Propagation of excitations induced by shear flow in nematic liquid crystals

P«yagation

of

excitations

induced

by

shear

flow

in

nematic

liquid

crystais

J.

T.

Gleeson'

AT&TBellLaboratories, Murray Hill, New Jersey 07974 and Physics Department, KentState University, Kent, Ohio 44242

P.

Palffy-Muhoray

Liquid Crystal Institute, Kent State University, Kent, Ohio 44242

W.van Saarloos

AT&TBellLaboratories, Murray Hill, New Jersey 07974 (Received 12October 1990)

Under shear, nematic liquid crystals can exhibit localized excitations that propagate much faster than

simple one-dimensional analysis can predict. We describe a two-dimensional theory inwhich the excita-tions are fronts between distinct solutions ofthe steady-state Ericksen-Leslie equations. We find that a front's speed isproportional toits width and also tothe difference in an effective free energy that we also

define. We consider the effects ofshear in Hele-Shaw cells on the director field ofa nematic liquid crys-tal, and find distinct classes ofmultiple steady-state director-field configurations. The front between two solutions always moves inthe direction that decreases the total effective free energy. Although the speed cannot be calculated explicitly without knowing the details ofthe front region, it can be estimated

self-consistently using experimentally determined values for the observed front width. In this way we can obtain values for the front speed in reasonable agreement with observation.

I.

INTRODUCTION

In viscous fingering experiments using nematic liquid crystals

[1]

in Hele-Shaw cells [2,

3],

as well as in the ex-periments

of

Zhu, Liu, and Bai [4,5],propagating

excita-tions, or "halos,

"

(cf.

Fig.

1), have been observed that travel at very large speeds

—

typically 1 cm/s but as high as 20 cm/s, which is much larger than the average Qow speed (

-0.

1cm/s).

If

one assumes that the director

dis-FIG.

1. Photograph ofradial Hele-Shaw cell containing the

liquid crystal 8CB. Itshows adiffuse halo that isapproximately

0.4 cmwide and ispropagating outward at

-0.

7cm/s. gis ap-proximately 1500.

tortion occurs only over a length that scales with the plate separation d, dimensional analysis predicts a propa-gation speed

K

0=

where

K

is an elastic constant and y& isthe viscosity for

director rotation. Putting in values for a typical experi-ment, we obtain

co=0.

005 cm/s. Since co is much less than the observed speeds, we must look further than this one-dimensional picture

if

we are to understand this phenomenon.

We describe an approach in which the director varies in two dimensions: perpendicular to the plates and along the direction

of

Aow. Since in this case we must deal with lengths along both directions, and these may be quite different, dimensional analysis fails because it alone

cannot tell us how to combine these two lengths. In our

approach, the excitation isa front between two solutions

of

the steady-state Ericksen-Leslie [6] equations. The

propagation speed is proportional to the width

of

the transition region between the two solutions and to the difference in an effective free energy that we also define.

The Ericksen-Leslie equation for the director has a rich

multiplicity

of

solutions because

of

both topological

con-siderations and the inherent nonlinearity. We define for

each solution an effective free energy by considering the work needed to rotate the director in the presence

of

viscous and elastic torques. We obtain the result that the

front always moves in such a way as to reduce the total

effective free energy. Since the width

of

the front and hence its speed cannot be calculated apriori, we use

PROPAGATION OFEXCITATIONS INDUCED BYSHEAR

FLOW.

.

.

2589

surements

of

the front width and show reasonable agree-ment with observations

of

propagation speed.

II.

BACKGROUND

Theoretical approaches [7

—

13]attempting to describe these phenomena have been in terms

of

the motion

of

smooth, nonlinear kinkline excitations (referred to as soli-tons) between two states

of

the director,

n.

[n(r)

is a unit

vector field that describes the local direction

of

orienta-tional order. See

Ref.

[1].

]

These calculations are one-dimensional in that the structure

of

the director profile in

the direction perpendicular to the plates is neglected.

Furthermore, the state invaded by the kink is implicitly assumed to be linearly unstable. However, it is

unrealis-tic

toassume that the observed front isbetween alinearly stable state and an unstable state that is invaded by the stable state, because the unstable state would have to be prepared and then exist for at least the transit time

of

the

front. This would be difficult to realize even were it the specific goal

of

the experimenter. Moreover, in a nematic liquid crystal, where director-orientation Quctuations are soft, itis probably not possible toprepare such astate.

