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Natural convective effects in directional dendritic solidification of binary metallic alloys: Dendritic array primary spacing
Acta metall, mater. Vol. 40, No. 7, pp. 1791-1801, 1992 Printed in Great Britain. All rights reserved
0956-7151/92 $5.00 + 0.00 Copyright © 1992 Pergamon Press Ltd
NATURAL CONVECTIVE EFFECTS IN DIRECTIONAL DENDRITIC SOLIDIFICATION OF BINARY METALLIC ALLOYS: DENDRITIC ARRAY PRIMARY SPACING M. D. D U P O U Y t, D. C A M E L 2 and J. J. FAVIER 2 t LTPCM, ENSEEG, B.P. 75-38402 ST-Martin-D'H6res-Cedex and 2CEA/DTA/CEREM/DEM/SESC, Centre d'Etudes Nucl6aires de Grenoble, 85X, 38041 Grenoble Cedex, France (Received 18 July 1991) Abstract--This paper is concerned with the effect of natural convective patterns evidenced in a previous paper [Acta metall. 37, 1143 (1989)], on the dendritic primary spacings. Primary spacings in normal gravity environment have been found to be much smaller than those measured in a microgravity environment, the latter being in good agreement with the diffusion controlled theoretical predictions. The scaling analysis of convective effects developed in the first paper allows us to propose a relationship, which gives the primary spacing as a function of the experimental parameters, in convective transport conditions of the solute in the liquid. This correlation is in good agreement with our experimental results, and those of the literature. Rrsumr--Cet article traite de l'effet des cellules de convection naturelle---drcrit dans un article prrcrdent [Acta metall. 37, 1143 (1989)]--sur respacement des dendrites primaires. Les espacements primaires dans un environnement de gravit6 normale sont bien plus petits que ceux que ron mesure dans un environnement de microgravitr, et qui sont, eux, en bon accord avec les espacements contrrlrs par diffusion que prrvoit la throrie. L'analyse d'rchelle des effets de convection, qui a 6t6 drveloppre dans l'article prrcrdent, nous permet de proposer une relation qui donne l'espacement primaire en fonction des paramrtres exprrimentaux, pour des conditions de transport convectif du solut6 dans le liquide. Cette corrrlation est en bon accord avec nos rrsultats exprrimentaux et avec ceux de la littrrature. Zusanunenfassung--In dieser Arbeit wird der in einer frfiheren Arbeit [Acta metall. 37, 1143 (1989)] beschriebene EinfluB der natiidichen Konvektionsmuster auf den prim/iren Dendritenabstand behandelt. Diese prim/iren Dendritenabst~inde sind in normaler Gravitation viel kleiner als diejenigen, die bei Mikrogravitation gemessen werden. Letztere stimmen gut mit den theoretischen Voraussagen unter Annahme von Diffusionssteuerung fiberein. Die in der fr/iheren Arbeit entwickelte Skalierungsanalyse der konvektiven Einfliisse ermfglicht uns, einen Zusammenhang vorzuschlagen, der den prim~iren Dendritenabstand in Abh/ingigkeit yon den experimentellen Parametern auf der Basis der konvektiven Transportmechanismen des gelrsten Stoffes in der Schmelze beschreibt. Dieser Zusammenhang stimmt gut mit unseren Ergebnissen und denen in der Literatur iiberein.
1. INTRODUCTION Macroscopic effects of natural convection (in terms of segregations) on the directional dendritic solidification have been studied in a previous paper [1], by performing comparative directional solidification experiments in different orientations relative to gravity as well as under microgravity conditions. The effects on the morphologies are discussed in [2]. In this paper, we will see how the dendritic array spacing may be affected. In the case of the solidification of a dendritic array, the main geometrical characteristic is the primary interdendritic spacing 2, for which numerous experimental measurements have been performed [3-9] and several theoretical models proposed [10-12]. At low solidification rates, the measured 2 values are systemAU~/7-w
atically lower than predicted by the models. This has been interpreted as the result of the transition from a dendritic to a cellular morphology, when the solidification rate is decreased [4, 6-8]. But it has also been suggested that convective effects might perturb the measurements [3, 4, 13]. In addition, in a preliminary microgravity experiment performed during the Franco--Soviet program Elma 01 [14], 2 values were found 50% larger in the space samples. The scaling analysis of convective effects, presented together with the macro and microsegregation results in the first paper [1], gives a general criterion for the transition from diffusive to convective transport conditions. The effects of the derived scaling laws on solute transport and segregation have been described in terms of the ratio F of a characteristic interdendritic flow velocity (which depends on the density
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DUPOUY et al.:
NATURAL CONVECTIVE EFFECTS IN SOLIDIFICATION
differences in the interdendritic liquid, the thermal gradient, the gravity level and the primary spacing 2) to the solidification rate. We will now use this analysis to discuss our experimental primary spacings measurements, and provide a correlation for convective transport conditions. 2. EXPERIMENTAL
Our experimental conditions have been described in detail in [1]. Most of the ground based experiments have been performed in a furnace similar to the one used in the space experiment, i.e. with programmed cooling down of two heating zones. The thermal conditions chosen for the space and ground based reference samples gave nearly stationary solidification parameters during the first seven centimeters solidified, i.e. thermal gradient G = 30 K ClT1-1, and solidification rate R = 4.2.10-4cm s -~. The chosen concentrations were 20 and 26 wt% Cu for the hypoeutectic samples, and 40wt% Cu for the hypereutectic ones. Quenching could only be performed by passive cooling of the furnace, which was satisfactory for the longitudinal macrosegregation analysis, but does not allow to reveal the morphology of the dendritic array near the tips with a sufficient accuracy. Thus, primary spacings have mainly been ob-
served within the nominally solidified samples, a few complementary experiments being performed in a classical Bridgman furnace with quenching [15]. The dendrite spacings have been measured by hand and with an image analysis equipment, by determining the number of primary dendritic arms N within a surface A of a transverse section. This method is commonly used for such case [3-5]. The primary spacing 2 is defined by ). = B ( A / N )
1/2
where B is a constant which depends on the geometry of the network (B equals 1, 1.075, and 0.5 for a square, hexagonal and random array, respectively). Considering that a square network is generally observed in our experiments, we take B = 1. The accuracy of the measure is given by reporting the lowest and the largest 2 value for a given micrography. Only grains where the axis of primary arms are near to the solidification direction are considered. The accuracy with which primary dendritic arms can be identified, depends on the particular morphology encountered. Image analysis [16] is used in order to identify or reconstruct connex elements corresponding to primary dendritic trunks as illustrated with examples in the next section.
A
B
C
D
1ram
Fig. 1. Typical microstructures for the hypo and hyper-eutectic samples solidified both: on the ground, in the solutally destabilizing configuration (A and B) and under microgravity conditions (C and D).
