Directional solidification of Sn-Pb eutectic: Dynamics of local lamellar spacing selection

Scripta METALLURGICA et MATERIALIA

DIRECTIONAL

National

Vol.

SOLIDIFICATION

27, pp. 1223-1228, 1992 Printed in the U.S.A.

OF

Sn-Pb EUTECTIC: SPACING SELECTION

DYNAMICS

Pergamon Press Ltd. All rights reserved

OF

LOCAL

Jun-Ming Liu, Zhi-Guo Liu and Zhuang-Chun Wu Laboratory of Solid State Microstructures, Nanjing Nanjing 210008, P,R.China

LAMELLAR

University,

(Received May 28, 1992) (Revised September 8, 1992)

I. Introduction When a eutectic composition is submitted to directional solidification, the solidifying interface forms parallel lamellae of two coexisting solid phases: a and ~. This problem is of interest both for technological applications and as an example of spontaneous pattern formation in nature[l]. In a previous paper[2], we proposed a new, dynamic conjecture for the spacing selection: a eutectic system will select its operating point corresponding to the minimum value o f supercooling, ATo, at the three phase conjunction point(TPCP}. In this paper, we provide a physical basis for this conjecture by considering the dynamics of formation of local termination. In other words, we study the tilting features of the ~-~ boundary during propagation of the solidifying interface when the eutectic growth deviates initially from the operating point under the fixed solidifying rate V and the temperature gradient in liquid in the front of the solidifying interface, G. If the tilting angle is not zero, and the u-~ boundary deviates from the propagation direction of the solidifying interface for a given lamellar spacing k, the eutectlc system in directional solidification is unstable and will transit to its operating point by readjustment of the local lamellar spacing. Termination formation seems to be a reasonable mechanism for the readjustment[3]. The operating point will correspond to a zero tilting angle, with the ~-~ boundary parallel to the propagation direction of the solidifying interface. We will prove that this operating point in fact coincides with the conjecture of minimum supercooling at the TPCP by numerically solving the coupling equation of Sn-Pb eutectic in directional solidification. Some physical parameters of this euetctic are listed in Table I[3,4]. For details of the numerical calculation, please refer to Ref.[2]. II.

Tilting

of the

~-~ b o u n d a r ~

In the coordinate system as shown in Fig.l, the coupling equation lamellar eutectic in directional solidification can be written as[5]: .2n~ , AT°-G'f(x)=~n=Ima'R'c°s~--~-'x~-

AT0-G.E(X)~m~-K.cos(~-x).

r~.f"(x) [l+f,Z(x)]3/2 '

r~'g"(x)

[ l ÷ g ' Z ( X ) ] 3/2

,

of a

0~x~Sa

(la)

S
{Ib)

1223 0956-716X/92 $5.00 + .O0 Copyright (c) 1992 Pergamon Press Ltd.

1224

LAMELLAR

SPACING

sELECTION

Vol.

K=VD~- C°~- sin(-Z-~.S )

27, No.

(Zc)

n

where f(x) and g(x) represent, respectively, the profiles of the ~-L and P-L interfaces. For definitions of the other parameters, please refer to Fig.1. The b o u n d a r y conditions are: f(x)=g(x)=0

as x=Sc¢

(2a)

f'(x)=-tane

, E'(x)=tan@~ as x=S a

(2b)

f' (x=0)=g' (x=Sa+S~=Xl2)=0

(2c)

Table I. Some Physical

I

parameters

=jK/wtX)

I I

Co{wt~)

Parameters

II

o.ovss

69.55

I

At the operating point, be held at the TPCP: o ~ "cosO;

of Sn-Pb Eutectic[3,4]

II p a r a m e t e r s II m (Z/wtX)

value

II

the

I

I I l

r#(K-pm)

e(=Sp/S~)

following

mechanical

value

s.5

0.0475 0.594

balance

condition

~aL'cosea=o~L ~aL-sine~+~L'sine;=~

where

~aL'

~L

interfaces,

and

@~ s a t i s f y i n g

Eq.(3),

(3a)

