Further analysis of the analogy between cellular solidification and viscous fingering

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Further analysis of the analogy between cellular solidification and viscous fingering

M. Hennenberg, B. Billia

To cite this version:

M. Hennenberg, B. Billia. Further analysis of the analogy between cellular solidification and viscous

fingering. Journal de Physique I, EDP Sciences, 1991, 1 (1), pp.79-95. �10.1051/jp1:1991116�. �jpa-

00246305�

ciassificafion

Physics

Abstracts

61.50C 61.55H 61.70W

Further analysis of the analogy between cellular solidific4tion and viscous fingering

M.

Hennenberg (*)

and B. Billia

Laboratoire de

Physique Cristalline(**),

Facultd des Sciences de St Jdr6me, Case lsl, 13397 Marseille Cedex13, France

(Received 8 January J990, revised J4 Septenlber J990,

accepted

2J

Septenlber J990)

Rksumk. En l'absence de tension de surface _et dans la limite des

petits

nombres de Pddet

P~

= A

li~,

off A est la

pdriode

et i~ la

longueur solutale,

_nous rdexaminons

l'analogie

entre la solidification cellulaire et la

digitation visqueuse.

En

rdalitd,

la croissance cellulaire

ddpend

aussi d'un second nombre de Pdclet P~ bask _sun le ddcalage du soInmet par rapport fi la

position

de

l'interface

plane. Lorsque

P~ est

petit, l'analogie

avec la

digitation visqueuse

s'avdre fondde _ce

qui

conduit I _une

expression

de la sursaturation _au sommet Q,

qui

est en excellent accord _avec les donndes

expdrimentales disponibles

_pour les

alliages

succinonitrile-acfitone. De

fait,

il

apparait

que P~ est

grand

_pour les cellules rdelles _ce qui nous amdne I

ddvelopper

_une

approche

_{pas fi pas,}

jusqu'au

second ordre _en le

petit paramdtre

P~. ^Au premier ordre, _nous retrouvons de manidre

rigoureuse

la relation de

Brody-Flemings

_pour Q. Le second ordre aboutit I _une

dquation intdgrale

du

profil

cellulaire, dtablie de manidre

explicite

dans le _cas des cellules

d'amplitude

finie.

Abstlract.

Neglecting capillarity,

the

analogy

between cellular solidification and viscous

fingering

is revisited in the limit of small Pdclet numbers

P~

=

A

li~,

where A is the

periodicity

and i~ the solutal

length. Actually,

cellular

growth

also

depends

on a second Pdclet number P~ based _on the shift of the

tip

from the

position

of the

planar

front. When P~^is small, the

analogy

with viscous

fingering

is

legitimate

which

gives

an

expression

for the

tip supersaturation

Q which fits

nicely

with the available

experimental

data _on succinonitrile-acetone

alloys.

Nevertheless, it

turns out that P~is

large

for real cells which leads _us to

develop

_a

step-by-step approach,

_up to the second order in the small parameter

P~.

At the first order, the

Brody~flemings

relation for Q is recovered but in _a

rigorous

_way. The second order results in _an

integral equation

for the

profile

which is

explicitly

derived for finite

amplitude

cells.

1. Introduction.

In the directional solidification of _a

binary alloy

the control parameters are the

groAvth

rate

V,

the thermal

gradient

G and the solute concentration

C~

in the melt far away from the

(*)

Pernlanent address:

Ddpartement

de

Chimie-Physique,

Facultd des Sciences,

Campus

Plaine, C-P- 231, Universitd Libre de Bruxelles, Bd du

Triomphe,

1050 Bruxelles,

Belgique.

(**)

U.R.A. _au CRNS _n 797.

solid-liquid

interface. For

given

^{G and}

C~,

one follows the evolution of the

shape

of this surface

by increasing

the

velocity

V from _one

experiment

to the next so that the

morphology

evolves from _a

plane

to an array of cells and then _to dendrites which _are characterized

by secondary

_arms

[I].

This _paper is devoted to the

study

of

steady

cellular fronts.

The

complexity

to reach _a

complete

and

satisfactory description

of the cellular patterns stems from the interaction between the

tip region,

the array of

periodicity

A and the tail

region,

when it exists

(Fig. I). Recently,

great

improvements

have been

brought

forward

[2,

3] ^from a rich

analogy

between

Saffman-Taylor fingers [4,

5] ^and directional solidification

of

binary alloys. Neglecting

surface tension in the

Gibbs-Thompson

law

linking

the temperature to the solute concentration

along

the unknown surface

f

of the

cell,

Pelcb and Pumir used _a self consistent

approach

to show that the

Saffman-Taylor shape

is _an exact

solution _near the cellular cap

[2, 4].

The curvature and surface tension effects _are then added to

provide

in _a very

elegant

_way a

relationship

between

tip undercooling

and cell

spacing [3, 5].

To reach these results _one

systematically

_uses two main

hypotheses

which are that

I)

the Pbclet number

P~

₌ A V

ID,

in which D is the solute diffusion coefficient for the

liquid phase,

is _a small

quantity

which will be denoted _e and that

it)

the term linked to the variation of the

miscibility

_gap

along

the cell surface _can be

dropped

in the interface solute balance. Further

scrutiny

of this

question

^lead to the

analysis

of Dombre et Hakim who studied _a

binary

mixture with _a constant

miscibility

_gap

[6] and,

_very

recently,

to the work of Weeks and _van Saarloos

[7]. Considering

_a finite

P~

^and a very small

segregation

coefficient

K,

these last authors used _a

power-series expansion

in K to

investigate

cells with

deep

narrow grooves. We will not focus _on this finite-Pkclet number

region.

