Lamellar eutectic growth at large thermal gradient. II. Linear stability

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Lamellar eutectic growth at large thermal gradient. II.

Linear stability

B. Caroli, C. Caroli, B. Roulet

To cite this version:

Lamellar eutectic

_growth

at

_large

thermal

_gradient.

II.

Linear

stability

B. Caroli

_(*),

C. Caroli and B. Roulet

Groupe

de

_Physique

des

Solides (**),

Tour 23, Université Paris 7, 2

_place

Jussieu, 75251 Paris Cedex 05, France

(Received

on March _{9, 1990,}

_accepted

on _{May 16,}

₁₉₉₀₎

Résumé. 2014

Nous analysons

la stabilité linéaire des structures

_eutectiques

lamellaires

_périodiques

stationnaires étudiées dans l’article

_{précédent [I]}

vis-à-vis des

_{perturbations}

de

grande longueur

d’onde. Nous montrons _{que, dans la limite des}

_{grands gradients thermiques}

considérée dans

_[I],

et

pour les structures de

_longueur

d’onde de l’ordre de _{celle, 03BBmin,}

correspondant

au

surrefroidisse-ment minimum:

₁₎

il

_apparaît

une instabilité associée à une variation

_quasi

uniforme de la

proportion

de chacune des deux

phases

solides. Elle

apparaît

seulement pour les

_mélanges

dont l’écart u~, à la

composition eutectique

est du côté du

_diagramme

de

_{phase qui présente}

le

_plus

grand

« gap » de

température.

Dans ce cas, les structures stationnaires avec 03BB

_03BB0,

où

03BB0

03B1u~, sont _{instables ;}

₂₎

toutes les structures

_{avec 03BB/03BBmin}

= O

₍₁₎

sont stables vis-à-vis du mécanisme d’Eckhaus. Ceci entraîne que

l’hypothèse dynamique

utilisée dans des travaux

antérieurs,

selon

_laquelle

les interfaces solide-solide seraient localement normales au _front, n’est

pas

compatible

avec

_{l’approximation}

usuelle des

petites

déformations de front.

Abstract. 2014 We

analyze

the linear

stability

of the

periodic

eutectic lamellar patterns studied in the

preceding

article

_[I]

with _respect to

_{long wavelength perturbations.}

We show that, in the

large

thermal

gradient

limit considered in

[I],

and for patterns with

_{wavelengths comparable}

with the minimum

_undercooling

one, 03BBmin:

1)

an

_instability

with _respect to a

_{quasi-uniform}

variation of the

_phase

fraction occurs

_only

for mixtures with a

_departure

from eutectic

composition

u~ on the side of the

phase diagram

with the

_{larger liquid-solid}

temperature gap. In this case,

stationary

patterns with 03BB

_03BB0,

where

_03BB0

_03B1u~, are _{unstable ;}

₂₎

all _patterns with 03BB

/03BBmin

= O

(1)

are Eckhaus stable. This entails that the

dynamical

ansatz, used in

previous

works, that solid-solid interfaces are

_locally

normal to the _front, is not

_compatible

with the usual small front

deformation

_{approximation.}

Classification

Physics

Abstracts

61.50C - 64.60 - 64.70D - 81.30F

1. Introduction.

In the

_preceding

_paper

_[1],

hereafter referred to as

_[I],

we have calculated

_{analytically,}

in the

limit of

_large

thermal

_gradients,

the front

_profiles

associated with

_stationary

lamellar eutectié directional solidification

_patterns.

Although

this

_{analysis disproves}

the

_{equal undercooling}

ansatz of Jackson-Hunt

_{(JH) [2],}

our results _agree

_{qualitatively}

with their

_{predictions :}

one

such

_solution,

with lamellae

_{perpendicular}

to _{the average}

_front,

exists for each value of the

space

period A

of the

pattern

(with A

e

where t

is the chemical diffusion

_length).

The average

undercooling

âT(Ã)

exhibits a minimum

for à = Ãmin,

_{where Ãmin}

is of order

(d1)1/2,

where

J

is a

_{capillary length}

associated with

_liquid-solid

surface

_tension(s).

Since

f

_{D / V}

(where

D is the chemical diffusion coefficient in the

_{liquid, V}

the

_{pulling velocity),}

CC

V

1/2.

À min OC V .

It is found

_{experimentally}

that,

in a

_{given experiment,}

_lamellar

_spacing

_{distributions}

_[3]

are

rather narrow, and centered about an _average

value Ã

of the order of

_magnitude

of

À min.

