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Effect of growth rate dependent partition coefficient on the dendritic growth in undercooled melts
Acta metall.Vol. 35. No. 4, pp. 965-970, 1987 Printed in Great Britain. All rights reserved
Copyright
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OOOI-6160/87 %3.00 + 0.00 1987 Pergamon Journals Ltd
EFFECT OF GROWTH RATE DEPENDENT PARTITION COEFFICIENT ON THE DENDRITIC GROWTH IN UNDERCOOLED MELTS R. TRIVEDI,’ J. LIPTON’ and W. KURZ3 ‘Ames Laboratory-USDOE
and Department of Materials Science and Engineering, Iowa State University, Ames, IA 50011, U.S.A., rDr Miiller Feindraht AG, 8800 Thalwil-Zurich, Switzerland and 3Department of Materials Engineering, Swiss Federal Institute of Technology, 1007 Lausanne, Switzerland (Received 7 April 1986)
Abstract-The theory of alloy dendritic growth at large undercoolings is extended to include the effect of growth rate dependent partition coefficient on the growth rate, tip radius and composition of dendrites. Three distinct behaviors are observed depending on the value of the dimensionless rate 0).For small values of u, k z k,. For the intermediate range, when v is between 0.01 and 100, a significant effect of the velocity dependent partition coefficient is found. When v > 100, k + 1 and the dendrite behavior in an alloy is found to approach that for the pure material. In undercooled alloy melts segregation free zones could then be obtained by dendrite growth. RCsum~La theorie de croissance des dendrites dans des alliages en surfusion &levee est developpee pour inclure l’effet dun coefficient de partage, fonction de la vitesse. Cette theorie permet de dtduire la vitesse de croissance, le rayon de la pointe et la composition des dendrites. Trois comportements diKerents sont observes, d&pendant de la valeur de la vitesse adimensionnelle a. Pour des petites valeurs de a, k 1 k,, Pour des valeurs intermediaires (0.01 > u > 100) du coefficient de parage celui-ci a un effet important sur la croissance. Si u > 100, k -+ 1 et la croissance des dendrites approche celle dans un mat&au pur. Dans des fusions d’alliage en surfusion on pourrait alors obtenir des regions dendritiques sans segregation.
Zusammenfassung-Die Theorie des dendritischen Wachstums in Legierungen bei grol3er Unterkiihlung wird erweitert, urn den Einflug des von der Wachstumsrate abhangigen Verteilungskoeffizienten auf Wachstumsrate, Spitzenradius und Dendritenzusammensetzung zu erfassen. Drei unterschiedliche Verhaltensweisen werden beobachtet, je nach dem Wert der dimensionslosen Geschwindigkeit o. Bei kleinem o ist k g k,. Im mittleren Bereich, fur v zwischen 0,Ol und 100, findet sich ein deutlicher Einflul3 des geschwindigkeitsabhingigen Verteilungskoeffizienten. Fiir u > 100 geht k + 1 und das DendritenVerhalten in einer Legierung niihert sich dem in einem reinen Metall. In unterkiihlten Legierungsschmelzen kiinnten dann seigerungsfreie Zonen durch dendritisches Wachstum erhalten werden.
INTRODUCTION Dendritic structures are often observed during the rapid solidification of highly undercooled droplets or ribbons [l-4]. The fine scale of microstructure and the possible reduction in microsegregation that can be achieved under high undercooling conditions provide new possibilities for designing alloys with superior mechanical or corrosion properties. Consequently, a theoretical model is needed that can define rapid solidification processing parameters which will enable one to produce desired microstructures. Theoretical models have been developed in the literature [5-81 to understand the growth behavior of free dendrites which grow into undercooled melts. Also, careful experimental studies in the succinonitrile-acetone system [9] have confirmed the validity of these models. These results, however, are limited to small undercooling values since the stability criterion that is used in these models is valid only under small Peclet number conditions. In order
to extend the theoretical model to large undercooling values, two major modifications are required: (1) a proper stability criterion that is valid under high Peclet number conditions should be incorporated into the model, and (2) the large growth rates require the model to take into account the velocity-dependent distribution coefficient. Both these factors have been included in the theoretical model for constrained growth that was developed by Kurz et al. [lo], and these factors have been shown to alter significantly the dendrite characteristics at high growth rates. In the previous paper by the authors [I 11,a proper stability criterion that is valid under high Peclet number conditions [12] was incorporated into the theoretical model for free dendrite growth from an undercooled alloy melt. It was found that the proper stability criterion caused the dendrite growth rate to increase sharply with undercooling near the maximum undercooling value for the dendrite growth. For example, the theoretically predicted growth rate as well as the experimentally measured growth rate in
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the Ni-Sn system [3] is about l-10 m/s near the unit dimensionless undercooling conditions. At such velocities, the solute trapping effect should become quite important. Thus, in this paper, a theoretical model is developed which takes into account the proper stability criterion as well as the velocity dependent partition coefficient, so that it could be applied to study dendritic growth from highly undercooled alloy melts. DENDRITE GROWTH MODEL A detailed description of the dendrite growth model has already been presented [l 11,so that only the relevant equations will be summarized here. The theoretical treatment will be divided into the following three basic parts: (1) the total undercooling, (2) the stability criterion, and (3) the velocity dependent distribution coefficient. 1. Total undercooling
the distribution coefficient. The solute Peclet number, P, is related to the thermal Peclet number, P,, by the relationship, P, = VP,, in which t] = a/D. The capillary term in equation (3) contains Q*, the stability constant (” 1/47r2), and the stability function, F, which will be described below. For a given composition, G, equation (3) gives the value of the thermal Peclet number _’ P,, as a function - of the total bath undercooling, AT. Since P, = I/. R, equation (3) only predicts the values of the product - V. R, and the separate values of P and i? are obtained in this model by using the marginal stability criterion [12]. 2. The stability criterion The general stability criterion has been described earlier [l 1, 121, and it predicts the value of dimensionless dendrite tip radius as 3 = [a*P,F(A)]-’
(7)
where the function F(A) is given by [ll]
If an alloy with a liquidus temperature, TL, is undercooled to a temperature T,, then the total dimensionless undercooling, L\T, is given by AT = (T, - T&/H
(1)
where C, and H are the specific heat and the latent heat, respectively. The total undercooling can be divided into three parts as L\T = m, + m, + m,
-1)+2a*Z’,F(A).
