The oscillatory instability in rapid solidification

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The oscillatory instability in rapid solidification

Herbert Levine, Wouter-Jan Rappel

To cite this version:

Herbert Levine, Wouter-Jan Rappel. The oscillatory instability in rapid solidification. Journal de

Physique I, EDP Sciences, 1991, 1 (9), pp.1291-1302. �10.1051/jp1:1991207�. �jpa-00246412�

Classification Physics Abstracts

44.30 68.45 81.30F

The oscillatory instability in rapid solidification

Herbert Levine and Wouter-Jan

Rappel

Department

of

Physics,

Institute for Nonlinear Science,

University

of Califomia, San

Diego,

La Jolla, CA 92093, U-S-A-

(Received 4 January 1991, revbed 25

April

1991,

accepted

24

May 1991)

Rksumk. Les

bquations

habituelles de la solidification directionnelle sont modifibes dans le _cas de vitesses de

tirage

blevdes

(solidification rapide).

A

partir

de _ces

d(uations,

_nous ddcrivons _une mdthode

numdrique

_pour

analyser

la stabilitd des structures cellulaires obtenues

numdriquement.

Ces cellules

prdsentent

une instabilitd oscillatoire, ce

qui prddit

l'observation de

rdgions

_en forme de bandes dans

l'expbrience.

La

largeur

de _ces bandes, _{que nous avons} estimde, est I peu

prds

_en

accord _avec celle observde

expdrimentalement.

Abstract. We modify the usual directional solidification

equations

for the _case of

large pulling

velocities

(rapid solidification).

Using these

equations

_we describe

a numerical method _to analyze the

stability

for

numerically

obtained cellular structures. We find cells which _are

oscillatory

unstable which will lead to the observation of banded

regions

in the

experiment.

The width of these bands _are estimated and found to agree

roughly

with the

experimentally

observed _ones.

I. Inboduction.

It is well known that the process of

rapid

solidification _can lead _to various

interesting

microstructures of the

solidifying

materials. One of the

techniques

consists of the

scanning

of

a

high intensity

laser _or electron beam _{over a} surface

of,

for

example,

_a metal

alloy.

The laser will melt the surface up ^to a certain

depth

and after the

passing

of the laser the melt will _re-

solidify.

The

speed

of the solid-melt front

depends

_on the distance from the surface and is maximal at the surface

[I].

One _can model this type ^of solidification with the well-known

equations

of directional solidification

[2]. Here,

one

places

_a bath of _a two-component

alloy

in _an

externally

fixed

temperature gradient

and _moves the bath with _a certain

pulling velocity

towards the

(fixed)

cold contact.

Typically,

the

pulling

velocities _are of the order of micrometers per second. If

one assumes that the laser will create a fixed temperature

gradient

_we will have _a directional

solidification

experiment

with

pulling

velocities close to the

scanning velocity

of the laser. Of course, this allows for very

high pulling speeds.

The observed microstructures after the melt has solidified

depend

_on the

velocity

of the laser and the

depth

of the melt.

Typically,

one observes cellular and dendritic structures for small and moderate velocities

[I].

For

higher velocities,

_one observes banded

regions

and for

9

even

higher

velocities _one has

microsegregation

free structures. These banded

regions

_are the

ones we are

investigating

in this paper.

The

high

velocities will alter the usual set of

equations

for directional solidification since _we

can no

longer

_assume local

thermodynamical equilibrium

at the interface. It _can be shown that _a linear

stability analysis

of the standard

equations

of directional solidification modified for

high pulling

velocities exhibit _a

Hopf bifurcation,

I-e- _a bifurcation where the

eigenvalue

corresponding

to the

growth

rate of the

perturbation

has _{a non-zero}

imaginary

_part when the

real part crosses zero. This will result in the

growth

of _an

oscillating perturbation

of the

interface and will

give

rise to banded

regions.

In this _{paper, we} will

present

a numerical

approach

to the

problem

of

rapid

solidification.

We solve

numerically

the set of

equations

for directional solidification for

steady-state growth,

I-e- _we

neglect

the time

dependent

concentration field.

Afterwards,

_{we use} the obtained

steady

state cellular cells _as the

input

for exact

stability

calculations.

Although

this

approach

has been used for usual directional

solidification,

it has _never been

applied

to

rapid

solidification. Our main result is that the cellular solutions _can go unstable to a

Hopf bifurcation, presumably giving

rise to banded cellular structures.

