Directional solidification of a binary alloy with deformed melt-crystal interface and hydromagnetic effects

Int. 1. Engng Sci. Vol. 30, No. 5, pp. 551-559, 1992 Printed in Great Britain. All rights reserved

0020-7225/92 $5.00 + 0.00 Copyright @ 1992 Pergamon Press plc

DIRECTIONAL SOLIDIFICATION OF A BINARY ALLOY WITH DEFORMED MELT-CRYSTAL INTERFACE AND HYDROMAGNETIC EFFECTS D. N. RIAHI Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.A. Abatraet-Asymptotic analysis of unidirectional solidification of a binary alloy in the presence of buoyancy and magnetohydrodynamic effects is done for small segregation coefficient and long wave disturbances. We derive an evolution equation which governs the cellular structure of the melt-crystal interface and the binary alloy. This equation incorporates coupled morphological-convective instabilities, buoyancy and hydromagnetic effects. The presence of a vertical magnetic field can promote the onset of cellular structure due to morphological instabilities and leads to an effective segregation which is smaller than the corresponding one in the absence of the vertical magnetic field.

1. INTRODUCTION

The hydrodynamic effects in directional solidification are known to be important [l]. The flow affects the patterns and the critical conditions for the onset of the melt-crystal interface instabilities. The magnetohydrodynamic effects on the convective and the melt-crystal interface instabilities are of interest to the crystal growing community. In industrial crystal growth processes it is desirable to impose certain constraints, such as rotation and/or magnetic field(s), in an optimized manner, upon the system in order to suppress such instabilities which can lead to microdefect density in the crystal and thus reduce the quality of the produced crystal. Therefore, it is of interest to impose a magnetic field upon the system in order to determine an evaluation of the hydromagnetic effects on the possible instabilities. This paper studies the problem of hydromagnetic effects on directional solidification of a binary alloy in the presence of buoyancy, for small segregation coefficient k and for long wave disturbances, where k is defined as ratio of the concentration of solute in the solid to the concentration of solute in the liquid at the interface. It is an extension of the recent work by Young and Davis [2]. These authors investigated the problem in the absence of any hydromagnetic effects. Their weakly nonlinear analysis was based on a small parameter which measured the degree of the undercooling, so that morphological instabilities of the melt-crystal interface could be significant and the interface could be non-planar. They derived an evolution equation governing the cellular structure of the binary alloy and found that presence of buoyancy can inhibit the onset of cellular structure. In the present problem we find that the combined hydromagnetic and buoyancy effects lead to an effective segregation coefficient s smaller than the corresponding value in the case with no hydromagnetic effects [2] by an amount which is determined by the magnetic field parameter Q (Chandrasekhar number) and the diffusivity ratio r (ratio of the magnetic diffusivity coefficient ?I to solute diffusivity coefficient d). However, for zero gravity case, s is independent of Q and r. Consequently, hydromagnetic effects can enhance morphological instabilities only in the presence of buoyancy. The importance of buoyancy has recently been demonstrated by studies of convective instabilities of the melt during solidification of binary alloys with planar interface [3-51 and by studies of morphological instabilities of the deformed melt-crystal interface during solidfication of binary alloys [2]. In the case with no hydromagnetic effects [2], presence of buoyancy can inhibit the morphological instabilities. In the case with hydromagnetic effects (present study), a vertical magnetic field can be effective only through presence of buoyancy. 551

D. N. RIAHI

552

The present study of directional solidification of binary alloys with magnetohydrodynamic effects considers the case of k << 1. Systems with very small k (k < 0.01) have great practical interest since they become nearly pure substances upon solidification. The small k limit also provides an asymptotic method of approach which can be used to develop a nonlinear evolution equation for the interface shape whose solution can display cellular structure of the types observed experimentally [6,7].

2. GOVERNING

EQUATIONS

We consider the problem of dir~tional ~lidi~cation of a horizontal layer of a binary alloy melt under the influence of an external magnetic field in the vertical direction (Fig. 1). The governing system of non-dimensional equations and boundary conditions for the flow velocity II, perturbation magnetic field b and the deviation c of solute concentration from the basic concentration [4], under the usual Boussinesq approximation, is written in a coordinate system moving at velocity v. (crystal growth rate). It is [2,8] d ---+u+=-Vp+Rcz+V’u+Qr(;+b4’)b, az

(

&-;+a14

) b =(;+b++rV’b,

v*u=o,

(14

V*b=O,

(ld)

(i+u.V)c-exp(--z)u*z=(V2+$)c, subject to the boundary conditions at the interface z = h(x, y, t), u = 0,

)]c=(&+~~c.V~)h

[-&+(I-k)(l+$

Of)

+[I-exp(-z$l--(l-k)(l+g)],

c - h-f-‘h + 1 - exp(-z)

b*z=O,

+ TK = 0,

(lg) (lh, 9

and conditions as z + 03,

lbl < 00,

I4 -=c to,

c

=o.

