Stability of a plate crystal growing in a channel

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Stability of a plate crystal growing in a channel

E. Brener, D. Temkin

To cite this version:

E. Brener, D. Temkin. Stability of a plate crystal growing in a channel. Journal de Physique I, EDP

Sciences, 1993, 3 (11), pp.2199-2205. �10.1051/jp1:1993111�. �jpa-00246863�

Classification

Physics

Abstracts

44.30 68.45 81.30F

Stability of

_a

plate crystal growing in

_a

channel

E.A.

Brener(I,*)

and D.E.

Temkin(~,**)

(~) Laboratoire de

Physique Statistique

de L'Ecole Normale

Sup6rieure(*** ),

24 _rue

Lhomond,

75231 Paris Cedex 05, France

(°**)

(~) Laboratoire de

Physique,

Ecole Normale

Sup6rieure

de

Lyon,

46 Al16e

d'Italie,

69364

Lyon

Cedex 07, France

(Received

5

April

1993, revised 5

July1993, accepted

15

July1993)

Abstract. The

stability

of the diffusion

growth

of _a

plate

in _a channel with respect to the

long

_wave fluctuations

along

the

edge

of the

plate

is considered. The

growth

is found to be

unstable for

wavelengths larger

than _a critical value which is of the order of the channel width.

The

destabilizing

factor has the usual diffusive nature. The main

stabilizing

factor is connected to the

dependence

of the

growing plate

width _on the curvature of its

edge.

1. Introduction.

Many experiments

on solidification _are carried out in thin films whose width is fixed

by

the

cover

glasses (see

for

example

_11,

2]).

In this _case the

growth

of two-dimensional structures is

possible:

the solid

phase

fills the whole channel width and the scale of the structure is much

larger

than the channel width. Another

possible

_case is when the

growing crystal

fills the channel width

only partly (see Fig. 1),

_[3]. The _occurrence of the two-dimensional structures in this _case is also

possible

due to the

instability

of the

simple

forms like disks _or

plates.

The

aim of this _paper is the

investigation

^{of the}

stability

of _a

growing plate

which fills the channel width

only partly.

Let us consider the isothermal solidification of the

binary alloy

controlled

by

the diffusion in the

liquid phase.

The diffusion in the solid is

neglected.

The concentration field is

governed by

the diffusion

equation

)

=

Dv~u (I)

(*>Pennanent

address: 14432 Chemogolvka Institute for Solid State Physics of the Academy of Science, Russia.

(**)

Permanent address: I-P- Bardin Institute for Ferrous Metals, 2 Baumanskaya Str.

9/23,

107005 Moscow, Russia.

(***> assoc14 _au C-N-R-S- et aux Universit4s Paris VI et Paris VII.

2200 JOURNAL DE

PHYSIQUE

I N°11

Fig-I-

Schematic representation of

plate growing

in _a channel.

where

U #

(CLE C)/(CLE CSE)

Here C is the

concentration; CLE

and

CSE

are the

liquidus

and solidus concentration for _a

given temperature;

D is the diffusion coefficient. The concentration in the

liquid

far _away from the

crystal

is

C~

and

u(y

_-

oJ)

=

A,

A

=

(CLE C~O)/(CLE CSE) (2)

At the channel walls there is _no flux and

3u/3z(z=+w/~

= 0

(3)

where W is the channel width. At the

solid-liquid

interface the _mass conservation law _says

un =

D) ₍₄₎

where _vz~ is the normal

velocity

of the interface and

3/3n

is the normal derivative to the interface.

(We

consider the one-sided model with

locally parallel liquidus

and solidus lines and with _a constant

miscibility

_gap

equal

to

(CLE CSE).

For details _see

[4]).

^{At the interface the}

concentration is written _as

u =

-doK (5)

K is the curvature of the interface

(K

< 0 if the interface is _convex towards the

liquid phase);

do

"

'fTm /

_[L(

ml (CLE CSE))

is the

capillary length;

_~iis the surface _energy

(assumed isotropic)

T~z~ is the

melting temperature

of the _pure

material;

L is the latent heat and _m is the

slope

of the

liquidus.

The

problem

^formulated above has _a

stationary

solution

describing

_a

plate

which is infinite in the x-direction and is

growing

in the

y-direction

with _constant

velocity _[5].