(3)
The Ivantsov function, Z,, is given by [13] Z,(F,) = F, exp (F,) G (F,)
(4)
where the thermal Peclet number, F, = VR/2a, in which V and R are the growth rate and the tip radius of a dendrite and a is the thermal diffusivity. By defining the dimensionless velocity, r = Vd,/2a and the dimensionless radius, Z? = R/d,, the Peclet number can be written as P,= v.R
(5)
where d, = TCJH, r being the ratio of the surface energy to the entropy of fusion per unit volume. The solutal term in equation (3) contains a dimensionless concentration q = ]m 1C,C,/H, in which C, is the alloy composition and m the liquidus slope. The function A is the ratio of dendrite tip composition in liquid to the initial alloy composition, and its value is given by A =[l -(l
-k)Z,(P,)]-’
(6)
in which the solute Peclet number, P, = VR 120, and D is the solute diffusion coefficient in liquid and k is
(8)
where the stability functions 5, and 5, which influence the thermal and solute gradients at high undertoolings are given by the relationship [12] l
i;,=l-
J
1+1
(2)
where m,, n, and m, are the dimensionless thermal, solutal and curvature undercoolings, respectively. By substituting the appropriate values of these undercoolings, one obtains m=Z,(P,)+G((A
F(-4)=5,+5,2qC,A(l-k)
*
(9)
@P2 t 2k
1;,=1+ l-2k-
(10)
l+--& $_’
By substituting equation (8) into equa;on (3), one obtains the value of the Peclet number P, for a given bath undercooling. The value of the dendrite tip radius is then calculated by using equation (7). The growth- rate is then obtained from the definition of P,= V.R, so that P = Q* PfF(A).
(11)
The functions F(A) and A contain the solute distribution coefficient, k, which has a constant value (equal to the equilibrium value k,) at low growth rates. At high growth rates, k is a function of velocity, so that a functional relationship between k and V is required before equations (3), (7) and (11) can be solved to obtain the growth rate and dendrite tip radius as a function of undercooling. 3. Velocity dependent distribution coeficient The effect of velocity on the solute distribution coefficient, k, has been studied by Aziz [14] and Jackson et al. [15], and they predict a relationship of the form (12) where V is the growth rate, D the liquid diffusion
TRIVEDI el al.: DENDRITIC
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coefficient, and a, a length scale which is of the order of interatomic distance. The value of a,, is estimated to be 0.5-5.0 nm [16]. In terms of the dimensionless velocity, u = rlc r, equation (12) can be rewritten as
where E is the ratio a,,/h. A typical value of the capillarity length for metal will vary from lo-r0 to 10m9m, which is of the same order of magnitude as the parameter a,. Thus, the ratio L will be of the order of unity. Since the dendrite tip is determined by the marginal stability criterion, the stability criterion remains unchanged at high growth rates if the appropriate value of k is substituted in equation (7) [ 171.Thus, the effect of velocity dependent distribution on dendrite growth rate can be evaluated by substituting the value of k, given by equation (13), into equation (3) and (7). Since the functions A and F(A) in equation (3) and (7) contain k which is a function of velocity, an iterative procedure is required to obtain selfconsistent solutions of equations (3) (7) and (13).
DISCUSSION
The variation in the partition coefficient, k, with the parameter u is shown in Fig. 1 for different k, values. The results can be divided into three regions: (1) for D < 10-2, the k value is constant and equal to k,, (2) for 10e2 < Y < IO*,k depends very strongly on velocity and it changes from k. to 1, and (3) D > lo-*, k is essentially unity and independent of velocity. Note that the change in the k value is quite appreciable when k, is small. Since the distribution coefficient influences the solute gradient in liquid at the interface, a significant change in dendrite velocity with respect to the low Peclet number approximation is expected when u > 10e2. The effect of the velocity dependent distribution coefficient on the dendrite growth rate vs undercooling relationship is shown in Fig. 2 for k, = 0.1 and G = 0, 0.1, 0.5 and 5.0. For comparison, the results for the constant k = k, model are also shown
Fig. i . The variation in the partition coefficient, k, with the dimensionless velocity parameter, v = ‘ILV’,for different k, values.
