Diffusion-Coefficient Measurements in Liquid Metallic Alloys J.-H. LEE, SHAN LIU, H. MIYAHARA, and R. TRIVEDI The value of the diffusion coefficient in the liquid (Dl) is generally obtained from the measurement of composition profiles ahead of a quenched planar interface. The experimental results show significant scatter. The main reason for this scatter will be shown to be due to the presence of fluid flow in the liquid. Directional-solidification studies in the Al-Cu system have been carried out to first establish the experimental conditions required for diffusive growth. The composition profiles are then measured to obtain the values of Dl for alloy compositions ranging from 4.0 to 24.0 wt pct Cu. The value of Dl 2.4 10 3 mm2/s was obtained along the liquidus line, and this result is significantly smaller than the values reported in the literature, which vary from 3.0 to 5.5 103 mm2/s. It is shown that the scatter in the reported values can be correlated with the diameter of the sample used and, thus, with the fluid flow present in their experiments. Detailed experimental procedures to obtain and verify diffusivegrowth conditions are outlined, and appropriate analyses of the data are discussed.
I. INTRODUCTION
THE solute-diffusion coefficient in a liquid metal is a key parameter for quantitative prediction of many kinetic processes. In directional solidification it is directly related to the length scales of solidification microstructures, which, in turn, control the properties of the solidified product. Several methods have been used to determine this technologically significant parameter, that include solute-enriched thin-layer diffusion,[1,2] droplet movement in a temperature-gradient field,[3] thermal transport,[4] and the solute boundary-layer measurements in a quenched directionally solidified alloy.[5–12] The last method has been extensively used to evaluate the diffusion coefficient in metallic alloys. One of the alloy systems in which several measurements have been made is the Al-Cu system,[2–4,7–12] in which the reported values of the diffusion coefficient vary from (3.0 to 5.5) 103 mm2/s. This uncertainty in the value of the diffusion coefficient has, indeed, prevented the precise validation of theoretical models of solidification microstructures. A small part of the variation could be attributed to the different compositions used in these studies (from a very dilute to a eutectic composition). A major reason for this scatter in the data, however, is due to the influence of fluid flow on the composition profile in the liquid. Verhoeven[13] discussed the effect of possible convection patterns in the melt on the diffusion-coefficient measurements and categorized them into threshold and thresholdless convection, which depends upon whether the driving force for convection (i.e., density gradient in the melt) is parallel or perpendicular to the gravity vector. In a vertically upward growth, the fluid flow will be controlled by the density gradient in the melt at the interface. If this density gradient is positive (i.e., the rejected solute is lighter), the fluidJ.-H. LEE, Assistant Professor, is with the Department of Metallurgy and Materials Science, Changwon University, Changwon, South Korea 641-773. SHAN LIU, Scientist, is with Ames Laboratory, United States Department of Energy, Ames, IA 50011. H. MIYAHARA, Associate Professor, is with the Materials Science and Engineering Department, Kyushu University, Fukuoka, Japan. R. TRIVEDI, Senior Scientist, Ames Laboratory, is Professor, Materials Science and Engineering, Iowa State University, Ames, IA 50011. Contact e-mail:
[email protected] Manuscript submitted June 9, 2003. METALLURGICAL AND MATERIALS TRANSACTIONS B
flow effect can be significant. In contrast, when the axial density gradient is negative (i.e., heavier solute is rejected), fluid flow can still occur if any horizontal thermal gradient is present. In Bridgman growth, the sample is heated radially by a furnace, and a difference in thermal conductivity between the ampoule and the sample is present, which will give rise to a horizontal temperature gradient[14–19] so that some fluid flow will always be present. A detailed numerical analysis has been carried out by Trivedi et al.[18] for hypoeutectic Al-Cu alloys, and it is shown that the intensity of convection due to the radial temperature gradient is governed by thermal Rayleigh number (RaT), defined as RaT
gbT GH d 4 na
[1]
where is the kinematic viscosity, is the thermal diffusivity of the melt, T is the thermal-expansion coefficient of the melt, GH is the horizontal temperature gradient, g is the gravitational acceleration, and d is the diameter of the sample. The value of RaT should be minimized such that the convective transport is negligible compared to the diffusive transport. Under terrestrial conditions, the only variable that will reduce the Rayleigh number for a given alloy system is the diameter of the sample. Thus, experiments to determine the diffusion coefficient have generally been carried out in capillary tubes. However, no systematic study has been made to determine the size of the capillary tube needed to obtain diffusive growth in a given system in which the diffusion coefficient measurements are carried out. Thus, it has not been possible to evaluate the accuracy of the reported diffusion coefficient values. In this article, we shall present experimental results in the hypoeutectic Al-Cu alloys to obtain the value of the diffusion coefficient in liquid that is not influenced by convection. Experiments were carried out in several alloy compositions that ranged from 4.0 to 24.0 wt pct Cu. The following studies were made to obtain an accurate value of the diffusion coefficient. (1) A systematic experimental study was carried out by directionally solidifying samples of different diameters, and the effect of fluid flow on the microstructure and segregation profiles was evaluated. It was found that a sample diameter of 0.8 mm is required in Al-Cu alloys at the pulling velocity of V 1.0 m/s in all compositions studied, in order to have negligible fluid flow
