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Coarsening of Ni3Ge in binary Ni–Ge alloys: microstructures and volume fraction dependence of kinetics
Acta Materialia 51 (2003) 4073–4082 www.actamat-journals.com
Coarsening of Ni3Ge in binary Ni–Ge alloys: microstructures and volume fraction dependence of kinetics D.M. Kim 1, A.J. Ardell ∗ Department of Materials Science and Engineering, University of California-Los Angeles, 6531-G Boelter Hall Los Angeles, CA 90095, USA Received 30 January 2003; received in revised form 17 April 2003; accepted 21 April 2003
Abstract Coarsening of Ni3Ge precipitates in binary Ni–Ge alloys containing 12.15, 13.01 and 14.03 at.% Ge, aged at 724 °C (equilibrium volume fractions, fe, equaling 0.022, 0.105 and 0.202, respectively) was investigated using transmission electron microscopy and magnetic analysis. The rate constants for the kinetics of particle growth and depletion of supersaturation depend anomalously on fe, i.e. they both decrease slightly as fe increases. Average values of the chemical diffusion coefficient and the Ni3Ge/matrix interfacial free energy, derived from analysis of the data, are in reasonable agreement with previously reported values. Alignment of the Ni3Ge precipitates parallel to cube directions is strong, but coalescence into plate shapes is never observed. Precipitates of the solid solution γ phase nucleated in large, coherent, concave-cuboidal Ni3Ge precipitates. This behavior is expected, considering the phase boundary between the two-phase and Ni3Ge regions in a recently published phase diagram. 2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Coarsening; Ni–Ge alloys; Anomalous kinetics
1. Introduction The original theory of the diffusion-controlled coarsening of precipitates, published nearly simultaneously by Lifshitz and Slyozov [1] and Wagner [2] (the LSW theory), is valid strictly speaking in the limit of an infinitely dilute dispersion. In real terms this means that the equilibrium volume fracCorresponding author. Tel.: +1-310-8257011; fax: +1310-2067353. E-mail address: [email protected] (A.J. Ardell). 1 Present address: Samsung Informations Systems America, Inc., 75 West Plumeria Drive, San Jose, CA 95134, USA. ∗
tion, fe, of the dispersed phase is equal to zero. Numerous theories have attempted to predict how a finite value of fe influences the kinetics of coarsening and the particle size distributions (PSD); discussions of the principal assumptions of these theories, as well as comparisons with extant data, can be found in several review articles [3–5]. Irrespective of the magnitude of fe, all the theories predict that the average radius of the particles, 具r典, increases with time, t, according to the wellknown equation 具r典3⬇k(fe)t,
(1)
where k(fe) is the rate constant for coarsening,
1359-6454/03/$30.00 2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. doi:10.1016/S1359-6454(03)00227-1
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which increases as fe increases. The PSD is also predicted to broaden as fe increases. The basis for the increase in k with increasing fe is the inverse relationship between fe and the inter-particle spacing, which decreases as fe increases, thereby reducing the diffusion distance, hence the flux of solute atoms from shrinking to growing particles in the PSD. This sound qualitative theoretical reasoning is belied by the coarsening behavior of ordered, coherent γ⬘-type (Ni3X, X = Al, Si or Ti) precipitates in binary Ni–Al [6], Ni–Si [7] and Ni–Ti [8] alloys, wherein k actually decreases anomalously with increasing f when f is small (⬇0.04–0.05), and remains about constant for larger values of fe. We believe that the elastic interaction energy associated with coherency strains is responsible for the anomalous coarsening behavior in these alloys, though no theory of this phenomenon has been forthcoming. Since elastic energy is proportional to the square of the lattice misfit parameter e, where e = (ap⫺am) / am, and ap and am are the lattice parameters of the precipitate and matrix phases, respectively, the magnitude of e governs the elastic interaction energy and potentially influences coarsening behavior. The elastic interaction energy is also a strong inverse function of the center-tocenter spacing between neighboring particles, and is therefore expected to become increasingly significant at larger volume fractions. On a superficial level the magnitude of the anomalous coarsening behavior correlates reasonably well with the magnitude of e. Of the three alloys for which data have been published, Ni–Si has the smallest lattice mismatch (e ⬇⫺0.0023 [9]), Ni–Al has an intermediate value of e (⬇ 0.0047 [9]) and Ni–Ti alloys the largest (e⬇ ⫺0.0085 [10]).2 The decrease of k with f is quite rapid in Ni–Al [6] and Ni–Ti [8] alloys, but relatively mild in Ni–Si alloys [7]. To determine, at least in part, if this trend would be upheld in an alloy with a lattice mismatch intermediate in magnitude between those of Ni–Al and Ni–Ti, we
2
These are room-temperature values of e. They of course change with increasing temperature, but the change is small and the relative magnitudes for the three alloys are about the same.
investigated the coarsening behavior of γ⬘ (Ni3Ge) precipitates as a function of fe in binary Ni–Ge alloys; the value of e in this alloy system is 0.0063 [9]. Another objective of the present work was to characterize the shape transitions experienced by the precipitates. In particular, we have observed individual precipitates of Ni3Al [11], Ni3Si [12] and Ni3Ti [8] with concave interfaces in dispersions with small values of fe (⬍0.04) and we were interested to find out whether Ni3Ge precipitates with concave interfaces might also be found under certain conditions.