Recently,

Billia _et al.

[8, 9]

^made a

thorough analysis

of the available cellular

experimental

data _on succinonitrile-acetone

alloys [10,11]

and

compared

them to various theoretical models of

tip supersaturation

in cellular directional solidification

[2,12,13].

A rather

Liquid

:

Tip region

~ x* ^R*

Y*

^Tail

region

^Interface

(

solid

Fig. I. Schematic

representation

of

arrayed-cellular

solidification showing the various

regions

in the

liquid phase.

ambiguous

conclusion is reached since the best agreement ^between

theory

and

experiments

was obtained for the least restrictive

hypotheses

which _were

long

_ago formulated

by Brody

and

Flemings[12].

These authors assumed that the intercellular isotherms _are flat and

perpendicular

to the

growth

direction and that constitutional

supercooling

is

negligible

in the

cell grooves as well _as the

capillary

term in the

Gibbs-Thompson equation.

The

major

drawback of this

theory

_was

precisely

its failure to link the

shape

of the cellular

tip

to its _mere existence.

Leaving

this

question

open

explains

the number of

subsequent attempts

based _on ad hoc

shapes (see

_e.g. Refs.

[13, 14])

and

why

the

mathematically

self-consistent

approach

of Pelcb and Pumir _{was so}

timely.

Nevertheless,

the

analysis

of the succinonitrile-acetone data has shown that the

limiting assumption (it)

stated

above,

which _was essential to Pelcb and Pumir to

analytically

derive the solution

analogous

to the

Saffman~Taylor finger,

_may be not

physically

relevant in the cellular

regime [9].

This throws _some doubts _{on a} direct

correspondence

between the Hele-Shaw

problem

and the _one at hand.

Therefore,

in the

present

paper, the

analogy

between cellular solidification and viscous

fingering

will be revisited. The order of

magnitude

of the dimensionless

parameters

will be estimated from the cellular data in the succinonitrile- acetone

system gathered

in the

literature,

which will enable _us to

finally

handle the

pertinent

mathematical limit. Since these

experimental findings

^show

P~

to be smaller than

unity,

this information is

merged

with the

methodology

initiated

by Pelcb

and Puwir. It turns out that

one cannot limit oneself to a first order

approximation

to

explicitly

derive the cell

shape f

even for _a

simplified

case where

capillarity

^is

neglected.

2~ The basic

equatious~

For the

description

of cellular directional

solidification,

_we will

adopt

the scheme of Pelcb and Pumir

[2].

A 2D one-sided model is considered. The isotherms _are

straight

lines

perpendicular

to the

growth

axis x* and the Pdclet number

P~

^is a small

quantity.

The cells _are

growing

in the

positive

x* direction.

Concerning

the

phase diagram,

_we will describe _a

binary alloy

with _a constant

partition coefficient,

I-e- the

liquidus

and the solidus lines of the

phase diagram

are

straight

lines _so that the

miscibility

_gap AC

=

(l K) C(liquid)

~.

The diffusion of solute in the solid and the latent heat

generation

_are also

ignored.

In _a reference frame

moving

with

velocity V,

the

steady 2D-equation

for the diffusion of solute in the

liquid phase

reads

1°~

_~ ^{+ ~~}_~ ^{+ °}

~

C

=

0

(la)

ax *

ay*

^ax

where C is the solute concentration. The

boundary

conditions

along

the unknown surface

f

are

AC _cos 8

=

( fi ₍₁₆₎

an _i

T(

=

TM

+ mC

~

«TM/LR* (lc)

where 8 is the

angle

between the normal and the x* axis.

Equation (16)

_expresses the transfer of solute _across

f

where n* is the unit normal directed towards the fuid.

Equation (lc)

is the

Gibbs-Thompson

law where _« is the surface

tension,

_m the

algebraic

value of the

liquidus slope,

L the latent

heat,

_TM the fusion temperature of the solvent and R* the radius of

curvature. All these

equations

_are written

using

dimensional _space

quantities.

The

profile

of

temperature

is assumed to be linear _so that

T

=

To

+ Gx*

(2a)

where the reference temperature To ^{is the} solidification temperature ^{of the}

planar solid-liquid

interface. It follows from

equation (lc)

that the reference solute concentration

C~/K

is linked to the reference

temperature

_To

through

C~/K

=

(To TM) /m (2b)

If the interface

separating

the

liquid

and the solid

phases

is

flat,

the

miscibility

_gap is AC

=

C~(I K)/K

and the solute field reads

C

=

C~

+ AC exp

(-

x*

V/D). (3)

Directional solidification thus involves three fundamental

lengths _I)

the solutal

length

f~=D/V,

introduced

by

the Fick diffusion

equation (la), it)

the thermal

length

f~ = m AC

/G,

which compares the

temperature

^difference

corresponding

to the

miscibility

gap ^to the

temperature gradient

in the

liquid,

and

iii)

the

capillary length do

= «

TM/mL

AC. These

lengths

_are linked to the classical

analysis

of

stability developed

by

Mullins and

Sekerka[15]

since the dimensionless ratio _v

=i~li~

is the level of

morphological instability

and

doli~

=

A/K

where A is the absolute

stability

_parameter.

