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Rapid dendritic growth in undercooled Ag-Cu melts
Acta metall, mater. Vol. 43, No. 11, pp. 4007~013, 1995
~
Elsevier Science Ltd Copyright © 1995 Acta Metallurgica Inc. Printed in Great Britain. All rights reserved 0956-7151/95 $9.50 + 0.00
Pergamon 0956-7151(95)00092-5
RAPID D E N D R I T I C GROWTH IN U N D E R C O O L E D Ag-Cu MELTS
S. WALDERand P. L. RYDER Institut fiir Werstoffphysik und Strukturforschung, University of Bremen, D-28334 Bremen, Germany (Received 1 December 1994; in revised form 22 February 1995)
Abstract--The results of measurements of the reciprocal recalescencerise time (At)-l of undercooled melts of an Ag-Cu alloy with 65 at.% Cu are presented. It is shown that (At)-~ is a semiquantitative measure of the growth rate. In agreement with observations already made on alloys with other compositions in this binary system, the growth rate as a function of the undercooling rises sharply at an undercooling corresponding to the To temperature. This behaviour can be explained in the framework of the theories of rapid dendritic growth, when the kinetic displacements of the solidus and liquidus are taken into account. A simple empirical expression for this kinetic effect is proposed. For the Ag-65 at.% Cu alloy the predictions of the theory are expressed as a plot of growth rate against undercooling and also in the form of a kinetic phase diagram. The predictions are in good qualitative agreement with the experimental results.
1. INTRODUCTION Specimens of metallic melts with masses of the order of several grams may be undercooled to temperatures as low as approx. 0.75 Tm (Tm = melting point) at moderate cooling rates (of the order of 10 K s - l ) , provided that heterogeneous nucleation is suppressed with the aid of suitable experimental techniques such as the glass flux method [1] or the levitation technique [2]. In the metastable state thus produced nucleation of the crystalline phase eventually occurs spontaneously, but may also be initiated intentionally by external triggering. With increasing undercooling-and correspondingly increasing driving force for crystallization-the growth rates increase and may reach rates of the order of lOOms -1. In addition, the compositions and structures of the ensuing solid phases may differ considerably from those of the equilibrium phases. Because of the moderate cooling rates, the temperature of the melt remains effectively uniform up to the onset of crystallization and is easily accessible to pyrometric measurement. Further, several methods have been developed for measuring the crystal growth velocities [3-7]. Thus the solidification process under extreme non-equilibrium conditions may be investigated by in situ observation. In two previous papers the results of an investigation of the solidification behaviour of the eutectic Ag-Cu alloy [8] and alloys with 25, 65 and 87.5at.% Cu [9] were presented. The melts were successfully undercooled by the glass flux technique, and the crystallization occurred spontaneously by heterogeneous nucleation. Since the known methods for measuring the growth velocity were not applicable AM43/u~
to this experimental technique, the maximum rate of heating (recalescence rate) measured after onset of crystallization was used as a semiquantitative measure of the growth rate as a function of the undercooling. With the exception of the alloy with 65 at.% Cu, all Ag-Cu alloys investigated showed a sudden increase in the recalescence rate at a critical value of the undercooling. At the same time a change in the solidification microstructure was observed; this was particularly marked in the case of the eutectic alloy, in which a transformation from a lamellar eutectic to dendritic growth occurred. It was remarkable that the critical temperature was always equal to the To temperature. In this paper we present further results on the 65 at. % Cu alloy, which show that there is a critical undercooling for this composition, and that the agreement between the critical undercooling temperature and the To temperature is valid for the whole composition range of the Ag4Su system. Further, a modification of the recalescence evaluation method is presented, which shows that the observed critical behaviour is a consequence of the influence of phase compositions on the interface kinetics. 2. EXPERIMENTAL METHODS Pellets of the Ag-65 at.% Cu alloy with a mass of 1 g were prepared by melting the elements (purity 99.99%) in an arc furnace under argon. The glass flux method [1] was used for undercooling. For this purpose, the specimens were placed in a quartz crucible together with 0.2 g glass granules and melted by inductive heating under an inert gas atmosphere (Ar and/or He). In order to melt the glass, the
4007
4008
WALDER and RYDER:
RAPID DENDRITIC GROWTH
specimens were first heated to several hundred deglees above the melting temperature; the gas pressure was varied to remove bubbles from the glass. Repeated melting and solidifying cycles gave a statistical scatter in the undercooling attained, but with an increasing tendency until a maximum was reached. Various different glass compositions were tried. As in the previous work [8, 9], the greatest undercooling was obtained with the boron silicate glass used by Wei et al. [10] for undercooling experiments on N i - S n alloys. An increase in the undercooling over earlier attempts was eventually achieved by repeated experiments in w h i c h the individual specimens were melted up to 250 times. The amount of glass used was such that part of the specimen surface remained accessible to direct observation. With the aid of a convex lens a 1: 1 image of the free specimen surface was projected onto an aperture with a diameter of 1 mm, behind which a silicon photo-diode was situated, for pyrometric measurement of the specimen temperature. The diode with an amplifier had a time constant of less than 1 #s, thus enabling much more rapid temperature changes to be measured than with a more accurate quotient pyrometer. The diode was calibrated by measuring "black body" radiation from a small hole in a laboratory furnace heated to 1500 K and cooled slowly (in about 100 rain) to 850 K, so that the radiation may be assumed to be in thermal equilibrium. The results of the calibration of the photo-diode are shown in Fig. 1, where the diode signal is plotted against the temperature. An analytical function of the form S ( T ) oc ( T -
T') x
101
100 lO-t
(p "ID O
i
.
:0
i ¢ i
T v
®"
ii
~o
I
i
i'-..2
i
tlme/s
0
-10
:
3
--"
,
/
i
1
4
5
i
~"
.....
~
-20
-40
-~0 --'---~/~-- at
b)
1
I
I
I
I
0
100
200
300
400
timeI
ms Fig. 2. (a) Diode signal of the solidification of an undercooled Ag-65 at.% Cu Specimen. The deviation from the expected cooling curve (dashed line) is due to an artefact caused by the glass flux. (b) The recalescence curve on a magnified time scale, converted to a temperature difference (referred to the recalescence maximum) by means of the calibration function (1). For further analysis the relative temperature rise ATR and the rise time At were determined. the distribution of energy in black body radiation at about 1200 K, it may be shown that the main contribution to the diode signal comes from the wavelengths between 800 and l l00nm. The results of emissivity measurements by Krishnan et aL [11] on silver and copper melts show that the required temperature and wavelength independence is given with sufficient accuracy in the range of temperatures and wavelengths concerned. Since the proportionality factor in equation (1) is not known, it is also required to relate at least one point on the diode signal curve to a known temperature (e.g. the liquidus temperature). 3. RESULTS
10-2
lO-a 1400
g
(1)
was fitted to the experimental data. For the silicon photo-diode used (type BPW 24) the fit gave T ' = 540 K and x = 6.7. With the aid of this calibration curve, the diode may be used for the measurement of the temperatures of metal melts provided that the emission coefficients can be assumed to be constant. Taking into account the spectral sensitivity of the silicon photo-diode and
i
d
1300 1200 1100 1000 900 temperature I
K
Fig. I. Calibration of the photo-diode used for the temperature measurements with a cavity radiator (laboratory furnace with a small opening). The continuous line represents a fit of the function (1) to the experimental points (C)).
