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Nucleation ahead of the advancing interface in directional solidification
MATERIALS SCIENCE & EWGIWEERIWG
ELSEVIER
Materials Science and Engineering A226-228
Nucleation
A
ahead of the advancing interface in directional solidification M. Gkmann
b Ames Labomlory
(1997) 763-769
a,*, R. Trivedi b, TV. Kurz a
’ Departmenl of Materials Engineering, Swiss Federal Institute 0s Techology, Lamarme, Switzerland US Department of Energy and Department of Materials Science and Engineering, Iowa State Univemity,
Ames,
USA
Abstract During directional solidification of an alloy, it is possible to nucleate the growing phase or a new phase at or ahead of the interface. This is critical in the phase selection, in the columnar to equiaxed transition under casting, welding or rapid solidi6cation conditions and the formation of bands in peritectic systems.Following Hunt, an appropriate theoretical model is developedto determinethe conditions under which nucleation can occur in the liquid closeto a moving solid-liquid interface for both, low and high interface velocities. At high growth rates, non-equilibrium effects are shown to play an important role in
predicting such transitions. 0 1997 Elsevier Science S.A. Keywords:
Solidification; Interface velocities; Non-equilibrium
effects
1. Introduction Prediction of solidification microstructures requires the modeling of nucleation of various phases and growth of corresponding morphologies. One often considers only growth of all possible phases and growth morphologies and chooses the one which under steadystate growth conditions develops the highest interface temperature [l]. Nucleation, however, can in many cases also play a critical role in the phase selection process [2]. In a more general approach, therefore, both nucleation and growth have to be considered. This is the case in castings [3], in which nucleation of equiaxed crystals occurs in the melt ahead of the interface. Nucleation at or ahead of the moving solid-liquid interface is believed to be important in the formation of layered structures (bands) in peritectic systems [4]. Elmer et al. [5] have observed layers of dispersed particles in directionally solidified systems under rapid solidification conditions. Experimental studies by Pierantoni et al. [6], in the Al-Si system, have shown that nucleation of silicon particles occurs ahead of the eutectic interface. They have shown that this occurs when the undercoot ing required for columnar eutectic growth is larger than * Corresponding
author.
0921-5093/97/$17.00 0 1997 Elsevier Science S.A. Ail rights reserved. ~~7~'071-~~01~0~\1~7~~-0
that for the nucleation of silicon. Recently Han et al. [7] have examined in situ the process of nucleation of the primary phase ahead of the eutectic interface in the hypereutectic carbon tetrabromide-hexachloroethane system at low velocities. In this case, nucleation occurs when the alloy composition deviates significantly from the eutectic composition. As has been shown by Hunt [8], Lipton et a!. [3], Cockcroft et al. [9] and others, equiaxed grains can
nucleate and grow ahead of the moving solidification interface when there is a region of undercooled liquid. This can eventually lead to the transition from a columnar to an equiaxed dendritic morphology, referred to as CET (columnar to equiaxed transition). Hunt was the first to develop an analytical model of this phenomenon. This model contains the basic physics of the CET. However, it uses simplified relationships to describe the undercooling of dendritic growth. Cockcroft et al. use a more recent growth theory for dendrites but do not include high velocity effects and can therefore not be used under conditions of rapid solidification. The aim of this paper is to develop a general theoretical model that establishes critical conditions for nucleation ahead of the moving solid-liquid interface and describes the growth and the extent of these crystals using recent solidification models including high velocity effects. Although the concept can be applied to
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different types of interfaces and growth morphologies, this paper will concentrate on the columnar dendritic growth and its transition to the equiaxed dendritic growth morphology of the same phase ahead of the moving interface, at low, as well as high solidification velocities.
2. Theoretical
model
During directional solidification, the heat flow is opposite to the growth direction. That is, the rate of advance of the isotherms constrains the solid-liquid interface to grow in steady state at this velocity. During the solidification of an alloy, solute will pile up ahead of the interface, when the distribution coefficient is less than unity. This change of concentration ahead of the interface will affect the local equilibrium solidification temperature. The interface structures vary from planar at low growth rates to cells and to dendrites which become finer and finer until they give rise to cellular structures again when close to the limit of absolute stability. At velocities higher than the limit of absolute stability, a planar interface is stabilized [l]. 2.1. Columnas growth It has been shown [lo] that the tip undercooling low velocity cells can be approximated by:
for
AT= GDJV
(1)
where G is the temperature gradient, D the diffusion coefficient of solute in liquid and V the solidification velocity. According to this model, cell tips are growing close to the limit of stability and no constitutional undercooling occurs ahead of the cellular front. During dendritic growth, a constitutionally undercooled zone always exists at the tips, and therefore, there is a driving force for nucleation ahead of the solid-liquid interface. Generally, in analytical models, the growth of dendrites is modeled in two stages. First, the mass and/or heat transport problems have to be solved, and then an operating point for the dendrite tip has to be selected
l-111. There exist two criteria specifying the dendrite tip radius upon the growth conditions. The first criterion for the operating point is growth at the extremum [12]. In this case, the dendrite tip undercooling is given for example by Burden-Hunt (BH) [13]: AT= (V-C0/A)“2
(2)
where C, is the alloy composition and A is a constant depending on the alloy system. This relation has been used by Hunt to calculate the CET. The second, and more recent, criterion is growth at marginal stability
and Engineering
A226-228
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763-769
[14], which corresponds qualitatively to solvability rion (see for example discussion in 1155):
R = Kr/~*>lhLG, - GN’12
