Kinetic dendrite in the boundary-layer model

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Kinetic dendrite in the boundary-layer model

D. Temkin, J. Géminard, P. Oswald

To cite this version:

D. Temkin, J. Géminard, P. Oswald. Kinetic dendrite in the boundary-layer model. Journal de

Physique I, EDP Sciences, 1994, 4 (3), pp.403-409. �10.1051/jp1:1994147�. �jpa-00246916�

Classification Physics Abstracts

61.50C 61.50J 81.30F

Kinetic dendrite in the boundary.layer model

D. Temkin

(*),

J. C. Géminard and P. Oswald

Ecole Normale Supérieure de Lyon, Laboratoire de

Physique,

46 Allée d'ltalie~ 69364 Lyon Cedex 07, France

(Received 29

September

J993,

accepted

3 December1993)

Abstract. Within trie

boundary-layer

model and

by taking

into _account only interface kinetics

(isotropic

or

anisotropic),

_we find _a contmuous set of _« kinetic _» dendrites. This solution exists at

arbitrary supercooling

A in the 2D- and

3D-axisymmetric

_cases and reduces _to the usual dendrites with

parabolic

tails _at A _~ l, and _«

angular

_» dendrites _at A _~ l. In _contrast with classical Ivantsov dendrites, there _are upper limits for

growth

velocity and tip curvature at each

supercooling

A.

Ivantsov

Ii

_was the first to show that there exists _{a contmuous set} of

stationary

needle-like solutions _to the

problem

of the free

growth

of _a

crystal.

These solutions _are obtained

by neglecting capillarity

and interface kinetics and exist

only

at dimensionless

supercoohng

A smaller than 1. To find _a

smgle

dendrite

(experimentally, only

one is

observed),

one needs _to introduce _a

length

in the

problem. Usually,

the

capillary length

_d~ is introduced and the solution _is found

by disturbing

the Ivantsov

parabolic

solution

using

a

singular perturbation

method. In this _case, and in _two

dimensions,

the

tip velocity

v is

proportional

to

(D/d~ ^A~

at A « 1, where the _constant of

proportionality depends only

on the

anisotropy

of the surface _tension

[2].

D is the diffusion coefficient.

This choice of the

length

is _not unique because there _is another characteristic

length

connected with interface kinetics _: that _is, D/w where _w is constructed from the interfacial kinetic coefficient

p.

The _ratio

d~

₌ d~ ^{w/D of these} two

lengths

determines which of them _is

more important. ^In pure

materials,

D is the thermal

diffusivity

in the

hquid,

whereas _m

alloys

it

is the solute diffusion coefficient in the

liquid.

If

d~

» 1, one expects

capillanty

to be _more

important

than kinetics. In the opposite case,

dt

« 1, kinetics should dominate surface tension

effects. The parameter d~ is a matenal constant which _can be evaluated. For

example,

d~

^mû-1 m pure materials _as Ni

(by taking

p =160cm/s/1ç

[3])

or succinonitrile

(with

p

=

20cm/s/K [4]).

In contrast, d~ is

usually

much

larger

than _m

alloys

because the chemical diffusion coefficient _is much smaller than the thermal

diffusivity.

Nevertheless,

(*) Permanent address: I.P.Bardin Institute for Ferrous Metals, 2Baumanskaya ^Str. 9/23,

107005 Moscow, Russia.

404 JOURNAL DE PHYSIQUE I N° 3

d~ can also be small in impure materials such _as discotic columnar

liquid crystals.

For instance,

we measured

previously

in CBHET p

w 1.3 _x 10~ ^~ cm/s/K which

gives dt

- 0.5

[5].

In ail

these

examples

we thus expect ^kinetics to be

significant

_even at small

supercooling.

For these reasons, we found it

interesting

_to

study

the _case

dt

=

0

corresponding

to zero surface tension.

This

problem

^has

already

been discussed

by

Lemieux et ai.

[6]

_{and _by} Brener et ai.

[7].

Both calculations _are carried _out

using

a full non-local

approach

and start from the Ivantsov solution.

This

requires

that kinetic

undercooling

is small. Lemieux _et ai. found

analytically

for

isotropic

kinetics _{a contmuous}

family

of solutions _m the limit of small A, while Brener _et ai. found _a discrete _set of solutions which _are selected

by

^kinetic anisotropy ^{for A} ~ l. In this article, _we

analyze

this

problem

for

arbin-aiy supercooling

A and

arbitrary

kinetic

undercooling.