Kelly

and

Ungar [3]

^have also found _an

oscillatory instability,

however

they

have used _a different model for the interface

temperature

^{and the} modification of the

partition

coefficient

(see

Sect.

3)

for

high pulling velocity.

This paper is

organized

_as follows. In section 2 _we review the basic

equations

for directional solidification and

modify

these

equations

for the _case of

rapid

solidification. In section

3,

we

review the linear

stability analysis

and

plot

the

regions

of

oscillatory

and

non-oscillatory

instabilities for the system ^{of Sisb. We} describe how _{we can} obtain numerical

steady

state

equations

in section 4 and how _{we can}

analyze

their

stability. Finally,

in section 5 _we will present ^{and discuss} our results.

2. Basic

equations

and their modification.

The basic

equations

for directional solidification _are well known and _are described in _many papers. In this paper we will

only

consider the _one sided

model,

where _we

neglect

the solute diffusion in the solid. In the frame

moving

with the

pulling velocity

_uo we have for the

concentration field

c

~ 3c 3c

~~~~~°fi~&

uofi.y+u~

(fi Vc)~

=

(l k)

_c~

D

Here,

D is the diffusion constant in the

liquid,

k is the

partition coefficient, i~

=

m~/G

is _a

thermal

length (m~

is the

slope

of the

liquidus

and

T~

is the

melting temperature

of the pure

melt), do

=

«T~/Lm~

is _a

capillary length («

is the surface free energy and L is the latent

heat)

and _K is the curvature. The total interface

velocity

_{is uo}fi

y

_{+ u~} where fi is the normal to the interface. Of _{course, we} have u~ =

0 for _a

steady

state.

In

deriving

the above

equations

_we have assumed local

thermodynamic equilibrium

at the interface. This

assumption

is correct for small

pulling

velocities but fails for

large

velocities. In that _{case we} have to

modify

the above

equations.

As _we will _see, these modifications

give

rise to _a different linear

stability

behavior and in

particular

there will _now exist a

Hopf

bifurcation for

large

velocities.

Perhaps

the most

important change

within the

equations

is that the

partition

coefficient k has to be modified for

high

velocities. It will _now

depend

_on the

velocity

and should

approach unity

in the limit of

high speeds.

In this limit the interface _{moves so}

quickly

that the solute _can not diffuse out of the solid and will be

trapped

inside the

solid,

hence the _name

« solute

trapping».

In

addition,

_{one can} introduce the mechanism of attachment kinetics which will describe _at which rate the molecules will attach to the

crystal.

There _{are a} number of models for the

velocity dependence

of k. We believe that the actual functional

dependence

of k _on v will not matter

greatly

for _{our purpose,}

finding

the

oscillatory instability.

In

fact,

Coriell and Sekerka

[4]

^{have shown that for} an

oscillatory instability

_{a non-}

zero derivative

3k/3u

is sufficient. We have chosen _a model

proposed by Boettinger

_et al.

[5, 6]. They

introduce _a

parameter flo

which _measures the

departure

of k from its

equilibrium

(u

=

0 value and _a parameter

Vo

^which

mimici

_the effect of kinetic attachment. In this paper

we will

only

consider the _case

Vo

₌ tx~, which _means that the atoms attach

infinitely

fast and there is _no kinetic

undercooling, although

_an extension for different values of

Vo

^is

straightforward.

For the functional

dependence

of k

they

_propose

k~+flou

~~~~~

l+flou

^~~~

where

k~

is the

equilibrium

value for _u

= 0.

Here,

_u is the total normal

velocity

of the interface. As _{we can see,} k

approaches unity

_as u increases.

Boettinger

et al. showed that for _a flat interface

moving

with

velocity

_uo the

temperature

at the interface will be

given by

where

m(vo)

= mE

1 ikE k(I

In

(k/kE))i (4)

Here, Vo

measures the effect of attachment kinetics. As mentioned

before,

_we will choose Vo = tx~ in the remainder of this paper.

Equation (4)

describes the

change

in the

liquidus slope

due to

non-equilibrium segregation.

