Search for a solute-drag effect in dendritic solidification

Acta metall, mater. Vol. 42, No, 3, pp. 975--979, 1994

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SEARCH F O R A SOLUTE-DRAG EFFECT IN D E N D R I T I C SOLIDIFICATION K. ECKLERt, D. M. HERLACH1 and M. J. A Z I Z z qnstitut f/Jr Raumsimulation, Deutsche Forschungsanstalt fiir Luft- und Raumfahrt e.V., D-51140 K61n, Germany and 2Division of Applied Sciences, Harvard University, Cambridge, MA 02138, U.S.A. (Received 3 M a y 1993)

Abstract--We report the results of an indirect experimental test for the solute-drag effect in alloy solidification by fitting the data of Eckler et al. for Ni-B dendrite tip velocities vs undercooling to models in several ways. The unknown equilibrium partition coefficient, ke, was varied as a fitting parameter. When we combine the dendrite growth model of Boettinger et al. with the Continuous Growth Model (CGM) of Aziz and Kaplan "with solute drag", we cannot fit the data for any value of k~. When we combine dendrite growth theory with the CGM "without solute drag", we obtain a reasonable fit to the data for k~ = 4 x 10-6. When we combine dendrite growth theory with a new "partial-solute-drag" interpolation between the "with-solute-drag" and the "without-solute-drag" versions of the CGM, we obtain a still better fit to the data for k e = 2.8 x 10-4. This result points out the possibility of partial solute-drag during solidification and the importance of an independent determination of k~ in order to distinguish between models.

Zusammenfassung--Wirberichten fiber die Ergebnisse

einer indirekten experimentellen Suche nach dem Solute-drag-Effekt bei der Erstarrung yon Legierungen. Dazu wurden Daten von Eckler et al, (dendritische Wachstumsgeschwindigkeit als Funktion der Unterkiihlung in Ni-B) an verschiedene ModeUe angepaBt. Der Gleichgewichts-Verteilungskoeffizient ke, dessen Wert nicht bekannt ist, diente als Anpassungsparameter. Die Kombination des dendritischen Wachstumsmodells von Boettinger et al. mit dem Modell kontinuierlichen Wachstums (KWM) von Aziz und Kaplan "mit Solute-drag" erlaubt fiir keinen Wert yon ko eine Besehreibung der Daten. Bei Verkniipfung der dendritischen Wachstumstheorie mit dem KWM "ohne Solute-drag" erhalten wir eine verniinftige Anpassung an die Daten fiir kc = 4.10 -6. Wenn wir das ModeU dendritischen Wachstums mit einer neuen Interpolation zwischen der "mit-Solute-drag"- und der "ohne-Solute-drag"-Version des KWM kombinieren ("partieller Solutedrag"), ergibt sich eine noch bessere Beschreibung der Daten fiir k, = 2.8- 10-4. Dieses Ergebnis zeigt, dab Solute-drag bei der Erstarrung mrglicherweise partiell auftritt und dab fiir eine Entscheidung zwischen den verschiedenen Modellen eine unabh/ingige Bestimmung von k e wichtig ist.

It is well known that rapid solidification of undercooled melts leads to significant deviations from local equilibrium at the solid-liquid interface [1], the most obvious manifestations of which are solute trapping (suppressed partitioning of species across the interface) and interfacial undercooling below the equilibrium temperature. In some kinetic models, there is an additional source of undercooling called "solute drag" [2-4]. This effect was analysed for grain boundary migration by C a h n [2], and grain growth experiments seem to indicate its presence [5]. Theoretical arguments have been presented for and against a solute-drag effect in phase transformations [3, 4, 6]. However, at present there appears no way to determine a p r i o r i whether the solute-drag effect applies to alloy solidification. The consequences of a solutedrag effect would be: (i) very much more interfacial undercooling, during solidification in the velocity regime of partial solute trapping, than is presently expected; (ii) a concomitant reduction in interface 975

mobility, which affects all models for dendritic solidification and for interface instabilities, particularly those for oscillatory instabilities [7-10]. Attempts to resolve the question theoretically, e.g. by examining the models for self-consistency using Onsager's reciprocity relations [11], have been inconclusive [12]. To date, there has been no experimental evidence for or against solute-drag effects in phase transformations. Interface kinetic models, such as the Continuous Growth Model (CGM) of Aziz and K a p l a n [4], provide the boundary conditions for continuum heat and mass transport. Dendrite growth models, such as that of Boettinger et al. [13] take as input the boundary conditions at the solid-liquid interface, and solve the bulk diffusion equation as a free-boundary problem. In this work, dendrite growth velocities and undercoolings measured in dilute Ni-B alloys are reanalysed using the dendrite growth model incorporating the C G M both with and without solute drag.