DUPOUY et al.: NATURAL CONVECTIVE EFFECTS IN SOLIDIFICATION
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3. EXPERIMENTAL RESULTS
3.1. Ground samples Figure 1 shows typical microstructures for the hypo and hyper-eutectic samples solidified both on the ground and under microgravity conditions. As we have seen in [2], in hyper-eutectic samples solidified horizontally and vertically upwards, the primary dendrite trunks appear as well separated connex areas. Figure 2(a) shows the microstructure at the quenched front of a sample solidified horizontally. Little dendrites nucleated during the quench alter the original picture. Figure 2(b) gives the bimodal histogram of the perimeters of dendrite tips micrography. The first part corresponds to the small dendrites nucleated during the quench. The true primary dendrite trunks are those with perimeters larger than 0.3 mm. Primary spacings measured in this way are in good agreement with those measured inside the directionally solidified part of the samples. In the hyper-eutectic samples solidified downwards, the primary spacings are measured in the central part of the dendritic portion (see Fig. 7 in [1]). Similar results are obtained with the image analysis equipment or by hand, but in the first case, automatic cutting by morphological openings is needed to reconstitute separated primary trunks. In the hypo-eutectic samples, the primary arms cannot be separated so easily (Fig. 7 in [1]). M o r p h o logical criteria have to be used to identify primary arms. Particularly, when the characteristic quaternary symmetry of dendrites growing in a direction near to [100] is preserved, this can be used to reconstruct the dendritic network step by step. The primary spacings measured in the different configurations are summarized on Table 1. These configurations depend on the orientation of the density gradient in the interdendritic. Solutal stabilizing effect corresponds to upward solidification for the hypo-eutectic samples (resp. downward solidification for the hyper-eutectic ones). Solutal destabilizing effect corresponds to downward solidification for the hypo-eutectic (resp. upward solidification for the
a
b
C Fig. 2. AI2Cu dendrites at the quenched front of a sample solidified horizontally. (a) Before image analysis treatment. (b) Perimeter histogram. (c) After treatment.
Table 1. Primary spacings measurements for AI-Cu alloys solidifiedon the ground and under microgravity conditions Ground based data Space data Co = 40 wt% Cu CO- 40 wt% Cu R = 4.2 E-04cm/s R - 4.2 E-04cm/s G = 25°/cm G - 30~/cm Hypereutectic Upward Downward Horizontal~1,
Hypoeutectic
230 + 10/tm 330_+30#m 225 + 10~m Co = 20 wt% Cu R = 4.2 E-04cm/s G = 25°/cm Upward Downward Horizontal•
450+20/am 340+10#m {],Stabilizing thermal effect. lI,Destabilizing solutal effect. t~Horizontal solidification.
350-L-10pm
560 + 50/am Co = 26wt% Cu R = 4.2 E-04cm/s G = 30°/cm 1540+ 10/~m
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DUPOUY et al.: NATURAL CONVECTIVE EFFECTS IN SOLIDIFICATION
hyper-eutectic). Horizontal solidification is the third configuration. The less accurate primary spacing results are obtained in the solutal stabilizing configuration. As we will see in the following discussion, the accuracy of the measures is adequate to reveal significant discrepancies with the theoretical values.
A less uniform morphology is observed in the hyper-eutectic sample [Fig. l(d)], due to faceted growth [2]. For the same reason, the spacing is not uniform within one grain. This illustrates the limits of the concept of uniform dendritic network with a unique primary spacing in the present case. The measured primary spacings are compared in Table 1 with those obtained on the ground.
3.2. S p a c e samples
In the hypo-eutectic samples, dendrites are well separated [Fig. l(c) and Fig. 3]. This allows a detailed analysis of the microstructure as shown in [16]. Particularly, one can define the surface area corresponding to each primary trunk and thus obtain the corresponding histogram of equivalent diameters d (Fig. 3). This quantity is related to 2 by = J, 12 Bd.
4. DISCUSSION The difference in spacing between our space experiments and the ground based ones clearly demonstrates that in our case natural convection reduces 2. For example, this reduction is by a factor around 4 for the hypo-eutectic samples. Before studying the influence of convection, we will first analyse the predictions of diffusive models. 4.1. The reference diffusive case
We use an adimensional representation, to compare the results of our space experiment to the experimental and theoretical studies on the A1-Cu system, available in the literature [4, 7, 17]. When the solute transport in the liquid is purely diffusive, the dendritic growth is characterized by three main factors, that is thermal diffusion, solute diffusion, and capillarity. These effects can be characterized by their respective lengths [16], viz. solute diffusive length ls, thermal diffusive length It, and capillary length lc, defined as follows
N 6
v
i
v
(lb)
lc = t r / ( A S ATo k)
(lc)
/ -
,/ J
1.2
1.6
(2)
where m E is the slope of the liquidus. The characteristics of dendrite morphology are determined only by the relative magnitudes of the three length parameters defined in equation (1). The final result will need only two variables, lc/l t and l~/ls, denoted as ~ and ~/~, whose values are
/ /
-
.8
It = k A T o / G ,
A T O= m L Coo (1 - k ) / k
-
/ 2
(la)
where D is the diffusion coefficient of solute in liquid, k the equilibrium solute distribution coefficient, tr the solid/liquid interfacial free energy, AS the entropy of fusion per unit volume and ATo the solidification interval of the alloy, given by the relationship
/ 4
ls = D / R ,
2.0
2,4
dtmrn) Fig. 3. Image analysis on A1 dendrites (hypo-eutectic sample) solidified under microgravity conditions: treated image of a cross-section showing the domain of each dendrite, and corresponding histogram of equivalent diameters.
(# = trG /[k 2 (ATo)2AS],
(3)
= a R / ( D k AToAS ).
(4)
The chosen adimensional variables allow us to decouple the effects of the thermal gradient G and the growth rate R. With these definitions, the expression of the primary spacing 2 as a power law may be written ,~~to = b (~)x( C-)v
(5)
DUPOUY et al.:
NATURAL CONVECTIVE EFFECTS IN SOLIDIFICATION
1795
(6)
function of log ~ [Fig. 4(a)]. We draw the experimental points for the hypo-cutectic alloys of McCartney [4], Helawell [17], Miyata [7], Tosello [19] and ourselves [15], as well as the theoretical diffusion
For the A14Eu system, the variation of ). with the thermal gradient has been studied in the literature [4, 9]. The corresponding value of the exponent x is found to be very near of - 0 . 5 , in agreement with the theoretical models [10, ll]. Thus, in order to check the growth rate effect and compare the different results in this system, we assume that x = - 1/2 and we draw log(,~/lc~ 1/2) a s a
Spacelab DI, 1986) is in good agreement with the Trivedi law. We find again the strong discrepancy already noticed at low growth rates between the theoretical and experimental values (cellular and mixed data of Helawell [17] and McCartney [4], but also dendritic data of Miyata [7] and ourselves [15]). Excluding these points, a good linear fit is obtained
which leads to the following relationship 2 = b (a/AS) x÷y÷l x D - Y [ m L C ~ ( 1 - k ) ] - 2 x - y - 1 G X R y.
Trivedi law [11] (b = 6.5, x = - 1 / 2 , y = - 1 / 4 ) . The microgravity hypo-eutectic point (mission
(a)
• MC Cartney [-]Miyata
AI-Cu system
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hyper-microgravity
O Dupouy <>Dupouy : Hyperhypo-microgravity eutectic •Tosello
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/~Helawell 1.80
~
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~
_
XMorgand (microgravity)
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(b)
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~
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:
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I.OOE-07 I.OOE-06 I.OOE-05 1.00E-04 I.OOE-03 I.OOE-02 I.OOE-OI 1.00E+O0 I.OOE+OI I.OOE+02 F(Ath) Fig. 4. (a, b) Caption overleaf.
DUPOUY et al.: NATURAL CONVECTIVE EFFECTS IN SOLIDIFICATION
1796
(c)
AI-Cu system
~exp/Xth 1.4
1.2
• []
l
m
.........................................
m .......... L
i
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•
......................................
•
•
li •
•
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...........