In _an array of

cells,

the

primary spacing

defines _a channel

containing

_a cell which does _not

exchange

solute with its

neighbours,

I.e. _no solute escapes

through

the cell borders at

y*

= ± A

/2.

In the frame of the

analogy

with viscous

fingering,

cells _are characterized

by

the relative width

A,

_a hidden parameter ⁱⁿ directional solidification which _can be evaluated from the determination of the dimensionless

tip

radius R

=

R*

IA

'~

fi

_~~~

~wR~

Far ahead of the

tips,

the existence of the cells _can be

ignored

_so that the cellular front is _seen

as a blurred flat interface at the scale of this outer

region.

In the inner

region

about the

tips,

the cell

description

becomes the essential feature. This

tip region,

on which _{we are}

focusing,

may or not be followed

by

_a

deep

_groove.

For _two basic _reasons, the

capillary

term in the

right-hand

side of

equation (lc)

will be henceforth

neglected

_so that the level of

morphological instability

_v will be the

only

control parameter. ^{On the} one

hand,

_one should not take the surface tension into account for the

Pelck-Pumir

methodology

to be feasible

and,

_on the other

hand,

it is

legitimate

to

neglect capillarity

for the available cellular data _on succinonitrile-acetone

alloys [10]. Indeed,

these

experiments

are not in the limit A

-

I,

where surface tension effects _are

quite large

for viscous

fingers.

The maximum value is A

=

0.9 and A is about 0.6 for the

largest

deviation in the

negative

_range when the Pelcb-Pumir

prediction

is used

(see Fig.

8 in Ref.

[9]). Therefore,

although

somewhat _narrow grooves are observed in cellular

experiments,

the

Saffman-Taylor fingers

which

actually

fit the cell caps have rather

large liquid

channels _on the sides.

Moreover,

the present

knowledge

of cellular solidification indicates that surface tension

plays

almost _no role in the cellular range. For the _same succinonitrile-acetone

alloys,

the

dependence

_on

capillarity

of the

deep

cellular-dendritic transition

only

_comes from the dendritic

regime [17]. Conversely,

if surface tension _{were a}

major contribution,

it is very much

likely

that the

Brody-Flemings

relation for the

tip supersaturation

would

strongly disagree

with the cellular

data,

which is not the _case.

In the outer

region (x*

_> A

), by taking

into account the fact that there is _no flux

crossing

the walls _at

y*

= ± A

/2,

_{one can} show that the solute field _can be written

Cext

-

Cm

₊ A exP

I- I

+

~( ^An

^{C°S ~}

T~*

x

In the small

P~ limit,

the last summation in the R.H.S. of

equation (5a) only

contains

transcendentally

small _terms.

Therefore,

the

y* dependence

_can there be

ignored

and the solute field taken as one-dimensional

C~~~ =

C~

₊ A exp

(- x*li~) (5b)

Equation (5b)

is very similar to the

profile

valid for _a

plane

interface

(Eq. (3))

_as the detailed

picture

^of the cellular front

disappears altogether.

But the _constant A _now differs from AC and has to be determined

by

the

asymptotic matching

with the inner

region.

In the inner

region,

dimensionless variables should be used

X =

x*/A

,

Y

=

y* IA (6a)

which _are

meaningful provided

that x* is less than the

primary spacing

A. In these

variables,

the

equation

of the unknown cell surface is

f

=

f(Y~

where Ye

[-1/2,1/2]. Along

f,

from the

simplified

version of the

Gibbs-Thompson equation (lc)

and the temperature

profile (Eq. (2a))

_we know that C

~ ⁼

C~/K sf

AC

Iv (6b)

where _s

=

P~.

^{Pelcb and Pumir}

[2]

have shown

that,

in the inner

region,

it is _more

appropriate

to consider the renormalized solute field

w =

C

(X, Y) C~/K

₊ SK AC

Iv (6c)

which

disappears

all

along

the interface

f.

Expressing equation (la)

in _terms of _{w one} obtains

Aw + s 3w

lax

=

s~

AC

/v (7)

Let _us write down the conditions

expressing

the transfer of solute _across the interface

3C/3N

~ ⁼ AC

~ ^{s cos} 8

(8)

It is much _more

physical

to express w in terms of the concentration C~ at the

tip

of the cell located at

f~. Indeed,

_we

immediately

obtain from

equation (6b)

C

~ ⁼

C~

s

(f f~)

AC

Iv (9a)

so that

w =

C

(X, Y) C~

+

s(X

f_{~) AC}

/v (9b)

and

~°'

