Trans-interface-diffusion-controlled coarsening of γ′ precipitates in ternary Ni–Al–Cr alloys

Available online at www.sciencedirect.com

ScienceDirect Acta Materialia 61 (2013) 7828–7840 www.elsevier.com/locate/actamat

Trans-interface-diffusion-controlled coarsening of c0 precipitates in ternary Ni–Al–Cr alloys Alan J. Ardell Department of Materials Science & Engineering, Henry Samueli School of Engineering and Applied Science, University of California, Los Angeles, Los Angeles, CA 90095-1595, USA Received 28 April 2013; received in revised form 8 September 2013; accepted 13 September 2013 Available online 7 October 2013

Abstract Published data on the coarsening behavior of c0 precipitates in three ternary Ni–Al–Cr alloys are examined in light of the theory of trans-interface-diffusion-controlled (TIDC) coarsening, in which the kinetics is controlled by diffusion through the coherent precipitate– matrix interface. The experimental data are independent of the equilibrium c0 volume fraction, as expected for TIDC coarsening. Kinetics of the type hrin / t for the growth of precipitates of average radius hri, and X1Al and X1Cr / t–1/n for the variations of the far-field matrix solute concentrations, X1Al,Cr, with aging time, t, are characteristic of TIDC coarsening. The temporal exponent n  2.4 was obtained from the fitting of published particle size distributions. Based on correlation coefficients, the dependencies of hrin on t and X1Al,Cr on t1/n were comparable for n = 2.4 and n = 3 (the temporal exponent for matrix-diffusion-controlled coarsening). The dependencies of volume fraction, f, and number density, Nv, on t are also compared with theoretical predictions. Using a thermodynamic model of the Ni–Al–Cr phase diagram, the interfacial free energy, r, was estimated from analysis of the data; r varies from 14.5 to 19 mJ m–2 in the three alloys. Interfacial diffusion coefficients, also obtained from analysis of the data, are greater than those in the c0 phase but smaller than those in the c phase, which is consistent with the demands of the TIDC theory. Comparisons with the results from previously published work are noted and all discrepancies are discussed. Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Coarsening, Ternary alloys; Coherent (Ni,Cr)3Al precipitates; Interface diffusion-control

1. Introduction The kinetics of decomposition of supersaturated ternary Ni–Al–Cr alloys, including the stages of nucleation, growth and coarsening, has been the focus of research over the past two decades [1–8]. Ni–Al–Cr alloys are microstructurally similar to binary Ni–Al alloys in that the matrix phase is a disordered solid solution (the c phase) in equilibrium with the ordered c0 phase, based on Ni3Al with the Cu3Au (L12) crystal structure. An important difference between the binary and ternary alloys is that for many compositions there is virtually no lattice mismatch between the matrix and the coherent precipitates, due primarily to the E-mail addresses: [email protected], [email protected]

partitioning of smaller Cr atoms to the c0 precipitates [9]. For this reason the precipitates remain nearly perfectly spherical at all sizes [10], which is not the case in binary Ni–Al alloys. Some recent research has addressed the role of elemental alloying of W, Ru, etc. [11–14] on the decomposition kinetics and microstructural evolution in this class of alloys, because partitioning of the elements to one phase or the other can affect the lattice mismatch, the diffusion coefficients and the interfacial free energies. In this paper the influence of such alloy additions will not be considered; the principal topic of concern here is the kinetics of coarsening in the ternary alloys. Research using atom probe tomography (APT) [3,12,15], supported by high-resolution transmission electron microscopy [16], has clearly shown that the interface

1359-6454/$36.00 Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.actamat.2013.09.021