10-3
10-Z
10-l
100
10
’
8 Fig. 2. The effect of k(vlon the growth rate versus undercooling relationship for C, = 0.1,0.5 and 5.0. k, = 0.1, c = 1, and t] = 104. The limiting case for G = 0, and k = /co= 0.1 are also shown for comparison (dotted lines).
in the figure as dotted lines. The result for 2 = 0 corresponds to the case of pure thermal dendrite. At low undercooling values, where k N ko, no change in the result is observed. However, at some undercooling value, k increases very sharply and changes from k, to 1 within a very short interval of undercooling. This sharp increase can be readily understood since not only k increases rapidly with 7 at - a certain velocity, but r also increases rapidly with AT at this velocity (e.g. Va AT3(1 11). At higher undercoolings, k + 1, and the dendrite growth behavior approaches that for the pure thermal case (?$ = 0), i.e. segregation free zones in undercooled alloy melts might be formed by such thermal dendrites. Note that, for dilute alloys, the transition from solute and thermal dendrite to pure thermal dendrite occurs below unit dimensionless undercooling value, so that the maximum undercooling will be L\T = 1. Consequently, no dendrite formation will be observed in the hypercooled region (m r 1). A localized maximum in undercooling is observed when the effects of the velocity dependent distribution coefficient just become important. We may consider this maximum to occur at some specific undercooling, AT+. If careful experimental studies [3,4] are carried out to measure dendrite growth rate as a function of undercooling, it is possible that a con-tinuous increase in velocity up to AT = AT+ will be found, and then a sharp increase in the growth rate will be observed when m is increased just above AT+. For high concentration alloys, AT+ will also be the maximum undercooling for which dendritic growth can occur, and the value of AT+ in this case will lie above AT = 1 but below the maximum undercooling for absolute stability for k = k, which has been derived as m, = 1 + AT,, [1l] where AT0 is the dimensionless equilibrium melting range. The variations in k and in interface concentrations, c,* (in solid) and ??, (in liquid) with undercooling are shown in Fig. 3 for the case k, = 0.1 and G = 0.1. For
TRIVEDI er al.:
968
DENDRITIC
GROWTH IN UNDERCOOLED
/
,,’
- 0
0.2
0.L
-kc k, 0.6
0.6
1.0
1.2
I5
Fig. 3. The vajation in k, c and e with undercooling for k, = 0.1 and C,, = 0.1. The results for C: with k = k,, = 0.1 are shown as a dotted line.
-this case k changes sharply from k0 to 1 around -AT N 0.21, so that for higher undercooling values, C,* N C,,, and a segregation free dendritic solidification is predicted (at least for the dendrite trunk which solidifies at the rate of the tip). The dotted -line shows the value of c when kzk,, . The value of CT, when k = k,, will be equal to C:/k, and it will vary from 0.1 to 1.O as the undercooling is increased. When the velocity dependent k is considered, Cf increases initially with undercooling but it then drops suddenly towards c value around -AT z AT+, which in this case is about 0.21. An important prediction of the model is the sharp change in P versus m behavior (at AT = dT+) when the effects of the velocity dependent k become important. Since AT+ decreases with decreasing composition, experimental studies on dilute alloys will be helpful in comparing the results of experiments with the theory to ascertain the validity of the k(v) relationship given by equation (13). AT+ will also depend on the parameters E and q (v = cq. r). Systems with large q (a/D) values will have lower AT+ value (i.e. the solute trapping effect will become important at lower undercooling values). Thus, for systems such as Al-Si where q = 12,000 or Sn-Pb where q = 8000, the solute trapping effect will become important at low v values. On the other hand, for succinonitrile-acetone with q = 90 or Fe-C with q = 300, the effect of k(v) will be smaller and will become important only at very high undercooling values. In all calculations, the value of D is assumed to be constant. However, for large undercoolings as observed with large C,-values, D can change significantly so that one should use D(T*), i.e. D value which corresponds to the dendrite tip temperature. Thus, the parameters q = a/D(T) and
6 = u,,/D(T) will increase as the undercooling is increased. If appropriate variation in D is taken into
MELTS
account, then the experimental measurements of AT+ will also give some insight into the parameter 6 since the effect of 6 on p is quite strong. Thus, if p can be measured experimentally for different alloys, one can measure a, values for each composition and check the basic assumption of the solute trapping equations which considers a,, to be independent of the alloy composition. Figures 4-6 show the effect of k(v) on V-G and K-G plot for three different undercooling conditions. The main effect here is that k is higher (k c 1) when P is large, so that k varies from some large value (which is determined by the largest velocity value) to k. as the concentration of solute is increased. Thus, one would expect k(u) to influence the -. results significantly when AT 1s large, G is small (i.e. P is large), or k, is small. This is illustrated in Figs 4-6 for m = 0.1, 0.2 and 0.5. The effect of k(u) is to shift the curves to the right (which is equivalent to having an effectively larger k,,
0
.OL
.12
.06
.20
16
G (a)
I
\ C-’
‘\
---I-- y__‘l_ 0.1 CO
0
0.2
(b)
Fig. 4. The variation in (a) the dimensionless velocity, P and (b) the dimensionless radius, R, with the dimensionless concentration, q, for a constant dimensionless undercooling value, m = 0.1. The dotted lines are for k = k, = constant and the solid lines are for k = k(u) (q = 10“ and L = 1).
