Stability of permeative flows in 1 dimensionally ordered systems

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Stability of permeative flows in 1 dimensionally ordered systems

Jacques Prost, Y. Pomeau, E. Guyon

To cite this version:

Jacques Prost, Y. Pomeau, E. Guyon. Stability of permeative flows in 1 dimensionally ordered systems.

Journal de Physique II, EDP Sciences, 1991, 1 (3), pp.289-309. �10.1051/jp2:1991169�. �jpa-00247519�

ClassificaUon

Physics

Abstracts

47.20 61 30

Stability of pernleative flows in I dimensionally ordered

systems

J. Prost

('),

Y. Pomeau

f)

and E.

Guyon ₍₃₎

(~) Laboratoire de Physico-Chlmle

Thkonque

(*)

f)

Laboratoire de

Physique StaUstique (**)

(3)

Physique

de la Matidre

Hktkrogdne (***)

(Received 28 December 1989, revised 6 June 1990, accepted 5 December 1990)

Rksumk. On rencontre des structures _en couches dans des

systdmes thssipaUfs

tels que les rouleaux convectifs de

Rayleigh-Bdnard

et dans les cristaux

hquides (smectiques

et cholestkri-

ques)

Nous

prbsentons

_{in une}

description gbnbrale

de la stabihtb de _ces structures dans le cadre du forrnahsme de la diffusion de phase, lorsqu'elles sont sournlses I _un

champ

de force extbneur

(bcoulement, champ klectnque)

_aglssant I

angle

droit de la direction des rouleaux, _en fonction des conditions _aux hmltes La solution unidimensionnelle d'kquihbre avec des conditions _aux hmltes

ngides

_pour la

phase

conduit I _un effet dkcouvert par Pocheau et

Croquette

_~P C ) dans la

convecUon de R B. et mettant _en jeu la coexistence de _zones dilatke et

compnmke

Cet effet _{a un}

analogue

dans les

cholestknques

Avec (es mdmes conditions _aux hmltes, _nous

gknkrahsons

l'instabilitk bien _connue d'ondulation, obtenue dans les

smectiques

_sous l'effet d'une thlatation,

au cas d'une force transverse I la fois du point ^de vue de la stabihtk hnkaire et dans le

rdglme

hautement _non linkaire Nous

suggkrons

aussi la

possibilitk

d'obtenir des structures fractales Pour des condiUons _aux hmltes mixtes, on

prkvoit

l'existence de

rkgimes dkpendant

du temps et

mettant en jeu la nuclkation de nouvelles couches _{alnsi que} cela _{a aussi} ktk observk dans les

expknences

de PC

Abstract.

Layered

structures are met _in

dissipative

_systems, such _as

Rayleigh

Bknard rolls, _as well as m

liquid crystalline phases (smectics

and

cholestencs)

^We present here a

general

description, m the framework of

phase

dynamics, of the stability of these structures when

submitted to _an extemal force field

(flow,

electnc field) acting

perpendicular

to the roll _axis for

vanous

boundary

conditions The one-dimensional

equilibrium

soluUon with fixed

boundary

conthtions leads to _an effect, discovered

expenmentally by

Pocheau and Croquette on

Rayleigh-

Bknard rolls _in the presence of a transverse flow, and

involving

the coexistence of

compressed

and dilated rolls, this effect has _a known counterpart in cholesterics

Using

the _same

boundary

conthtions, _we

generalize

the well known undulaUon

instability

obtained under _a dilative _stress to the _case of the action of _a transverse force both from the point of view of linear

stability

and _m the

highly

nonlinear limit. The

possibility

of

observing

fractal structures _is mthcated. For mixed

boundary

conditions, it is

possible

to have _a sustained Ume

dependent

behavior

involving

the nucleation of _new

layers

_as also observed _m the above mentioned expenments

(*)

URA 1382, ESPCI, 10 _rue

Vauquehn,

F-75231 Pans Cedex05, France

(**)

URA 731, ENS, 24 _rue Lhomond, F-75231 Pans Cedex 05, France

(***)

URA 857, ESPCI, 10 _rue

Vauquehn,

F-75231 Pans Cedex05, France.

1. lntToducfion.

It _{is a} priori ^not obvious that _{one can use} the _same

equations

for

describing

the

dynamics

of

out-of-equilibrium

and

thermodynamic _systems. However, spatial symmetry

sets require-

ments _on the _structure of the

equations

which _are the _{same m} both _cases The absence of

Onsager

relations and of _a

thermodynamic

function _m the _case of

out-of-equilibrium systems

allows for _a

larger variety

of terms than with

thermodynamic

systems, ^{but those terms which} exist with the latter also exist with the former. ThJs _is

particularly

true with

layered

structures

(smectic

_or cholestenc

liquid crystals, Rayleigh-Bknard

rolls

.)

for which the linearized

equations

^{of motion} are

essentially

similar

[I].

One of the most remarkable char>ctenstics of the

dynamics

of

layered systems

is the existence of _a mode called permeation. ^{In its}

simplest form,

it tells that _a flow _{m a} direction

orthogonal

to the

layers

tends to carry the whole

structure

along.

