Lamellar eutectic growth at large thermal gradient: I. Stationary patterns

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Lamellar eutectic growth at large thermal gradient: I.

Stationary patterns

K. Brattkus, B. Caroli, C. Caroli, B. Roulet

To cite this version:

Lamellar eutectic

_growth

at

_large

thermal

_gradient:

I.

_Stationary

_patterns

K. Brattkus

(~),

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 étudions les

profils

de fronts stationnaires des

eutectiques

lamellaires en

solidification directionnelle. Nous _{montrons que,} dans la limite des

_grands

_{gradients thermiques,}

on _peut calculer

_{analytiquement}

ces formes de front sans avoir recours à

_{l’hypothèse d’égalité}

des surrefroidissements de front moyens des deux

phases

faite par Jackson et Hunt

_[7].

Nous trouvons _que la différence entre ces

_quantités

est

_comparable

au surrefroidissement

moyen

global

0394T, mais que 0394T

présente

un minimum à une valeur de la

_longueur

d’onde lamellaire du même ordre de

_grandeur

que celle

prédite

par J. H. Nous cherchons s’il existe des solutions à lamelles inclinées, et montrons

_qu’elles

ne _peuvent, en

_général,

exister que pour des

valeurs isolées de

_{l’angle d’inclinaison,}

mais

_qu’elles

sont absentes à

_{grand gradient,}

en accord avec l’observation

_{expérimentale}

d’un seuil de vitesse pour les « ondes d’inclinaison ».

Abstract. 2014 We

study stationary

front

profiles

of

directionally

solidified lamellar eutectics. We show that, in the limit of

_large

thermal

_gradients,

front

_shapes

can be calculated

_analytically

without

_resorting

to the Jackson-Hunt ansatz

_[7]

of

equal

average front

undercooling

of both

phases.

We find that the difference between them is

_comparable

with the

_global

average one

0394T,

but that the variation of 0394T with the lamellar

_period

03BB exhibits a minimum for a value of

03BB of the same order of

magnitude

as that

predicted by

J. H. We look for

stationary

tilted solutions. These can occur in

_general

for isolated values of the tilt

_angle,

but are absent in the

large gradient

limit, in _agreement with the

experimental

observation of a

_velocity

threshold for « tilt waves ».

Classification

Physics

Abstracts

61.50C - 64.60 - 64.70D - 81.30F

1. Introduction.

Directional

_{solidification,}

when

_applied

to a

_binary

metallic

_alloy

with a concentration

C 00

close to

_that,

_CE,

of its eutectic

_{point, produces}

a solid

organized, basically,

as a

periodic

arrangement of the two stable solid

_phases

a,

/3

(Fig. 1).

Two

_types

of such structures are

(t )

Present address :

_Department

of

Applied

Mathematics, California Institute of

_Technology,

Pasadena, CA _91125, U.S.A.

(*)

Also : UFR Sciences Fondamentales et

_Appliquées.

Université de Picardie, 33 rue Saint Leu, 80039 Amiens

_(France).

(**)

Associé au Centre National de la Recherche

Scientifique.

Fig.

1. -

Phase

diagram

of a eutectic

_alloy

with constant concentration gaps.

observed in bulk

_{samples : alternating}

lamellae and

_hexagonal

rod

_patterns

_parallel

to the

common

_(Oz)

direction of the

_{pulling velocity V}

and of the external thermal

_gradient

G. These

_{metallurgical}

studies

_{[1] have}

been

_complemented

with

_experiments

on

_transparent

eutectics

_[2]

with

_{microscopically rough solid-liquid}

interfaces

_(so

that the

_growth

kinetics

be,

as in

metals,

_{quasi-instantaneous).}

In these

experiments, performed

on thin

samples

in order

to allow for continuous

_optical

observations of the

_patterns

- and,

in

_particular,

of their

dynamics

- the standard

_{stationary growth}

mode appears to be lamellar. Most

_recently,

interest has focused on the

_problem

of

wavelength

selection in

_{out-of-equilibrium periodically}

structured

_systems

and on the nature of the

_dynamical

mechanisms involved in this selection. The

_experiments

of Faivre et al.

_[3]

on the

_CBr4-C2C16

eutectic have shown that among such

dynamical

defects,

«

_solitary

tilt waves »

_play

an

_important

role. These consist in domains in

which the

_lamellae,

which continue those of the « normal »

_patterns,

are

_parallel

but tilted at a

well-defined finite

_{angle 0}

with

_respect

to the

_pulling

direction. These domains are defects of

the

_{mesoscopic growth}

_pattern,

which are not correlated with defects of the

_underlying

microscopic

lattice. Their width is much

_larger

than the

_wavelength

k of the lamellar

structure, and for

_{given V}

and

_G,

domains of various widths are observed. Such

_dynamical

defects have also been identified in two different

_experiments

on

_systems

with

_periodic

front

patterns,

namely

directional

_growth

of a nematic

_{liquid crystal [4]}

and directional viscous

fingering [5].

Coullet et al.