2. Experimental Alloys of three different compositions, 12.15, 13.01 and 14.03 at.% Ge (14.61, 15.61 and 16.79 wt% Ge) were investigated; for simplicity they will be referred to as alloys 122, 130 and 140. The alloys were prepared by arc-melting and drop-casting at the Oak Ridge National Laboratory to overcome the considerable segregation of Ge which we observed in alloy buttons produced by conventional arc-melting in our laboratory. The cast alloys were supplied in the form of rods approximately 15 mm in diameter and 40 mm long. The asreceived rods were cut lengthwise using a lowspeed diamond wheel saw, solution-treated at 1150 °C for 45 min, quenched in refrigerated water, and subsequently cold-rolled to final sheet with thicknesses ranging from 200 to 250 µm. The coldrolled sheet specimens were then re-solutiontreated at 1150 °C for 45 min and water quenched prior to aging. The experiments were performed at an aging temperature of 724 °C for times up to 168 h. A protective atmosphere of Ti-gettered argon was used for all the heat treatments. Temperature control was maintained to within ±0.5 °C using a calibrated K-type thermocouple. Specimens for transmission electron microscopy (TEM) were prepared by punching disks 3 mm in diameter from the aged sheet, grinding them to 150 µm thickness using 600 grit SiC emery paper, then dimpling them electrochemically using a solution of 20% HNO3, 15% CH3COOH, 20% H3PO4 and 45% H2O at 90 V. Final electropolishing was done in a solution of 10% HClO4 and 90% C2H5OH at
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room temperature using a DC voltage of 10–20 V. The alloys were examined using a JEOL model 100CX TEMSCAN operating at an accelerating voltage of 100 kV. Superlattice reflections from the Ni3Ge phase of the type {100} were used to take dark-field TEM images from thin foils in [001] orientation. The sizes of the precipitates were measured from scanned images of the negatives using the commercial image-analysis software application Image-Pro PlusTM for Windows. The procedures for measuring the sizes from the TEM images, were identical to those in previous work [7]. Magnetic analysis was used to obtain the ferromagnetic Curie temperature, ⌰c, as described previously [13]. The solute concentration in the matrix, XGe, was evaluated from measurements of ⌰c. To obtain XGe from ⌰c we used a new calibration curve relating the variation of ⌰c (in K) to the weight fraction of Ge, WGe. The calibration curve is given by the equation ⌰c ⫽ 650.37⫺3238 WGe,
(2)
which differs slightly from the equation used by Wimmel and Ardell [13]. Their calibration curve tended to slightly underestimate the value of WGe, hence it was corrected to rectify the problem.
3. Results 3.1. Morphology and its evolution Fig. 1 shows the evolution of the precipitate microstructures in the three alloys for aging times of 1, 8 and 16 h. On visual inspection of these photographs it is evident that the effect of volume fraction on the kinetics of coarsening is not very strong. Nearly all the Ni3Ge precipitates seen in Fig. 1 are fully coherent, but at aging times longer than 65 h the precipitates started to lose coherency. Fig. 2 shows an example of semicoherent precipitates in alloy 140 aged for 168 h, where interfacial dislocations are readily visible. The shapes of the semicoherent γ⬘ precipitates are irregular shape and the γ/γ⬘ interfaces are non-crystallographic in orientation, as coherent interfaces tend to be. The Ni3Ge precipitates lost coherency after 65 h of aging only in alloy 122. The precipitates in the
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other two alloys remained fully coherent at this aging time, and started to lose coherency between 65 and 168 h. This is attributed to the slightly slower kinetics of coarsening in the more concentrated alloys, but it is conceivable that the larger elastic interactions in the more concentrated alloys promote the stability of the coherent precipitates. It is difficult to pinpoint the size at which loss of coherency commences, but it is between 80 and 100 nm and depends on fe. Concave-cuboidal precipitates (i.e. cuboidalshaped precipitates with concave interfaces), have been observed in Ni–Al [11], Ni–Si [12] and Ni– Ti [8] alloys. We did not observe concave-cuboidal Ni3Ge precipitates in alloy 122 (fe⬇0.02), or in any of the other alloys for that matter, because the large precipitates lost coherency. However, we did see such precipitates in an alloy we had prepared ourselves using conventional arc-melting, followed by annealing and cold-rolling rolling into sheet form. This alloy had a nominal composition of 14.03 at.% Ge, but our initial experiments indicated that its composition was quite heterogeneous. In particular, on subjecting the alloy to what would have been a solution-treatment anneal at 1150 °C, for an alloy of this composition, we observed quite large concave-cuboidal Ni3Ge particles in the microstructure. They are shown in Fig. 3(a). The matrix also contained fine-scale Ni3Ge precipitates, easily visible in Fig. 3(a), which most probably nucleated and grew to a limited extent during cooling. According to the older phase diagram of Dayer and Feschotte [14], there is no two-phase Ni(Ge) + Ni3Ge region at 1150 °C for an alloy containing 14% Ge. Instead, there is a eutectic at 1124 °C. There was no indication that the sheet specimen used in the experiment had melted, so our result suggests that the eutectic temperature is higher than 1150 °C and that the alloy was in the two-phase Ni(Ge) + Ni3Ge region of the phase diagram. At 1150 °C the Ni3Ge precipitates would be expected to coarsen quite rapidly. Coherency was not lost because of the difference in composition, hence lattice constants, between the solid solution and Ni3Ge phases are much smaller at higher temperatures; a crude estimate, based on extrapolated values of the equilibrium compositions at 1150 °C [14] and the data on lattice constants vs. compo-
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Fig. 1. Dark-field transmission electron micrographs of the Ni3Ge precipitate microstructures in binary Ni–Ge alloys aged at 724 °C. All the thin foils are in [001] orientation. The concentrations of Ge (in at.%) are indicated on the left and pertain to each row of images. The aging times in h are shown below each column of images and the approximate values of the equilibrium volume fractions are shown on the right.
Fig. 2. A bright field TEM image of typical semicoherent precipitates in alloy 140 aged for 168 h.
sition reported by Kamara et al. [9], places the value of e at 0.0034. A specimen of this same alloy was aged for 2 h at 714 °C. We found that the small Ni3Ge precipitates in the matrix, Fig. 3(a), grew and that some of the large Ni3Ge precipitates contained small particles which we believe are precipitates of the disordered Ni–Ge solid solution; an example is shown in Fig. 3(b). The observation of precipitates of the solid solution inside the large Ni3Ge precipitates is consistent with the recently reported slope of the phase boundary between the two-phase and Ni3Ge regions of the Ni–Ge phase diagram [15]. Since the solubility of Ni in Ni3Ge increases with increasing temperature the Ni3Ge phase can become supersaturated at lower temperatures. This behavior is similar to that in the Ni–Al alloy system, wherein precipitates of the disordered Ni(Al) solid solution can nucleate and grow from supersaturated Ni3Al [16,17]. Despite their large size, the large Ni3Ge precipitates in Fig. 3(b) did not lose coherency.
