Smectics : A model for dynamical systems?

SMECTICS

J.

: A MODEL

Prost+,

+ Centre

SYSTEMS

E. D u b o i s - V i o l e t t e ++,

de R e c h e r c h e

++ L a b o r a t o i r e X ERA

FOR DYNAMICAL

i000

Paul

- Universit~

E. G u a z z e l l i

Pascal

de P h y s i q u e

des

?

- 33405

Solides

de P r o v e n c e

, M.

Clement

TALENCE

- Bat

-Dept.

510 - 9 1 4 0 5

ORSAY CEDEX

de P h y s i q u e

des

Syst~mes

MARSEILLE

CEDEX

13 (*)

#

D6sordonn6s X~ERA

i000

- Centre

- ESPCI

- Laboratoire

i0, rue Vauquelfn

i.

patterns

(systems smectic

above

in o n e

Crystals

CEDEX

number

/1/.

et M 6 c a n i q u e

%

Physique

05

going

The

equivalent

description A

has

given

reference between

of t h e

call

in s m e c t i c s .

Rayleigh-B@nard

Siggi%

Cross

recovers

/2/.

our

and

This

In this

there

with

the c o u p l i n g

phase

3, e x p e r i m e n t s

In the

the

layers. with dynamics")

2) w e

(for m o r e

shall

details,

is a n a t u r a l

velocity

field

see

coupling v

in m o d e l

exception

, called equations

of Z i p p e l i u s

to the v e r t i c a l

l i m i t of l a r g e

relax

describes

of the c o n s i d -

A phase

introduced

of

a large

(relaxation

(section

development

commonly

with

by analogy

"smectics

the h y d r o d y n a m i c

behavior

which

smectics

equation

analogy

type /3/,/4/,

%

variable

paper

descrip-

Of the r o l l s w i l l

variable

of the

following

of this

It is n o t

instability

u

and translational

box"

of the w a v e l e n g t h

dynamics

smectic

/5 a,b,c/(where

In s e c t i o n reported.

in the

to the p e r m e a t i o n ) .

the c l a s s i c a l

the d y n a m i c a l

for a " l a r g e

translation

some power

features

Within

with

is a h y d r o d y n a m i c a l

with

of

a phenomenological

the p h a s e

to the p o s i t i o n

the p h a s e m o t i o n

equivalent

We give

situations

parameter)

the b r e a k i n g

is v a l i d

case,

in r e f e r e n c e

the mean

/2/).

permeation

space.

of t h e p h a s e

(we s h a l l

recall

of s o m e c o n t r o l

by analogy

In t h a t

of the r o l l s

smectics been

systems

to i n f i n i t y

e r e d mode)

of

from equilibrium

in c o m m o n

(where a h o m o g e n e o u s

slowly).

the p o s i t i o n

value

Such a description

of r o l l s

infinitely

to far

share

direction

of d i s s i p a t i v e

smecfics

only

d'Hydrodynamique

PARIS

corresponding

a threshold

A Liquid

symmetry

time

- 75231

- 13397

INTRODUCTION Roll

tion

de St J ~ r 6 m e

vorticity

permeation,

and

is

one

equation~4a/, on

the

is a g o o d

shear

instability

example

in n e m a t i c s

of a l a r g e b o x

system

(*~he present work has been performed at the ESPCI - Laboratoire d'Hydrodynamique et m6canique Physique.

are (num-

216

her of r o l l s ~ 2 0 0 )

where

f i x e d b y the e x t e r n a l tion and dislocation performed

The

interactions

are

analyzed.

4, o u r

(thickness

/6/,/7/

s t u d y of

the

static

to d e t e r m i n e

the

two d i f f u s i o n

We f u r d ~ r slmw (sect.5) f o r c e /8/,

effective

well

coefficients

t i o n to the a p p l i e d Although

our

in this

paper,

stability The

of

failure

of d e f e c t s i n

our

of t h e

a wedge

of a s m e c t i c

tical

instability

tilted

with

respect

(defined by±

~)

in a s m e c t i c

C.

to s p e c i f y tation sense

of

the

between

layers.

of the e l l i p t i c a l

(Vx÷ -Vx). smectic

This

C phase

We also

change

dynamics

introduce

due

to the p r e s e n c e

2.