_[6]

have

_proposed

a mathematical

_{phenomenology}

_describing

these defects.

This

_puts

to the fore the fact that

_they

are associated with the

_breaking

of the reflection

symmetry

about the Oz

_axis,

_{i.e. with the presence of} a finite

_{antisymmetric}

_component

of the front

_profile.

This

_{phenomenology predicts,}

that,

above some

velocity

threshold,

there should

exist

_stationary

solutions of the eutectic

_{growth problem}

with

_{periodic parallel}

tilted lamellae

filling

the whole

_sample.

The

question

thus arises of

_{substantiating}

this

_{description by directly solving}

the

_growth

equations

for

_specific

_systems,

so as to determine their domain of existence in _parameter

The basic theoretical

_description

of

_stationary

eutectic

growth

in due to Jackson and Hunt

[7]

(JH).

The basic

ingredients

of their

theory

of untilted

_periodic

lamellar

_{(and rod)}

structures are the

following :

(i)

the solute diffusion field in the

_liquid

is

_{approximated by}

that of a

_planar

front

(with,

of

course, a

_periodic

_arrangement

of

( « )

and

(,B ) regions) ;

(ii)

the ansatz is then made that the average front

undercoolings I1Ta,

_I1Tp

along

a and

_f3

lamellae are

_equal.

This enables them to

_predict

a

_{wavelength dependence}

of the _average front

_undercooling

AT(A )

which exhibits a minimum for

(where do

and

f

are,

respectively,

the

capillary

and solute diffusion

length),

which is of the

order of

_{experimentally}

observed

_wavelengths.

In order to

_study

tilted lamellar

solutions,

one is therefore

_naturally

led to

_try

and extend the JH

_approach.

On can then

immediately,

for an

_arbitrary

value of the tilt

angle

0,

calculate an _average

undercooling I1T( A,

cp

),

which

_suggests

the

existence,

for each value

of A,

of a continuum of

stationary

tilted solutions. This

clearly

contradicts the

experimental

observation

_[3]

that.0

is

_sharply

selected.

The source of this

_discrepancy

can be traced-back to be the

_{equal undercooling}

ansatz.

Indeed,

there are two

_steps

in the JH

_{theory :}

in the first one, averages

_{I1Ta, f3}

are calculated

with the

_help

of the

_planar

front

_{approximation (i), by integrating along}

the front of each lamella the Gibbs-Thomson

_{equation (i.e.}

of the condition of local

_{thermodynamic}

equilibrium).

This can be

performed formally

without

calculating explicitly

the front

profile.

JH then

_apply

their ansatz and obtain

_{WT- (À ).}

The front

_profile

is later calculated without

using

the ansatz as a

_{by product}

of this

_approach.

It can be

_checked,

as we will

_show,

that the

value of

AT(A )

which _may then be deduced from the

_shape

of this

_profile

is not consistent with that

_resulting

from the

_{equal undercooling}

ansatz -

although

the

_predicted

Amin’

is

_{only changed by}

a factor of order one, as can be

expected

from a

_qualitative

evaluation

of the order of

_magnitude

of the

_capillary

and diffusive contributions to W.

This

_flaw,

which has

_only

a minor effect on the

_description

of non-tilted

_patterns,

becomes

crucial when

_looking

for tilted ones :

indeed,

we will see

that,

at

_finite,

_profiles

calculated

by solving

the Gibbs-Thomson

_{equation separately}

in the

( a )

and

(/3)

regions

cannot, in

general,

be matched to fit the

_equilibrium

conditions at thé

_a-,6-liquid

_{triple points.}

These remarks lead us to

_modify

the JH

_approach

and make it consistent in the

following

way : we

first calculate a front

_{profile satisfying}

the

_{triple point}

conditions.

_This,

of course, due to the non-linearities of the

_problem,

we cannot

_{perform exactly.}

We therefore retain the

_planar

front

_{approximation (i)}

in the calculation of the diffusion field to calculate the

_resulting

front

profile.

We show

that,

as can be

_expected,

this

_{approximation}

is

_valid,

in the

_experimental

range A -

A min’

only

in the

_large

thermal

_{gradient regime}

-

more

_precisely,

for

FT/f «

1,

where

_PT

is a

thermal length

oc

G - 1,

and f

the diffusion

length

oc

v -1

(see §

_2).

The validation of the

_planar

front

_{approximation}

in this

_regime

was

_originally

discussed

_by

Brattkus and Davis

_{[8] although}

error estimates

appropriate

for

A - Ã

_{min were} not derived. In this

_{large gradient}

limit we calculate

à T(,k )

and the

precise

value

of Amin

for untilted lamellae.

We then

_investigate

the existence of tilted solutions. We first

show,

by

an exact

_counting

argument,

that

_they

can exist

_only

for discrete values of the tilt

_angle.

We solve the front

equation

separately

in the

_{( a )}

and

_{(f3) regions}

and obtain from the

_matching

conditions at

gradient

limit where our

_approach

is

valid,

this has no solution

width 0 :0

0,

in

agreement

with the

_experimental

observation,

at fixed

G,

of a lower

_velocity

threshold

_Vt

for

_solitary

tilt

waves. It _{appears in} our calculation that the relevant

_parameter

for the existence of tilted lamellae is

_’T/e.

