Planar to cellular transition during solidification in ternary systems

Scripta METALLURGICA et MATERIALIA

Vol.

26, pp. 1157-1161, 1992 Printed in the U.S.A.

Pergamon Press Ltd. All rights reserved

PLANAR TO CELLULAR TRANSITION DURING SOLIDIFICATION IN TERNARY SYSTEMS J.J. Hoyt Department of Mechanical and Materials Engineering Washington State University Pullman, WA 99164-2920 (Received January

24, 1992)

Introduction

The classic work of Tiller et al. [1,2] used the concept of constitutional supercooling (CS) to examine the stability of a planar solid-liquid interface during unidirectional solidification of a binary alloy. In subsequent years, Mullins and Sekerka (MS) [3] established a more complete stability criterion by investigating the growth or decay in amplitude of a sinusoidal perturbation in the planar interface. In 1970, Wollkind and Segel (WS) [4] presented a more mathematically rigorous derivation of the interface stability condition and, through a nonlinear perturbation analysis, discussed the existence of a subcritical instability. A number of studies have examined the extension of the stability criterion to the case of ternary systems. Chalmers [5] arrived at the simplest possible generalization of the CS criterion by simply adding together two terms found in the binary criterion. The Chalmers suggestion neglects all interspecies liquid diffusion effects. Cole and Winegard [6] measured the planar to cellular transition in the Sn-Pb-Sb system. These authors were unable to explain their results in terms of the Chalmers prediction and concluded that off-diagonal diffusion coefficients could not be neglected. In 1968, Coates, Subramanian and Purdy [7] extended both the CS and MS stability criteria to ternary solutions. In the case of the CS extension diffusional interactions were considered but in the MS extension such interactions were neglected. The purpose of the present note is to examine interface stability in ternary alloys via the WS method and include off-diagonal diffusivities in the liquid. Planar Steadv State Solutions

To determine a stability criterion it is convenient to first solve for the steady state concentration distribution in the liquid and the planar steady state temperature distribution in both the liquid and solid. The following conditions will be assumed: 1) both convection in the liquid and diffusion of solute in the solid can be neglected, 2) solute concentration in the liquid is small and 3) the extent of both solid and liquid is infinite. Let C 1 and C 2 be the solute concentrations in the liquid, Djk be the four elements of the ternary liquid diffusivity matrix and V be the constant velocity of the planar solid-liquid interface. The necessary diffusion equations to solve become [8]: °c1

~

Dn~C l + Dn~C 2

+

VVCI

&

(1)

OaC2 = D 2 1 ~ C ! + D 2 2 ~ C 2 + VVC 2

&

To solve the above system of equations, let P be a matrix which diagonalizes the diffusivity matrix D ie,

Here ~.1 and k 2 are the two necessarily positive eigenvalues of D. In addition, let the new concentration variables

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be: [~l : e-t[c'J

(3)

where the brackets denote vectors. The above transformation yields a diagonal system of equations for the planar interface case: - x 02~1 + v d~'~ a~

' azm

az-

(4)

where z-is the direction of motion of the planar interface. (The superscript bars on the various quantities of eqs. 4 are used here because eventually the results will be rewritten in terms of scaled variables.) The boundary conditions of eqs. 4 are determined from solute conservation. They are written as: vc;(k,-,) : o,, vc: +o,,vc~ vc~%-,) : o=,vcl ÷o.vcl

(5)

where kj=CjS°'/Ci'iq at the interface. Since it is assumed that equilibrium is maintained at the interface, to determine kj one must know the tie-lines in the solid-liquid region of the ternary phase diagram. The superscript i is used to denote the value of a quantity at the interface. Rewriting the boundary conditions 5 in terms of the new variables ~j one obtains. V - - -

(6)

V

~',cK,-,) t ~ )'

'~'20~-,) ~ az j

where the K~ are given by: 1

-1

'

-i

i

(7)

The relevant differential equations for heat flow are:

o,,v~, = --=a~'

(8)

at

for the solid and

/~,~

= ~a~

(9)

at

in the liquid. The quantities Dms and Dth L a r e the thermal diffusivities in the solid and liquid respectively and the prime denotes the solid phase. The interracial boundary conditions for the planar steady-state solutions of eqs. 8 and 9 are: = T'

and

T = r ~-ta,'~ + n ~

(10)

where Tm is liquidus temperature. Equation 10 expresses the fact that the temperature of the interface depends on, not one, but two solute concentrations. For dilute solutions the f~i can be viewed as a Taylor expansion ie.,

°

Physically the % are the slopes of the liquidus surface with respect to concentration measured along the eigenvectors of D.

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Conservation of heat yields, an additional expression for the velocity:

Here L is the latent heat of fusion and K s and K L are the thermal conductivities in the solid and liquid phases. At this point it is convenient to transform to scaled, dimensionless variables. Following WS these are expressed as:

'~j

z-v

~/ =-z =-~F~ )h

tv 2

t =--

T=--

)'1

~

rv

y =-Tm ~'1

Ks

--i

a~j

n =-M~ = . KL T•

03)

where r' is the solid-liquid surface energy divided by the latent heat of fusion. With the above definitions, the steady state planar interface solutions are given by: ~F~ = I+('Kc1)(1-e -~)

~e =

l+(i~_l)(l_e-X,/x:)

(14)

T e = I +MI +M2+Gz T i e = I +MI +Me +Gn-t z

where the superscript p denotes the planar case and G is given by G'~.I/VTM, G" being the temperature gradient in the liquid at the interface. In the above solutions it has been assumed that either eigenvaluc of D is much less than either thc liquid or solid thermal diffusivity. We are ultimately interested in the stability of the above planar solutions and this problem is addressed in the next section.

