On the phenomenology of tilted domains in lamellar eutectic growth

(1)

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On the phenomenology of tilted domains in lamellar eutectic growth

B. Caroli, C. Caroli, S. Fauve

To cite this version:

B. Caroli, C. Caroli, S. Fauve. On the phenomenology of tilted domains in lamellar eutectic growth.

Journal de Physique I, EDP Sciences, 1992, 2 (3), pp.281-290. �10.1051/jp1:1992143�. �jpa-00246483�

(2)

J.

Phys.

I France 2 (1992) 281-290 _MARCH 1992, PAGE 281

Classification

Physics

Abstracts

81.30F 03.40 64.70D

On the phenomenology of tilted domains in lamellar eutectic

growth

B. Caroli

(I, 2),

C. Caroli

(I)

and S. Fauve

(3)

(1)

Groupe

de

Physique

des Solides

(*),

Universitds Paris 7 et Paris 6, 2 Place Jussieu, 75005 Paris, France

(2) Facult6 des Sciences Fondamentales et

Appliqudes,

Universit6 de Picardie, 33 _rue Saint Leu, 80039 Amiens, France

(3) Ecole Norrnale

Supdrieure

de Lyon, 46 allde d'ltalie, 69364 Lyon Cedex 07, France

(Received ^I

July

1991, revised 5 Npvember 199I,

accepted

13 November1991)

Rdsum4. Nous montrons que, du fait du

couplage

entre les

dynamiques

d'inclinaison

(amplitude

de la

partie impaire

du

profil

de front) et de

phase,

la

ph6nom6nologie

des domaines d'inclinaison de

largeur

finie

propos6e

_par Coullet et al.

pour1e

cas d'une bifurcation d'inclinaison

homogbne

sous

critique garde

les mEmes

caract6ristiques

qualitatives

quand

cette bifurcation est

directe, _comme c'estle _cas pour la croissance

eutectique

lamellaire.

Abstract. We show that, due to the

coupling

between tilt

(amplitude

of the

antisymmetric

_part

of the front

profile)

and

phase dynamics,

the

phenomenology

of tilt domains of finite width

proposed

by Coullet _et al. within the assumption of _a subcritical homogeneous ^tilt bifurcation retains the _same

qualitative

features when this bifurcation is direct, _as is the _case for1ame11ar

eutectics.

It has

recently

_come to

light

that out of

equilibrium

_systems

exhibiting

one-dimensional

stationary

front pattems ^with ^small deformation

amplitudes undergo

_a

generic parity- breaking

_« tilt _» bifurcation. This

phenomenon

has been observed and studied in directional

solidification of

liquid crystals ill,

lamellar eutectic

growth [2]

and directional viscous

fingering [3].

The basic

pattem

loses its reflection

symmetry

^about ^the ^cell

(or lamella)

center and the appearance ^of ^this

asymmetric

distorsion is

accompanied by

_a drift of the

pattem along

the front at _a constant

velocity.

In eutectic

growth,

where the front

dynamics

is

registered by

the frozen space

arrangement

^of ^the two immiscible

phases

in the

solid,

this results in the formation of

regions

of

periodic

lamellae tilted at a constant

angle

with respect

to the

growth

^direction.

Coullet et al.

[4]

have

proposed

_a

phenomenological description

^of this

bifurcation,

based

on a

(truncated) amplitude equation

for the asymmetry

parameter A,

which _can be

understood _as the

amplitude

of the

antisymmetric

_part of the front deformation

(Qr,

(*) Assoc16 _au CNRS.

(3)

equivalently,

the lamellar tilt

angle).

The drift of

asymmetric

_pattems

implies

that the evolution of A is

strongly coupled

to the

phase dynamics

of the

periodic

_pattem.

Contained in such _a

phenomenology is,

of _course, the

description

of

homogeneously

tilted states of _an

infinitely

extended front. In the absence of _a _«

microscopic

_»

description

of this

homogeneous bifurcation,

_one has _to make _an

assumption

about its super or subcritical

character. In all

experiments reported

_up to now, tilted _states have been observed _as

inclusions of finite lateral size

[5],

limited

by

_a front and _a _rear _« wall

»

traveling along

the front at constant

velocity(ies).