(3b)

are

-~

respectively;

the

interface

8 ° and

i.e.

will

the

tensions

e~0 d e n o t e

values

at

the

of

the

respectively

the

operating

point.

u-L,

~-L

and

a-~

of

8

and

However,

if

the

values

eutectic system deviates from its operating point f o r g i v e n V a n d G, t h e mechanical balance condition Eq.(3) is broken and the resultant force acting on t h e TPCP d e v i a t e s from the propagation direction of the solidifying interface, which compels tilting o f t h e u-~ b o u n d a r y . Here, we d e f i n e a tilting a n g l e Ae a s :

~o=(%-e~)-(e~-op),° o~>eo~

(4)

o

For

Sn-Pb

eutectic,

the

calculations

e>e~ a n d e~
show t h a t

the

=-~

boundary tilts toward the ~ lamella side(right), tilts toward the u lamella side(left), we have

s o AS>0. I f ~ h e b o u n d a r y @ e;,_ and hence

AS<0. The t w ~ casis of AS>0 addition to GUL, U~L and __ ~,

in to

resultant

is

~d'

there

is

and Ae<0 are shown which are assumed

a driving

force,

~,

which

Figs.(2a} and (2b). In be isotropic and whose drives

the

solidifying

interface p r o p a g a t i n g alone the z axis at the rate V, no matter whether the ~-~ boundary tilts. Mote that this force does not break the balance condition Eq.(3) since it acts on each ~oint of the solidifying interface. The resultant force acting on the TPCP is F G, which deviates left from the z axis as de<0

9

Vol.

27, No.

9

LAMELLAR

and d e v i a t e s right as Ae>O. as Ae0.

SPACING

SELECTION

Correspondingly,

the

1225

u-~ b o u n d a r y

will

tilt

left

Now we will c a l c u l a t e the tilting angle Ae as a f u n c t i o n of the spacing k. Note that the coupling condition for p r o p a g a t i o n of the solidifying interface r e q u i r e s that AT terms in Eqs.(la) and (Ib) are e q u i v a l e n t to each other.

If

we

boundary), become

and then of

the

two

calculate

it

calculated

~I is the spacing

e~te~0

and

condition For

an

by

for

AT0~.

Ae as

a

of

As

Eqs.(la)

can

not

be

and

means

no

X,

we

denoted

of

to o b t a i n

k is

shown

to the m i n i m u m

we

o

@uw9 u

in

and

take

eu and

the

u-~

@u and and

@6

@~,

o

AT ° in terms

ATo=(ATou+ATo~)/2

@6 as

Fig.3,

position

of

Therefore,

assume

by ATou,

approximation, (Ib)

tilting

satisfied.

spacing

Eq.(la), an

function

corresponding

(which

arbitary

AT ° in terms

denoted

substitute

A. The

coupling •

varlables.

Eq.(lb),

and

e uw9~ u

assume

a

where

function X:k/kl,

of s u p e r c o o l i n g

of and

at the

TPCP. Interestingly, for all the cases, we have A ~ O as k:l.O, which means that the u-~ b o u n d a r y has no tilting when s u p e r c o o l i n g at the TPCP reaches its m i n i m u m value. Moreover, A@O as l>l means that the u-~ I 1 b o u n d a r y will tilt left (toward the u lamella side) as A
the ~ lamella

III. If

we

side)

The profile

suppose

that

as A>~ . 1

and k

splitting is

the

of

the

operating

solidifying, point

for

interface both

k
and

k>kl,

tilting of the u-~ boundary will result in readjustment of the local spacing. This dynamics depends obviously on the profile of the solidifying interface. As a f u n c t i o n o f k, t h e c a l c u l a t e d profiles of the solidifying interface for V=O.5~/s a n d G=IOOK/om a r e s h o w n i n F i g . 4 . The midpoint A(x--O, z - - f ( O ) ) o f t h e u-L interface initially moves toward the liquid side and then toward the solid side relative to the TPCP. The midpoint B(x=k/2, z=g(k/2)) of the ~-L interface always moves toward the solid side relative to the TPCP. Note that f'(O)O (g'(~/2)>0) as the interface is c o n c a v e into the s o l i d side. There is a c r i t i c a l point Af (kg), at w h i c h f"(O}=O (g"(k/2)=O). When x increases from

will into into

a value smaller k f ( kg) and s u r p a s s e s k f (k g }, the u-L (B-L) interface experience the critical splitting and evolve from a non-splitting profile a split one, because the middle range of the interface becomes concave the solid. Our calculations show that Ag=11 a n d A f > ~ l . T h i s indicates

that the g-L interface is in the critical splitting interface still has a non-splitting profile when reaches its minimum value.