A.J. Ardell / Acta Materialia 61 (2013) 7828–7840

between the c and c0 phases in various alloys is not sharp, and computer simulations of the atomic structures of planar coherent interfaces in binary Ni–Al alloys [17,18] support the experimental observations. Instead of an atomically sharp interface, there exists a transition region, 2 nm in width, over which the compositions of the various components vary more or less smoothly between the c and c0 phases. Moreover, Srinivasan et al. [16] have presented evidence that the degree of long-range order decreases from its maximum value in c0 to zero in c over roughly half this distance. This is significant because diffusion through the interface is expected to be slower in the ordered region, consistent with the experimental findings that diffusion in ordered Ni3Al is significantly slower than diffusion in the disordered c phase solid solution [19–23]. Since the interfaces cannot move unless atoms diffuse through them, it is reasonable to assert that diffusion through the interface, rather than matrix diffusion to the interface, can control its movement. The theory of transinterface-diffusion-controlled (TIDC) coarsening for binary alloys [18] is a by-product of this reasoning. Just as the theories of matrix-diffusion-controlled (MDC) coarsening, of which the Lifshitz–Slyozov–Wagner (LSW) theory is the iconic standard [24,25], must be modified for the presence of a third alloying element in ternary alloys [26–29], so must the TIDC theory be modified. This has been accomplished in a recent publication [30]. As is the case for binary alloys, the TIDC theory for coarsening in ternary alloys (TIDC-T) is consistent with the absence of an effect of volume fraction on the kinetics of coarsening. Other predictions of the TIDC-T theory have recognizable counterparts in the theory for coarsening in binary alloys, especially the form of the equations, but there are three important differences: (i) two equations are required to describe the variation with aging time of the two solute atom species; (ii) two diffusion coefficients are needed to describe the diffusion of the two solute atom species through the interface; (iii) the diffusion coefficients play an important role in the Gibbs–Thomson equation during coarsening, which as pointed out by Kuehmann and Voorhees [29] is completely different from the situation for binary alloys. As in the TIDC theory for binary alloys, the temporal exponent, n, in the TIDC-T theory, which governs the kinetics of all the time-dependent variables, also dictates the shapes of the particle size distributions (PSDs). In this paper relatively recent data on the coarsening of c0 precipitates in ternary Ni–Al–Cr alloys [4–8] are re-evaluated in light of the TIDC-T theory [30]. The work is justified not only by the convincing observations on the widths of interfaces in a variety of Ni-base c/c0 alloys, but also by the fact that coarsening of c0 precipitates in ternary Ni–Cr–Al alloys is independent of volume fraction, as will become evident on examination of the published data.

7829

2. Analyses of data on coarsening of c0 precipitates in ternary Ni–Al–Cr alloys 2.1. Particle size distributions The kinetics in TIDC theory is governed by a temporal exponent, n, which can vary over the range 2 6 n 6 3 [30]. To proceed with an analysis of data on the kinetics of coarsening, it is imperative to know the value of n, which can be obtained from the fitting of experimentally measured data on PSDs, as well as data on kinetics. Unfortunately, PSDs were not reported by Booth-Morrison et al. [6,7] or Sudbrack et al. [4,5], so the results of Chellman and Ardell [31] and Jayanth and Nash [32] (a total of six PSDs) are used for this purpose. Their PSDs were fitted to the data using a Mathematica subroutine, as in previous work [18,33,34]. The fitting involves the functions: hðzÞ ¼ 3f ðzÞ expfpðzÞg;

ð1Þ

where p(z) is given by Z z pðzÞ ¼ f ðxÞdx;

ð2Þ

0

and f(z) is the function: f ðzÞ ¼

½zðn  1Þðn1Þ nn ðz  1Þ  zn ðn  1Þ

ðn1Þ

ð3Þ

The variable z is a dimensionless particle radius defined by z = r/r, where r is a critical radius at aging time t that is neither growing nor shrinking; particles larger than r grow, while particles smaller than r shrink and ultimately disappear from the population of precipitates. Since the average radius hri of the PSD, not r, is the experimentally measured quantity, it is important to express the dimensionless particle radius as u = r/hri and write the theoretical PSD as a new function, g(u), which is expressed by the equation: gðuÞ ¼ hzihðzÞ:

ð4Þ

The theoretical Eqs. (1)–(4) were fitted to the data of Chellman and Ardell [31] and Jayanth and Nash [32] in a least-squares sense using trial values of n until the value of n producing the smallest deviation was found. The value of n for the six PSDs is 2.397 ± 0.083; n = 2.4 was chosen for use in all subsequent analyses of the data. The experimentally measured PSDs are shown in Fig. 1 along with the PSD of the TIDC theories calculated using n = 2.4, and the LSW theory (n = 3). It is evident that the computed PSD is displaced slightly with respect to the data, and has a practical cut-off at u  1.6, whereas the data extend to values of u up to 2. Overall, however, the agreement is excellent. There is also no question that the theoretical PSD of the LSW theory is a much poorer fit to the data.

7830

A.J. Ardell / Acta Materialia 61 (2013) 7828–7840

Fig. 1. The particle size distributions measured by Chellman and Ardell [31] (filled circles) and Jayanth and Nash [32] (open symbols). The data of Chellman and Ardell represent PSDs in a Ni–3.4Al–16.9Cr alloy (wt.%) aged for 144 h at 750 °C. The data of Jayanth and Nash represent PSDs in a 9.7Al–17.6Cr (wt.%) alloy aged at 800 °C for 25 h (squares), 49 h (circles), 75 h (diamonds), 100 h (triangles) and 125 h (inverted triangles). The heavy solid curve is the PSD of the TIDC theory for n = 2.4 and the heavy dashed curve is the PSD of the LSW theory (n = 3).

2.2. The kinetics of growth of the average particle The growth rate of the average precipitate, hri, in the TIDC theory is described by the equation [30]: n

n

hri  hro i ¼ k T t;

ð5Þ

where hroi is the average radius at the onset of coarsening, kT is a rate constant related to the thermophysical parameters of the alloy system (see Eq. (18) in Ref. [30]). Chellman and Ardell [31] and Jayanth and Nash [32] also measured the kinetics of particle growth during coarsening. Their results were fitted to plots of hrin vs. t for n ranging from 2 to 3 to seek best fits to the data, determined from the magnitudes of the correlation coefficients, R2, for each plot. The results are shown in Fig. 2. The plot of R2 inset in Fig. 2a passes through a maximum at n between 2.4 and 2.5 (Fig. 2a), whereas R2 increases monotonically with decreasing n even at n = 2 (Fig. 2b). The purpose of plots inset in Fig. 2 is to show that the iconic temporal exponent n = 3 does not come close to providing the best fit to these data, not necessarily to support the choice of n = 2.4. Nevertheless, relying primarily on the best fit to the data on the PSDs in Fig. 1, it is evident that n = 2.4 is a sensible value for the temporal exponent. On examining the results in Figs. 1 and 2 it is evident that the temporal exponent n = 2.4 is a much more compelling choice than n = 3. With this in mind, the data on the kinetics of coarsening of c0 precipitates in the three ternary Ni–Al–Cr alloys studied by Sudbrack et al. [4,5] and Booth-Morrison et al. [6,7] are re-evaluated using n = 2.4. The compositions of the three alloys and their designations are presented in Table 1.

Fig. 2. Plots of average particle radius, hri, raised to the 2.4 power vs. aging time, t, for the data of Chellman and Ardell [31] (a) and Jayanth and Nash [32] (b). The alloys and experimental conditions are the same as those specified in the caption to Fig. 1. The curves inset in each figure are plots of the correlation coefficient, R2, vs. the temporal exponent, n, in plots of hrin vs. t.

Table 1 Compositions in at. frac. of the ternary Ni–Al–Cr alloys investigated and equilibrium volume fractions, fe, therein. The nominal compositions of Alloys 1, 2 and 3, respectively, in at.% are 5.2 Al, 14.2 Cr [4], 6.5 Al, 9.5 Cr [7] and 7.5 Al, 8.5 Cr [6]. Alloy

1 2 3

Compositions

fe

Al

Cr

Reported

Measureda

0.0524 0.0624 0.0756

0.1424 0.0964 0.0856

0.156 ± 0.004 0.096 ± 0.029 0.164 ± 0.006

0.1383 ± 0.0073 0.1040 ± 0.0034 0.1461 ± 0.0068

a The values of fe in this column were obtained from the intercepts of the plots in Fig. 6 (see Section 2.4).