This « selected »

_{wavelength À}

oc V

_with,

for _many material

[4], a t.12:21

2

These observations raise the

_question

of

_wavelength

selection within the continuum of

stationary

patterns,

and

therefore,

as a first

_step,

the

_problem

of their local

_stability.

Several

approaches

to this

_problem

have been

_proposed.

The first ones

_{[5, 6]}

did not, in

_particular,

allow for variations of lamellar

_spacings.

This a

_{priori excludes,}

for

_{example, studying}

a

possible

Eckhaus

_instability.

Later

_{approaches [7, 8]}

included the

_possibility

of such

motions,

but assumed a flat front

_profile.

More

_recently,

_Datye

and

_Langer

_performed

an extensive

stability analysis

of the Jackson-Hunt

_stationary

_{[9] solutions,}

_involving.

(i)

an

_{approximation}

of small front deformations and

(ii)

a

_dynamical

_ansatz, first

_{suggested by}

Cahn and JH

[2].

This assumes that the solid-solid interfaces

_{(lamella boundaries) always}

remain

_{perpendicular}

to _{the local average}

solid-liquid

front orientation.

They

find that the lamellar

_system

is Eckhauss unstable when its

_{wavelength A Ik min,}

in

agreement with the

_{following simple qualitative}

_argument

_{[2] :}

a convex front deformation

(bump)

increases the

_{local A,}

if À

_{Ã min}

_{(resp. A}

>

>i min)

the local

undercooling

decreases

(resp. increases),

thus

_{amplifying (resp. reducing)}

the deformation.

_They

also

_predict

--an

oscillatory instability

at twice the basic

_wavelength

for

_sufficiently

off-eutectic

_systems

in a

small thermal

_gradient.

We have shown in

_[I]

that the small front deformation

_{approximation,}

used to _compute thé

diffusion

field,

is

_{justified only}

in the

_{large gradient}

_limit,

i.e. for

_{FT «}

_f,

where the thermal

length

FT

(Eq. (1.11))

is

_{inversely proportional}

to the thermal

_gradient

G.

_Now,

since it is the thermal field which

_primarily

controls front

deformations,

it can be

suspected

that,

the

larger

G

_(the

more valid

_(i)),

the more difficult front orientational

_adjustement,

and thus the more

doubtful ansatz

_(ii).

In the order clear _up this

_point,

_{in this paper} we

_study

the local

_stability

of the

_{large gradient}

stationary

solutions calculated in

_[I]

with

_respect

to deformations of

_wavelength

2

_7r/

much

larger

than the basic one À

_{(kÀ « 1 )}

assumed to be of the same order of

magnitude

as

¡min.

This ansatz-free calculation is

_performed

with the

_help

of an

_{approximation}

which

extends to the

_{non-stationary}

_system

that

_justified

for the

_stationary

one.

_Namely,

the

diffusion field is

_{approximated by}

that of the

_{« locally-averaged »}

front

_{(i.e. only profile}

modulations on the scale of

k-1

1 are

_retained).

I n this

_{approximation,}

three branches _{of modes appear,} one of which

_corresponds

to

_phase

diffusion. We find that lamellar

_patterns

at

_large

G are Eckhaus-stable _{in the whole range} of

Ã

_{(0 (X min)}

covered

_by

our

_calculation,

in contradiction with the

prediction

based on the

The other two modes are, at small

k,

mixtures of

_{optical phase}

motions

(i.e.

variations of the local

_{phase fraction)}

_{and average front}

_profile

modulations. One is

_always

stable,

while the second one _may be

_unstable,

at small

_enough

values of the basic

_pattern

_wavelength

X,

if the

_system

is

_sufficiently

off-eutectic. This

only

occurs for a

_departure

from eutectic

composition

of a

_{given sign,}

defined

_by

the

asymmetry

of the

_{phase diagram}

of the mixture.

2. Formulation of the

_{dynamical problem.}

The

_equations

_describing

the

_dynamics

of the front

profile

of a

growing

lamellar eutectic have

been written in

_[I].

We have shown

_that,

since front deformation

_amplitudes

_{(, - t),}

are

small

_(due

to the small value of

_{X If)}

it is

_legitimate

to linearize the

_expression

of the concentration field in

_{(, - [).}

These

_equations

read :

where u is the dimensionless

_departure

from eutectic

_{concentration,}

and :

All the notations are those defined in

_[I].