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•
[]
•
OD! 0.2 I '( I I t I I I 1.00E-07 1.OOE-06 1.OOE-05 1.00g-04 1.00g-03 1.00E-02 I.OOg-o1 1.OOE+00 I.OOE+O1 1.OOE+02 r(Xexp) Fig.4.A~usystem:gr~wthratee~ct~nprimaryspacings.(a)Log(~/2/~)vs~g~.(b)2~xp/2~vs
between our space experiment points and the other at higher growth rates. The linear regression gives - 0 . 3 2 for the value of the growth rate exponent, which is significantly higher than the theoretical value (-0.25); this is in accordance with the experimental observations on other systems [4-8]. Besides, the same discrepancy is to be found between our hyper-eutectic values in ground environment (230 ___10/~m) and the theoretical one (925/~m). The space hyper-eutectic point is significantly below the regression curve. This discrepancy might be due to the inability of this faceted system to develop highly branched dendrites. This would explain why the difference between the spacing in space and ground based samples is lower than for the hypoeutectic alloy. 4.2. Influence o f convection According to the whole experimental results collected in the Fig. 4(a), the observed maximum in the variation of 2 with solidification rate does not correspond to the cellular-dendritic transition. In fact, the discrepancy with the diffusion law at these low growth rates, for dendritic microstructures, corresponds to a convective pattern generated in the mushy zone [1, 15]. In this case, the dendritic growth is not fully described by the two adimensional variables ~ and ~ above defined. A third parameter, F, which characterizes the solute transport by solutal convection [1], must be taken in account, in order to correlate the experimental results. In the previous paper [1], we have demonstrated that the solutal convection, generated by the density
differences in the interdendritic liquid, plays a major role on the macrosegregation. The characteristic solutal convection parameter F, is the ratio of the mean interdendritic flow velocity (obtained by the scaling analysis of the equations of motion and solute conservation) to the imposed growth rate. The mean value 1 for F corresponds to the diffuso-convective transition [1]. Solutal convection has then a significant influence on the solute redistribution. This influence is all the greater as the composition is near the eutectic one, and the growth rate as well as the thermal gradient are weaker. In opposition with the usual conceptions [6], these effects cannot be damped, even in the solutal stabilizing configuration, contrary to the planar front solidification case [20]. Indeed, this is linked with the ability of the dendritic front to get out of planar shape, inducing an important macrosegregation [1]. We will now show that these convective patterns directly influence the local distribution of one phase with respect to the other, that is the primary dendrite spacing, for the studied dendritic solidification domain, and we will propose a correlation, which characterizes this effect. 4.2.1. Evaluation o f the convective parameter F. According to [1], F involves the primary spacing 2 as follows F = V*/R = B ~ g G X * l m L l - ~ v - ~ R - ~ f ~ 7 2.
(7)
Here, B is a constant found equal to 3.10 -4 [1], X* is a characteristic dimension of the mushy zone, and fL is a characteristic liquid fraction in the mushy zone.
DUPOUY et al.: NATURAL CONVECTIVE EFFECTS IN SOLIDIFICATION In the general case, X* and fL are affected by convection. In [1], the interdendritic flow was considered as a small perturbation so that X* andfL can be estimated at first order by the classical diffusive model i.e. fL =fL(X*) is liquid fraction at a distance X* from dendrite tips, and is given by
fL = [GX*/ImLI(C ~ - C * ) + 1] 1/0 ~)
(8)
in the case of negligible diffusion in the solid. Here, C* is the value of the bulk concentration at the cellular to planar transition. X* is the smallest among the following characteristic lengths (1) X I = I m L I ( C ~ - C * ) G
~(1-k)/(i+k)
(9.1)
the characteristic distance across which the liquid fraction decreases (X~--*0 when one approaches the cellular to planar transition at C~ = C * ) , (2) X2 = ImLI(C** -- C~)G ~
(9.2)
the distance from dendrite tips to the eutectic front (X2--,0 when one approaches the dendrite to eutectic transition at C~ = C**), and (3))(3 = HI2
(9.3)
the half width of the crucible. F may be evaluated either with the primary spacing 2tu obtained in pure diffusive transport conditions [11], or with the measured primary spacing 2exp. The values of the other physical data used in subsequent calculations are summarized in Table 2.
4.2.2. Variation of the primary spacing with the convective parameter F. The values 2exp/2th are reported on the Fig. 4(b) in function of F(2th). Here, ~'th corresponds to the fitting curve of Fig. 4(a), i.e. is given by (6) with x = - 0 . 5 , y = - 0 . 3 2 , b = 2.7. Two domains clearly appear, separated by a transition which corresponds to F = 1. For F less than or equal to 1, the J-expvalues are in good agreement with those )[th calculated; this is particularly true for the space data, which correspond to values of F much less than 1 (according to [1], F is directly proportional to the gravity level g, which had a typical value of 10 _4 go in the space experiment). For F greater than 1, the ratio ,~,exp/,~th is always less than 1. The experimental 2 value strongly departs from the theoretical value in pure diffusive transport conditions. Besides, the estimation of/'(J-exp) from the measured spacing [Fig. 4(c)], shows that F(2exp) generally remains of order 1, and never exceeds 10.
1797
Thus, we verify that the practical /'exp parameter never reaches high values. This experimental limit corresponds to a physical one. As a matter of fact, the spacing 2 cannot possibly reach such values that F ( 2 ) becomes much greater than 1. The analysis of the solute balance equations [1, 15], shows that this would lead to the dendritic network disappearing, and thus the driving force for the interdendritic motion (that is the concentration gradients imposed in the mushy zone). Neglecting the dependence of X* andfL on convection, the condition F(2exp)= 1 can simply be written
2exp/•th = /"(/~th) 1/2.
(10)
The corresponding curve is shown on Fig. 4(b). Equation (10) gives an analytical expression for the dependence of 2 on R and G in the convective regime. We below discuss this dependence by assuming that 2th is given by the theoretical law (i.e. x = - 0 . 5 and y = -0.25). • Growth rate effect. Equation (10) predicts that 2¢xpincreases as R 1/2 in the convective regime. This means that it goes through a maximum at the convecto-diffusive transition. Besides, the 1/2 exponent value shows that the growth rate parameter has a stronger influence in this domain, compared to the - 1 / 4 exponent value in diffusive conditions. • Thermal gradient effect. Equations (10) and (7) show that the gradient effect will depend on the characteristic dimension of the mushy zone X*. If X* = X, or )/2, equation (10) predicts that 2expno longer depends on G in the convective regime. If X* is the half of the sample diameter (i.e. X3), equation (10) predicts that 2exp increases as G-~/2, as in the diffusive regime.
4.3. Generalization: analys& of data & other alloys To confirm our results, we have analysed the ground based experiments on other binary alloys available in the literature [3, 5, 6], where a solutal effect might be expected at low growth rates. We have compared the experimental 2 measurements with the corresponding theoretical 2 values (obtained by the same method previously exposed). The alloys chemical parameters needed to estimate J-th and F are summarized in Table 2. P b - S n results are presented on Fig. 5, P b - A u and P b - P d results on Fig. 6 and Fig. 7.
Table 2. Values of the physical data used in the evaluation of 2tb and F System AI (Cu) A12Cu (AI) Pb (Sn) Pb (Au) mL (K) 340 300 233 732 CE (wt/wt) 0.332 0.619 0.153 k 0.17 0.05 0.3 0 /~c (K-~) 1 1 0.4 0.4 v (cm2/s) 5 10-3 5 10-3 2 10-3 [3] 2 10-3 [3] D (cm2/s) 4 10-5 4 10-5 0.7 10-5 [3] 1 10-5 [3] a/AS (K-cm) 2.41 10-5 [22] 5.5 10-6 [22] 4.75 10-6 [22] 1 10-5
Pb (Pd) 1240 0.05 0 0.13 [211 2 10-3 1 10-5 1 10-5
NATURAL CONVECTIVE E F F E C T S I N SOLIDIFICATION
DUPOUYetaI.:
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2
00E+8
X,x~At
h
1 80E+O I BOE+O I 4OE+O 1 20E+0 1.80E+8
i
• ,.." .
