(80)K(80))g,z

^{>, D = <}

^P(80)

^>,

E "

)(-K'(i1)

_<

P(80)

> + <

P(80)K'(80)

₊

(P'(80)K(80) ^>),

~

~~'

~ ~~

)i1)

~'~~~~

_~

~'~~~ ~~'

_~~~

Note added in

proof

_: We

recently

learned that experiments under the

plane oscillatory

shear flow in

homeotropically

oriented nematic

layers

considered here have also been

performed

by

Belova G-N- and Remizova

E-N-,

Akust. Zh. 31

(1985)

289

(Sov. Phys.

Acoust. 31

(1985) 171).

A theoretical

analysis

of the transition to rolls found there has been

presented by

Kozhemikov

E.N.,

Zh.

Eksp.

Teor. Fiz.

91(1986)1346 (Sov. Phys.

JETP 64

(1986) 793).

Similar theoretical and experimental results for low

frequencies if

= 20 200

Hz)

and

planar alignement

_were

published

_very

recently by Hogan S-J-,

Mullin T. and Woodford

P.,

Proc. R.

Sov. London A 441

(1993)

559.

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(1979)

1.

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1583.