Much of the work done by these researchers was concerned with the early and intermediate stages of aging. Since the work on coarsening is of sole concern here, only the measurements for times of 4 h and greater are analyzed. The

A.J. Ardell / Acta Materialia 61 (2013) 7828–7840

results are not much affected data at if the shortest of these aging times, 4 h, is eliminated. A brief discussion of the proper handling of the published data is in order. The data reported by Sudbrack et al. [4,5] and Booth-Morrison et al. [6,7] are accompanied by standard deviations of the tabulated measured quantities. In most (but not all) cases these represent statistical errors in the measurements, as in any normal experiment. When the variables of the experiments are related by a linear equation and one of the two variables has an uncertainty assigned to it (i.e. an error bar), the “best” values of the slopes and intercepts are obtained using a weighted regression analysis for the fitting of straight lines. The correct formulas for the slope and intercept under these conditions, as well as their variances, are given by York [35].

7831

Cantrell [36] presents explicit equations for the slope and intercept for weighted regression analysis that are perhaps a bit more transparent. If both variables are “exact”, in the sense that they are measured or reported without errors (aging times, for example), the best values of the slopes and intercepts and their variances are obtained using unweighted, or standard, regression analysis. The data on the kinetics of particle growth in the three alloys are considered first. The results are presented in Fig. 3. The column of figures on the left shows the plots of hri3 vs. t, while the column on the right shows the same data plotted as hri2.4 vs. t. The values of R2 are also shown in each plot. For two of the three alloys R2 is larger for n = 2.4, while for the 7.5% Al–8.5% Cr alloy R2 is larger for n = 3. The main point here is simply to demonstrate

Fig. 3. Plots of the average particle radius, hri, raised to the nth power vs. aging time, t. The open circles show the data for n = 3 (left column of figures) and the filled circles show the same data for n = 2.4 (right column of figures). The plots for Alloys 1 (5.2% Al), 2 (6.5% Al) and 3 (7.5% Al) are taken from the tabulated data of Sudbrack et al. [4] and Booth-Morrison et al. [7] and [6], respectively. The numbers in the bottom right corner of each figure represent the correlation coefficient, R2.

7832

A.J. Ardell / Acta Materialia 61 (2013) 7828–7840 Table 2 Values of the rate constants k and kT obtained from analyses of the data in Fig. 3. The data published by Sudbrack et al. [4] on Alloy 1 and Booth-Morrison et al. [6] on Alloy 3 are shown in column 2. The values of k and kT from this work are shown in columns 3 and 4. Alloy

k (m3 s–1)

k (m3 s–1) (n = 3)

kT (m2.4 s–1) (n = 2.4)

1 2 3

8.80 ± 3.30  1032 [4] Not reported 1.84 ± 0.43  1031 [6]