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3.2. Kinetics of coarsening and the effect of volume fraction The data on the three alloys, plotted in the form 具r典3 vs. t for consistency with Eq. (1), are shown in Fig. 4. Since the precipitates in all three alloys lost coherency at the longer aging times, only the data on coarsening of the coherent precipitates are shown. It is apparent that the slope of the curve, i.e. the magnitude of k in Eq. (1), for alloy 122 is slightly larger than the slopes for the other two alloys. The values of k for the three alloys are tabulated in Table 1 and plotted vs. fe in Fig. 5. The dependence of k on fe in Ni–Ge alloys is anomalous, which is similar to the behavior observed in Ni–Al [6], Ni–Si [7] and Ni–Ti [8] alloys. The PSDs are shown in Fig. 6. In general, the PSDs are broader than that predicted by the LSW theory. The PSDs in Fig. 6 are generally consistent with those reported by Wimmel and Ardell [13]. Despite the scatter in the data, which was generally due to the relatively small numbers of images examined, it would appear that scaling behavior is observed, which is consistent with observations of PSDs reported previously [3,4]. The standard deviations of the PSDs were calculated and tended to
Fig. 3. Dark-field TEM images of Ni3Ge precipitates observed in a highly inhomogeneous Ni–Ge alloy. The microstructure in (a) was from a specimen aged for 30 min at 1150 °C and quenched. The microstructure in (b) was obtained after re-aging a similarly treated specimen of the same alloy for 2 h at 714 °C.
Fig. 4. Plots of the cube of the average particle radius 具r典 vs. aging time t for the three alloys examined in this investigation.
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Table 1 Summary of data on the rate constant, k, for the kinetics of precipitate growth, the rate constant, , for the kintics of solute depletion and the equilibrium solubility, XGe,eq, during coarsening of Ni3Ge precipitates at 724 °C in the three alloys investigated at.% Ge
fe
k (nm3/s)
(105 s⫺1)
XGe,eq (at. frac.)
12.153 13.012 14.029
0.022 0.105 0.202
2.447 ± 0.204 2.157 ± 0.103 2.230 ± 0.098
8.605 ± 6.146 1.060 ± 0.562 0.789 ± 0.573
0.1197 ± 0.0001 0.1195 ± 0.0001 0.1190 ± 0.0002
difficult to decide on how many data points should be included in the determination of ⫺1/3. Since there is no reliable criterion for eliminating data points in this instance, all of them were included in the analysis, leading to the values of and XGe,eq, which are also shown in Table 1. The data on are shown in Fig. 8, where it is seen that decreases with increasing fe somewhat more rapidly than does k (Fig. 5), and its dependence on fe is clearly anomalous, similar to what is found in Ni–Si alloys [7]. 4. Discussion
Fig. 5. The variation of the rate constant for growth of the average precipitate, k, as a function of the equilibrium volume fraction of Ni3Ge, fe. The curve is drawn as a guide to the eye.
increase slightly with aging time, but the scatter was such that there would not appear to be any significance to this trend. There were no systematic variations of the PSDs with volume fraction. The kinetics of solute depletion are predicted to vary with aging time according to the equation [1,2] XGe⫺XGe,eq⬇(t)⫺1/3,
(3)
where XGe,eq is the equilibrium coherent solubility of Ge in Ni and is a rate constant. According to Eq. (3) plots of XGe vs. t⫺1/3 should be linear, with a slope of ⫺1/3 and an intercept of XGe,eq. The data obtained from the magnetic measurements of ⌰c, with the help of of Eq. (2), are shown in Fig. 7. It is evident that the scatter is considerable and it is
Despite the much larger room-temperature lattice mismatch in this system (e = 0.0063) compared to that in Ni–Al alloys, (e = 0.0047) [9], the influence of elastic energy on the kinetics of coarsening and the evolution of morphology and spatial correlations in Ni–Ge alloys is evidently significantly smaller than in Ni–Al alloys. The decrease in both k and (Figs. 5 and 7) shows that the dependence of these parameters on fe is indeed anomalous, but k and both depend much more weakly on fe in Ni–Ge alloys than in Ni–Al alloys [6]. The spatial correlations among Ni3Ge precipitates are also weaker than in Ni–Al alloys. In fact, the coarsening behavior of Ni3Ge precipitates is unusual in that there is no tendency for Ni3Ge precipitates to coalesce into plates. In this sense Ni3Ge precipitates behave more like Ni3Si precipitates, which have a much smaller lattice mismatch (e = ⫺0.0027 [9]. Indeed, in our experience Ni3Si precipitates never coalesce into plates no matter how large they become. The anomalous dependencies of k and on fe are also weak in Ni–Si alloys, as they are in Ni–Ge alloys. Many of these issues have been discussed in a recent paper [18].
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Fig. 6. Particle size distributions for all the Ni3Ge microstructures examined plotted in scaled form; g(u) is the probability density function and u is the scaled particle size defined by u = r / 具r典. The aging times are shown at the top and the alloy contents on the right-hand side of the array. The upper and lower numbers inset in each histogram represent the number of particles measured and the average radius, 具r典, in nm respectively. The LSW distribution is inset in each figure as a dashed curve.
Fig. 7. Plots of the solute concentration in the matrix, XGe, vs. aging time to the ⫺1/3 power, t⫺1/3, for the three alloys examined in this investigation.
Fig. 8. The variation of the rate constant for solute depletion, , as a function of the equilibrium volume fraction of Ni3Ge, fe. The curve is drawn as a guide to the eye.