DYNAMICAL

given

defined

reveals

of

The

velocity

dependent. we s h a l l

shear

in-

account

This

of

and motion

a "SmecticsCdynamof the e l l i p -

velocity

defines

field

a direction

to the m o l e c u l a r

in r e f e r e n c e ~%/~x

change

a change

illustrated

the e q u i v a l e n t

the P e a c h -

of the d i s l o c a -

analysis

equivalent

in s e c t i o n

from

of d e f e c t s

a hydrodynamic

the c o m p r e s s i o n

excitation

a l l o w s one

/9/,/10/.

a proper

to propose

An experimental

is w e l l

field

of e l l i p t i c a l

to g i v e

theoretical

Synunetry p r o p e r t i e s

field

is g i v e n .

instabilities

(interaction

rolls

strain

structure.

in g e n e r a l

of the rolls.

p a i r of

disloca-

performed.

sample)~adsus

to t h e a x i s

for o n e

the

the v e l o c i t y

the c a s e

A analogy

The

experiments

results

t y p e of

have b e e n

A one.

disloca-

indicate

strain the

motion

to b e

on

direction

results

state physics

for a n y

/Ii/ predicts

the c o u p l i n g

~%/~y

found

observations

shape

of

linking

attention

smectic

ics"instead shear

are

experiments

s o m e of the e x p e r i m e n t a l

the d y n a m i c

in s o l i d

is v a l i d

d ~y)

to d e t e r m i n e

coefficients

friction

stress

focus

and

Moreover

experimental

that dislocation known

description

for w h i c h

with

defined

of a m o v i n g

and c l i m b .

is u s e d

A comparison

field

d~ x a n d

to b o t h g l i d e

description

a dislocation.

Koehler

in a w e l l

strain

corresponding

In s e c t i o n

appear The

in a w e d g e d s a m p l e

tion motions

around

the r o l l s

excitation.

in the

/ll/are

and

axis

used

the o r i e n -

in the r o t a t i o n a l of the g l i d e

framework

of

velocity

the

6.

the

thermomechanical

coupling

of a w e d g e .

EQUATIONS

a- S m e c t i c s A dynamic of /I/.

We give

corresponds direction follows

smectics

close

the d y n a m i c a l

to e q u i l i b r i u m

equations

to the d i m e n s i o n l e s s (qo: 2~/ao'

all lengths

a ° is the

will

in t e r m of

displacement layer

be s c a l e d

is d e s c r i b e d

of

thickness)

the p h a s e % the

layers

in r e f e r e n c e :-qo u which in the x

(see f i g . l ) .

to the p e r i o d i c i t y

of the

In w h a t

structure.

217

•

e,w-en

/0

0

~O-~2~

/L----~----_~=vective

layers (a) and conrolls (b)

/oea~oe~ ~ --/:- 71-- ~ / ',~ IX 1 I I~ (a) Iz

(b)~ z

= qo x and for s i m p l i c i t y P (DVi/$t)

where

= -qo ViP + qo2

/ ~t) + V x :

(~

the e l a s t i c

_

free

6F/6%

: B

A2

2 =

process

is ° (3)

(4)

~2: K /B 1

for i n c o m p r e s s i b l e

(i) is the m o d i f i e d

which

fluid

(5)

Navier-Stokes

equation,

behavior

(viscosity)

of f l o ws

appears

in e q u a t i o n

(i), o r i g i n a t e s

as d e s c r i b e d

the p e r m e a t i o n

(2)

+ i 2 (A~) 2 ~)

'

the a n i s o t r o p i c f o r c e 6F/6%

(I)

2 + i 2 (Al%)2 } dx dy

~2

qo

Equation

(6F / 6%) ~ ix + qijkl qo2 ?j Vk Vl

energy

(-~2~/~x2

div ~ : 0

shall o m i t the tilt s u p e r s c r i p t s

(~/~)

Ip qo2

m : 1/2 / { B ( ~ % / ~ x )

and

we

in e q u a t i o n

qijkl e x p r e s s e s

in smectics.

The e l a s t i c

from the p e r m e a t i o n

(2).Ip is the p a r a m e t e r

characterizing

process.