TRIVEDI et al.:
DENDRITIC
GROWTH IN UNDERCOOLED
MELTS
969
‘x’o-2 4 k, = 0.1, 0.5. 0.8 -_
-_
I\
I>
-_ --__ ‘1 \ \\ .\ ',O.S \ \. -\ .
\O.’
\ 0
0.2
0.'
0
-0.9
’ 5x10-3
\
\
-_ --_
‘. \ .-
-.
-.
- - _.
--,_
0
0.'
G (a)
0.2
co (4
2x’o3x
5r'o2q
3
k,
=
IIZ:
/--.
I_\ I’
2
I _I
\
0 0
0.' CO
0.2
W
\ ___-_--
/I
0.5,
H'
,-
_-----
I
\ -
0
k, =
\ \ 1' _sc-
1
I
0.8 __---0.1.0.5. 0.8
01
.___--_--L_-_-I
0
0.' CO
0.2
,ib)
Fig. 5. The variation in (a) the dimensionless velocity, P, and (b) the dimensionless radius, R, with the dimensionless concentrat&n, C,, for a constant dimensionless undercooling, AT = 0.2. The dotted lines are for k = k, = constant and the solid lines are for k = k(o) (n = lo4 and t = 1).
Fig. 6. The variation in (a) the dimensionless velocity, P and (b) the dimensionless radius i?, with the dimensionless concentratiop C,, for a constant dimensionless undercooling, AT = 0.5. The dotted lines are for k = k, = constant and the solid lines are for k = k(u) (q = lo4 and L = 1).
alloy). The higher velocities observed at larger undertoolings show a more pronounced effect of k(u). A significant difference in the results for k = k, and k = k(v) is seen for small k, values at low concentrations. When the undercooling becomes large, i.e. 6T = 0.5, as shown in Fig. 6, the velocities are sufficiently high that k -+ 1. Consequently, the results are independent of the k, value, since all k + 1, and the results approach those for the pure thermal case, i.e. C, = 0.
in the model which describes the growth rate and tip radius of a dendrite which grows from an undercooled alloy melt. The effect of k(u) becomes important when the material parameters and the growth rates are such that u = ~6 ris larger than 10m2.When the effect of k(u) becomes important, a localized maximum in AT is predicted at AT = r. It is suggested that one may observe a sharp discontinuous increase in r at AT+ such that for m > AT+, k = 1 and the growth of dendrites is controlled by thermal diffusion. Only under these conditions, segregation free dendritic structures may be obtained for dilute alloys.
SUMMARY
At high melt undercoolings when the dendrite growth rate becomes large, local equilibrium cannot be assumed any more. Therefore a growth rate dependent partition coefficient, k(v), was introduced
Acknowledgemenrs-This study was supported financially by the “Kommission zur Forderung Wissenschaftlicher Forschung”, Bern, and Ames Laboratory, which is operated for the USDOE by Iowa State University, under contract
970
TRIVEDI et al.:
DENDRITIC
GROWTH IN UNDERCOOLED
No. 3-7405-ENG-82, supported by the Director of Energy Research, Office of Basic Energy Sciences.
REFERENCES 1. H. Jones, J. Mater. Sci. Engng 65, 145 (1984). 2. C. G. Levi and R. Mehrabian. Metall. Trans. A13. 13 (1983). 3. M. C. Flemings and Y. Shiohara, Mater. Sci. Engng 65, 157 (1984). 4. Y. Wu, T. J. Piccone, Y. Shiohara and M. C. Flemings, MeiaN. Trans. To be published. 5. J. S. Langer, Physicochem. Hydrodyn. 1, 41 (1980). 6. A. Karma and J. S. Langer, Phys. Rev. A30, 3147 (1984). 7. J. Lipton, M. E. Glicksman and W. Kurz, Muter. Sci. Engng 65, 57 (1984).