As an example for a typical measurement, Fig. 2(a) shows the diode signal as a function of time during the solidification of an undercooled specimen of the alloy with 65 at.% Cu. After the onset of spontaneous crystallization (t = 0), the temperature of the specimen rises to a maximum (recalescence phase) and then falls along the liquidus to the eutectic plateau, which should give a diode signal indicated by the dashed line in Fig. 2(a). Experimentally, however,
WALDER and RYDER:
RAPID DENDRITIC GROWTH
a further rise of the diode signal is generally observed. This is indeed not a second temperature rise but due to a glass film spreading over the metal surface during cooling, as can be clearly visually observed. Glass has a higher emission coefficient than metal, thus giving rise to a higher diode signal at a given temperature. This artefact is of no consequence, however, since, as explained below, only the recalescence phase was evaluated. The recalescence phase is shown with an expanded time scale in Fig. 2(b), where the diode signal has been converted to temperature relative to the maximum (e.g. the liquidus temperature) with equation (1). For each undercooling experiment the two quantities indicated in Fig. 2(b) were measured: ATR, the undercooling below the maximum, and At, the rise time defined by extrapolating the maximum slope section of the recalescence curve to the minimum and maximum temperatures. The temperature of the maximum in the diode signal corresponds exactly to the liquidus at the given alloy composition only at the limit of low growth rates (ATR---*0). With increasing growth rate, the maximum is displaced to lower temperatures. In the present investigation, the deviations of the maxima from the true liquidus temperature were calculated using a method due to Piccone et aL [12] in order to convert the relative undercoolings ATR to the absolute undercoolings AT. After the recalescence a fraction f R = ( c p / A H f ) A T R of the sample is solidified (with the specific heat Cp and the heat of fusion AHf) and the simple balance equation (~sfR + (~1(1 _ f R ) = Co
(2)
must be satisfied, where (~s and 6"1 are the average solid and liquid composition respectively and Co the alloy composition. In a first approximation Piccone et al. used the equilibrium phase diagram to find values of t~s and 6"1 which satisfy equation (2), thus obtaining the temperature after recalescence. The maximum temperature rise achieved for the 65at.% Cu alloy was about 130K. After correction by the method of Piccone et al., the value of the maximum undercooling below the liquidus temperature was found to be ATm~x= 190 + 20K. The reciprocal of the rise time, i.e. (At) -~, was used as a measure of the dendritic growth rate. This may be justified by a comparison with direct measurements of the growth velocity V as a function of the undercooling AT by Suzuki et al. [3], as shown in Fig. 3 and Table 1. Both Vand (At) -t follow a power law, i.e. V = A ( A T ) ~,
(At)-] = B(AT) ~
(3)
with ~ ~ f l (see Table 1). Thus (At) -l is a good semiquantitative measure of the growth rate apart from an undetermined geometrical factor which does not vary with the undercooling. The relatively large scatter in our measurement may be due to the fact that the point of nucleation, which should have an
4009
10s~
10t
102
i0 o
6
O
,
101
0
510
i
100
>
10 "t
150
AT/K
Fig. 3. Comparison of the experimentally determined reciprocal reealeseencetimes (At)-l for Cu ((3) with the dendritic growth rates V (BI) measured by Suzuki et al. [3]. A fit is obtained with the same vertical logarithmic scale for (At)and V, showing that they follow the same power law. Table 1. The pre-factor .4 and the exponent ~t experimentally determinedby Suzukiet al. [3] for the relation V = .4(ATybetween the growth rate V and the undercoolingAT of Cu and Ag melts, compared to the corresponding values in the relation (At)-i = B(AT)Pdeterminedin the presentinvestigationfor Ag, Cu and Ag-65at.% Cu A (ms-')
Ag 9.5 x l0-3 Ca 5.3 x 10-3 Ag-65at.%Cu --
ct
B
(S -])
fl
1.7 l.l _+0.2 1.55+ 0.05 1.8 0.8 + 0.2 1.66+ 0.06 -- 0.28_+0.07 0.75+_0.06
influence of the recalescence rise time, is not known and varies statistically. The variation of (At)-' with the relative undercooling ATR for the alloy with 65 at.% Cu is shown in Fig. 4. For comparison the curve fitted to the measurements on pure Cu is also plotted in the figure. Up to a certain critical undercooling AT* ~ 115 K, the measurements for the alloy may also be fitted to a power law, but with a much lower value of B and fl (see Table 1). At the critical temperature there is a sudden rise in the growth rate, and the experimental values of (At)-l approach those of pure Cu. Applying the correction due to Piccone et al. [12], the value of the critical undercooling below the liquidus temperature for the 65at.% Cu alloy was found to be A T * = 150_+ 20 K. A similar critical undercooling behaviour was
10' ~,~
lOa
~-
10 2
101 10o 0
50
100
150
ATR Ik(
Fig. 4. The inverse recalescence time (At)-I as a function of the relative temperature rise ATR for the Ag-65 at.% Cu alloy. The experimental curve for Cu is also plotted for comparison. At a critical value of AT* ~ 115 K there is a sudden rise in the value of (At)-t, i.e. in the growth rate.
4010
p
WALDER and RYDER: RAPID DENDRITIC GROWTH criterion is also required. In order to take into account deviations of the compositions of the solid and liquid phases from the equilibrium values, Trivedi et al. [16] extended their theory to include a growth-rate-dependent partition coefficient k ( V ) in the form proposed by Aziz [17]
1000 go0
Q.
\ -. ~ /
800
.,
"'4
700 I
o
20
.......