Recent detailed studies in eutectic

growth

show that such domains may either grow

(I.e.

increase their width

linearly

with

time),

_or

shrink,

_or

keep

a

constant

width, depending

on the values of the

growth velocity

V and _on the

wavelength

A of the basic

periodic

structure which is

being

_swept

[6].

^In

liquid crystals

^and directional viscous

fingering, these, usually repeated, sweepings by

_« tilt _waves _» in

general

result in _a decrease of the

wavelength

of the

emerging

untilted pattem, ^which ^relaxes ^towards a

fixed, dynamically

selected value. In

eutectics,

the _same trend is

observed,

but it _can be

complicated by

the

occurrence of other instabilities

[6]. Quite naturally,

the observation of finite size domains led

Coullet _et al.

[4]

to _assume that the

homogeneous

tilt bifurcation is subcritical.

They

then build _a

description

of tilt inclusions which accounts for all the

qualitative

features of domain

behavior.

They find,

in

particular,

that the

wavelength

selected

by

tilt _wave

sweeping

should

correspond

to the

« Maxwell

point

_» of the inverted

homogeneous

bifurcation.

Since then, Kassner and Misbah

[7]

have been able to solve the full _«

microscopic

_»

equations describing

eutectic

solidification,

and _to

investigate

the existence of

homogeneously

tilted solutions.

They

thus locate

directly

the

position

of the tilt bifurcation and show that it is

supercritical.

This is also

predicted [8-11]

in the _cases where the tilt bifurcation _can be described from the resonant interaction of _a basic mode and its second

harmonic,

close to _a

primary

bifurcation of cellular type.

On the other hand, tilted domains in eutectics _are

qualitatively

well described

by

^the

phenomenology

^of Coullet _et

al.,

which _assumes _a subcritical

bifurcation,

_as indeed

suggested by

the intuition based _on

equilibrium phase

transitions.

In this article, _we show that this

paradox

is

only

_apparent and

that,

due to the strong

coupling

between tilt and

phase dynamics,

_a

system

^with a

supercritical

^tilt bifurcation _can also sustain

traveling

tilt inclusions with the _same

qualitative

^behaviour as that

produced by

_a

subcritical _one.

As discussed

by

Fauve et al.

[12, 13],

in the

vicinity

of the tilt

bifurcation,

the

dynamics

of

our system ^should ^be ^described

by

the

following coupled equations

A~ ₌

pA-aA~+pA~+A~+aitb~+eAtb~+yAA~+bitb~tb~+. _(la)

tb~ =

wA+Dtb~+a~A~+b~A~+C~AA~+d~tb~tb~+. (lb)

Here A stands for the

amplitude

of the

antisymmetric

part ^of the front

profile

of basic wavevector qo =

2

w/Ao (or, equivalently,

for the lamellar tilt

angle e).

tb ^is ^the

phase

of the

profile.

For

example,

if the front deformation is

(as

_can be done _to describe tilted cellular fronts

[8-10]) approximated by

the

superposition

of two Fourier components ^with wavevectors

qo, 2 qo of

respective complex amplitudes

_ai

e~~',

_{a~ e~~~}

tb ₌

(2tbi+tb~)/2

and Ax

(2tbi-tb~).

Equations (I)

_are

expansions

ⁱⁿ _powers of A, tb~ ^and ^their space

derivatives,

the functional

form of which is chosen _so _as to

satisfy

the symmetry

properties

of _our system,

namely

translational invariance

(tb

_- tb + cst. and space reflection symmetry

(x

- x, tb -

tb,

A -

A).

(4)

N° 3 TILTED DOMAINS IN LAMELLAR EUTECTIC GROWTH 283

Note that the model used

by

Coullet et al.

corresponds

to

choosing

_a

~

0, p

~ 0

and,

arbitrarily imposing

in

equations (I)

_ai ₌

bi

= a~ =

b~

= c2 =

d~

= 0.