We _use the

boundary-layer

model

(BLM)

which has been shown _to

give quahtatively

_correct results in

many situations

[8].

Another

important point

_is that _we do _{non use} A>antsov's solution _as the starting approximation.

In the two-dimensional one-sided model

(no

diffusion in the

sohd)

the BLM

equation

for isothermal and

steady-state

solidification of _a dilute

alloy

reads

k( V(1 U)

cos o

~~~

K _sm o ~

l~~

[1 +

U(1/k~ -1)]

^d° cos o [1 +

U(1/k~ -1)]

K d KU dU

~

o

(1)

~ V do

cos o [1 +

U(1/k~ 1)]

^d°

where V

=

u/w and K

=

J~D/w _are

respectively

the dimensionless

velocity

in the z-direction

and the curvature of the interface. The

angle

between the z-axis and the normal _to the front _is

9 and K is the curvature defined

by

^K

=

d9/ds where _{s is} the

arclength along

the interface measured in units of D/w. The dimensionless

undercooling

at the interface U is defined to be U

= (C,~~

C~)/[C~(1/k~-1)]

where C,~~ ^and

C~

_are concentrations of solute at the interface and in the

hquid

far from the interface _;

k~

^{is the}

equilibrium

distribution coefficient.

Equation (1)

_is still valid for pure materials and thermal

growth.

In this _case, U

= (T~~~

T~)/(L/c)

where L is the latent heat per ^unit volume and _c the heat capacity.

k~

must also be set

equal

to in

equation (1)

which grues the _same

equation

_as that

previously

used

by Langer

^and

Hong [9].

The second

important equation

_comes from the Gibbs-Thomson relation. Without anisotropy

effects,

it is

U

=

A

d~

K V _cos 9

(2)

with

d~

= d~ ^{w/D. In the chemical} case, 4

=

(C~~ C~ )/[C~(1/k~ 1)]

where

C~~

is the

equilibrium

solute concentration m the

liquid

at the temperature ^{chosen. The}

capillary length

is d~ =

yT~/L )/ [mC

~

l/k~ )]

and _w

=

pmc

~

l/k~

where _{y is} the surface stiffness and

m the

slope

of the

hquidus

fine. In the thermal _case A

=

(T~ T~ )/(L/c ),

d~ ₌

yT~ c/L~

and

w =

pL/c.

One _sees

immediately

that in both _cases

d~

can be wntten m the

general

form

dt

=

°'~j~ ⁽³⁾

Equations (1)

and

(2)

_can be

generahzed

for 3D axisymmetnc

growth

as follows

[8]

k( _{V(1 U)}

_cos 9