As in

[7]

we suppose that the temperature at the interface is

given by

the above

expression

for the flat interface

plus

_a Gibbs-Thomson term _:

The above modifications will

change

the standard

equations

for the directional solidifi- cation. As in

[8]

_{we go} into the frame

moving

with the

pulling velocity

_uo and _measure

length

in units of A

/2 (the wavelength

of the

cell),

concentration in units of c~ and time in units of

A~/4D. Then,

for the _one sided

model,

_we have

~ ~ ^{3c 3c}

~~~

~~fi"A

(fi Vc)~

=

(l k(u))(2 pelf y

₊

u~)

_c~

~~

k()) f~ ^"*

~ ~~~

where _y *

=

2

«Tp/Lm(u)

_{c~ A, pe}

=

~~

,

the Peclet number and

f*

=

2 _m

(u)

_c~

Tp/GA.

4 D In _our rescaled units the above

equations

read

k~

+

flu

~~~~

l ₊

flu

^~~~

where

p

=

po

2

Dj> (8)

3. Linear

perturbation theory.

In this section _we will review linear

perturbation theory

for the _set of

equations (6)

as

described

by

Merchant and Davis

[7]

and

apply

the results to _an actual system. ^As our system

we will choose the Sisb system described in

[3].

We will _see that for this system we will _not have _an

oscillatory instability

if _we do not

modify equations (I),

I-e- when _we choose

po

=

0.

It is easy ^to see that

equations (6)