Far

from the

edge

the

plate

has the relative width

= A

(6)

By assumption,

the

profile

of the

plate

is

parametrised by

the

equation

y =

(I/s) lncos(7rz/A) (7)

with _an unknown

shape parameter

s

just

_as the

profile

of the

Saffman-Taylor finger

with

s =

7r/(1 A). (Here

and below the

lengths

_are reduced

by

the channel width

W) Then,

^the

shape parameter

s is found to be [5]

S = b~~ +

7r~/(1 A)~l~/~

_-p

(8)

where _p

=

uW/2D

is the Peclet

number,

and the

velocity

_u is determined

by

the selection condition

a +

doD/(UW~)

=

Ii (7r/(AS))~i~/~/(~IS~) (9)

where /~ is a numerical factor of order _one.

Equations (8)

and

(9) give

two branches of the

dependence

_{u on at} fixed channel width [5]. ^These solutions exist

only

for _>

1/2

and W

larger

than _some critical value

Wc/do

~ (A

-1/2)~~/~,

_{for (A}

_-1/2)

_{< 1.}

₍₁₀₎

On the lower branch of the solution the

velocity

_u decreases with

increasing

and _on the _upper branch _u increases with A. Note that

parameter

_a in

equation (9)

cannot be

large. Being

small

on the whole _upper branch and also _on the lower branch at close to

1/2

,

it increases up ^to the value of order _one at close to 1.

2. Formulation of _an effective

problem.

Pelce [6] has formulated the idea that for small Peclet

numbers,

when the diffusional

length

is much

larger

than the channel width W

,

the one-dimensional concentration field in front of the

plate

is

decoupled

from the

complicated

field in the

vicinity

of the

plate

which

changes

in the scale of W. Let _{us use} this idea to formulate the effective

equations

which describe the

growth

of the

plate taking

into account _a small distortion

along

the x-direction.

The concentration field in front of the

finger

has _a scale much

larger

than W and does not

depend

on z. In the frame

moving

with _{a constant}

velocity

_u the concentration field is

governed by

the

equation

~i I

"

~

~

~)

~

~Pi' ^~~~'~

"

°°~

_" ~ ~~~~

For this field the effective

boundary

condition _at the interface _y

=

f(x, t), corresponding

to the

edge

of the

plate,

is

~

(u

+

W3f/3t)

= au

/3y(y-~ (12)

D

Here _{we assume} that behind the

edge,

the

solidifying plate

has the relative width A.

Equation

(12)

is written for _a small deformation of the

edge profile f(x, t).

We do _not consider the details of the real

shape

of the interface which

changes

in the scale of W In the _same

approximation

u(y=~

= 0.

(13)

To

complete

the formulation of the

problem

_one has to _use the selection conditions

(8)

and

(9) which,

_as it is

shown,

_can be

presented

in the form

=

A(u

+

W3f/3t) a3~f/3x~ (14)

where

A(u)

^is

given by equations (8)

and

(9)

and _a is _a numerical factor In order to find the

parameter

a one should find the linear correction with

respect

to the curvature

3~f/3x~

to the

parameter

_s in

equation (7).

The solution of this

problem

is

presented

in the next section and

gives

_a

=

1/(167r)

for A close to

1/2.

The motivation for the

phenomenological equation

(14)

is due to the smallness

off

and due to the

long

_wave

approximation

which allows _us to

2202 JOURNAL DE

PHYSIQUE

I N°11

neglect

the

higher

derivatives with respect ^to x. We should note that

equation (14)

contains

an adiabatic _ansatz in the _{same sense as} in references

[6, 7].

Considering growth

^of _a

plate,

_{we can} divide the whole _space in the

y-direction

into three

regions:

the outer

region

in front of the

edge,

the

region

close to the

edge

with _a characteristic

(dimensionless)

size

equal

to

I,

^and the inner

region

behind the

edge.

The formulation of the

stability problem, given by equations (11-14), decouples

the first

region

^{from the} other two.

This _can be done when the Peclet number and the _wave number of the

perturbation

_are small:

p <

I,

^k < I. In this _{case we can} exclude the intermediate

edge region

from consideration and

replace

it

by

_some ^effective

boundary

conditions at the

boundary

_{y =}

f

between the outer and the inner

regions.

At this

boundary

the field _u should be continuous and

equal

to _some value connected with _{a mean} curvature K of the interface in the

edge region: ~1(y=~-o

"

u(y=~+o

=

-doK.

At the _same

time,

the diffusion fluxes should be discontinuous because solid

phase

with _a relative thickness is formed _at this

boundary: 3u/3y(y=~+o (I A)3u/3y(y=~-o

₌

A

§ (u

+

W3f/3t).

For the

growth

of _an

unperturbed plate,

the field _u in the inner

region

and the flux au

/3y(~-o equal

_zero. For the

perturbed plate

the concentration field in the inner

region

and the flux from this

region

are caused

by capillary

effect _{on a}

corrugate

interface and

by

the relaxation of the

corrugations.