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Estimates of the interfacial free energy, s, and the chemical diffusion coefficient in the Ni-rich ˜ , can be obtained from the expersolid solution, D imentally measured values of k and using the equations [19] k⫽
˜ lγ 4D 9(Xγ⬘e⫺Xγe)
(4)
Table 2 ˜ from the data on coarsening at 724 °C in Values of s and D the three alloys investigated at.% Ge
fe
s (mJ/m2)
˜ (10⫺17 m2/s) D
12.153 13.012 14.029
0.022 0.105 0.202
10.0 ± 2.4 19.3 ± 3.4 21.7 ± 5.3
4.77 ± 1.17 2.19 ± 0.39 2.03 ± 0.50
and k ⫽ 3, lγ
(5)
where Xγe and Xγ⬘e are the equilibrium concentrations of the matrix and γ⬘ phases, respectively, and lγ is the so-called capillary length, which is given by the equation lγ ⫽
2Vmγ⬘s , G⬙mγ(Xγ⬘e⫺Xγe)
(6)
where Vmγ⬘ is the partial atomic volume of Ge in the γ⬘ phase and G⬙mγ is the second derivative of the free energy of mixing of the γ phase, evaluated at the composition Xγe. The interfacial free energy is readily obtained from the data on k and by substituting Eq. (6) into Eq. (5) and solving for s, ˜ is obtained from the equation while D ˜ ⫽ D
9(Xγ⬘e⫺Xγe) 2 1/3 (k ) . 4
(7)
The rate constant k is not normally expressed in ˜ , and terms of the chemical diffusion coefficient, D indeed was not expressed this way by Calderon et al. [19] or in the original LSW theories [1,2]. However, Ma et al. [20] have argued that it is more appropriate to describe diffusion in terms of the chemical diffusion coefficient because it is necessary to transport both solute and solvent atoms during diffusion-controlled growth, under conditions in which the concentration of vacancies is everywhere in equilibrium. Taking Vmγ⬘ = 6.85 × 10⫺6 m3 / mol, Xγ⬘e = 0.242 (estimated by extrapolating the phase boundary reported by Ikeda et al. [15] to 724 °C) and the values of Xγe = XGe,eq in Table 1, the values ˜ calculated from the data on k and of s and D are shown in Table 2. The estimates of s were obtained assuming that the Ni–Ge solid solution is
ideal, in which case G⬙mγ = RT / Xγe(1⫺Xγe). The standard deviations reported in Table 2 reflect the errors in k, and Xγe, and were calculated using the procedures described by Mandel [21]. The average value of s (weighted by the standard deviations in Table 2 [21]) is 14.1 ± 3.9 mJ / m2, which is about 25% smaller than the similarly-weighted value of 19.5 ± 5.3 mJ / m2 obtained from re-analysis of the data of Wimmel and Ardell [13], using Eq. (2) to evaluate the kinetics of solute depletion. There appears to be a systematic increase of s and ˜ as fe increases. There is no physical decrease of D reason underlying this apparent behavior, however, and we believe that the trends are manifestations of the relative paucity of data and the relatively large scatter, particularly in the data on XGe vs. t⫺1/3 (Fig. 6). ˜ A comparison of the average value of D obtained from the data on coarsening (weighted by the standard deviations in Table 2), i.e. 2.30 ± 0.77 × 17 m2 / s, with data from conventional diffusion experiments [22,23] is shown in Fig. 9; the diffusion coefficient re-evaluated from the data of Wimmel and Ardell [13] is also shown. The values ˜ from data on coarsening are generally higher of D than those corresponding to an infinitely dilute solution of Ge in Ni [22,23] (which are just the tracer diffusion coefficients of Ge in Ni at 0% Ge). However, the chemical diffusion coefficient in Ni–Ge solid solutions increases with increasing Ge con˜ centration [23], and it is evident in Fig. 9 that D obtained from the data on coarsening is in quite good agreement with the data on chemical diffusion in the solid solution containing 12% Ge. If the two data points from the coarsening experiments were included in the fitting, the two lines in Fig. 9 would be parallel and the deviations from
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Fig. 9. Arrhenius plots of the chemical diffusion coefficients, ˜ , obtained from analyses of the data on coarsening as well as D data from conventional diffusion experiments. The datum from this investigation and that from Wimmel and Ardell [13] were not included in the fit to the data of Komai et al. [23] on the alloy containing 12% Ge.
that line would be comparable to the deviations of the data on diffusion in the infinitely dilute alloy.
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˜ obtained from analysis of the data on coarsenof D ing is in very good agreement with extrapolated values of the chemical diffusion coefficients in a solid solution containing 12% Ge. The value of s is somewhat smaller than the value of 19.5 ± 5.3 mJ / m2, obtained from re-analysis of the data of Wimmel and Ardell. Coherent concave-cuboidal Ni3Ge particles were observed in an alloy of inhomogenous composition which was heat-treated at a very high temperature. The concave-cuboidal particles remained fully coherent, most likely due to the considerably smaller lattice mismatch expected at very high temperatures. Particles of the γ phase (the Ni–Ge solid solution) were seen inside large Ni3Ge precipitates in the compositionally inhomogeneous Ni–Ge alloy after it was aged at a lower temperature. This observation is consistent with a recently published phase diagram, which shows that the solubility of Ni in Ni3Ge increases with increasing temperature.
Acknowledgements The authors are grateful to the National Science Foundation for financial support for this research under grant #DMR 9900714.
5. Summary References Ni3Ge precipitates in supersaturated Ni–Ge alloys aged at 724 °C grow by diffusion-controlled coarsening. The cube of the average particle size increases linearly with t and the supersaturation decreases linearly with t⫺1/3 (though the data exhibit considerable scatter), as predicted by the LSW theory. The rate constants k and decrease slightly with increasing precipitate volume fraction, and thus behave anomalously. The PSD are broader than the theoretical distribution of the LSW theory, but there is no correlation with aging time or volume fraction. Larger cuboidal-shaped particles of Ni3Ge lose coherency when they grow to have “diameters” somewhere between 80 and 100 nm. ˜ and s derived from the data on The parameters D coarsening have the values 2.30 ± 0.77 × 17 m2 / s, and 14.1 ± 3.9 mJ / m2, respectively. The value
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
Lifshitz IM, Slyozov VV. J Phys Chem Solids 1961;19:35. Wagner C. Z Elektrochem 1961;65:581. Voorhees PW. J Stat Phys 1985;38:231. Ardell AJ. In: Lorimer GW, editor. In phase transformations’87. London: Institute of Metals; 1988. p. 485. Voorhees PW. Ann Rev Mater Sci 1992;22:197. Maheshwari A, Ardell AJ. Acta Metall 1992;40:2661. Cho J-H, Ardell AJ. Acta Metall 1998;46:5907. Kim DM, Ardell AJ. Scr Mater 2000;43:381. Kamara AB, Ardell AJ, Wagner CNJ. Metal Mater Trans A 1996;27A:2888. Sass SL, Mura T, Cohen JB. Philos Mag 1967;16:679. Maheshwari A, Ardell AJ. Scr Metall Mater 1992;26:347. Meshkinpour M, Ardell AJ. Mater Sci Eng 1994;A185:153. Wimmel J, Ardell AJ. Mater Sci Eng 1994;A183:169. Dayer A, Feschotte P. J Less-Common Met 1980;72:51. Ikeda T, Nose Y, Korata T, Numakura H, Koiwa M. J Phase Equilib 1999;20:626.