^have a solution

corresponding

to _a flat interface

~ ~

l

~-2pejy-yoj

°

k

~~~

Yo "

~

k where

k~

₊ 2

pep

~

l ₊ 2

pep

^~~~~

We

perturb

the flat solution

c = co + eci e~~~ e~ ^QY e~~

(I I)

y = yo + eyi e~~~e~~

(12)

where

Q

= Pe +

~

(13)

and em I.

Substituting

these

expansions

in the above

equations

for

k(u)

and

m(u)

and

linearizing

for

e we get

k(u)

₌

k

_{+ eA}

j wyi + o

(e2)

m

(U)

= ni +

8A2

_{WYI +} O

(8~) ₍₁₄₎

where

p (I k~) Ai

=

2Pe(1

₊

fl

~

Ai

In

(k/k~)

~~

ni

(k~ )

^~~~~

ni=ni~(1-_~[~ (k~-k(i-in(k/k~)))j.

Next,

_we substitute

equations (14)

in

equations (6)

and _we find

k~°~~l~i~~'~)~~~~j ^~~

+

[~-2pe ~~-2pe ^~~ w-2pe(1-k)( _+yq~) ^{+4pe~~ ~~=0} (16)

k

k k f k

where

2

«T~

~'

LmC~

A

(17)

2

nic~ TM f

"

~>

To find the neutral

stability

_{curve we} set _{w =} 0. This will

give

_{us a} closed _curve in

parameter (A

vs. u

)

_space inside which the flat interface is unstable with _{a pure} real

eigenvalue.

However,

it is also

possible

that _an

w with _non-zero

imaginary part

satisfies

equations (16).

If _we write

Q

as

Q

= pe ⁺

R,

_we get a third order

equation

for R from which _{we can} solve for

w. For each

wavelength

we get three solutions for _w which the determine the nature of the

instability.

If the real part of

w crosses zero and its

imaginary

_part is _{non-zero we} will have _an

oscillatory instability

and _we will encounter _a

Hopf

bifurcation. As _{we can see, a non-zero}

Aj

or

equivalently

_{an non-zero}

3k/3u, gives

_us a cubic

equation

and for realistic parameter values _we will find a

Hopf

bifurcation.

As _an

example,

_we have chosen the Sisb system as described in

[3].

The relevant material parameters are

given

in table I. We have

plotted

the boundaries of the

stability regions

for three different values of

flo

ⁱⁿ

figures

1, 2 and 3. In

figure

I

fl

o ⁼ 0 and _{we see} that there is _no

oscillatory instability.

The

instability

in the unstable

region

is the Mullins-Sekereka

instability

and _we expect to find cells

and/or

dendrites inside this

region.

For

flo

= I

x10~?

_and

flo

₌ I _x

10~~ _however,

we do find _an

oscillatory instability

for

high

velocities

(see Figs.

2 and

3).

There is _a fundamental difference between the

figures

2 and 3. In

figure

²

flo

₌ I _x 10~

?)

maximum of the neutral

stability

curved lies below the minimum of the

oscillatory

unstable

region. Therefore,

if _one increases the

velocity,

_one

expects

^{that the cellular} structures which form inside the neutral

stability

_curve bifurcate back to a

planar

interface. This

planar interface,

in turn, will

undergo

_a

Hopf

bifurcation if _one increases the

velocity

_{even more.}

Since the

planar

interface will go unstable for _a

perturbation

with

= tx~, the

planar

interface will oscillate in the direction of the

growth velocity.

The concentration left behind in the solid

Table I. Parameter values

of

the Sisb system ^used in the numerical calculation.

do

=

7.72 _x lo ^~~ _m G

= 1.33 _x

166 K/m T~

=

427 K mt =

452

Klat9b

c~ = I _x

10~~

atfb D

=

1.5 _x lo ^~~

m~/s

k

=

0.023

~=0

lo ~

~°

stable

j~

~

(

io 6

~/

unstable

>

io 5

lo 4

lo .I lo ° lo I lo 2

~w

(~llll)

Fig. I. The

regions

of

instability

for po 0.

J=lx10~~

lo 8

~° ~

oscillatory ^unstable

~

E

~ lo 6

~

>

lo 5

io 4

10 .1 10 ° 10 10 2 10 3 10 4

~w

(~llll)

Fig. 2. The

regions

of instability for po ₌ ^I x 10~~

depends

_on the

velocity (k

is _a function of

u)

and _we expect to see banded

regions.

In

figure

3 however

(po

= I _x

lo~~)

the cellular structures should _not bifurcate _{to a} stable

planar

interface since the

top

^{of the} neutral

stability

_curve is

already

in the

oscillatory

unstable

region. Instead,

_one should find _a

composite pattern

^of bands of cells.

Although

_we need _a

J=lX10~~

oscillatory unstable

lo ~

~

~s

~

^{~~ ~}

unstable

>

lo 5

~~/o

'i lo ° lo I lo 2 lo 3 lo 4

~w

(~Lm)

Fig.

3. The

regions

of

instability

for

po

=

I _x 10~~

nonlinear

analysis

to examine these

patterns,

a banded cellular structure _seems to be _a resonable guess.

4. Numerical method.

Next,

we will describe how _{we can} obtain numerical solutions of

equations (6).

Since this has been described elsewhere

[9, 10]

in great

detail,

we will be very brief. Once _we have found _a

steady

state

solution,

_we

perturb

this solution with _a normal

shift,

linearize about the

amplitude

of this shift and solve the

resulting eigenvalue problem.

The trick to solve

(6)

in _a numerical efficient way is to rewrite the

