The stability of cells in directional solidification

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The stability of cells in directional solidification

Wouter-Jan Rappel, E. Brener

To cite this version:

Wouter-Jan Rappel, E. Brener. The stability of cells in directional solidification. Journal de Physique

I, EDP Sciences, 1992, 2 (9), pp.1779-1790. �10.1051/jp1:1992244�. �jpa-00246659�

Classification Physics Abstracts

44.30 68.45 81.30F

The stability of cells in directional sofidilication

Wouter-Jan

Rappel

and E.A. Brener

(*)

Laboratoire de Physique Statistique

(*°),

24 _rue Lhomond, 75231 Paris Cedex 05, France

(Received

6 August 1991, revised 27 February 1992, accepted 15 May

1992)

Abetract. The stability of numerically obtained cells in the _one sided model of directional solidification is investigated, using _a numerical approach. In particular, the shapes of the cells

corresponding

to the Saffman-Taylor branch _are discussed. The possibility for _an oscillatory instability is investigated and the results _are compared with _a recent approximate analytical stability calculation. For large thermal length _we find _an instability which is oscillatory for

some parametervalues. The results _agree partially with the analytical stability calculation. The

possible limit cycle arising from this instability is discussed.

1. Introduction.

In directional

solidification,

_one

pulls

_a

binary alloy through

_an

externally imposed

fixed tern~

perature

gradient.

For

high enough pulling velocities,

the

planar interface, separating

the

liquid phase

from the solid

phase,

becomes unstable and forms cells with _a well defined

wavelength ill.

For

metals,

where the diffusion of

impurities

in the solid _can be

neglected,

the

resulting

cells _are

typically

_very

deep

_[2].

The

problem

of the determination of the

resulting

cells has received

considerably

attention.

Previous work includes numerical calculations

[3-6]

^and

approximate

theories [7, 8].

Contrary

to the _case of the

symmetric

model [9], ^where one assumes

equal dilfusivity

in the

liquid

and the

solid,

and _to the _case of

rapid

solidification

[10],

^{where the}

pulling velocity

becomes _so

large

that the

assumption

of local

thermodynamic equilibrium

at the interface is violated, _a linear

stability analysis

of these cells in the _one sided model has not been

performed.

In this _{paper, we} examine the linear

stability

of cells in the _one sided model. We focus

on the _case of small Peclet

numbers,

for which _{one can} make _an

analogy

between directional solidification and viscous

fingering

_[7]. For this _case, Karma and Pelcd

ii Ii

have

recently argued

that the cells should exhibit _an

oscillatory instability.

The

goal

of this _paper is to introduce the

stability

calculation and

apply

this to find _an

oscillatory instability.

This _paper is

organized

_as follows: in section 2, the basic

equations

of directional solidification

as well _as the numerical

techniques

used in this paper are reviewed. Section 3 describes the (*) Pennanent address _: Inst. for Solid State Physics, Chemogolovka, U.R.S.S.

(**) assoc16 _aux universit£s Paris VI, VII _et l'Ecole Nomlale Sup6lieure.

shapes arising

from _our numerics and _compare the results from the numerics with

analytical predictions.

In section

4,

the

stability

of the obtained cells is

investigated

and in section 5 _we discuss the limit

cycle

which results from the

instability

of

large

thermal

length.

2. Basic

equations

and numerical

technique.

The standard set of

equation

_are well described elsewhere

iii.

As is often

done,

_{we go} in the frame

moving

with the

pulling velocity

_vp and _measure

lengths

in units of half the

wavelength A/2

,

the concentration _c in units of _cn~

(the

concentration far away from the

interface),

and time in units of l~

/4D. Then,

for the _one sided

model,

_we have

v ~C +

2Pe)

=

Ii

(fi

V

cL)

_"

-(I k)(2pefi

§ +

vn)cL

c~ =

j

= -7~c

(1)

Here,

D is the diffusion constant in the

liquid,

k is the

partition coefficient, f

is the dimensionless thermal

length,

₇ is the dimensionless

capillary length

and _pe is the Peclet number: _pe

=

(~.

vn is the contribution from the interface other than the _constant

pulling velocity

_up ^which is

2pe

in _our

units).