MELTS
8. J. Lipton, M. E. Glicksman and W. Kurz, Metall. Trans. To be published. 9. M. Chopra, Ph.D. thesis, RPI, Troy, N.Y. 10. W. Kurz, B. Giovanola and R. Trivedi, Acta metall. 34, 823 (1986). 11. J. Lipton, W. Kurz and R. Trivedi, Acta metall. 35,957 (1987). 12. R. Trivedi and W. Kurz, Acta metall. 34, 1663 (1986). 13. G. P. Ivantsov, Dokl. Akad. Nauk. S.S.S.R. 58, 567 (1947). 14. M. J. Aziz, J. Appl. Phys. 53, 1158 (1982). 15. K. A. Jackson, G. H. Gilmer and H. J. Leamy, Symp. Mater. Res. Sot., p 104. Academic Press, New York (1980). 16. W. J. Boettinger, S. R. Coriell and R. F. Sekerka, Mater. Sci. Engng 65, 27 (1984). 17. S. . Coriell and R. F. Sekerka, J. Crystal. Growth 61,499 (1983).
Copyright
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OOOI-6160/87 %3.00 + 0.00 1987 Pergamon Journals Ltd
EFFECT OF GROWTH RATE DEPENDENT PARTITION COEFFICIENT ON THE DENDRITIC GROWTH IN UNDERCOOLED MELTS R. TRIVEDI,’ J. LIPTON’ and W. KURZ3 ‘Ames Laboratory-USDOE
and Department of Materials Science and Engineering, Iowa State University, Ames, IA 50011, U.S.A., rDr Miiller Feindraht AG, 8800 Thalwil-Zurich, Switzerland and 3Department of Materials Engineering, Swiss Federal Institute of Technology, 1007 Lausanne, Switzerland (Received 7 April 1986)
Abstract-The theory of alloy dendritic growth at large undercoolings is extended to include the effect of growth rate dependent partition coefficient on the growth rate, tip radius and composition of dendrites. Three distinct behaviors are observed depending on the value of the dimensionless rate 0).For small values of u, k z k,. For the intermediate range, when v is between 0.01 and 100, a significant effect of the velocity dependent partition coefficient is found. When v > 100, k + 1 and the dendrite behavior in an alloy is found to approach that for the pure material. In undercooled alloy melts segregation free zones could then be obtained by dendrite growth. RCsum~La theorie de croissance des dendrites dans des alliages en surfusion &levee est developpee pour inclure l’effet dun coefficient de partage, fonction de la vitesse. Cette theorie permet de dtduire la vitesse de croissance, le rayon de la pointe et la composition des dendrites. Trois comportements diKerents sont observes, d&pendant de la valeur de la vitesse adimensionnelle a. Pour des petites valeurs de a, k 1 k,, Pour des valeurs intermediaires (0.01 > u > 100) du coefficient de parage celui-ci a un effet important sur la croissance. Si u > 100, k -+ 1 et la croissance des dendrites approche celle dans un mat&au pur. Dans des fusions d’alliage en surfusion on pourrait alors obtenir des regions dendritiques sans segregation.
Zusammenfassung-Die Theorie des dendritischen Wachstums in Legierungen bei grol3er Unterkiihlung wird erweitert, urn den Einflug des von der Wachstumsrate abhangigen Verteilungskoeffizienten auf Wachstumsrate, Spitzenradius und Dendritenzusammensetzung zu erfassen. Drei unterschiedliche Verhaltensweisen werden beobachtet, je nach dem Wert der dimensionslosen Geschwindigkeit o. Bei kleinem o ist k g k,. Im mittleren Bereich, fur v zwischen 0,Ol und 100, findet sich ein deutlicher Einflul3 des geschwindigkeitsabhingigen Verteilungskoeffizienten. Fiir u > 100 geht k + 1 und das DendritenVerhalten in einer Legierung niihert sich dem in einem reinen Metall. In unterkiihlten Legierungsschmelzen kiinnten dann seigerungsfreie Zonen durch dendritisches Wachstum erhalten werden.