=

sC~(I K) 1 ~~~

~~~$

^~~~ _cos 8

(9c)

3N f NO

VCt

where

Ct*

= KC

~/C~. (9d)

To solve this

problem,

_one should

anticipate

the order of the square bracket in the R.H.S.

of

equation (9c),

which

depends

_on the

magnitude

of I

vC~* )~

^'

Indeed,

to the first order in _s, it is this term which determines the normal balance of _w.

Mathematically,

several situations _a

priori

are conceivable. Which _{ones are}

physically meaningful?

The

original

Pelcb-Pumir limit

corresponds

to I

(vC~*)~

=

O

(I). However,

the

comparison

with the data obtained for succinonitrile-acetone

alloys,

for which K

=

0.

I,

shows that this mathemati- cal limit may, for small

P~,

lead to

negative

values for the relative width A

(Eq. (4)).

As

0 WA _w

I,

these

negative

A values _are

non-physical

and this has motivated _our present

attempt

to reconcile

theory

and

experiments.

For these succinonitrile-acetone

alloys,

the

negative

deviation is maximum for _v

=

4, P~

= 0.2

(see Figs. 8,

9 in Ref.

[9]). Figure

2 shows that

genuine

cellular solidification then verifies I

(vC~*)~~

=

O(s)

_so that the direct

analogy

^with

Saffman-Taylor fingers

is

actually

lost _as the normal balance of

w is _{zero up to}

the first order in _s.

10 loI

Cells 0.5 % .

~~

~

~°.. _. ^*

fl

^{~~"~ ~'~ ~} _{_}

'"

.. ~ .

.

° °

t~

a . o

° > o

~ o

U Cells 0.5 %

~ a

Cells 1.3

10 ^~

~

10

1~0

_V 10

i~-2

10 p ^lo

~

Fig.

2. Variation of a) the Pkdet number P~ with wand b) vC~* ^{I with} P~, in the cellular range and

the beginning of dendritic

growth.

By using

_a

phenomenological

law based _on cellular

experimental

data and

applying

_a method based _on

expansions

in power series of the small

parameter

_s, ^the

problem

will be solved in section 4 up ^to the second order in _s, where _a

relationship f(Y)

will be obtained.

Before,

the Pelcb-Pumir

analysis

will be modified

by considering

_a

slightly

different mathematical limit for the square bracket in

equation (9c),

which will

give

_some

interesting

results and show the

key

role of _a second Pbclet number

P~

based _on the shift of the cell

tip

from the

position

^{of the}

planar

solidification front.

Through

the

tip supersaturation

n

=

(Ct C«)/Ct(i K), (lo)

we will put an

emphasis

_{on C~}

which, assuming equilibrium solidification,

_can be deduced either from measurements of the

tip temperature

and radius

(transparent alloys)

_or from

measurements of the

tip

solute concentration in the solid

(metallic alloys).

3. Modified Pelck~Pumir

theory.

Pelcb and Pumir have assumed that I

(vC~*)~

^' is _a finite

quantity

in

equation (9c)

so that

only

^the term

proportional

to

f f~

is

neglected

in the square bracket. In the absence of

experimental

data for the solute concentration _{at a} cellular

tip,

this

assumption

_was natural when

seeking

for the

analogy

between cellular solidification and viscous

fingering. Since,

it

tumed out that this limit _case may

disagree

with

experiments.

It is obvious that the

analogy

between _a cell and _a

Saffman-Taylor finger

is

preserved by considering

the mathematical limit where the square bracket in

equation (9c)

reduces to I I

Iv

to the

dominating order,

which

implies

that the thermal

length

is somewhat greater ^{than the} solutal

length.

As _we have _a

small parameter s, we can write

«

w =

£ s'w,(X, Y). (ll)

<=1

Neglecting

terms in

El

_{we get} from

equation (9c) 3wj/3N

~ ⁼

C~(I K)(I

I

Iv)

_cos 8

(12a)

where _wj has to

satisfy

Awj

= 0

(12b)

and

3w j

la

Y

= 0 at Y

= ±

1/2. (12c)

As the

equations

_are similar to those derived

by

Pelcb and

Pumir,

it is _a trivial matter to solve the

problem.

The cellular

shape

is

given by

the well known formula obtained

by

Saffman and

Taylor

for viscous

fingering

in _a Hele-Shaw cell

[4]

f(Y)

=

~ ln ^{~ ~°~}

~"~~~~

(13)

AT 2

Using

the

matching

conditions

C~~~(f~)

₌ C~ ₌

C~

+ A

(14a)

i~ 3C~~~/3x*(

= A

(14b)

the relative width A is also

given by

^the ratio of the value of the normal

gradient

of wj for X- _{+ co} to its value at the

tip

^{of the} cell where 8

= 0

From

equation (Isa),

_{one gets}

~j~i~ ^K

I-1/~/~

i-K

i-i/v ^(lsb)

This result differs

by

_a factor C~* ^{in the} denominator from the

expression

obtained

by

Pelcb and

Pumir,

which reads

K I I

/C~*

~ _~

l K I I

/vC~*

^~~~~~

A very

satisfactory

agreements is obtained when

equation (lsb)

is confronted with the cellular

experiments (Eq. (4))

for K

=

0.I

(Fig. 3).

^It turns out that the

non-physical negative

cel~. 1.3 % ~l

, o Dendntes, Q

qp

~~~~~.5

%

~ ~~~~' ~

j Denddtes. 1.3% ^~

£~ ^*

~ x X.

~ + + . ,.

2p . #O ~O

U$ l ~

a- -a-O- -o- ~", ^~ _{, o} ~$~

- a O

.

+ + +~ _~ ~

j .O

, ^O *O O

O

m U o, m

~

~l$ntes 1$0f~r

~ ~ ~2

10°

°

v ^° °

v

Fig.

3.

Fig.

4.

Fig.

3. Variation with the level _v of the ratio of the value of the relative width A

given by

the modified Pelck-Pumir

theory (Eq.(lsb))

to the _one obtained from the geometrical definition (Eq.(4j).