In

_{particular lengths}

are measured in units if the

diffusion

_length.

Equations (1)

and

₍₂₎

must be

_supplemented

with

_boundary

conditions at the

_{triple points}

of abscissae

_{Xnl (t) (,B --+ a contact)}

and

_{X,,2 (t) (a --+ 8 contact) (see Fig. 1) :}

-

continuity

of (

- front

equilibrium :

where the surface tension forces

_point

out of the

_{triple point.}

We

_expand

the front

_profile

about the

_stationary

one,

eo(x),

as :

From the

_{Floquet-Bloch}

_theorem,

we know that the

eigenmodes

that we are

_looking

for are

Fig.

1. Dashed line : basic

_{periodic stationary}

pattern. Full line : deformed

time-dependent

front.

where

_Zk(X)

is a

_periodic

function with the

_period,

_A,

of the basic

_stationary

_pattern.

This

implies

that the

_{corresponding}

lateral

_{displacements}

of the

_{triple points}

have the form :

The

_geometrical

factor

_L1(x,

_{t )}

of which

_{4 (Q, t )}

in

_{equation (2)}

is the Fourier transform is

given by :

where 0 is the usual

_step

function and 8 = ,

the deviations from the

_stationary

_{pattern :}

where

_{Ao (x) corresponds}

to the basic

_periodic

pattern.

We have checked in

_[I]

_that,

in the

_large

_gradient

limit,

up to corrections of order

iT

=

FT/t,

the concentration _on the front _can be

approximated by

that

corresponding

to the

planar

average e

of the true

_profile.

The

_profile

deformation

₍₆₎

has Fourier

components

at

q = k

and k + nK

(n #= 0),

where

_{K = 2 7r /,A > 1.}

Since we

_only

consider here

long

wavelength

modes with k «

_{1, k}

+ nK > 1 and in order to be consistent with the

stationary

case, we

only

retain,

when

_calculating

the first order variation of

u (x, C (x, t), t),

the q = k

component

of the front deformation. That

_is,

we set, in

_{equation (2)}

where :

with :

Expressions (11-12)

must then be

_plugged

into the Gibbs-Thomson

_{equations (1).}

In the

large-G

limit that we are

_considering,

these could then be solved

_by

the

_{asymptotic matching}

procedure

used in

[I].

However,

in order to

_simplify

somewhat the rather

_heavy

_forthcoming

algebra,

we assume that the

_solid-liquid

surface

_{tensions 0.L,}

_UPL

are much

_larger

than the

solid-solid

_{one, U aP’}

so that contact

_angles

at the

_{triple points (and}

thus

_{profile slopes}

everywhere)

are small

_enough

for the curvatures _{Ka, {j} to be linearized :

That this is a

_physically

unessential

_assumption

can be inferred from the solution of the

stationary problem.

The solution of the

_fully

linearized version of

_{equations (1)}

can be

written,

in the front

region,

0 :: x :: À : .

wi th : .

2013

where :

The deformed front

_{profile {(x,}

t )

must, moreover,

_satisfy

the

_boundary

conditions at the

triple points

which we also linearize in

_{Z(x, t ), Xnl (t), Xn2 (t).}

2.1 CONTINUITY OF THE PROFILE. 2013 It

reads,

for

_example,

at

_{the a --+ f3 triple point}

:4>2 (t)

Linearizing equation (17)

and

_{using = - tg 0 « t---e - 0 «,}

_{£i m 0,,,}

this becomes :

Similarly, continuity

at

_{the f3 -+}

a

triple point yields :

(J a’ (J {3

are the contact

_angles

for the

stationary

front

_{(see Fig.}

2 of

[1]).

2.2

_EQUILIBRIUM

CONDITIONS. - Since the

_{triple points}

_{abscissae x,,i} are

_{time-dependent,}

the solid-solid contact lines make an

_{angle .0,,, (t)}

with the

_pulling

direction Oz

_{(see Fig. 1),}

given by :

The

_equilibrium

condition

_{(3) imposes}

that the

_angles

between the three interfaces at each

triple point

are

fixed,

so that the variations of the contact

_angles

_{& 0 ,i,}

measured from the

x-axis

_{(Fig. 1), satisfy :}

The

_slope

condition on, for

example,

the a side of the %i

triple point,

which reads :

is

_easily

linearized into :

Similarly,

we

_get

from the other three

_slope

conditions :

Finally,

the k-Fourier coefficient of the front

_{profile (14a), (15a)}

must

_satisfy

the

self-consistency

condition :

We then solve

_{equations (22a-d)}

for the four

_integration

constants _{t i, v i,} and

plug

the

resulting

values into the

_continuity

conditions

₍₁₈₎

and the

_{self-consistency}

_{equation (23).}

This results into three

_equations

which are

_homogeneous

and linear in the three

dynamical

variables

_X1,

_X2,

Z. Each of these

_equations

is linear in CIJ. The

_solvability

condition of this

system

yields

the

_{dispersion equation}

of the lamellar

_pattern.