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effect. The same conclusion was reached earlier in Al-4.0 wt pct Cu alloys.[18,19,20] This critical diameter is for hypoeutectic Al-Cu alloys and is for the growth conditions and the thermal characteristics of the directional-solidification apparatus used in this study. (2) A very fast quench was used to ensure that the motion of the interface during quenching was negligible compared to the thickness of the solute boundary layer. (3) Since steady-state growth was not achieved in the Al-4.0 wt pct Cu alloy due to the length of the sample required, a rigorous timedependent model of the transient regime was used to obtain the value of the diffusion coefficient. The experimental results for 4.0, 22.0, and 24.0 wt pct Cu show that Dl 2.4 103 mm2/s, which is smaller than the values reported in the literature. We shall show that all the reported values were obtained for sample diameters that varied from 2.0 to 7.0 mm, in which significant convection effects were present. A systematic increase in the reported value of Dl was observed when all the available data were plotted vs the sample diameter used in these studies.
II. EXPERIMENTAL Directional solidification experiments were carried out in a Bridgman directional solidification system at low solidification rates in hypoeutectic alloys of Al with 4.0, 15.0, 20.0, 21.0, 21.5, 22.0, and 24.0 wt pct Cu. Hypoeutectic alloys were selected to obtain a long-range solute profile in which the rejected solute (copper) is heavier than the solvent. An alumina tube of 5.5 mm i.d. was used as a bulk sample ampoule. In order to examine the effect of convection, the thin-tube solidification technique, described by Trivedi et al.,[20] was used, in which one thin tube of diameter 0.8 mm was inserted in the center of the bulk ampoule in each experiment. The thin tube was aligned with the central axis of the large ampoule by using two graphite holders at both ends of the thin tube. A 100 -oriented single-crystal seed of the alloy composition was placed at the bottom of the large ampoule to obtain the same growth orientation of the interface in the thin sample and in the bulk region, so that the interface shape and its location inside and outside the thin tube could be quantitatively compared. The effect of convection in an Al-4.0 wt pct Cu alloy was examined earlier by Trivedi and co-workers,[18,19,20] who showed that diffusive growth is attained at V 1.0 m/s in our experimental setup only when the sample size was 0.8 mm. We have carried out similar experiments with different diameter samples for other alloy compositions (20.0 to 25.0 wt pct Cu) used in this study and found the diffusive growth to be present in samples of 0.8 mm diameter. We also carried out experiments in larger-diameter samples, in which a significant effect of convection was observed. In this study, the thermal gradients in the liquid at the interface were first measured from thermal profiles obtained by using a thermocouple in the sample for different values of translation velocity, alloy composition, and furnace temperature. The values of the thermal gradient at the interface location in the adiabatic zone were then obtained by using the regression analysis of the temperature data with a secondorder polynomial fit that gave the correlation coefficient 0.99. The ampoule with the thin tube was directionally solidified for about 4 to 6 cm (where the solid fraction ( fs) was about 910—VOLUME 35B, OCTOBER 2004
0.3 to 0.5) and quenched to preserve the growing solid/liquid interface. The sample was quenched by quickly moving the sample into the cold zone that contained liquid metal cooling, which gave rapid quenching at the interface location. Solidification microstructures were investigated by optical metallography and scanning electron microscopy (SEM). Composition profiles in the quenched liquid ahead of a planar interface, and in the solid along the interface, were measured by an electron-probe microanalyzer (EPMA). Since the quenched microstructure is composed of fine eutectic and dendrites, the composition measurements were averaged on a 100 m line scan (parallel to the quenched interface) with a beam size of about 2 m in diameter. The standards with Al-4.0, 23.0, and 32.7 wt pct Cu, which were first determined by wet chemical analysis, were used to improve the accuracy of the composition-profile measurements. III. EXPERIMENTAL RESULTS We shall first show that convection effects are present in bulk samples of diameter 1.0 mm and then present experimental results in thin samples (0.8 mm i.d.) to establish the presence of diffusive growth. Next, we shall present experimentally measured composition profiles ahead of planar singlephase and planar eutectic interfaces in thin samples to obtain the values of the diffusion coefficient. A. Attainment of Diffusive Growth Conditions Figure 1(a) shows the microstructure of an Al-21.0 wt pct Cu alloy, directionally solidified at G 10.0 K/mm and V 0.4 m/s, where a coupled growth is predicted by the diffusive model. The diameter of the sample was 5.5 mm and the solid fraction was 0.6. The microstructure in the central region of the sample is composed of leading cells/dendrites with intercellular/dendritic lamellar eutectic, while a coupled growth was present in the region closer to the periphery. Since the temperature gradient (G) and the velocity (V) in the growth direction were the same for every point on the transverse cross section, the change in the growth morphology should result from the variation in composition in the radial direction caused by fluid flow. The competitive growth theory of dendrites/cells and coupled eutectics indicates that the composition at the periphery lies within the coupled-growth regime, whereas that in the center is outside the coupled-growth regime.