/o
I
6o
8'o
ioo
at.gCu Fig. 5. A g ~ u equilibrium phase diagram with the critical temperatures of the undercooled melts and the To curve due to Murray [13]. found in previous investigations [8, 9] on Ag~2u alloys with 25, 40 and 87.5 at.% Cu. In contrast to these alloys, which show in addition a change of the microstructural morphology at the critical undercooling, there was no change observed for the 65 at.% Cu alloy. The critical undercooling temperatures for all these alloys are plotted in a phase diagram of the Ag-Cu system in Fig. 5. A striking feature of these results is the agreement between the critical undercooling temperatures and the To temperatures calculated by Murray [13]. A possible interpretation of this result is discussed below. 4. DISCUSSION The growth of dendrites in pure element melts at high thermal Prclet numbers is well described by the theory developed by Lipton et al. [14]. There are three contributions to the undercooling required to maintain a finite growth rate: the temperature gradient needed for the transport of the latent heat, the Gibbs-Thompson effect due to the curvature of the dendrite tip and the kinetics of atom transfer across the solid-liquid interface. The first effect can be calculated by solving the heat conduction equation, taking into account a suitable stability criterion to determine the tip radius, which is then used to calculate the Gibbs-Thompson effect. For the kinetic effect, Turnbull [15] developed a "collision limited growth model" and derived a linear relationship between the kinetic undercooling ATk and the growth rate V ATk = r/l#o
(4)
where the so-called linear growth coefficient go is given in terms of the speed of sound (V0) in the melt, the heat of fusion per mole (AHf), the melting point (Tin) and the universal gas constant (R) by #o = VoAHf/RT2m • In alloy melts, there is a further contribution to
AT: the constitutional undercooling, resulting from the concentration gradients required for material transport. The solution of the material diffusion problem is mathematically equivalent to that for heat conduction, but a modification of the stability
ko+-~l V k(V) = - (5) l+a°v DI where/co is the equilibrium partition coefficient, D~ the diffusion coefficient in the melt and a0 is a characteristic length for the diffusion process, of the order of magnitude of the interatomic distances. The function k ( V ) defined by equation (5) has the property k ~ 1 for V ~ oo, i.e. in the limit of high growth rates the liquid and solid phases have the same composition. In the growth theory of Trivedi et al. [16], the slope of the liquidus is taken to be a constant. This causes a problem in applying equation (5), because the solidus would then converge towards the liquidus, crossing the To line on the way, leading to the thermodynamically impossible situation that both solidus and liquidus are above the To line. A physically sounder description of the growth velocity dependence of the compositions of the solid and liquid phases is provided by the "continuous growth model" of Aziz and Kaplan [18]. They consider two different cases: with and without "solute drag", and calculate "kinetic" phase diagrams for the Ag-Cu system showing the phase compositions for different given growth rates, using the free energies of the liquid and solid phases calculated by Murray [13]. In both cases the iiquidus and solidus first converge towards the To line and are displaced to lower temperatures at higher growth rates due to the kinetic effect expressed by equation (4). The theory of Aziz and Kaplan [18] applies strictly only to growth with a planar interface, but consideration of the Gibbs-Thompson effect allows it to be applied also to dendritic growth. A direct test of the theory with the experimental results presented above is, however, difficult for two reasons. Firstly, multiple solutions and instabilities are found in the central regions of the calculated kinetic phase diagrams. Secondly, the inclusion of the numerical calculations of the displacements of the liquidus and solidus in the dendritic growth theory would involve very complex computations. In the dilute solution approximation and in other regions where the liquidus, solidus and To lines can be approximated by straight lines, an analytical expression can be found for the growth rate dependence of the liquidus and solidus slopes without [19] or with [20] solute drag. Eckler et al. [21] used this to extend the dendritic growth theory and compared the predictions with and without solute drag with the experimental results obtained from undercooled Ni-B melts. For low growth rates the assumption of no
WALDER and RYDER:
RAPID DENDRITIC GROWTH
solute drag fitted the results best; with increasing growth rates a transition to the continuous growth model with solute drag was observed. Kittl et al. [22] also found agreement with the model without solute drag for planar growth in Si-As melts with growth velocities up to 2 ms -t. Since the assumption of linear liquidus, solidus and To lines cannot be applied over the whole range of the A g - C u system, the following empirical approach was adopted: the kinetic undercooling [equation (4)] was extended by a function which approached T m - - T o (T m = equilibrium liquidus temperature) for infinite growth rates. This ensured that the liquidus and solidus approached the T O line, except for the additional term V/#o. The function describing the variation of the additional undercooling from 0 at low growth rates to T m - To at high rates was chosen to be similar to that given by Aziz [17] for the growth rate dependence of the partition coefficient, i.e. the total kinetic undercooling was taken to be a0 V V Dl A Tk = - - + - #0 l+a0v DI
(T m - To).
(6)
Figure 6 shows a comparison of the additional undercooling calculated from equation (6) with the analytical solutions of the continuous growth model with and without solute drag, calculated under the assumption of straight-line solidus and liquidus. It is seen that the curve calculated from equation (6) lies between the other two. For k0--~0 (i.e. for complete separation of the components) the curve for the continuous growth model without solute drag approaches that derived from equation (6). As has already been done for the case of directional dendritic solidification [23, 24], the concentrationdependence of the liquidus temperature TI(C) and the temperature-dependence of the equilibrium partition coefficient k o ( T ) were expressed as polynomials (see Table 2). The constitutional undercooling is then given directly from the solute concentration C* at the 1 with solute d r a g - / / / /
/ / /
,
0
~
=11-1"-
0.01
0.1
I
I
I
1
10
100
ao
D--~• V Fig. 6. Comparison of different models for the additional kinetic undercooling as a function of the growth rate (for k 0 = 0.17): the continuous growth model with [20] or without [19] solute drag, and the empirical function of equation (6).
4011
Table 2. The thermo-physicalparameters of the Ag-65 at.% Cu alloy used in the calculation of the dendritic growth rates. The linear growth coefficientwas determinedfrom a fit to the Cu measurements of Suzuki et al. [3], and the characteristic diffusion length by fitting to the Ag~55at.% Cu measurements. The polynomial functions for the liquidus and the partition coefficientwere fitted to the Ag-Cu phase diagram Alloy composition CO= 47.8 wt. % Ag Melting point Tm= 1151m To = 993 K To temperature AHf= 1.62/ 104Jmol l Heat of fusion c/= 31.6 j mol-i K t Specific heat of the liquid cq=5× 10-5 m2s Temperature diffusivity of the melt Do= 1.22× 10 7m2s-~ Parameters of the diffusion Q=4.19×104jmol t coefficientin the melt F = l . 4 x 10 7Km Gibbs-Thompson coefficient Linear growth coefficient #o= 0.6ms-~ K i Characteristic diffusion length ao= 1.5 nm Liquidus line TI(K) [C(wt.% Ag)] TI(C) = 1360- 6,534C + 0.1561C2 - 4.964 x 10 3C3 +7,534 x 10-5C4-4.206 × l0 7CS Partition coefficient k0 IT(K)] ko(T ) = -7.084 + 2.883 x 10-2T--4.355 x 10-ST2 +2.891x l0 8T3-7.019x 10-t2T4
dendrite tip: A T c = Tm - TI(C~'). In the stability function (see e.g. [14, 16]), the derivative d T ~ ( C ) / d C is used instead of a constant liquidus slope. As suggested by Trivedi et al. [25], the temperaturedependence of the diffusion coefficient, which enters into the dendritic growth theory via both the chemical P+clet number and equations (5) and (6), can be taken into account by using an Arrhenius function: DI(T) = Do e x p ( - Q / R T ) . The parameters D Oand Q are given, together with the other thermo-physical data for the A g - 6 5 at.% Cu alloy in Table 2. F o r the linear growth coefficient the same value was taken as for pure copper [9]. This means that the growth o f the interface is still collision limited, which seems to be plausible because of the absence of chemical order in the A g - C u alloy. In the case of chemical order the growth is diffusion limited and the linear growth coefficient will increase about three orders of magnitude [20], as e.g. for the intermetallic compounds FeSi and CoSi [26]. Figure 7 shows the dendritic growth rate as a function of the melt undercooling calculated using the dendritic growth theory [16] with the modifications described above. In the undercooling range 190-220 K the growth rate is not uniquely defined, and a discontinuous rise, e.g. along the path indicated by the dotted line and arrow in the figure, is to be expected. This corresponds to the experimentally observed sudden rise in the inverse recalescence time (see Fig. 4). The To temperature, at which this rise is observed experimentally, is indicated by an arrow on the temperature axis in Fig. 7. The theoretical curve indicates a somewhat higher value for the critical undercooling. The deviation between the observed and calculated critical temperatures (approx. 40 K) may be due, at least in part, to erroneous assumptions about the appropriate value of the latent heat of fusion. The heat released during the solidification of
4012
WALDER and RYDER:
RAPID DENDRITIC GROWTH resulting in the growth of a solid phase with the same composition as the melt. Figur e 8 illustrates the significance of the additional kinetic undercooling. Without the additive term in equation (6) the temperature of the dendrite tip would rise above the To temperature, which is thermodynamically impossible.