crite(3)
where r is the Gibbs-Thomson coefficient and g.* is a stability constant given by 1/47c*. The parameter m, is the velocity dependent liquidus slope [16], [, is a stability parameter and G, is the solute gradient at the dendrite tip [17]: m,=m{l
-I<,(1 -ln[lc,/r’c]))/(l
-1~)
(4)
[, = 1 - 2k,/([l + (2?t/P,)Z]1’* - 1 + 2/C”)
(5)
G, = - (1 - 7cJ VCT/D
(6)
where k and m are the equilibrium partition coefficient and liquidus slope respectively, lc, is the velocity dependent partition coefficient [18], Cy is the composition of the liquid at the dendrite tip and P, is the solutal P&let number. They are given by: k, = (k + a, V/Q/( 1 + n, V/D)
(7)
CT = Co/( 1 - (1 - JG,)IU[PJ)
(8)
P, = RVj2D
(9)
where a, is a characteristic length scale for solute trapping and IV is the Ivantsov function [11,19]. The diffusion coefficient is assumed to follow an Arhenius type relation: D = Do evt -
Q&r]
(10)
where D, is a pre-exponential factor, R, the gas constant, T the temperature and Q an activation energy. In this paper we will consider growth at marginal stability, and use Ivantsov’s model to determine the composition profile in the liquid ahead of the dendritic interface. With the KGT model [17] we will determine the dendrite tip temperature in order to obtain the actual temperature profile in the liquid. The tip undercooling for columnar dendrites can be expressed by a sum of different contributions: AT= AT,+
AT,+ AT,
(11)
where AT, is the chemical undercooling, AT, is the curvature undercooling and AT, is the attachment kinetic undercooling. Expressions for them are: AT, = m,(C, - Cf)
(12)
AT, = 2rlR
(13)
ATk = V/p
(14)
where p is the linear kinetic coefficient. The dendrite tip temperature can then be calculated: Ttip= T,Sm,C,-
AT
where T, is the melting temperature
(15) of the pure metal.
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the liquid. This leads to an additional thermal undercooling which has to be added to the dendrite tip undercooling calculated in Eq. (ll), to relate dendrite growth characteristics to the total melt undercooling: AT, = Iv[P,] AH/c,
(16)
where AH is the latent heat, cp is the specific heat and P, the thermal P&let number, which is given by: 895 f 3 ‘1”“‘; 10’7 10’6
“,““i
P, = VR]Za
“+ 10‘5
lo”
Fig. 1. Liquid composition at the dendrite tip and resulting tip temperature as a function of velocity calculated for the Al-& system. co = 3 wt.%, G = 0.
Values of Ttip and Cf are plotted against velocity in Fig. 1. The values presented have been calculated using the physical properties of the Al-3 wt.% Cu (Table 1). One clearly sees, that an increase of the solidification velocity leads to an increase of the composition of the liquid at the interface. At high velocity, close to 1.0 m s -I, a plateau appears due to solute trapping, followed by a drastic increase close to absolute stability. The temperature of the dendrite tip decreases correspondingly, until it reaches a minimum value, then starts to increase again. This corresponds to absolute stability: the dendrite tip radius increases towards infinity and the plane front is stabilized again. 2.2. Equiaxed growth
Up to this point, we have been concerned with the columnar dendritic growth. The growth of equiaxed dendrites has been treated by Lipton et al. [ZO]. Contrary to the constrained growth where the latent heat is dissipated through the solid, during the free growth of dendrites the latent heat has to be dissipated through Table 1 Relevant physical parameters used for calculation (largely inspired from [8] and [23]) Nucleation pre-exponential factor Diffusion pre-exponential factor Activation energy Gibbs-Thompson coefficient Solidification constant Characteristic length scale Linear kinetic coefficient Polynomial phase diagram 09 (K a%) (K a%*) (K a”#) (K a%4) (K a”@) (K a”P) (K a”/b7)
I,: le*O [l s-l) Do: l.lew7 [mZ s-‘) Q: 23 800 (J mol-‘) l?: 2.4em7 (m K) A: 3em4 (m s-’ x wt.%pC’) a,: 7e-‘O (m) p: 2.88 (K s-* m-‘) Solidus Liquidus 933.60 933.60 - 5.3639 -62.231 -0.3165 9.6617 0.03036 -0.86953 - 192.64e-G 0.04696 - 104.70es6 -0.01509 4.61em6 26.64ee6 -56.26~~ - 198.90ewg
(17)
where a is the thermal diffusivity. Compared to the other contributions to the dendrite tip undercooling, the thermal undercooling is often negligible in metals with high thermal diffusivity. The calculation of the dendrite tip radius, however, requires the temperature gradient which is positive in the case of constrained growth (columnar) and negative in the case of free growth (equiaxed). However, in both cases, it is usually negligible compared to the contribution of the solutal gradient. Thus, it is possible to neglect the thermal gradient and use the same model for both cases of dendritic growth to characterize conditions at the dendrite tip. 2.3. NucZention
Nucleation is the dominant process at the beginning of solidification and leads very rapidly to the establishment of the final grain population, with each nucleus forming one equiaxed grain. The nucleation rate per unit volume can be written as [8]: I = (NO- N)I,, ‘exp[ - AG,/R,T]
(18)
where No is the total number of heterogeneous nucleation sites per unit volume, N is the number of sites that have already nucleated and I0 is a constant. The exponential term contains AG,, which is the free energy change associated with the formation of a critical nucleus (AG a l/AT’, where AT is the undercooling) and T is the temperature. The number of activated nucleation sites at a given undercooling is obtained by integration: rAT
N=
T
-?- dAT J o GV
Fig. 2 displays the nucleation rate and the number of nuclei as a function of undercooling. One can see that the total number of nuclei is rapidly equal to the number of nucleation sites available, once the nudeation undercooling is reached, The nucleation rate exhibits a sharp peak close the nucleation undercooling. 2.4. Undercooling ahead of the solid-liquid interface
During the advance of the columnar dendritic front, solute is rejected ahead of the solidification interface. The composition profile in the liquid for one dendrite
166
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,
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,
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0:o
200.0
40b.o
1 1.4 109
- 1.2 109
1.2 log--
of No
Number Nuclei,
1.0 log--
-s--z _- 1.0 109
8.0 108--
j- 8.0 108 z
6.0 lOB--
j- 6.0 10%$
2.0 lop-0.0 4.0 log-100
Nuclealion
2::
13 2
R&Z
-2.0 108 0.60
, o.;o
,4 0.50
, oh0
I 1.00
, 1.10
/
/ 1.30
1.20
1 -2.0 108 Lb0
AT [“Cl
Fig. 2. Nucleation rate and number of nuclei as a function of undercooling for the Al-3 wt.% Cu system. C, = 3 wt.%, V= 32 pm s-l, G= 1000 K m-l, No = 1 (1 mm3), AT, = 0.75”C.