Whether it

really

can or cannot

depends

_on

boundary

conditions It is clear

that this

tendency

to

drag

the _structure

along

exists irrespective ^{of the fact that} one is

dealing

with _a

thermodynamic

_{or an}

out-of-equilibrium _system [2]

^More

generally,

the

permeation phenomenon

_expresses the fact that _an external force drives _a difference between

layer

and

barycentnc velocitiis.

In the _case of

cholesterics,

it _is associated with the rotation of the local

axis as observed

long

_ago

by

Lehmann _{m a} temperature

gradient

at small

amplitude [3]

and revived _m the nice

quantitative

_expenments of Madhusudana and Pratiba

[4]

using a D-C electric field. A natural framework for

understanding

these observations _is

provided by

the Leslie

hydrodynamic theory [5].

The connection made

by

Helfnch

[6]

^{between the local}

rotation m a cholesteric and permeation ^makes it

possible

to

incorporate

the Lehmann effect

m the framework of the

hydrodynamics

of

layered systems [7,

8]: a

permanent

^bulk rotation of the order

parameter (I.e.

local

optical

_{axis m} cholestencs _or the

phase

of the

layers

m the

smectic

case)

takes

place

with _an

angular velocity proportional

to the

_extemal

field at small

amplitude

m the _case of free

boundary

conditions. Direct observations of permeation in either

case have been found

extremely

difficult for _reasons which will be discussed in the conclusion of'thJs _paper. However _a clear

experiment by

Clark

[9] provided

_{a convincing} indirect evidence for the existence of

permeation

in smectics.

The _case of out of

equihbnum

roll structures has been studied

by

Pocheau and

Croquette (P.C [10].

Their

analysis

of their

expenment

on the «compression » of

Rayleigh-Bdnard (R.B.)

rolls under the effect of _{a transverse}

flow,

uses a

generahiation

of the

phase dynamics equation (

II whJch _is identical to that

descnbmg thermodynamic systems.'The

_very existence of the

non-homogeneous

compression m the P-C-

expenments

is a direct

proof

of the

validity

of the permeation concept in out of

equilibnum

systems

However thJs remark _is not

only

of interest for what _concems the

similarity

of the

equations,

^{but it has also} some flesh _{on it as one}

might

_{expect some} s1mllarities in the

physics

too. In thJs respect, ^{it should be noticed}

that,

in their expenment, ^{Pocheau and}

Croquette

observed distorted but

steady _patterns

_as well _{as a}

periodic

nucleation and destruction of the rolls at the ends of the convection cell. The

present

^work was simulated

by

the above

experimenjs

It _{is a}

general stability analysis,

in the framework of the

phase

diffusion

equations,

of the

stability

of

layered

structures submitted to a transverse extemal force field We introduce the

general _equations

which will be used

throughout

the paper m section 2. In section

3,

_{we review} the basic solutions

corresponding

to a flow normal _to the

rolls, insisting

on the

importance

^of

boundary

conditions

In section

4,

_we show that the static deformation obtained

by

Pocheau and

Croquette

_may

become

linearly unstable,

if _a transverse modulation is allowed The

interpretation

^{of thJs} effect is _as follows _m the

equation

of _{motion, some} terms represent a kind of

elasticity

and express the

property

that the structure has _a

preferred

wavenumber

[11]

^In the P-C-

effect,

part

^{of the} structure _is

compressed

and part ^of it _is under extension. Therein the wavenumber

may increase

locally

and _{so can} opt1mlze ^the energy

by adding

_a modulation _{m a} transverse

direction. The

corresponding instability generahses

the well known undulation

instability

obtained under _a dilative stress and _m the absence of flow _m smectics

[12, 13].

It _is

analysed

_m

section 3 both from the

point

^{of view of the linear}

stability

and in the limit of _{a very}

large

constraint A

key

feature of _our

analysis

_is that all

interesting phenomena

_occur at

vamshJngly

small velocities thJs

corresponds

to the

large

box limit

(I.e.

_a

large

number of

penods

N _m the

system)

We

study

both the linear

stability

close to threshold V l

/N~)

and the

strongly

nonlinear regime

(keeping

V « I

IN).

The

instability

takes

place

when the

phase

_is fixed at

both ends of the structure

(it

should also be

possible

to observe it with mixed

boundary

conditions)

In section 5 _we will _see that _a sustained time

dependent

behavior _is

possible

with different-

boundary conditions,

_m the absence of transverse modulation of the

rolls,

it

corresponds

to the creation-annihilation of rolls _as observed

ixpenmentally _by

_Pocheau

ind Croquette.

More

complicated phenomena

_can also be observed and have been

analysed

_using different

boundary conditions

on both sides In the presence of _a flow whJch forces _a

winding

of the

phase,

new rolls have to be

continuously

added inside the structure, thJs cannot be described within the framework of the

phase dynamics

which excludes fast events such _as the nucleation of _new

rolls,

in the

Rayleigh-Bdnard terminology

» that _we shall _use

throughout

this paper.