Succinonitrile-acetone alloys (experimental data from Ref. [10]).

Fig.

4. Variation with the level _v of the

tip supersaturation

12 given

by

the modified Pelck-Pumir theory

(Eq. (16)) together

with the value

directly

obtained from the definition of12. Succinonitrile-

acetone

alloys (experimental

data from Ref.

[10]).

peak

obtained

by

Billia _et al,

using equation (lsc) (see Fig,

in Ref.

[9])

is _now

suppressed.

Surprisingly,

_no

significant

deviation from

unity

is

observed,

_even in the dendritic

region

for which

P~

»

I,

which is out of the scope of the

present analysis.

By reversing equation (Isa),

the

tip supersaturation

reads il

=

[I

₊

A(v

I

)] Iv (16)

which is

plotted

in

figure

4

together

with the

experimental

data. It should be noticed that, in the cellular range, the correction _to the

Brody-Flemings approach

is _now in the

right

direction

(compare

with

Fig. 6),

which leads _{to a}

good fit,

whereas the deviation is enhanced in the Hunt and Pelcb-Pumir models

(see Figs. 4,

7 in Ref.

[9]).

The

tip undercooling AT*,

which is used in reference

[9],

is linked to il

by

the relation AT*

=

Ci*

il.

Although

the difference between

expressions (lsb)

^and

(15c) giving

A is

formally minute,

it has _an

important repercussion. Indeed,

^from

equation (9d) giving Ci*,

the condition

(lc)

of

equilibrium

solidification and

equation (2a),

it results that

vci*

= v

Pi(I K). (17)

This last

equation

shows

that,

for

given

values of K and _v, the

magnitude

of

vci* directly depends

on the Pbclet number

Pi,

^I-e- on the dimensionless shift of the

tip

from the

position

of the

planar

solidification front.

Therefore,

it follows from

equation (17)

^that

assuming

^that

I

(vci*)~

reduces _to I

II

v

implies

that

Pi

^{is assumed} to be of order

B in the square bracket of

equation (9c).

Unfortunately,

it turns out that

P~ /P~

is much greater ^{than I for} the cells which _are

actually

selected in

experiments (Fig. 5). Therefore, despite

the

good

fit which is obtained for the cellular

points,

the _new mathematical limit _we have considered

again

does not

correspond

to real cellular solidification

(this

will be

analysed

in the next

section)

and the

origin

of the agreement

presently

is rather

puzzling.

^It

happens

in

physics

that _an

expression

derived under restrictive

assumptions

can still be used _{as an}

empirical relationship

outside the domain in which it is

rigorously proved

_so

that,

in _our

opinion,

it would be worth to determine the range

io~

+ +

~

~

. ~. +p

©~ _{a o} ^~ *

' ~~l

~ ^{. .}

"

fl~~

~ Cells, o-S %

cells- .3%

. Dendrites, o-S % Dendfites, 1.3 %

1

o°

i i o~

v

Fig.

5. Variation of

P~/P>

_{versus v} for succinonitrile-acetone

alloys (experimental

data from Ref.

[10]).

of

validity

of

equation (16) experimentally,

in order to be able to check whether _{we are} in such _a situation _or not.

Indeed,

two limit _{cases are}

supporting

this

suggestion I) figure

4 shows that

equation (16)

_can be used for cells when K

=

0,I,

I-e- when the variation of the

miscibility

_{gap is} rather

large,

and

it)

for the _case of _a constant

AC,

which also

corresponds

to the limit K

-

I,

the square bracket in the R-H-S- of

equation (8c)

^is

exactly

I I

Iv

_so that the

analogy

with viscous

fingering

is ensured. This

again

establishes

equation (16)

as the

right

relationship

for il

=

(C~ C~)/AC

when

P~

is small.

4. A realistic linlit for cellular solidification.

From

experiments [9],

it stems that in the cellular range I

(vC~*)~~

is

actually

_a small

quantity

of order

B

(see Fig. 2)

_so that

~ ⁼

( p;

^B'

(18)

NO

,=1

with

p

about I for the succinonitrile-acetone

alloys. Thus,

the R-H-S- of

equation (9c),

which is the normal balance of _w, is of order B~. It is _more convenient to consider the function

w = w +

C~/K,

which derives from

equation (6c)

w = C

(X, Y)

+

BXC~(I K)/Kv (19a)

which has to

satisfy

Aw _{+ B} %w

/%X

= B~

C~(I K) /Kv (19b)

and

%w

/%N

~ ⁼ %w

/%N

~.

(19c)

Let _us also

expand

_w and C in power series of _B

~

w ₌

~j

w, e'

(20a)

=o

~

C

=

£ i~,

e'.

(20b)

, =o

From

equation (19a),

_we deduce

wo =

i~o (21a)

wi =

i~,

₊ XC

~(l K) /Kv (21b)

w, =

i~,,

I m2

(21c)

4-1 TIP SUPERSATURATION. From

equation (19b),

_{wo is a} harmonic function.

Furthermore,

its normal derivative is _zero

along

^the wall of the cell and

along

the unknown surface

f,

since the R-H-S- of

equation (19c)

is of order e~. It results that

%wo/%X

^{is also} zero at

infinity

in the inner _zone due to the

asymptotic matching

with the _outer solution which does not

depend

_upon

y*.