[7,21] The interface composition near the periphery is, thus, higher than that near the center, and this radial solute segregation can be directly related to the melt flow in the bulk liquid.[18,20] In this case, the morphological evolution of the solidification interface is governed by the local interface composition rather than the nominal alloy composition. When a capillary tube of 0.8 mm i.d. was inserted in the center of the bulk sample, the microstructure in the thin sample was significantly different from the surrounding bulk, as shown in Figure 1(b). The bulk region shows radially varying microstructures similar to Figure 1(a), whereas the interface morphology in the thin sample shows / coupled growth only. The growth interface in the thin sample was flat except at the ampoule wall, where a small curvature was present due to the surface tension (contact angle) effect. The microstructure and the volume fraction of eutectic were found to be uniform over the cross section of the thin sample, indicating METALLURGICAL AND MATERIALS TRANSACTIONS B
Fig. 1—(a) Microstructure transition in the radial direction from primary phase to eutectic in the presence of fluid flow in 5.5-mm-i.d. tube, V 0.4 m/s, Al-21.0 wt pct Cu alloy. (b) Microstructure differences in the thin sample of 0.8-mm diameter and in the outside bulk region in Al-22.0 wt pct Cu alloy, V 0.5 m/s. A eutectic planar interface is formed in the thin sample, whereas the outside bulk region shows a cellular microstructure in the middle and the eutectic at the edge. (c) Coupled growth in Al-24.0 wt pct Cu alloy solidified at V 0.5 m/s and G 10.0 K/mm. A lamellar eutectic is present in the thin sample, whereas a rod eutectic in the middle and a lamellar eutectic at the edges are formed in the outside bulk region.
that no radial segregation occurred and, thus, no significant melt flow was present. Thus, the solute transport was controlled by diffusion only. Similar results were obtained in other offeutectic alloys in the hypoeutectic region. Figure 1(c) shows a longitudinal section of Al-24.0 wt pct Cu alloy grown at V 0.5 m/s and G 10.0 K/mm. Coupled growth was present both inside the thin sample (0.8 mm i.d.) and in the outer bulk region. A higher-magnification observation of a transverse cross section revealed that the eutectic was lamellar inside the thin sample, and the eutectic structure was uniform across the entire cross section. In contrast, the eutectic morphology in the bulk region changed in the radial direction. Near the thin sample the eutectic was rodlike, near the periphery of the bulk the eutectic was mainly lamellar, and in the region in between, a mixed structure of rods and lamellae was present. A detailed study of the radial composition METALLURGICAL AND MATERIALS TRANSACTIONS B
variation in the thin sample and in the bulk region was carried out to quantitatively examine the effect of this composition variation on eutectic microstructures. The variation in microstructure was found to directly correlate with the radial-composition variation caused by the melt flow in the bulk region. B. Solute-Profile Measurements in off-Eutectic Alloys Solute profiles in the axial direction were measured in thin samples and in the bulk region (i.e., outside the thin sample). All composition profiles were measured at the center of the region in the thin sample to avoid any wall effect.[12] In the bulk region, the measurements were made in the radially middle region. Figure 2(a) shows composition profiles in the liquid in both the thin sample and the bulk region. In both cases, the interface composition was equal to the eutectic VOLUME 35B, OCTOBER 2004—911
thin sample with a planar eutectic interface, the measured composition in the solid at the interface was found to be equal to the alloy composition, indicating that the steady-state growth had been reached. Equation [2] can, thus, be used to describe the liquid composition profile, and the plot of In ((ClC0)/ (CEC0)) vs Vz should give a straight line with the slope of 1/Dl. Figure 2(b) shows such a plot for both Al-22.0 and 24.0 wt pct Cu off-eutectic alloys, and all the data were found to fall on a single straight line. The regressed slope was 428.6 s/mm2, which gave the diffusion coefficient Dl as 2.4 103mm2/s. In order to check the validity of the aforementioned value of the diffusion coefficient, the entire composition profile in the liquid in the thin sample was fitted with the decaying parameter (V/Dl) in the exponential term. A good match was obtained with the diffusion coefficient of Dl 2.4 103mm2/s, as shown in Figure 2(a). (a) C. Diffusion Coefficient Measurements in Al-4.0 Wt Pct Cu Alloy
(b) Fig. 2—(a) Solute profiles in the solid and in the quenched liquid in Al24.0 wt pct Cu alloy. The liquid profile coincides with the theoretical profile for a diffusion coefficient of 2.40 103 mm2/s. (b) A comparison of experimentally measured composition data in Al-22.0 wt pct Cu and Al24.0 at. pct Cu with Eq. [2]. The results for the two alloys overlap and give a slope that is an inverse of the diffusion coefficient.