Of _{course, we} have _vn

= 0 for _a

steady

_state.

To solve the above set of

equations

in _a numerical efficient _{way we} rewrite the

equations

as a

integrc-differential equation

and solve for the

(unknown

and

free) boundary

_[3] We will do this for both the

quasi-static approximation,

where _we

neglect

the time derivative _term in the

diffusion

equation

and for the full

equation.

In the

quasi-static approximation

we find

-(

+

7~t(S))

₌

/(@

+

7~t(S')) (h'

?

'G(S> S'))

^dS'

+2Pek /(@

+

7~t(S')) niG(S> S')

^dS'

+

/ _vn(S') (k(f

+

7~t(S')) (f

+

~t(S'))j

G(S>

S')

^dS'

(2)

where the Green's function G is the solution of

V

~G

₊

2pe )

=

-6~(x x') (3)

The

explicit

form of G _can be found in reference [3].

For the _non

quasi-static

_{case we go} back to the rest frame and _we find

Y~S>t)~~ 2P~t) ~~c(s,i)

₌ i

f

_ds,

_f~

_di,

~Y~S'>t')~- ^2P~t') ~c(s,,i,))

-«

A~ v

'G(~(s,1), ~'(/,i,);i,i,)

+

f

_ds~

_f~

_di'

~~~~"~'~j

_~~~~'~

~(s',i~))

-~

G(~(s,t), ~'(s',t');t,t')(2pen(

+ _vn

(s',t')) (4)

where G is _a solution of the full diffusion

equation

T7

~G ~

=

-6~(x x')6(t t') (5)

To find

steady

state solutions _we

parameterize

^the

(unknown) boundary

with N

points

^of

equal arclength.

Our

independent

variables _are then the normal

angles

at the

midpoints

of the

equal ardength

sections and the

position

of the

tip.

We solve the

integrc-difLerential equation

on the

points

of

equal arclength,

except at the

endpoints (~

₌ 0 and _~

=

l)

and

require

that the

slope

vanishes _at both the

tip

and the tail. In

general,

_we have chosen _a discretization rate of l10

points.

To examine the

stability

of the

resulting cells,

we

perturb

the

steady

state solution

by adding

the normal shift:

x(S)

₌

xo(S)

+

iio(S)6(S>t) (6)

where _{we assume} that

6(s, t)

₌

b(s)e~~

^and

b(s)

_« I. We substitute this

perturbation

in

(2)

and linearize in

b,

which

gives

_us

Ids, ^(7~o

+

(° )(6(s')41

v

'+ b(s)ho

_T7

)hl

T7 'G +

~(~'~

~(6"(s')

₊

c(b(s')))

hi

V 'G

f

+

?~to

+

~°) (-b'(S')il

'v 'G

+1C0b(S')fit

?

G)j

f -2pek /

_ds'

^fly

(71Co ⁺

( _)(6(S')fi~

_V ' +

6(~)~°

?

)~

+ ji

~(~'~ 7(b"(s')

₊

c(6(s')))

G'

~

f

+

7~o

+

(° (-b'(/)iiG

₊

iy~to6(s')G)j

+

~~~'~ 7(6"(S')

+

C(b(S')))

f

= w

/ dsJ(I k)

^~° + 7no

6(s')G ₍₇₎

f

The discretization of the

integrals gives

_{us a} set of

equation

of the form

M M

L _Aij6j

= w

L _B"j

b>

(8)

>=o j=o

where M is the number of discretization

points.

This

ordinary eigenvalue problem

for _{w can} be solved with _one of the standard numerical

packages.

For the _non

quasi-static approximation

_we

again

substitute the normal shift in the

integrc-

difLerential

equation (4).