tip assuming a parabolic Ivantsov’s solution [21]:
tip geometry
is given by
60b.O
8i0.0
lOdO.
z Mm1
Fig. 4. Equilibrium liquidus and local temperature profile ahead of a moving solid-liquid interface calculated for the Al-Cu system. C, = 3 wt.%, V=32 pm s-t, G= 1000 K m-l.
where E, is the integral exponential function and z is the distance in the liquid from the tip of the dendrite parallel to the dendrite axis. Correspondingly, a liquidus temperature profile can be calculated in the liquid, assuming a linear phase diagram in the solidification range:
Fig. 3 shows the concentration profile ahead of a moving dendritic solidification front for the Al-3 wt.% Cu system, at a solidification velocity of 32 pm s-i, while Fig. 4 shows the profile of the equilibrium liquidus and of the local temperature for the same conditions. By combining Eqs. (21) and (24), one gets the local undercooling prof?.le ahead of the moving solid-liquid interface:
WI = TI + ve, -
AT[z] = T[z] - T,[z]
WI
= C0 + (CT - G)-W,(2z
+
JQI~lIwc1
(20)
+ w~IIa~cl
~J-wd2~
(21)
where Tr is the equilibrium liquidus temperature at composition C, and Teq is the equilibrium liquidus temperature at the tip composition CT: T, = T,+mC,
(22)
Teq= T, i- rnCt
(23)
It is assumed for simplification, that the composition and liquidus temperature profiles are constant along the whole dendritic front and corresponds to those of the tips. The local temperature in the liquid is controlled by the heat extraction through the solid. It is given by: Tq[z] = Ttip + GZ
010
(24)
20b.o
40b.o
sob.0
80b.0
(25)
As shown in Fig. 5, the liquidus and temperature profiles lead to a constitutionally undercooled region ahead of the dendritic front. In this undercooled region, nucleation may happen if the maximum growth undercooling is larger than the undercooling required for nucleation. 2.5. Columnar to equinxed tn-msition (CET) Once nucleated, equiaxed grains will develop with various volume fractions, depending for instance on the thermal gradient or the solidification velocity. They will be incorporated into the columnar zone if their volume fraction is too small, or will supersede the columnar zone if their volume fraction is large. Hunt proposed that a volume fraction of equiaxed grains of 0.49 is the
lOdO.
2 [WI Fig. 3. Concentration profile ahead of a moving dendritic solid-liquid interface cakulated for the Al-0.1 system. C, = 3 wt.%, V= 32 urn s-r.
Fig. 5. Schematic representation of the undercooled region ahead of the moving solidification interface.
M. Giiumam et al. /Materials
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(1997) 763-769
767
limit beyond which one has to talk about an equiaxed microstructure and a limit of 0.0066 under which one has to talk about a columnar microstructure. In order to simplify the discussion, we propose to take a unique limit of 0.5. The volume fraction is obtained by calculating the size of each equiaxed grain when it gets entrapped by the columnar front. Numerically, this is obtained by integration: -6.0
I’,=
n -Y,kl & 0 v s
(26)
where z, is the distance from the interface in the undercooled liquid, where the undercooling is equal to the nucleation undercooling. The growth velocity for the equiaxed grains, V&Z], is calculated using the local composition and undercooling as obtained with Eqs. (20) and (25). When the calculation assumes a spherical growth of unhindered dendritic grains, then the extended volume fraction is: (6, = iQn]3r~
(27)
The actual volume fraction, removing the hypothesis of unhindered growth, is then obtained by applying Avrami’s equation [22]:
~=~--~~phLl
(28)
3. Discussion In order to compare between the present numerical model and Hunt’s analytical model for the CET, all calculations were made for the Al-Cu alloy as in Hunt’s original paper. The most relevant parameters are presented in Table 1. The composition profile varies along the isotherm of the dendrite tips, and is maximum at the dendrite tips and minimum in-between. This leads to a corresponding fluctuation of the liquidus temperature profile. In other words, the assumption of constant composition along the solidification front minimizes mathematically the undercooling, thus underestimates the CET.
a / 1.0
/ 2.0
1
, 3.0
log Gradient
4 / 4.0
! ’ 1 ’ 1 p 5.0
fK/m]
Fig. 6. Comparison of Hunt’s analytical model and the present numerical model for the AI-Cu system.