Another important issue that has not been fully ad-dressed previously is the effect

of

the director boundary condition at the glass plates. First, the director profile perpendicular to these planes cannot be homogeneous.

Second, because

of

the boundary conditions, as well as

the inherent nonlinearity

of

the problem, there is a large multiplicity

of

possible steady-state director profiles.

These different director profiles cannot in general be

con-tinuously transformed into each other,

i.e.

, they may be topologically inequivalent. Wepropose that the observed halo is a front propagating between different steady-state

director profiles; we will show that there is a large

num-ber

of

solutions corresponding to different deformation modes. Three

of

these modes have been previously ob-served

[14],

and their linear stability has been calculated

Given multiple solutions for the director pro61e and

that the halos are fronts moving between them, what mechanism determines the propagation speed and the selection mechanism,

i.e.

, which state does the system

"prefer'"? We are able to provide qualitative and intui-tively appealing results by studying the Ericksen-Leslie [6]equations (which describe the coupling

of

the director to the fiow field) in the approximation that the director

remains in the shear plane and that the Qow field isthat

of

an isotropic Quid. We believe that this approach

cap-tures the essential physics responsible for the state selec-tion and front speed problem. One attractive feature is

that we can associate a "fiow free energy,

"

Vs,

„,

with the viscous torque. The motion

of

fronts between different states is governed largely by the difference in the effective free energy:

V—

:

V,

&+9'„,

„(where

9',& is the Frank free

energy, which is the energy cost

of

spatial variations in n) between the states, so that the higher-energy state gives way

to

the lower. This picture allows us to solve the selection problem from an analysis

of

V alone. We find

that, for Qow speeds comparable to those used in the ex-periments, states that are topologically different from the

undistorted state become the lowest-effective-free-energy

state.

In our approach, the direction

of

motion

of

a front be-tween inequivalent states is determined by the difference in effective free energy

of

the two states, but the magni-tude

of

the velocity depends also on the details

of

the

director profile in the front region. However,

if

we as-sume the director distortions

of

the front extend over dis-tances along the Qow direction much larger than the

plate separation, as appears to be the case in the experi-ment, we 6nd that our analysis is consistent with front

speeds that are much larger than c0or the Quid velocity, and

of

the same order as seen experimentally.

In order to understand the front dynamics completely, one must relax the assumption that the director remains in the shear plane; we expect some director profiles to

es-cape. In this more general case, however, one cannot

as-sociate a unique Qow energy with each state, since the

work necessary to rotate the director from one state to

another depends on the way in which the director

dis-torts into the third dimension. We present arguments, however, that the picture developed here remains qualita-tively the same even in this case. The full dynamical

be-havior

of

the fronts can be studied only with two-dimensional numerical solutions

of

the Ericksen-Leslie

equations, which are beyond the scope

of

this paper. This paper proceeds as follows: in

Sec.

III

we discuss

the Ericksen-Leslie equations that describe this problem, as well as the simplifications we have made and their

im-plications. Section

III

A presents our results for

steady-state solutions

of

the director profile, including the multi-plicities referred to above. In

Sec.

IV we discuss the

relevant energies and their calculation. Section V de-scribes how the theory

of

front propagation applies to

this problem and how one may calculate halo speed;

Sec.

VI

gives a comparison between our results and experi-ments, and

Sec. VII

is our discussion and conclusions.

III.

THE ERICKSEN-LESLIE EQUATIONS Shear exerts viscous torque on the director

of

a

nemat-icliquid crystal. The balance

of

this torque with that due

to

elasticity

[1]

and the corresponding backfiow are de-scribed by the Ericksen-Leslie equations. Assuming that the director always lies in the plane

of

shear and that

there is translational invariance perpendicular to this plane, n is described uniquely by its angle

8(x,

z,

t)

with

the direction

of

the Qow; the coordinates

x

and z are

shown schematically in

Fig. 2.

In this geometry,

cosO=n. x

and the torque balance equation is

a8

a8

a'8

a'8

yi

_dt

+uz

_{Bx Qx~ (jz2}

+

au(z)

(a2sin

8

—

a3cos

8),

(2) az

where u isthe Qow velocity in the

x

direction, the

a's

are

ma-u(z)

FIG.

2. Schematic ofthe geometry considered.

0

_—0.

5

0.

0

zjd

0.

5

terials u2&&cx3, so we take

+3=0.

We have also verified

that this affects the results very little.