976

ECKLER et al.: SOLUTE-DRAG EFFECT IN DENDRITIC SOLIDIFICATION

The Ni-B system was chosen because it shows a sharp transition from diffusion-controlled to thermally controlled growth when the undercooling approaches a critical value [14]. According to chemical rate theory [15-17], the velocity, v, of the advancing interface for a single component system is given by v = vo(T)" [1 - exp(Al~/RT)]

(1)

where R is the gas constant, T is the absolute temperature, and A/t, termed the driving force for crystallization, is the chemical potential change on solidification. For materials exhibiting short-range diffusion-limited growth [17], the kinetic prefactor vo(T ) is approximately the atomic diffusive speed vo (the ratio of the diffusivity at the interface to the interface width). For materials exhibiting collisionlimited growth [18, 19], vo(T) is approximately the speed of sound. The maximum solidification rates measured in metals are much higher than predicted by equation (I) with short-range diffusion-limited growth. Therefore, the collision-limited form of equation (1) is commonly used for metals and alloys. In the CGM it was proposed that the solidification rate in alloys is also governed by equation (1), except that A/~ must be replaced by a term appropriate for alloys, which is called AG. Several possibilities for AG have been pointed out. In the form of the CGM "without solute drag", A/~ is replaced by AGoF, the Gibbs free energy change per mole of alloy solidified AGDF = c ~'A/.t B+ (1 - c ~')A/I A

(2)

where A # A'B=/.t A'B-/gLA'B is the difference between the chemical potentials of the species A and B in the solid and the liquid; c is the mole fraction of species B, the subscript S or L indicates the solid or the liquid phase, respectively, and the asterisk (*) indicates the value at the interface. Alternatively, in the form of the CGM "with solute drag", it was argued that a part of AGoF must be consumed by the solute-drag effect in driving solvent-solute redistribution and is therefore not available to drive interface motion. In this case, A~( in equation (1) is replaced by AGe = c*A/~ B+ (1 - c*)A/~ A.

(3)

The CGM was developed for an atomically sharp interface, within which the composition is not defined. Aziz and Kaplan have shown [4] that equations (2) and (3) correspond to the limiting cases of a treatment of the diffusive fluxes across a solid-liquid interface of nonzero width, where the fluxes and compositions vary continuously from the solid to the liquid. The limiting cases are reached when the diffusive fluxes become step functions, where in the "without-solute-drag" model the step occurs at the liquid side of the interface layer and in the "with-solute-drag" model the step occurs at the solid side. Interestingly, a new continuum "phase field" model by Wheeler et al. [21], involving sol-

utions to the time-dependent Ginzburg-Landau equation, predicts undercooling behavior very similar to the CGM "with solute drag". In the solute-drag model there is no solute drag present both at V<>vo. But solute drag may become important in an intermediate velocity regime around v ~ v o, where solute trapping sets in. In semiconductors [22] and metals [23] the velocity dependence of solute trapping has been shown to behave according to (4)

k (v) = (ke + V/VD)/(1 + V/VD)

where k = Cs/c L is the partition coefficient whose equilibrium value is ke. For a given composition, the difference between the predicted undercooling behavior in the models with and without solute drag becomes greater with decreasing values of ke. Since it possesses a very small equilibrium partition coefficient ke<<1, the Ni-B system is quite suitable for a re-examination of measurements of dendrite growth velocities vs undercooling, AT, in search for a solute-drag effect. Figure 1 shows the experimental data for pure Ni and two Ni-B alloys [14]. The dashed lines represent the calculated v (AT)-relationships according to the dendrite growth model of Boettinger et al. [13] (incorporating the CGM "without solute drag"). As ke for the Ni-B system is not known, it has been treated as a fitting parameter. An extensive interpretation of Fig. 1 can be found in Ref. [14]. Suffice it to say that the drastic enhancement of the growth velocities in the alloys at a concentration-dependent undercooling, AT ct, is caused by the reaching of the critical velocity for absolute solutal stability, (va)c, accompanied with a sharp increase of the partition coefficient due to solute trapping. Thus, AT ¢t marks the transition from diffusion-controlled to thermally "~ \

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Fig. 1. Dendrite growth velocities, v, measured as a function of undercooling, AT, for pure Ni and two dilute Ni-B alloys. Theoretical prediction according to the dendrite growth model when combined with the Continuous Growth Model (CGM). Dashed lines: CGM "without solute drag", and k, = 4 x 10-6; solid lines: interpolation between the "withsolute-drag" and the "without-solute-drag" versions of the CGM, and ko = 2.8 x 10-4 (k~: equilibrium partition coefficient).