composition given by the equilibrium-phase diagram. However, a significant difference in the composition in the solid and in the axial composition profiles between the thin sample and the bulk was observed. The solid composition at the interface in the bulk region was about 4.0 wt pct Cu lower than that in the thin sample, and the composition profile in the liquid in the bulk was below that in the thin sample. Both of these observations reflect the difference in the masstransfer mechanism in the thin sample and in the bulk. The profile in the thin sample was controlled by diffusion, while significant fluid flow was present in the bulk region. For the steady-state eutectic growth, the composition in the solid at the interface must be equal to the alloy composition (C0), and the solute profile in the liquid (Cl), for diffusive growth, is given by[7,21] Cl C0 (CE C0) exp (Vz /Dl )
[2]
where z is the distance from the eutectic interface into the quenched liquid and CE is the eutectic composition. For the 912—VOLUME 35B, OCTOBER 2004
The composition profiles ahead of a single-phase interface were also measured in Al-4.0 wt pct Cu alloys. The presence of a significant fluid flow effect in the single-phase solidification is illustrated in Figure 3. The interface is highly curved in bulk sample of 5.5 mm in diameter. The interface is smooth in the middle, but becomes unstable near the periphery and even shows the presence of a eutectic layer near the wall of the ampoule, as shown by an enlarged view in Figure 3(a). This variation in microstructure is due to the lateral flow of solute caused by fluid flow. In contrast, in a thin sample of 0.4 mm in diameter (Figure 3(b)), the interface is flat over the entire cross section, so that no lateral solute variation is present and the growth was diffusive. Another important consideration was to select the growth condition for a planar interface growth, since conditions for steady-state growth are often not feasible. For Al-4.0 wt pct Cu, the critical velocity for G 11.0 K/mm is 0.179 m/s. Such a low velocity is not only difficult to maintain constant over a long period of time, but it also requires the sample to be much larger than what can be accommodated in most experimental systems. For example, at the growth velocity of 0.179 m/s, the transient distance (Dl /Vk) predicted by the Tiller et al. model[22] is about 95.7 mm. However, this transient distance is based on the criterion that the composition at the interface has reached about 67 pct of the steady-state value. A detailed numerical model of the transient, described in this section, shows that about a 3 times larger distance is required to reach 90 pct of the steady-state composition. In addition, to avoid end effects, one needs additional 2Dl /V length of the sample, so the length of the sample should be about 400 mm. Since part of the sample is in the cold zone initially, and thermal conditions alter near the end of the sample, a significantly longer sample is required. Consequently, we carried out directional solidification at V 0.4 m/s but quenched the sample when the length of the solid reached about 4.0 cm or the solid fraction was about 0.3. In this case, the interface was still planar and did not have time to build a sufficient solute boundary layer to initiate instability. We then analyzed the composition profile ahead of the transient planar interface to obtain the value of the diffusion coefficient. METALLURGICAL AND MATERIALS TRANSACTIONS B
Fig. 4—Line-scan EPMA result of the composition profiles in the thin sample and in the bulk region of Al-4.0 wt pct Cu sample directionally solidified at V 0.4 m/s and G 11.0 K/mm
(a)
(b) Fig. 3—(a) A quenched S/L interface of Al-4.0 wt pct Cu growing at G 7.0 K/mm, V 0.4 m/s in 5.5-mm i.d. ampoule. The interface becomes highly curved at the center and shows instabilities at the edges. A thin layer of eutectic is observed at the ampoule wall, as shown on the side. (b) A flat interface is observed in a thin sample of 0.4-mm diameter indicating a negligible fluid flow effect.