Xexp/Ath
• *" " . , , . •
m
r(Xth)-l/2
B.80E-I 8.08E-I 4.88E-I 2 .80E-I r(%th)
I
I Iiiiiii
l
i iiiiii I
I W
i
I W •
i iliiil[
I + W
•
7
i iiililj
J-'J'illlllj + W
•
Ii')"
+ W $
7
+
I
•
7
7
Fig. 5. Pb-Sn system (data from J. T. Mason et al. [5]): 2=p/Ath VS F (2th).
In all cases, for the solutal destabilizing systems (Pb-Sn, NH4C1-H20), as for the solutal stabilizing ones (Pb-Au, Pb-Pd), the departure from the pure diffusion law models becomes sensible [i.e. (2cxp -- 2tb)/2exp t> 1] when F(2cxp) becomes of order 1.
(a)
Together with this transition, macroscopic deformation of the solidification front for the stabilizing systems (Pb-Au, P b - P d ) is to be seen as well as longitudinal macrosegregation (Pb-Sn) and freckles (NH4CI-HzO) for the destabilizing systems. P b - S n
Pb-Au System (data from J.T. Mason et al.[6])
log(), ~
1/2 / l c )
\
2.10
1.90
/
,lop.
-° 375L'to~3 x - o, y -
m @ m m~ m" • • •m~
1.70
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-6.00
-5.50
-5.00
-4.50
-4.00
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-3.00
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-2.50
-2.00
log ~ " Fig. 6. (a) Caption on facing page.
D U P O U Y et al.:
1799
N A T U R A L CONVECTIVE EFFECTS IN SOLIDIFICATION
(b)
Pb-Au system Xexl>/Ath
m~ m •
1.20
~
i
1.00
m- mr •
•
•
Aexp/Ath - r ( X t h ) " I / 2
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o.so
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0.20
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(
i
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i
1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04 r(Xth)
(c) Xexp/~th
Pb-Au system
1.20 l
• ~•iLml~m• •
1.00
• 0.80
•
mmi /
•
)• 0.60 •
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O. 40
ill mm 0.20
im
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0.00 I I I t J [ I.OOE-05 I.OOE-O4 I.OOE-03 I.OOE-02 I.OOE-OI I.OOE+O0 I.OOE+OI I.OOE+02 F(%exp) Fig. 6. P b - A u system (data from J. T. Mason et aL [6]). (a) Log (2~t/2/~) vs log ~e-. (b) 2exp/2thVS F (2,h). (C) 2¢xp/2th VS F (2exp).
results (Fig. 5) (destabilizing solutai effect) as well as P b - A u results (Fig. 6) (stabilizing solutal effect) remarkably confirm our theory. Pb--Pd results (Fig. 7) show that even when the fl0 coefficient is lowered (0.1 against 1 for AI--Cu system), a significant convective effect is to be seen. 5. CONCLUSION The difference in spacing between the space samples and the ground based ones clearly demonstrates that natural convection reduces (by a factor of
about 4) the primary spacings for the AI(Cu) system at the low growth rate studied. The analysis of our data as well as other literature data showed that the departure of 2 from theoretical diffusive laws is correlated to the convective parameter F defined in our previous paper [1] (ratio of the interdentritic flow rate to the solidification rate). This departure was found to occur when F becomes of order 1. Our analysis shows that the maximum of 2 vs R observed in several alloys in the literature (AI-Cu, Pb-Sn, Pb-Au, Pb--Pd) is associated to the transition from a diffusive to a convective regime.
1800
(a)
NATURAL CONVECTIVE EFFECTS IN SOLIDIFICATION
et al.:
DUPOUY
log(~Wl/2
PbPd S y s t e m ( d a t a
/i c)
from J.T.
....,.
2.10
slope : - 0.41 ; at x - O, y 0.09
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: -3.00
-2.50 log ~
(b)
Pb-Pd system ~exp/Ath 1.20 I.I0
•
1.00
if
I
0.90
|i
0.80 0.70
,,,
0.60
,
0.50 0.40 0.30 0.20
i
I
I
I
I
I.OOE-06 1.00E-05 I.OOE-04 1.00E-03 1.00E-02 1.00E-01 I.OOE+00
t 1.00E+OI
r(~th)
(c) PB - Pd system "
Aexp/Ath 1.20 I.i0 1.00
• •
~|ll niBi
• nil •
0.90 0.80 0.70 0.60 0.50
0.40 0.30 0.20 1.00E-05
I
i
I
P
t
1.00E-04
I.OOE-03
1.00E-02
I.OOE-01
I.OOE+O0
1.00E+01 r(~exp)
Fig. 7. Pb--Pd system (dam Horn J. T. Mason et al. [~). (a) Log (~/2/~) vs log ~. (b) 7~p/~ vs F (~tD. (c) ~p/~th vs r (~o~p).
DUPOUY et al.: NATURAL CONVECTIVE EFFECTS IN SOLIDIFICATION M o r e o v e r we show that in convective transport conditions, the primary spacing adjusts itself, so that the interdentritic flow rate, driven by the solutal convection, remains of order of the solidificatio~ate. In the convective regime the primary spacing increases as R 1/2 and thus goes through a m a x i m u m vs R at the convecto-diffusive transition. The thermal gradient dependence is a function of the characteristic length of the mushy zone (either 0 or - 1/2 exponent gradient value). At last, for metallic alloys in bulk samples, transitions in microstructures (for instance cellulardendritic transition) occurring at low solidification rates cannot in general be studied in pure diffusive transport conditions under normal gravity. Acknowledgement--This work was conducted within the framework of the GRAMME agreement between C.N.E.S. and C.E.A. REFERENCES
1. M. D. Dupouy, D. Camel and J. J. Favier, Acta metall. 37, 1143 (1989). 2. M. D. Dupouy, D. Camel and J. J. Favier, J. Cryst. Growth. Submitted. 3. C. M. Klaren, J. D. Verhoeven and R. Trivedi, Metall. Trans. llA, 1853 (1980).
1801
4. D. G. McCartney and J. D. Hunt, Acta metall. 29, 1851 (1981). 5. J. T. Mason, J. D. Verhoeven and R. Trivedi, J. Cryst. Growth 59, 516 (1982). 6. J. T. Mason, J. D. Verhoeven and R. Trivedi, Metall. Trans. 15A, 1665 (1984). 7. Y. Miyata, T. Suzuki and J. I. Uno, Metall. Trans. 16A, 1799 (1985). 8. K. Somboonsuk, J. T. Mason and R. Trivedi, Metall. Trans. 15A, 967 (1984). 9. A. N. Geying and Liu Lixin, J. Cryst. Growth 80, 383 (1987). 10. J. D. Hunt, Solidification and Casting of Metals, Vol. 192, p. 3. The Metals Soc., London 0979). ll. R. Trivedi, Metall. Trans. 15A, 977 (1984). 12. W. Kurz and D. J. Fisher, Acta metall. 29, 11 (1981). 13. M. H. Burden and J. D. Hunt, Metal. Sci. 10, 156 (1976). 14. J. J. Favier, J. Berthier, P. Arragon, Y. Malmejac, V. T. Khryapov and I. V. Barmin, Acta Astronautica 9, 255 (1982). 15. M. D. Dupouy, thesis, Grenoble (1986). 16. J. E. Mazille, J. Microsc. Spectrosc. Electron. 12, 57 (1987). 17. R. M. Sharp and A. Hellawell, J. Cryst. Growth 11, 77 (1971). 18. R. Trivedi, J. Cryst. Growth 49, 219 (1980). 19. I. Tosello, DEA, Grenoble (1984). 20. A. Rouzaud, thesis, Grenoble (1984). 21. L. D. Lucas, Techniques de l'ingdnieur 7 (1984). 22. M. Gfindfiz and J. D. Hunt, Acta metall. 33, 1651 (1985).