We can therefore

_reasonably

infer from this that the threshold

_velocity

Vt

should be

_proportional

to G.

2. The front

_equation.

The

_growing

eutectic is described in the

_{following simple}

model which _preserves the

essential

physical

features of the

_{system :}

- We

assume that the thermal

_gradient

G is constant and uniform. That

_is,

we assume that the thermal diffusivities are

_equal

in the three

_{(a, /3,}

_{L ) phases,}

and that thermal

_transport

(evacuation

of latent

_heat)

is much faster than chemical diffusion.

- We

assume that the kinetics of atomic attachment at

_{the a /L}

and

_/3

_/L

interfaces is instantaneous on the time scale of the

_phenomena

of

_interest,

i.e. that the front is

_everywhere

at local

_{thermodynamic equilibrium.}

This

_implies,

in

_particular,

that the interfaces are

microscopically rough.

- We

neglect

chemical diffusion in the solid

_{phase (one-sided model) :}

Da,

_{D’a «}

D,

where D is the solute diffusion coefficient in the

_{liquid phase}

at near-eutectic

_composition.

We moreover make the

_simplifying

_(but

_{unessential) assumption}

_that,

for

_liquid

compo-sitions C close to the eutectic one,

CE,

the two concentration gaps on the

_{phase diagram}

of the

_binary

_system are constant

_{(*). They}

will be denoted

_by

_{AC,,, =}

_{CE -}

_{C a,}

I1C /3

=

c 0 -

CE

(see

Fig. 1),

and we set :

(with

AC =

OC a

₊

AC,6)

and call _ma,

_mp

the absolute values of the

_slopes

_{(dT/dC )}

of the

liquidus

lines at the eutectic

_point.

We define the dimensionless

_{composition :}

and express

length

and time in units

_of,

_{respectively,}

the solute diffusion

_length

and time :

We are interested here

_only

in lamellar structures, i.e. in 1-D front deformations. The

solidifying

system

is then

_described,

in the

_laboratory

frame - where the

_sample

is

_pulled

at

b)

On the front

(z

= ((x,

t ) ).

(i)

Solute conservation :

where n is the unit vector normal to the interface

_pointing

into the

_liquid

and :

(ii)

local

_{equilibrium :}

At each

_{triple (a -}

_{/3 -}

L )

point (see Fig.

2),

it is

_{expressed by}

the condition that the

capillary

forces balance. When the

_anisotropies

of the three surface tensions _{(T aL’}

(T f3L, (T af3

can be

_neglected

- which we will from now on assume, this

simply

reduces to :

where each Q vector

_points

out of the

_{triple point}

and is

tangent

to the

_{corresponding}

interface.

Fig.

2. - Periodic lamellar _pattern.

On all other

_points

of the

_solid-liquid

front local

_equilibrium

is

_{expressed by}

the Gibbs-Thomson

_equation

for the relevant

_{(a -}

L or

_{/3 -}

_L)

_interface,

_{namely :}

position

of the T =

_TE

_{isotherm. di}

and

_{$Tl (i}

= _a,

8 )

_are,

respectively,

the dimensionless

capillary

and thermal

_lengths

associated with the

_{(i) phase :}

and the

_Li

are the

_specific

latent heats.

Equations

(5)

to

_{(9), supplemented}

with the

_boundary

condition :

provide

a

_{complete description,}

of the

_{growth problem.}

As is well known it is

_possible

in such a one-sided free

_boundary

_problem,

to _express

_exactly,

with the

_help

of

_equations

_{(5), (6)}

and

(12),

the value of the

_diffusing

field u at the interface as a functional of the front

shape.

When

this is

_done,

_{equations (9), together}

with the

_{triple point}

conditions

_(8),

become a closed set of

equations

for the

single

unknown

function ((x,

t).

We will now

_perform

this

reduction,

which will prove

handy

for

_qualifying

our

approximation

and for the linear

_{stability analysis performed}

in the

companion

paper

[10].

As shown in reference

_{[11] from}

_equations

₍₅₎

and

_(12),

one can _express

_exactly,

in the 2-D

one-sided

_model,

the value of u on the front as :

where :

and

One can then eliminate

u (x, C, t )

and

_{(tî . Vu}

t from

équation (13)

with the

help

of

equation (6)

and of the Gibbs-Thomson

_{equations (9).}

This results in two

_{integrodifferential}

equations

for the front

_{shape £ (x, t),}

valid in the

_{( a - L )}

and

_{(,6 - L )}

front

_regions

respectively.

These are to be solved

_together

with the

_boundary

conditions at the

_{triple points,}

namely :

-

continuity

off

-

equilibrium

conditions

(8).

These

_equations

are -

as usual in such free

boundary problems

- _non linear in

£,

which appears in

particular

within the diffusion

propagator

G.

However,

an

_{important simplification}

can be

performed.