Figure 4 shows the EPMA line-scan result along the axial direction for both the thin sample and the bulk region. The line scan in the bulk sample was taken at the center of the sample. Two basic differences between these two profiles are observed. (1) The liquid composition at the interface is higher for the thin sample than for the bulk sample; (2) the solid composition at the interface in the thin sample is higher. Note that steady-state growth conditions were not attained in the thin sample, which is quite evident from the value of the solid composition at the interface that is significantly below the alloy composition. Consequently, the interface velocity (Vi) will be different from the pulling velocity (V), METALLURGICAL AND MATERIALS TRANSACTIONS B
and a rigorous transient model of the planar interface growth is required to analyze the results of composition profiles. Accurate models of planar interface growth have been developed by numerical modeling[23] and by using the boundary integral method.[24] Warren and Langer[25] developed a model of initial transient in terms of a coupled pair of differential equations, and the results of this model were found to match accurately with the results of the numerical model.[24] In this model, the interface dynamics in the transient regime were shown to be controlled by the interaction between the two processes: the rate of rejection of solute from the accelerating interface and the rate of diffusion in the liquid to build up the boundary layer. They formulated the problem in terms of two time-dependent variables, i(t) and zi(t), that represent the boundary-layer thickness and the position of the interface in a frame of reference that is moving in the z direction with a constant velocity of V, whose origin is taken at the isotherm corresponding to the melting point of pure solvent. To obtain the values of these two variables as a function of time, the following pair of equations was derived by using the flux balance at the interface and the global mass balance: mC0 2 Dl dzi azi bV dt d (1 k)zi G
[3a]
mC0 4Dl
zi d
d a kzi b a b
t d (1 k)zi G (zi (m/G)C0) t [3b] where G and m are the thermal gradient in the liquid and the slope of the liquidus, respectively. A numerical solution of the previous set of equations was obtained by using the procedure outlined by Warren and Langer[25] for the alloy and growth conditions of the experimental profile shown in Figure 4. Since the value of the solute diffusion coefficient is needed in the calculation, we carried out an iterative procedure by first assuming a value of Dl, and then carried out calculations until the interface composition in the liquid reached the experimentally measured VOLUME 35B, OCTOBER 2004—913
value of 17.4 wt pct Cu to determine the predicted interface velocity V and the concentration gradient at the quenched interface. The diffusion-coefficient value was then calculated from the flux balance at the interface. If this value was different from the initially assumed value, the procedure was repeated with a different value of the diffusion coefficient until the assumed value coincided with the value calculated from the flux balance. These values were found to coincide when Dl 2.4 103 mm2/s. The transient model with this value of the diffusion coefficient was now used to obtain the composition profile in the liquid when the interface composition in the liquid was 17.4 wt pct Cu. The theoretical profile was found to match the experimental profile, as shown in Figure 5(a). This value of Dl is the same as the value obtained in higher copper content alloys. The results of the time evolution of composition in the solid and the interface velocity, for this value of the diffusion coefficient, are shown in Figure 5(b) for V 0.4 m/s and G 11.0 K/mm. The composition profile increases somewhat gradually, whereas the interface velocity is found to increase sharply during the initial period. The interface loca-
tion is near t 105 s, which is very close to the characteristic time (Dl /V 2k 1.071 105 s) predicted by the model of Tiller et al.,[22] although the composition in the liquid at the quenched interface is 17.3 wt pct Cu, which is significantly lower than the steady-state value of C0/k 24.0 wt pct. The interface velocity is 0.37 m/s, which is very close to the pulling velocity of 0.4 m/s. Our calculations show that one requires t 3Dl /V 2k for the composition in the solid to reach about 90 pct of the steady-state value. D. The Effect of Quenching on Solute Profiles To analyze the solute profile in the liquid ahead of the quenched interface, it is necessary to ensure that the motion of the interface during quenching was negligible. In order to examine the quenching effect, we carried out a series of experiments with different quenching rates and measured the displacement of the interface through composition measurements in the solid near the interface. As the interface accelerates during quenching, the composition in the solid will increase. By measuring the region of this composition increase we found that, for the quenching method used in our present study, the interface displacement was about 10 to 15 m. This distance is negligible compared to the diffusive-boundary layer in our experiments, which is Dl/V 5.9 mm for V 0.4 m/s. The composition profile in the liquid, away from the interface, will not be affected significantly since diffusion in this time interval will be negligible. Consequently, all composition data were taken from a distance of about 0.5 mm from the interface.