For

simplicity we also neglect the effect

of

the director on u and approxi-mate it by the Qow

of

an isotropic Quid determined solely by the externally imposed pressure gradient Bp/Bx. This,

with no-slip boundary conditions gives

( ) Bp/Bx d 2 u d 2 )

4 y2 4

where

a'

isan average viscosity and u is the average Qow speed. We have checked the validity

of

this assumption by calculating the true fiow field u(z) using the

Ericksen-Leslie theory. The relative deviations from the assumed parabolic profile are less than

20%,

which isnot significant at the level

of

this analysis.

A. Steady-state solutions

Using the above assumptions,

Eq.

(2)reduces to

FIG.

3.Solutions ofEq.(4)for various winding numbers and g

=400.

8(z'=

—

—

') =

—

IT

₊

mm.

_.

2

2

Traversing the cell from

z'=

—

—,' to

z'=

—,

',

the director

rotates through an angle mm, where m is the "winding

number"

_[18].

Classes

of

solutions are distinguished by their winding number.

It

_{is important} to note that two solutions that differ in winding number by Am are

topo-logically inequivalent and must be separated by a line

de-fect (ofstrength s

=b,

m

/2).

This immediately raises the question

of

which winding number (if any) is preferred.

Figure 3 shows solutions

of Eq.

(4) for

rI=400

and vari-ous winding numbers.

In addition, we have found that for each winding num-ber, a rich multiplicity

of

solutions exists. Unlike previ-ous researchers, we have not assumed that d

0

+rlz'sin

(8)=0,

Qz (4)

where

z'=z/d

and

g—

:

—

12uda2/X

isthe Ericksen

num-ber

Er.

Er

is a dimensionless number obtained by multi-plying a characteristic shear rate (in our case 12u/d )by the characteristic relaxation time for director distortions,

—

_a2d

_/K.

_{Typically, g is}

_of

_{the order}

_of

₁₀₀₀

_[16].

Equation (4) issolved using fourth-order Runge-Kutta in-tegration and the shooting method

[17];

the bisection root-finding algorithm converges reliably to the correct shot. There is alarge multiplicity

of

the solutions

of

this equation, largely unaddressed by other researchers, that

plays a dominant role in determining the speed

of

the lo-calized excitations referred to in the Introduction

—

this will be expanded on in Sec.

IV.

Boundary conditions consist

of

specifying the angle 0 on the planes

z'=+

—,

'.

The angle

0

can be defined to within an additive integral multiple

of

~,

giving rise to distinct classes

of

possible solutions. We assume strong anchoring for the

homeo-tropic alignment used in our experiment, that is, the boundary conditions at the glass walls are

—

1

—0.

5 l

0.

0 z cl

0.

5

PROPAGATION OFEXCITATIONS INDUCED BYSHEAR

FLOW.

.

.

2591 50 g)40 C:

0

o

30 M

creased, undulations

of

this curve increase in both fre-quency and magnitude, giving rise to many more

solu-tions. The complexity

of

this curve illustrates the in-herent richness

of

this problem.

Each solution represents a configuration with zero torque on the director everywhere and hence is a possible steady state.

0

~20—

~ 0 ~

~ 0 IV. ENERGY CALCULATIONS

Given a solution

of Eq.

(4), its Frank free energy per unit area

_P«

is [20] 2 0I ~ ~ ~ ~

'0

500 I 1000 1500I 2000]

E

«2

BO dz

.

2 —d/2 BZ

FIG.

5. The number ofsolutions ofEq.(4)increases dramati-cally as _gis increased. This isforwinding number 0.

Essential-ly the same behavior isseenfor other winding numbers. The er-ror bars arise because at very large g,more solutions might be found bydecreasing the mesh sizeused.

8(z)

—

(m+1)m/2

is an odd function

of

z

[19].

These

multiple solutions arise because

Eq.

(4)isnonlinear. Mul-tiple solutions appear for

g)

_g,

(m ), where r),

(0)=475.

Figure 4 shows nine different solutions

of

this equation

for

q=900

and m

=0.

The number

of

solutions, shown in

Fig.

5, increases dramatically with

g.

Furthermore,

this number increases in an interesting way. In

Fig.

6we plot

8(z'=

—

—,' )

—

m/2, obtained from the shooting algo-rithm, versus

d8/dz'~,

_&z when

r1=1000.