The WA _term in

equation (16)

describes the drift of the tilted

pattem along

the average

front. The bifurcation towards

homogeneously

tilted pattems

corresponds

to p =0.

Following

Kassner and Misbah

[7],

_we take it to be

direct,

_so that _a ~0. The

PA

^~ term in

equation (la)

then becomes irrelevant and _we

will,

from _now _on,

forget

it.

In the

homogeneously

^tilted state of _wavevector qo, A

= ±

(pla)~~~

and

tb~

₌ ^0. ^The ^tilt threshold

corresponds

to V

~

V~(qo) (where

V is the

growth velocity),

and _we know from references

[7, 14] that,

in the usual

experimental

_range,

V~(qo)

_cc

q(,

I.e.

homogeneously

tilted states occur for _:

A² v _~

(A

²v

)~. (2)

So, increasing

_p in

equation (la) corresponds

to

increasing

the

growth velocity

at constant qo, while

increasing

the

wavelength

at constant V

corresponds

to the shift _p

- p +

etb~ (with

tb~ = q qo =

cst.).

Since _p must increase with the

wavelength,

this at once indicates that

e ~ 0.

Finally,

D is the

phase

diffusion coefficient in the untilted state, which _we _assume to be Eckhaus-stable

(D

~ 0

).

Following

references

[12, 13],

it _can be checked

immediately by linearizing equations (I)

about the

homogeneously

tilted solution :

Ao

= ±

(pla

_)~'~

_tbo

=

wtAo (3)

that the

stability

of this solution

against long wavelength perturbations (oz

_exp

(

at + ikx

))

is

govemed by

the

dispersion

relation _:

«~

₊

«

(2

_p

ikAo(y

+

b~)

+

tJ(k~))

ikA

o(2 pb~

+

em)

+

d(k~)

=

0.

(4)

The _term ikA

o(2 pb~

₊ _em is

destabilizing

_: it

gives

rise _to _an unstable mode with _:

Re _«

(k )

= ~

~ j ^{e2 w2}

+

o(» )j (5)

So,

due to the

coupling

between tilt and

phase, immediately

above the bifurcation

(where

pm

I),

not

only

is the non-tilted state

unstable,

but _so

is,

_as

well,

the

bifurcating

homogeneously

tilted _one. This poses the

question

of the _nature of the

(time-dependent)

structure into which the system ^then

restabilizes,

and in

particular, again,

of the existence of

traveling

tilt inclusions.

1. Tilted domains of constant width,

We first want to examine whether the

phenomenological coupled equations

for tilt and

phase

can sustain, below the

homogeneous

tilt

bifurcation,

I.e. for _p

~ 0, solutions

corresponding

to

drifting

domains of

large

^but finite width _: A

=

A

(x

vt

)

_; _~b

~ ⁼

~b~(x vt) (6)

with _:

A(Qc)=A(-Qc)=o; ~x(Qc)=o. (7)

JOURNAL DE PHYSIQUEI T 2, N' 3, MARCH 1992 12

(5)

Equations (I)

contain 12 parameters, among which

only

_one, the

« distance _» from the

homogeneous

bifurcation _p, is

necessarily

small. On the other

hand,

from the direct

study

of

homogeneously

tilted states

[7, 14],

it is

possible

to deduce the values of the three parameters

a, e, w.

Moreover,

_on the basis of

approximate

calculations

[15]

of the

phase

diffusion coefficient in untilted structures, one can

reasonably

expect that, for

eutectics,

D « I

(in

units of the chemical diffusion

coefficient),

due _to the smallness of the relevant Peclet numbers.

However, equations (I)

contain _seven _more coefficients about

which,

_up to now, no

information is available I.e. neither their

signs

_nor their orders of

magnitude

are known.