The fact that this

_spectrum

contains

_only

three branches of modes results from the

_{approximation}

we use to calculate the

diffusion field : the

_{only degrees}

of freedom retained in this small-k

_{approximation,}

are the

positions

of the

_{triple points}

and the local average

(on

a basic

_period)

of the

_{profile height.}

That

_is,

we cannot describe modes which would involve

_profile

deformations on a scale

A.

These are

_likely

to be

_{strongly relaxing}

in the

_{large gradient}

limit which we

_study

here.

Writing

down the three above-mentioned

_equations

is

_{totally straightforward,}

but their final detailed form is

_{exceedingly heavy,}

due to the

_large

number of material

_parameters

(q«, qo,

f Ta’

e Tp).

In order to

_simplify

somewhat the

_{resulting algebra,}

we assume from now on that our eutectic

_system

is what we call «

_{pseudo-symmetric}

».

_Namely

we set :

or,

_{equivalently :}

d is the

_{(common) capillary length.}

The

_{only remaining}

_asymmetry

is that _{of the concentration gaps, reflected in the}

_{a priori}

finite

_parameter

_{(,6 - 1/2).}

Taking advantage

of the fact that in the

_{large gradient}

limit,

and for A values of the same

order of

magnitude,

d 1/2

,

as Amin, qA » 1,

we

neglect

terms

0[exp - qqk

]

and

0[exp - (1 - q ) qk ].

The zeroth-order curvatures

_Õa’

_op

are

_readily

obtained from the

fully

linearized version of the results of

_{(1) (e.g. Eqs. (1.9)}

where K

₀

For

_{example :}

where :

We then obtain the 3 x 3

_{system :}

The first two

_equations

of

₍₂₆₎

are obtained

_{by respectively adding}

and

_substracting

equations (18a)

and

_(18b).

Then :

We will see below that we do not

_need,

when

_studying

the

_{long wavelength}

_spectrum,

the

full

_expressions

of the four other

a’s,

but

_only

their k = 0

limit,

which reads :

where :

3. Discussion.

As discussed

_above,

our

_{approximation (Eq. (10))}

for the diffusion field restricts our

_analysis

to

_long

_wavelength

modes

_{(kA « 1 ).}

So,

we will first

_investigate

the k = 0 limit of the

1) k

= 0 modes.

- Noticing

that II

(Eq. (12f))

is an even

_function,

it is

_immediately

seen on

_{equations (28)-(31)}

that :

The k = 0

_spectrum

therefore

decomposes,

as

_expected,

into :

(i)

a neutral mode with

_Xl

=

X2,

Z =

0,

i.e. uniform translation of the basic

pattern

along

the

_{x-direction ;}

(ii)

two modes with

_{XI = - X2}

and

_{frequencies .given}

_{by :}

These modes therefore mix front

_profile

modulations and

_{« optical » phase}

motions in which two consecutive

_{triple points}

move out of

_phase,

i.e. variations of the

_phase

fraction 71.

Equation (38)

can be

written,

with the

_help

of

_{equations (32)-(35) :}

where :

since

1

and that b :::» 0.

Finally,

the

_{compatibility}

condition

_{(Eq. (40))}

for the

_stationary

_pattern

_reads,

for our

pseudo-symmetric

model :

For a finite

_asymmetry

of the _{eutectic concentration gaps}

_AC,,,,,6

_(finite

(1 -

_15»

this reduces 2

to :

a is

always positive.

Then the two roots of

_{equation (39)}

have

_negative

real

_parts.

Both the non-neutral modes are

_relaxing

_(with

or without

_{oscillations,}

_depending

on the values of

_A,

uOO’

s,

...)

.

a is

_negative

for

_stationary

_patterns

with

_{wavelengths :}

In this case, one of the roots of

₍₃₉₎

is

_positive,

the

_stationary

_pattern

is unstable. So we find that the near-eutectic

_{pseudo-symmetric}

_system

exhibits an

_asymmetric

behavior,

depending

on its concentration.