ECKLER et al.:

Table 1. Material parameters as used in the calculations of dendritic growth Melting point (Ni) Heat of fusion (Ni) Specific heat of the liquid Slope of the liquidus line Solid-liquid interface energy Gibbs-Thomson coefficient Thermal diffusivity of the liquid Thermal diffusivity of the solid Speed of sound in the liquid Diffusion velocity 0.7 at.% B 1.0 at.% B Diffusion coefficient 0.7 at.% B 1.0 at.% B Equilibrium partition coefficient "without solute drag" "interpolation"

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controlled solidification. The parameters used in the calculations are listed in Table 1. D L and vD have also been treated as fitting parameters. The fit, however, is rather insensitive to the values chosen for these two parameters provided that they remain in a physically reasonable range. According to the dendrite growth model, the velocity-dependence of the interface temperature, T~, is given for dilute solutions by [13] T i = T m + taLC * {1 -- k + f "

977

SOLUTE-DRAG EFFECT IN DENDRITIC SOLIDIFICATION

(5)

The symbols are defined in Table 1, except r, which denotes the radius of curvature of the solid-liquid interface at the dendrite tip. Equation (5) describes the constitutional, kinetic and curvature undercooling of the interface. If the CGM "without solute drag" is incorporated, f = k ( v ) [13]; if the CGM "with solute drag" is incorporated, f = 1 [24]. The relationship between velocity and total undercooling is accomplished via Invantsov's solution of the diffusion equation [25] and the marginal stability hypothesis [26-28]. The combination of the dendrite growth model and the CGM "with solute drag" does not fit the experimental data well at all, no matter what value for k, is chosen (Fig. 2). On the other hand, the combination of the dendrite growth model and the CGM "without solute drag" fits the experimental data quite well if the best fit value of ke = 4 x 10 -6 is used (Fig.l); however, the rise of the measured growth velocities around AT ct is sharper than its theoretical predictionS', especially for the N i - I at.%B sample (the isolated data point at AT = 270 K, v = 8 m/s is

i'The parameters given in Table I differ somewhat from those in Table 1 of Ref. 14. In the present work we have used an experimentally determined value for the speed of sound [29]; also, we have taken into account now the difference between the thermal diffusivities of solid and liquid Ni [35], which enters the marginal stability criterion [27]. However, the conclusions drawn in Ref. [14] still remain valid.

Fig. 2. Dendrite growth velocites, v, measured as a function of undercooling, AT, for a Ni-I at.%B sample. Theoretical prediction according to the dendrite growth model when combined with the Continuous Growth Model "with solute drag", for different values for the unknown equilibrium partition coefficent, ke.

perhaps an outlier). This discrepancy might possibly be attributed to the model's assumption of smooth concentration profiles ahead of the solidification front in a velocity regime where the atomic structure of the liquid necessarily causes deviations from mathematical continuity, as discussed below. Alternatively, note that in the model "with solute drag", the rise of the velocities with undercooling in the calculated curves is sharper than in the model "without solute drag". Hence the possibility exists that some kind of solute-drag effect may be responsible for the steep rise, but in a manner that becomes important only at high growth rates. Although as v ---*0 there is no solute-drag effect present in either model, it is evident that in the model "with solute drag" the effect "turns on" too soon. Aziz and Kaplan [4] have shown how the models with and without solute drag might be limiting cases of models with partial solute drag. In an attempt to find a better fit to the steep velocity rise exhibited by the data, here we try an interpolation between the two models with no a p r i o r i justification: we replace A/~ in equation (1) by A G = w I " A G o F + w2 " A G e

(6)

with W1 "~ W2 ~-

1.