As g is

in-The Frank energy is the free-energy cost

of

producing spatial variations in the director orientation. We define

theow

energy Vs,„

V„.

„=

—

_t

_—

"'

j"',

d8d

d/2 m/2

=

_f

_J

@~~Bu _(a2sin

₈

—

_a3cos

_{8)d 8}

_dz —_{d /2 /2} BZ d/2 Qg

[(a2

a3)8(z

) ₂

(a2+a3)sin28(z

)

—d/2 BZ

+

const

]dz,

(7)

where v isthe director torque due to shear. The integra-tion constant does not affect the calculation and will be ignored. We stress that this isnot an energy in a thermo-dynamic sense, but is the integrated work done by the torque due to shear in rotating the director through 8(z ). In the general case in which the director escapes out

of

the shear plane, this work depends on the

"path"

of

the director rotation from the undistorted case,

i.

e.

, on the detailed behavior

of

this rotation. However, by restrict-ing the director tothe shear plane, the work necessary to rotate the director is, according

to Eq.

(7), completely

20—

0 I

—

20 II N 250 500 ~

_9~

9'f]ow ~ ~ 0 ~

~I

~

8

~

0—

~ ~ ~ —

_40—

—

₈₀

—

₆₀

—

₄₀

88/Bz

at

z=d/2

I

—

₂₀ ~ -250 C —_{500 —10} I —5 I 0 winding number 10

FIG.

6.

8(z'=

—

—

')

n/2 vs 88/Bz'~—

_,

=,zz using the

algo-rithm for Eq. (4);

g=1000.

Each zero ofthis function corre-sponds to a solution. As _q increases, the shape of the curve near the center becomes more intricate, leading to more solu-tions.

FIG.

7. Elastic energy and flow energy asa function of wind-ing number for

q=400.

Note that the How energy is nearly an-tisymmetric about m

=0

while the Frank energy is nearly

100

—200

100 200 300 400 500

er solutions vs g.

mber ofminimum energy

FIG.

8. Winding number o m

amounts to z the restriction

.

We can think

of

_Vs,„ h

f

the rotation. We

ca

the pat o the director specifying s ociated with

t

e s e . m

of

these two energy free en _y .

T

the eff'ective r

1from which

Eq.

a unov functiona

eded to change

t

e u

( . 11

V is also

t

e o

=m._/2,

_to

_{some soluttion}

_{of Eq.}

state, 8=m. for solutions o i iiow o ortional to

m,

w

ores

_{o di g} This m. Solutions free energy.

0 therefore have i eren

m&

ve i ere

"

_solution

can be seeen in

Fig.

ost stable S,

wesow

mailer win ing has smaller and sma

(

thth

lo st

9)

b'

lin'"1

un't'b

q. w be in qualitative agr which seems to be in suits. V. FRONT PPROPAGATION nde

e

a'6j

a'e

—

_{y, (c}

—

_{u )} 2

+

a&sin 8

—

a3cos

8),

az

in the lab).

Expen-is the frame spec measure in

which we are inte

rest-dofth

fro

tin

ic

h

6

1

ed is much larg

r

er than

in the above exprression.'

.

uu

b BO/3

d'

t

p1yinging both sides by 8

x

'2 BO d

gp

(9) s a ropagating soolution

of

h h h d' We assume

t

a re ions in w ic at

.

(2)ta

c _g

(4) such that in aa reference the director e

E

.(2)b h' moving frame nt' in this corno depe 2

f

icd l (10)

all,

list

e nt

a,

' h spatial extent p which BO/Bx deviates _pp

difference in effective ree e

c

—

=b,

P,

and the speed is given by

IbP

lb,

C=

=

_O

id (12)

s eed for the

ma-haracteristic spec

71

where

co=

the d

oto

1 b

t

een the two s is determine states

t

at r h effective-free-erence ewe e-energy i g-bl ol tio between

t

E/d

or more. We e

of

order 10 en states with m can e o

that the trans'sition between s

ries

of

Am

=

=2

transitions. th

thP'

h g. 10—

30)K/d.

As note ea We as-ail in Appendix

).

lt is derived m detai in the front ( ~x~

))x

the case where t e ir

h . N

thl

as

dis-ins in the plane

of

shear

al effective-free-energy e 0 o states ar ' n e left

(x

_=xo)

of

d.