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[16] Cornwell LR, Purdy GR. Metall Trans 1974;5:780. [17] Ma Y, Ardell AJ. Z Metallkde, in press. [18] Ardell AJ, Kim DM. In: Turchi EA, Gonis A, editors. Phase transformations and evolution in materials. Warrendale, PA: TMS; 2000. p. 309. [19] Calderon HA, Voorhees PW, Murray JL, Kostorz G. Acta Metall Mater 1994;42:991. [20] Ma Y, Joshi J, Ardell AJ. Proceedings of the UEF Third
International Alloy Conference. Estoril, Portugal, in prepration. [21] Mandel J. The statistical analysis of experimental data. New York: Dover, 1964. [22] Mantl S, Rothman SJ, Nowicki LJ, Lerner JL. J Phys F 1983;13:1441. [23] Komai N, Watanabe M, Horita Z. Acta Metall Mater 1995;43:2967.
Coarsening of Ni3Ge in binary Ni–Ge alloys: microstructures and volume fraction dependence of kinetics D.M. Kim 1, A.J. Ardell ∗ Department of Materials Science and Engineering, University of California-Los Angeles, 6531-G Boelter Hall Los Angeles, CA 90095, USA Received 30 January 2003; received in revised form 17 April 2003; accepted 21 April 2003
Abstract Coarsening of Ni3Ge precipitates in binary Ni–Ge alloys containing 12.15, 13.01 and 14.03 at.% Ge, aged at 724 °C (equilibrium volume fractions, fe, equaling 0.022, 0.105 and 0.202, respectively) was investigated using transmission electron microscopy and magnetic analysis. The rate constants for the kinetics of particle growth and depletion of supersaturation depend anomalously on fe, i.e. they both decrease slightly as fe increases. Average values of the chemical diffusion coefficient and the Ni3Ge/matrix interfacial free energy, derived from analysis of the data, are in reasonable agreement with previously reported values. Alignment of the Ni3Ge precipitates parallel to cube directions is strong, but coalescence into plate shapes is never observed. Precipitates of the solid solution γ phase nucleated in large, coherent, concave-cuboidal Ni3Ge precipitates. This behavior is expected, considering the phase boundary between the two-phase and Ni3Ge regions in a recently published phase diagram. 2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Coarsening; Ni–Ge alloys; Anomalous kinetics
1. Introduction The original theory of the diffusion-controlled coarsening of precipitates, published nearly simultaneously by Lifshitz and Slyozov [1] and Wagner [2] (the LSW theory), is valid strictly speaking in the limit of an infinitely dilute dispersion. In real terms this means that the equilibrium volume fracCorresponding author. Tel.: +1-310-8257011; fax: +1310-2067353. E-mail address: [email protected] (A.J. Ardell). 1 Present address: Samsung Informations Systems America, Inc., 75 West Plumeria Drive, San Jose, CA 95134, USA. ∗
tion, fe, of the dispersed phase is equal to zero. Numerous theories have attempted to predict how a finite value of fe influences the kinetics of coarsening and the particle size distributions (PSD); discussions of the principal assumptions of these theories, as well as comparisons with extant data, can be found in several review articles [3–5]. Irrespective of the magnitude of fe, all the theories predict that the average radius of the particles, 具r典, increases with time, t, according to the wellknown equation 具r典3⬇k(fe)t,
(1)
where k(fe) is the rate constant for coarsening,
1359-6454/03/$30.00 2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. doi:10.1016/S1359-6454(03)00227-1
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which increases as fe increases. The PSD is also predicted to broaden as fe increases. The basis for the increase in k with increasing fe is the inverse relationship between fe and the inter-particle spacing, which decreases as fe increases, thereby reducing the diffusion distance, hence the flux of solute atoms from shrinking to growing particles in the PSD. This sound qualitative theoretical reasoning is belied by the coarsening behavior of ordered, coherent γ⬘-type (Ni3X, X = Al, Si or Ti) precipitates in binary Ni–Al [6], Ni–Si [7] and Ni–Ti [8] alloys, wherein k actually decreases anomalously with increasing f when f is small (⬇0.04–0.05), and remains about constant for larger values of fe. We believe that the elastic interaction energy associated with coherency strains is responsible for the anomalous coarsening behavior in these alloys, though no theory of this phenomenon has been forthcoming. Since elastic energy is proportional to the square of the lattice misfit parameter e, where e = (ap⫺am) / am, and ap and am are the lattice parameters of the precipitate and matrix phases, respectively, the magnitude of e governs the elastic interaction energy and potentially influences coarsening behavior. The elastic interaction energy is also a strong inverse function of the center-tocenter spacing between neighboring particles, and is therefore expected to become increasingly significant at larger volume fractions. On a superficial level the magnitude of the anomalous coarsening behavior correlates reasonably well with the magnitude of e. Of the three alloys for which data have been published, Ni–Si has the smallest lattice mismatch (e ⬇⫺0.0023 [9]), Ni–Al has an intermediate value of e (⬇ 0.0047 [9]) and Ni–Ti alloys the largest (e⬇ ⫺0.0085 [10]).2 The decrease of k with f is quite rapid in Ni–Al [6] and Ni–Ti [8] alloys, but relatively mild in Ni–Si alloys [7]. To determine, at least in part, if this trend would be upheld in an alloy with a lattice mismatch intermediate in magnitude between those of Ni–Al and Ni–Ti, we
2
These are room-temperature values of e. They of course change with increasing temperature, but the change is small and the relative magnitudes for the three alloys are about the same.
investigated the coarsening behavior of γ⬘ (Ni3Ge) precipitates as a function of fe in binary Ni–Ge alloys; the value of e in this alloy system is 0.0063 [9]. Another objective of the present work was to characterize the shape transitions experienced by the precipitates. In particular, we have observed individual precipitates of Ni3Al [11], Ni3Si [12] and Ni3Ti [8] with concave interfaces in dispersions with small values of fe (⬍0.04) and we were interested to find out whether Ni3Ge precipitates with concave interfaces might also be found under certain conditions.