1.244 ± 0.074  1031 2.697 ± 0.214  1031 1.526 ± 0.037  1031

8.956 ± 0.027  1027 1.286 ± 0.068  1026 1.039 ± 0.068  1026

that both temporal exponents yield comparable results, not to suggest that the values of R2 indicate a clear preference for TIDC kinetics over MDC kinetics. The rate constants k and kT, obtained from the slopes of the fitted straight lines in Fig. 3, are presented in Table 2, along with the values of k reported by Sudbrack et al. [4] for Alloy 1 and Booth-Morrison et al. [6] for Alloy 3; there is no value for Alloy 2 reported by Booth-Morrison et al. [7]. The rate constant k, without the subscript T, is simply the rate constant expected from analysis of the data assuming MDC coarsening (n = 3). According to the ternary alloy section for 600 °C reported by Booth-Morrison et al. [7], Alloys 2 and 3 lie on essentially the same tie line, so the larger equilibrium c0 volume fraction in the latter alloy Table 1 should be accompanied by a measurably larger rate constant, k, according to any theory of the effect of volume fraction on coarsening kinetics [37–39]. The data in Table 2 completely belie such predictions, k being smaller for Alloy 3 than for Alloy 2. It is emphasized that the straight lines in Fig. 3 and the values of k and kT in the last two columns of Table 2 were obtained using standard regression analysis even though the tabulated data on hri of Sudbrack et al. [4] and Booth-Morrison et al. [6,7] ostensibly include the “errors” in their measurements. However, these errors are actually standard deviations of the PSDs and do not originate from statistical errors in the measured particle sizes. This is why there are no error bars shown in Fig. 3; the same considerations apply to the data in Fig. 2. It is likely that the discrepancies between the values of k (n = 3) for Alloys 1 and 3 in this work and those previously published [4,6] (cf. k in columns 2 and 3 in Table 2) arise from the alternative ways of treating and fitting the data. 2.3. The variation of solute concentrations with aging time Coarsening of c0 precipitates in ternary Ni–Al–Cr alloys is accompanied by the depletion of Al and the enrichment of Cr in the matrix c phase. Unlike the case for binary alloys, there are two equations that govern the kinetics of solute redistribution in the matrix phase during coarsening. These are [30]: X 1Al  X eAl ¼ jTAl t1=n ; and

X 1Cr  X eCr ¼ jTCr t1=n ;

ð6bÞ

where X1Al,Cr represents the far-field average concentration of Al,Cr in the matrix, XeAl,Cr are the concentrations of these species in the c phase at thermodynamic equilibrium, and jTAl,Cr are rate constants that depend on the thermophysical parameters of the alloy (see Eqs. (20a) and (20b) in Ref. [30]). The kinetics of Al solute depletion and Cr enrichment are shown in Figs. 4 and 5, respectively; the plots are arranged using the same scheme as in Fig. 3. The data were subjected to weighted regression analysis [35,36], using the tabulated values reported by Mao et al. [8]. The slopes and intercepts of the plots of XAl vs. t1/3 or t1/n differ by at most 5% using weighted and standard regression analysis. Once again, the values of R2 (inset) are comparable for both temporal exponents, being slightly larger for Alloys 1 and 2 for n = 2.4 (Fig. 4) and Alloys 2 and 3 (Fig. 5). The values of ji and jTi are summarized in Table 3. The rate constants ji, without the subscript T, are the rate constants expected from analysis of the data assuming MDC coarsening (n = 3). The rate constants reported by Sudbrack et al. [5] and Booth-Morrison et al. [6] are also shown in Table 3. The analyses of their data in this work suggest that the ji are slightly larger than those previously reported, the biggest discrepancy being about 25% for jCr of Alloy 3. The plots in Figs. 4 and 5 provide values of XeAl and XeCr, respectively, by extrapolation of the data to t–1/n = 0. The results are summarized in Table 4. It is apparent that for n = 2.4 (column 6) the values of XeAl are slightly larger than they are for n = 3 (column 4) and that the differences are generally <2%. The values of XeCr are slightly smaller for n = 3 than for n = 2.4 (cf. columns 5 and 7), though the differences are <1%. The agreement with the equilibrium solubilities of Al and Cr reported by Mao et al. [8] (columns 2 and 3) for the three alloys is excellent. 2.4. The variation of volume fraction with aging time The TIDC theory predicts that the volume fraction, f, varies with aging time according to the equation [30]:

ð6aÞ f ¼ fe  ð1  fe Þ

jTAl;Cr t1=n c=c0

DX eAl;Cr

;

ð7Þ

A.J. Ardell / Acta Materialia 61 (2013) 7828–7840

7833

Fig. 4. Plots of the Al concentration in the matrix, XAl, of Alloys 1 (5.2% Al), 2 (6.5% Al) and 3 (7.5% Al) (top to bottom) vs. aging time, t, raised to the 1/n power. In the column of figures on the left n = 3, while in the column to the right n = 2.4. The data are from the compilation of Mao et al. [8]. The numbers in the bottom right corner of each figure represent the correlation coefficient, R2.