0956-7151/92 $5.00 + 0.00 Copyright © 1992 Pergamon Press Ltd
NATURAL CONVECTIVE EFFECTS IN DIRECTIONAL DENDRITIC SOLIDIFICATION OF BINARY METALLIC ALLOYS: DENDRITIC ARRAY PRIMARY SPACING M. D. D U P O U Y t, D. C A M E L 2 and J. J. FAVIER 2 t LTPCM, ENSEEG, B.P. 75-38402 ST-Martin-D'H6res-Cedex and 2CEA/DTA/CEREM/DEM/SESC, Centre d'Etudes Nucl6aires de Grenoble, 85X, 38041 Grenoble Cedex, France (Received 18 July 1991) Abstract--This paper is concerned with the effect of natural convective patterns evidenced in a previous paper [Acta metall. 37, 1143 (1989)], on the dendritic primary spacings. Primary spacings in normal gravity environment have been found to be much smaller than those measured in a microgravity environment, the latter being in good agreement with the diffusion controlled theoretical predictions. The scaling analysis of convective effects developed in the first paper allows us to propose a relationship, which gives the primary spacing as a function of the experimental parameters, in convective transport conditions of the solute in the liquid. This correlation is in good agreement with our experimental results, and those of the literature. Rrsumr--Cet article traite de l'effet des cellules de convection naturelle---drcrit dans un article prrcrdent [Acta metall. 37, 1143 (1989)]--sur respacement des dendrites primaires. Les espacements primaires dans un environnement de gravit6 normale sont bien plus petits que ceux que ron mesure dans un environnement de microgravitr, et qui sont, eux, en bon accord avec les espacements contrrlrs par diffusion que prrvoit la throrie. L'analyse d'rchelle des effets de convection, qui a 6t6 drveloppre dans l'article prrcrdent, nous permet de proposer une relation qui donne l'espacement primaire en fonction des paramrtres exprrimentaux, pour des conditions de transport convectif du solut6 dans le liquide. Cette corrrlation est en bon accord avec nos rrsultats exprrimentaux et avec ceux de la littrrature. Zusanunenfassung--In dieser Arbeit wird der in einer frfiheren Arbeit [Acta metall. 37, 1143 (1989)] beschriebene EinfluB der natiidichen Konvektionsmuster auf den prim/iren Dendritenabstand behandelt. Diese prim/iren Dendritenabst~inde sind in normaler Gravitation viel kleiner als diejenigen, die bei Mikrogravitation gemessen werden. Letztere stimmen gut mit den theoretischen Voraussagen unter Annahme von Diffusionssteuerung fiberein. Die in der fr/iheren Arbeit entwickelte Skalierungsanalyse der konvektiven Einfliisse ermfglicht uns, einen Zusammenhang vorzuschlagen, der den prim~iren Dendritenabstand in Abh/ingigkeit yon den experimentellen Parametern auf der Basis der konvektiven Transportmechanismen des gelrsten Stoffes in der Schmelze beschreibt. Dieser Zusammenhang stimmt gut mit unseren Ergebnissen und denen in der Literatur iiberein.
1. INTRODUCTION Macroscopic effects of natural convection (in terms of segregations) on the directional dendritic solidification have been studied in a previous paper [1], by performing comparative directional solidification experiments in different orientations relative to gravity as well as under microgravity conditions. The effects on the morphologies are discussed in [2]. In this paper, we will see how the dendritic array spacing may be affected. In the case of the solidification of a dendritic array, the main geometrical characteristic is the primary interdendritic spacing 2, for which numerous experimental measurements have been performed [3-9] and several theoretical models proposed [10-12]. At low solidification rates, the measured 2 values are systemAU~/7-w
atically lower than predicted by the models. This has been interpreted as the result of the transition from a dendritic to a cellular morphology, when the solidification rate is decreased [4, 6-8]. But it has also been suggested that convective effects might perturb the measurements [3, 4, 13]. In addition, in a preliminary microgravity experiment performed during the Franco--Soviet program Elma 01 [14], 2 values were found 50% larger in the space samples. The scaling analysis of convective effects, presented together with the macro and microsegregation results in the first paper [1], gives a general criterion for the transition from diffusive to convective transport conditions. The effects of the derived scaling laws on solute transport and segregation have been described in terms of the ratio F of a characteristic interdendritic flow velocity (which depends on the density
1791
1792
DUPOUY et al.:
NATURAL CONVECTIVE EFFECTS IN SOLIDIFICATION
differences in the interdendritic liquid, the thermal gradient, the gravity level and the primary spacing 2) to the solidification rate. We will now use this analysis to discuss our experimental primary spacings measurements, and provide a correlation for convective transport conditions. 2. EXPERIMENTAL
Our experimental conditions have been described in detail in [1]. Most of the ground based experiments have been performed in a furnace similar to the one used in the space experiment, i.e. with programmed cooling down of two heating zones. The thermal conditions chosen for the space and ground based reference samples gave nearly stationary solidification parameters during the first seven centimeters solidified, i.e. thermal gradient G = 30 K ClT1-1, and solidification rate R = 4.2.10-4cm s -~. The chosen concentrations were 20 and 26 wt% Cu for the hypoeutectic samples, and 40wt% Cu for the hypereutectic ones. Quenching could only be performed by passive cooling of the furnace, which was satisfactory for the longitudinal macrosegregation analysis, but does not allow to reveal the morphology of the dendritic array near the tips with a sufficient accuracy. Thus, primary spacings have mainly been ob-
served within the nominally solidified samples, a few complementary experiments being performed in a classical Bridgman furnace with quenching [15]. The dendrite spacings have been measured by hand and with an image analysis equipment, by determining the number of primary dendritic arms N within a surface A of a transverse section. This method is commonly used for such case [3-5]. The primary spacing 2 is defined by ). = B ( A / N )
1/2
where B is a constant which depends on the geometry of the network (B equals 1, 1.075, and 0.5 for a square, hexagonal and random array, respectively). Considering that a square network is generally observed in our experiments, we take B = 1. The accuracy of the measure is given by reporting the lowest and the largest 2 value for a given micrography. Only grains where the axis of primary arms are near to the solidification direction are considered. The accuracy with which primary dendritic arms can be identified, depends on the particular morphology encountered. Image analysis [16] is used in order to identify or reconstruct connex elements corresponding to primary dendritic trunks as illustrated with examples in the next section.
A
B
C
D
1ram
Fig. 1. Typical microstructures for the hypo and hyper-eutectic samples solidified both: on the ground, in the solutally destabilizing configuration (A and B) and under microgravity conditions (C and D).