Stability Let the shape of the planar interface be perturbed by a function E~(x,t) where e is a small quantity and describes the shape of the interface in a direction x which is perpendicular to z. Furthermore, let =~1 cos ~ x e * : .

Following, WS the linear stability problem can be expressed in vector form as:

~g2 T T/

~2 = Tp

+ e T, [ cos~xe a*t

T/el

(15)

1"'1

0

~

'

The procedure at this point is to first substitute the above vectors into the following scaled differential equations Oz

--%X:~ 2 + O~F2 kt

Oz

&

O~Fz at

0

~ T = 0 and ~ T / = 0 1

retain all terms to first order in e and solve for 'Fll, '/I2 ... ~x.

(16)

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Before solving the above equations, a few changes in the boundary conditions given in the previous section must be noted. First, one must account for the Gibbs Thomson effect, i.e., the temperature of the interface changes with its curvature. And second, the derivatives with respect to z appearing in eqns. 6 and 12 must be replaced by derivatives with respect to the direction normal to the curved interface. The necessary boundary conditions which hold at the interface z = e ~ can be written as

&

c~

~

- (K:I)V 11+~-~

ov__.~ _ ~ d~ ova. = 7 ' 1 ~ _ 1 ) ( v : 1 ) , r . ( l + d ~ )

dx Ox

7'2

"~

at)

(17)

T = 1 +MtVl +M2,2+y(E (~(~2

0T / /I--

aT

aTl^.

d~(aT /

In addition to the above boundary conditions one also has that the perturbed functions (~11 etc) must approach

zero as tzl'**. The solutions to eqs. 16 are:

I +(I +~2 +a.)i/2 A,I

2

4

TI

B,I

7", =

C, I exp(- 27'2 [t27.2)

r<

D,[

I,,,I

E,I

-foj

--+II--I

+o

+--ao/

7'2 ]

Z)

(18)

0 By Taylor expanding the boundary conditions 17 about z = ¢ ~ = 0 and substituting into the above solutions, one obtains a linear system of five equations in the five unknowns A 1 through E 1. This system is given by:

C t -M 1Al -M2Bt +YEI =0 C1-DI +EI~ n-~nl)=o Ct +riD1=0

(19)

-RIA 1+(1-K1)(K 1+ao)E 1=0 7'1 7'1

-, BI +(I -vO-r

A2

=o

A2

where

Y = G-(KI-I)MI-(I~-I)~M2+¥~ 2 ~2

Rt = -l +°~2+a2 +(-I 4 o)Ir~+(K1=l)

(20)

7"1 [/7.1~2 2 ;'1 l~r~ 7"1 R2---~2+[/-~2 ) +to +-~2ao] +-~2(K2-1) The system of eqs. 19 has a nontrivial solution only if the determinant of the coefficient matrix equals

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After considerable algebra one then obtains:

f

K, ~

X,:

X! K2 / (unstable) X2 ~(o") )

(21)

The stability of the system is determined by the sign of a 0 in eq. 15; a0<0 implying stability. In eq. 21 above a 0 has been set equal to zero thus defining the planar to cellular transition. Expression 21 yields the stability criterion for a given wavelength of perturbation through the ¢02 dependence. To determine stability for all wavelengths one must compute the value ~c2 for which the RHS of 21 is an absolute minimum. With ~c2 substituted back in 21, the complete extension of the MS stability criterion to ternary systems is accomplished. A modified CS criterion can be obtained by neglecting capillarity (y =0 in 21). In this case the minimum of the RHS of 21 occurs at ~2,**. Thus one obtains: 7,1

o~- 2o +MltXl_I)+M2_0~_I) n+l X2

(22)

Inequalities 21 and 22 are the main result of this work. The important point being that off-diagonal liquid diffusivities have been included in the planar-cellular stability criterion. Acknowledgement

This work was supported by the National Science Foundation under contract number DMR-8919193. Reference 1.

2. 3. 4. 5. 6. 7. 8.

W.A. Tiller, J.W. Rutter, K.A. Jackson and B. Chalmers, Acta. Met., 1, 428 (1953). V.G. Smith, W.A. Tiller and J.W. Rutter, Canad. J. Phys., 33 723 (1955). W.W. Mullins and R.F. Sekerka, J. AppL Phys., 35, 444 (1964). D.J. Wollkind and L.A. Segel, Proc. Roy. Soc. (London), 268, 351 (1970). B. Chalmers, "Principles of Solidification," Wiley: New York, (1964). G.S. Cole and W.C. Winegard, J. Inst. Metals., 92, 322 (1964). D,E. Coates, S.V. Subramanian and G.R. Purdy, Trans of AIME, 242, 800 (1968). J.S. Kirkaldy and D.J. Young, "Diffusion in the Condensed State," Institute of Metals: London (1987).