"ng

f--

"

tion describ-fo o h 'nterface between s otion

of

the inter

u formalism

b -L d

hases in a inz

f't-hand side (lhs n

of c

is determined _y as in the case where

1

t

t ' ad th

ve in such away t a

r isalways decrease

.

h tthe tota1 energy

(9) or

te

interested in,

.

(9) b

PROPAGATION OFEXCITATIONS INDUCED BYSHEAR

FLOW.

.

.

2593 crystals,

co=10

cm/s. Moreover, we observe that the

spatial extent l

of

afront is

of

the order

of

0.

1cm, sothat

l/d

=20.

Thus with these estimates we see that

Eq.

(12)

isconsistent with front speeds up toseveral cm/s.

We stress that the estimate above shows only that it is natural to expect large front speeds.

To

predict the speed one must solve the front structure explicitly to compute the rhs

of Eq.

(9) (or its generalization to the case where the director escapes the shear plane). This would require a full two-dimensional numerical solution

of Eq.

(2) that

is beyond the scope

of

this paper.

We can only make speculations about effects caused by

the director escaping out

of

the plane

of

shear; other

researchers have found that solutions

of Eq.

(4) can be linearly unstable

to

this

_[15].

A stationary disclination between two states with a difference in winding number

of

2 corresponds to a defect

of

strength

1.

It

is well known [23] that this will reduce its strain energy (and

cease to be singular) by escaping into the third dimension (out

of

the shear plane in the geometry considered). We similarly expect a moving defect

to

distort out

of

the shear plane but do not expect this to change the qualita-tive picture. Quantitatively, it appears that the effect would be to shift the transitions to higher values

of g,

since the torque is decreased by cos _{P (where P is the} director angle with the normal

to

the shear plane) when

the director no longer remains in the shear plane.

It

is also well known that static disclinations with s &1 will

break up into anumber

of

weaker disclinations such that the sum

of

their strengths is the strength

of

the original

defect

[24],

and there is no reason moving defects should

not do the same.

For

example,

if

the director field is in an m

=3

state and the Aow is suddenly reversed so that the m

= —

3 state becomes stable, we expect that three

s=1

defects will be generated; this is perhaps why in

director waves, multiple fronts are often seen when the flow isreversed

[4].

It

has also been observed by us as well as others [4] that the propagation

of

these excitations ismost dramatic

when the source

of

the Bow is abruptly reversed. This

also is in good qualitative agreement with our theory; it can be seen in

Fig.

7 that when the sign

of

the driving

Aow is changed, there is an exchange

of

stability between classes with m and

—

m.

VII.

CONCLUSIONS

In summary, we have presented a two-dimensional theory that predicts the existence

of

propagating fronts in

a nematic liquid crystal subject to shear Row. These

fronts are transition regions between steady-state solu-tions

of

the director-torque-balance equation. Using this theory, we have obtained an expression for the front

ve-locity; it isproportional to the width

of

the front and the

difference in the effective free energy

of

two states

separated by the front. Calculating the width

of

the front

is straightforward in principle, but requires the solution

of

a nonlinear partial differential equation (which may be singular), which is beyond the scope

of

this paper. The

velocity dependence on front width is in reasonable agreement with experimental results. Our expression also exhibits semiquantitative agreement with experimental results on front speeds in two different systems.

ACKNOWLEDGMENTS

We thank

P.

E.

Cladis and

R. G.

Larson for helpful discussions.

J.

T.G.

acknowledges support from Natural Sciences and Engineering Research Council (Canada).

APPENDIX A: DERIVATION OF

EQ.

(5)

The first term on the rhs

of Eq.

(2) is

T

(Al)

VI. COMPARISON WITH EXPERIMENTAL RESULTS

In our experiments on radial Hele-Shaw cells filled with the liquid crystal octylcyanobiphenyl (8CB),we have observed halo speeds up

to

-0.

7cm/s with an apparent

front width

of

-0.

4cm. The plate separation was

0.

0025 cm and

g=1800.

This diffuse halo is likely a front be-tween two topologically equivalent states. In this range

of g,

we expect alarge number

of

solutions, sowe cannot

say specifically which ones are present. The order

of

magnitude

of

hP

between these types

of

states, how-ever, is

=

10K

/d.

This our estimate for the halo speed is

=

1600co

=

0.

8 cm/s. Other experiments on

"director

waves" in the nematic phase

of

N-(0-methoxybenzylidene)-p-butylaniline (MBBA) [4] showed speeds

of

up to 20 cm/s. In addition, the width

of

the

front in the direction

of

motion can be inferred from the photographs to be

1.