INTRODUCTION Dendritic structures are often observed during the rapid solidification of highly undercooled droplets or ribbons [l-4]. The fine scale of microstructure and the possible reduction in microsegregation that can be achieved under high undercooling conditions provide new possibilities for designing alloys with superior mechanical or corrosion properties. Consequently, a theoretical model is needed that can define rapid solidification processing parameters which will enable one to produce desired microstructures. Theoretical models have been developed in the literature [5-81 to understand the growth behavior of free dendrites which grow into undercooled melts. Also, careful experimental studies in the succinonitrile-acetone system [9] have confirmed the validity of these models. These results, however, are limited to small undercooling values since the stability criterion that is used in these models is valid only under small Peclet number conditions. In order
to extend the theoretical model to large undercooling values, two major modifications are required: (1) a proper stability criterion that is valid under high Peclet number conditions should be incorporated into the model, and (2) the large growth rates require the model to take into account the velocity-dependent distribution coefficient. Both these factors have been included in the theoretical model for constrained growth that was developed by Kurz et al. [lo], and these factors have been shown to alter significantly the dendrite characteristics at high growth rates. In the previous paper by the authors [I 11,a proper stability criterion that is valid under high Peclet number conditions [12] was incorporated into the theoretical model for free dendrite growth from an undercooled alloy melt. It was found that the proper stability criterion caused the dendrite growth rate to increase sharply with undercooling near the maximum undercooling value for the dendrite growth. For example, the theoretically predicted growth rate as well as the experimentally measured growth rate in
965
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the Ni-Sn system [3] is about l-10 m/s near the unit dimensionless undercooling conditions. At such velocities, the solute trapping effect should become quite important. Thus, in this paper, a theoretical model is developed which takes into account the proper stability criterion as well as the velocity dependent partition coefficient, so that it could be applied to study dendritic growth from highly undercooled alloy melts. DENDRITE GROWTH MODEL A detailed description of the dendrite growth model has already been presented [l 11,so that only the relevant equations will be summarized here. The theoretical treatment will be divided into the following three basic parts: (1) the total undercooling, (2) the stability criterion, and (3) the velocity dependent distribution coefficient. 1. Total undercooling
the distribution coefficient. The solute Peclet number, P, is related to the thermal Peclet number, P,, by the relationship, P, = VP,, in which t] = a/D. The capillary term in equation (3) contains Q*, the stability constant (” 1/47r2), and the stability function, F, which will be described below. For a given composition, G, equation (3) gives the value of the thermal Peclet number _’ P,, as a function - of the total bath undercooling, AT. Since P, = I/. R, equation (3) only predicts the values of the product - V. R, and the separate values of P and i? are obtained in this model by using the marginal stability criterion [12]. 2. The stability criterion The general stability criterion has been described earlier [l 1, 121, and it predicts the value of dimensionless dendrite tip radius as 3 = [a*P,F(A)]-’
(7)
where the function F(A) is given by [ll]
If an alloy with a liquidus temperature, TL, is undercooled to a temperature T,, then the total dimensionless undercooling, L\T, is given by AT = (T, - T&/H
(1)
where C, and H are the specific heat and the latent heat, respectively. The total undercooling can be divided into three parts as L\T = m, + m, + m,
-1)+2a*Z’,F(A).
(3)
The Ivantsov function, Z,, is given by [13] Z,(F,) = F, exp (F,) G (F,)
(4)
where the thermal Peclet number, F, = VR/2a, in which V and R are the growth rate and the tip radius of a dendrite and a is the thermal diffusivity. By defining the dimensionless velocity, r = Vd,/2a and the dimensionless radius, Z? = R/d,, the Peclet number can be written as P,= v.R
(5)
where d, = TCJH, r being the ratio of the surface energy to the entropy of fusion per unit volume. The solutal term in equation (3) contains a dimensionless concentration q = ]m 1C,C,/H, in which C, is the alloy composition and m the liquidus slope. The function A is the ratio of dendrite tip composition in liquid to the initial alloy composition, and its value is given by A =[l -(l
-k)Z,(P,)]-’
(6)
in which the solute Peclet number, P, = VR 120, and D is the solute diffusion coefficient in liquid and k is
(8)
where the stability functions 5, and 5, which influence the thermal and solute gradients at high undertoolings are given by the relationship [12] l
i;,=l-
J
1+1
(2)
where m,, n, and m, are the dimensionless thermal, solutal and curvature undercoolings, respectively. By substituting the appropriate values of these undercoolings, one obtains m=Z,(P,)+G((A
F(-4)=5,+5,2qC,A(l-k)
*
(9)
@P2 t 2k
1;,=1+ l-2k-
(10)
l+--& $_’
By substituting equation (8) into equa;on (3), one obtains the value of the Peclet number P, for a given bath undercooling. The value of the dendrite tip radius is then calculated by using equation (7). The growth- rate is then obtained from the definition of P,= V.R, so that P = Q* PfF(A).
(11)
The functions F(A) and A contain the solute distribution coefficient, k, which has a constant value (equal to the equilibrium value k,) at low growth rates. At high growth rates, k is a function of velocity, so that a functional relationship between k and V is required before equations (3), (7) and (11) can be solved to obtain the growth rate and dendrite tip radius as a function of undercooling. 3. Velocity dependent distribution coeficient The effect of velocity on the solute distribution coefficient, k, has been studied by Aziz [14] and Jackson et al. [15], and they predict a relationship of the form (12) where V is the growth rate, D the liquid diffusion
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coefficient, and a, a length scale which is of the order of interatomic distance. The value of a,, is estimated to be 0.5-5.0 nm [16]. In terms of the dimensionless velocity, u = rlc r, equation (12) can be rewritten as
where E is the ratio a,,/h. A typical value of the capillarity length for metal will vary from lo-r0 to 10m9m, which is of the same order of magnitude as the parameter a,. Thus, the ratio L will be of the order of unity. Since the dendrite tip is determined by the marginal stability criterion, the stability criterion remains unchanged at high growth rates if the appropriate value of k is substituted in equation (7) [ 171.Thus, the effect of velocity dependent distribution on dendrite growth rate can be evaluated by substituting the value of k, given by equation (13), into equation (3) and (7). Since the functions A and F(A) in equation (3) and (7) contain k which is a function of velocity, an iterative procedure is required to obtain selfconsistent solutions of equations (3) (7) and (13).