We will _see that _an

analytic

treatment of tilt inclusions is valid

only

under

quite

restrictive

conditions

(namely,

accidental smallness of _some combinations of coefficients of the

A

tb~ coupling terms). So,

such _a calculation _can

help

to elucidate

qualitatively

how tilt inclusions _can coexist with the

supercritical bifurcation,

but it cannot

yield quantitative predictions.

In this

situation,

and for the sake of

simplicity,

_we choose to

perform

the

analysis

_on the truncated version of

equations (I)

used

by

Coullet et al. The

qualitative

effect of

reinserting

non-zero values for ai,

...,

d~

will be

briefly

discussed at the end of

paragraph

2.

The truncated

equations

read

[4]

_:

A~= pA-aA~+A~+eAtb~+yAA~ (8a)

~~

₌ ^WA +

Dtb

~

(8b)

and _we choose to look for _a domain

moving

towards

increasing

x

(v

_~

0).

Formal

integration

of the x-derivative of

equation (8b) yields

_:

i w

A~,(x', t') (x x')~

~~'

_~°

_w ~'~-w

~~~

~'

[4 wD(t t')]~'~~ (~ ~'~

The infinite limit in the time

integration implies

that initial conditions

(details

of the nucleation

event)

have been

forgotten,

I.e. that the domain has been

existing

for _a time much

longer

than the

appropriate phase

diffusion time-here

D/v~.

For _a

stationary

domain _as defined

by equations (6)

and

(7), equation (9)

becomes _:

~~~ l

^~'

~~'~ ^~ ^ ~'~if~ ^~'~

=

) ~- ^A(X~

⁺

j~

^dX'

j'~

exp

~ ~'

( lo)

-w

where X

= x vt and

f~

= D/v is the diffusion

length of

the

phase

in the basic state.

From

equations (8)

and

(10),

we see that the space

dependence

of A is

govemed by

two

length parameters, namely

the

Landau-Ginzburg length [p[~~'~,

which

diverges

at the

bifurcation,

and

f

~, which remains finite. This entails that the local _wavevector tb~ ^is

spatially

slaved

by

the tilt

amplitude.

The lowest order of the

corresponding multiple

scale

expansion gives

:

~bx(X~ =

) (- A(S~

+

f~ @ _(i1)

Such _a truncation is valid

only

^if :

f~

₌ ^~ « p ^~'~

(12)

v

(6)

Equation (8a)

becomes _:

~~~+~v+(y+@)A)~+~~=0 (13)

dX _v dX &A

where

U(A )

₌

~ )

_A ^~

(

A^~

(14)

That

is,

the

coupling

to the

phase gives rise,

in the

«

potential

_»

U(A),

associated with _a

stationary

domain

drifting

at

velocity

_v, to a

v-dependent

cubic term : the initial

pitchfork

bifurcation of

homogeneous

structures becomes _an effective transcritical _one.

In the

region

_:

~2 ~2

~ ^< ^~1 ^~ 0

(15)

4 _au

U(A)

has three extrema, among which _a maximum _at A

=

0,

U

=

0.

As usual in such

problems,

and since _we _are

only looking

for

qualitative behaviours,

_we will treat the _«

dissipative

_» term

(proportional

to

dA/dx)

in

equation (13) perturbatively.

A wide

stationary

domain

(I.e.

_a domain with

negligible

interaction between its

walls)

is _a mechanical

trajectory

of the

particle

of

position

A at time X in

potential U(A) ending

at A

= 0 for X

= + oa after

having

started from _some A at X

= oa and

having

spent an

arbitrarily long

time _at _some finite A* The

only

_way for such _a process to be

possible

is that _:

(I)

the two maxima of U have the _same

height,

I.e.

U(A*)

= 0

(16)

where A* is the second maximum of U.

(ii) A(x- oa)

= 0 and the total

dissipation integrated along

_any half of the

trajectory joining

A

=

0 with A

=

A* vanishes.

Calling A(X~

₌

A*Z(X~

^this

half-trajectory,

this

condition reads _:

v

j~ dX(~ _)~+ (y+@)A* _j~ dXZ(~)~=0. (17)

_w

dX v _-w dX

Conditions

(16)

and

(17) provide

_two relations between

the,

still

floating,

_parameters _p and

v, the solution of

which,

if it

exists,

defines both the location in

parameter

space,

p

*,

of _an _« effective Maxwell

point

_» and the daft

velocity

v* of tilt domains of constant width.

It is

immediately

seen that condition

(16)

entails that

~~~~

~

~.