Both

cappilary effects,

in balance

equation (12)

and in

equation (13),

_are considered _to be small and _are

neglected.

This allows us to

decouple

the

outer

region

from the inner _one. Both of these effects turn out to be small because of the

long

wavelength approximation

and because of the smallness of the parameter a. We will return to this

point

^later.

3. Calculation of

parameter

a.

We introduced the

phenomenological parameter

a into

equation (14).

Let us calculate it. Ac-

cording

to

equations (7-9)

the relative width of the

plate

is connected to the

shape parameter

s. In the _non

perturbed problem

this

parameter

was found _[5] from the

analysis

of the _asymp- totic behaviour for _z close to

A/2 (see Eq.(8))

Let _us

present

the

perturbed shape

of the

interface in the form

z(x, v,t)

=

A/2 exP(8v)

+

Bf(x,t) exPl(8

+

b8)vl, (15)

where

f(x,t)

=

to exp(uJt) cos(kx).

We look for the

perturbation

of the concentration field in the form

~~ ~ ~~~~~°~ ~ ~~ ~ ~~~~~

~"~~~~

_~"

(~ )~'

~~~~

which satisfies the conditions

3bu/3z(z=1/2

₌ 0 and bu

= 0 at _z

=

A/2.

The relation between A and

Bfo

_can be found from the conservation law

(4),

but it is not

important

for the

following

From the

stationary

diffusion

equation

in the

moving

frame

i7~bu

₊

2p3bu/3y

= 0 _we find

(s

₊

bs)~

^k~ ~r~

/(I

_A)~ +

2p(s

+

bs)

= 0.

(17

For the _non

perturbed _system (k

=

0,

^bs ₌

0)

this

equation gives

the result

(8).

For k _«

we

get

~2

~~

2(s

+

p)

^~~~

In order to find the correction to the

solvability

condition

(9)

_we have to know the

shape

the interface not

only asymptotically

but

everywhere

in the _{same sence as} ansatz

(7).

Let

solve

equation (15)

for _{y as} function of _z in the linear with respect to

f approximation

y =

ljlu(A/2 _z)

₊

Btexpjbs lu(A/2 z)/sjj. ₍₁₉₎

The basic idea _[5] of ansatz

(7)

for the

steady-state

solution is to

replace In(A/2 z) by lncos(7rz/A).

It

gives

_an exact

Saffman-Taylor

solution for _p

= 0. We _use the _same model

for

equation (19). Moreover,

not very close to the

asymptotic regime

_{we can}

expand

the

exponential

term in

equation (19)

for small bs

Is.

This is correct for

k2

_{« 1.}

Finally

_we

get

y =

f(x,t)

₊ ^~

^~~~ )ln cos(7rz/A) (20)

s

which for

f

= 0

gives

the initial

shape (7)

and for _z

= 0

gives

_{y =}

f (the

latter

gives

B

=

s).

Equation (20)

differs from

equation (7) by

_a factor _s

/( I+fbs)

instead of _s and

by

the translation of

f

in the

y-direction (the

latter is not

important

for the selection condition

(9)). Taking

into

account the mentioned

reparametrization

_{s, one} finds from

equation (8)

and

equation (9)

for

p « I and close to

1/2

> ₌

>(u)

+

tbs/4 (21)

where

A(u)

is a non

perturbed

selection condition.

Collecting equations (8, 14,

18 and

21)

_we

find for _{p «} I and (A

-1/2)

_« 1

a =

1/(167r) (22)

We note

again

that the calculation of the

parameter

a contains the model

step,

but the

impor-

taut

point

is that the parameter a is

just

a

positive

number.

4.

Stability analysis.

For the

steady-state growth

of the infinite

unperturbed plate (f

=

0)

we find from

equations (11-13)

u(y)

₌ A A

exp(-2py),

and

= A.

(23)

Equation (14) gives

the

velocity

_u. Pelce [6] examined the

stability

of this solution

(for

the lower

Saffman-Taylor branch)

with

respect

to the

homogeneous perturbation

in the

x-direction, f

₌

to

_exp

(cot).

Because of the translation invariance _one of the solutions is _uJ

= 0. Another _one which is

proportional

to the derivative

(-al /3u)

leads to the

instability

_on the lower branch where

3A/3u

_< 0. The _upper branch is stable because

3A/3u

_> 0. It is easy ^to understand these results

qualitatively

from

equation (12).

Parameter

plays

the role of the effective _source of the material. If the

velocity

increases and the

strength

of the _source decreases

(al /3u

<

0)

it leads to further increase of _the

speed

and therefore to

instability.

Let _us consider the

perturbation

in the form

t(x,t)

=

to exp(uJt) cos(kx) (24)

The solution of

(II

and

13)

is

u = A A

exp(-2py) 2pAfo exp(uJt

_QY)

cos(kx) (25)

where

q = p +

(p~

+ k~ ₊

2pfl)~/~,

fl

=

UJW/u. (26)

From

equation (12), taking

into account

equation (14)

in the form

~ = A ₊

u(3A/3u)Qf ab~f/3x~, ₍₂₇₎

2204 JOURNAL DE

PHYSIQUE

I N°11

n

,

Q

l~~~

_Q

~

0

Fig.2.

Schematic spectrum

~1(k).

The hatched

region corresponds

to the stable continuous _spec- trum.

we find

fl(A

+ u3A

/3u)

₌

QA ak~ 2pA. (28)

Equations (25), (26)

and

(28)

_are correct at

k2

₊

2pn

> o.

(29)

Under the

opposite

condition there is _a continuous stable

spectrum (hatched region

_on

Fig.

2).

Let _us look at the limit of small k

,

k « p « I. For al

/3u

> 0

fl =

k~A/(2pu3A/3u) (30)

For

3A/3u

< 0

~