-._e.-e

- MELT ._-.---

Fig. 1. Crystal growing configuration.

OH

Directional solidification of a binary alloy

553

Here p =$ + cp

I(Bz + b)12

is the modified pressure, ~1is the magnetic permeability, p is the reference density (a constant), B is the strength of the external magnetic field, R = /3g(l- k)c,d2/(vk$ is the Rayleigh number, Y is the kinematic viscosity, /3 is the fractional change in density due to a change in the solute concentration, g is acceleration due to gravity, z is a unit vector in vertical direction, in the solid at the interface, c, is the solute k = c,Ic,, ccc is the solute concentration concentration in the melt at the interface, M = mG,/G is the morphological parameter, m is the liquidus slope, G is the imposed temperature gradient, G, is the basic solute gradient at the interface, S, = v/d is the Schmidt number, Q = pB2d/(4n&v) is the Chandrasekhar number, r = q/d is the diffusivity ratio of magnetic field to solute concentration, T’= TMyv$/(LmGcd2) is the surface free energy parameter, TM is the melting point of the pure substance, y is the surface free energy, L is the latent heat, K is twice the mean curvature of the interface K = [h,(l

+ h;) - 2h,h,h,

+ h,(l

+ hf)](l

+ h; + h;)-3’2,

(3)

a subscript such as x means a/& and V2 is the horizontal gradient. The boundary conditions for b given in (li-j) are based on the assumptions that the adjoining medium at z = ~0is a non-conductor medium, while the crystal is a perfect conductor. For details on magnetohydrodynamic equations and the boundary conditions for b and u the reader is referred to Chandrasekhar [8]. For details on the equation for c and its boundary conditions at the liquid-crystal interface and at z = ~0 the reader is referred to Young and Davis [2]. Non-dimensionalization procedure is also given in Ref. 2. We shall assume that k<
(4)

and the crystal growth is assumed to be promoted vertically downward (in the positive z-direction) so that the solidification process generates destabilizing density gradient when heavier solute is being rejected. As in the work by Young and Davis [2], we have neglected the thermal contribution to the density gradient since solute gradient dominates over that of temperature [2]. The governing system of equations and boundary conditions (la-f) and (li-k) is simplified further, in the limit of S, = ~0, by considering the vertical components of double curl of (la, b) and using (lc, d). The system (la-f) and (li-k) can then be converted to the following form: A2(V4@- Rc) = -Qt[A,V’DE A2

[(

-t-6(&E - V6E) - z],

tV4 + V2D - 5 V2)E + V2D$] = 6(6q5 . WE - 6E . V&5). z, V2 + D - & c - exp(-z)A2# > at

$=D#=E=O )$I
IEI
(5b)

= 6@ . Vc,

z=h(x,y,t),

as

(54

z---*~,

(54

(5e7 f)

where D = d/az, AZ is horizontal Laplacian and # and E are the poloidal components of u and b given by (u,b)=6(&

E),G=VxVxz.

(6a, h)

As in the case with no magnetic field [2], the results of the present study indicate that the toroidal components of u and b are insignificant and thus are not included in (5,6). The

D. N. RIAHI

554

simplification of the limit of SC= tQis assumed here since the results given in [3] for the regime of practical interest SC>> 1 indicate that the limit of 5, = tQis expected to be achieved so long as SC3 10. Equations (5a-c) must then be solved subject to (lg, h), (11) (3) and (5d-f). We shall obtain the solutions using perturbation, asymptotic and multi-scales techniques similar to those of Young and Davis [2].

3. LINEAR

STABILITY

ANALYSIS

The stability of the basic state is studied by using the infinitesimal perturbation method. Our method of approach is similar to that carried out by Young and Davis [2] in the case with no magnetic field. We consider the linear version of the governing system of equations and boundary conditions given in the previous section for the dependent variables c, u - z = ug and b - z = f+. The resulting system is analyzed using normal modes in the x, y and t variables and by assuming that the critical wave number approaches zero as k-, 0 [3, 41. Hence, if (ug, c, &) = ]fi(z), E(z),

b)lexp(~ + k - 4, r = (x, Y),

the growth rate CTof the disturbance with wave number (Y(magnitude an eigenvalue of the following system:

(7% b)

of the wave vector k) is

(0’ - (~~)~fi+ Qr(D” - a2)D6 = cx2RE,

@a)