DISCUSSION
The variation in the partition coefficient, k, with the parameter u is shown in Fig. 1 for different k, values. The results can be divided into three regions: (1) for D < 10-2, the k value is constant and equal to k,, (2) for 10e2 < Y < IO*,k depends very strongly on velocity and it changes from k. to 1, and (3) D > lo-*, k is essentially unity and independent of velocity. Note that the change in the k value is quite appreciable when k, is small. Since the distribution coefficient influences the solute gradient in liquid at the interface, a significant change in dendrite velocity with respect to the low Peclet number approximation is expected when u > 10e2. The effect of the velocity dependent distribution coefficient on the dendrite growth rate vs undercooling relationship is shown in Fig. 2 for k, = 0.1 and G = 0, 0.1, 0.5 and 5.0. For comparison, the results for the constant k = k, model are also shown
Fig. i . The variation in the partition coefficient, k, with the dimensionless velocity parameter, v = ‘ILV’,for different k, values.
10-3
10-Z
10-l
100
10
’
8 Fig. 2. The effect of k(vlon the growth rate versus undercooling relationship for C, = 0.1,0.5 and 5.0. k, = 0.1, c = 1, and t] = 104. The limiting case for G = 0, and k = /co= 0.1 are also shown for comparison (dotted lines).
in the figure as dotted lines. The result for 2 = 0 corresponds to the case of pure thermal dendrite. At low undercooling values, where k N ko, no change in the result is observed. However, at some undercooling value, k increases very sharply and changes from k, to 1 within a very short interval of undercooling. This sharp increase can be readily understood since not only k increases rapidly with 7 at - a certain velocity, but r also increases rapidly with AT at this velocity (e.g. Va AT3(1 11). At higher undercoolings, k + 1, and the dendrite growth behavior approaches that for the pure thermal case (?$ = 0), i.e. segregation free zones in undercooled alloy melts might be formed by such thermal dendrites. Note that, for dilute alloys, the transition from solute and thermal dendrite to pure thermal dendrite occurs below unit dimensionless undercooling value, so that the maximum undercooling will be L\T = 1. Consequently, no dendrite formation will be observed in the hypercooled region (m r 1). A localized maximum in undercooling is observed when the effects of the velocity dependent distribution coefficient just become important. We may consider this maximum to occur at some specific undercooling, AT+. If careful experimental studies [3,4] are carried out to measure dendrite growth rate as a function of undercooling, it is possible that a con-tinuous increase in velocity up to AT = AT+ will be found, and then a sharp increase in the growth rate will be observed when m is increased just above AT+. For high concentration alloys, AT+ will also be the maximum undercooling for which dendritic growth can occur, and the value of AT+ in this case will lie above AT = 1 but below the maximum undercooling for absolute stability for k = k, which has been derived as m, = 1 + AT,, [1l] where AT0 is the dimensionless equilibrium melting range. The variations in k and in interface concentrations, c,* (in solid) and ??, (in liquid) with undercooling are shown in Fig. 3 for the case k, = 0.1 and G = 0.1. For
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/
,,’
- 0
0.2
0.L
-kc k, 0.6
0.6
1.0
1.2
I5
Fig. 3. The vajation in k, c and e with undercooling for k, = 0.1 and C,, = 0.1. The results for C: with k = k,, = 0.1 are shown as a dotted line.
-this case k changes sharply from k0 to 1 around -AT N 0.21, so that for higher undercooling values, C,* N C,,, and a segregation free dendritic solidification is predicted (at least for the dendrite trunk which solidifies at the rate of the tip). The dotted -line shows the value of c when kzk,, . The value of CT, when k = k,, will be equal to C:/k, and it will vary from 0.1 to 1.O as the undercooling is increased. When the velocity dependent k is considered, Cf increases initially with undercooling but it then drops suddenly towards c value around -AT z AT+, which in this case is about 0.21. An important prediction of the model is the sharp change in P versus m behavior (at AT = dT+) when the effects of the velocity dependent k become important. Since AT+ decreases with decreasing composition, experimental studies on dilute alloys will be helpful in comparing the results of experiments with the theory to ascertain the validity of the k(v) relationship given by equation (13). AT+ will also depend on the parameters E and q (v = cq. r). Systems with large q (a/D) values will have lower AT+ value (i.e. the solute trapping effect will become important at lower undercooling values). Thus, for systems such as Al-Si where q = 12,000 or Sn-Pb where q = 8000, the solute trapping effect will become important at low v values. On the other hand, for succinonitrile-acetone with q = 90 or Fe-C with q = 300, the effect of k(v) will be smaller and will become important only at very high undercooling values. In all calculations, the value of D is assumed to be constant. However, for large undercoolings as observed with large C,-values, D can change significantly so that one should use D(T*), i.e. D value which corresponds to the dendrite tip temperature. Thus, the parameters q = a/D(T) and
6 = u,,/D(T) will increase as the undercooling is increased. If appropriate variation in D is taken into
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account, then the experimental measurements of AT+ will also give some insight into the parameter 6 since the effect of 6 on p is quite strong. Thus, if p can be measured experimentally for different alloys, one can measure a, values for each composition and check the basic assumption of the solute trapping equations which considers a,, to be independent of the alloy composition. Figures 4-6 show the effect of k(v) on V-G and K-G plot for three different undercooling conditions. The main effect here is that k is higher (k c 1) when P is large, so that k varies from some large value (which is determined by the largest velocity value) to k. as the concentration of solute is increased. Thus, one would expect k(u) to influence the -. results significantly when AT 1s large, G is small (i.e. P is large), or k, is small. This is illustrated in Figs 4-6 for m = 0.1, 0.2 and 0.5. The effect of k(u) is to shift the curves to the right (which is equivalent to having an effectively larger k,,
0
.OL
.12
.06
.20
16
G (a)
I
\ C-’
‘\
---I-- y__‘l_ 0.1 CO
0
0.2
(b)
Fig. 4. The variation in (a) the dimensionless velocity, P and (b) the dimensionless radius, R, with the dimensionless concentration, q, for a constant dimensionless undercooling value, m = 0.1. The dotted lines are for k = k, = constant and the solid lines are for k = k(u) (q = 10“ and L = 1).