where fe is the equilibrium volume fraction and 0 0 c=c0 DX eAl;Cr ¼ X ceAl;Cr  X eAl;Cr , where X ceAl;Cr is the equilibrium concentration of Al,Cr in the c0 phase, is the jump in concentration of Al,Cr across the c/c0 interface at thermodynamic equilibrium. The dependencies of f on aging time, taken from the data of Sudbrack et al. [4] and Booth-Morrison et al. [6,7], are shown as plots of f vs. t–1/2.4 in Fig. 6; the data look very similar when plotted as f vs. t–1/3. For these data the slopes and intercepts obtained using weighted and standard regression analyses are different. Both are shown in Fig. 6. The linearity predicted by Eq. (7) is moderately well obeyed, as evidenced by the correlation coefficients inset in each figure, which all exceed 0.9. The intercepts of the fitted straight lines in Fig. 6 represent the equilibrium volume fractions, fe. They are shown in

Table 1 (measured) where they are compared with the values calculated by the authors [4,6,7] using the lever rule. The agreement between the reported and measured values of fe is reasonably good. To compare the measured slopes of the curves in Fig. 6 with their calculated counterparts it is necessary to know c=c0 DX ei for each alloy, which means that the equilibrium concentrations of the c0 phase in each alloy must be known. 0 0 To this end the values of X ceAl and X ceCr reported by Sudbrack et al. [5] and Booth-Morrison et al. [6,7] are used in all subsequent calculations; they are shown in Table 5. The slopes were calculated using the measured values of fe in c=c0 Table 1, jTAl,Cr (n = 2.4) in Table 3 and DX eAl;Cr in Table 5. The experimentally measured slopes and their calculated values are shown in Table 5. The agreement is generally fair,

7834

A.J. Ardell / Acta Materialia 61 (2013) 7828–7840

Fig. 5. Plots of the Cr concentration in the matrix, XCr, of Alloys 1 (5.2% Al), 2 (6.5% Al) and 3 (7.5% Al) (top to bottom) vs. aging time, t, raised to the 1/n power. In the column of figures on the left n = 3, while in the column to the right n = 2.4. The data are from the compilation of Mao et al. [8]. The numbers in the bottom left corner of each figure represent the correlation coefficient, R2.

Table 3 Values of the rate constants jAl,Cr and jTAl,Cr obtained from the slopes of the curves in Figs. 4 and 5. The rate constants jAl,Cr reported by Sudbrack et al. [5] and Booth-Morrison et al. [6] are also shown. Alloy

jAl (s1/3)

jCr (s1/3)

jAl (s1/3)

jCr (s1/3)

jTAl (s1/2.4)

jTCr (s1/2.4)

1 2 3

0.186 ± 0.022 [5] Not reported 0.180 ± 0.050 [6]

0.140 ± 0.050 [5] Not reported 0.060 ± 0.010 [6]

0.1958 ± 0.0129 0.1851 ± 0.0143 0.1871 ± 0.0131

0.1409 ± 0.0227 0.0684 ± 0.0131 0.0805 ± 0.0040

0.4217 ± 0.0216 0.3932 ± 0.0235 0.3889 ± 0.0331

0.2968 ± 0.0529 0.1472 ± 0.0241 0.1649 ± 0.0074

Table 4 Values of the equilibrium solubility limits, XeAl and XeCr obtained from the intercepts of the curves in Figs. 4 and 5. The equilibrium solubility limits reported by Mao et al. [8] in their Table 4 are also shown. Alloy

XeAl [8]

XeCr [8]

XeAl (n = 3)

XeCr (n = 3)

XeAl (n = 2.4)

XeCr (n = 2.4)