This

gives

us:

Ids' ^(7~co

+ ~°

)(b(s')fi[

V ^' ₊

b(s)fro

V

)fi[

V

'G(w)

3~~,

+

7(b"(s')

₊

n(b(s')) _hi

V

'G(w) f

+

71Co

₊

() _(-6'(S')i~

'V

'G(W) +'~06(S')fi~

'~7

G(~°)j

-2pek /

_ds'

^fly

(7~co ^{+ ~°}

)(6(s')fi[

V

'+ 6(s)fro

^V

)G(w)

f

fi( J)

+

~v (f 7(b"(S') '~(6(S')))

G(~°)

+

7'~o

+

() (-6'(S')I$G

₊

ylCo6(S')G(W))j

+

~~l'~

7(b"(S')

+

C(b(S')))

" ~k°~

/

_d~'

((

+

~0) b(S')G(W) (9)

where _an

explicit

form of

G(w)

is

given

in reference [9].

Upon discretizing

the

integrals

_we get

an

equation

like

£

M

_Cjj(W)bj

# 0

(10)

j=0

where C is _a function of _w. and therefore in

general complex.

To solve the above

equation,

_we

have to vary w until C has _{a zero}

eigenvalue.

This is done with _a Newton's

algorithm

which will _converge to the

right

_w after _we have

guessed

_an initial value for _w.

The

technique

for

finding

^the

growth

rate of the

perturbation

is the

following.

We will first examine the

stability

of the pattern in the

quasi-static

limit. We will not expect to find the

oscillatory instability

in this _way but it is _a useful first estimate. We then calculate the

largest eigenvalue

for

a number of

growth

rates. This will

give

_{us an} indication where the

eigenvalue

becomes _zero. We then _{use a} Newton's solver _to

pin

down the value of the

growth

rate for which the determinant, or the

largest eigenvalue

is _zero. For further details _we refer to

[10].

3.

Steady

state cell

shapes.

In this

section,

_we will

investigate

the cell

shapes corresponding

to the

region

in parameter _space where _{one can compare} these cells with the

fingers

in the

Salfman-Taylor (ST) experiments

[7].

We have taken the diffusion constant to be D

= 2.5 _x 10~~ m~

Is

,

the

capillary length

to be

do _" I. _x 10~~ _m and the

wavelength

₌ 20. _x 10~~ _m. For the

partition

coefficient k _we have

taken two

values,

k ₌ 0.2

corresponding

to _a subcritical

bifurcation,

and k

=

0.8, corresponding

to _a

supercritical

bifurcation. The thermal

length

lT is used _as the control parameter, but is

always large compared

to the difLusion

length 2D/vp.

This is

important,

since this enables _us to check the

stability

_program in _a non-trivial _way

(see below).

Depending

_on the distance from the neutral _{curve, we} obtain either _a

deep

cell

(see Fig.

la)

or a shallow cell

(see Fig-16).

From

figure

la _{we see} that the cell is indeed _very

deep (notice

the difference in scales _on the

axis)

and has _a little bubble at its tail. This _seems to be characteristic of these

deep

cells and has been

reported

in earlier numerical work in both the one-sided and

symmetric

model [3, 5]. ^{This bubble is also} present ^{for the values of}

lT

close to the

marginal stability

_curve

(see Fig. lb).

We think that this bubble does not have

a

significant physical meaning,

in the one-sided

model,

for two _reasons: first of

all,

since the cells _are

propagating

into the

liquid,

the solid

just

above the bubble will melt first and then

resolidify,

_as the cell _{moves on.} This _{seems to} be rather

unphysical

in the one-sided

model,

where _one

neglects

the diffusion of

impurities

in the solid. A second argument ^{is the numerical}

result

plotted

in

figure

2 where _we have

plotted

the cell

shapes

for fixed

velocity

and

wavelength

but different discretization rates. We _see that the

tip

of the cells remain

essentially unchanged

and that the bubble propagates ^down as we increase the number of discretization

points.