leads to a smaller undercooling in front of the moving interface, which reduces the driving force for the growth of equiaxed grains. Thus, equiaxed grains occur only at higher velocities compared to the present numerical model. In the middle region of Fig. 6, the transition lines predicted by the two models cross. The difference of slopes of G-V is explained by the weaker dependence of the dendrite tip undercooling on the growth velocity in KGT than in BH. At high velocity, the numerical model predicts an increase of the slope of the transition line, which stabilizes the columnar dendritic morphology. This has to be related to Fig. 1, where at high velocity, when the absolute stability is about to be reached, the tip temperature (Ttip) increases, thus, decreasing the undercooled region and reducing the driving force for the equiaxed dendritic growth. 3.2. Influence of different parameters on the CET
Fig, 7 displays the CET lines for different nucleation undercoolings. At low velocity, the dendrite tip undercooling is weak, thus, the nucleation undercooling plays a selective role. The lower the nucleation undercooling is, the easier it is to form equiaxed grains. If no undercooling is required in order to form a new grain
3.1. Comparison of Hunt’s and the present numerical model
Fig. 6 displays the transition line between the columnar structure and the equiaxed structure for the analytical solution by Hunt, and for the numerical solution. In Fig. 6, one can see that at low temperature gradients the CET occurs at lower velocities than in Hunt’s description. This is linked to the different growth criteria that are used. The criterion of growth at the extremum as considered in Hunt’s publication underestimates the dendrite tip undercooling. This in turn
1.0
2.0
3.0 log Gradient
4.0
5.0
[K/m]
Fig. 7. Influence of the nucleation undercooling
on the CET.
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4. Conclusions
~~~~~~r 4I / 1.0
2.0
3.0
4.0
5.0
log Gradient [K/m]
Fig. 8. Influence of the number of nucleation sites on the CET.
(like in the case of dendrite arm detachment due to convection), then, there is no change of slope for the CET line in the lower velocity part. Fig. 8 shows the influence of the number of nucleation sites on the columnar to equiaxed transition. In the higher velocity region, the important parameter is the, number of nucleation sites. Indeed, once nucleation occurs, the volume fraction of equiaxed grains is mainly linked to the number of nucleation sites. Therefore, if the number of nucleation sites is reduced, then the columnar microstructure is stabilized, as can be seen in Fig. 8. The number of nucleation sites and the nucleation undercooling are parameters that are specific to the equiaxed grains and control the formation of nuclei. On the other hand, the nominal composition is a parameter that is rather specific to the growth of the dendritic solidification front, in the sense that it controls the amount of solute rejection. Thus, the composition is a factor that has a strong influence on the shape and magnitude of the undercooled region. This is verified in Fig. 9, where increasing the nominal composition leads to an increased tendency to form equiaxed grains. Indeed, in most cases, the dendrite tip undercooling e increases with increasing alloy composition.
In this paper a modification of Hunt’s model for the columnar to equiaxed transition is proposed. It considers a more recent dendritic growth model, which is more accurate for the calculation of the .dendrite tip undercooling. The model takes into account the local concentration and,, r&ore importantly, the local undercooling that the equiaxed grains are exposed to in front of the solid-liquid interface at low, as well as high velocity. It has been shown for the Al-Cu system, that the numerical model proposed in this paper predicts, at low gradients, the equiaxed transition at lower velocities than in Hunt’s model, but that at higher gradients there is less driving force to nucleate equiaxed grains. This is linked to the growth model used in this paper which also considers high velocity effects on the partition coefficient and the liquidus slope. Thus, the present model is appropriate for predicting the microstructures in casting as well as in various welding situations or other processes involving rapid solidification, Finally, the proposed model is suitable to predict the columnar to equiaxed transition in a wide range of velocities or gradients, but also to predict phase selection in the undercooled region ahead of the interface.
Acknowledgements This work was carried out in part (RT) at Ames Laboratory which is operated for the US Department of Energy at Iowa State University under Contract No. W-7405-Eng-82, supported by the Director of Energy Research. The help of P. Gilgien and 0. Hunziker at the Swiss Federal Institute of Technology, Lausanne, in preparing the numerical model is appreciated.
References 111W. Kurz and R. Trivedi, Mater. 121G. Shao, and P. Tsakiropoulos, [31
Sci. Eng., A179-160 Acta Metall. Mater.,
(1994) 46. 42 (1994)
2937. J. Lipton, W. Heinemann and W. Kurz, Ax/z. Eiselzhii~te~nlllcs., 55 (1984)
195.
[41 R. Trivedi, Metall. and Ma&F. Turns. A, 26A (1995) 1583. 151 J.W. Elmer, SM. Allen and T.W. Eagar, Metall. Trans. A, 20A (1989)
-5.0 -(jolt
!a
-P
-j----7 1
1.0
I
I1
210
s
2117.
[61 M. Pierantoni, M. Gremaud, P. Magnin, D. Stall and W. Kurz, IC~LUMNAR +--j-
-----~ /
3.0
I
,
4.0
log Gradient [Wm]
Fig. 9. Influence of composition on the CET.
dcta 4 I*,4
5.0
Metall.
Mater.,
40 (1992)
1637.