However this fast

dynamics

_may be considered _as instantaneous

(compared

to the

phase dynamics).

This allows _us to include it m a consistent fashJon into the

phase dynamics

equations, as shown _m the last section, where _we also sketch _an

analysis

of_the behavior when

nucleation of _new

wavelengths

_occurs

2. General

equations.

It _is worth

starting

with the smectic

hydrodynamic equations. They

_are valid _m the

long wavelength,

low

frequency,

small distortions

limit,

and _can be derived using standard

procedures [8].

^In the isothermal

incompressible

_case, which _is relevant to our current

considerations, they read,

in their hneanzed _version

~"

V~

₌

Ao

^~~ +

pE~ (I)

at _8u

p

~~~

=

V~P

_8~~ ^~~ +

V~«]~ (2)

(

=

«E~

+ p

~~

(3)

Div V

=

0

(4)

~~

+ Div J

=

0

(5)

u is the

layer displacement

variable _as

defined'in

_the

_appendix,

z the

unperturbed layers

normal _axis, V the

barycentnc velocity,

F the elastic free energy of the smectlc

[7, 8]

F

=

I B ^°"

~

+ K

~~

+

~~

^~

dr

(6)

2 3z ax

by

The first _term describes the

compressional

_energy of the _smectic

layers,

^{and the second their}

bending

_energy. The ratio A

=

(K/B)~'~

has the dimensions of _a

length.

It _{is a «} microscopic »

quantity

^{of the order of the}

penod

of the structure and _is called the de Gennes

screening length [7].

This is the existence of thJs

microscopic length

which _is the _source of the

originality

of one-dimensional order in two _or three dimensions.

E~

is the _z

component

^of any extemal force

having

the sytnmetry of _an electric field

(it

could be _a

temperature gradient

for

instance),

_p

the _mass

density,

P the pressure,

«]~

^the

dissipative

_part of the stress tensor and

£

the flux conjugate ^{to the force}

E~ A~,

p and _{« are}

dissipative

coefficients. Note that because of

Onsager relations,

_p enters both

equations (I)

and

(3).

Equation (I)

_is the permeation

equation:

_a

velocity

field V~ or an extemal field E- may either set the structure into motion _via

~",

or distort it vta

~~. Equation (2)

_{is a}

at 8u

generalization

of Namer Stokes

equations.

The

dissipative

stress tensor involves three viscosities _m the

incompressible

limit. Note that ~~

is _a force acting

along

the _z direction 8u

Equation (3)

_expresses the _current

and/or

the

layer

distorsion

resulting

from the action of the field.

Equation (4)

_gives the

incompressibility

condition and equation

(5)

_is the conservation law for the

extenjive

quantity

Q (charge

if

E~

_{is an} electnc

field)

the flux of which is

£.

As discussed _m the

introduction,

the

expenmental

relevance of these

equations

has been

reasonably

well established

[14].

What about out of

equihbnum

structures ? A

frequent description

involves the _use of _a

complex

order parameter :

#

=

#o e'4 (7)

This _is

really

not different from _a smectic A

phase

for which such _an order

parameter

^{has been}

widely

used to describe the nematic-smectic A transition

[7]. Equations (1)-(5)

_are valid in the so-called

hydrodynamtc ltmit,

where

#o

may be considered constant, that is for

wavelengths large compared

to correlation

length charactenzlng

the modulus fluctuations and

controlling

the size of the _core of dislocations. In _out of

equihbnum

_systems it _is called the

large

box

limit, similarly

to the smectic _case, the _use of _a roll

displacement

vanable is

meaningful

_in

what _is called the

phase

approximation

[11]

fb ₌

qo(z

₊

U) (8)

Can _{one use} equations

(1)-(5)

without any precaution m the latter case? As

already announced,

expressions which result from

spatial

_{symmetry are}

necessanly

s1mllar

However, Onsager

relations have _{no reason} to be satisfied. As _a

result,

instead of _one coefficient _{p, one} has to _use two different coefficients _p and

p'

m

(I)

and

(3) Similarly,

attention has to be

paid

to the _viscous term _in

(2)

_{m an} infinite smectic it reads (1~, are

viscos1tles)

°~~j ^" ~ 'i2

~ij ~l'i3

_'i

2)(Viz ~jz

+

~jz ~iz)

I'll+'i2~~'i3~~'i5+'i4)~iz~jz~zz~1'i5~'i4+'i2)~ij~zz. (~)

The _same expression ^cannot be used _m the _case of roll systems ^with

rigid boundary

conditions,

such _{as m} standard

Rayleigh-Bdnard

cells.

Indeed,

fnction results from the existence of _an

averaged

fluid

velocity

with respect ^{to that of the walls.}

Thus,

_m the reference

frame,

where the walls _are at rest, _one has the

relationships Vjtr~j

=

iii

_U~

Vjtr~j

= ~ U~

~~~~

In which

iii

and

A~

have the dimensions of _a

viscosity

_per

length squared iii

=1~ max

(d~~, (A~I~)~~)

A~ =l~d~~.