Indeed,

in lamellar eutectic

solidification,

the observed lamellar

_{wavelengths A}

are

_typically,

of the order of 10

_microns,

while diffusion

_lengths

lie in the 500 _{m range}

_{(D - 10 - 5}

_{CM2/S@ V -}

a few

_J..Lmjs).

So

The dominant contribution to the

_integrals

on the r.h.s. of

equation (13)

comes from the

regions lx - x’l 5

1, t

- t’ 5 1

_{corresponding}

to the diffusive range of our

_problem.

On the other

_hand,

_slope

variations of the front

_profile

are restricted

by

the

triple point equilibrium

conditions.

_So,

deformation

_amplitudes

are at most of order k «

1,

and one can

_legitimately

linearize the diffusion

_propagator

G in

_{(C -}

_(’).

_Moreover,

we assume the

quasi-stationary

approximation [12]

to be valid. This amounts to

_{setting C (x’,}

_{t’ )}

_{(x’, t ) everywhere}

in the r.h.s. of

_{equation (13).}

This

_{approximation}

is

_valid,

for

_{time-dependent phenomena,}

when the relevant time scales are short

_compared

with the time of

_propagation

of the diffusive

signal

on the

_{typical length}

scale of the front

deformation,

which will be the case for the

phenomena

we

_investigate.

Then,

using

equation (6)

and

_performing

the time

_{integration, (13)}

reduces to :

where

_Ko

is the modified Bessel function

_[13]

and :

d(x, t)

is the

_{step-function}

defined

_{by equation}

_(7).

For

_{non-stationary}

front

_profiles,

it

depends

on time via the

_positions

of the

_{triple points.}

At this

_stage,

one may of course

_{simply replace}

u on both sides of

equation

(16)

by

expressions (9).

However,

note that the r.h.s.

_{integration swèeps}

the whole

x-axis,

including

triple points,

where

_slope

discontinuities

_give

rise to 5-function

_{singularities}

of the curvature

K which must be

_explicitly

taken into account as an

_{inhomogeneous driving}

term in the front

equation [14].

This

_complication

can be avoided in the

_following

_{way :}

_{equation (16)}

can be rewritten

formally

as :

which can be inverted into :

Using

the

_{expression [13]}

of the Fourier transform of

_{Ko :}

one finds

_{immediately :}

where :

3. The standard

_stationary

lamellar

_pattern.

We first

_study

untilted

_{stationary periodic}

lamellar

_patterns

_(Fig.

_2).

Let

be the abscissae

of,

respectively,

the

_{6 /a}

and

_{a / f3}

_{triple points,}

where the

_a-phase

fraction

q is a

_priori

unknown

_(except

in the

_special

case of vertical solidus lines

_{[ 15],}

which we do not

consider

_{here, where q =}

1 -

C 00).

The values of the

_{pinning angles}

_{() a’}

_0,6,

are

_uniquely

determined

_{by equation (8),}

which reads in the _present

_{geometry :}

So,

the value of the

_{profile slopes}

are constrained to be

_opposite

at the two ends of a

_given

lamella.

1. THE LARGE

GRADIENT

LIMIT. - The

_stationary

front

_equations

are obtained from the

time

_independent

version of

_{equations (9), (22) together}

with the

_matching

conditions

Although

their diffusive term has been

linearized,

due to the

_{non-homogeneity}

of the solid

phase,

their are

_highly

_non-trivial,

and no exact solution is

_presently

known.

However,

the

_following

remark can be made : the diffusion field ahead of the eutectic front

exhibits two

_types

of

_{space-varying}

contributions : a diffusion

_layer

of thickness -

f,

associated with

_global

solute

_rejection,

and modulations due to the

_periodic

structure of the solid

front,

which extend on a _range of order

X (X f).

When the

_amplitude

of front deformations is much smaller than this _range, it is

_certainly

reasonable to

_approximate

the diffusion field

_by

that of the _average

_planar

front

_{(with alternating a}

and 6

parts).

This is

_precisely

the central

_{approximation}

of the JH

_{approach [7]}

as well as of the

_stability

analysis

of

_Datye

and

_Langer

_[15].

However,

the

question

of the range of

validity,

in

_parameter

space, of these

approxima-tions,

was left open. It is

qualitatively

reasonable to

_expect

that,

the

larger

the thermal

gradient

G,

the

_higher

_{the energy} cost of front excursions and thus the better this

_planar

approximation.

That

_is,

for small

_enough

reduced thermal

_lengths

_ÎT,

we assume

that £(x)

in

Gibbs-Thomson

_equations

are then

_easily

checked to reduce to :

where use has been made in

_{equation (22)}

of the relation :

0

(x)

is the JH function

_{given by :}

where :

In our small

wavelength

limit

·

How small should

_{e r}

be for this

_{approximation}

to be valid will discussed

_below,

where it will appear that

equations

(27)

are the lowest order in an

expansion

in the

_parameter

4=

FTIÎ-2. STATIONARY FRONT PROFILE. - Note

that,

unlike the diffusion term, the

_capillary

term in

equations (27)

cannot in

_general

be linearized : the

_{pinning angles}

- determined

by

the relative

_magnitudes

of surface tensions - are in

_general

_{finite, and,}

at least close to the

_triple

points, £ ’

cannot be

_neglected.