It therefore _seems

likely

that the cell

prefers

to be infinite

deep

and that the bubble is _an artifact

-30300

-30400

1

-30500

~

-30600

-30700

O.O 2.O 4.O 6.0 8.O lO.O

x

~m) a)

-450

-soo

-550

-600

O.O 2.O 4.O 6.O 8.O lO.O

x

@m)

^~~

Fig-I. a)

A typical example of _a deep cell

(k

= 0.8, _{~ =} 144

pm/s,

IT

= 25300pm; ^{other material} parameters _as given in

text). b)

An example of _a cell close to the marginal stability _curve, for _a supercritical bifurcation

(k

= 0.2, ~ = 134

pm/s,

IT ₌ 400

pm).

of _{our zero}

slope

condition at the tail. This result is

typica1for

the one-sided

model;

in the

symmetric

model _we do observe _a similar bubble at the tail of the

cell,

but its

position

does not

change

_{as a} function of the discretization rate.

Before _we discuss the

stability

of the

cells,

let _us

verify

that _{we are} indeed in the ST

regime.

In

figure

3 _we have

plotted

the

position

of the

tip

_{vi vs.} the

pulling velocity

_vp. As _{we can}

see, this _curve is non-monotonic. For vp ^< 600

pm/s

the

position

of the

tip

increases

by

increasing velocity.

This part ^{of the} _{curve we} ^{will call the ST}

branch,

while the part on which the

tip position

decreases for for

increasing velocity

we will call the second branch

(see

also the discussion section

below).

In this _{paper we} will focus _on ST

branch,

for which _we expect

-25

[

₁₂₅

/~

~

/ 100 Points

-225

250 points

325

2 4 6 8

x

(~m)

Fig.2.

The cell shapes for fixed material parameters but different discretization _rates.

-2500

_

-3000

~

<

c

° -3500

aI

+

~ -4000

-4500

O 200 400 600 BOO IOOO

Velocity ~ptn/s)

Fig.3.

The position of the tip _{Vt as a} function of the pulling velocity _~p, for IT

" 1100 pm and

k

= 0.2. The circles represent the numerical data points, with the solid line _{as a} guide to the eye.

the

tip position

to increase for

increasing velocity ill].

We _can also

directly

_compare the

steady

state cells with _a ST

finger.

^{We know} that in the limit of

large

IT

compared

to the diffusion

length,

the cell should have _a width A of

ill]:

i

A= ~~

(II)

I-(I-k))

T

where

§

is the

position

of the

tip

of the cell relative to the

position

of the

planar

interface.

We have verified the above relation with _our numerics. We find that the width of the

finger

agrees within 5$lo of the value

given

above.

Furthermore,

the width remains

roughly

constant for thermal

lengths typically

_an order of

magnitude larger

than the difLusion

length.

We _can

therefore conclude that _{we are on} the ST branch. In other

words,

the results of Karma and Pelcd should

apply

to _our cells.

There _{are some} checks _{on our}

stability

calculation. First of

all,

_{we can} check that the

planar stability

calculation

gives

the

right

_answer. The other check _uses the fact that the nature of the bifurcation of the

planar

interface _can be either _{super- or} subcritical. For

metals,

with small

k,

the onset to cells from the

planar interface,

is

typically through

_a subcritical bifurcation.

Indeed,

_{as we} decrease the

wavelength

for fixed

velocity

for k

=

0.2,

_we encounter _a saddle- node bifurcation. From

elementary

bifurcation

theory,

_we know the

stability assignments

of the two

resulting branches;

the _upper branch should be

stable,

while the lower branch should be unstable. Our numerics also agree with this

stability picture, although

in the

vicinity

of the

saddle node _we need _more

points

_on the interface _to resolve the

stability.

The last check of the _program, is _a

highly

non-trivial _one. If _we take the thermal

length

to be much

bigger

than the diffusion

length,

we

approximate

the _case of _a dendrite in

a channel.

There,

_we have _a

single

cell

growing

in _{a narrow} channel, in the absence of _a temperature

gradient (I.e. lT

"

oo).