These

boundary

conditions define _a classical Neumann

problem

whose

solution is

unique

and constant. From

equation (19a),

_we have

wo[

~ ⁼

Ci (22)

so that wo =

C~ everywhere

inside the inner

liquid

volume. As _a consequence, wi is

again

a

harmonic function that _can be written

wi =

fi(X, Y) C~(I K) /K. (23a)

From

equations (9a)

and

(19a), fi

verifies

fi1~

=

fi/vci* (23b)

Along

the walls the normal derivative of

fi

is still _zero and condition

(19c)

_now becomes

fi /%N

=

0

(23c)

so that _a

reasoning

sinfilar to that used to deternfinate wo leads to

fi

=

ii Iv Ci*. Thus,

_up to

B terms _we deduce from

(19a)

and

(23b)

that C

=

Ci eC~(I K)(X- fi)/Kv (24)

By

_an

asymptotic matching

between

equation (24)

and the _outer

solution,

_one

readily

^obtains

K(Ci C~ /C~ (I K)

=

I

Iv,

I-e- AT*

=

I

Iv (25a)

which leads to

il

=

II vC~* (25b)

The

comparison

of

equation (25b)

with the

experimental

data for succinonitrile-acetone cells

(Fig. 6)

shows _a fit which is

again acceptable

while

being slightly

_worse than for

equation (16), given by

the modified Pelcb-Punfir

theory (Fig. 4).

It is

likely that, despite

^{the fact that it} is obtained in _a mathematical limit which does not

exactly

coincide with the

physics

of

cells,

the latter

expression

is in better agreement because it introduces a correction which has the

right dependence

on the cellular

shape.

In the dendritic range, the

expression (25b)

of the

tip supersaturation

shows _a

systematic

deviation which increases with _{v, a} fact which has been

recognized

for _a

long

time.

Equations (25)

_were

already

found

by Brody

and

Flemings who,

for

deep cells,

built _a

phenomenological

model based _on

rough

ad hoc

approximations [12]

whereas it should be stressed that the

present

^{derivation is} self-consistent in the limits _we have considered. Our

1.0

« ©~~i~vC

~ Denirites, Q

0.8

°"~~~~' ~~~~

~ _

+ x

~~

f

_ ~

o °

0 4

+

I.

o)$o

oo

Cl

° #~.