IV. DISCUSSION A. The Effect of Fluid Flow on the Diffusion Coefficient
(a)
We shall now analyze the composition profile in both the solid and the quenched liquid in bulk samples to determine how the value of the calculated diffusion coefficient is influenced by convection when the data are analyzed by using the diffusive-growth model. We shall determine the value of the diffusion coefficient when fluid flow is present and obtain the value of the effective diffusion coefficient (Dle) under the following assumptions. (1) The curvature of the interface is small so that normal concentration gradient does not deviate from the axial concentration gradient. (2) Use the flux balance at the interface, Vi (Cli Csi) Del
(b) Fig. 5—(a) A comparison of the experimental concentration profile with the results of the numerical model of Warren and Langer[25] for ƒ Dl = 2.4 103 mm2/s. (b) The time variation of the interface velocity and interface composition in the liquid in the transient regime. The dotted vertical line shows the location of the quenched interface. 914—VOLUME 35B, OCTOBER 2004
Cl `
z z0
[4]
in which the convective transfer is ignored, so that the diffusion coefficient is the effective diffusion coefficient in the liquid, and we have differentiated this with the symbol Del . (3) The interface velocity is assumed to be equal to the pulling velocity, since it is not significantly different from the external pulling velocity as found in our numerical calculations. The axial composition profiles in the bulk were measured at the center of the concentric region. For the bulk composMETALLURGICAL AND MATERIALS TRANSACTIONS B
Table I. A Summary of the Experimentally Determined Solute Diffusion Coefficient in the Melt in Al-Cu Alloys Sample (Diameter Length) in mm
Composition, Wt Pct Cu
Dl (mm2/s)
Reference
0.4 130 0.8 130 2.0 120
4.0 22.0, 24.0 2.0
2.42 103 2.4 103 3.0 103
2.0 120
27
3.26 103
3.2 65 5 ( 100) 8.7 131
32.7 0.2 to 1.8 0.27
3.57 103 4.76 103 5.5 103
present studies present studies Sharp and Hallawell[8] Jordan and Hunt[7] Bhat[4] Miyata et al.[26] Tensi and Mackrodt[27] Sato and Ohira[10]
5 3 225*
0.1
5.36 103
*Rectangular cross section.
ition profile in Figure 2(a), the liquid and the solid compositions at the interface are 33.2 and 19.4 wt pct Cu, respectively. The growth velocity, Vi 0.5 m/s, and the composition gradient at the interface, ∂Cl /∂z 0 z0 2.23 wt pct/mm, were obtained from curve fitting of the composition profile in the liquid. This gave the value of the diffusion coefficient as Dle 3.1 103 mm2/s. It is higher than the value derived from the pure diffusive growth process, but very close to the value obtained by Jordan and Hunt[7] and Sharp and Hellawell,[8] who used a 2.0-mm-diameter sample to determine the diffusion coefficient in which some convection effect would be present. Since the composition profile varies with the radial locations, the derived effective diffusion coefficient varied from (3.0 to 4.2) 103 mm2/s depending on the location where the axial profile was measured. Therefore, it is not possible to accurately determine the effective diffusion coefficient if melt flow exists, which causes radial solute segregation. We note, however, that when convection is present, the effective diffusion coefficient value is larger than the true diffusioncoefficient value, since the effective diffusion coefficient includes the convective transport present in the liquid. We have shown that convection increases as the sample diameter increases, so the effective diffusive coefficient should also increase with the increase in the sample diameter. We may, thus, examine different values of the diffusion coefficient in the liquid in the Al-Cu system presented in the literature,[4,7,8,10,26,27] in which different-diameter samples were used. All the measured values of the diffusion coefficient in hypoeutectic Al-Cu alloys, presented by different research groups using vertically upward solidification in the Bridgman apparatus, are summarized in Table I. Since different sizes of samples were used in these studies, we have compared the results with the diameter of the sample used in each study. Figure 6 shows that a systematic increase in the value of the diffusion coefficient is observed with an increase in the diameter of the sample. It is seen that the value of the diffusion coefficient becomes constant only when the sample diameter is 0.8 mm, for which diffusive growth was established in this study through microstructure and radial composition measurements. METALLURGICAL AND MATERIALS TRANSACTIONS B
Fig. 6—Variation in the diffusion coefficient values reported in the literature with the diameter of the sample used in these experiments. A systematic correlation is observed indicating that the data in the literature were influenced by the presence of fluid flow. A constant value is observed in thin samples used in the present study.