DUPOUY et al.: NATURAL CONVECTIVE EFFECTS IN SOLIDIFICATION
1793
3. EXPERIMENTAL RESULTS
3.1. Ground samples Figure 1 shows typical microstructures for the hypo and hyper-eutectic samples solidified both on the ground and under microgravity conditions. As we have seen in [2], in hyper-eutectic samples solidified horizontally and vertically upwards, the primary dendrite trunks appear as well separated connex areas. Figure 2(a) shows the microstructure at the quenched front of a sample solidified horizontally. Little dendrites nucleated during the quench alter the original picture. Figure 2(b) gives the bimodal histogram of the perimeters of dendrite tips micrography. The first part corresponds to the small dendrites nucleated during the quench. The true primary dendrite trunks are those with perimeters larger than 0.3 mm. Primary spacings measured in this way are in good agreement with those measured inside the directionally solidified part of the samples. In the hyper-eutectic samples solidified downwards, the primary spacings are measured in the central part of the dendritic portion (see Fig. 7 in [1]). Similar results are obtained with the image analysis equipment or by hand, but in the first case, automatic cutting by morphological openings is needed to reconstitute separated primary trunks. In the hypo-eutectic samples, the primary arms cannot be separated so easily (Fig. 7 in [1]). M o r p h o logical criteria have to be used to identify primary arms. Particularly, when the characteristic quaternary symmetry of dendrites growing in a direction near to [100] is preserved, this can be used to reconstruct the dendritic network step by step. The primary spacings measured in the different configurations are summarized on Table 1. These configurations depend on the orientation of the density gradient in the interdendritic. Solutal stabilizing effect corresponds to upward solidification for the hypo-eutectic samples (resp. downward solidification for the hyper-eutectic ones). Solutal destabilizing effect corresponds to downward solidification for the hypo-eutectic (resp. upward solidification for the
a
b
C Fig. 2. AI2Cu dendrites at the quenched front of a sample solidified horizontally. (a) Before image analysis treatment. (b) Perimeter histogram. (c) After treatment.
Table 1. Primary spacings measurements for AI-Cu alloys solidifiedon the ground and under microgravity conditions Ground based data Space data Co = 40 wt% Cu CO- 40 wt% Cu R = 4.2 E-04cm/s R - 4.2 E-04cm/s G = 25°/cm G - 30~/cm Hypereutectic Upward Downward Horizontal~1,
Hypoeutectic
230 + 10/tm 330_+30#m 225 + 10~m Co = 20 wt% Cu R = 4.2 E-04cm/s G = 25°/cm Upward Downward Horizontal•
450+20/am 340+10#m {],Stabilizing thermal effect. lI,Destabilizing solutal effect. t~Horizontal solidification.
350-L-10pm
560 + 50/am Co = 26wt% Cu R = 4.2 E-04cm/s G = 30°/cm 1540+ 10/~m
1794
DUPOUY et al.: NATURAL CONVECTIVE EFFECTS IN SOLIDIFICATION
hyper-eutectic). Horizontal solidification is the third configuration. The less accurate primary spacing results are obtained in the solutal stabilizing configuration. As we will see in the following discussion, the accuracy of the measures is adequate to reveal significant discrepancies with the theoretical values.
A less uniform morphology is observed in the hyper-eutectic sample [Fig. l(d)], due to faceted growth [2]. For the same reason, the spacing is not uniform within one grain. This illustrates the limits of the concept of uniform dendritic network with a unique primary spacing in the present case. The measured primary spacings are compared in Table 1 with those obtained on the ground.
3.2. S p a c e samples
In the hypo-eutectic samples, dendrites are well separated [Fig. l(c) and Fig. 3]. This allows a detailed analysis of the microstructure as shown in [16]. Particularly, one can define the surface area corresponding to each primary trunk and thus obtain the corresponding histogram of equivalent diameters d (Fig. 3). This quantity is related to 2 by = J, 12 Bd.
4. DISCUSSION The difference in spacing between our space experiments and the ground based ones clearly demonstrates that in our case natural convection reduces 2. For example, this reduction is by a factor around 4 for the hypo-eutectic samples. Before studying the influence of convection, we will first analyse the predictions of diffusive models. 4.1. The reference diffusive case
We use an adimensional representation, to compare the results of our space experiment to the experimental and theoretical studies on the A1-Cu system, available in the literature [4, 7, 17]. When the solute transport in the liquid is purely diffusive, the dendritic growth is characterized by three main factors, that is thermal diffusion, solute diffusion, and capillarity. These effects can be characterized by their respective lengths [16], viz. solute diffusive length ls, thermal diffusive length It, and capillary length lc, defined as follows
N 6
v
i
v
(lb)
lc = t r / ( A S ATo k)
(lc)
/ -
,/ J
1.2
1.6
(2)
where m E is the slope of the liquidus. The characteristics of dendrite morphology are determined only by the relative magnitudes of the three length parameters defined in equation (1). The final result will need only two variables, lc/l t and l~/ls, denoted as ~ and ~/~, whose values are
/ /
-
.8
It = k A T o / G ,
A T O= m L Coo (1 - k ) / k
-
/ 2
(la)
where D is the diffusion coefficient of solute in liquid, k the equilibrium solute distribution coefficient, tr the solid/liquid interfacial free energy, AS the entropy of fusion per unit volume and ATo the solidification interval of the alloy, given by the relationship
/ 4
ls = D / R ,
2.0
2,4
dtmrn) Fig. 3. Image analysis on A1 dendrites (hypo-eutectic sample) solidified under microgravity conditions: treated image of a cross-section showing the domain of each dendrite, and corresponding histogram of equivalent diameters.
(# = trG /[k 2 (ATo)2AS],
(3)
= a R / ( D k AToAS ).
(4)
The chosen adimensional variables allow us to decouple the effects of the thermal gradient G and the growth rate R. With these definitions, the expression of the primary spacing 2 as a power law may be written ,~~to = b (~)x( C-)v
(5)
DUPOUY et al.:
NATURAL CONVECTIVE EFFECTS IN SOLIDIFICATION
1795
(6)
function of log ~ [Fig. 4(a)]. We draw the experimental points for the hypo-cutectic alloys of McCartney [4], Helawell [17], Miyata [7], Tosello [19] and ourselves [15], as well as the theoretical diffusion
For the A14Eu system, the variation of ). with the thermal gradient has been studied in the literature [4, 9]. The corresponding value of the exponent x is found to be very near of - 0 . 5 , in agreement with the theoretical models [10, ll]. Thus, in order to check the growth rate effect and compare the different results in this system, we assume that x = - 1/2 and we draw log(,~/lc~ 1/2) a s a
Spacelab DI, 1986) is in good agreement with the Trivedi law. We find again the strong discrepancy already noticed at low growth rates between the theoretical and experimental values (cellular and mixed data of Helawell [17] and McCartney [4], but also dendritic data of Miyata [7] and ourselves [15]). Excluding these points, a good linear fit is obtained
which leads to the following relationship 2 = b (a/AS) x÷y÷l x D - Y [ m L C ~ ( 1 - k ) ] - 2 x - y - 1 G X R y.