Han and Trivedi, unpublished work, Ames Laboratory, Ames IA USA, 1996. Sci. Bg., 65 (1984) 75. 181 J.D. Hunt, Mater. [91 S.L. Cockcroft, M. Rappaz, A, Mitchell, J. Fernihough and A.J. Schmalz, Materials for Aduorzced PONW Engbzeeriug, Pat’t II, Kluwer Academic Publishers 1994, Netherlands, p. 1145. [71
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[lo] J.D. Hunt and S.-Z. Lu, Metall. Mater. Trunx. A, 27A (1996) 611 [ll] W. Kurz and D.J. Fisher, Funciamentals of SoZidijkation, third edition, Trans Tech Publications 1989, Switzerland. [12] C. Zener, Trans. TMS-AIME, 167(1946) 550. 1131 M.H. Burden and J.D. Hunt, J. CrJst. Growth, 22 (1974) 109. 1141 J.S. Langer and H. Miiller-Krumbhaar, J. Cryst. Growth, 4.2 (1977) 11. [15] W. Kurz and R. Trivedi, Acta Metall. Mater., 38 (1990) 1. [16] W.J. Boettinger and S.R. Coriell, in P.R. Sahm, H. Jones, CM. Adams (eds.), Science and Technology of the Undercooled Melt, NATO AS1 Series E-Noll4, Martinus Nijhoff, Dordrecht,
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Netherlands, 1986, p. 81. 1171 W. Kurz, B. Giovanola and R. Trivedi, Acta Metall., 34 (1986) 823. [18] M.J. Aziz, J. Appl. P/zys., 53 (1982) 1158. [19] R. Trivedi and W. Kurz, 6~. Mater. Rev., 39 (1994) 49. [20] J. Lipton, W. Kurz and R. Trivedi, Acta Metall., 35 (1987) 957. [21] D.E. Temkin, Sov. Phys. Cry&, 7 (1962) 354. [22] D.A. Porter and K.E. Easterling, Phase Transformations in IMetals and Alloys, second edition, Chapman and Hall 1992, London. [23] S.C. Gill and W. Kurz, Acta Metall. Muter., 43 (1995) 139.
ELSEVIER
Materials Science and Engineering A226-228
Nucleation
A
ahead of the advancing interface in directional solidification M. Gkmann
b Ames Labomlory
(1997) 763-769
a,*, R. Trivedi b, TV. Kurz a
’ Departmenl of Materials Engineering, Swiss Federal Institute 0s Techology, Lamarme, Switzerland US Department of Energy and Department of Materials Science and Engineering, Iowa State Univemity,
Ames,
USA
Abstract During directional solidification of an alloy, it is possible to nucleate the growing phase or a new phase at or ahead of the interface. This is critical in the phase selection, in the columnar to equiaxed transition under casting, welding or rapid solidi6cation conditions and the formation of bands in peritectic systems.Following Hunt, an appropriate theoretical model is developedto determinethe conditions under which nucleation can occur in the liquid closeto a moving solid-liquid interface for both, low and high interface velocities. At high growth rates, non-equilibrium effects are shown to play an important role in
predicting such transitions. 0 1997 Elsevier Science S.A. Keywords:
Solidification; Interface velocities; Non-equilibrium
effects
1. Introduction Prediction of solidification microstructures requires the modeling of nucleation of various phases and growth of corresponding morphologies. One often considers only growth of all possible phases and growth morphologies and chooses the one which under steadystate growth conditions develops the highest interface temperature [l]. Nucleation, however, can in many cases also play a critical role in the phase selection process [2]. In a more general approach, therefore, both nucleation and growth have to be considered. This is the case in castings [3], in which nucleation of equiaxed crystals occurs in the melt ahead of the interface. Nucleation at or ahead of the moving solid-liquid interface is believed to be important in the formation of layered structures (bands) in peritectic systems [4]. Elmer et al. [5] have observed layers of dispersed particles in directionally solidified systems under rapid solidification conditions. Experimental studies by Pierantoni et al. [6], in the Al-Si system, have shown that nucleation of silicon particles occurs ahead of the eutectic interface. They have shown that this occurs when the undercoot ing required for columnar eutectic growth is larger than * Corresponding
author.
0921-5093/97/$17.00 0 1997 Elsevier Science S.A. Ail rights reserved. ~~7~'071-~~01~0~\1~7~~-0
that for the nucleation of silicon. Recently Han et al. [7] have examined in situ the process of nucleation of the primary phase ahead of the eutectic interface in the hypereutectic carbon tetrabromide-hexachloroethane system at low velocities. In this case, nucleation occurs when the alloy composition deviates significantly from the eutectic composition. As has been shown by Hunt [8], Lipton et a!. [3], Cockcroft et al. [9] and others, equiaxed grains can
nucleate and grow ahead of the moving solidification interface when there is a region of undercooled liquid. This can eventually lead to the transition from a columnar to an equiaxed dendritic morphology, referred to as CET (columnar to equiaxed transition). Hunt was the first to develop an analytical model of this phenomenon. This model contains the basic physics of the CET. However, it uses simplified relationships to describe the undercooling of dendritic growth. Cockcroft et al. use a more recent growth theory for dendrites but do not include high velocity effects and can therefore not be used under conditions of rapid solidification. The aim of this paper is to develop a general theoretical model that establishes critical conditions for nucleation ahead of the moving solid-liquid interface and describes the growth and the extent of these crystals using recent solidification models including high velocity effects. Although the concept can be applied to
164
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Giizrmann
et 01, /Materials
Science
different types of interfaces and growth morphologies, this paper will concentrate on the columnar dendritic growth and its transition to the equiaxed dendritic growth morphology of the same phase ahead of the moving interface, at low, as well as high solidification velocities.