(D* - a2)[r(D2 - a2)6 + DB + (D - a)61 = 0,

(8b)

(D2 + D - a2 - a)E + exp(-z)fi

= 0,

ti=D~=~=[D+1-a2k-(a+cu2k)/(M-‘-1+~2~)]E=0 lfi]
(8~) at

as

z--,m.

z=O,

(8d) @e-g)

The interface boundary condition for c given in (8d) is determined by using (7) in the boundary conditions (lg, h), assuming h is of order E (amplitude of melt motion) and by eliminating h between these two boundary conditions. Guided by the previous works [2-41, we anticipated to rescale k such that k = a2f.

(9)

We next assume a power series expansion for dependent

variable,

(;)=@++)+...,

(10)

and use (10) into (8). At leading order we find Co = 6, = 0, E0= g, exp( -z),

(11)

where go is a constant of integration. The solutions given in (11) are for the case of the most dangerous disturbances which have the highest growth rate. This result is also consistent with the fact that every term of the boundary condition for c given in (8d) is at most of order one for M-+l,M>l

(12a, b)

[2]. Following Young and Davis [2], we anticipate that the conditions (12) can be necessary for morphological instabilities to occur. At order a2 we obtain El= El = -go[g,

g~[foexp(-z)+f,exp(-r~z)+f,l,

exp( -22) + g2 exp( -r,z

(134

- z) + g,z exp( -z)],

6, = go[eo exp( -z) + e, exp(-r,z)

+ e2],

(13b) (13c)

Directional solidification of a binary alloy

555

where r, = [l + (I+ 4Qt2)“*]/2t,

(14)

and the expressions for the coefficients fn, e,(n = 0, 1, 2) and g,(m = 1, 2, 3) are given in the Appendix. Using (13b) in the lower boundary condition for E,, we find the following expression for u:

w-'+ dr I[-N(Q) 6 I

(1-M-‘)cC2-Ly4r

u=

[

1_

R

_k

’

where N(Q, t) = 2(Qr + 1 - r)(l + ro)/[(l - t)(l - q,)].

(15b)

The result (15a) implies that CJis real, It is of interest to note that the condition of thermal diffusivity less than q is amply satisfied for terrestrial materials [9]. Also d, >> d [9]. Hence we shall assume that the following condition holds r>> 1.

(16)

The results (15) and condition (12) imply that it is impossible to have non-negative cr for t d 1, provided that l? is not too small which we assume to be true as in the case with no magnetic field [2]. Hence, all the solutions decay to zero for t s 1 as t increases. The non-trivial solutions, therefore, correspond to cases where t > 1 and, in particular, where the condition (16) is valid. For marginal stability, cr = 0 and (15a) is used to minimize R with respect to LY.We then find the following critical values for R and CY R, = N(Q, r)[l - kI%(M”2 a: =

- l)-‘1,

kf-T-l(bP - I).

(17a)

UW

We find that R, increases with Q, while LY,is independent of Q. The inhibiting effect of the magnetic field on the onset of motion is apparent from the former result. In a steady state, the energy released by the buoyancy force acting on the fluid must balance the energy dissipated by viscosity and Joule heating. This can be achieved only at higher value of R than is sufficient in the absence of Joule heating. In the limit of Q + 0, N(Q, r) + 2 and (17a, b) agree with the results for non-magnetic field case [2] in the limit of SC---*00. Figure 5 of the stability diagram given in Ref. 2 can then be used, after replacing 2 by N(Q, r), for the interpretation of our results for non-zero magnetic field. There are stable regions for both (M < 1, R < N(Q, z) and (M > 1, R, < R < iV(Q, z)), Morphological instability region corresponds to M > 1 (mostly in the region where (12) holds) and R -CR,, while convective instability region corresponds to R > N(Q, t). A sustained convection is possible only in this later range for R.

4. WEAKLY

NONLINEAR

ANALYSIS

Following Young and Davis [2] for the weakly nonlinear behavior of the system for small k, we let k=c*k,,M=l+s,O
(lSa-c)

where parameter E measures the degree of the undercooling. Similar to the non-magnetic field case [2], our linear results indicated that it is reasonable to consider the limit of small k near M = 1. Young and Davis [2] gave a reasonable example for this situation when silicon or germanium with trace contaminants are processed. Following Young and Davis [2], we

D. N. RIAHI

556

introduce resealed variables: (X’, y’) = &lB(X,y), z’ = 2, t’ = E?,

(19a-f)

