THE DISTRIBUTION OF IMPURITY IN A SEMI-INFiNITE SOLIDIFIED MELT

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1 R 294 Philips Res. Rep. 11, , 1956 THE DISTRIBUTION OF IMPURITY IN A SEMI-INFiNITE SOLIDIFIED MELT by O. W. MEMELINK : Summary The transient distribution of impurity in a solidifying melt is calculated for the case of a homogeneous initial impurity concentration and constant velocity of the solid-liquid interface. The assumptions involved are: (1) the fraction of impurity concentration segregatingfrom the liquid at the interface is constant, (2) the transport of solute from and to the interface occurs by diffusion in the liquid of semiinfinite length. The transient distribution for various values of the segregátion constant is evaluated numerically. Résumé La distribution transitoire des impuretés dans une coulëe en cours de solidification est calculëe dans Ie cas d'une concentration initiale homogène d'impuretés et une vitesse constante de la surface de sëparation solide-liquide. Le calcul est basë sur les hypothèses suivantes: (1) la fraction de la conèentration d'impuretés se sëparant du liquide à la surface de séparation est constante, (2) Ie transport de produit dissous vers la surface de séparation a lieu par diffusion dans Ie liquide qui est considéré comme étant de longueur semi-infinie. La distribution transitoire pour diverses valeurs de la constante de ségrégation est êvaluëe numériquement. Zusammenfassung Die übergangsverteilung von Verunreinigungen einer erstarrenden Schmelze wird berechnet für den Fall einer homogenen Anfangskonzentration der Verunreinigungen und konstanter Geschwindigkeit der Erstarrungsfront. Folgende Annahmen werden zugrunde gelegt: (1) der Anteil der Verunreinigungskonzentration, der sich aus der flüssigen Phase an der Erstarrungsfront ausscheidet, ist konstant; (2) der Zu- und Abtransport des gelösten Stoffes auf die Erstarrungsfront erfolgt durch Diffusion in der halb-unendlichen Schmelze. Die übergangsverteilung für verschiedene Werte der Ausscheidungskonstanten wird numerisch bestimmt. 1. Introduetion Upon freezing of a melt the impurity solved in the liquid tends to segregate in the growing solid. At very low growth velocities equilibrium exists between the solute concentrations in the two phases. The ratio of the two concentrations is a constant k which is independent of the coneentration magnitudes so long as very small amounts of solute are considered. The segregation constant keff met in practice often differs appreciably from the value k, as the situation of equilibrium is upset by the finite velocity of the advancing solid-liquid interface. Two different mechanisms for the dependence of keff on growth rate have been proposed.

2 ! ' I 184 o. W. MEMELINK The influence of solute transport in the liquid on the rate of segregation is stressed upon by Burton et al. 1). Owing to turbulent flow the impurity concentration in the melt is homogeneous except for Ir thin layer ahead of the interface, where laminar flow should occur. In the laminar region transport of solute takes place by diffusion. The diffusion gradient, and consequently the concentration adjoining the solid (which determines the segregation rate), is strongly dependent upon the growth rate of the crystal. Opposite to this view, Hall 2,3) ignored the influence of transport and interpreted the behaviour of keff as a growth-rate-dependent impurityexchange process at the crystal surface. No attempt will he made to justify one or both of the mechanisms described; instead, a mathematical procedure will be presented for the derivation of the transient distrihution of impurity in a crystal pulled from an impure melt. The assumptions implied are: (1) No dependence of impurity exchange at the solid-liquid interface on growth rate; (2) The rate of segregation is exclusively determined by diffusion of the solute in the liquid; (3) No stirring; that is, the diffusion region is of semi-infinite length. The last assumption implies that in the stationary state keff will always be equal to unity,. ' Tiller et al. 4) havederived an approximate expression based on the same model. The treatment of Pohl 5) yielded a series expression for the impurity distribution which is rather difficult; to handle. As is shown below, it is possible to obtain an exact expression in closed form of the impurity distribution. 2. Mathematical formulation If z = Zs is the position of the solid-liquid interface moving at a constant speed v in the positive z-direction, all material for z < Zs will he solidified and all material for z > Zs will be liquid. The impurity' concentration in the liquid will he governed by the diffuion equation (Z> zs),(i) where D is the diffusion constant and CL the solute eoncentration in the melt. At the interface the gradient of CL is determined by the rate of remov:a:l of impurity from the recrystallized material, so'that ÖCL (1- k)vcl D ö;-' (z = zs, k > 0) (2)

3 DISTRIBUTION OF IMPURITY IN A SEMI-INFINITE SOLIDIFIED MELT 185 Furthermore, and C = Co for z 0 at t = 0, (3) CL --+ Co for all t as z (4) The required solution of (1) has to satisfy (2), (3) and (4). Once this solution has been found, the impurity concentration in the crystallized material is obtained by multiplying CL(Zs) by the equilibrium segregation coefficient k. 3. Solution of the mathematical problem By substituting Z = x + vt in eq. (1), which means physically that we choose the 'origin of the new coordinate system in the solid-liquid interface, the differential equation (1) is transformed into OCL = D 02CL + v ocl ot ox2 ox "(5) Introducing one obtains with the boundary the dimensionless variables y = vxj2d, (6) conditions ocl ö 2 CL öcl --=--+2--, ö-r oy2 oy OCL 2(1 - k)cl = -_ oy' (y = 0, -r 0, k > 0) (7) (8) CL = Co, (-r = 0, y 0) CL --+ Co. (-r 0, y ) (9) (10) After an initial increase of the impurity concentration in front of th e moving interface, a stationary distribution of impurity in the liquid will he reached, as the concentration in the solid approaches Co. Putting öcl/ö-r = 0 in eq. (7) and solving eq. (7) under the conditions (8) and (10) one obtains as the stationary distribution:.' l-k CL(y, 00) = -- Co e- + 2y Co' " k Returning now to the time-dependent differential equation (7) we introduce a new function f(y, r] by substituting l-k. CL(Y -r) = -- Co e-y-t f(y, r) + Co' k " (11) (12)