~~~ ~

~~~

~

Kj

sin 9 ~

~~

[l +

U(1/k~ )]

^d9 cos 9[1 +

U(1/k~

1)]

K d

~~i

^~ _dU

1=

^o

^(1~)

~ VR d9

cos 9 [1 +

U(1/kE 1)]

^{d9 '}

U

= A

d~(Ki

+

K~)

V _cos 9

(2~)

where

Kj(9)=d9/ds

and

K~(o)

₌

(sin9(/R(9)

are the two

principal

curvatures and

R(9

₌

o dt(cos

t

)/Kj (t)

_is the radius of the circular section normal to the z-axis

(growth

direction).

Let _{us now} discuss the solutions _to

equations (1)

and

(2)

ⁱⁿ the

limiting

_case

dt

=

0

corresponding

to _zero surface tension

(2D case).

To

simplify,

we further _assume that

à « and k~ =

1. In this

limit,

the BLM is _not

stnctly

vahd but it

gives qualitatively

the

right physics

as shown

by Langer [10].

We _{can now} substitute U

by

A in ail terms of

equation (1)

besides

dU/d0=V

_sm 9 in the last _term of

equation (1).

We thus obtain

ÎÎ

^~

k

~n 9111~

^~ ~~~

where k

= K/A, _v

=

V/A~.

_The

problem

now is to determine whether this

equation

has

symmetric

needle-like solutions

corresponding

to an even function

k(9)

which _{starts at} o

=

0 with _a fimte curvature ko ^{and which tends} to k

=

0 when 9

- gr/2.

Equation (4)

has _a

singular point

at o

=

0.

Thus,

for any

physical solution,

the numerator of the r-h-s- of equation

(4)

must also vanish _at 9

=

0. This

regulanty

condition gives ^a relation between

velocity

_v and tip curvature ko:

v =

ko k(. ₍₅₎

We _can

expand

the curvature

k(9)

close _to o

= 0 in the

followmg

form _:

k(o

=

jj ~

_C,~

,~(o~)~~+~

for _a # (or

ko

#

IN) (6a)

n o m o

k o

=

IN ₊

~ ~ Ci

~

o^~'~

(In

o

)~

for ko = Il

(6b)

n i m o

with

Co

o =

ko. Inserting (6a)

into

(4)

and equatmg coefficients of

(o~)""

+"', _we find

a =

1/2ko

-1

(7)

and _recursion relations which define coefficients C,~,

~

in _ternis of

Co

= 3 VI

[2 (4 ko )]

and of the

arbitrary

coefficient C

m C

o. Constant C _must be evaluated

by integratmg equation (4)

with

boundary

^{condition k} ₌ 0 at o

=

gr/2. For

example,

for _a

= 1/2

(or

_ko

=

1/3),

_we find

C - 0.39. The _same

procedure

can be used when ko =

Il for

calculating C,[~

m terms of C j*1 =

3 v/2

ko

₌ 9/8 and of the

arbitrary

constant C ^*

m C _i*

o which agam ^must be calculated

by integration.

Relations

(5-7)

define _a contmuous set of needle-like solutions which _can be

parametrized

by

_a. The

only

limitation _is a>0

(to

avoid

divergence

of the curvature

k(o)

_at

o

=

0).

Note that for 0

~ a ~

l/2,

we have

k'(o

= 0

)

= m, whereas for

a > 1/2, we have

k~(o

=

0)

=

0.

The _main difference between these solutions and lvantsov's solutions

[Ii

_is that the

tip

curvature and

velocity

_now have upper limits which _came from the

k(

term m

equation (5)

(lvantsov's

solution

corresponds

to v =

ko

m the BLM

approximation), Indeed, equation (5)

imposes ko ~ l and _v

~ Il while condition _a mû

gives

and additional stronger ^limitation

ko _~ ^{l/2 and} v ~ Il. The dimensional

velocity

and

tip

curvature are given

by

v ₌

ko(1 ko) wA~, _J~o

=

ko(w/D

4

,

ko

_~ ^l/2

(8)

JO~RNAL DE PHYSIQUE T 4 N' 1 &IARCH 19g4 Ii

406 JOURNAL DE PHYSIQUE I N° 3

The

problem

_{now is to} understand what

happens

when surface tension and

anisotropy

_are

mcluded. In this _case

equations (2)

and

(4)

_are

generalized

_as follows

U

= A

Kdt(1

_{y,~ cos} mo V _cos o

(1 p

~

cos m o

) (9)

v COS~ ~ k COS~ ~

ù

~ol

_~

ù _{ikô (i}

Ym COS m~ +

+ v cos o

(1 p~

_cos m9

= 0

(10)

where à

=

d~/A~.

y~ is the surface stiffness

anisotropy

while

p~

_is the amsotropy ^{of the kinetic} coefficient

p

for m-fold symmetry. ^{We also} assume that directions of minimal surface stiffness and of maximal kinetic coefficient coincide with the