state (CSS) and the u-L supercooling at the TPCP

We then a n a l y s e d the s t a b i l i t y of the s o l i d i f y i n g interface. For the i n t e r f a c e w i t h a n o n - s p l i t t i n g profile, the p o s i t i v e thermal g r a d i e n t and the G i b b s - T h o m p s o n e f f e c t ( t h i s effect is strong b e c a u s e of small spacing at this stage} s t a b i l i z e the interface, a l t h o u g h the mass d i f f u s i o n in liquid will d e s t a b i l i z e the interface. Therefore, the n o n - s p l i t t i n g p r o f i l e is stable for the f l u c t u a t i o n s and is n a m e d as the s u p e r - s t a b l e profile. However, for the i n t e r f a c e with a split profile, a small local p e r t u r b a t i o n to the solid side of the p r o f i l e will result in a local c o n c e n t r a t i o n p i l e - u p and raising of the

1226

L A M E L L A R SPACING S E L E C T I O N

Vol.

27, No.

constitutional supercooling in the liquid at the front of this local region of the interface. Since the Gibbs-Thompson effect i s now w e a k d u e t o t h e l a r g e spacing in the positive thermal gradient field, this local region will propagate further into the solid side, and a pocket range will appear and will be deepened further with propagation of the interface. Obviously, this profile is unstable for a fluctuation. Naturally, the critical splitting profile is marginally stable. For the present case, the a-L interface is superstable and the ~-L interface is marginally stable when ~=kg=l , which follows the marginal stability principle[6]. of minimum supercooling at the IV. local

Dynamics

According spacing

g"(k/2)>O

and

of

the

local

This TPCP.

argument

spacing

to the above discussion, selection: formation of dO>0.

The

local

half

seems

selection: a

to

support

termination

the

conjecture

formation

we c a n now s t u d y the dynamics of the termination. Suppose k>kl{kg), then

width

of

~ lamella,

S B,

will

decrease

with

propagation of the solidifying interface because the a-~ boundary tilts to the right side. From Fig.4 we c a n s e e t h a t the ~-L interface will move to the liquid range relative t o t h e TPCP, w h i c h w i l l restrain overlapping of the lamella by the neighbour two ~ lamellae, although in the initial stage, the middle region of ~-L interface is concave to the solid side(g'(k/2)>O). Contrarily, owing to an increase of the width of the a lamella, the middle region of the u-L interface will move to the solid side relative t o t h e TPCP (f"(O) transits from negative to positive). With propagation of the s o l i d i f y i n g interface, a deep pocket range forms in the front of the u-L interface. Finally, a new ~ lamella will n u c l e a t e and grow in this pocket range, and a p o s i t i v e t e r m i n a t i o n forms. This e v o l u t i o n is s c h e m a t i c a l l y shown in Fig. Sa. By this dynamic mechanism, the local spacing will decrease. For

the

case

of

~
the

a-~ boundary

will

tilt

to

the

left

side

because

AS<0. A t t h i s stage, both the u-L interface and the ~-L interface are of non-splitting profiles. With propagation of the solidifying interface, the u-L interface will move along the -z direction and the ~-L interface will move along the z direction. Finally, the local ~-L interface will be overlapped by the two neighbour ~ lamellae and a negative termination forms. The local spacing will increase. This evolution is sketched in Fig.5b. to

be

From the dynamic more difficult

point of view, than formation

formation of of a positive

a negative termination seems termination since Sa>S B a n d

f"(0)
solidified terminations

sample will

of be

a physical basis for the The calculated scaling law from present experimental range(V~0.5~m/s) further open.