($‘, E’) = E(& E), c’ = c, h = oh’. In terms of these resealed variables and parameters the governing system of equations and boundary conditions (5a-f), (lg, h) and (11) yield, to the leading order terms, the following system, after dropping the primes, A&A,

-i- D*)“# - ERC] = -Qt[A,(sA,

+ D2)DE -I-6(&E . V&E) . z],

Az(sA2 + D2)( [ r(eA, -I-D’) -t- D - ~~$1 E f D@ } = S(S@ . V&E - &E V6@) . z, l

D2+D+~Az--s2~

dt >

c - exp( -z)A,@

= SC$. Vc,

(2Oa) (20b)

(204

E = Q,= DC#I= (D f 1 - E~/c~)c- ~~(h,c, + h,c,) + g3(h, -t hk,) = c - (1 - E i- e’)eh + s21’A$ + 1 - exp(-eh) ]IpI]
as

=0

.z*~,

at

z = sh,

(20d) (2Oe-g)

where D = d/& and terms of order less than e3 in (20d) are neglected since they are not needed in the present analysis which is based on systems of equations and boundary conditions up to third powers of E. We now seek a solution in terms of a series in powers of E:

(21) In the order E (20) becomes D4@, + QzD3E1 = 0,

(224

D3(tD + l)E1 + D3$, = 0,

(22b)

D(D + l)cl - exp(-z)A,@,

= 0,

#I=D~I=El=(Dfl)cI=c,=O

(22c)

at

I&]
as

z=O,

z--,00

(22d) (22e-g)

(22) yield ~l=E1=e,=O.

(23)

In the order s2 (20) becomes D4+2 + QrD3E2 = 0,

(2W

D3(rD + l)Ez + D3@, = 0,

(24b)

D(D + 1)~~- exp(-z)A,#, #2=D~z=Ez=(D+l)~~=cCZ-H=0, ]&I < to, JE21c CQ,c2 = 0

= 0,

(24c) at

as

z-+ M,

z=O,

(24d) (2&-g)

where H=;h;-ho-i-.A2ho.

(25)

Directional solidification of a binary alloy

557

(24) yield G2 = E2 = 0, c2 = H exp( -z),

(26a, b)

In the order e3 (20) becomes D4#3 + QtD3E3 - Rc2 = 0,

(274

D3(rD + 1)E2 + D3$3 = 0,

Wb)

(D2+ D)c3 + A2c2 - exp(-z)A2#3

= 0,

@s=D@3=E3=(D+l)~3-($+k,)h,=O

at

1@31< m, (E31< 00, c3 = 0

as

(27~) z=O,

z + 00.

(274 (27e-g)

Using (26) (27) yield 43 = u~Hlg,>

E3 = W/g,,

(28a-c)

~3 = -cd2Hlgm

where u,, bI and c, are defined in (13) for u = 0. Using (25) and (28~) in (27d) for c3, we find the following evolution equation for h,,: k,r + kho + W%o

1 - &)

+ V2. [O - W’2h,l~(

= 0,

where V2 is the horizontal gradient. The expression for N(Q, t) given in (15b) increases with Q. For Q + 0, N(Q, t)+ 2. For Q+m and under the condition (16), N(Q, t)-,2Q. Similar to the work carried out by Young and Davis [2], we restrict our analysis to two-dimensional case only where dlay = 0. For R < N(Q, z), sustained convection is absent and transformation h, = A, r = IS/[1 - R/N(Q,

t)], x = l?g

(30a-c)

convert (29) to the following equation At + sA + A,,,,

+ [(l - A)A&

= 0,

(31)

where s = kI’l{s2[1 - R/N(Q,

z)]}.

(32)

Equation (31) is effectively the same amplitude equation as that derived in [2] for non-magnetic case where s equals a constant times ki. Here s increases with R but decreases with increasing Q both in the regime R l/4, initial disturbances decay to zero and the zero basic state value (A = 0) is regained. For s < l/4, the system is unstable and a cellular structure forms. This cellular structure is, however, unstable in the following sense. Numerical integration of the evolution equation as time increases leads to runaway of the roots of the cells. This is documented by Hyman et al. [lo]. For $ << l/4, the results reported in [2] indicate an apparent secondary instability where the tips of the cells show a tendency to split. It is clear from the above results and discussion that the presence of a vertical magnetic field can enhance the onset of cellular structure resulted from morphological instabilities only if R # 0. For zero gravity case (R = 0), the results are independent of magnetic field. Similarly sufficient large

558

D. N. RIAHI

Q(R/N << 1) has little effect on the morphological instabilities. Due to magnetohydrodynamic analog of Taylor-Proudman constraint [8], hydromagnetic force acting on the heavy solute near the interface reduces the diffusion of solute away from the interface. This increases the concentration gradient, thus increasing the tendency for the liquid to become supercooled. The result is an enhancement of the morphological instability.