2. Theoretical
model
During directional solidification, the heat flow is opposite to the growth direction. That is, the rate of advance of the isotherms constrains the solid-liquid interface to grow in steady state at this velocity. During the solidification of an alloy, solute will pile up ahead of the interface, when the distribution coefficient is less than unity. This change of concentration ahead of the interface will affect the local equilibrium solidification temperature. The interface structures vary from planar at low growth rates to cells and to dendrites which become finer and finer until they give rise to cellular structures again when close to the limit of absolute stability. At velocities higher than the limit of absolute stability, a planar interface is stabilized [l]. 2.1. Columnas growth It has been shown [lo] that the tip undercooling low velocity cells can be approximated by:
for
AT= GDJV
(1)
where G is the temperature gradient, D the diffusion coefficient of solute in liquid and V the solidification velocity. According to this model, cell tips are growing close to the limit of stability and no constitutional undercooling occurs ahead of the cellular front. During dendritic growth, a constitutionally undercooled zone always exists at the tips, and therefore, there is a driving force for nucleation ahead of the solid-liquid interface. Generally, in analytical models, the growth of dendrites is modeled in two stages. First, the mass and/or heat transport problems have to be solved, and then an operating point for the dendrite tip has to be selected
l-111. There exist two criteria specifying the dendrite tip radius upon the growth conditions. The first criterion for the operating point is growth at the extremum [12]. In this case, the dendrite tip undercooling is given for example by Burden-Hunt (BH) [13]: AT= (V-C0/A)“2
(2)
where C, is the alloy composition and A is a constant depending on the alloy system. This relation has been used by Hunt to calculate the CET. The second, and more recent, criterion is growth at marginal stability
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[14], which corresponds qualitatively to solvability rion (see for example discussion in 1155):
R = Kr/~*>lhLG, - GN’12
crite(3)
where r is the Gibbs-Thomson coefficient and g.* is a stability constant given by 1/47c*. The parameter m, is the velocity dependent liquidus slope [16], [, is a stability parameter and G, is the solute gradient at the dendrite tip [17]: m,=m{l
-I<,(1 -ln[lc,/r’c]))/(l
-1~)
(4)
[, = 1 - 2k,/([l + (2?t/P,)Z]1’* - 1 + 2/C”)
(5)
G, = - (1 - 7cJ VCT/D
(6)
where k and m are the equilibrium partition coefficient and liquidus slope respectively, lc, is the velocity dependent partition coefficient [18], Cy is the composition of the liquid at the dendrite tip and P, is the solutal P&let number. They are given by: k, = (k + a, V/Q/( 1 + n, V/D)
(7)
CT = Co/( 1 - (1 - JG,)IU[PJ)
(8)
P, = RVj2D
(9)
where a, is a characteristic length scale for solute trapping and IV is the Ivantsov function [11,19]. The diffusion coefficient is assumed to follow an Arhenius type relation: D = Do evt -
Q&r]
(10)
where D, is a pre-exponential factor, R, the gas constant, T the temperature and Q an activation energy. In this paper we will consider growth at marginal stability, and use Ivantsov’s model to determine the composition profile in the liquid ahead of the dendritic interface. With the KGT model [17] we will determine the dendrite tip temperature in order to obtain the actual temperature profile in the liquid. The tip undercooling for columnar dendrites can be expressed by a sum of different contributions: AT= AT,+
AT,+ AT,
(11)
where AT, is the chemical undercooling, AT, is the curvature undercooling and AT, is the attachment kinetic undercooling. Expressions for them are: AT, = m,(C, - Cf)
(12)
AT, = 2rlR
(13)
ATk = V/p
(14)
where p is the linear kinetic coefficient. The dendrite tip temperature can then be calculated: Ttip= T,Sm,C,-
AT
where T, is the melting temperature
(15) of the pure metal.
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the liquid. This leads to an additional thermal undercooling which has to be added to the dendrite tip undercooling calculated in Eq. (ll), to relate dendrite growth characteristics to the total melt undercooling: AT, = Iv[P,] AH/c,
(16)
where AH is the latent heat, cp is the specific heat and P, the thermal P&let number, which is given by: 895 f 3 ‘1”“‘; 10’7 10’6
“,““i
P, = VR]Za
“+ 10‘5
lo”
Fig. 1. Liquid composition at the dendrite tip and resulting tip temperature as a function of velocity calculated for the Al-& system. co = 3 wt.%, G = 0.