102 lOt 100 > 10-1 10-2
, 50
a 100
5. SUMMARY
=
i~ 150
200
6T/K
Fig. 7. The dendritic growth rates of the alloy Ag--65at.% Cu as a function of the melt undercooling calculated from the model of Trivedi et aL [16], but taking into account the variation of the functions Tj(C), ko(T) and DI(T) and the modified kinetic undereooling according to equation (6). A sudden increase in the growth rate occurs approximately along the dotted path. The curve calculated for Cu is shown for comparison. The To temperature, at which the sudden rise in the growth rate is observed experimentally, is indicated on the temperature axis.
the supersaturated phase is expected to be smaller than the equilibrium latent heat. Since the latter was used in the theoretical calculations, the calculated temperature of the dendrite tip, and hence the undercooling, will be too high. The compositions of the solid and liquid phases at the dendrite tip and the interface undercooling may also be calculated from the dendritic growth theory. In Fig. 8 the compositions of the two phases have been plotted as a function of the interface temperature in the A g - C u phase diagram. As the undercgoling is increased from zero, the compositions are~ first displaced along the solidus and liquidus lines. With increasing growth rate (right axis of Fig. 8) the partition coefficient deviates more and more from the equilibrium value according to equation (5). Finally, there is a discontinuous change along the dotted line, corresponding to the path indicated in Fig. 7,
.'/,
1000
'
~"
•
s
900
,__'_~_- . . . . . . . . . . . ,,_. . . . . . . . " " " ' - - -'"
800
70O l : , : ~,, , 0
20
"-
40
60
'
' 0.005
~, ',', ,0.05 ,
,,,(. 50
80
E
>
'
100
wt.%Cu
Fig. 8. The change of the compositions of the solid and liquid phases with increasing interface undereooling, plotted in the Ag--Cu phase diagram (with metastable extensions of the solidus and liquidus line and the To temperature due to Murray [13]). The corresponding growth velocity is shown on the fight axis. The discontinuous change in the composition (dotted line) corresponds to the path indicated in Fig. 7.
In summary it may be concluded that a simple phenomenological modification of the kinetic undercooling described above may be used in alloy systems where the dilute solution approximation is not applicable. This modification [e.g. equation (6)] accounts very well, at least qualitatively, for the experimentally observed sudden increase in the growth rate at temperatures near To for the Ag-65 at.% Cu alloy without thermodynamic inconsistency. A complete understanding of the fact that the critical undercooling agrees with the To temperature over the whole range of compositions of the Ag-Cu system would require a theory which takes better account of thermodynamic considerations from the beginning. Some recent papers [21, 22] suggest that this could be possible in the framework of the continuous growth model without solute drag. Acknowledgement--The authors are pleased to acknowl-
edge support of this research by the Deutsche Forschungsgcmeinschaft (DFG-SP Unterkiihlte Metallschmelzen, Ry 7/8-2). R E F E R E N C E S
I. P. Bardenheuer and R. Bleckmann, Stahl u. Eisen 61, 49 (1941). 2. D. M. Herlach, R. Willnecker and F. G. Gillissen, Proc. 5th Eur. Syrup. on Materials Science under Microgravity,
p. 399, ESA SP 222. Schloss Elmau (1984). 3. T. Suzuki, S. Toyoda, T. Umeda and Y. Kimura, J. Cryst. Growth 38, 123 0977). 4. Y. Wu, T. J. Piccone, Y. Shiohara and M. C. Flemings, Metall. Trans. 18.4,, 915 (1987). 5. E. Schleip, R. Willnecker, D. M. Herlach and G. P. G6rler, Mater. Sci. Engng 98, 39 (1988). 6. K. Eckler, M. Kratz and I. Egry, Rev. Sci. Instrum. 64, 2639 (1993). 7. B. T. Bassler, W. H. Hofmeister, G. Carro and R. J. Bayuzick, Metall. Mater. Trans. 25A, 1301 (1994). 8. S. Walder and P. L. Ryder, J. appl. Phys. 73, 1965 (1993). 9. S. Walder and P. L. Ryder, J. appL Phys. 74, 6100 (1993). 10. B. Wei, G. Yang and Y. Zhou, Acta metall, mater. 39, 1249 (1991). l I. S. Kfishnan, G. P. Hansen, R. H. Hauge and J. L. Margrave, High Temp. Sci. 29, 17 (1990). 12. T. J. Piccone, Y. Wu, Y. Shiohara and M. C. Flemings, MetalL Trans. 18A, 925 (1987). 13. J. L. Murray, Metall. Trans. 15A, 261 (1984). 14. J. Lipton, W. Kurz and R. Trivedi, Acta metalL 35, 957 (1987). 15. D. Tumbull, MetalL Trans. 12A, 695 (1981). 16. R. Tfivedi, J. Lipton and W. Kurz, .4eta metall. 35, 965 (1987). 17. M. J. Aziz, J. appl. Phys. 53, 1158 0982).
WALDER and RYDER:
RAPID DENDRITIC GROWTH
18. M. J. Aziz and T. Kaplan, Acta metall. 36, 2335 (1988). 19. W. J. Boettinger, S. R. Coriell and R. Trivedi, in Rapid Solidification Processing: Principle and Technology I V (edited by R. Mehrabian and P. A. Parrish), p. 13. Claitor's, Baton Route, La (1988). 20. M. J. Aziz and W. J. Boettinger, Acta metall, mater. 42, 527 (1994). 21. K. Eckler, D. M. Herlach and M. J. Aziz, Acta metall. mater. 42, 975 (1994).
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22. J. A. Kittl, M. J. Aziz, D. P. Brunco and M. O. Thompson, Appl. Phys. Lett. 64, 2359 (1994). 23. M. Carrard, M. Gremaud, M. Zimmermann and W. Kurz, Acta metall, mater. 40, 983 (1992). 24. Shu-Zu Lu, J. D. Hunt, P. Gilgien and W. Kurz, Acta metall, mater. 42, 1653 (1994). 25. R. Trivedi, P. Magnin and W. Kurz, Acta metall. 35, 971 (1987). 26. D. M. Herlach, Mater. Sci. Engng A 179/180, 147 (1994).