~*

=

~ ~ ~

(j8)

w2 ~2 9'

3 _au

so that the

corresponding potential

_:

U*(A

=

flA ~(l ^~~

~

(19)

The

half-trajectory joining (A*,

X

=

oa)

to

(A

=

0,

X

= +

oa)

is _:

z(x)

=

((~

=

ii

₊ _exp x

11 ₍₂₀₎

(7)

so that condition

(17) yields

:

~~

~il

_~~

~j

^~ ^~~~~

which

always

^has

(whatever

the

sign

of

yw)

the

single

solution _:

*2 W~ E~ _Y

y~

₊ 12 aD

'2j

~ _"

~ £j

⁺ _~ _~

⁽²²⁾

w e

and the

corresponding

values of _p

, A,

_are related _to v*

by equations (18).

This result _can be

expressed

_as follows _:

given

_a

wavelength _Ao

of the basic untilted front

pattern,

there exists _a

growth velocity

V*

(Ao)

^such ^that :

A(V*(Ao)

~

(A~V)~

at which the system can

support drifting

tilted domains of _constant

width,

I.e. _« tilt _waves _» with

parallel

walls. In this

situations,

since

tb(X

_-

oa)

=

0,

the

wavelength

of the untilted

pattem

^is ^unaffected

by

^the _passage ^of the tilt _wave.

Conversely (since adding

a constant to

tb~ simply

shifts

p),

this

defines,

for each

growth velocity Vo,

_a basic

wavelength A(V~)[A~(Vo)Vo~(A~V)~]

_for _the

propagation

of

stationary

tilt _waves.

Moreover,

from

equation (10),

the _wavevector q~ inside the tilted

region

is such that _:

q~-qo=~x=-ii* =lllli«0.

_~23~

So,

the

wavelength

is

larger

ⁱⁿ ^the tilted

region

than outside of it. This

prediction

agrees

qualitatively

with

experimental

observations.

However,

in order for the

corresponding

solution _to be

valid,

v* _must

satisfy

condition

(12).

Moreover, ^the ^truncation of the

amplitude expansion leading

to

equations (I)

or

(8) implies that,

for _p ^~'~

= O (1~ ^«

1,

A*

= O (1~

). Using equations (18)

and

(22),

_one

easily

checks that this

imposes

_:

we=o(~2)=o(i»i);

_D«1

₍₂₄₎

together

with

ewy~0,

I.e., for

eutectics,

since

e<0,

_my ~0.

(When ewy~0, v*,

_as

given by equation (22)

cannot

satisfy

the condition for

slaving

of the

phase

to the

amplitude, _fw

_« _»

~~.

Note that condition

(24)

_on how the

coupling _parameter

_we should scale with _p _can also be obtained if _one

imposes

that all terms in

equation (8a) (resp. (8b))

scale

coherently

as 1~~

(resp. 1~~.

This

imposes

that

w =

d(1~)

and _e

=

tJ(1~).

As mentioned

above,

if the condition _on D is _most

probably

satisfied for

eutectics,

it is _not

necessarily

_so for the other

physical systems

^where ^tilt ^has been

observed,

and in which

phase

diffusion is known not to be very slow. We believe that the condition _on the _«

off-diagonal coupling

parameter » we can be

satisfied,

in real systems,

only accidentally.

2.

GroM4ng

and

shrinking

domains.