Values of Ttip and Cf are plotted against velocity in Fig. 1. The values presented have been calculated using the physical properties of the Al-3 wt.% Cu (Table 1). One clearly sees, that an increase of the solidification velocity leads to an increase of the composition of the liquid at the interface. At high velocity, close to 1.0 m s -I, a plateau appears due to solute trapping, followed by a drastic increase close to absolute stability. The temperature of the dendrite tip decreases correspondingly, until it reaches a minimum value, then starts to increase again. This corresponds to absolute stability: the dendrite tip radius increases towards infinity and the plane front is stabilized again. 2.2. Equiaxed growth
Up to this point, we have been concerned with the columnar dendritic growth. The growth of equiaxed dendrites has been treated by Lipton et al. [ZO]. Contrary to the constrained growth where the latent heat is dissipated through the solid, during the free growth of dendrites the latent heat has to be dissipated through Table 1 Relevant physical parameters used for calculation (largely inspired from [8] and [23]) Nucleation pre-exponential factor Diffusion pre-exponential factor Activation energy Gibbs-Thompson coefficient Solidification constant Characteristic length scale Linear kinetic coefficient Polynomial phase diagram 09 (K a%) (K a%*) (K a”#) (K a%4) (K a”@) (K a”P) (K a”/b7)
I,: le*O [l s-l) Do: l.lew7 [mZ s-‘) Q: 23 800 (J mol-‘) l?: 2.4em7 (m K) A: 3em4 (m s-’ x wt.%pC’) a,: 7e-‘O (m) p: 2.88 (K s-* m-‘) Solidus Liquidus 933.60 933.60 - 5.3639 -62.231 -0.3165 9.6617 0.03036 -0.86953 - 192.64e-G 0.04696 - 104.70es6 -0.01509 4.61em6 26.64ee6 -56.26~~ - 198.90ewg
(17)
where a is the thermal diffusivity. Compared to the other contributions to the dendrite tip undercooling, the thermal undercooling is often negligible in metals with high thermal diffusivity. The calculation of the dendrite tip radius, however, requires the temperature gradient which is positive in the case of constrained growth (columnar) and negative in the case of free growth (equiaxed). However, in both cases, it is usually negligible compared to the contribution of the solutal gradient. Thus, it is possible to neglect the thermal gradient and use the same model for both cases of dendritic growth to characterize conditions at the dendrite tip. 2.3. NucZention
Nucleation is the dominant process at the beginning of solidification and leads very rapidly to the establishment of the final grain population, with each nucleus forming one equiaxed grain. The nucleation rate per unit volume can be written as [8]: I = (NO- N)I,, ‘exp[ - AG,/R,T]
(18)
where No is the total number of heterogeneous nucleation sites per unit volume, N is the number of sites that have already nucleated and I0 is a constant. The exponential term contains AG,, which is the free energy change associated with the formation of a critical nucleus (AG a l/AT’, where AT is the undercooling) and T is the temperature. The number of activated nucleation sites at a given undercooling is obtained by integration: rAT
N=
T
-?- dAT J o GV
Fig. 2 displays the nucleation rate and the number of nuclei as a function of undercooling. One can see that the total number of nuclei is rapidly equal to the number of nucleation sites available, once the nudeation undercooling is reached, The nucleation rate exhibits a sharp peak close the nucleation undercooling. 2.4. Undercooling ahead of the solid-liquid interface
During the advance of the columnar dendritic front, solute is rejected ahead of the solidification interface. The composition profile in the liquid for one dendrite
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,
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0:o
200.0
40b.o
1 1.4 109
- 1.2 109
1.2 log--
of No
Number Nuclei,
1.0 log--
-s--z _- 1.0 109
8.0 108--
j- 8.0 108 z
6.0 lOB--
j- 6.0 10%$
2.0 lop-0.0 4.0 log-100
Nuclealion
2::
13 2
R&Z
-2.0 108 0.60
, o.;o
,4 0.50
, oh0
I 1.00
, 1.10
/
/ 1.30
1.20
1 -2.0 108 Lb0
AT [“Cl
Fig. 2. Nucleation rate and number of nuclei as a function of undercooling for the Al-3 wt.% Cu system. C, = 3 wt.%, V= 32 pm s-l, G= 1000 K m-l, No = 1 (1 mm3), AT, = 0.75”C.
tip assuming a parabolic Ivantsov’s solution [21]:
tip geometry
is given by
60b.O
8i0.0
lOdO.
z Mm1
Fig. 4. Equilibrium liquidus and local temperature profile ahead of a moving solid-liquid interface calculated for the Al-Cu system. C, = 3 wt.%, V=32 pm s-t, G= 1000 K m-l.
where E, is the integral exponential function and z is the distance in the liquid from the tip of the dendrite parallel to the dendrite axis. Correspondingly, a liquidus temperature profile can be calculated in the liquid, assuming a linear phase diagram in the solidification range:
Fig. 3 shows the concentration profile ahead of a moving dendritic solidification front for the Al-3 wt.% Cu system, at a solidification velocity of 32 pm s-i, while Fig. 4 shows the profile of the equilibrium liquidus and of the local temperature for the same conditions. By combining Eqs. (21) and (24), one gets the local undercooling prof?.le ahead of the moving solid-liquid interface:
WI = TI + ve, -
AT[z] = T[z] - T,[z]
WI
= C0 + (CT - G)-W,(2z
+
JQI~lIwc1
(20)
+ w~IIa~cl
~J-wd2~
(21)
where Tr is the equilibrium liquidus temperature at composition C, and Teq is the equilibrium liquidus temperature at the tip composition CT: T, = T,+mC,
(22)
Teq= T, i- rnCt
(23)
It is assumed for simplification, that the composition and liquidus temperature profiles are constant along the whole dendritic front and corresponds to those of the tips. The local temperature in the liquid is controlled by the heat extraction through the solid. It is given by: Tq[z] = Ttip + GZ
010
(24)
20b.o
40b.o
sob.0
80b.0
(25)
As shown in Fig. 5, the liquidus and temperature profiles lead to a constitutionally undercooled region ahead of the dendritic front. In this undercooled region, nucleation may happen if the maximum growth undercooling is larger than the undercooling required for nucleation. 2.5. Columnar to equinxed tn-msition (CET) Once nucleated, equiaxed grains will develop with various volume fractions, depending for instance on the thermal gradient or the solidification velocity. They will be incorporated into the columnar zone if their volume fraction is too small, or will supersede the columnar zone if their volume fraction is large. Hunt proposed that a volume fraction of equiaxed grains of 0.49 is the
lOdO.
2 [WI Fig. 3. Concentration profile ahead of a moving dendritic solid-liquid interface cakulated for the Al-0.1 system. C, = 3 wt.%, V= 32 urn s-r.
Fig. 5. Schematic representation of the undercooled region ahead of the moving solidification interface.