equations

as a

integro-

differential

equation

and solve for the

(unknown

and

free) boundary.

We will do this for both the

quasi-static approximation

and for the full

equation.

In the

quasi-static approximation

_we

neglect

the time derivative term in the diffusion

equation.

In the

quasi-static approximation

_we find

(fl+y*K(s))

= i-

@+y*K(s'))(4"v'G)ds'+

+ 2

pek(u @

+ y ^* K

(s'i nj

G ds'

+

I

[k(u) @

+ y ^* K

(s') (()

+ y ^* K

(s')

G ds'

(18)

where the Green's function G is the solution of

V~G

+ 2 pe ~~

= 3

~(x x') (19)

°Y

The

explicit

form of G is

given

in

[8].

For the _non

quasi-steady

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

(Y(S,

t

)

2 Pet

)

~ ^~ ~ ~~ ~~ ~

j d~,

~

d~i

(Y(S', t')

2

Pet')

~ ^~ ~ ~~, ^~,

f*

^'

-~

f*

^'

fi'. V'G

(x(s,

_t

), x'(s', t')

t,

t')

₊ ds'

~ dt'( ^{(~(~" ~') j ~P~~'~}

y ^* K

(s',

t'

)

x

#

-w

where G is _a solution of the full diffusion

equation

V~G

^~~

=

3

~(x x')

3

(t t') (21)

To find

steady

state solution _we

parameterize

the

(unknown) boundary

with N

points

of

equal ardength.

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

integro-differential equation

on the

points

of

equal ardength,

except ^{at the}

endpoints (x

=

0 and _x

=

I)

and

require

that the

slope

vanishes at both the

tip

and the tail. Once _we have found _a solution _{we can} fit _a

spline through

the solution to obtain _a smaller discretization

step.

We

perturb

the

steady

solution

by adding

the normal shift

x(s)

=

xo(s)

+

ho (s)

3

(s,

t

(22)

where _{we assume} that 3

(s,

_t

= 3

(s)

e~~ and 3

(s)

_« I. We substitute this

perturbation

in

(18)

and linearize in 3. This

gives

_us the rather messy

looking equation

for the

quasi steady

state

approximation

ids'

yKo ⁺

~°

(3(s') fi(

V' ₊ 3

(s) b0'

V

b~'

V'G ₊

f

+

@

y

(3"(s')

+

Kj

₃

_{(s')) hi}

V'G

+

lYKo+1°1(-3(S)to.vG+Ko3(S)no.vG)1

-2pek

ds'

b~ yKo+~° (3(s')fi(.V'+3(s)bo.V)G f

+

(yKo +9

(- 3'(s') ijG

₊

_4~

Ko

3(s') G)j

+

j- Y(3"(S')+ K13(s'))j

=w

ids'((I-k)~~°+yKo~ ^3(s')G- ~9+yKo) bo.VGA~3(s')

f f

2

Pen)- °

+ YKo GA 3

(S)

₊ 2

Pekni1 ^°

^{+ YKo GA 2 3}

^{(S 1 (23)}

Once _we have chosen M

points

with

equal arclength ds,

the discretization of the

integrals gives

_{us a} set of

equation

of the form

M M

£

_A~j 3

j ^{= w}

£

_B~y 3

j.

(24)

j o j o

This is

just

_an

ordinary eigenvalue problem

for

w and _can be solved with _one of the standard numerical

packages.

For the _non

quasi-static approximation

_we

proceed along

the _same lines. Substitution of the

perturbation

ⁱⁿ

(20)

and

linearizing gives

_us

ids' ~ ^yKo+~° (3(s')b(.V'+3(s)fro.V)b(.V'G(w)+

f

+

fi y(3"(s')

₊

Kj3(s'))j ^{hi V'G(w)}

+ yKo +

) _{(- 3'(s') j}

_V'G

_(w)

+ Ko 3

(s') hi

V'G

(w ))j

-2pek ds'fi~ yKo+~° f~ (3(s')fi(.V'+3(s)fro.V)G(w)

+

4~( @

y(3"(s')+ Kj3(s'))j ^G(w)

+

(yKo+9 ^{(- 3'(s') ijG+}

fi~ Ko

3(s') G(w))j

+

j- Y(3"(s'i

₊

K13(S'ii)

= W

ids~lkli°+ ^Kol 3(S~IG(Wi- li°+ ^Kol no.vG(WiA23(S~i

2

pen(

^~° +

yKo)

^G

^{(w ) Ai}

³

^(s')

^{+ 2}

peknj

^~° ₊

yKo)

^G

^{(w ) A~}

³

(s')j ⁽²⁵⁾

f f

where _an

explicit

form of

G(w )

is

given

in

[8]. Upon discretizing

the

integrals

_{we get an}

equation

like

£

M

Cq(w

_{3~ =} ⁰

⁽²⁶⁾

j =o

where C is _a function of _w and therefore in

general complex.

To

satisfy

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. In

fact,

_{we can} look for the

eigenvalues corresponding

to different modes.

S. Results and discussion.

The

quasi-static stability

calculation _can be checked in several ways. First of

all,

_we know from the linear

stability analysis exactly

for which

velocity

and

wavelength

the flat interface goes unstable. We have checked that _our

stability

_program does

predict

the

right

onset for

instability

of the flat

interface,

both for small velocities and

large

velocities.

Furthermore, carrying

out the

expansion

described in section 3 to third order

gives

_{us an}

equation

for the

amplitude

A of the cell of the form

~ _{=aiA+a~A~ (27)}

Depending

_on the

sign

_{of a~, the} so-called Landau

coefficient,

_we have _a sub- _or

supercritical

bifurcation. It can be shown that for the _one sided model and for values of k _~ 0.45 _we will have _a subcritical bifurcation

[11]. Indeed,

_we do find _a subcritical bifurcation at onset. The lower branch of the subcritical branch has _to be

unstable, regaining stability

when it bends in the forward direction

again.

We have checked this behavior for the system at hand.

The full

stability

_{program can} be checked _as

well, using