+~

o.2

2 1o

~~~

~o ~

~

Fig.

6.

Comparison

of I

Iv

_C~* with the value of the

tip supersaturation given by

the definition of12

showing

the _range of

validity

of the

Brody-Flemings

relation. Succinonitrile-acetone

alloys (experimen-

tal data from Ref. II

0]).

demonstration also shows that the cell

shape f Y),

which

actually exists,

does _{not enter} into the

problem

before the second order in the Pkclet number

P~.

It should be noticed that the coefficient

pi,

which _appears in relation

(18),

is determined at this

step. Indeed,

from

equations(10)

and

(25b),

it follows that

vci*= I+K(v- I)

_so that

pi

=

K(v

I

/[P~(I

+

K(v

I

))],

which will enable _us to obtain _an

integral equation

for the cell

shape (see

next

section).

4.2 CELLULAR _SHAPE. This

mathematically

self-consistent

approach

shows that _we have

to go

beyond

_a first-order

analysis

in

P~

to get a relation which involves the

shape explicitly.

Let _us formulate the

B~-problem.

From

equations (18)

to

(21),

_we

get

Aw~

₌

C~(I K) /Kv (26a)

with the

boundary

conditions

%W~/%N

₌

c~(i K)jpi (i K)(f tjj

cos o

(26b)

and %w~

Ii

Y

= 0 at Y

= ±

1/2.

From relations

(9a), (19a), (22)

and

(24),

it is easy ^to show that

w~(

= 0.

(26c)

From the

asymptotic matching

between the inner and outer

solutions,

we get

lim w~

ix

~ ~ ⁼

C~ (

I

K) X~/2

Ku

(26d)

so

that,

instead of w~, it is

preferable

to

study

m~ = w~

C~(I K)(X- fi)~/2

Ku

(27a)

which is _a harmonic function

Am~

= 0

(27b)

satisfying

the

boundary

conditions

lim _w

~ x

~ ~ ⁼

0

(27c)

%m~ Ii

Y

= 0 _at Y

= ±

1/2 (27d)

~"~

~ ⁼

C~

^~~

^~~ lpi

₊

K(f ii)]

cos 0

(27e)

m~ =

(l K) C~(f fi)~/2

_Ku

_(27f~

The

advantagi

_of

considering

_m~ instead of w~ is that it

depends only

_on the distance from the

tip

_so ^that there is _{no more} reference _to the

arbitrary origin

^{of the} coordinate

system

we

are

using.

At this

point, ^different

_cases should be

distinguished according

to the

magnitude

of

Pi.

Let _us notice

that,

from

equation (27b) giving

the

tip supersaturation

il and the condition of

equilibrium solidification,

the

displacement Pi

of the

tip

is

given by

Pi

= v

(28)

This relation is verified

experimentally by

the succinonitrile-acetone

alloys

_{over a} wide range

(Fig- 7).

io~

( _.'

ioi

~

"

«

a

~~^o Cells, o-S%

ce%, 1.3 %

~ Dendntes. o.5 %

Dendrites 3 %

1gi

i i o~

v-1

Fig.

7.

Experimental

variation of P~ with the distance _v I from the onset of

morphological instability

of _a

planar

front. Data from reference [10].

P~

₌

O(B).

The

planar

front is embedded into the inner _zone

(Figs. 8a,b).

^It thus follows that

Ci*

=

I ₊ O

(B)

_so that

vC~*

I of order _B

implies

v I

=

O

(B). Therefore,

such cells _are restricted to _v ~2 _or thereabout. Two subcases should be differentiated

depending

_on the maximum

depth

of the intercellular

liquid

_I) finite

amplitude

cells

(Fig. 8a)

when this

depth

is also of order _B

(this actually happens just

above the onset of

morphological instability

of the

planar

front

II1, 16])

and

iii deep

cells

(Fig. 8b)

when the

depth

of the

liquid

grooves is at least of order I. The latter situation has _never been observed for either succinonitrile-acetone _or

pivalic

acid-ethanol

alloys [11]

but the

general

character of these observations is _up to _now

questionable.

Pi

₌ O

(I

_{or more.} The inner _zone is far ahead of the flat front. In this _case,

only deep

cells _are

possible (Fig. 8c). Indeed,

the

global

conservation of solute then

imposes

that the cells must be

separated by deep liquid

channels

plunging

well below the

position

of the

planar

solidification front.

Deep

cells

correspond

to a three-zone

problem,

which has been

extensively analysed

in the literature

(see

_e.g,

6, 7, [18-21]).

For finite

amplitude cells,

the tail

region,

which otherwise should be matched the

tip region,

_no

longer

exists _so that the

solid-liquid

interface is

fully

into the inner _zone.

Therefore,

the deternfination of the cellular

shape

then leads to a two-zone

Liquid Liquid Liquid

Planar Front

solid

solid

a. b. _c.

Fig.

8. Schematic

representation

of

Pi

= O

(B)

: case

a)

finite

amplitude

cells _: 2 _{zones, no}

liquid

tail, _case

b) deep

cells _: 3 _zones,

liquid

tail below the

planar

front

Pi

=

O(1)

Case

c) deep

cells 3 _zones,

liquid

tail above and below the

planar

front.

problem

which will _now be handled

by using

the Green method. To derive _an

integral equation

for _{m~, we} first need the

expression

of the solute field in the solid

phase.

From

equation (9a),

the solute concentration

C~

^{in the solid reads}

C~

₌

KCI[I e(I K)(f ii) /vci*] (29)

which is _a function of Y

only

and which contains _no second-order term in _B.

Therefore,

it is natural to take

m~ ~ =

0

(30)

as a continuation of the _m~ field inside the solid.

The

equations (27)

for _m~

together

with the continuation m~,~ define _a harmonic

problem which, by periodicity,

_can be restricted to _an infinite channel of width

unity

in inner coordinates

(Fig. 9).

The unknown cellular

shape

is obtained

by integrating

_over the whole

strip

the

quantity [m~(X, Y)

AG

(to,

Yo;

X,

Y

)

G

(fo,

_Yo

X,

^Y

Am~(X, Y)],

where

(to, Yo)

is _a

point

on the interface and G _a Green function for the free

Laplacian

with

periodic boundary

conditions. As _m~

~ is

identically

_zero, the surface

integral

leads to _a contour

integral

_on the

liquid

side.

Frim

_the

_boundary

_{conditions at the cellular walls and at}

infinity

and _a proper choice of the Green function

G,

this last

integral

reduces to _an interface

equation

"~~~°' ~°~

_"

~~°' ~°~

=

j~ _ds(m~

^{~~ G}

^{~"~ (31)}

2 ar ~ %N %N

where ds

=

[(dX)~

₊

_(dY~~]~'~

is the

length

element

along

the front and the

tip

is henceforth taken _as the

origin

^of the coordinates. For finite

amplitude cells,

_a

tangential

contact with the walls at

points

A and B

(see Fig. 9)

cannot be assumed _so that _one should introduce the

angle

a between the left and the

fight

tangents at any

point

_on the

solid-liquid interface,

whose

y

m

~=0 Am~=0

solid ^~

Liquid

N

A "1/2

Fig.

9. Sketch of the harmonic

problem

obtained from the

analysis

of the second order terms in

P>,

which leads to the determination of the cellular

shape.

value is _ao at

points

A and B and _ar elsewhere. A convenient Green function G is obtained

by adding

the harrnonic contribution

(X Xo)

to the function

G~

used

by

Karma

[19]

(X-Xo( (X-Xo)

_j

G

= In £

(32)

with

£

=1+exp(-4ar(X-Xo()

-2cos

[2ar(Y- Yo)]exp(-2ar(X-Xo(). (33)

The main

advantage

is that the first terra _on the R-H-S- of

equation (32)

is _{now zero} for all

X

Xo

_m 0. This _ensures that there is _no contribution to the

integral

from

infinity

whatever

the linfit of

%m~/%X

^for X- _co.

After substitution of

m~(

~ and

%m~/%N

~ in

equation (31) by equations (27e,

f~, a more

elaborate form of the

integral equation giving

the cell

shape

is obtained

j ~

Cf(fo(Yo), Yo)

1~°~~~

ar

=

~~~

dY(G ^(P

^{i +}

^{f )}

⁺

^{~~~~ ~ ~~° ~}

^~ ~

~~ ~~

_~ ^~

~~ ~~

_~~ ~~~~

where 0 is the step function.