As has been

shown,

both

analytically

_[12] and

numerically [13],

^{in the}

case of _a dendrite in _a channel the ST branch is unstable. This

provides

_us with _an excellent check for _our

stability

_program: the cell should become unstable for _very

large lT.

We have

indeed found that this is the _{case, as} will be discussed in detail below.

4. The

oscillatory instability.

As mentioned

before,

Karma and Pelcd

ill]

have

recently argued

that in the limit of small Peclet

numbers,

the cells _can

undergo

_a

Hopf

bifurcation.

Using

_an

approximate

interface

formalism, they

showed that the

growth

rate _can have _a

positive

real part ^with a non-zero

imaginary

part, which would _{mean an}

oscillatory instability.

If _one would increase the thermal

length,

_one will _{encounter a} first critical thermal

length

for which the real part ^{of the}

growth

part ^becomes

positive.

If _one increases the thermal

length

even more, the

complex conjugate

root

pair

will

collapse

into two pure real

positive

roots at _a second critical thermal

length.

One of the roots is

going

to _a finite

value, corresponding

to the

instability

^{of the dendrite} in _a

channel,

and _one is

going

to zero,

corresponding

to the _zero mode which is present ^{in the} case

of _zero temperature

gradient.

To

verify

the result of Karma and

Pelcd,

Kessler and Levine

performed

_a linear

stability analysis

like _we have

presented

in this _paper, for the

symmetric

model [9]. ^That

is, they

obtained

steady

state solutions with the

integrc-difLerential method, perturbed

this solution with _a normal shift and solved the

resulting eigenvalue problem

for the

growth

rate of the

perturbation.

Their conclusion _was that there _{was no} evidence for _an

oscillatory instability.

These

contradictory

results have been controversial _ever since. There _{are a} few

possibilities

for the

disagreement,

apart from the fact that _one of the two groups

might

have made _a mistake.

One

possible

difference between the two

approaches

is that Karma and Pelcd

analyzed

the _one sided model while Kessler and Levine used the

symmetric

model.

However,

_we will argue below that _we don't believe that there will be _an essential difference between the models.

Nevertheless,

_we have

performed

all _our calculations in the _one sided model.

For the parameter ^{values of} Kessler and

Levine,

_we have not been able to find _an

instability.

All the cells _are

linearly stable,

consistent with the results of Kessler and Levine.

However,

as we have

pointed

out

above,

for

large

IT ^the

problem

of directional solidification

approaches

JOURNAL DE PHYSIOUEI T 2, N'9. SEPTEMBER 1992 64

O.3

O.2

o-1 5

j

^o-o

-o.i

-O.2

SOD 1000 1500 2000 2500

~~wn)

FigA.

The real and the imaginary part of the eigenvalue _{as a} function of the thermal length IT Here, k

= 0.2 and _pe

= 0.268.

that of the dendrite in _a channel.

There,

_as has been shown before

[12],

the cell

corresponding

to the ST branch is unstable. It is therefore

logical

to extend the range of parameters to

larger

IT. Note that the ST branch in _a channel is also unstable in the _two sided model, ^and we

therefore expect to see the _same

qualitative

behavior in that _case [14].

We have

performed

the calculation for two different values of the

partition

coefficient: k

= 0.2

and k

= 0.8. For _a fixed Peclet number _{pe we} have varied the thermal

length

IT ^{and have} examined the

stability

of the

resulting

cells.

For all Peclet numbers _we found that if _we increased the thermal

length

above _a critical

value,

_say

lT,c

^{the cells become unstable. For}

large

values of lT

(I.e.

lT _»

lD),

this

instability

is

always

_pure real. This is in

agreement

^{with the}

stability

calculation for _a dendrite in _a

channel,

_as well _as Karma and Pelcd.

The results of the

eigenvalues

for k

= 0.2 and _{pe =} 0.268 _are

presented

in

figure

4.

Here,

the solid line

correspond

to the real part of the

largest eigenvalue,

and the dotted line

correspond

to the

imaginary

part ^{of this}

eigenvalue.