2. Experimental Alloys of three different compositions, 12.15, 13.01 and 14.03 at.% Ge (14.61, 15.61 and 16.79 wt% Ge) were investigated; for simplicity they will be referred to as alloys 122, 130 and 140. The alloys were prepared by arc-melting and drop-casting at the Oak Ridge National Laboratory to overcome the considerable segregation of Ge which we observed in alloy buttons produced by conventional arc-melting in our laboratory. The cast alloys were supplied in the form of rods approximately 15 mm in diameter and 40 mm long. The asreceived rods were cut lengthwise using a lowspeed diamond wheel saw, solution-treated at 1150 °C for 45 min, quenched in refrigerated water, and subsequently cold-rolled to final sheet with thicknesses ranging from 200 to 250 µm. The coldrolled sheet specimens were then re-solutiontreated at 1150 °C for 45 min and water quenched prior to aging. The experiments were performed at an aging temperature of 724 °C for times up to 168 h. A protective atmosphere of Ti-gettered argon was used for all the heat treatments. Temperature control was maintained to within ±0.5 °C using a calibrated K-type thermocouple. Specimens for transmission electron microscopy (TEM) were prepared by punching disks 3 mm in diameter from the aged sheet, grinding them to 150 µm thickness using 600 grit SiC emery paper, then dimpling them electrochemically using a solution of 20% HNO3, 15% CH3COOH, 20% H3PO4 and 45% H2O at 90 V. Final electropolishing was done in a solution of 10% HClO4 and 90% C2H5OH at
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room temperature using a DC voltage of 10–20 V. The alloys were examined using a JEOL model 100CX TEMSCAN operating at an accelerating voltage of 100 kV. Superlattice reflections from the Ni3Ge phase of the type {100} were used to take dark-field TEM images from thin foils in [001] orientation. The sizes of the precipitates were measured from scanned images of the negatives using the commercial image-analysis software application Image-Pro PlusTM for Windows. The procedures for measuring the sizes from the TEM images, were identical to those in previous work [7]. Magnetic analysis was used to obtain the ferromagnetic Curie temperature, ⌰c, as described previously [13]. The solute concentration in the matrix, XGe, was evaluated from measurements of ⌰c. To obtain XGe from ⌰c we used a new calibration curve relating the variation of ⌰c (in K) to the weight fraction of Ge, WGe. The calibration curve is given by the equation ⌰c ⫽ 650.37⫺3238 WGe,
(2)
which differs slightly from the equation used by Wimmel and Ardell [13]. Their calibration curve tended to slightly underestimate the value of WGe, hence it was corrected to rectify the problem.
3. Results 3.1. Morphology and its evolution Fig. 1 shows the evolution of the precipitate microstructures in the three alloys for aging times of 1, 8 and 16 h. On visual inspection of these photographs it is evident that the effect of volume fraction on the kinetics of coarsening is not very strong. Nearly all the Ni3Ge precipitates seen in Fig. 1 are fully coherent, but at aging times longer than 65 h the precipitates started to lose coherency. Fig. 2 shows an example of semicoherent precipitates in alloy 140 aged for 168 h, where interfacial dislocations are readily visible. The shapes of the semicoherent γ⬘ precipitates are irregular shape and the γ/γ⬘ interfaces are non-crystallographic in orientation, as coherent interfaces tend to be. The Ni3Ge precipitates lost coherency after 65 h of aging only in alloy 122. The precipitates in the
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other two alloys remained fully coherent at this aging time, and started to lose coherency between 65 and 168 h. This is attributed to the slightly slower kinetics of coarsening in the more concentrated alloys, but it is conceivable that the larger elastic interactions in the more concentrated alloys promote the stability of the coherent precipitates. It is difficult to pinpoint the size at which loss of coherency commences, but it is between 80 and 100 nm and depends on fe. Concave-cuboidal precipitates (i.e. cuboidalshaped precipitates with concave interfaces), have been observed in Ni–Al [11], Ni–Si [12] and Ni– Ti [8] alloys. We did not observe concave-cuboidal Ni3Ge precipitates in alloy 122 (fe⬇0.02), or in any of the other alloys for that matter, because the large precipitates lost coherency. However, we did see such precipitates in an alloy we had prepared ourselves using conventional arc-melting, followed by annealing and cold-rolling rolling into sheet form. This alloy had a nominal composition of 14.03 at.% Ge, but our initial experiments indicated that its composition was quite heterogeneous. In particular, on subjecting the alloy to what would have been a solution-treatment anneal at 1150 °C, for an alloy of this composition, we observed quite large concave-cuboidal Ni3Ge particles in the microstructure. They are shown in Fig. 3(a). The matrix also contained fine-scale Ni3Ge precipitates, easily visible in Fig. 3(a), which most probably nucleated and grew to a limited extent during cooling. According to the older phase diagram of Dayer and Feschotte [14], there is no two-phase Ni(Ge) + Ni3Ge region at 1150 °C for an alloy containing 14% Ge. Instead, there is a eutectic at 1124 °C. There was no indication that the sheet specimen used in the experiment had melted, so our result suggests that the eutectic temperature is higher than 1150 °C and that the alloy was in the two-phase Ni(Ge) + Ni3Ge region of the phase diagram. At 1150 °C the Ni3Ge precipitates would be expected to coarsen quite rapidly. Coherency was not lost because of the difference in composition, hence lattice constants, between the solid solution and Ni3Ge phases are much smaller at higher temperatures; a crude estimate, based on extrapolated values of the equilibrium compositions at 1150 °C [14] and the data on lattice constants vs. compo-
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Fig. 1. Dark-field transmission electron micrographs of the Ni3Ge precipitate microstructures in binary Ni–Ge alloys aged at 724 °C. All the thin foils are in [001] orientation. The concentrations of Ge (in at.%) are indicated on the left and pertain to each row of images. The aging times in h are shown below each column of images and the approximate values of the equilibrium volume fractions are shown on the right.