~
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Pergamon 0956-7151(95)00092-5
RAPID D E N D R I T I C GROWTH IN U N D E R C O O L E D Ag-Cu MELTS
S. WALDERand P. L. RYDER Institut fiir Werstoffphysik und Strukturforschung, University of Bremen, D-28334 Bremen, Germany (Received 1 December 1994; in revised form 22 February 1995)
Abstract--The results of measurements of the reciprocal recalescencerise time (At)-l of undercooled melts of an Ag-Cu alloy with 65 at.% Cu are presented. It is shown that (At)-~ is a semiquantitative measure of the growth rate. In agreement with observations already made on alloys with other compositions in this binary system, the growth rate as a function of the undercooling rises sharply at an undercooling corresponding to the To temperature. This behaviour can be explained in the framework of the theories of rapid dendritic growth, when the kinetic displacements of the solidus and liquidus are taken into account. A simple empirical expression for this kinetic effect is proposed. For the Ag-65 at.% Cu alloy the predictions of the theory are expressed as a plot of growth rate against undercooling and also in the form of a kinetic phase diagram. The predictions are in good qualitative agreement with the experimental results.
1. INTRODUCTION Specimens of metallic melts with masses of the order of several grams may be undercooled to temperatures as low as approx. 0.75 Tm (Tm = melting point) at moderate cooling rates (of the order of 10 K s - l ) , provided that heterogeneous nucleation is suppressed with the aid of suitable experimental techniques such as the glass flux method [1] or the levitation technique [2]. In the metastable state thus produced nucleation of the crystalline phase eventually occurs spontaneously, but may also be initiated intentionally by external triggering. With increasing undercooling-and correspondingly increasing driving force for crystallization-the growth rates increase and may reach rates of the order of lOOms -1. In addition, the compositions and structures of the ensuing solid phases may differ considerably from those of the equilibrium phases. Because of the moderate cooling rates, the temperature of the melt remains effectively uniform up to the onset of crystallization and is easily accessible to pyrometric measurement. Further, several methods have been developed for measuring the crystal growth velocities [3-7]. Thus the solidification process under extreme non-equilibrium conditions may be investigated by in situ observation. In two previous papers the results of an investigation of the solidification behaviour of the eutectic Ag-Cu alloy [8] and alloys with 25, 65 and 87.5at.% Cu [9] were presented. The melts were successfully undercooled by the glass flux technique, and the crystallization occurred spontaneously by heterogeneous nucleation. Since the known methods for measuring the growth velocity were not applicable AM43/u~
to this experimental technique, the maximum rate of heating (recalescence rate) measured after onset of crystallization was used as a semiquantitative measure of the growth rate as a function of the undercooling. With the exception of the alloy with 65 at.% Cu, all Ag-Cu alloys investigated showed a sudden increase in the recalescence rate at a critical value of the undercooling. At the same time a change in the solidification microstructure was observed; this was particularly marked in the case of the eutectic alloy, in which a transformation from a lamellar eutectic to dendritic growth occurred. It was remarkable that the critical temperature was always equal to the To temperature. In this paper we present further results on the 65 at. % Cu alloy, which show that there is a critical undercooling for this composition, and that the agreement between the critical undercooling temperature and the To temperature is valid for the whole composition range of the Ag4Su system. Further, a modification of the recalescence evaluation method is presented, which shows that the observed critical behaviour is a consequence of the influence of phase compositions on the interface kinetics. 2. EXPERIMENTAL METHODS Pellets of the Ag-65 at.% Cu alloy with a mass of 1 g were prepared by melting the elements (purity 99.99%) in an arc furnace under argon. The glass flux method [1] was used for undercooling. For this purpose, the specimens were placed in a quartz crucible together with 0.2 g glass granules and melted by inductive heating under an inert gas atmosphere (Ar and/or He). In order to melt the glass, the
4007
4008
WALDER and RYDER:
RAPID DENDRITIC GROWTH
specimens were first heated to several hundred deglees above the melting temperature; the gas pressure was varied to remove bubbles from the glass. Repeated melting and solidifying cycles gave a statistical scatter in the undercooling attained, but with an increasing tendency until a maximum was reached. Various different glass compositions were tried. As in the previous work [8, 9], the greatest undercooling was obtained with the boron silicate glass used by Wei et al. [10] for undercooling experiments on N i - S n alloys. An increase in the undercooling over earlier attempts was eventually achieved by repeated experiments in w h i c h the individual specimens were melted up to 250 times. The amount of glass used was such that part of the specimen surface remained accessible to direct observation. With the aid of a convex lens a 1: 1 image of the free specimen surface was projected onto an aperture with a diameter of 1 mm, behind which a silicon photo-diode was situated, for pyrometric measurement of the specimen temperature. The diode with an amplifier had a time constant of less than 1 #s, thus enabling much more rapid temperature changes to be measured than with a more accurate quotient pyrometer. The diode was calibrated by measuring "black body" radiation from a small hole in a laboratory furnace heated to 1500 K and cooled slowly (in about 100 rain) to 850 K, so that the radiation may be assumed to be in thermal equilibrium. The results of the calibration of the photo-diode are shown in Fig. 1, where the diode signal is plotted against the temperature. An analytical function of the form S ( T ) oc ( T -
T') x
101
100 lO-t
(p "ID O
i
.
:0
i ¢ i
T v
®"
ii
~o
I
i
i'-..2
i
tlme/s
0
-10
:
3
--"
,
/
i
1
4
5
i
~"
.....
~
-20
-40
-~0 --'---~/~-- at
b)
1
I
I
I
I
0
100
200
300
400
timeI
ms Fig. 2. (a) Diode signal of the solidification of an undercooled Ag-65 at.% Cu Specimen. The deviation from the expected cooling curve (dashed line) is due to an artefact caused by the glass flux. (b) The recalescence curve on a magnified time scale, converted to a temperature difference (referred to the recalescence maximum) by means of the calibration function (1). For further analysis the relative temperature rise ATR and the rise time At were determined. the distribution of energy in black body radiation at about 1200 K, it may be shown that the main contribution to the diode signal comes from the wavelengths between 800 and l l00nm. The results of emissivity measurements by Krishnan et aL [11] on silver and copper melts show that the required temperature and wavelength independence is given with sufficient accuracy in the range of temperatures and wavelengths concerned. Since the proportionality factor in equation (1) is not known, it is also required to relate at least one point on the diode signal curve to a known temperature (e.g. the liquidus temperature). 3. RESULTS
10-2
lO-a 1400
g
(1)
was fitted to the experimental data. For the silicon photo-diode used (type BPW 24) the fit gave T ' = 540 K and x = 6.7. With the aid of this calibration curve, the diode may be used for the measurement of the temperatures of metal melts provided that the emission coefficients can be assumed to be constant. Taking into account the spectral sensitivity of the silicon photo-diode and
i
d
1300 1200 1100 1000 900 temperature I
K
Fig. I. Calibration of the photo-diode used for the temperature measurements with a cavity radiator (laboratory furnace with a small opening). The continuous line represents a fit of the function (1) to the experimental points (C)).