Equation (10)

holds _as well for smectics bounded

by parallel plates

with the

layers perpendicular

to the walls

[9].

Note

that, conversely,

_m the _case of _an

instability

with free

boundary

conditions

equation (9) holds,

because of Gahlean invariance

(i

_e.

only gradients

of

v can

play

_a

role, and,

because of

spatial _sytnmetry, they

have to be second

denvative).

The last point concems the difference ^~"

v~)

^{The occurrence of this term in}

^(I)

at

results from Gahlean _invanance

looking

at the

system

^from a moving frame of reference should not

change

the

physics.

In _an

out-of-equilibnum system,

one can construct Gahlean

invariant

expressions involving

not

only

the

background

fluid

velocity,

but also the walls

velocity.

TbJs

implies that,

_{m a} frame in which the walls _are at rest, the coefficient

affecting

u~ is not

necessarily

_one. The

tendency

for _a flow to

drag

the structure is

certainly

there

(and

allowed

by symmetry) but,

in the ideal _case, the limit

drag velocity

needs not be

u~.

The lmeanzed

equations

^read now :

~-AvUz=-Ap( ~( +HEz (")

3u~

~ SF

~ _~

(12)

P

$

^"

~'~

'z

$

" ~

£

=

«E~

₊

p'

^~~

(13)

3u~

3v~

G

^~

G

^{~ ~ ~~~~}

~~+DivJ=0.

(15)

Equation (14)

results from the fact that

u~ ^{is zero at} the walls. F _{is a}

Lyapunov

functional which _is identical to

(6)

because of

spatial

_sytnmetry Note that the expression

(6)

would not be relevant for

Taylor-Couette

Tolls _or convection _m

planar

nematics

[20] because,

_m these two

problems,

there _{is an} intnnsic direction of the rolls Note that the _x

component

^of

(12)

could exhibit _a term

~~"

It does exist in smectics, ^but is included _m the pressure

[7].

^In

ax 3z

what

follows,

its inclusion would

simply change

the elastic constant B. In order to discuss the instabilities of section

4,

_we will need _a nonlinear _version of equations

(I1)-(15).

For the _same

reason as m smectics

[7, 12],

^the essential _non

linearity

_comes from the

rotationally

invariant

expression

of F _as described _in the

appendix.

The

rotationally

invariant F involves

E(u)

=

~"

+

(Vu

_)~ instead of ^~"

),

^{and the covanant} expression of the curvature term

3z 2 3z

Div _{n, can} be

replaced by A~u [7, 19]

_:

F

=

( _(E(u)

+ A

~(Ai u)~)

dr.

(16)

There _{are many} other nonlinear terms to be added _to equations

(I I)-(15),

the most obvious of which _expresses convection in equation

(11), namely

:

A

~ u~

V~u (17)

It has to be

compared

to u~ in order to estimate its

relevance,

that _is

V~u

^{has to be}

compared

to unity. ^{We will show}

that,

_{except m}

boundary layers,

the

description

of which is

beyond

the

scope of this

article, V~u (A IL

Since A is _{a «}

microscopic length,

L _can

always

be chosen

large enough

for~ that ratio to be small

compared

to

unity.

It _is worth

giving

another

example illustrating

the role of the small

parameter AIL

equation

(11), together

with the defi-

nition(16)

of

F,

involves the

product _A~B

=

D,

whJch has the dimension _a of diffusion

constant. In _a

complete

nonlinear _{treatment one} should

keep

track of the

dependence

of D,on

u From

sytnmetry,

one should write expressions of the

type.

'

D(~)

~

D0

+

Dl ~]

₊

('8)

Dimensional

analysis implies

Dj ~/Do

^'

(19)

We will show that the critical

velocity

^for

setting

the

instability

_{is given}

by

_:

u~ ~

DA

IL

^~

(20)

which

immediately yields

D(u Do

+

~

(21)

L

Again

in the _«

large

box _» limit thJs _{term is}

entirely negl1glble.

ThJs

possibility

of

keeping only

the nonlinear _terms due to rotational invariance in

F,

_{is a unique} feature of _one

dimensionnally

ordered systems.

3. Onedimensional

equilibrium

solutions.

In the present section _we consider one-dimensional deformations _u

(z,

t

)

which

correspond

to

translation,

dilation _or

compression

of rolls. The

incompressibility

cond1tlon

requires

_a

uniform

velocity

field _u~

= uo the flow _is then

impoied by

the inlet and outlet fluxes at the

extremities of the system as in the P-C- expenment. The _z component ^of

(12)

defines the

pressure,

whilq

_u~ + 0 satisfies the _x component.

Steady

state solutions involve _constant currents

~"

^find

^£

^{As a}

^result,

^{we are left with just two equations}

t

J~

₌

«E~

D' ^~

~ (23)

(with

D'

=

lip').

Equation (23)

defines the flux due to the extemal force

(heat

flux if

E~

^is a temperature

gradient,

electric _current flux if

E~

is _an electric

field)

_{we assume} that the system is

impedance

matched for J~.