(7)

Equation (6) is equivalent to AG = cenA~ta + (1 - cen)A# A

(8)

where c~ = wI • c * + w2" c *. The factor f i n equation (5) then becomesf = (wl" k + w:). As the solute-drag effect is assumed in this case to appear at higher growth velocities only, and as k (v) can be regarded as a quantity that naturally divides the velocities into higher and lower regimes, the weights Wl and w2 are chosen arbitrarily as wl = (1 - k)/(1 - k~) and wE= (k - k,)/(1 - k~) with k (v) given by equation (4).

(9)

978

ECKLER et al.: SOLUTE-DRAG EFFECT IN DENDRITIC SOLIDIFICATION

For the Ni-B system (k~<>VD, W~---,0 and w2---* 1. Note that for v>>vD our interpolation becomes identical to both of the other models, because in this case k ~ l , c*~c*, f---~ w2---) 1, AG ~ AGc---~AGDF. This interpolation yields v(AT)-relationships that look like the solid lines in Fig. 1. The measured data are described quite well, including the sharp increase of the growth velocities. Note, however, that for a good fit, k, must be set to 2.8 x 10 -4, that is two orders of magnitude greater than the best fit in the calculations without solute drag. We conclude that if the "real" value of ke would be known through an independent experiment, it should become unambiguous which expression best accounts for the data. Although our "interpolating model" describes the steep rise in the measured data very well, one must keep in mind its limitations. First, it is based on the most current dendrite growth model [13] which is still incomplete in that it only deals with the very tip of the dendrite and ignores any effect due to the lower values of v and k (v) along the sides of the dendrite tip. Additionally, all aspects of the modelling assume the spatial variation of the concentration field ahead of the interface to be a continuous, smooth function. As the velocity increases, the solute boundary layer shrinks, and its computed thickness approaches physically meaningless values of the order of 0.1 nm. Thus, one could imagine that the transition from "still-partitioning" to "non-partitioning" at the interface (owing to solute trapping) is not as smooth as the calculated curves indicate. Nevertheless, the assumed solution to the diffusion problem should be not too far from reality for velocities up to somewhat above vo. Below vo, it should still be possible to represent the concentration profile ahead of the interface with a smooth function. Above vD, growth is purely thermally controlled, and the temperature boundary layer in front of the interface is much thicker than the vanishing solutal boundary layer, for the thermal diffusivity is around three orders of magnitude larger than the solute diffusion coefficient (Table 1). The computed v (AT)-curves around AT ct are very steep indeed, and the position of the sheer rise on the undercooling scale sensitively depends upon the assumed value ofke (cf. Fig. 2). As the sharp enhancement covers quite a wide range of the velocities investigated, it should become clear that even although the theory may have some difficulties with the thickness of the solutal boundary layer around AT ct, the good description of the data may not be regarded as accidental. A remark concerning the deviations of the measured growth velocities from the theoretical v (AT)-curves for the Ni-0,7 at.%B sample at undercoolings A T > ~ 2 2 5 K is in order. In almost all measurements of dendrite growth velocities in Ni-

based alloys reported this sort of deviation has been observed [30-33]. Generally, it is accompanied with a grain refinement process, and the morphology of the solidified material changes from a dendritic to a fine equiaxed one. The mechanisms involved in this process are not yet totally clear. There are hints that in this undercooling regime rapid fragmentation of the growing primary dendrites occurs [34], which, in turn, seemingly influences the growth velocity of the dendrite tip. For the Ni-l.0at. % B sample, the maximum undercooling reached was probably not high enough to observe this effect. In summary, we have performed the first experiment to test, albeit indirectly, for the solute-drag effect by fitting the data for Ni-B dendrite tip velocities vs undercoolings in several ways. The fitting parameters allowed to vary were ke, % , and D E. When we combine dendrite growth theory with the C G M "with solute drag", we cannot fit the data for any value of ke. When we combine dendrite growth theory with the CGM "without solute drag", we obtain a reasonable fit to the data for ke = 4 x 10-6; v o = 1 5 . 6 or 18.9m/s, and D L = I . 4 or 1.7x 10-gm2/s, depending on concentration. When w e combine dendrite growth theory with a particular interpolation between the "with-solute-drag" and the "without-solute-drag" versions of the CGM, we obtain a still better fit to the data for ke = 2.8 x 10 -4 and values for vo and D L as before. The large difference in "best-fit" values for ko for these latter two fitting techniques indicates that with an independent measurement of ke, one should readily be able to eliminate one or the other of the models. Acknowledgements--One of us (M.J.A.) was supported by

NSF-DMR-92-08931. Continuous support by Professor B. Feuerbacher is gratefully acknowledged.

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