8 mm (the distance between planes was

0.

050mm). These experiments were at much higher values

of

q(

=

1.

5X 10 ). The effective-free-energy difference must be

=

10

K/d

in this case

to

arrive at the

correct order

of

magnitude for the observed speed. This

isnot unreasonable atsuch large

g.

K

fBO

_—

+

BO

BOddz dx d/2 o Bx Bz (A2) z=d/2 Front Region / z=-d/2 X = X₀ X = X_o

FIG.

9. Schematic ofregion ofintegration for Eq. (8). The distance xoofthe boundaries from the front ismuch larger than l.

where xo issome length larger than I and we have placed the center

of

the front at

x

=0.

The first term in the in-tegrand can be rewritten as

T

(A3)

Combining the two terms gives Eq. (9).

APPENDIX

B:

GENERALIZATION TO INCLUDE ESCAPE

(A11)

which can be integrated once; since by definition

ag/ax—

:

0

at

x

=+xo,

this term vanishes. The second term is rewritten as

K

dn

f

0 ag a _ag 2 —d/2

-.

ax az az (A4)

fdn

_f

o _ag a~g —d/2 —&p Bz BZBX

which canbe rewritten as

(A5) and integrated by parts:

ag/ax

vanishes on the lateral boundaries

(z=+d/2)

where strong anchoring is as-sumed; we thus have

The results presented so far are for the specific case

where the director always remains in the shear plane. Since this is almost certainly not the case experimentally, we present arguments here that the essential physics remains the same and that our method should still be val-id in this more general case. When the director escapes out

of

the shear plane, forming an angle P with this plane,

Eq.

(2)becomes

ao

ae

80

BL9

y,

cosP

+

u

=K

cosP

+

Bt Bx Bx Bz

ae

ay ay

ae

ax ax az az

ICf«—

2

f

o _a

ae

2 —d/2 xp Bx Bz (A6)

+

Bu cosp(a2sin 2

8

—

a3cos2

8)

or

(A7) using the definition

of

V„of

Eq.

(6).

The second term on the rhs

of Eq.

(2),when multiplied by BO/Bx and integrated over the same region as we

con-sidered above, is

f f

_— Bu (a2sin.

8

—

a3cos

8)

BOdz

dx,

d/2 xp Bz Bx which we rewrite as

The torque balance equation that describes the behavior

of

_{the escape angle P} is

ay

ae

au

y,

+

u

=

(a&+

a3)cosg sine cospsing

Bt Bx Bz

+z. '&+

"&

X2 _gz i

ae'

+

cosPsing 0(z, xp)

f

_—

f

(a2sin

8

—

a&cos

8)dgdz,

d/2 @ —o~ Bz which becomes (A9)

_ao

+

az 2 (B2) d/2 Qu y

[8(z,

—

xo)

—

8(z,

xo)]

—d/2 BZ

—

_{(a2+a3)[sin28(z,}

—

_xo)

—

_sin28(z,

_xo)]dz

_{. (A10)}

That is, recalling the definition in

Eq.

(7),

We make the transformation to the comoving frame as

before. The first equation is multiplied by

cosp(ae/ax)

and the second by

ap/ax

and both are integrated over the same region as before. The procedure outlined in Ap-pendix A is carried out and the two resulting equations

are added term by term. The result is

2

cy,

f f

_—d/2 0

cosy

ag

+

ap

d/2 ~p Bx Bx

2

d/2 o Qu ₂ . 2 2 Qg ay

=b.

_V„+

_— cos p(a@sin 8

—

a3cos

8)

+(a2+a3)sinecosgcospsinp

dx

dz,

d/2 o Bz Bx X (B3)

where

9,

&is now

ae

ay

PROPAGATION OFEXCITATIONS INDUCED BYSHEAR

FLOW.

. .

2595

The last integral in

Eq.

(B3)isrewritten as

f

_—d/2 Qg

dzf

p

a

cos

/sin 8

BO

d/2 BZ ~p Bx

sin20 8 _cos2 2 2

00

sin20 8

f

ct3 cos tPcos 8

+

cos _{P dx}

Bx Bx 4 Bx

which becomes, after some manipulation,

d/2 t)u 0 (~2

+3),

ae

dz cos (t

—d/2 az —o 2 Bx

(a&+ a3)

(sin20cos P)dx . (B6)

The second term can be integrated once, leaving

d/2 t)u (Ct2+~3) . d/2 t)u o (+2

~3),

c)e

a

—

f

_— dz

sin29cos

y+

f

dz

f

cos(t dx

.

d/2 BZ 4 —d/2 Bz +p 2 Bx (B7)

In

Eq.