Equation (22)

_is

interesting

in that it illustrates _very

simply

how

boundary

conditions _on

u are important in

determining

the type ^of behavior which will be observed

expenmentally Equation (22)

can be rewntten

= V ₊ D ^~

(24)

t

~

in which- _we have _set V

= A

~

Vo

+

pE~,

which insists on the

equivalent

role

played by

_a flow and _-an extemal field

(althougll

their

physical

_ongm and time reversal behavior is

different).

Equation (24)

_{is a} linear version of the convection diffusion

equation

^for the

phase dynamics [11, 15-17]

and has also been discussed _in the context of cholestencs and smectics

[5-9].

Let _us consider _a

sample

of thJckness L

(0

_{< z <} L

)

We _can calculate

easily

_a number of

unperturbed

solutions for _vanous

boundary

conditions.

They

_are obtained quite

naturally

in

experiments.

In

liquid crystals,

free

bouqdary

conditions

correspond usually

to free

liquid

surfaces and

ngld

B-C- to the material in contact with _a solid surface with

appropriate

^surface

treatment. In R.B.

rolls, ngld

B.C.

correspond

to solid lateral walls whereas free B-C- _can be obtained with

smoothly

_varying

properties (18).

i)

For free _or Neuman

boundary

conditions

~"

(z

=

0)

=

~"

(z

=

L)

= 0

(25)

3z 3z

we

get

^{the solution of the Lehmann} rotation _:

u ₌ Vi

(26)

ii)

For

ngld

_or Dinchlet

boundary

conditions _:

u(z

=

0)

=

0,

_u

(z

=

L)

= uo

(27)

the solution _{is static :}

v zuo

"

2 D ^~

~~

_{~ ~}

L ~~~~

~

It describes the P-C- effect. An

equivalent

effect had been

predicted by

Leshe

[5l'for

cholestencs lvhen L is

equal

to _an

integer

number of 2

ar/qo,

the

equation (28)

descnbes _a compression of rolls _m the downstream region

(z

_<

L/2)

and _a dilation _m the

upstream

one as

observed in the P.C.

expenments.

Note

that,

_m these

expenments,

^the

change

_m

wavenumber induced

by

the flow _was not

always

small.

ii)

For mixed conditions _:

u(z=0)

=0

)(z=L)=0 ₍₂₉₎

a

steady

solution _is obtained

u =

~'

z(2

L _z

(30)

In the

following paragraphs,

_we will

study

the

stability

of _some of those

simple

solutions

by considering

two dimensional

perturbations

m case

ii) (chapter 4),

defect nucleation _{in one} dimension _{in case}

iii) (chapter 5)

4.

Beyond

the one-dimensional situation.

4.I FORMULATION. In this section _we consider the _same

problem

_{as m} the

preceding

_one

with

rigid boundary conditions,

_{m a system} of infinite extent m the _x direction. To look for the

stability

of the solution

(28) (called

_uo in the

following),

_we allow for the

possibility

of fluctuations

depending

_on coordinates _x and _z. For the sake of

simplicity

_we do _not add the extemal field

E~.

The

equations

reduce to _:

~

~

~~ ^~

P

~~

^~~~~

~

~i $ ^~i _f

~~ ~~ ~~~~

P

(I

"

$

_~

xx ~x ~~~~

3v~

_3U~

~

⁺

~

^{= o.}

(34)

In _view of translational _invariance and the autonomous character of

equations (31)-(34)

with respect to time, we choose to

study perturbations

of the form

u(z,

_x

t)

=

uo(z)

_{+ w}

(z) Cxp(iqx

+

)~

([ii i i iiiiiii~~iii ii~i ^~~~

.-

(35)

P

(Z,

x t

)

=

Po

₊ 8P

(Z,

x

,

t

)

Since _we will be

looking

at the onset of

instability

and at nonlinear

steady

_{states, we can}

neglect

the acceleration terms _in

(32)

and

(33)

We then eliminate _p and

V~

^{and transform}

equation

(32)

_in

O=-~~-A~~V~(z)+~)~~ ₍₃₆₎

8u q

z

Again anticipating

^~

~

(

,

q~

~

(

AL

)~

_{~, one} obtains _: 3z

V~(z)

=

~~

^l + O ^~ ~~

(u)

^~~

(uo) (37)

L 8u 8u

That is for

(31)

_:

~'~

= D

~~ ~"°

q~

+ A^~

q~)

^w

⁽³⁸⁾

at az °z

IA

_~

where D

=

A~

⁺

B,

^and terms of order

AIL

smaller than those retained have been

Azz again omitted.