Fig. 2. A bright field TEM image of typical semicoherent precipitates in alloy 140 aged for 168 h.
sition reported by Kamara et al. [9], places the value of e at 0.0034. A specimen of this same alloy was aged for 2 h at 714 °C. We found that the small Ni3Ge precipitates in the matrix, Fig. 3(a), grew and that some of the large Ni3Ge precipitates contained small particles which we believe are precipitates of the disordered Ni–Ge solid solution; an example is shown in Fig. 3(b). The observation of precipitates of the solid solution inside the large Ni3Ge precipitates is consistent with the recently reported slope of the phase boundary between the two-phase and Ni3Ge regions of the Ni–Ge phase diagram [15]. Since the solubility of Ni in Ni3Ge increases with increasing temperature the Ni3Ge phase can become supersaturated at lower temperatures. This behavior is similar to that in the Ni–Al alloy system, wherein precipitates of the disordered Ni(Al) solid solution can nucleate and grow from supersaturated Ni3Al [16,17]. Despite their large size, the large Ni3Ge precipitates in Fig. 3(b) did not lose coherency.
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3.2. Kinetics of coarsening and the effect of volume fraction The data on the three alloys, plotted in the form 具r典3 vs. t for consistency with Eq. (1), are shown in Fig. 4. Since the precipitates in all three alloys lost coherency at the longer aging times, only the data on coarsening of the coherent precipitates are shown. It is apparent that the slope of the curve, i.e. the magnitude of k in Eq. (1), for alloy 122 is slightly larger than the slopes for the other two alloys. The values of k for the three alloys are tabulated in Table 1 and plotted vs. fe in Fig. 5. The dependence of k on fe in Ni–Ge alloys is anomalous, which is similar to the behavior observed in Ni–Al [6], Ni–Si [7] and Ni–Ti [8] alloys. The PSDs are shown in Fig. 6. In general, the PSDs are broader than that predicted by the LSW theory. The PSDs in Fig. 6 are generally consistent with those reported by Wimmel and Ardell [13]. Despite the scatter in the data, which was generally due to the relatively small numbers of images examined, it would appear that scaling behavior is observed, which is consistent with observations of PSDs reported previously [3,4]. The standard deviations of the PSDs were calculated and tended to
Fig. 3. Dark-field TEM images of Ni3Ge precipitates observed in a highly inhomogeneous Ni–Ge alloy. The microstructure in (a) was from a specimen aged for 30 min at 1150 °C and quenched. The microstructure in (b) was obtained after re-aging a similarly treated specimen of the same alloy for 2 h at 714 °C.
Fig. 4. Plots of the cube of the average particle radius 具r典 vs. aging time t for the three alloys examined in this investigation.
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Table 1 Summary of data on the rate constant, k, for the kinetics of precipitate growth, the rate constant, , for the kintics of solute depletion and the equilibrium solubility, XGe,eq, during coarsening of Ni3Ge precipitates at 724 °C in the three alloys investigated at.% Ge
fe
k (nm3/s)
(105 s⫺1)
XGe,eq (at. frac.)
12.153 13.012 14.029
0.022 0.105 0.202
2.447 ± 0.204 2.157 ± 0.103 2.230 ± 0.098
8.605 ± 6.146 1.060 ± 0.562 0.789 ± 0.573
0.1197 ± 0.0001 0.1195 ± 0.0001 0.1190 ± 0.0002
difficult to decide on how many data points should be included in the determination of ⫺1/3. Since there is no reliable criterion for eliminating data points in this instance, all of them were included in the analysis, leading to the values of and XGe,eq, which are also shown in Table 1. The data on are shown in Fig. 8, where it is seen that decreases with increasing fe somewhat more rapidly than does k (Fig. 5), and its dependence on fe is clearly anomalous, similar to what is found in Ni–Si alloys [7]. 4. Discussion
Fig. 5. The variation of the rate constant for growth of the average precipitate, k, as a function of the equilibrium volume fraction of Ni3Ge, fe. The curve is drawn as a guide to the eye.
increase slightly with aging time, but the scatter was such that there would not appear to be any significance to this trend. There were no systematic variations of the PSDs with volume fraction. The kinetics of solute depletion are predicted to vary with aging time according to the equation [1,2] XGe⫺XGe,eq⬇(t)⫺1/3,
(3)
where XGe,eq is the equilibrium coherent solubility of Ge in Ni and is a rate constant. According to Eq. (3) plots of XGe vs. t⫺1/3 should be linear, with a slope of ⫺1/3 and an intercept of XGe,eq. The data obtained from the magnetic measurements of ⌰c, with the help of of Eq. (2), are shown in Fig. 7. It is evident that the scatter is considerable and it is
Despite the much larger room-temperature lattice mismatch in this system (e = 0.0063) compared to that in Ni–Al alloys, (e = 0.0047) [9], the influence of elastic energy on the kinetics of coarsening and the evolution of morphology and spatial correlations in Ni–Ge alloys is evidently significantly smaller than in Ni–Al alloys. The decrease in both k and (Figs. 5 and 7) shows that the dependence of these parameters on fe is indeed anomalous, but k and both depend much more weakly on fe in Ni–Ge alloys than in Ni–Al alloys [6]. The spatial correlations among Ni3Ge precipitates are also weaker than in Ni–Al alloys. In fact, the coarsening behavior of Ni3Ge precipitates is unusual in that there is no tendency for Ni3Ge precipitates to coalesce into plates. In this sense Ni3Ge precipitates behave more like Ni3Si precipitates, which have a much smaller lattice mismatch (e = ⫺0.0027 [9]. Indeed, in our experience Ni3Si precipitates never coalesce into plates no matter how large they become. The anomalous dependencies of k and on fe are also weak in Ni–Si alloys, as they are in Ni–Ge alloys. Many of these issues have been discussed in a recent paper [18].
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Fig. 6. Particle size distributions for all the Ni3Ge microstructures examined plotted in scaled form; g(u) is the probability density function and u is the scaled particle size defined by u = r / 具r典. The aging times are shown at the top and the alloy contents on the right-hand side of the array. The upper and lower numbers inset in each histogram represent the number of particles measured and the average radius, 具r典, in nm respectively. The LSW distribution is inset in each figure as a dashed curve.
Fig. 7. Plots of the solute concentration in the matrix, XGe, vs. aging time to the ⫺1/3 power, t⫺1/3, for the three alloys examined in this investigation.
Fig. 8. The variation of the rate constant for solute depletion, , as a function of the equilibrium volume fraction of Ni3Ge, fe. The curve is drawn as a guide to the eye.