However,

as

_argued

_above,

a

_large

thermal

_gradient

counteracts the

_pinning

effect

_by

flattening

the

_profile.

As is

_apparent

on

equations (27),

the meniscus

shape resulting

from this

competition

has a

_typical

_range of _space variation :

On the other

hand,

the r.h.s. diffusive terms _vary on the scale of the lamellae

_thickness,

of

order A.

_Thus,

if the

_{e T’S}

are small

_enough

for the condition :

to be

_satisfied,

it becomes

_possible

to

_separate,

within each lamella :

-

edge regions

where space variations

of 0

can be

_neglected

while the

_{non-linearity}

of thé

curvature is retained.

- _a central

part

in

_which

_l"

₁

1 and

₍₂₇₎

can be

_fully

linearized.

the JH minimum

_undercooling

one

_{(Eq. (1)).}

We will from now on assume that

À lies in this _range, so

_that,

in order of

_{magnitude :}

So,

condition

(31)

becomes :

with

_respect

to lamellae centers

and to that of the

_pinning

_conditions,

the front

profile

of each lamella is itself

_symmetric.

Let us first build the solution

₍₂₇₎

in a a lamella

_{(0 -}

x «

_7yA

_/2).

a) Edge

region

(0 --

x «

_’rJÀ/2)

In this

_{region qb (x) c-= 0 (0). Equation (27.a)}

then

_exactly

reduces to that for the meniscus formed

_by

a semi-infinite

_liquid

submitted to

_{gravity [16]}

with a

slope

at its

_edge

(x

=

0)

fixed

by

condition

(26) :

b)

Lamella center

In this

_region,

the

_capillary

term in

_(27.a)

reduces to

_{da (;.}

One obtains from the linearized version of

_equation

_(27.a)

Taking

into account the

_profile

_symmetry

_{(’ (,qA /2)}

_{= 0)}

and the

_{expression (29.c)}

of

0 (x) :

:

where _{IL a} is a

_(free)

constant of

_integration.

c) Asymptotic

matching

Such a

_matching

is

possible

provided

that the ranges of

validity overlap,

i.e.,

when condition

₍₃₁₎

holds. The

_expansions

of

₍₃₄₎

and

₍₃₅₎

in the common

_{asymptotic region}

( q;;

1 « x

«

’T/2À )

2

both are of the

form ? a =

A + B exp

(- q a x). Matching

the

exponential

Compatibility

between the

two-already

known A constants

_imposes

that :

Using expression (29.b)

for

0,

one

_easily

checks that :

that

_is,

_up to the order at which our calculation is

_performed,

condition

_(37.a)

is indeed

satisfied.

The front

_{equation (Eq. (9.b)) appropriate}

to

_{(B )}

lamellae is solved

by

the same

procedure.

We must

_impose

the

_continuity

condition

_(25.a)

at

_{the 6 - a}

_triple

_point

x = 0

(due

to the reflexion

_symmetry

of the

profile

with

_respect

to each lamella center, this

also ensures

_continuity

at the a -

_/3

_ones).

From

_expression

_{(34) :}

Similary,

from the solution in the

_/3

lamella :

Continuity

therefore results in the

_{following compatibility}

_condition,

which determines the

a-phase

fraction 71

(À )

for a

_pattern

of fixed

_{wavelength À :}

where :

The r.h.s. of

_{equation (40)}

is of order

_{(fTq)- 1 -}

_(d/fT)1/2.

_{Typically, capillary lengths}

à-

50

Â.

For

_{CBr4 - C2}

_{CII, =}

26

Â,

_{df3 =}

47

_Â,

while ma AC = 15

K, mf3

AC = 22 K.

So it is seen

_that,

even in the

_{large gradient regime (defined below),}

ÏT -

_20/G,

and

â

_4.

Since,

moreover, A

1,

equation (40)

can be solved

_by

iteration,

and one

finds, using

conditions

₍₃₂₎

and

_(33),

which entail

(q’T)

-

0( IêT) «

1

: ’

That

_is,

the fractional amount of

_a-phase

is _very close of what it would be for a

_system

with a front at the eutectic

_temperature.

3. AVERAGE FRONT UNDERCOOLING. -

_Following

_JH,

we define the average front

undercooling

as measured from the eutectic

temperature

TE :

where,

e.g. :

This we write as :

where the cut-off X lies in the

_region

of

_{asymptotic matching (ql,}

1 «

_{X « qk}

_/2).

We then

obtain,

with the

_help

of

_{equations (34), (35),}

_{(36) :}

Similarly :

So that :

This

_{expression, together}

with

_{equation (40)}

which

détermines

(A )

is to be

_compared

with the JH results. Let us recall their

_{procedure : they}

_{compute ta,}

_t{3

_by

_{directly integrating}

equations

(27)

across each

_{lamella, which,}

of course,

precisely yields expressions (45).