4 186 o. W. li1eliielink so that (7) reduces to of 02f = oy2' (13) The corresponding boundary conditions for f (y,.) are: of (1-2k)f= - oy - 2k e", (y = 0, r 0, k ;» 0) (14) f=o, f-+ O. (. = 0, y 0) (. 0, y-+oo) (15) (16) In view of the fundamental solution 1. (y2) fo(y,.) =, exp - -,. 2yn. 4. (17) we attempt to solve (13) under the conditions (14) to (16) by a linear combination of functions (17) as follows: 00 1 f [ (Y_Y')2] f(y,.)= -r= f(y',o)exp- dy', 2yn which approaches f(y, 0) as,.. O. For f(y, 0) a suitable expression has to be found for all real values of y between - 00 and 00. In the region y 0, the functionf(y, 0) must be zero according to (15). In the region y < 0, however, some function f (y, 0) has to he chosen such that the integral (18) obeys the conditions (14) and (16). On inspection, it is suggestive to try (18) where AI' A 2, A3, A4 and m are constants yet to be determined. Substituting (19) in (18), recallingf (y, 0) = 0 (y 0), and carrying out the integration, we obtain 2f(y, ) = AI.e T + Y erfc (2 + -y) + A2 et-y erfc (2i - f)+ + A3 em't+my erfc( 2i +.mi)+ A4 e m ' T - my erfc(2i - mi). (20) It is easy to verify that this function satisfies (16) since (. 0) (21)

5 DISTRIBUTION OF IJlIPURITY IN A SEMI.INFINITE SOLIDIFIED MELT 187 We note that erfc u = 1 - erf u, 2 u >tf-'".'- erfu =.r: f e- w1 vno dw. (22) Finally, application of the boundary condition (14) to eq. (20) leads to the values 1-2k A 2 = 1, Aa = 0, A4 = -, l-k m= 1-2k. (23) Some doubt may exist regarding the uniqueness of the solution of the differential equation with boundary conditions as given. Under certain assumptions concerning quadratical integrability and similar regularity conditions, which may be left out of the physical picture, the uniqueness of f(y, 0') was mathematically verified. 4. The impurity distribution in the crystàllized material The impurity concentration in the solid is given by Cs = kcl (0, 0'), which can be obtained from eqs (12), (20) and (23). The final result is found to be Cs = tc o [1 + erf [lr;] - (1-2k) e-4k(l-k)t P + erf [(1-2k) 1"-;H], (24) where 0' is the dimensionless variable v 2 t/4d and z = vt the positional coordinate in the solid. Expression (24) is identical with that of Hulme 6) and Smith et al. 7) wbo recently derived it by a different mathematical procedure. 5. Discussion of (24) Numerical values of CS/Co as a function of 0' with k as parameter have been calculated, and the results are shown in table I and fig These numerical values illustrate that for k < 0 5 and 0' 3 the accuracy is within one per cent if the expression (24) is approximated by the simpler formula Cs = Co1- (1-2k)e-4k(1-k)T. (25) In cases wbere the segregation coefficient k is smaller than unity, the deviation of impurity concentration Cs from Co is negligible beyond 0' = l/k. For k-values exceeding' unity, the ratio CS/Co is approximately equal to 1 beyond 0' = 1. The limiting form of (24) for (2k - 1)y-:; 1 is Cs = tco 1 + erf (1"-;) +,;-T. (26) vno'

6 ... <Xl ce TABLE I Impurity eencentratien Cs/ Co ' I' l !=>, ;<! l'l

7 DISTRIBUTION OF IMPURITY IN A SEMI INFINITE SOLIDIFIED MELT 189 From fig. 1 we notice that Cs/Co for k = 10 does not differ appreciably from the curve obtained by evaluation of (26) in the region of 7: considered, CD = l r.1 i- V 0.05 Q 1/ '1J 0.02 I- I) v 1'$" I :il r Fig. 1. Impurity concentratien in the solid. Acknowledgement The encouragement of Mr Penning in the analysis presented is gratefully acknowledged. His calculations on the same problem by a different method made it possible for us to check the numerical results. Eindhoven, February 1956 REFERENCES 1) J. A. Burton, R. C. Prim and W. P. Slichter, J. chem. Phys. 21, , ) R. N. Hall, Phys. Rev. 88, 139, ) R. N. Hall, J. phys. Chem. 57, , ') W. A. Tiller, K. A. Jackson, J. W. Rutter and B. Chalmers, Acta metal. 1, , ) R. G. Pohl, J. appl. Phys. 25, , ) K. F. Hulme, Proc. phys. Soc. 68 B, , ') V. G. Smith, W. A. Tiller and J. W. Rutter, Canad. J. Phys. 33, , 1955.