5. DISCUSSION

The present investigation demonstrates that asymptotic analysis of directional solidification with magnetohydrodynamic effects are indeed possible for small segregation coefficient because the most dangerous disturbances have long wave length. In the linear theory we find that steady solutions are possible only for t > 1. The critical value R, increases with Q, while cu, is independent of Q and r. Both R, and a, are in exact agreement with the results given in [2] in the limit of Q ---, 0. Presence of a vertical magnetic field has stabilizing effects for convective instability region. For given R, sustained convection is not possible for sufficiently large Q. Linear growth rate of disturbances is independent of Q only for R = 0. Effects due to presence of a vertical magnetic field are destabilizing for morphological instability region since this region expands with Q. However, for R = 0, results are independent of Q. The result of the weakly nonlinear analysis of the cellular structure of the solid-liquid interface is a single partial differential equation which governs the weakly nonlinear behavior of either two or three-dimensional cells as a function of s. Buoyancy and magnetic field effects lead to an effective buoyancy-magnetic fields parameter B = R/N(Q, z). Non-zero B leads to an effective segregation s larger than k determined by R, Q and t. B decreases with increasing Q, while it increases with R. Hence, R and Q have opposing roles. For M > 1 and R < N(7’, z), magnitudes of flow velocity, perturbation magnetic field and solute concentration increase with Q, while they decrease with increasing B. For sufficiency large B (s > l/4), morphological instabilities are suppressed. For moderate values of B and k,[s < l/4, s = O(l)], as well as for sufficiently small values of B and k,(s << l/4), stable cellular structure is not possible. Young and Davis [2] presented a physical explanation that an increase in s increases the tendency for morphological instability to be suppressed. Their two-dimensional cellular structure result for s < 1/4[s = O(l)] 1s . similar in appearance to those of the experiment [7]. It is of interest to note that the onset of three-dimensional cellular structure is delayed when the effects of an inclined uniform magnetic field is considered. It is well known [8] that the dynamics of two-dimensional convection rolls parallel to the horizontal component of the magnetic field is not influenced by these additional effects, but three-dimensional disturbances are inhibited. The two-dimensional results presented in this paper thus has a direct application to the case of directional solidification with two-dimensional cellular structure on liquid-crystal interface in the presence of an inclined magnetic field. Coriell et al. [9] investigated the linear regime of the convective and interfacial instabilities during directional solidification of a binary alloy and in the presence of a vertical magnetic field. Their result that application of a vertical magnetic field increases the critical concentration for convective instability is in agreement with the results presented in Section 4 that R, increases with Q.

REFERENCES [l] S. H. DAVIS, J. Fluid Mech. 2l2, 241-262 (1990). [2] G. W. YOUNG and S. H. DAVIS, Phys. Rev. B S&3388-3396 (1986). [3] D. T. J. HURLE, E. JAKEMAN and A. A. WHEELER, Phys. Fluids X,624-626

(1983).

Directional solidification of a binary alloy

559

D. N. RIAHI, Phys. Fluids 31,27-32 (1988). D. S. RILEY and S. H. DAVIS, Physica D 39,231-238 (1989). L. R. MORRIS and W. C. WINEGARD, j. Inst. Met. !37,220-222(1969). K. A. JACKSON, in Solidification (Edited by T. J. HUGHEL and G. F. BOLLING), p. 133. American Society for Metals, Metal Park, Ohio (1971). [8] S. CHANDRASEKHAR, Hydrodynamic and Hydrornagnetic Stability. Oxford Univ. Press (l%l). [9] S. R. CORIELL, M. R. CORDES, W. J. BOETTINGER and R. F. SEKEKKA, J. Cryst. Growth 49, 13-28

[4] [5] [6] [7]

(1981).

[lo] J. M. HYMAN, A. NOVICK-COHEN

Phys. Rev. E 37,7603-7608

and P. ROSENAU,

(Revision received 11 September

1991; accepted 29 September

(1988).

1991)

APPENDIX Theexpressions

for coefficients f., e,,(n = 0, 1, 2) and g,(n = 1, 2, 3) are given below: fo =

R[l - Qr/(Qz

h = -hhfi= g, =h/Z eO= -R/(Qr

g, =X/C4 +

+ 1 - 91,

(Al) (Ah. b)

-&-A, + rd, g, = 1 -fi+

1 - T), e, = -&/Qr,

0,

e, = R/et.

(A3a-c) (A4a-c)