We _see that the cell becomes unstable with for lT _~' l125 _~tm, and that the

instability

has _{a non-zero}

imaginary

part. In other

words,

_we do

have _an

oscillatory instability

here.

Our numerics

show, however,

that _an

oscillatory instability

_occurs

only

for

larger

Peclet number

(I.e,

pe >

0.I).

For small Peclet number

we have

never found _an

oscillatory instability.

There,

the

eigenvalue

becomes pure real _so close the

lT,c

^that we are not able to observe _a

non-zero

imaginary

part for

positive

real part. ^{Note however that the}

imaginary

part of the

growth

rate at the critical thermal

length

in Karma and Pelc6's

theory

_goes to zero as pe goes

to _zero. It is

possible

that _our numerics _cannot handle such _a small

imaginary

part.

In

figures

5a and 5b _we have summarized _our results for the two values of k. The solid line

correspond

to the

predicted

value of

lT,c.

We _see that of the _{two curves} the _one for k

= 0.8

agrees better with the

prediction

of Karma and Pelc4. For this value of k, our numerics agree better with the solid _curve for small values of _pe. This is

expected

since their

analysis

is valid for small Peclet numbers. For

higher

values of _pe the

analytical _lT,c

becomes constant, while the

numerically

obtained value increases

again.

This _can be

explained by noting

that

3500

2500

soo

SOD

~~

a)

isoo

iooo

soo

O

0.OO O.05 O.lO O,15 O.20

pe ~~

Fig.5. a)

Our numerical results for lT,c

(circles)

and the prediction Irom the analytical theory

(solid line).

Here k

= 0.8,

b)

Same _as

figure

5a but _now for k

= 0.2.

for

increasing

Peclet number _we will _move towards the stable second

branch,

which will be discussed in _more detail in the

following

discussion

(see

also

Fig. 6).

As _{one moves} toward the transition

point

between the two branches

lT,c

- oo since the second branch is stable.

The

analysis

of Karma and Pelcd is of _course

only

valid for the ST branch.

Also, according

to Karma and

Pelcd,

the thermal

length

for which the

instability

becomes _pure real is much

larger

than

lT,c.

Our numerics

however,

find _no second critical thermal

length,

_{or one} which is much closer to

lT,c.

~t

limit C

ST Second

branch branch

~ V

P

Fig.6.

A schematic drawing of the limit cycle arising Irom the instabfiity for large thermal length.

The solid line corresponds _to the steady state solution branch while the dashed _curve corresponds to the limit cycle. The ST branch

(or

at least the part on which point A is

located)

is unstable while the second branch,

corresponding

to the decreasing part of the non-monotonic _curve is stable

(compare Fig.3)

The limit cycle will consist ofthe trajectory ^D - E

- C _- D. For

a detailed explanation _see the text.

5. Discussion.

To summarize _our numerical

results,

_{we can say} that in the ST limit the cells will become unstable for

large

IT- If _{we are} in the unstable

region

the

instability might

_or

might

not be

oscillatory

of nature,

depending

_on the

parametervalues

of the system. ^The agreement ^with the

theory

of Karma and Pelcd is at most

partially.

Let _{us now}

discuss,

^what _we think this

instability

will lead to. We

emphasize

that the scenario

we are

giving

here is

only

valid for the _case when

wavelength

selection is not available. In other

words,

_we envision _a situation where the cell cannot

adjust

its

wavelength. Experimentally,

this would

correspond

to directional solidification in _a channel. If the

wavelength

_can be selected it is _very well

possible

that the

instability

described in this _{paper can} be avoided

altogether.

The limit

cycle

which will arise is sketched in

figure

6. We _{assume a}

large

but not infinite thermal

length (I.e.

the temperature

gradient

is small but

non-zero).

Here _we have

sketched,

as a solid

line,

the two

existing steady

state

branches,

_as in

figure

3. Note that the

dependence

of vi on up ^is non

monotonic,

and

corresponds,

in the limit of

large lT,

to the _one discussed in reference

[14].