In order _to decide whether

non-stationary

tilt inclusions may ^exist ^for ^values ^of p different from _p

*,

_as well as to check whether these _can

give

Rse to

dynamical selection,

_we _now look for solutions

describing

tilted domains which _grow _or shrink _at _a constant rate, I.e, _we _assume

(8)

that _as is observed in

experiments

the front

(right)

and _rear

(left)

walls of _a domain

propagating

towards x _~ 0 _move at constant but different velocities v~, v~.

The

positions

at time t of these walls _are _:

x~

(t)

= x~ + v~ t ; x~

(t)

= x~ + v~ t

(25)

and x~ x~ ^» p ^~'~

A~

^is ^therefore a sum of two

(quasi)

_non

overlapping

terms

peaked

around x~,

x~, so

that,

in the formal

expression

^of

tb~ (Eq. (9)),

the space

integration

_can be

separated

into the _two

regions (R)

and

(L) corresponding respectively

to x _~

x~(t)

and _x _<

x~(t),

where

x~(t)

^lies far from both walls

(I.e.

at _a distance much

larger

than _p _~'~) inside the tilted domain. Then _:

for _x _e

(R ),

A and tb~ are functions of

X~

= x x~

(t) only

and, from

equations (9)

and

(lo) together

with A

(X~

- oa

)

= 0 _:

tb~ =

I ~-

^A ⁺ ^~ ^~

⁽²⁶⁾

VR VR

dXR

with _oa

<X~

_< oa

for _x _e

(L),

A and

tb~

are functions of

X~

₌ _x

x~(t)

with _:

tb~ =

wA*( ⁺ⁱ (-A

⁺ ^~ ^~

⁽²⁷⁾

VL VR VL

VLdXL

(-

_oa _<

X~

_~ _oa and A * is the

(quasi constant)

value of A inside the domain.

Equations (26)

and

(27),

when

plugged

into

equation (8a),

^define two disconnected mechanical

problems

^with

potentials

U~(A )

₌

U(A iv

₌

v~) ^(28a)

u~(A )

=

u(A iv

=

v~)

₊ ^°~

j

^*

) (

_A2

(28b)

where

U(A iv)

is defined

by equation (14).

For

(- w~ e~/4 avj)

~ p ~

0, UR

^and

U~

both have three extrema

corresponding

to the

same values of A. The _common

position

of their second maximum is _:

~ ~

4 _p _a

vi

1'2

A*

= +

~ ~

(29)

2 _a v~ _w _e

A tilted domain with wall velocities

(vR, v~) corresponds

to a set of two

trajectories TR, T~

of the

particle

ⁱⁿ

UR

^and

U~ respectively,

^such that _:

T~

connects A

=

0 _at

X~

= oa with A* at

X~

= + oa

TR

connects A

=

A* at

XR

= oa with A

=

0 _at

XR

= + oa.

TR

is _a solution of _:

~~+~v~+ (y+@)A~~ +(=0. ₍₃₀₎

dXR

_VR _R

(9)

In order to be able _to solve _our

problem analytically,

_we _assume that _:

(H H)/H'

~

(~H/H*(

~ l

(VR,L V)/V(

₌

(BVR,L~~*'

^"1

where _p

*,

v* _are the values

corresponding

to

stationary

domains obtained in

paragraph

I.

We then

expand UR

and A in

equation (30)

to first order about

U*(A)

and A* Z

(XR) respectively,

and also treat,

again, dissipation

to first order. The

corresponding

Fredholm condition

(see

for

example

reference

[4])

then reads _:

l~ ~j~ ^~dZ

^we ~~

&VR~2dZ

H

@+q _q @+

+v~(~)~+(y+@)A*Z(~)~)=0. ₍₃₁₎

dX vR ^~~°

Using

the fact that _p

, v,

A *

satisfy

conditions

(17)

and

(18),

_as well _as

expression (20)

for

Z(X),

we can rewrite

(31)

_as _:

2[~* ~~

6

~~[

i12

l~

~ ^* +

Y~ *1

° ~~~~

An

analogous

calculation for

trajectory T~ yields

:

~j~*j ~~~

~

~ ^~

~

~j

~~jI/2

~

~~ _~

~~~~

~' ~~~~

From

equations (32)

and

(33)

_we then find _:

av~ _{a ~}

/

^~ ₂₁

» ^*

(1

₊

P )

^~~~~~

3v~ 3vR

2

p

(34b)

3VR

p

+

where _:

p ₌

~

- _i

+

w~e~ j"~~ ^)~/~

~ 0.

(35)

12av[p[

^' ³ _w _e

So

3v~

and

(3v~ 3v~)

both have the _same

sign

_as

3p.

That

is,

close in

parameter

space ^to the _« Maxwell

point

_» _p ^*

(I.e., equivalently,

^for A^~ V close to (A^~

V) *)

_we _can state that _: if

the basic untilted

pattern

^is ^stable ^with respect ^to

homogeneous

tilt but such that

A~V ~(A~V)* _(resp. ~(A~V)*)

_a

_growing _(resp. _shrinking)

domain with _a

uniquely

defined

opening angle

_can _propagate ^into ^the untilted state. Its forward wall _moves faster

(resp. slower)

than that of the

stationary

tilt _wave

existing

^for

^A2V

=

(A~V)*

Inside the

domain,

from

equation (26), tb~~ 0,

I.e. the

wavelength

^is

always larger

than that of the pattem ^which ^is

being

invaded. The

symmetric _pattem emerging

after the

system

has been swept

by

such _a tilt _wave has _:

tbj~~

₌ f 3p

(36a)

(10)

where _:

~

² ^w

~

e

P

(36b)

3£YV~j~j(i+p) _~2 ~+fl

is

(since

e ~

0)

_a

positive

number.

So, provided

^that conditions

(24)

_are

satisfied,

_we _can conclude that the

wavelength

of untilted

pattems evolves,

under the action of tilt _waves, in the _same way as when the tilt bifurcation is subcritical _: _as

analyzed by

^Coullet et al.

[4], equations (36)

entail

that,

whatever the

sign

of

3p

for the initial

pattem, repeated sweepings by

tilt domains should result in the

relaxation of the

emerging symmetric _pattem

towards the _« Maxwell

point

_»

p*, I.e.,

at

constant

growth velocity V~

ⁱⁿ a relaxation of the

wavelength

towards the

«

dynamically

selected

» value A *

(Vo).

The

question

now arises of whether

using

the

complete amplitude expansion (I)

rather than the truncated

dynamics (Eqs. (8))

leads to

qualitative

modifications of the above results. The

analysis

of

paragraph

I _can be

repeated,

either

by using

the

expansion

method

leading

to

equation (11)

_or with the

help

of the non-linear transform used in references

[12, 13].

The

study

of inclusions of constant width follows the _same lines _as that of

paragraph I, yielding

for

v*,

instead of

e@uation (21),

the

equation

_:

v~

Ai

^v~

A~

₌ 0

(37)

where

Ai

and A~ are combinations of the 12

parameters appealing

in

equations (I)

the

sign

of

which remains unknown.

So,

two cases may occur

[16]

_:

I) Equation (37) has, formally,

_one _or two

positive

roots for

v*~.

In

general, only

_one of them is

acceptable,

and

this, only

when _some

appropriate

smallness conditions _on the

coefficients of the

amplitude expansion

_are satisfied. These in fact reduce to conditions

(24)

derived above _on the truncated model.

Indeed,

if _w

=

tJ(1~),

e =

d(1~),

^D «

I, retaining

only

terms d (1~ ~) in

equation (16)

amounts to

neglecting

_{a~ A}

~, c~ AA

~

and

d~ tb~ tb~.

^On ^the

other

hand, since,

in

equation (16), _b~

=

?°'~~~

,

in

general d(b~)

₌

d(w ),

and the

aq

q =q~

b~Atb~

term is

negligible. Finally,

the

(ai

+

bi tb~) tb~

terra is

generated by

the _same

diffusion processes which contribute _to

Dtb~,

so

assuming

that D«I entails that

ai,

bi

« I as well.

So,