As an example for a typical measurement, Fig. 2(a) shows the diode signal as a function of time during the solidification of an undercooled specimen of the alloy with 65 at.% Cu. After the onset of spontaneous crystallization (t = 0), the temperature of the specimen rises to a maximum (recalescence phase) and then falls along the liquidus to the eutectic plateau, which should give a diode signal indicated by the dashed line in Fig. 2(a). Experimentally, however,
WALDER and RYDER:
RAPID DENDRITIC GROWTH
a further rise of the diode signal is generally observed. This is indeed not a second temperature rise but due to a glass film spreading over the metal surface during cooling, as can be clearly visually observed. Glass has a higher emission coefficient than metal, thus giving rise to a higher diode signal at a given temperature. This artefact is of no consequence, however, since, as explained below, only the recalescence phase was evaluated. The recalescence phase is shown with an expanded time scale in Fig. 2(b), where the diode signal has been converted to temperature relative to the maximum (e.g. the liquidus temperature) with equation (1). For each undercooling experiment the two quantities indicated in Fig. 2(b) were measured: ATR, the undercooling below the maximum, and At, the rise time defined by extrapolating the maximum slope section of the recalescence curve to the minimum and maximum temperatures. The temperature of the maximum in the diode signal corresponds exactly to the liquidus at the given alloy composition only at the limit of low growth rates (ATR---*0). With increasing growth rate, the maximum is displaced to lower temperatures. In the present investigation, the deviations of the maxima from the true liquidus temperature were calculated using a method due to Piccone et aL [12] in order to convert the relative undercoolings ATR to the absolute undercoolings AT. After the recalescence a fraction f R = ( c p / A H f ) A T R of the sample is solidified (with the specific heat Cp and the heat of fusion AHf) and the simple balance equation (~sfR + (~1(1 _ f R ) = Co
(2)
must be satisfied, where (~s and 6"1 are the average solid and liquid composition respectively and Co the alloy composition. In a first approximation Piccone et al. used the equilibrium phase diagram to find values of t~s and 6"1 which satisfy equation (2), thus obtaining the temperature after recalescence. The maximum temperature rise achieved for the 65at.% Cu alloy was about 130K. After correction by the method of Piccone et al., the value of the maximum undercooling below the liquidus temperature was found to be ATm~x= 190 + 20K. The reciprocal of the rise time, i.e. (At) -~, was used as a measure of the dendritic growth rate. This may be justified by a comparison with direct measurements of the growth velocity V as a function of the undercooling AT by Suzuki et al. [3], as shown in Fig. 3 and Table 1. Both Vand (At) -t follow a power law, i.e. V = A ( A T ) ~,
(At)-] = B(AT) ~
(3)
with ~ ~ f l (see Table 1). Thus (At) -l is a good semiquantitative measure of the growth rate apart from an undetermined geometrical factor which does not vary with the undercooling. The relatively large scatter in our measurement may be due to the fact that the point of nucleation, which should have an
4009
10s~
10t
102
i0 o
6
O
,
101
0
510
i
100
>
10 "t
150
AT/K
Fig. 3. Comparison of the experimentally determined reciprocal reealeseencetimes (At)-l for Cu ((3) with the dendritic growth rates V (BI) measured by Suzuki et al. [3]. A fit is obtained with the same vertical logarithmic scale for (At)and V, showing that they follow the same power law. Table 1. The pre-factor .4 and the exponent ~t experimentally determinedby Suzukiet al. [3] for the relation V = .4(ATybetween the growth rate V and the undercoolingAT of Cu and Ag melts, compared to the corresponding values in the relation (At)-i = B(AT)Pdeterminedin the presentinvestigationfor Ag, Cu and Ag-65at.% Cu A (ms-')
Ag 9.5 x l0-3 Ca 5.3 x 10-3 Ag-65at.%Cu --
ct
B
(S -])
fl
1.7 l.l _+0.2 1.55+ 0.05 1.8 0.8 + 0.2 1.66+ 0.06 -- 0.28_+0.07 0.75+_0.06
influence of the recalescence rise time, is not known and varies statistically. The variation of (At)-' with the relative undercooling ATR for the alloy with 65 at.% Cu is shown in Fig. 4. For comparison the curve fitted to the measurements on pure Cu is also plotted in the figure. Up to a certain critical undercooling AT* ~ 115 K, the measurements for the alloy may also be fitted to a power law, but with a much lower value of B and fl (see Table 1). At the critical temperature there is a sudden rise in the growth rate, and the experimental values of (At)-l approach those of pure Cu. Applying the correction due to Piccone et al. [12], the value of the critical undercooling below the liquidus temperature for the 65at.% Cu alloy was found to be A T * = 150_+ 20 K. A similar critical undercooling behaviour was
10' ~,~
lOa
~-
10 2
101 10o 0
50
100
150
ATR Ik(
Fig. 4. The inverse recalescence time (At)-I as a function of the relative temperature rise ATR for the Ag-65 at.% Cu alloy. The experimental curve for Cu is also plotted for comparison. At a critical value of AT* ~ 115 K there is a sudden rise in the value of (At)-t, i.e. in the growth rate.
4010
p
WALDER and RYDER: RAPID DENDRITIC GROWTH criterion is also required. In order to take into account deviations of the compositions of the solid and liquid phases from the equilibrium values, Trivedi et al. [16] extended their theory to include a growth-rate-dependent partition coefficient k ( V ) in the form proposed by Aziz [17]
1000 go0
Q.
\ -. ~ /
800
.,
"'4
700 I
o
20
.......