4 2 LINEAR STABILITY With the chosen form

(35),

_« becomes _an

eigenvalue

of _a linear

differential operator ^with q as a parameter :

(« M~)

wo = o

(39)

with the

boundary

^conditions _wo

= 0 at _z

=

0 and

L,

and the

following

definition for the linear

operator M~

:

M~

₌ ^D

(3~/3~ q~[-

V

(z L/2 ID

₊

uo/L]

A^~q~)

(40)

m which

again

V

= A~

Vo

As equation

(39)

has _a coefficient linear _{m z,} it _can be solved

by Laplace's

method Tlus

yields A1ry

^functions that become

simple

circular functions if the external _constraint V _{is zero.}

4.2.I V

= 0 _case. The

quantized

_« values _are

"n "

Dl'T~lL(2~+1)l~~+ "0qiL

₊

A~q~),

where _{n is a} natural integer.

The less

damped

_or most unstable mode _{is at n}

=

0 and for _a wavevector

q~" ~"0/(2LA~)>

with the

eigenvalue

"0 ~ D

l'TiL~ Uil(4 L~ ~)j

,

which shows that the

parallel layered

_{system can} become unstable _in the absence of _an extemal force field V if _a

large enough imposed

dilation _uo is

putting

the

systeit~

out of

equihbnum («

~ 0

).

The

marginal stability

_is reached with _uo

= 2 vi and

corresponds

to _a dilation with respect to

equilibrium (the

definition of the _sign of uo is discussed in the

appendix)

The

corresponding instability

tends to restore the

optimal

and shorter

wavelength by building

an undulation _as

permitted

in the _x direction This situation has been well studied

in smectlcs

(m particular

_see

(12)).

4.2.2 V # 0.

Coming

back _now to the

original problem,

that _is to

perturbations

around the

basic solution _{given in}

(28),

_we shall _{use as a new} vanable

z'=z-zo

with zo=

L/2

₊

D(uo/L

₊ A

~q~+ «/Dq~/V

The coordinate zo

corresponds

to _a

turning point

^{for the}

problem

and defines the domain of existence of the dilation

instability.

This

change

of coordinates allows _us to _recover the familiar

Airy equation

_:

d~w/dz'~

₊ bz' _w

= 0

with b

=

Vq~/D.

It has the

general

solution.

w ₌ z

~'iaJjj~(2 b~'~z'~'~/3)

₊

flNjj~(2 b~'~z'~'~/3)]

where _a and

fl

_are

arbitrary

constants for the _moment and J and N _are Bessel functions.

Putting

_zj

= L zo, one finds that the

boundary

conditions for _w impose a transcendental

equation

to be satisfied

by

_«.

~l/3(~

^b

~~~Z)13) fi~l/3[~

^b~~~(~20)~~~/3) "

fi~l/3(~

^b

~~~z/~13) ~l/3[~

b ~~~(~20)~~~/3)

(41) Indeed,

this last expression allows _{us m}

pnnciple

to find

numencally

the unknown

quantity

«.

Let _us consider for the moment the onset of

instability.

ThJs is reached when

« = 0 _{is a} solution of

(41)

which then gives a relation between the

quantities

_uo, L, V and _{q at}

threshold. We know

already that,

if

V=0,

the onset of

instability

_is reached at

uo = 2 vi and

q$

=

ar/(AL)

We

expect

thJs to _remain true as

long

as V _is

negligible.

In scaled quantities,

by comparing

terms in

M~

we can express this cond1tlon

(V negligible)

_as

V _« V~ = «DA

IL

^~ _{or as} C « _ar

,

the dimensionless number C

=

VL~/(DA

will be used in

the

following

to charactenze the relative

efficiency

of the

dnving

force

(it plays

the role of _a Pedet number for the

problem).

If thJs

inequality

is true, the _source of the

instability

is _m the

boundary

cond1tlons

imposed

_{upon u.}

On the other

hand,

if V

~ V~ or C

~

l,

the _source of

instability

is the field V. In thJs

limit,

and _at threshold _«

=

0,

_one has zo m

L/2

+ DA ^~

q~/V

and zj m

L/2

DA

~q~/V.

If

Vi _V~,

_{the last term}

on the

right

hand side of the expressions of zo and zj is

negligible

_as

compared

to the first _one.

Thus,

_{one can}

replace

the

equation (41) by

_:

~l/3(Q~~~) Nl/3(~ lQ~~~)

"

~l/3(~ lQ/~ _{fi~l/3(Q~i (~2)}

where

Q~

is defined

by _Q~

=

q~(VL~/D)"~

The _pure number

Q~

is the smallest root of

(42), corresponding

to the onset of linear

instability.

One _can check

that,

m the _crossover region

(V

_m

V~),

the expressions q~~ and q~o

coincide,

_up to a numencal factor In the absence of

imposed

dilation

(uo=0),

thJs

implies

that the

instability

sets up for

VmV~

with q~ m

(ar IA L)"~

_One

_{expects that,}

_{if V is much}

_bigger

_than this threshold

value,

_a whole band

of wavenumbers will become unstable.

Let _us estimate the

shape

of those _two borders. Let _{us assume} first that the argument of the Bessel functions _m

(42)

remains finite _{on one} side of the band of unstable wavenumbers. ThJs

yields the,long wavelength

limit _q

m

Q~(D/VL~)"~

_as obtained above.