From the previous results, one may consider using an effective diffusion coefficient for experiments in which fluid flow is present. However, it is not quite accurate, since its value depends on the fluid flow intensity, which is controlled not only by the size of the ampoule, but also by the alloy system, processing conditions, and thermal characteristics of the apparatus. As we have seen in our experiments, the value of the effective diffusion coefficient also varies along the radial direction. A good correlation with the sample diameter, however, seems to indicate that the size of the sample may play a dominant role in this system, although an accurate quantitative comparison may not be obtained by using a single effective diffusion-coefficient value. B. Diffusion-Coefficient Results The values of the diffusion coefficients in 4.0, 22.0, and 24.0 wt pct Cu alloys were found to be the same, and a value of 2.4 103 mm2/s was obtained. This may not imply that the diffusion coefficient is independent of composition. The measured diffusion coefficient through the directional solidification process gives a value of the diffusion coefficient that is a weighted average value over the composition range in the profile and, for the present case, it can be approximated to correspond closely to the composition and temperature at the interface,[28] which are related by the liquidus line for the local equilibrium condition at the interface. Thus, the diffusion-coefficient values in 4.0 and 22.0 wt pct Cu correspond to the composition-temperature values of 17.4 wt pct Cu–883.0 K and 33.2 wt pct Cu–821.2 K, respectively. Thus, the increase in the diffusion coefficient value with composition could be balanced by the decrease with temperature. A similar observation was made by Pearson and Verhoeven[29]for the diffusion coefficient of carbon in austenite, which varies significantly with composition, but was found to be nearly constant for the composition-temperature conditions along the ferrite-austenite phase boundary. It would be interesting to examine whether the constancy of the diffusion coefficient along the phase boundary in these systems is a mere coincidence or is valid in general. VOLUME 35B, OCTOBER 2004—915
Table II. Experimental Conditions for Directional Solidification of Near-Eutectic Alloys and the Observed Interface Morphology
Composition (Wt Pct Cu) 15.0
20.0 21.0 21.5
22.0 25.0
V (m/s)
G (K/mm)
Solid Fraction, fs
Interface Microstructure in the Thin Sample (0.8-mm Diameter)
0.4 0.5 0.8 2.0 0.5 0.4 0.5 0.45 0.4 2.3 0.9 0.5 0.4 0.3 0.7 0.5 0.4 1 0.9 0.7 0.4 0.3
12.5 12.5 12.5 12.5 10.0 10.0 10.0 11.0 10.0 10.0 10.0 10.0 10.0 10.5 9.5 9.5 9.5 10.0 10.0 10.0 10.0 10.0
0.6 0.6 0.6 0.6 0.60 0.45 0.65 0.50 0.56 0.5 0.5 0.45 0.51 0.48 0.60 0.51 0.69 0.50 0.62 0.60 0.45 0.70
coupled growth cellular dendritic dendritic dendritic coupled growth dendritic coupled growth coupled growth dendritic dendritic dendritic coupled growth coupled growth dendritic coupled growth coupled growth dendritic coupled growth coupled growth coupled growth coupled growth
We also checked the results of the diffusion coefficient values through independent measurements of the condition for transition from a single phase to a coupled growth, which is given by[30,31,32] G/V ml (C0 CE)/Dl
[5]
A series of experiments was carried out in thin samples at different velocities in alloys of compositions 15.0, 20.0, 21.0, 21.5, 22.0, and 25.0 wt pct Cu to examine the transition conditions between the primary phase and eutectic. All the experimental conditions and observations are listed in Table II, and the results are plotted as a microstructure map (G/V vs alloy composition) in Figure 7. The transition conditions given by Eq. [5] are plotted for two different diffusion-coefficient values that correspond to the value obtained in the present study (i.e., Dl 2.4 103 mm2/s) and the value generally used in the literature (i.e., Dl 3.2 103 mm2/s). Note that a good agreement is observed when the diffusion coefficient value determined in this study is used. In the determination of the liquid diffusion coefficient from the concentration profile, we have ignored the Soret effect, i.e., solute transport by thermal gradient effects. For Al-Cu alloys, Bhat[4] has evaluated the Soret coefficient s 1.74 103 K1; the contribution from the Soret effect can be estimated to be[3,4,5] Dl s x G Dl Dl V (1 k)
[6]
where x is the solute composition in atom fraction, which is equal to 0.17 for a eutectic alloy, and G is the tempera916—VOLUME 35B, OCTOBER 2004
Fig. 7—Microstructure map for Al-(15.0 to 32.7) wt pct Cu showing the regimes of a single phase (triangles) and a coupled growth (circles). The theoretical transition lines for two values of the diffusion coefficient are shown and the experimental data show good fit with the value of Dl 2.4 103 mm2/s.