Trivedi law [11] (b = 6.5, x = - 1 / 2 , y = - 1 / 4 ) . The microgravity hypo-eutectic point (mission
(a)
• MC Cartney [-]Miyata
AI-Cu system
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DUPOUY et al.: NATURAL CONVECTIVE EFFECTS IN SOLIDIFICATION
1796
(c)
AI-Cu system
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between our space experiment points and the other at higher growth rates. The linear regression gives - 0 . 3 2 for the value of the growth rate exponent, which is significantly higher than the theoretical value (-0.25); this is in accordance with the experimental observations on other systems [4-8]. Besides, the same discrepancy is to be found between our hyper-eutectic values in ground environment (230 ___10/~m) and the theoretical one (925/~m). The space hyper-eutectic point is significantly below the regression curve. This discrepancy might be due to the inability of this faceted system to develop highly branched dendrites. This would explain why the difference between the spacing in space and ground based samples is lower than for the hypoeutectic alloy. 4.2. Influence o f convection According to the whole experimental results collected in the Fig. 4(a), the observed maximum in the variation of 2 with solidification rate does not correspond to the cellular-dendritic transition. In fact, the discrepancy with the diffusion law at these low growth rates, for dendritic microstructures, corresponds to a convective pattern generated in the mushy zone [1, 15]. In this case, the dendritic growth is not fully described by the two adimensional variables ~ and ~ above defined. A third parameter, F, which characterizes the solute transport by solutal convection [1], must be taken in account, in order to correlate the experimental results. In the previous paper [1], we have demonstrated that the solutal convection, generated by the density
differences in the interdendritic liquid, plays a major role on the macrosegregation. The characteristic solutal convection parameter F, is the ratio of the mean interdendritic flow velocity (obtained by the scaling analysis of the equations of motion and solute conservation) to the imposed growth rate. The mean value 1 for F corresponds to the diffuso-convective transition [1]. Solutal convection has then a significant influence on the solute redistribution. This influence is all the greater as the composition is near the eutectic one, and the growth rate as well as the thermal gradient are weaker. In opposition with the usual conceptions [6], these effects cannot be damped, even in the solutal stabilizing configuration, contrary to the planar front solidification case [20]. Indeed, this is linked with the ability of the dendritic front to get out of planar shape, inducing an important macrosegregation [1]. We will now show that these convective patterns directly influence the local distribution of one phase with respect to the other, that is the primary dendrite spacing, for the studied dendritic solidification domain, and we will propose a correlation, which characterizes this effect. 4.2.1. Evaluation o f the convective parameter F. According to [1], F involves the primary spacing 2 as follows F = V*/R = B ~ g G X * l m L l - ~ v - ~ R - ~ f ~ 7 2.
(7)
Here, B is a constant found equal to 3.10 -4 [1], X* is a characteristic dimension of the mushy zone, and fL is a characteristic liquid fraction in the mushy zone.
DUPOUY et al.: NATURAL CONVECTIVE EFFECTS IN SOLIDIFICATION In the general case, X* and fL are affected by convection. In [1], the interdendritic flow was considered as a small perturbation so that X* andfL can be estimated at first order by the classical diffusive model i.e. fL =fL(X*) is liquid fraction at a distance X* from dendrite tips, and is given by
fL = [GX*/ImLI(C ~ - C * ) + 1] 1/0 ~)
(8)
in the case of negligible diffusion in the solid. Here, C* is the value of the bulk concentration at the cellular to planar transition. X* is the smallest among the following characteristic lengths (1) X I = I m L I ( C ~ - C * ) G
~(1-k)/(i+k)
(9.1)
the characteristic distance across which the liquid fraction decreases (X~--*0 when one approaches the cellular to planar transition at C~ = C * ) , (2) X2 = ImLI(C** -- C~)G ~
(9.2)
the distance from dendrite tips to the eutectic front (X2--,0 when one approaches the dendrite to eutectic transition at C~ = C**), and (3))(3 = HI2
(9.3)
the half width of the crucible. F may be evaluated either with the primary spacing 2tu obtained in pure diffusive transport conditions [11], or with the measured primary spacing 2exp. The values of the other physical data used in subsequent calculations are summarized in Table 2.
4.2.2. Variation of the primary spacing with the convective parameter F. The values 2exp/2th are reported on the Fig. 4(b) in function of F(2th). Here, ~'th corresponds to the fitting curve of Fig. 4(a), i.e. is given by (6) with x = - 0 . 5 , y = - 0 . 3 2 , b = 2.7. Two domains clearly appear, separated by a transition which corresponds to F = 1. For F less than or equal to 1, the J-expvalues are in good agreement with those )[th calculated; this is particularly true for the space data, which correspond to values of F much less than 1 (according to [1], F is directly proportional to the gravity level g, which had a typical value of 10 _4 go in the space experiment). For F greater than 1, the ratio ,~,exp/,~th is always less than 1. The experimental 2 value strongly departs from the theoretical value in pure diffusive transport conditions. Besides, the estimation of/'(J-exp) from the measured spacing [Fig. 4(c)], shows that F(2exp) generally remains of order 1, and never exceeds 10.
1797
Thus, we verify that the practical /'exp parameter never reaches high values. This experimental limit corresponds to a physical one. As a matter of fact, the spacing 2 cannot possibly reach such values that F ( 2 ) becomes much greater than 1. The analysis of the solute balance equations [1, 15], shows that this would lead to the dendritic network disappearing, and thus the driving force for the interdendritic motion (that is the concentration gradients imposed in the mushy zone). Neglecting the dependence of X* andfL on convection, the condition F(2exp)= 1 can simply be written
2exp/•th = /"(/~th) 1/2.
(10)
The corresponding curve is shown on Fig. 4(b). Equation (10) gives an analytical expression for the dependence of 2 on R and G in the convective regime. We below discuss this dependence by assuming that 2th is given by the theoretical law (i.e. x = - 0 . 5 and y = -0.25). • Growth rate effect. Equation (10) predicts that 2¢xpincreases as R 1/2 in the convective regime. This means that it goes through a maximum at the convecto-diffusive transition. Besides, the 1/2 exponent value shows that the growth rate parameter has a stronger influence in this domain, compared to the - 1 / 4 exponent value in diffusive conditions. • Thermal gradient effect. Equations (10) and (7) show that the gradient effect will depend on the characteristic dimension of the mushy zone X*. If X* = X, or )/2, equation (10) predicts that 2expno longer depends on G in the convective regime. If X* is the half of the sample diameter (i.e. X3), equation (10) predicts that 2exp increases as G-~/2, as in the diffusive regime.
4.3. Generalization: analys& of data & other alloys To confirm our results, we have analysed the ground based experiments on other binary alloys available in the literature [3, 5, 6], where a solutal effect might be expected at low growth rates. We have compared the experimental 2 measurements with the corresponding theoretical 2 values (obtained by the same method previously exposed). The alloys chemical parameters needed to estimate J-th and F are summarized in Table 2. P b - S n results are presented on Fig. 5, P b - A u and P b - P d results on Fig. 6 and Fig. 7.
Table 2. Values of the physical data used in the evaluation of 2tb and F System AI (Cu) A12Cu (AI) Pb (Sn) Pb (Au) mL (K) 340 300 233 732 CE (wt/wt) 0.332 0.619 0.153 k 0.17 0.05 0.3 0 /~c (K-~) 1 1 0.4 0.4 v (cm2/s) 5 10-3 5 10-3 2 10-3 [3] 2 10-3 [3] D (cm2/s) 4 10-5 4 10-5 0.7 10-5 [3] 1 10-5 [3] a/AS (K-cm) 2.41 10-5 [22] 5.5 10-6 [22] 4.75 10-6 [22] 1 10-5
Pb (Pd) 1240 0.05 0 0.13 [211 2 10-3 1 10-5 1 10-5
NATURAL CONVECTIVE E F F E C T S I N SOLIDIFICATION
DUPOUYetaI.:
1798
2
00E+8
X,x~At
h
1 80E+O I BOE+O I 4OE+O 1 20E+0 1.80E+8
i
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• *" " . , , . •
m
r(Xth)-l/2
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I
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i iiiiii I
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I + W
•
7
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J-'J'illlllj + W
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+ W $
7
+
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7
7
Fig. 5. Pb-Sn system (data from J. T. Mason et al. [5]): 2=p/Ath VS F (2th).
In all cases, for the solutal destabilizing systems (Pb-Sn, NH4C1-H20), as for the solutal stabilizing ones (Pb-Au, Pb-Pd), the departure from the pure diffusion law models becomes sensible [i.e. (2cxp -- 2tb)/2exp t> 1] when F(2cxp) becomes of order 1.