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Estimates of the interfacial free energy, s, and the chemical diffusion coefficient in the Ni-rich ˜ , can be obtained from the expersolid solution, D imentally measured values of k and using the equations [19] k⫽
˜ lγ 4D 9(Xγ⬘e⫺Xγe)
(4)
Table 2 ˜ from the data on coarsening at 724 °C in Values of s and D the three alloys investigated at.% Ge
fe
s (mJ/m2)
˜ (10⫺17 m2/s) D
12.153 13.012 14.029
0.022 0.105 0.202
10.0 ± 2.4 19.3 ± 3.4 21.7 ± 5.3
4.77 ± 1.17 2.19 ± 0.39 2.03 ± 0.50
and k ⫽ 3, lγ
(5)
where Xγe and Xγ⬘e are the equilibrium concentrations of the matrix and γ⬘ phases, respectively, and lγ is the so-called capillary length, which is given by the equation lγ ⫽
2Vmγ⬘s , G⬙mγ(Xγ⬘e⫺Xγe)
(6)
where Vmγ⬘ is the partial atomic volume of Ge in the γ⬘ phase and G⬙mγ is the second derivative of the free energy of mixing of the γ phase, evaluated at the composition Xγe. The interfacial free energy is readily obtained from the data on k and by substituting Eq. (6) into Eq. (5) and solving for s, ˜ is obtained from the equation while D ˜ ⫽ D
9(Xγ⬘e⫺Xγe) 2 1/3 (k ) . 4
(7)
The rate constant k is not normally expressed in ˜ , and terms of the chemical diffusion coefficient, D indeed was not expressed this way by Calderon et al. [19] or in the original LSW theories [1,2]. However, Ma et al. [20] have argued that it is more appropriate to describe diffusion in terms of the chemical diffusion coefficient because it is necessary to transport both solute and solvent atoms during diffusion-controlled growth, under conditions in which the concentration of vacancies is everywhere in equilibrium. Taking Vmγ⬘ = 6.85 × 10⫺6 m3 / mol, Xγ⬘e = 0.242 (estimated by extrapolating the phase boundary reported by Ikeda et al. [15] to 724 °C) and the values of Xγe = XGe,eq in Table 1, the values ˜ calculated from the data on k and of s and D are shown in Table 2. The estimates of s were obtained assuming that the Ni–Ge solid solution is
ideal, in which case G⬙mγ = RT / Xγe(1⫺Xγe). The standard deviations reported in Table 2 reflect the errors in k, and Xγe, and were calculated using the procedures described by Mandel [21]. The average value of s (weighted by the standard deviations in Table 2 [21]) is 14.1 ± 3.9 mJ / m2, which is about 25% smaller than the similarly-weighted value of 19.5 ± 5.3 mJ / m2 obtained from re-analysis of the data of Wimmel and Ardell [13], using Eq. (2) to evaluate the kinetics of solute depletion. There appears to be a systematic increase of s and ˜ as fe increases. There is no physical decrease of D reason underlying this apparent behavior, however, and we believe that the trends are manifestations of the relative paucity of data and the relatively large scatter, particularly in the data on XGe vs. t⫺1/3 (Fig. 6). ˜ A comparison of the average value of D obtained from the data on coarsening (weighted by the standard deviations in Table 2), i.e. 2.30 ± 0.77 × 17 m2 / s, with data from conventional diffusion experiments [22,23] is shown in Fig. 9; the diffusion coefficient re-evaluated from the data of Wimmel and Ardell [13] is also shown. The values ˜ from data on coarsening are generally higher of D than those corresponding to an infinitely dilute solution of Ge in Ni [22,23] (which are just the tracer diffusion coefficients of Ge in Ni at 0% Ge). However, the chemical diffusion coefficient in Ni–Ge solid solutions increases with increasing Ge con˜ centration [23], and it is evident in Fig. 9 that D obtained from the data on coarsening is in quite good agreement with the data on chemical diffusion in the solid solution containing 12% Ge. If the two data points from the coarsening experiments were included in the fitting, the two lines in Fig. 9 would be parallel and the deviations from
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Fig. 9. Arrhenius plots of the chemical diffusion coefficients, ˜ , obtained from analyses of the data on coarsening as well as D data from conventional diffusion experiments. The datum from this investigation and that from Wimmel and Ardell [13] were not included in the fit to the data of Komai et al. [23] on the alloy containing 12% Ge.
that line would be comparable to the deviations of the data on diffusion in the infinitely dilute alloy.
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˜ obtained from analysis of the data on coarsenof D ing is in very good agreement with extrapolated values of the chemical diffusion coefficients in a solid solution containing 12% Ge. The value of s is somewhat smaller than the value of 19.5 ± 5.3 mJ / m2, obtained from re-analysis of the data of Wimmel and Ardell. Coherent concave-cuboidal Ni3Ge particles were observed in an alloy of inhomogenous composition which was heat-treated at a very high temperature. The concave-cuboidal particles remained fully coherent, most likely due to the considerably smaller lattice mismatch expected at very high temperatures. Particles of the γ phase (the Ni–Ge solid solution) were seen inside large Ni3Ge precipitates in the compositionally inhomogeneous Ni–Ge alloy after it was aged at a lower temperature. This observation is consistent with a recently published phase diagram, which shows that the solubility of Ni in Ni3Ge increases with increasing temperature.
Acknowledgements The authors are grateful to the National Science Foundation for financial support for this research under grant #DMR 9900714.
5. Summary References Ni3Ge precipitates in supersaturated Ni–Ge alloys aged at 724 °C grow by diffusion-controlled coarsening. The cube of the average particle size increases linearly with t and the supersaturation decreases linearly with t⫺1/3 (though the data exhibit considerable scatter), as predicted by the LSW theory. The rate constants k and decrease slightly with increasing precipitate volume fraction, and thus behave anomalously. The PSD are broader than the theoretical distribution of the LSW theory, but there is no correlation with aging time or volume fraction. Larger cuboidal-shaped particles of Ni3Ge lose coherency when they grow to have “diameters” somewhere between 80 and 100 nm. ˜ and s derived from the data on The parameters D coarsening have the values 2.30 ± 0.77 × 17 m2 / s, and 14.1 ± 3.9 mJ / m2, respectively. The value
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