1 2 3

0.0313 ± 0.0008 0.0549 ± 0.0005 0.0542 ± 0.0009

0.1560 ± 0.0020 0.1030 ± 0.0005 0.0939 ± 0.0009

0.0311 ± 0.0002 0.0549 ± 0.0003 0.0542 ± 0.0002

0.1562 ± 0.0005 0.1031 ± 0.0003 0.0939 ± 0.0001

0.0318 ± 0.0002 0.0554 ± 0.0002 0.0549 ± 0.0002

0.1557 ± 0.0004 0.1029 ± 0.0002 0.0936 ± 0.0001

A.J. Ardell / Acta Materialia 61 (2013) 7828–7840

7835

but incorrect, long-time behavior Nv / t1 is recovered. Despite the fact that it has been shown convincingly [37,40,41], using the best experimental data available, that the product Nvt is never constant in the limit t ! 1, it has been difficult to disabuse the idea that the asymptotic behavior Nv / t1 is incorrect. For the TIDC-T theory applied to coarsening in Ni–Al–Cr alloys, the predicted asymptotic behavior is Nvt1.25 (3/2.4 = 1.25) should vary linearly with t1/2.4. The data of Sudbrack et al. [4] and Booth-Morrison et al. [6,7] are plotted accordingly in Fig. 7, the straight lines having been fitted using standard regression analysis. Two aspects of the asymptotic behavior of the product Nvt1.25 are immediately apparent. The first is that the agreement with the predictions of Eq. (8) is poor, which is why no attempt was made to compare the measured slopes and intercepts with their calculated counterparts. The poor agreement is evident visually and also from the values of R2 included in each figure. The second aspect is that Nvt1.25 clearly increases substantially as t1/2.4 decreases (i.e. as t increases). This increase is expected, and even though it is not shown because there is no need to do so, the product Nvt behaves the same way. Reasons for the discrepancy between theory (both MDC and TIDC) and experiment are discussed later. 2.6. The interfacial free energy, r The interfacial free energy can be calculated from the equation [30]: c=c0

r¼

1=n k T jTAl

c=c0

DX eAl G00 DX eCr G00 1=n ¼ k T jTCr ; 2V m gP hzi 2V m hzi

ð9Þ

where the parameters P and g are given by the equations: 0

Fig. 6. Plots of the volume fraction of c precipitates, f, in Alloys 1 (5.2% Al), 2 (6.5% Al) and 3 (7.5% Al) vs. aging time, t, raised to the 1/2.4 power, taken from the tabulated data of Sudbrack et al. [4] and BoothMorrison et al. [7] and [6], respectively. The numbers in the bottom left corner of each figure represent the correlation coefficient, R2.

the measured slopes exceeding the calculated slopes by up to 43%. 2.5. The variation of precipitate number density with aging time The number of precipitates per unit volume, Nv, according to the TIDC theory obeys the equation [30]: ( ) 3 jTAl;Cr t1=n Nv ¼ fe  ð1  fe Þ ; ð8Þ c=c0 4pwðk T tÞ3=n DX eAl;Cr where w = hr3i/hri3. According to Eq. (8) the product Nvt3/n is predicted to vary linearly with t1/n in the asymptotic limit t ! 1. If n = 3 (MDC coarsening) and the dependence on t1/n is ignored, the commonly accepted,

P¼

DX c=c eAl

0

ð10Þ

c=c0

DX eCr

and g¼P

DICr ; DIAl

ð11Þ

and G00 is defined by the equation: G00 ¼ gPG00AlAl þ ðg þ P ÞG00AlCr þ G00CrCr ; G00ij

ð12Þ

2

where ¼ @ Gm =@X i @X j are the curvatures of specific contributions to the Gibbs free energy of mixing, evaluated at the equilibrium compositions of the c phase. The other parameters in Eqs. (9) and (11) are the trans-interface diffusion coefficients, DIAl,Cr, of Al and Cr (i.e. average diffusion coefficients in the concentration gradient that constitutes the interface between the disordered c and ordered c0 phases) and Vm is the partial atomic volume of Al and Cr in the c0 phase (assumed to be equal). The parameter g can also be expressed as [30]: g¼

jTAl : jTCr

ð13Þ

7836

A.J. Ardell / Acta Materialia 61 (2013) 7828–7840 Table 5 Comparison between the measured slopes of the curves in Fig. 6 and their values calculated using Eq. (7). The calculations 0 require usage of the jumps in concentration across the equilibrium c/c0 interfaces, DX c=c eAl;Cr , which were calculated using the values c0 of X eAl;Cr shown in the second and third rows of this table and the values of XeAl and XeCr shown in columns 6 and 7 of Table 4. The units of the measured and calculated slopes are s1/2.4. 0