(a)
Together with this transition, macroscopic deformation of the solidification front for the stabilizing systems (Pb-Au, P b - P d ) is to be seen as well as longitudinal macrosegregation (Pb-Sn) and freckles (NH4CI-HzO) for the destabilizing systems. P b - S n
Pb-Au System (data from J.T. Mason et al.[6])
log(), ~
1/2 / l c )
\
2.10
1.90
/
,lop.
-° 375L'to~3 x - o, y -
m @ m m~ m" • • •m~
1.70
1.50
1.30
1.10
0.90 -6.50
I
I
I
I
I
I
I
-6.00
-5.50
-5.00
-4.50
-4.00
-3.50
-3.00
I
-2.50
-2.00
log ~ " Fig. 6. (a) Caption on facing page.
D U P O U Y et al.:
1799
N A T U R A L CONVECTIVE EFFECTS IN SOLIDIFICATION
(b)
Pb-Au system Xexl>/Ath
m~ m •
1.20
~
i
1.00
m- mr •
•
•
Aexp/Ath - r ( X t h ) " I / 2
•
o.so
mm
•
0.60
f~
0.40
0.20
0.00
)
#
f
(
i
i
i
i
i
1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04 r(Xth)
(c) Xexp/~th
Pb-Au system
1.20 l
• ~•iLml~m• •
1.00
• 0.80
•
mmi /
•
)• 0.60 •
I
O. 40
ill mm 0.20
im
mm
0.00 I I I t J [ I.OOE-05 I.OOE-O4 I.OOE-03 I.OOE-02 I.OOE-OI I.OOE+O0 I.OOE+OI I.OOE+02 F(%exp) Fig. 6. P b - A u system (data from J. T. Mason et aL [6]). (a) Log (2~t/2/~) vs log ~e-. (b) 2exp/2thVS F (2,h). (C) 2¢xp/2th VS F (2exp).
results (Fig. 5) (destabilizing solutai effect) as well as P b - A u results (Fig. 6) (stabilizing solutal effect) remarkably confirm our theory. Pb--Pd results (Fig. 7) show that even when the fl0 coefficient is lowered (0.1 against 1 for AI--Cu system), a significant convective effect is to be seen. 5. CONCLUSION The difference in spacing between the space samples and the ground based ones clearly demonstrates that natural convection reduces (by a factor of
about 4) the primary spacings for the AI(Cu) system at the low growth rate studied. The analysis of our data as well as other literature data showed that the departure of 2 from theoretical diffusive laws is correlated to the convective parameter F defined in our previous paper [1] (ratio of the interdentritic flow rate to the solidification rate). This departure was found to occur when F becomes of order 1. Our analysis shows that the maximum of 2 vs R observed in several alloys in the literature (AI-Cu, Pb-Sn, Pb-Au, Pb--Pd) is associated to the transition from a diffusive to a convective regime.
1800
(a)
NATURAL CONVECTIVE EFFECTS IN SOLIDIFICATION
et al.:
DUPOUY
log(~Wl/2
PbPd S y s t e m ( d a t a
/i c)
from J.T.
....,.
2.10
slope : - 0.41 ; at x - O, y 0.09
•
2.00 1.90
Mason e t a 1 . [ 6 ] )
-!
1.80 1.70 1.60 1.50 1.40 1.30 -6.00
I
I
I
i
I
-5.50
-5.00
-4.50
-4.00
-3.50
m~_
: -3.00
-2.50 log ~
(b)
Pb-Pd system ~exp/Ath 1.20 I.I0
•
1.00
if
I
0.90
|i
0.80 0.70
,,,
0.60
,
0.50 0.40 0.30 0.20
i
I
I
I
I
I.OOE-06 1.00E-05 I.OOE-04 1.00E-03 1.00E-02 1.00E-01 I.OOE+00
t 1.00E+OI
r(~th)
(c) PB - Pd system "
Aexp/Ath 1.20 I.i0 1.00
• •
~|ll niBi
• nil •
0.90 0.80 0.70 0.60 0.50
0.40 0.30 0.20 1.00E-05
I
i
I
P
t
1.00E-04
I.OOE-03
1.00E-02
I.OOE-01
I.OOE+O0
1.00E+01 r(~exp)
Fig. 7. Pb--Pd system (dam Horn J. T. Mason et al. [~). (a) Log (~/2/~) vs log ~. (b) 7~p/~ vs F (~tD. (c) ~p/~th vs r (~o~p).
DUPOUY et al.: NATURAL CONVECTIVE EFFECTS IN SOLIDIFICATION M o r e o v e r we show that in convective transport conditions, the primary spacing adjusts itself, so that the interdentritic flow rate, driven by the solutal convection, remains of order of the solidificatio~ate. In the convective regime the primary spacing increases as R 1/2 and thus goes through a m a x i m u m vs R at the convecto-diffusive transition. The thermal gradient dependence is a function of the characteristic length of the mushy zone (either 0 or - 1/2 exponent gradient value). At last, for metallic alloys in bulk samples, transitions in microstructures (for instance cellulardendritic transition) occurring at low solidification rates cannot in general be studied in pure diffusive transport conditions under normal gravity. Acknowledgement--This work was conducted within the framework of the GRAMME agreement between C.N.E.S. and C.E.A. REFERENCES
1. M. D. Dupouy, D. Camel and J. J. Favier, Acta metall. 37, 1143 (1989). 2. M. D. Dupouy, D. Camel and J. J. Favier, J. Cryst. Growth. Submitted. 3. C. M. Klaren, J. D. Verhoeven and R. Trivedi, Metall. Trans. llA, 1853 (1980).
1801
4. D. G. McCartney and J. D. Hunt, Acta metall. 29, 1851 (1981). 5. J. T. Mason, J. D. Verhoeven and R. Trivedi, J. Cryst. Growth 59, 516 (1982). 6. J. T. Mason, J. D. Verhoeven and R. Trivedi, Metall. Trans. 15A, 1665 (1984). 7. Y. Miyata, T. Suzuki and J. I. Uno, Metall. Trans. 16A, 1799 (1985). 8. K. Somboonsuk, J. T. Mason and R. Trivedi, Metall. Trans. 15A, 967 (1984). 9. A. N. Geying and Liu Lixin, J. Cryst. Growth 80, 383 (1987). 10. J. D. Hunt, Solidification and Casting of Metals, Vol. 192, p. 3. The Metals Soc., London 0979). ll. R. Trivedi, Metall. Trans. 15A, 977 (1984). 12. W. Kurz and D. J. Fisher, Acta metall. 29, 11 (1981). 13. M. H. Burden and J. D. Hunt, Metal. Sci. 10, 156 (1976). 14. J. J. Favier, J. Berthier, P. Arragon, Y. Malmejac, V. T. Khryapov and I. V. Barmin, Acta Astronautica 9, 255 (1982). 15. M. D. Dupouy, thesis, Grenoble (1986). 16. J. E. Mazille, J. Microsc. Spectrosc. Electron. 12, 57 (1987). 17. R. M. Sharp and A. Hellawell, J. Cryst. Growth 11, 77 (1971). 18. R. Trivedi, J. Cryst. Growth 49, 219 (1980). 19. I. Tosello, DEA, Grenoble (1984). 20. A. Rouzaud, thesis, Grenoble (1984). 21. L. D. Lucas, Techniques de l'ingdnieur 7 (1984). 22. M. Gfindfiz and J. D. Hunt, Acta metall. 33, 1651 (1985).