If its

_{representative point}

in the

_{phase diagram (see Fig. 2)}

is on the side of the eutectic

point corresponding

to the smaller concentration gap, it is

stable,

whatever

k,

with

_respect

to

uniform variations

_{of 71}

and of the _average

_{undercooling.}

If,

on the

_contrary

it is on the

_larger

_gap

_side,

_only

those

_stationary

_patterns

with

,k >» k _{o are} stable with

_respect

to these deformations.

The

_question

then

_naturally

arises of whether and how such an

_asymmetric

behavior is

modified when one relaxes the

_{pseudo-symmetric approximation}

and deals with the more

realistic

general asymmetric

model

_{(u aL =1=}

_{U /3 L’ ma =1=}

_{m {3).}

In the k = 0

limit,

this

generalization, although heavy,

is

_relatively

easy to

_perform.

We find that the

_instability

condition

₍₄₃₎

extends into :

So,

the

_{asymmetry parameter}

which decides on the existence of a

_stability

threshold

,k ()

for the basic

_pattern

is now :

where

_{à T,,,,,6}

are defined in

_figure

2. So the k = 0

_{instability only}

occurs when the

representative point

A

_(Fig.

2)

is on the side of

_CE

with the

larger

AT.

Of _course, since the two

_modes,

studied

here,

have finite relaxation rates for k =

0,

it is clear that these

stability

results extend _to

long

but finite

wavelengths.

2)

Phase

_{diffusion coefficient.}

- We

are left with

studying,

for the

pseudo-symmetric

model,

the behavior of the acoustical

_phase

mode which continues at small but finite k the uniform neutral one

_(i).

At finite

_k,

the full

_dispersion

_equation

is

_{given by:}

where Det is the determinant of the 3 x

_{3 { a ij}}

matrix

and,

as seen

_above,

_{ao (0)}

= 0. The relaxation rate 0)

a.p.

(k)

of the acoustical

phase

mode is therefore

given,

to lowest order

in

k,

by :

where Det

_{(k, 0 )}

itself must be calculated to lowest order in k. It is

_immediately

found that :

Expanding

in powers of k the

expressions (28)-(3 1)

for the

_{ai, (k,}

_{0 )}

and

_{a 1 i (k, 0 ),}

and

neglecting

corrections

_O(qÀ)

=

O(f¥2)

we find :

So,

Det

_{(k, 0 )}

= 0

(k2),

and,

to this

order,

one must use the k = 0 values

(Eqs. (32)-(35))

of the

_remaining

_{a ij ’s.}

Using expression (52)

of

[1]

]

for the minimum

_{undercooling wavelength, namely :}

we then find that the relaxation rate of the acoustical

_phase

mode is

_{given by :}

where the

_phase

diffusion coefficient :

That

is,

in the range of

wavelengths

of

_physical

interest,

all

_periodic

lamellar

patterns

are

Eckhaus-stable at

_large

thermal

_gradients.

The

_{only possible long wavelength instability,}

in this

_regime,

is the

_{« optical phase »}

one discussed above.

Our result about the

phase

diffusion coefficient contradicts the

_{Datye-Langer [9] prediction}

that

patterns

with A A min should be Eckhaus unstable. This

clearly

means

that,

in the

only

parameter

region

-

namely large

thermal

_gradients

- where the diffusion field

can be

calculated in _{the average}

_planar

front

_{approximation,}

Cahn’s

_dynamical

ansatz about lamellar orientation is not valid.

As we discussed in

_{[1], unfortunately,}

the

large gradient

limit in which the above

_analytic

treatment is

_justified

is restricted to values of the

_gradient

G which are difficult to reach

experimentally.

So,

there is an obvious need for an extension of our calculations to lower G s.

It is reasonable to

_expect

_that,

the smaller G

_(i.e.

the less confined front

_{deformations),}

the better the

_dynamical

ansatz should become.

_However,

it is clear

that,

at the same

_time,

the

less

_justified

it is to use the

_planar

front

_{approximation}

for the diffusion field. In this

perspective

numerical studies of both the

_stationary

front

_profile

and of its

_stability

are

_clearly

necessary.

Acknowledgments.

We are

_grateful

to K. Brattkus for numerous

_stimulating

discussions.

References

[1]

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_Phys.

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JACKSON K. A. and HUNT J. D., Trans. Metall. Soc. AIME 236

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