(B3)the first term

of

the coefficient

of

con the lhs is changed (from the solvable case where there is no escape) only by a positive factor that isnot greater than unity, and there is an additional positive term, sothat the sign on the lhs isthe same. The last two terms in the integrand

of

Eq. (B3)differ only from the flow energy

of

the no-escape case by

apositive factor not greater than unity. However, detailed knowledge

of

the front isnecessary

to

evaluate the last term.

Nevertheless, we donot expect qualitatively different behavior from the case where the director stays in the shear plane.

*Present address: Department ofPhysics, Princeton Uni-versity, Princeton, NJ 08544.

~Present address: Instituut Lorentz, University ofLeiden, Nieuwsteeg 18,2311SBLeiden, The Netherlands.

[1]Fora general treatment ofliquid crystal physics, see P.

G.

deGennes, The Physics

of

Liquid Crystals (Clarendon, Ox-ford, 1974).

[2]A. Buka,

J.

Kertesz, and

T.

Vicsek, Nature 323, 424 (1986)~

[3] A. H. Buka, P.Palffy-Muhoray, and Z. Racz, Phys. Rev. A 36, 3984 (1987).

[4] Ghozhen Zhu, Phys. Rev.Lett. 49, 1332 (1982).

[5]Guozhen Zhu, Xingzhou Liu, and Naibing Bai, Phys. Lett.A117,229(1986).

[6]

J.

L. Ericksen, Archs. Ration. Mech. Analysis 4, 231 (1960);

F.

M. Leslie, Quart.

J.

Mech. Appl. Math. 19,357 (1966).

[7]Lin Lei, Shu Changqing, Shen Juelian, P.M. Lam, and

Huang Yun, Phys. Rev. Lett. 49,1335(1982).

[8]Z. C.Liang,

R.

F.

Shao, C. Q. Shu, L.Y.Wang, and L.

Lei, Mol. Cryst. Liq.Cryst. Lett.3,113 (1986).

[9]

R.

F.

Shao, S.Zheng, Z.C.Liang, C. Q.Xhu, and L.Lei, Mol. Cryst. Liq.Cryst. 144, 345(1987).

[10]C. Q. Shu and L. Lei, Mol. Cryst. Liq. Cryst. 131,47 (1985).

[11]L.Lei, C.Q.Shu, and G.Xu,

J.

Stat.Phys. 39, 633 (1985).

[12]

G.

Xu, C. Q.Shu, and L.Lei, Phys. Rev.A36,277(1987). [13]Y.Guo and Z.C.Ou-Yang, Phys. Rev. A40,2810 (1989). [14]

K.

Hiltrop and

F.

Fischer, Z.Naturforsch. 31,800(1976). [15]

I.

Zuniga and

F.

M Leslie,

J.

Non-Newtonian Fluid

Mech. 33,123(1989).

[16]We arrive at this estimate using the material constants for MBBA, a Quid velocity

-0.

1 mm/s, and a plate separa-tion of

-0.

025 mm.

[17] W. H. Press,

B.

P.Flannery, S. A. Teukolsky, and W.

T.

Vetterling, Numerical Recipes (Cambridge University, Cambridge, 1986),p. 582.

[18]Note that this is not tumbling, in which there are no configurations with zero torque on the director and hence no steady-state solutions. During tumbling, the director

winding up isa time-dependent solution.

[19]P.Palffy-Muhoray and

J.

T.Gleeson (unpublished). [20]SeeRef.[1]foradetailed discussion ofnematic elasticity. [21]P.

E.

Cladis, W.van Saarloos, D.Huse,

J.

S. Patel,

J.

W.

Goodby, and P.L.Finn, Phys. Rev. Lett.62, 1764(1989). [22]M. A. Anisomov, P. E.Cladis, E.

E.

Gorodetskii, D. A. Huse, V.

E.

Podneks, V. G. Taratuta, W. van Saarloos, and V.P.Voronov, Phys. Rev.A41, 6749 (1990). [23]P.

E.

Cladis and M.Kleman,

J.

Phys. 33,591(1972). [24] The energy in the strain field associated with a defect of

strength sis proportional to s and thus is decreased by