The other side of the unstable band _is reached when the two terms

m'zo

and

zj

( =,L zo)

are of the _same order of

magnitude.

ThJs

implies

_{q m}

(LV/DA ~)"~

and allows to use the

asymptotic

WKB-like expression for the Bessel functions _m

(41).

However it is

more, transparent ^to come back to the

onginal

equations ^{With the}

scaling

Z

=

z'/L

and

q =

Q(LV/DA

_{~)~'~, one} transforms

(38)

into

d~wjdZ~

₊

Q~( VjV~)~ (Z lj2 Q~)

_w

=

0,

with the

boundary

conditions _w

= 0 at Z

=

0,1.

This has solutions at

large (V/V~)~

if _a

« classical _»

region

of the

potential Q~( VI

_V~)~

(Z 1/2 Q~

_exists, that _js if thJs

potential

_is

attractive somewhere~ for

particles

w~th energy zero. Otherwise _one would have

purely

exponential (non oscillating)

solutions which would not fit both

boundary

conditions. As the

largest

value of the

potential

_is reached for Z

=

I, having

_{a pos1tlve} value of the

potential

imposes

Q~<1/2.

This condition defines

completely

the short

wavelength

border of the

stability

domain at

large

C's with _a variation

of_the

cntical wavevector _as V~'~

ThJs ends the linear

stability analysis

of _our

equatioii (36).

Below We shall consider the domain of very

strong nonlineanties,

reached at

large

C's

(but

still _m

agreement

^{with the}

inequality

C «

L/A ).

4.3

I~ONLINEAR

_{REGIME. The}

study

of the nonlinear domain _is made

formally simpler by keeping

the order of

magnitude

_coming from the above

developments:

_u~c

A,

_xx

(AL)~'~,

z~cL Then the intensity ^{of the extemal} constraint V _is measured

bf

the

dimensionless number

C,

and the nonlinear equation ^{for the}

steady

solution

reads,

after

making

_a

change

of variables uA

- u,

x/(AL)~'~

- x,

z/L'$z~

c

=

(ui

₊

U]/2)z

+

lUx(Uz

+

U]/2)lx

_Uxxxx

(43)

In the

following,

_we restrict _our

analysis

to the

boundary

conditions _u

= 0 at z ₌ 0 and I such

that _-no dilation is

imposed by

the

boundary

conditions _.~

We

already

established that _a threshold value of C exists such that the

x-independent

solution of

(43)

bifurcates to a

sqlution depending

_{on x m a non} tnvial fashion. Moreover thJs

equation

is the

Euler-Lagrange copdition

_expressing that _a functional G is

stationary Indeed, relaxing

the e'~~

dependence (but

still

looking

at the _same

scale),

the

equation

for _u may be

wntten

au SF

at ^~

~P

8u

m which _§

=

V(I

_{+ A~/A==}

A~)

^and

i~

=

A~(I

+

A~/A-= A~),

or

au 8G

I

_8u

where G

= F +

l'dr

^flu.

Thus it makes _sense to look at the

optimal

solution with the lowest « energy » As G _contains

a term which is linear _{m u m} thJs energy and _is

multiplied by

the

large

_quantity

C,

_one

expects

that this

optimal

solution has the

largest possible

order of

magnitude

_m C.

We _{can use a} dimensional argument to evaluate the order of

magnitude

of the different

terms _m

(43)

More

§recisely

_we look for solutions of the form

u«c«, z«cfl,

x«cY.

(44)

If _we look first for extended solutions

along

_z, the

scaling

of the reduced _z variable _as I should

correspond

to

fl

= 0. It _is

possible

to

satisfy

the

scaling

of the four first terms m equation

(43)

with _{u ~c}

C,

_{z ~c} I and _xx

C~'~ (a

=

I, p

=

0,

_y

=

).

ThJs choice makes

negligible

the 2

higher

denvative in the

right

hand side of

(43)

which is of order

C~~.

It also

yields

the

parameterless

equation, deduced from

(43)

with the

scalings

_given

by (44)

but with the

change

of notations _{u m}

Cu,

f m z, x m

C~'~x

:

I

=

(uz

+

ul/2)z

₊

lux(uz

+

u]/2)lx ₍₄₅₎

with the b.c. _u

=

0 _at

z = 0 and I The

scaling (44)

with the above choice of

exponents

indicates that the

preferred wavelength

of the

optimal

structure _increases like

C~'~

_at

large C's, although

the

amplitude

of the

perturbation

_{m u varies as} C. This _seems to be

incompatible

at first with the

general

assumptions ^{of small}

gradients

_u made _m the

calculations,

_m

particular,

the

phase

fluctuations has to be small _{m some sense m} order to

permit

to write the

equations

^{in the} coordinates of the

unperturbed

system. ^However this _can be done

consistently

even in the limit where C goes ^to

infinity

because there is _a smallness

parameter

independent

_on

C,

which _is

again

the ratio

AIL

The tilt of the

equiphase

lines

remains

small,

_as

required

if the dimensionless quantity u~

(resp

_{u~) is}

small,

_m the

large

C

limit,

this

gradient

scales as

(

CA

IL

_)~'~

(resp

CA

IL ))

and thus _{can remain} small _even with _a

large

C _as

long

_as

AIL

is smaller than C~

This

scaling

does not give much information _on the detailed structure of the solution of

(45)