Conclusion

In conclusion, the tilting of the ~-~ boundary and the profile of the solidifying interface have been studied by numerically solving the nonlinear coupling equation of Sn-Pb eutectic during directional solidification. When

9

Vol.

27,

No.

9

LAMELLAR SPACING SELECTION

1227

the tilting angle o f t h e cx-~ b o u n d a r y equals zero, the profile o f t h e ~-L interface reaches its marginally stable state(the critical splitting state), with the ~-L interface being of a super-stable profile. Correspondingly, the supercooling at the TPCP r e a c h e s its minimum value as a function of the lamellar spacing at the zero-tilting position of the ~-6 boundary. This analysis provides a physical basis for the conjecture that the system will select the minimum position of supercooling at the TPCP a s its operating point.

References l.V.Dayte,J.S.Langer,Phys.Rev.B,24,4155(1981) 2 J.M.Liu,Scripta.Met.& Mat.,26,179(1992) 3 K.A.Jackson,J.D.Hunt,Trans.Metall.AIME.,236,1129(1966) 4 M.Gunduz,J.D.Hunt,Acta.Met.,33,165](1985) 5 J.M.Liu,Y.H.Zhou,B.L.Shang,Acta.Met.& Mat.,38,1625(1990) 6 For e x a m p l e , s e e J . S . L a n g e r , R e v . M o d . P h y s . , 5 2 , 1 ( 1 9 8 0 ) 7 W . K u r z , P . R . S a h m , G e r i c h t e t erstarrte e u t e k t i s c h e W e r k s t o f f e , S p r i n g e r - V e r l a g , Berlin,(1975) 8.P.Haasen,Physical Metallurgy,Cambridge Univ.Press,Cambridge,p75(1978)

ZI%liquidilheinterfcepr°file I ~'~

°~L

;

in d i r e c t i o n a l s o l i d i f i c a t i o n and the c o o r d i n a t e system for analysis. 1:2(S +S~), 0 and e~ are

~L

0

~

~ /f(x)

liquidus slopes respectively, D

of is

the the

~x

and solute

~ phases diffusion

[

I

{

the groove angles of ~-L interface and the ~-L interface at the TPCP, respectively. In Eq.(1), m and m~ are the absolute values of the

a

~o ~

~

I

S~ "'

~-

~

"~

coefficient in liquid, F~ and F~ are the G i b b s - T h o m p s o n c o e f f i c i e n t s of the u-L and ~-L I interfaces respectively, C is the d i f f f e r e n c e 0 { between the m a x i m u m solubilities of the two solid phases. 60

o

~G

%,

40

o L

20

d

D

d

0

0.5

-20

(a )

r o~,,,8

X

Fig.2 The sketch of the forces the TPCP. (a).Ae>O (b),A8
~o~ acting

(b )

on

-40

1,0 1.5 2 . 0

k

S

-

Fig.3 The calculated tilting angle Ae as a function of k. l - - V = 0 . 5 ~ m / s , G = l O O K / c m , l I=3.5~m 2 - - V = 3 0 ~ m / s , G = 1 0 0 K / c m , k i=0.5~m

1228

LAMELLAR SPACING

0,4 0.3 ~o.

SELECTION

Vol.

27, No. 9

4~

1

1 o.4

.6

0.8

4

x/(Sa+S ~ ) -

0

.

2

~

I

-o.3

l

Fig.4 The calculated profiles of the the lamellar spacing k. V=30~m/s,

3:X=O.50~a,

4:k=O.70~m,

5:k=O.9~m,

solidifying G=100K/cm,

interface as a function of 1:k=0.15~m, 2:k=0.40~m,

6:k=l.lO~m.

tv

tv

Fig.5 The readjustment of local {a).Ae>O as k>k1' the a-~ boundary

spacing by the tilts toward the

termination mechanism. ff lamella side, and a

pocket range will appear in liquid in the front of the ~-L interface with finally a new ~ laaella growing in the pocket and a positive termination forming. {b).Ae<0 as t