To avoid _any

possible

confusion _{we note} that in the

problem

of _a dendrite in _a channel _one

typically plots

the

velocity

_{as a} function of the

undercooling, resulting

ⁱⁿ a double valued _curve.

Here,

_we

plot

the

tip position

_{as a} function of the

velocity resulting

in _{a non-} monotonic function. In reference

[14],

^it was shown that the ST branch

(where

_yt increases for

increasing

_{vp, compare}

Fig. 3)

is unstable while the second branch is stable

(we

have verified that the cells

corresponding

to the second branch of

Fig.3

_are indeed

stable). Thus,

the

steady

state solution _we start with

(point

A in

Fig. 6)

is unstable. If _we start from this unstable solution _on the ST

branch,

it is

possible

to

jump

to

point

B _on the second branch. The time for this

jump

is

quite

small:

2jump

-~

1/Re

_w where _w is the

growth

rate of the

perturbation.

At B, the

tip position

of the cell will be

essentially unchanged

but the

tip velocity

_{vtip is}

larger

than the

pulling velocity

_vp.

Next,

the

tip

of the cell will advance in the channel towards _a hotter

region,

while the

velocity

of the

tip

^decreases

(B

_-

C,~following

the dashed

curve).

At

point C,

the

velocity

of the

tip

is

equal

to the

pulling velocity:

_{vtip " vp.} There is _no stable

steady

state solution

however,

so that the

velocity

continues to decrease and becomes smaller than the

pulling velocity.

As

a result, the

tip position

decreases _as well and the cell will follow the

trajectory

indicated

by

the _arrow in

figure

6 _on the dashed _curve

(C

_-

D).

The time for this _process is of the order

of

lT/vtip,

which is

quite large.

Once _vi has decreased _so much that the

tip position

is smaller than the

largest possible

_{vi on} the

branches,

_we have

again

the

possibility

of _a

jump

to the second branch

(D

_-

E).

This will

finally give

_{us a}

limiting cycle

D _- E _- C _-

D,

_as sketched in

figure

6

by

the dashed _curve. The _average

velocity

of the cell

during

this limit

cycle

should be the

pulling velocity

_vp and the

period

of the

cycle

will of the order of

lT/@.

The selection of the

points

D and E _on the solution branches _are not known at the _moment

(one

should solve the full non-linear

problem),

but

perhaps they

will be selected

by

the condition ₌ vp. Note

that,

since _yt is related _to

it,

the width of the cell will

(slowly)

oscillate.

Let _us conclude with _a final word _on

sidebranching

in dendrites. Karma and Pelc6 have _ar-

gued

that their

oscillatory instability

could be _a mechanism for the side

branching

in dendrites.

This idea of _a linear

Hopf instability

from cells _to dendrites which leads to

sidebranching

does not agree,

however,

with the

general accepted phenomena

of free _space dendrites. First of

all,

_one has found that the free _space needle is

linearly

stable. Sidebranch

generation

_can be

explained

if _one introduces _a noise-driven mechanism.

Secondly,

unlike the _case of cellular structures, ^dendrites

depend

_very

strongly

_on the

anisotropy

of the

crystal.

We therefore do not believe that the

oscillatory instability

could lead to

sidebranching.

In

conclusion,

_we have

presented

^{numerical results} _on the

stability

of cells in the one-sided model of directional solidification. We show that _{our program}

produces

_an

instability

for

large

values of the thermal

length.

We also show

that,

for _some values of the parameter, we find _an

oscillatory instability.

Acknowledgements.

One of _us

(E.A.B.)

has been

supported by

the Minist4re de la Recherche et de la

Technologie.

Part of the work of E.A.B. _was done

during

_a visit at IFF-KFA Jiilich. We would like _to

acknowledge

useful discussions with H. Levine.

References

ill

For _a review, _{see e.g.}

Langer

J., Rev. Mod. Phys. 52

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1987);

Kessler D-A-, Koplik J, and Levine H., Adv. Phys. 37

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[3] Kessler D-A- and Levine H., Phys. Rev. A 39

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