ture gradient in the liquid. The gradient varies for different alloys, but the value ahead of the solidification interface in our experiments was 10.5 to 11.0 K/mm. With V 0.5 m/s and k 0.16 at the eutectic temperature, we obtain ∆Dl /Dl 1.73 102, which shows that the Soret effect can be ignored for these low-thermal-gradient experiments in the Al-Cu system. C. Diffusive-Growth Condition To obtain a reliable value of the diffusion coefficient in the liquid through the solute-profile measurements in a quenched directionally solidified alloy, it is shown that one of the techniques that can be used for some alloy systems is the use of very-small-diameter samples in which fluid-flow effects are negligible. The size of the sample is a strong function of the alloy system and the thermal characteristics of the directional solidification apparatus. We have used an experimental method in which samples of different diameters were directionally solidified to find the critical size below which fluid-flow effects on the microstructure and on solute segregation were negligible. For vertically upward directional solidification of alloys in which the solute is heavier, fluid flow causes a radial variation in the solute profile at the interface. We, thus, suggest that before the axial concentration profile is used to determine the diffusion coefficient, a radial composition profile in the same sample should be measured to ensure that it is constant and diffusive growth is present in the experiment and that the interface is flat. For eutectic growth, the presence of a planar eutectic interface does not ensure diffusive growth, since the radial variation in composition can lead to the variation in volume fraction of the eutectic. Consequently, the radial composition in the solid at the interface, as well as the radial variation in the volume fraction, needs to be measured in the sample to establish diffusive growth before the axial composition profile in the same sample is used to evaluate the value of the diffusion coefficient. METALLURGICAL AND MATERIALS TRANSACTIONS B
If the rejected solute in a binary alloy is less dense than the solvent, or the density gradient in the melt at the interface is positive, the convective flow can be quite intense.[18,33,34] In this case, the interface will be generally planar except at the ampoule wall, and the composition variation in the radial direction will be negligible even though significant fluid flow is present in the melt.[18] The presence of diffusive growth in such systems can only be established by measuring the composition profile in the axial direction for different solidification lengths to ensure that no macrosegregation is present in the system. Experimental studies in Pb-Sb and Pb-Bi alloys[35] have shown that diffusive growth was not present in these systems even when the capillary size was 0.4 mm. It should be noted that when the sample size becomes very small, of the order of tens of microns, the curvature of the interface at the wall due to the contact-angle effect can lead to a significant lateral diffusion, which can give rise to a curved interface in the diffusive-growth regime. This suggests that accurate measurements of the diffusion coefficient for these systems may require low-gravity experiments to suppress fluid flow. V. CONCLUSIONS Diffusion-coefficient measurements have often been carried out by analyzing the concentration profile ahead of a planar interface in the quenched liquid in directionally solidified alloys. It is shown that significant error can be present if any fluid flow is present in the system. We have shown that in alloy systems in which the density gradient in the liquid decreases with the distance from the interface, diffusive growth could be obtained by using very-small-diameter samples. Through systematic study of the directional solidification of hypoeutectic Al-Cu alloys in different-diameter samples, we have established that a diffusive growth is present at low velocities when the ampoule sizes are 0.8 mm i.d. The axial solute profiles in the quenched liquid ahead of the planar single-phase and planar eutectic interface in thin samples were analyzed to obtain the value of the intrinsic diffusion coefficient in the Al-Cu alloys. All these results in alloys of Al-(4.0 to 24.0) wt pct Cu have been found to be consistent with each other and give the diffusion coefficient as 2.4 103 mm2/s. All the previously reported diffusion coefficient values in the Al-Cu system showed a significant scatter in the results. Since samples with an i.d. 2 mm were used in these experiments, melt convection should be present. We have shown that the scatter in the data is not random, but the measured values increased systematically as the diameter of the sample used in the experiments was increased. ACKNOWLEDGMENTS This work was supported, in part, by the Office of Physical and Biological Sciences, NASA, and funded through the Marshall Space Flight Center. This work was also supported, in part, by the Basic Energy Sciences, Division of Materials Science, Department of Energy, and carried out
METALLURGICAL AND MATERIALS TRANSACTIONS B
at Ames Laboratory, which is operated by Iowa State University under Contract No. W-7405-ENG-82.
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