However,

since

_they

do not solve

_{for Ç (x),}

_they

miss the

_{compatibility}

condition

_(40).

In

order to

_{determine q(A ), they}

make _up for this with the

_{complementary}

ansatz

ta

=

t{3o

We are now in a

_position

to check the value of this ansatz : from

_{equations (45),}

we may

calculate ta - t{3’

which is

_immediately

seen to be of the same order as the average

undercooling

itself. We should now _{compare the}

shape

of the

A-dependence

ofATas

given by

(46), (40)

with the JH one, which exhibits a minimum for A = A

m n :

where from

_{equation (40) :}

Then,

up to corrections of order À and

_{(die r)1/2,}

we find

_(using

P (,q ) = P ( 1 - q »

where

U(,q)

is

_{easily computed numerically}

and checked to be

_positive

for

all q’s

(0

: _q --

1 ).

So,

à T(À )

also exhibits a

minimum,

at a

wavelength Amin

*

So,

our ansatz-free calculation

_yields

a value of

_{À min}

different from the JH one, but with the

same order of

_magnitude

_{(Ik min -(,j f ) 1/2)}

. This results from the fact

that,

even

though

the

equal undercooling

ansatz is not correct, the error on _q

_(of

order

_{A )}

that is

_produces

does not

change

the order of

_magnitude

_{(f T À )}

of the diffusive contribution to the

undercooling.

4. RANGE OF VALIDITY OF THE LARGE GRADIENT APPROXIMATION. - We must now

estimate the

_validity

of our central

approximation,

i.e. of the

replacement,

in the front

equation,

of the true diffusion field

_{u(x) by}

that of its

planar

average

UO(X) = U. + 71 + + 0

(x).

Setting

we

_{estimate 6 u (x) by replacing,}

in

_equation

_(22),

the

_{true £}

_by

its

_large

_gradient

approximation

calculated above

Separating

out the p = 0 and n + p = 0 terms, and

neglecting everywhere

_{p *} 0 terms such

as

_{77(/?Â:) -}

1

with

_respect

to

_1,

we obtain :

The second term in the r.h.s. of

_{equation (54)}

is of

_order,

at _most,

5l =

_{1 [a - [/31 ;}

_{; the}

first one is

_easily

checked to be

_everywhere

_(including

the

_{triple point}

_regions),

of order

1 (3 + q - 1 ) & e-

1

«--

15 [1.

In order to estimate the third term, we

_{replace l (nK) by}

the Fourier coefficients of an

_{approximate profile.}

The

_simplest

choice is a crenel with

_heights

[a

8

which

_yields

a contribution

O A

8 ) 6 e.

We have also tried a finer

_{approximation}

where,

in each

lamella,

}

is

_replaced,

close to the

_{triple points, by}

a

_straight

line of

slope,

e.g.,

tg

B a,

and checked that this does not

_change

the above order of

_magnitude.

__

From

_equations

_{(40), (42),}

and with the

_help

of conditions

_{(32), (33), uo(x)}

=

ü(Vd/£r).

On the other

_hand,

from

_{equation (45),}

5 t = 0 ( Jdfr ).

So the condition of

_validity,

,6u « u o, of our

approximation

on the diffusion term :

is

_precisely

the same as that for the

_{asymptotic matching}

used to calculate the front

_profile.

So,

we can conclude that :

- The JH

approximation

for the diffusion field is valid

_only

in the

_{large gradient regime}

defined

by

equation (55) ;

- in this

regime,

their ansatz is not

_valid,

since the difference between the front

undercooling

of the two

_phases

is

_comparable

with the

_global

one, AT.

_However,

it is

unnecessary, since the

large gradient problem

can be solved

_{completely, yielding}

the same

qualitative

behavior for

_{AT(,A )}

but a different value for À

min-4. Search for tilted

_stationary

_patterns.

Let us now

_try

to find whether it is

_possible

to find modes of

_growth

with

_parallel

lamellae tilted at an

_{angle 0}

with

respect

to the

_pulling

direction Oz and with space

period

,k

_along

_{the Ox average front direction}

_{(Fig. 3).}

Such a

_pattern

_corresponds

to a front

_profile

(x -

v t )

which drifts

_along

Ox at

_velocity

From the

_{triple point equilibrium}

condition

_(8),

the

_{pinning angles (Fig. 3)}

measured from the x-direction are

_{given by :}

So,

in contradistinction with the untilted case, the

_{pinning angles}

at the two ends of a

_given

lamella now have different values. This

_immediately

entails that the

_profile

within a lamella is no

_{longer symmetric}

with

_respect

to its center : it

_necessarily

_develops

an odd

_component,

the

amplitude

of which. can be identified with the order

_parameter

of Coullet et al.

[6].

One may be

tempted,

at least in order to

_get

a

qualitative description

of such states, to resort to the JH

_approach.

As seen

above,

this amounts to

_computing

the concentration field of the average alternate

planar

front,

i.e.