/o
I
6o
8'o
ioo
at.gCu Fig. 5. A g ~ u equilibrium phase diagram with the critical temperatures of the undercooled melts and the To curve due to Murray [13]. found in previous investigations [8, 9] on Ag~2u alloys with 25, 40 and 87.5 at.% Cu. In contrast to these alloys, which show in addition a change of the microstructural morphology at the critical undercooling, there was no change observed for the 65 at.% Cu alloy. The critical undercooling temperatures for all these alloys are plotted in a phase diagram of the Ag-Cu system in Fig. 5. A striking feature of these results is the agreement between the critical undercooling temperatures and the To temperatures calculated by Murray [13]. A possible interpretation of this result is discussed below. 4. DISCUSSION The growth of dendrites in pure element melts at high thermal Prclet numbers is well described by the theory developed by Lipton et al. [14]. There are three contributions to the undercooling required to maintain a finite growth rate: the temperature gradient needed for the transport of the latent heat, the Gibbs-Thompson effect due to the curvature of the dendrite tip and the kinetics of atom transfer across the solid-liquid interface. The first effect can be calculated by solving the heat conduction equation, taking into account a suitable stability criterion to determine the tip radius, which is then used to calculate the Gibbs-Thompson effect. For the kinetic effect, Turnbull [15] developed a "collision limited growth model" and derived a linear relationship between the kinetic undercooling ATk and the growth rate V ATk = r/l#o
(4)
where the so-called linear growth coefficient go is given in terms of the speed of sound (V0) in the melt, the heat of fusion per mole (AHf), the melting point (Tin) and the universal gas constant (R) by #o = VoAHf/RT2m • In alloy melts, there is a further contribution to
AT: the constitutional undercooling, resulting from the concentration gradients required for material transport. The solution of the material diffusion problem is mathematically equivalent to that for heat conduction, but a modification of the stability
ko+-~l V k(V) = - (5) l+a°v DI where/co is the equilibrium partition coefficient, D~ the diffusion coefficient in the melt and a0 is a characteristic length for the diffusion process, of the order of magnitude of the interatomic distances. The function k ( V ) defined by equation (5) has the property k ~ 1 for V ~ oo, i.e. in the limit of high growth rates the liquid and solid phases have the same composition. In the growth theory of Trivedi et al. [16], the slope of the liquidus is taken to be a constant. This causes a problem in applying equation (5), because the solidus would then converge towards the liquidus, crossing the To line on the way, leading to the thermodynamically impossible situation that both solidus and liquidus are above the To line. A physically sounder description of the growth velocity dependence of the compositions of the solid and liquid phases is provided by the "continuous growth model" of Aziz and Kaplan [18]. They consider two different cases: with and without "solute drag", and calculate "kinetic" phase diagrams for the Ag-Cu system showing the phase compositions for different given growth rates, using the free energies of the liquid and solid phases calculated by Murray [13]. In both cases the iiquidus and solidus first converge towards the To line and are displaced to lower temperatures at higher growth rates due to the kinetic effect expressed by equation (4). The theory of Aziz and Kaplan [18] applies strictly only to growth with a planar interface, but consideration of the Gibbs-Thompson effect allows it to be applied also to dendritic growth. A direct test of the theory with the experimental results presented above is, however, difficult for two reasons. Firstly, multiple solutions and instabilities are found in the central regions of the calculated kinetic phase diagrams. Secondly, the inclusion of the numerical calculations of the displacements of the liquidus and solidus in the dendritic growth theory would involve very complex computations. In the dilute solution approximation and in other regions where the liquidus, solidus and To lines can be approximated by straight lines, an analytical expression can be found for the growth rate dependence of the liquidus and solidus slopes without [19] or with [20] solute drag. Eckler et al. [21] used this to extend the dendritic growth theory and compared the predictions with and without solute drag with the experimental results obtained from undercooled Ni-B melts. For low growth rates the assumption of no
WALDER and RYDER:
RAPID DENDRITIC GROWTH
solute drag fitted the results best; with increasing growth rates a transition to the continuous growth model with solute drag was observed. Kittl et al. [22] also found agreement with the model without solute drag for planar growth in Si-As melts with growth velocities up to 2 ms -t. Since the assumption of linear liquidus, solidus and To lines cannot be applied over the whole range of the A g - C u system, the following empirical approach was adopted: the kinetic undercooling [equation (4)] was extended by a function which approached T m - - T o (T m = equilibrium liquidus temperature) for infinite growth rates. This ensured that the liquidus and solidus approached the T O line, except for the additional term V/#o. The function describing the variation of the additional undercooling from 0 at low growth rates to T m - To at high rates was chosen to be similar to that given by Aziz [17] for the growth rate dependence of the partition coefficient, i.e. the total kinetic undercooling was taken to be a0 V V Dl A Tk = - - + - #0 l+a0v DI
(T m - To).
(6)
Figure 6 shows a comparison of the additional undercooling calculated from equation (6) with the analytical solutions of the continuous growth model with and without solute drag, calculated under the assumption of straight-line solidus and liquidus. It is seen that the curve calculated from equation (6) lies between the other two. For k0--~0 (i.e. for complete separation of the components) the curve for the continuous growth model without solute drag approaches that derived from equation (6). As has already been done for the case of directional dendritic solidification [23, 24], the concentrationdependence of the liquidus temperature TI(C) and the temperature-dependence of the equilibrium partition coefficient k o ( T ) were expressed as polynomials (see Table 2). The constitutional undercooling is then given directly from the solute concentration C* at the 1 with solute d r a g - / / / /
/ / /
,
0
~
=11-1"-
0.01
0.1
I
I
I
1
10
100
ao
D--~• V Fig. 6. Comparison of different models for the additional kinetic undercooling as a function of the growth rate (for k 0 = 0.17): the continuous growth model with [20] or without [19] solute drag, and the empirical function of equation (6).
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Table 2. The thermo-physicalparameters of the Ag-65 at.% Cu alloy used in the calculation of the dendritic growth rates. The linear growth coefficientwas determinedfrom a fit to the Cu measurements of Suzuki et al. [3], and the characteristic diffusion length by fitting to the Ag~55at.% Cu measurements. The polynomial functions for the liquidus and the partition coefficientwere fitted to the Ag-Cu phase diagram Alloy composition CO= 47.8 wt. % Ag Melting point Tm= 1151m To = 993 K To temperature AHf= 1.62/ 104Jmol l Heat of fusion c/= 31.6 j mol-i K t Specific heat of the liquid cq=5× 10-5 m2s Temperature diffusivity of the melt Do= 1.22× 10 7m2s-~ Parameters of the diffusion Q=4.19×104jmol t coefficientin the melt F = l . 4 x 10 7Km Gibbs-Thompson coefficient Linear growth coefficient #o= 0.6ms-~ K i Characteristic diffusion length ao= 1.5 nm Liquidus line TI(K) [C(wt.% Ag)] TI(C) = 1360- 6,534C + 0.1561C2 - 4.964 x 10 3C3 +7,534 x 10-5C4-4.206 × l0 7CS Partition coefficient k0 IT(K)] ko(T ) = -7.084 + 2.883 x 10-2T--4.355 x 10-ST2 +2.891x l0 8T3-7.019x 10-t2T4
dendrite tip: A T c = Tm - TI(C~'). In the stability function (see e.g. [14, 16]), the derivative d T ~ ( C ) / d C is used instead of a constant liquidus slope. As suggested by Trivedi et al. [25], the temperaturedependence of the diffusion coefficient, which enters into the dendritic growth theory via both the chemical P+clet number and equations (5) and (6), can be taken into account by using an Arrhenius function: DI(T) = Do e x p ( - Q / R T ) . The parameters D Oand Q are given, together with the other thermo-physical data for the A g - 6 5 at.% Cu alloy in Table 2. F o r the linear growth coefficient the same value was taken as for pure copper [9]. This means that the growth o f the interface is still collision limited, which seems to be plausible because of the absence of chemical order in the A g - C u alloy. In the case of chemical order the growth is diffusion limited and the linear growth coefficient will increase about three orders of magnitude [20], as e.g. for the intermetallic compounds FeSi and CoSi [26]. Figure 7 shows the dendritic growth rate as a function of the melt undercooling calculated using the dendritic growth theory [16] with the modifications described above. In the undercooling range 190-220 K the growth rate is not uniquely defined, and a discontinuous rise, e.g. along the path indicated by the dotted line and arrow in the figure, is to be expected. This corresponds to the experimentally observed sudden rise in the inverse recalescence time (see Fig. 4). The To temperature, at which this rise is observed experimentally, is indicated by an arrow on the temperature axis in Fig. 7. The theoretical curve indicates a somewhat higher value for the critical undercooling. The deviation between the observed and calculated critical temperatures (approx. 40 K) may be due, at least in part, to erroneous assumptions about the appropriate value of the latent heat of fusion. The heat released during the solidification of
4012
WALDER and RYDER:
RAPID DENDRITIC GROWTH resulting in the growth of a solid phase with the same composition as the melt. Figur e 8 illustrates the significance of the additional kinetic undercooling. Without the additive term in equation (6) the temperature of the dendrite tip would rise above the To temperature, which is thermodynamically impossible.