For w e a k p e r m e a t i o n ,

I + 0, the fluid m o t i o n is i d e n t i c a l to the layer P (molecules do not flow a c r o s s the layers) and 3~ / ~t = - V x

displacement

On the contrary, in the o t h e r I p + ~ , % and V x are i n d e p e n d a n t then r e c o v e r s

the c l a s s i c a l

instabilities

/3/,/4/,/5/

phase

limit corresponding (eqs.

(I) and

equation

to strong p e r m e a t i o n

(2) are d e c o u p l e d ) .

:

~ /St : -Ip qo2 (6 F/6%)

b- Roll

One

c o m m o n l y u s e d to d e s c r i b e

(6)

instabilities

Equations

similar

to the s m e c t i c

ones can now be s u g g e s t e d

for dis-

218

sipative

structures.

describe

the p h a s e

First

in the x d i r e c t i o n ) , thickness.

The

quantities

averaged

with

implies

that

boundary all

term

qA[

a Darcy

law

conditions,

sample

leads

behavior

of

sample

long

to

structure

of

depend

conditions

the

with wave

vec-

as we are con-

the p h a s e

(q >

vx

i/2

%,I ))

+

~Vo_~_

o_m_~_

2D,,)

Voxx

~/~ e~ 2D~ + - ~ f my=

~((-k/r~

)+(Vo/2(Dm D/l) ))

(31)

)-r/~

I/2 /(r/ 0)

the m o t i o n

in t h e

y < 0

(or x < 0)

the m o v i n g

dislocation

is

with

u s e of

my (y,x) dy

the F o u r i e r

transform

iqxX

is that, in,

/2~

once

there

the m o t i o n

appears

all V

a glide

is no 1 0 n g

is d i r e c t e d

in t h e w a k e

For

values,

oy

A¢ : 0

V

x > 0

all V

component

range

butwon

phase

in t h e m o t i o n shift

the contrarylthe

of t h e d i s l o c a t i o n

as

of the

in r e g i o n s

shown

whole

phase

on fig.

4

V

x > 0

O n the c o n t r a r y

A¢ :- ~

The whole

the

has

evaluated

in the

static curves

Vox

A¢ = ~r A%(x)

pattern

been

x < 0

climb motion

for d i f f e r e n t

D~/D/f

Theoretical

A% = 0

,

distorsion

kept

=

x < 0 A¢(-x)

has

been

values

(fig.

14)

fixed

as

s h o w n o n fig.

calculated

of the

constant

limit (section

has been

x < 0

0

for p u r e

the ratio

the velocity

<

ox

symmetry

glide motion

h% = +2~

,

x > 0

recovers

> 0

ox

,

values,

oy

A¢ = - 2 ~

pure

of

( Voy : 0

(screening

c.

For

one

around

dqx my (qx,~)e

result

difference

and glide

l a w [Y ~i/2( °r I x I-I/2 ).

%(y : - ~,x) :

f .~ lim qy-~ 0

A main

:0)

decreasing

for y >0

In the w a k e

of t h e p a t t e r n

is c a l c u l a t e d

dislocation

Vox

the phase difference :

: ¢ (Y = + ~,x) -

h# :

(Voy # 0,

is e x p o n e n t i a l l y

(or g l i d e ) .

decays

enlightened £4

The

4b.

in the c a s e of

screening

l e n g t h Dll / V o x ;

at the e x p e r i m e n t a l

value

0.06

2000,

where

3b)

correspond

to

]311=20,

to the e x p e r i m e n t a l

200,

value

V o x = 0.16

layers/sac. The

asymmetry

pronounced

between

for s m a l l

D@

the

front

(or l a r g e

and

the w a k e

V o x ). T h e r e

of t h e m o t i o n

is w e l l

is a c h a r a c t e r i s t i c

non

234

a

Ix

D:20

,I

i

"i.

"_'_'m-_. ........

b

1,x

-'-

2 ..................

D=200

The phase has been calculated for the experimental value Vx= 0.16 layers per second (corresponding t o f i g . 4 (a)) a n d p : ( D ~ /DIj )I/2= 0 . 2 5 (found in the"static case) and for different values of m~ : (a) D # : 20 s e c -~

}

' .... = : < . : ; . _ ~ 1 2 1 2 - Z

......

_2 ....................... ------, ¢: ; ;.._._2_; 2222-_-/.~_i-i-

c

Ix

.....................

~_;. : . ~

(b)

D#:

200

(c)

D# =

2000

O=2000

...........................

-C-{~2_