_setting

in

_equation

_{(22) {(x - vt) = [}

_(and

j (x -

vt)

=

0),

and

à (x, t)

=

à (o)(x -

vt).

One

immediately

obtains

(see

Eq.

(27))

Fig.

3. Schematic

_periodic

eutectic

growth

pattern with lamellae tilted at an

_{angle 0}

with _respect to

the

_pulling

direction Oz.

Then,

averaging

the Gibbs-Thomson

_equations

over

_{( a )}

and

₍₁₃₎

lamellae,

and

_using

the

equal undercooling

ansatz,’ one

immediately

computes

an _average

_{undercooling AT(A,}

_{0 )}

which

_{only depends}

on 0 via the values of the

_{pinning angles (57).}

This

_suggests

that there should

_be,

for a

_given

_A,

a continuum of tilted lamellar

_solutions,

in contradiction with the

observation of tilt waves with well-defined (/J’s.

What has been missed in this

_{simple-minded}

extension of the JH

_approach

can be

understood with the

_help

of the

_{following counting}

_argument.

Consider the Gibbs-Thomson

equations (9)

for our

_problem.

In the

_drifting

frame

_{(x -}

v t

_{--+ x)}

_they

become

time-independent.

u {x, (x)}

is a functional of the

_{profile (x),}

_i.e.,

a function of its

N(N --> oo )

Fourier coefficients

_ln.

The solutions of the two

_{(second-order differential)}

Gibbs-Thomson

_{equations (9)}

therefore

_depend

on N + 4

parameters,

namely

the

7V s

and 4

_integration

constants. Since the «

fraction q

is not fixed a

_priori,

one thus has N + 5 unknowns. These must be determined

_{by :}

- 2

continuity

conditions

_{for £}

at

_{the a - 13}

and

_{/3 -}

«

_{triple points}

- 4

pinning

conditions on the

_{profile slopes}

N self

_consistency

conditions

_{for C 1, - - -, N,}

i.e.,

N + 6

_equations.

Eliminating

the N + 5 unknowns therefore

_yields

a

_{compatibility}

condition for the tilt

_angle

cP,

which can

_only

have discrete solutions.

It is now

_apparent

that

_{directly computing}

a

1-T is

insuficient to ensure that it is

possible

to

build an

_acceptable

front

profile.

That e = 0 is a trivial solution of this

_equation

can be checked

immediately :

in this case,

the

_pinning

conditions are

_symmetric

_{(p, f3 = P a, (3)’}

the

_profile

within each lamella is

Note,

however,

that we have been

_considering

here a _system with

_isotropic

interfacial tensions. When

_{capillary anisotropies}

are taken into _account, two cases must be

dis-tinguished :

- if the

pulling

direction is a direction of

_high

_symmetry

for both the a and

/3

phases,

the

_lTiL’S

are

_symmetric,

and the

_argument

still works. There

_always

is an untilted

solution ;

- if _one is not

pulling along

a common

_principal

_axis,

the

_symmetry

_argument no

_longer

holds, 0

= 0 is no

_longer

a solution. That

_is,

as is well known from

_experiments,

in a

misoriented

_grain

lamellae are tilted. Let us however insist

_that,

in

general,

the observed l/J has no reason to be

_{strictly equal}

to the misorientation

_angle

_(how

much

_they

_may differ in

practice

of course

_depends

on the

_strength

of the

_anisotropy

of

_0-,,,,,).

Let us now come back to the

_{isotropic problem}

in the

_{large gradient}

_régime.

We set :

where x is now the horizontal coordinate in the frame

_drifting

at

_velocity

_v, and

F(x)

is defined

_by

_{equations (22),}

₍₂₃₎

in which

_{Ç(q, t ) + - iqv£ (q).}

We then solve

formally

the Gibbs-Thomson

_equations

with the same

_procedure

as in §

(111.2a-c).

In the x = 0

_{edge region}

and in the center of the « lamella for

example, (a(x)

is

given by

equations (34),

(35.a)

where

_B"

-

l/J a; l/J (x) -+ F (x).

The

asymptotic

matchings

at the two

edges

(x c2e 0, x =-- q À )

determine the

_integration

constants _{J.L a’ Va}

_(and

_analogously,

J.L 13’

v f3).

We then must

impose

the

continuity

conditions

(25.a). Using

the

analogues

of

expressions (39)

for

_{j(O), i (,q À),}

we

_get

two

_{coupled equations}

for q and 0 :

In the 0 = 0

limit,

where lamellae

profiles

are

_symmetric

and

v = 0, F (0) = F (,q À)

and,

while

_(60.a)

determines q,

equation (60.b)

is

trivially

satisfied,

as

_already

seen from the

counting

argument.

When cP =1= 0,

we must

_compute

F

_by

_plugging

into

_{equation (22)}

an

_approximate

expression ap (x)

for ((x) depending

on a finite number

of parameters,

which must then be determined

_{by self-consistency}

conditions.

The

_{simplest approximation}

_is,

of course, as in the untilted case, the JH one :

iap (X) = 1 .