M. Giiumam et al. /Materials
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limit beyond which one has to talk about an equiaxed microstructure and a limit of 0.0066 under which one has to talk about a columnar microstructure. In order to simplify the discussion, we propose to take a unique limit of 0.5. The volume fraction is obtained by calculating the size of each equiaxed grain when it gets entrapped by the columnar front. Numerically, this is obtained by integration: -6.0
I’,=
n -Y,kl & 0 v s
(26)
where z, is the distance from the interface in the undercooled liquid, where the undercooling is equal to the nucleation undercooling. The growth velocity for the equiaxed grains, V&Z], is calculated using the local composition and undercooling as obtained with Eqs. (20) and (25). When the calculation assumes a spherical growth of unhindered dendritic grains, then the extended volume fraction is: (6, = iQn]3r~
(27)
The actual volume fraction, removing the hypothesis of unhindered growth, is then obtained by applying Avrami’s equation [22]:
~=~--~~phLl
(28)
3. Discussion In order to compare between the present numerical model and Hunt’s analytical model for the CET, all calculations were made for the Al-Cu alloy as in Hunt’s original paper. The most relevant parameters are presented in Table 1. The composition profile varies along the isotherm of the dendrite tips, and is maximum at the dendrite tips and minimum in-between. This leads to a corresponding fluctuation of the liquidus temperature profile. In other words, the assumption of constant composition along the solidification front minimizes mathematically the undercooling, thus underestimates the CET.
a / 1.0
/ 2.0
1
, 3.0
log Gradient
4 / 4.0
! ’ 1 ’ 1 p 5.0
fK/m]
Fig. 6. Comparison of Hunt’s analytical model and the present numerical model for the AI-Cu system.
leads to a smaller undercooling in front of the moving interface, which reduces the driving force for the growth of equiaxed grains. Thus, equiaxed grains occur only at higher velocities compared to the present numerical model. In the middle region of Fig. 6, the transition lines predicted by the two models cross. The difference of slopes of G-V is explained by the weaker dependence of the dendrite tip undercooling on the growth velocity in KGT than in BH. At high velocity, the numerical model predicts an increase of the slope of the transition line, which stabilizes the columnar dendritic morphology. This has to be related to Fig. 1, where at high velocity, when the absolute stability is about to be reached, the tip temperature (Ttip) increases, thus, decreasing the undercooled region and reducing the driving force for the equiaxed dendritic growth. 3.2. Influence of different parameters on the CET
Fig, 7 displays the CET lines for different nucleation undercoolings. At low velocity, the dendrite tip undercooling is weak, thus, the nucleation undercooling plays a selective role. The lower the nucleation undercooling is, the easier it is to form equiaxed grains. If no undercooling is required in order to form a new grain
3.1. Comparison of Hunt’s and the present numerical model
Fig. 6 displays the transition line between the columnar structure and the equiaxed structure for the analytical solution by Hunt, and for the numerical solution. In Fig. 6, one can see that at low temperature gradients the CET occurs at lower velocities than in Hunt’s description. This is linked to the different growth criteria that are used. The criterion of growth at the extremum as considered in Hunt’s publication underestimates the dendrite tip undercooling. This in turn
1.0
2.0
3.0 log Gradient
4.0
5.0
[K/m]
Fig. 7. Influence of the nucleation undercooling
on the CET.
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4. Conclusions
~~~~~~r 4I / 1.0
2.0
3.0
4.0
5.0
log Gradient [K/m]
Fig. 8. Influence of the number of nucleation sites on the CET.
(like in the case of dendrite arm detachment due to convection), then, there is no change of slope for the CET line in the lower velocity part. Fig. 8 shows the influence of the number of nucleation sites on the columnar to equiaxed transition. In the higher velocity region, the important parameter is the, number of nucleation sites. Indeed, once nucleation occurs, the volume fraction of equiaxed grains is mainly linked to the number of nucleation sites. Therefore, if the number of nucleation sites is reduced, then the columnar microstructure is stabilized, as can be seen in Fig. 8. The number of nucleation sites and the nucleation undercooling are parameters that are specific to the equiaxed grains and control the formation of nuclei. On the other hand, the nominal composition is a parameter that is rather specific to the growth of the dendritic solidification front, in the sense that it controls the amount of solute rejection. Thus, the composition is a factor that has a strong influence on the shape and magnitude of the undercooled region. This is verified in Fig. 9, where increasing the nominal composition leads to an increased tendency to form equiaxed grains. Indeed, in most cases, the dendrite tip undercooling e increases with increasing alloy composition.
In this paper a modification of Hunt’s model for the columnar to equiaxed transition is proposed. It considers a more recent dendritic growth model, which is more accurate for the calculation of the .dendrite tip undercooling. The model takes into account the local concentration and,, r&ore importantly, the local undercooling that the equiaxed grains are exposed to in front of the solid-liquid interface at low, as well as high velocity. It has been shown for the Al-Cu system, that the numerical model proposed in this paper predicts, at low gradients, the equiaxed transition at lower velocities than in Hunt’s model, but that at higher gradients there is less driving force to nucleate equiaxed grains. This is linked to the growth model used in this paper which also considers high velocity effects on the partition coefficient and the liquidus slope. Thus, the present model is appropriate for predicting the microstructures in casting as well as in various welding situations or other processes involving rapid solidification, Finally, the proposed model is suitable to predict the columnar to equiaxed transition in a wide range of velocities or gradients, but also to predict phase selection in the undercooled region ahead of the interface.
Acknowledgements This work was carried out in part (RT) at Ames Laboratory which is operated for the US Department of Energy at Iowa State University under Contract No. W-7405-Eng-82, supported by the Director of Energy Research. The help of P. Gilgien and 0. Hunziker at the Swiss Federal Institute of Technology, Lausanne, in preparing the numerical model is appreciated.
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