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969
‘x’o-2 4 k, = 0.1, 0.5. 0.8 -_
-_
I\
I>
-_ --__ ‘1 \ \\ .\ ',O.S \ \. -\ .
\O.’
\ 0
0.2
0.'
0
-0.9
’ 5x10-3
\
\
-_ --_
‘. \ .-
-.
-.
- - _.
--,_
0
0.'
G (a)
0.2
co (4
2x’o3x
5r'o2q
3
k,
=
IIZ:
/--.
I_\ I’
2
I _I
\
0 0
0.' CO
0.2
W
\ ___-_--
/I
0.5,
H'
,-
_-----
I
\ -
0
k, =
\ \ 1' _sc-
1
I
0.8 __---0.1.0.5. 0.8
01
.___--_--L_-_-I
0
0.' CO
0.2
,ib)
Fig. 5. The variation in (a) the dimensionless velocity, P, and (b) the dimensionless radius, R, with the dimensionless concentrat&n, C,, for a constant dimensionless undercooling, AT = 0.2. The dotted lines are for k = k, = constant and the solid lines are for k = k(o) (n = lo4 and t = 1).
Fig. 6. The variation in (a) the dimensionless velocity, P and (b) the dimensionless radius i?, with the dimensionless concentratiop C,, for a constant dimensionless undercooling, AT = 0.5. The dotted lines are for k = k, = constant and the solid lines are for k = k(u) (q = lo4 and L = 1).
alloy). The higher velocities observed at larger undertoolings show a more pronounced effect of k(u). A significant difference in the results for k = k, and k = k(v) is seen for small k, values at low concentrations. When the undercooling becomes large, i.e. 6T = 0.5, as shown in Fig. 6, the velocities are sufficiently high that k -+ 1. Consequently, the results are independent of the k, value, since all k + 1, and the results approach those for the pure thermal case, i.e. C, = 0.
in the model which describes the growth rate and tip radius of a dendrite which grows from an undercooled alloy melt. The effect of k(u) becomes important when the material parameters and the growth rates are such that u = ~6 ris larger than 10m2.When the effect of k(u) becomes important, a localized maximum in AT is predicted at AT = r. It is suggested that one may observe a sharp discontinuous increase in r at AT+ such that for m > AT+, k = 1 and the growth of dendrites is controlled by thermal diffusion. Only under these conditions, segregation free dendritic structures may be obtained for dilute alloys.
SUMMARY
At high melt undercoolings when the dendrite growth rate becomes large, local equilibrium cannot be assumed any more. Therefore a growth rate dependent partition coefficient, k(v), was introduced
Acknowledgemenrs-This study was supported financially by the “Kommission zur Forderung Wissenschaftlicher Forschung”, Bern, and Ames Laboratory, which is operated for the USDOE by Iowa State University, under contract
970
TRIVEDI et al.:
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No. 3-7405-ENG-82, supported by the Director of Energy Research, Office of Basic Energy Sciences.
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8. J. Lipton, M. E. Glicksman and W. Kurz, Metall. Trans. To be published. 9. M. Chopra, Ph.D. thesis, RPI, Troy, N.Y. 10. W. Kurz, B. Giovanola and R. Trivedi, Acta metall. 34, 823 (1986). 11. J. Lipton, W. Kurz and R. Trivedi, Acta metall. 35,957 (1987). 12. R. Trivedi and W. Kurz, Acta metall. 34, 1663 (1986). 13. G. P. Ivantsov, Dokl. Akad. Nauk. S.S.S.R. 58, 567 (1947). 14. M. J. Aziz, J. Appl. Phys. 53, 1158 (1982). 15. K. A. Jackson, G. H. Gilmer and H. J. Leamy, Symp. Mater. Res. Sot., p 104. Academic Press, New York (1980). 16. W. J. Boettinger, S. R. Coriell and R. F. Sekerka, Mater. Sci. Engng 65, 27 (1984). 17. S. . Coriell and R. F. Sekerka, J. Crystal. Growth 61,499 (1983).