However

_since,

as seen

in § 3,

the

_{corresponding F (x)}

is

_simply

the

_symmetric

JH

field

₀

_(x),

₎

_{équation (60.b) simply yields :}

_q

sin 0 =

0,

i.e. the

_planar

front

_{approximation}

2

produces

no tilted solution.

We must then

_try

to

_improve

_upon this

_{by choosing}

a

_shape

for

_(ap(x)

which describes more

precisely

the true

_profile,

_which,

as discussed

above,

contains within each

lamella,

an odd

component

driven

_by

the

_asymmetry

of the

_pinning

conditions. This we have done

by

choosing

_{for e,, , p (x)}

the

_shape

shown in

_figure

4. The

_straight

line

_{edge profiles}

are the

tangents

to the meniscus ones at the

_{triple points ;}

the central

_straight

lines :

Fig.

4. -

Approximate shape

of the

_liquid-solid

interface used to calculate the

liquid

concentration field

on the front.

A tedious but

_{straightforward}

calculation

_analogous

to that of §

(3.4) yields,

to lowest order

in

_eT,

_A,

the

_expression

of

_F(x)

in terms of the

_’i,

_pi. These are then made self

consistent,

from which the final

_expression

of

_{F(0) -F(7yÀ)}

is obtained. We do not

_give

here the

corresponding

detailed result

_since,

as will now _appear, its

_{only important}

feature is its order

of

_magnitude.

This we find to be :

which can be

_easily

understood

_{qualitatively. Since k - k}

Jd" q- l "" À

Jfr.

So

(61)

simply

expresses the fact the

amplitudes

of odd

profile

deformations

(and

therefore of odd

concentration

_modulations)

are

_squeezed

_by

the

_large

thermal

_gradient.

The r.h.s. of

_equation

_(60.b),

_O(fTlq),

is of the order of terms which have been

_neglected

in our

approximation

and must itself be

_neglected.

_So,

in the

_{large gradient}

limit,

the

_only

solution of

_{equation (60.b)}

is the untilted one.

5. Conclusion.

The conclusion of our

_analysis

are therefore somewhat

_{disappointing :}

it is

_only

in the

_large

gradient

limit that the JH

_approach,

_improved

so as to become

_consistent,

can be

_justified.

In this

limit,

due to the fact that front deformations are restricted to small

_values,

no tilted

lamellar

_stationary

state can exist.

From

_{(Eq. (55», together}

with

(11),

this

_{regime corresponds}

to

In

_{practice, pulling}

velocities in lamellar

_{growth experiments}

are at least of the order of

1

_.tm/s,

while

D -10- 5

_cm2/s.

For

_CBr4-C2C’6

for

_{example m}

AC - 20 K.

So,

in

_practice,

the

large gradient regime corresponds

to

G > 200

K/cm .

So,

it will

_{unfortunately}

be very difficult for

experiments

to reach the

_{large gradient}

_regime.

So if one wants for

_example

to compare in detail measured front

profiles

or

_undercooling

with

On the other

hand,

for values of

G, -

100

_K/cm,

comparable

to those in the

_{experiments [3]}

where « tilt waves » have been

observed,

condition

_{(62) implies :}

V « 0.5

_{J.Lm/s .}

So,

our above

_negative

conclusion is

_compatible

with the

_experimental

observation of a

velocity

threshold below which tilt waves do not exist.

It is clear that

_again

more

_{precise predictions}

about these

_patterns

will demand numerical studies :

_indeed,

as shown

_by

the discussion of §

_4,

the relevant

parameter

which controls their

_possible

existence is

_ÎT.

It is clear that it is

_only

for

_ÎT

1 that the

_system

can tolerate

(odd)

front deformations

_{large enough}

for such solutions to become

_possible.

At this

_stage,

we can

_only

infer from our

_analysis

_that,

most

_likely,

the threshold for a tilted

_growth

mode to

appear should be

given by

FTIÎ

= cst, i.e. the threshold

velocity

should then be

proportional

to the thermal

_gradient.

Quantitative

experimental

studies of threshold velocities which are

presently

in progress

[17]

should

permit

to check on this

_conjecture.

Acknowledgements.

We are

_grateful

to G. Faivre for numerous

_stimulating

discussions.

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[1]

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₍₁₉₈₀₎

154.

[2]

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

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843. SEETHARAMAN V. and TRIVEDI R., Metall. Trans. A 19A

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1.

[13]

GRADSHTEYN I. S. and RYZHICK I. M., Table of

_integrals,

series and

_products,

4th Edition, Academic Press Ed.

(1980).

[14]

CAROLI B., CAROLI C. and ROULET B., J.

_Crystal

Growth 76

₍₁₉₈₆₎

31.

[15]

DATYE W. and LANGER J. S.,

Phys.

Rev. B 24

₍₁₉₈₁₎

4155.

[16]

See for instance LANDAU L. D. and LIFSCHITZ E. M., Fluid Mechanics 2nd Ed. § 61,

p. 243,

Pergamon Press

_(1987).