The

following _{arguments suggest}

that it results from _a cascade of instabilities down to scales where _a different

scaling corresponding

to

boun"dary layers

takes _over The cascade

results from the fact

that,

_as V increases,

undulating

rolls become

again

unstable when

they

are

sufficiently

dilated. Thus

they

tend to

develop

_an undulation withJn the undulation. ThJs process can

only stop

^when a

sensitivity

to curvature is recovered In order to demonstrate this

possibility,

_we show that any solution _u of

(45)

_is at least

locally

unstable. Since there _{is no}

length

scale _m

(45),

it _is also

globally

unstable.

First,

_we know that the solution

uo(z)

=

z(f _z)/2

_{does not}

_represent

_the

minimum of the

energy, since we have

already

shown its

instability.

Let again w be thJs fluctuation with the wavenumber q m the _x direction. If

neutral,

this fluctuation has to be the solution of

w~~ +

q~

_w

(z 1/2)/2

=

0 with _w

=

0 and _z

= 0 and

This

Airy equation

_imposes

quantized

values of _q~ and those values exist, as can be shown in the WKB limit for _instance. Since the solution

uo(z)

is

neutrally

stable

against

_some

perturbations,

_a

weakly

nonlinear

analysis

would show that _some small

amplitude pertur-

bation with _a wavenumber close to the _ones given

by

the

abovej eigenvalue problem yields

_a

solution with _a lower energy.

Let _{us come now} to the smoothness of the solution of

equation (45).

Th1s

equatiin

is

the Euler-Lagrange

cond1tlon for the statlonanty ^{of the functional} :

£(u)= dr(-u+ (u~+uj/2)~), _together

with the

boundary

conditions u=0 and

z ₌ 0 and I.

Suppose

that _one has found _a smooth solution

U(x, z) making

thJs functional

stationary

and _as

negative

as

possible

with _our choice of

sign

for the definition of

£(u).

Consider _now the second variation of

£(u)

around thJs solution. Let

w(x, y)

be the

corresponding

fluctuation of _{u near}

U,

then thJs second variation has the form _:

d£

(w

=

ldr ^(w)

+ 2 w~ w~

U~

+ w

j(

_{U~ +}

_3/2 Uj)/2) (46)

This expression is

quite

remarkable m the _sense that it is

homogeneous

_as far _as the denvation order of _{w is} concerned. Tins

implies

that the

sign

of d£

(w)

_is

completely

determined

by

the

one of the

quadratic

form _m the variable w~ and w~ that appears in the

integrand

of

d£(w).

This

quadratic

form may get

positive

values

(and

the initial solution may become

unstable)

if the determinant

(U~

+

1/2 Uj

_is

_positive

somewhere _in the

physical

domain.

Before going ^to the

physical meaning

of thJs last

condition,

let _us show that thJs

happens

for

a smooth solution U. Let _us consider the

following change

of

vanables, kindly suggested by

Haklm _:

dz=dT, dx=dT.UJ2.

In terms of the _new variable _{T, we can express the} variation of U _as

dU

=

Uz

^dz +

U~

dx

=

dT(U~

₊

Uj2)

The vanation of U between _z

=

0,1

_{is given m} the _new running variable _{T as}

lgradU.dT=o

since the

boundary

cond1tlons U

= 0 holds at both ends.

Thus,

either U is _zero

everywhere,

_or

it has _a maximum _m

the mte~val.

In such _{a case,} the quantity

(U~

₊

1/2 Uj

_must be positive somewhere _in the

interval

_z

=

0,

On the

other,hand,

U _cannot be _zero

everywhere

_on the line indexed

by

T

IS.

Near _z

=

0

(or

_z

=

I,

but with _a

slight change

of sign and of

vanable),

the

Taylor

_expansion of _a solution of equation

(45)

should include _a

quadratic

term

(- z~/2), and,

as the line under consideration merges with the _z

=

0 line at

nght angle,

U cannot vanish

everywhere

_on it.

The

physical

_meaning of thJs result may be understood _as follows _: the determinant of the

quadratic

form _is

precisely equal

to the local variation of the

wavelength expressed

_m intrinsic coordinates _as shown in the

appendix

Thus the

possibility

of

having locally

_an unstable fluctuation is j manifestation that _a

locally

dilated roll will tend to reduce this dilation

by

_a local modulation _in the transverse direction. Here _we have the remarkable situation that thJs

process may be continued down to

arbitrary

small scales

This

unphysical

result

implies

that

equation (45)

cannot be

uniformly valid,

because the

onginal equation (43)

had

actually

_a built-in small space

scale, represented by

the fourth denvative _m

(43),

that has

precisely disappeared

_m

(45)

after the

rescahngs

motivated

by

the