It should be

emphazised

that the

solid-liquid

interface

f( Y)

is not

uniquely

deterrnined

by equation (34). Indeed,

for any value Yo, the distance

to Yo),

^which is measured in units of the

periodicity

A, is

actually

_a function of

pi,

which _was shown in section 4,I to

depend

on the

partition

coefficient

K,

the level _v of

morphological instability

and

Pi. Therefore,

for the finite

amplitude

cells under

consideration,

it is very much

likely that,

for

given

K and _v,

equation (34)

will

give

_{us a}

shape

at least _{over a} band of

P~,

which would _not be in contradistinction with the current

understanding

of the _array

growth

of cells. No

computation

of the cell

shape

from

equation (34)

will be

attempted

here.

Indeed,

there is first _a need of

complementary experimental

work in the finite cellular

regime

for _a sound

comparison

to be feasible. Since _no selection mechanism is included in the

integral equation, P~

^{will have} to be taken

directly

from the

experiments.

5. Discussion and conclusions.

The

present

^{work is} an

attempt

to

recognize

the fact

that,

for cellular

solidification,

the constitutional

supercooling

at the

tip,

of which

vci*

^{I is} a measure, may

actually

be _a small

quantity

_: this has been shown in well controlled

experiments

_on succinonitrile-acetone

alloys (K

= 0.I

)

_grown in _a Hele-Shaw

cell,

I-e- in the absence of convection in the

liquid phase.

Then,

_even in the small

Pi linfit,

it becomes _{a more} subtle task to handle the cellular

shape properly,

in _an

approach

based _{on an}

expansion

in the _powers of the small

parameter Pi (= B).

Beforehand,

we have identified _a situation in which the

analogy

between the

equations governing

viscous

fingering

and those

goveming

cellular solidification is

preserved.

This mathematical limit for the square bracket in

equation (9c) clearly

shows the

key

role of _a second Pbclet number

P~

^based on the shift of the

tip

from the

position

of the

planar

front.

This domain is limited to

Pi

=

O

(e ),

_v I

= O

(

I

)

and

P~

= O _B

). Despite

^the fact that this last condition is not fulfilled for

deep

cells and dendrites of succinonitrile-acetone

alloys,

the

relation which is obtained for the

tip supersaturation

^il

nicely

fits the cellular data.

For these

alloys,

real solidification

experiments

show that cells

correspond

to

large

values of

Pi

so that %w

/%N(

~ ⁼

O(B~). Consequently,

the

O(e) problem provides

_a

relationship

between the

tip

concentration

Ci*

and the level _v of

morphological instability

^which involves

no

shape

characteristic. This relation leads to il

=

I

/vC~*

^{which is the well known}

^Brody- Flemings expression,

which _was found to work for cells but the _reason of the fit _was not understood. This

point

is _now

clarified,

the

agreement

results from the

peculiarities

of the mathematical limit which

corresponds

to the cells which _are selected in directional solidification.

Nevertheless,

it should be

emphasized

that the selection mechanism has still to be elucidated. From relation

(10) giving

the

tip supersaturation il,

it follows that

vci*

=

I ₊

K(v

I

)

_so

that,

_as

vC~*

I

=

O

(e), equation (25b)

is restricted to

K(

_v I

= O

(B ). Thus,

_v

=

(I

₊

K) /K

can be taken _{as an} order of

magnitude

estimate of the upper limit of the domain _over which the

Flenfings expression

is

valid,

I,e. _over which

vci*-

I

=O(B).

The

O(B~) _problem,

which

gives

an

integral equation,

enables the obtention of the cellular

shape.

As both

expressions

_were found to fit the cellular data

satisfactorily people

interested in _an

empirical prediction

of the cell

shape

characteristics

might

be

tempted

to combine the modified Pelcb-Pumir

expression

for il with

Flemings'

this would allow _a

prediction

of the relative width A _{as a} function of _v

A

= I Kv

/[I

₊

K(v 1)] (35)

For about 2 _{w v w}

(I

+

K)/K,

this

expression

_compares well with the

experimental

value obtained for cells

by using equation (4) (Fig,10).

For the succinonitrile-acetone system, K= 0,I _so that this substitution is

acceptable

_{over a} rather

large

interval in _v. It is

nevertheless clear that the smaller the

partition

coefficient

K,

the better the utilisation of

equation (35)

for

practical

_purposes. In

metallurgy

_or in

designing

3D

microgravity experiments

in which the

growth velocity

is varied in order to determine the variation of

Pi

with _v for _a

given

A

~ i,

equation (35)

could for instance be

helpful

_{as a} criterion for the

a

priori

^{choice of the values of the control}

parameters.

In

conclusion,

we would like to stress that _we do _not claim here for

generality

_as the limit

cases we have considered _were retained from observations _{on a}

single

_system with

K

=

0.I. Even at low

K,

these observations

correspond

to the low

velocity

side

(low

v) of the

diagram

of

morphological instability

of the

planar front,

where the

dependence

of the cellular patterns on the

parameter A,

which contains

capillarity,

is weak.

Obviously,

_a

complete

JOURNAL DE PHYSIQUE I T I, M I,JANVIER 1991 7