the flat interface _{as our}

steady

state solution. The linear

stability predicts

for which values of _u and A the flat solution will lose

stability

to a cellular structure, will

regain stability

and lose

stability again

to an

oscillatory instability.

We _are of _course

especially

interested in the last

instability.

Let _us take the _case of

po

= I _x

10~?

_{In table II}

we have listed the

predicted eigenvalues

from the linear

stability

calculation of section 3 and the obtained values from _our numerical

analysis.

We _see that the agreement is excellent. We have also checked the lower branch of the subcritical bifurcation at onset and have found that the lower branch is unstable while the

upper branch is

stable,

which is _as

expected.

Table II.

Eigenvalues for d#ferent flat interfaces (flo

=

I _x

10~?).

~

(Jll/Sl

A

(iLJlll

_{t°numencal t°}

linearstabil,ty

0.023 5 3.51 3.52

0.32 5 89.8 ₊ I 54.3 89.9 ₊ I 56.0

0.9 5 15.3 ₊ I 1673.7 9.I ₊ I 1681.6

Next, we have examined the

stability

of the cellular structures for

high velocities,

I-e- _at the

top

^{of the neutral}

stability region.

For

flo

=

I

x10~~

we have indeed found _an

oscillatory instability

for the cellular solution. In

figure

4 _we have

plotted

two

steady

state solution with

u ₌

2.3m/s

and _u

=2.32m/s.

Here _x and _{y are} the

position

of the solution in the

computational

box

(the wavelength

of the solution is thus 0.15 ~Lm). We _can search for the value of _w

corresponding

to any mode. We expect ^the « zero mode which

corresponds

to _a translation of the interface in the direction of the

growth velocity

to be the first mode to go unstable. The results of our numerics for this mode _are

presented

in table III and the modes

are

plotted

in

figure

5. We have calculated _w for different number of discretization

points

of the interface to check that the _answer converges. We _see that the real part ^{of the}

growth

rate

w crosses zero somewhere between _u

=2.3m/s

^and u

=2.32m/s. Thus,

^the solution

corresponding

to an u ₌ 2.3

m/s

is stable while the solution

corresponding

to an u ₌ 2.32

m/s

is unstable with _an

eigenvalue having

a non-zero

imaginary

part. ^This solution is thus

oscillatory

unstable The other modes _were found to stable.

The cellular structures with

flo

= I _x 10~ ^? however _are found to be

stable,

_even at the

top

of the unstable

regime. This,

of _course, is related to the fact that in the _case of

larger

flo

the

planar

interface can restabilize before it

undergoes

the

Hopf

bifurcation.

1.34

v=2.32x10~

v=2.3x10~

~

/s

~

-l.35 i'>

~~'~~.00

_1.50 3.00 4:50 6.00 7.50x10.2

X

(~m)

Fig. 4. The steady state solutions for _v

=

2.3 m/s and u ₌ 2.32 m/s

(po

=

I _x

10~~).

Table III.

Eigenvalues for

^{the cells} in

figure

4

for

_a

d%ferent

number

of

discretization

points of

the

interface (A

₌ ^0.15 _~Lm,

fl

=

I

x10~~).

v

(m/s)

_w

(50 points)

_w

(100 points)

_w

(150 points)

2.3 0.I I ₊ I 1.46 0.15 ₊ I 1.41 0.15 ₊ I 1.40

2.32 ₊ 0.041 ₊ I 1.43 ₊ 0.036 ₊ I 1.44 ₊ 0.035 ₊ I 1.44

We _can also estimate the width of the

resulting

bands. This is

just

the

growth velocity

times the

period

of oscillation. For

w we take the value

corresponding

to the

marginal

stable mode

(I.e.

where the real

part

^is

equal

to

zero).

Since _{we measure} time in units of ~/4 ^D we have

aruo ^A~ W

=

(28)

In the above _{case we} have

u=2.32m/s,

A

=0.l5~Lm

and w~l.5. Thus _we find

W 3 _~Lm which is close to the

expeRmentally

observed bandwidths. Of _course this is

just

_a

rough

estimate since _we have chosen _a

quite arbitrary

value of

flo. Nevertheless,

the result

makes _sense in _a

physical

_way.

In

conclusion,

we have

developed

_a numerical _way to examine the

stability

of

steady

state cellular _structures for the one-sided model. We have found that modifications due to

large pulling

velocities _can result in

oscillatory

unstable

cells, depending

_{on our} modification

parameter

flo. Also,

^the

resulting

bandwidths agree

roughly

with the

experimentally

observed

ones. Of _course this

approach

cannot

explain

the

alternating

bands with and without structure

as observed in the

experiment.

For the exact

dynamics

of the cells _we have to

perform

_a

nonlinear

analysis.

xlo.I -o.70

0.80

o.90

-1,oo

V=2.32X10~

~ v=2.3x10~

i>~ ^~'~°

-l.20

1.30

-1.40

0.00 1.50 3.00 4.50 6.00 7.50x10.2

X

(~m)

Fig.

5. The modes corresponding to the tigenvalues in table III.

Acknowledgments.

This work _was

supported

in

part by

^DARPA Grant No. N00011-86-K-0758.

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