102 lOt 100 > 10-1 10-2
, 50
a 100
5. SUMMARY
=
i~ 150
200
6T/K
Fig. 7. The dendritic growth rates of the alloy Ag--65at.% Cu as a function of the melt undercooling calculated from the model of Trivedi et aL [16], but taking into account the variation of the functions Tj(C), ko(T) and DI(T) and the modified kinetic undereooling according to equation (6). A sudden increase in the growth rate occurs approximately along the dotted path. The curve calculated for Cu is shown for comparison. The To temperature, at which the sudden rise in the growth rate is observed experimentally, is indicated on the temperature axis.
the supersaturated phase is expected to be smaller than the equilibrium latent heat. Since the latter was used in the theoretical calculations, the calculated temperature of the dendrite tip, and hence the undercooling, will be too high. The compositions of the solid and liquid phases at the dendrite tip and the interface undercooling may also be calculated from the dendritic growth theory. In Fig. 8 the compositions of the two phases have been plotted as a function of the interface temperature in the A g - C u phase diagram. As the undercgoling is increased from zero, the compositions are~ first displaced along the solidus and liquidus lines. With increasing growth rate (right axis of Fig. 8) the partition coefficient deviates more and more from the equilibrium value according to equation (5). Finally, there is a discontinuous change along the dotted line, corresponding to the path indicated in Fig. 7,
.'/,
1000
'
~"
•
s
900
,__'_~_- . . . . . . . . . . . ,,_. . . . . . . . " " " ' - - -'"
800
70O l : , : ~,, , 0
20
"-
40
60
'
' 0.005
~, ',', ,0.05 ,
,,,(. 50
80
E
>
'
100
wt.%Cu
Fig. 8. The change of the compositions of the solid and liquid phases with increasing interface undereooling, plotted in the Ag--Cu phase diagram (with metastable extensions of the solidus and liquidus line and the To temperature due to Murray [13]). The corresponding growth velocity is shown on the fight axis. The discontinuous change in the composition (dotted line) corresponds to the path indicated in Fig. 7.
In summary it may be concluded that a simple phenomenological modification of the kinetic undercooling described above may be used in alloy systems where the dilute solution approximation is not applicable. This modification [e.g. equation (6)] accounts very well, at least qualitatively, for the experimentally observed sudden increase in the growth rate at temperatures near To for the Ag-65 at.% Cu alloy without thermodynamic inconsistency. A complete understanding of the fact that the critical undercooling agrees with the To temperature over the whole range of compositions of the Ag-Cu system would require a theory which takes better account of thermodynamic considerations from the beginning. Some recent papers [21, 22] suggest that this could be possible in the framework of the continuous growth model without solute drag. Acknowledgement--The authors are pleased to acknowl-
edge support of this research by the Deutsche Forschungsgcmeinschaft (DFG-SP Unterkiihlte Metallschmelzen, Ry 7/8-2). R E F E R E N C E S
I. P. Bardenheuer and R. Bleckmann, Stahl u. Eisen 61, 49 (1941). 2. D. M. Herlach, R. Willnecker and F. G. Gillissen, Proc. 5th Eur. Syrup. on Materials Science under Microgravity,
p. 399, ESA SP 222. Schloss Elmau (1984). 3. T. Suzuki, S. Toyoda, T. Umeda and Y. Kimura, J. Cryst. Growth 38, 123 0977). 4. Y. Wu, T. J. Piccone, Y. Shiohara and M. C. Flemings, Metall. Trans. 18.4,, 915 (1987). 5. E. Schleip, R. Willnecker, D. M. Herlach and G. P. G6rler, Mater. Sci. Engng 98, 39 (1988). 6. K. Eckler, M. Kratz and I. Egry, Rev. Sci. Instrum. 64, 2639 (1993). 7. B. T. Bassler, W. H. Hofmeister, G. Carro and R. J. Bayuzick, Metall. Mater. Trans. 25A, 1301 (1994). 8. S. Walder and P. L. Ryder, J. appl. Phys. 73, 1965 (1993). 9. S. Walder and P. L. Ryder, J. appL Phys. 74, 6100 (1993). 10. B. Wei, G. Yang and Y. Zhou, Acta metall, mater. 39, 1249 (1991). l I. S. Kfishnan, G. P. Hansen, R. H. Hauge and J. L. Margrave, High Temp. Sci. 29, 17 (1990). 12. T. J. Piccone, Y. Wu, Y. Shiohara and M. C. Flemings, MetalL Trans. 18A, 925 (1987). 13. J. L. Murray, Metall. Trans. 15A, 261 (1984). 14. J. Lipton, W. Kurz and R. Trivedi, Acta metalL 35, 957 (1987). 15. D. Tumbull, MetalL Trans. 12A, 695 (1981). 16. R. Tfivedi, J. Lipton and W. Kurz, .4eta metall. 35, 965 (1987). 17. M. J. Aziz, J. appl. Phys. 53, 1158 0982).
WALDER and RYDER:
RAPID DENDRITIC GROWTH
18. M. J. Aziz and T. Kaplan, Acta metall. 36, 2335 (1988). 19. W. J. Boettinger, S. R. Coriell and R. Trivedi, in Rapid Solidification Processing: Principle and Technology I V (edited by R. Mehrabian and P. A. Parrish), p. 13. Claitor's, Baton Route, La (1988). 20. M. J. Aziz and W. J. Boettinger, Acta metall, mater. 42, 527 (1994). 21. K. Eckler, D. M. Herlach and M. J. Aziz, Acta metall. mater. 42, 975 (1994).
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22. J. A. Kittl, M. J. Aziz, D. P. Brunco and M. O. Thompson, Appl. Phys. Lett. 64, 2359 (1994). 23. M. Carrard, M. Gremaud, M. Zimmermann and W. Kurz, Acta metall, mater. 40, 983 (1992). 24. Shu-Zu Lu, J. D. Hunt, P. Gilgien and W. Kurz, Acta metall, mater. 42, 1653 (1994). 25. R. Trivedi, P. Magnin and W. Kurz, Acta metall. 35, 971 (1987). 26. D. M. Herlach, Mater. Sci. Engng A 179/180, 147 (1994).