~~~~~~~~~2

~

pu~~~3u(

~~~~

The result

(31)

_agrees ^with the result of _[7] for k

= 0. For p « k _« I for both _{cases one}

finds from

equations (26

and

28)

fl

=

(Ajkj ak~) /(A

₊ u3A

/3u) (32)

Note that the denominator in the

equation (32)

is

always positive

even at the lower branch

[6].

For

3A/3u

> 0 both

limiting

_cases

(30

and

32)

_can be described

by

the formula

fl

=

A

p2

+

k2

1/2

~ ^~

~~~

~~~~~ ~~~~~~l~~ ~/3u

~~~~

The

spectrum fl(k)

is

schematically presented

_on

figure

2.

Let _us look

formally

at the

system

^{where is}

independent

of the

velocity

and curvature

(I.e.

^al

/3u

=

0,a

=

0).

From

equation (32)

_we

get

^{the usual} diffusional

instability

fl

r~

(k(.

In this

system

^the

stabilizing

^factor is the surface

tension,

which should

give

^the well known correction to the

spectrum

^{of the order of}

a(k(k2.

In _our system for small _a and k the main

stabilizing

^{factor is}

provided by

the influence of the curvature of the

plate

_on the width of the

plate.

It

gives

the correction to the

spectrum ak2 (Eq.(32) ).

This effect of stabilization is caused

by

the fact that _on the _convex

part

^{of the front the}

plate

width is

bigger (a

>

0),

which

corresponds

to _a

bigger

^effective source of the material in

equation (12).

It leads _to the decrease of the

velocity

of this

part,

I-e- to the stabilization.

Finally,

we can make _an estimate of the

marginal

_wave number

(see Eq. (32))

jkoi

"

Ala (34)

which

corresponds

to _a dimensional

wavelength

I

=

W(27r/ko)

"

W/(8A)

which is of the order

of the channel width W. Note that all the

analysis

has been carried out for small k.

Therefore,

the result

(34)

is

only

an estimate of the order of

magnitude.

In

conclusion,

the

steady-state growth

of the

plate

^is

possible only

for channel widths

larger

than the critical value

Wc (see Eq. (lo)).

In this _case the

instability

leads _to the

development

of the three-dimensional structure with _a characteristic scale of order W. For the channel width W <

Wc

the solid

phase

fills the whole width of the channel. The characteristic scale of the

two-dimensional structure in this _case is of the order of

Wc [8].

Acknowledgments.

We _are

grateful

to P-Oswald and J.C.Geminard for fruitful discussions which stimulated this work. This work _was

partly supported by

_a

grant

of the D.R.E.T.

(France)

and

by

the American

Physical Society.

One of _us

(D.T.) acknowledges

financial

support

^{from the C-N-R-S-}

References

ill

Flesselles J-M., Simon A., Libchaber A-J-, Adv.

Phys.

40

(1991)

1- [2] Oswald P., Malthete J., Pelce P-, J-

Phys-

IFance 50

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2121.

[3] ^Janiaud B., Bouissou Ph., Perrin B.,

Tabeling

P., Phys. Rev. A 41

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7059.

[4]

Langer J-S-,

Rev- Mod.

Phys.

52

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[5] ^Brener E-A-, Geilikman M-B-, Temkin D-E-, JETP 67

(1988)

1002.

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[7] ^Karma

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6741- [8] Brener

E.,

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