growth

direction.

In the

simplest

_case, we include

only

^kinetic

anisotropy (with

zero surface tension, à

=

0). Thus,

for

sufficiently

small

anisotropy p~,

there exists _a continuous _set of solutions given

by

v =

ko k(/B

,

a =

(B/2 ko

1) _> 0

(11)

with B

=

Il

[1- p~(m~ _+1)]

_and

_hmiting

_values of

velocity

and

tip

curvature given

by

v ~ BM and ko ^{~B/2. The solutions}

disappear

when

p~(m~

₊

1)

m1

(Fig. I).

> OE=

~

fi,=0.03 fi,=

0.15

4£j _£~4

,~ ~~~

_{0 X~ 0}

X~

'

0.10 0.15

fi4

Fig.

l. Dimensionless

velocity

_{as a} function of kinetic

anisotropy.

Trie sobd line grues trie maximal

velocity

of _« kinetic dendrites

>>. Dots correspond to velocities v~ of dendrites selected by anisotropy (see

also mset m

Fig.

2). Two dendrite

profiles

(with trie _unit of length D/(w. A. _v)) corresponding to two different values of P4 and à

=

10~ ^~ _are given m trie insets. Trie profiles are compared with trie parabolic

profiles havmg

trie _same

tip

curvature (dashed lines; _m trie nght mset trie _two profiles are

indistinguishable

_on trie choosen scale).

When _we include

isotropic

surface _tension without any

anisotropy (y~

₌

p~

₌

0)

all

solutions

disappear

_as usual

[10].

This result is obtained from numerics and reflects the

impossibility

of

finding physical

solutions

k(

9

) starting

from 9

=

gr/2 with k

=

0 and

arriving

at 9

=

0 with

k'(0)

=

0

iii (in

all _cases,

k'(0)

is found to be

negative).

When amsotropy ^is

included,

we find

numerically

some values of _v which

provide k'(0)

=

0,

thus

giving

_a discrete spectrum ^{of solutions for each}

given

value of à

[12].

In

figure 2,

_we

plot

maximum selected

velocity

_{v as a} function of dimensionless parameter à

=

d~/A~.

These _curves have been

numerically

calculated for different values of

p~

and

y~. Two different _{cases can} be considered

depending

_on the _{existence or not} of surface

stiffness amsotropy :

if y~ #

0,

there _are two

asymptotic regimes

_: in the

capillary regime,

_at à

- m, the

dimensionless

velocity

_v scales hke

y/'~/ô

in agreement ^{with the} calculations of

Langer

and

Hong

for the BLM

[9].

In the kinetic

regime,

at à

- 0

(or d~

_-

0), velocity

_v tends _{to a} limit vo

(which

is obtained

by extrapolation) depending only

_on

p~ (see

mset in

Fig. 2).

At small

anisotropy

_vo

PI

_{~~, in agreement with}

a

previous analytical prediction [7].

This selected

velocity

_is

plotted again

m

figure1

in order _to compare it with the maximal

velocity

of

« kinetic _» dendrites

corresponding

to

d~

₌ ^0. Two

points

must be

emphasized

:

first,

the selected dendrite still exists when

p~

>1/17 and _« kinetic _» dendrites

disappear.

Second,

when

p4

m

(0.02-0,04 ),

the selected

velocity

is close to the maximal

velocity

of _« kinetic _» dendrite.

~

o , p4"0.01,

o

v

=

x. ,,v=150 m

~

' ~

> ',, ',

»

O

>

v~ =

1 ~ j

j

> j

i

j (

jj

4

°'°~

Fig.

2. _-

v

^VIA

stiffness and

kinetic

P~. Trie

velocity v~ found by

to

à = 0 are given m

trie

mset.

-

Wheny~ =

0

_two symptotic regimes

at _à

-

0 _is

independent of

y~ and _thus

coincides

_with the inetic regime discussed just

before, _while the _second ne, at

à

^- _m, ^is haracterized by v ~ p ('~ ô

~

m agreement with the

analytical of

Brener. We can agam phasize

that esult

is easier ^{to find}

with the

_BLM than by _solvmg the

full

nonlocal problem [14].

SO far, we have _assumed that AM 1 and

k~

= 1. We

now

_discuss

rbitrary

4

_and k~ without

amsotropy and _tension. ^sing _the ^ame _procedure as that

408 JOURNAL DE PHYSIQUE I N° 3

used

previously

to denve

equations (4)-(7),