in the limit defined

by

conditions

(24),

the full

equations (I)

_can

legitimately

be reduced to the truncated model.

ii)

It may also

happen,

in _some range of values of the

coefficients,

that

equation (37)

has _no solution. In this _case _we _are left with _two

possibilities

_: either tilted domains of constant width do _not exist

(and

the system

probably

evolves into _a

quasi-chaotic behavior),

_or such domains

exist with _a

velocity

v ^* which does

satisfy

the

phase-slaving

condition. A conclusion about this could

only

be reached with the

help

of _a

complete time-dependent

numerical

study.

From the

foregoing analysis,

we can thus draw the

following qualitative

conclusion _: in systems

susceptible

of spontaneous

symmetry breaking

of their front

profile

via _a

supercritical bifurcation, (which

_seems to be the _case in all systems ^studied

experimentally

_up to

now)

the

nature of the structures to be

expected

in the

vicinity

of the

homogeneous

tilt bifurcation

exhibits unusual features due _to the

couplings

between the tilt order

parameter

^and

wavelength

modulations. It is this

coupling

which is

responsible

for the

possible long wavelength instability

of the

homogeneously

tilted state. It

is, also,

the _same

coupling

which

allows

(in

the extented space

including configurations

with variable

tb~)

^for ^the ^existence ^of

(11)

drifting

tilted domains of finite size below the bifurcation threshold. Our

analysis

does not

permit

to conclude

definitely

about whether such domains exist whatever the value of the coefficients in the

amplitude expansion.

It is not excluded that systems

might

exist in

which,

close to

threshold,

front

profiles

would restabilize into _a _more

complicated time-dependent

structure.

References

[ii SIMON A. J., BECHHOEFER J., LIBCHABER A.,

Phys.

Rev. Lent. 61(1988) 2574 _; FLESSELLES J. M., SIMON A. J., LIBCHABER A. J., Adv.

Phys.

(to be

published).

[2] FAIVRE G., _DE CHEVEIGNt S., GUTHMANN C., KuRowsKI P., Europhys. ^Lett. 9 (1989) 779.

[3] RABAUD M., MICHALLAND S., COUDER Y., Phys. Rev. Lent. 64 (1990) 184.

[4] COULLET P., GOLDSTEIN R. E., GUNARATNE G. H.,

Phys.

Rev. Lent. 63

(1989)

1954 GOLDSTEIN R. E., GUNARATNE G. H., GIL L.,

Phys.

Rev. A 41 (1990) 5731.

[5]

Only

_very

recently

have

space-filling

tilted states been observed in eutectic

growth

(FAIVRE G., MERCY J. _:

private

communication).

j6] FAIVRE G., MERCY J. _: submitted for

publication

to

Phys.

Rev. A.

[7] ^KASSNER K., MISBAH C., Phys. Rev. Lett. 66 (1991) 522 and 65 (1990) 1458.

[8] PROCTOR M. R. E., JONES C., J. Fluid Mech. 188 (1988) 301.

[9] DANGELMAYR G., Dyn. ^Stab.

Syst,

1(1986) 159.

[10] ARMBUSTER D., GUCKENHEIMER J., HOLMES P.,

Physica

29D (1988) 257.

[I Ii LEVINE H., RAPPEL W. J., RIECKE H.,

Phys.

Rev. A 43 (1991) l122.

[12] FAUVE S., DOUADY S., THUAL O., J.

Phys.

II France

1(1991)

311.

[13] FAUVE S., DOUADY S., THUAL O.,

Phys.

Rev. Lent. 65 (1990) 385.

[14] KASSNER K., MISBAH C.,

Phys.

Rev. Lent. 66

(1991)

445 and to be

published.

[15] CAROLI B., CAROLI C., FAIVRE G., MERCY J., to be

published

in J.

Cryst.

Growth.

[16] Note that,

strickly speaking,

_as far _as _one does not know beforehand the order of

magnitude

of v*, _one should also consider the

possibility

of

adding

to

equations

(I) _terms of the form A4

]

(n