we find _:

k(V(1-A+V) _[1+ (A-V)(1/k~-1)]-Ko(A-V)~+K((A-V)=0 ₍₁₂₎

a =

[(A-V)/2Ko-1]>0. (13)

Similar

equations

_con be obtained for 3D

axisymmetnc

dendrites

using equations (1')

and

(2~)1

k(V(1-4+V) _[1+ (A-V)(1/k~-1)]-2Ko(A-V)~+2K((A-V)=0 ₍₁₄₎

and the _same relation for _{a as} that

given by equation (13).

In this _case, the two curvatures at the

tip

of the dendrite _are

equal (Kj(0)

=

K~(0)

=

Ko)

and the

tip

radius

equals 1/Ko.

Equations (12-14) only

_grue the relation between

growth velocity

and

tip

curvature as well _as their upper limits

(Fig. 3).

On the other hand, to find dendrite

profiles (see

msets in

Fig. 3),

_we need to solve

numerically

the

corresponding

differential

equations (1)

or

(1')

in

K(9)

with

condition

K(gr/2)

=

0 at A~1

(usual

dendrite with _a

parabolic tait)

and with condition

K(90)=0

at ~ ml

(angular dendrite[15, 16]).

In the latter _case,

90

is

given by

V _cas

90

₌

(A 1).

à = o.5 ^3D

>

à ₌ 1.5

2D

o-s à

Fig. ^3.

Tip

velocity V (in unit of w) and tip curvature Ko (in unit w/D) _versus supercoolmg ^for two and three-dimensional dendrites (k~

=

1). Solid lines correspond to « kinetic _» dendrites of maximal velocity (d~ = 0 and _a

- 0). Trie thick sohd lines V

=

0 _at A _~ l, V

= (A 1) for A _~ l, and K~ = 0

correspond

to

steady-state growth

of trie

planar

front.

They

also

correspond

to kinetic dendrites _m trie hmit _a

- co. Two

profiles

for 2D _« kinetic

» dendrites (d~ ₌ ^{0 and} a - 0) _are shown _m trie _msets.

In

conclusion,

_we have found _{a new} basic continuous

family

of solutions which takes mto

account interface kinetics but _not surface _tension. This solution descnbes _a needle-like

dendrite at

arbitiaiy supercoolmg

A smaller _or

larger

than 1.

Consequently,

it should be _more

appropnate

^for

describing

dendrites _at

large supercoohng

than the usual Ivantsov solution

only

vahd at A

~ l. It could also _{serve as a} startmg

approximation

m a more

complete theory.

Acknowledgments.

This work _was

supported by

CNES _contract No. 93.0215. D. Temkin _is

grateful

for the Poste

Rouge provided by

the CNRS.

References

Ii Ivantsov G. P., Dokl. Akad. Nauk SSSR 58 (1947) 467.

[2] For _a review _see Brener E. A., Mel'nikov V. I., Adv. Phys. 40 (1991) 53.

[3] Willnecker R., Herlach D. M., Feuerbacher B., Phys. Rev. Lent. 62 (1989) 2707.

[4] Glicksman M. E., Schaefer R. J., Ayers J. D., Menai Tians. A 7A (1976) 1747.

[5] Géminard J. C., Oswald P., Temkin D., Malthête J., Europhys. Lent. 22 (1993) 69.

[6] Lemieux M.-A., Liu J. and Kothar G., Phys. Rev. A 36 (1987) 1849.

[7] Brener E. A., Geilikman M. B., Temkin D. E., Sov. Phys. ^JETA 67 (1988) 1002.

[ii Ben-Jacob E., Goldenfeld N.,

Langer

J. S., Schôn G., Phys. Rev. A 29 (1984) 330.

[9] Langer J. S., Hong ^D. C., Phys. ^Rev. A 34 (1986) 1462.

[10] Langer J. S., Phys. Rev. A 33 (1986) 435 and also Ref. [ii.

l Ii Condition k'(0 0 _comes from symmetry ^{and from trie fact that trie} differential equation is now of second order (see Refs. [8-10]).

[12] Ben-Jacob E., Garik P., Mueller T. and Grier D.,

Pliys.

Rev. A 38 (1988) 1370.

[13] Brener E. A., J. Ciyst. Growth 99 (1990) 165.

[14] There _is nevertheless _a restriction _to this conclusion concemmg trie scalmg with A. Indeed, _m the limit A

- 0, exact scaling for dimensional velocity and tip curvature reads _u

yÎ'~(D/d~)

A~

and Ko

yl'~(1/d~)

A~ [2] whereas BLM grues v

yÎ"(D/d~)

~~ _and Ko

yÎ~~(1/d~)

A~. We suggest that this difference in trie power of A exists in ail trie asymptotic regimes we have

discussed.

[15] Brener E. A., Temkin D. E., Europhys. ^{Lent. 10} (1989) 171.

[16] Classen A., Misbah C., Müller-Krumbhaar H., Saito Y., Phys. Rev. A 43 (1991) 6920.