X ceAl 0 X ceCr 0 c=c DX eAl0 c=c DX eCr Slope (meas) Slope (Al, calc) Slope (Cr, calc)

Alloy 1

Alloy 2

Alloy 3

0.1669 ± 0.0022 [5] 0.0677 ± 0.0015 [5] 0.1351 ± 0.0022 0.0881 ± 0.0015 4.6763 ± 0.4455 2.6896 ± 0.1457 2.9077 ± 0.5080

0.1753 ± 0.0033 [7] 0.0606 ± 0.0036 [7] 0.1197 ± 0.0033 0.0423 ± 0.0036 4.3657 ± 0.1898 2.9388 ± 0.1972 3.1168 ± 0.5809

0.1782 ± 0.0015 [6] 0.0585 ± 0.0012 [6] 0.1233 ± 0.0015 0.0351 ± 0.0012 4.7268 ± 0.4653 2.6937 ± 0.2416 4.0144 ± 0.2282

In binary alloys the rate constant jT is always positive because X1 always decreases towards its equilibrium value. In ternary alloys this is not necessarily the case for both solute atom species. During c0 coarsening in Ni–Al–Cr alloys in particular, X1Al decreases during coarsening while X1Cr increases so that jTCr is negative. For this situation DX1Cr is also negative, as are both parameters g and P. To proceed, it is necessary to calculate G00 using the terms in Eq. (12) and to keep in mind that g and P are both negative. The thermodynamic model of the ternary Ni–Al– Cr alloy system of Dupin et al. [42] is used here. The molar Gibbs free energy of mixing is the sum of two terms, Gm-id and Gm-exc, which are the ideal and excess free energies of mixing, respectively. In principle, there is a magnetic contribution to the free energy of mixing, but it is ignored here since it is negligible at the high temperatures used in the experiments on coarsening [43]. Details of the calculation of G00 are presented in Appendix A and all the parameters needed to calculate G00 , including g; P ; G00AlAl ; G00CrCr and G00AlCr , are summarized in Table 6. The values of XeAl and XeCr used in the calculations are those in the last two columns of Table 4 (n = 2.4). 0 P was calculated using Eq. c=c c=c0 (10) and the values of DX eAl and DX eCr in Table 5, while g was calculated by substituting the rate constants jTAl,Cr in Table 3 into Eq. (13). As for the other parameters required to calculate r, kT c=c0 was taken from Table 2, DX eAl;Cr from Table 5, Vm = 6.7584  106 m3 mol–1 [5] and hzi = 0.951; the results are summarized in Table 7. Since g = jTAl/jTCr, there is no difference between the values of r obtained using the data on Al and Cr; this is reflected in the results shown in Table 7. For comparison, the values of r obtained from analyzing the data using n = 3 are also presented in Table 7. These were calculated using the values of k in Table 2, jAl,Cr0 in Table 3 (n = 3), XeCr,Al in Table 4 c=c (recomputing DX eAl;Cr for the slight differences among the equilibrium solubilities measured when n = 3 cf. n = 2.4) and hzi = 1 in Eq. (9). The interfacial free energies calculated using the equations of TIDC coarsening (n = 2.4) are consistently smaller than those for MDC coarsening

(n = 3), and consistently smaller than those reported by Sudbrack et al. [5] and Booth-Morrison et al. [6,7]. It is possible to calculate error estimates for all the values of r using the uncertainties associated with all the parameters involved in the calculation. This has not been done here because there are no published error estimates for the Redlich–Kister [44] coefficients that contribute to Gm and G00 [42], hence the error estimates would not be statistically meaningful. 2.7. The diffusion coefficients The interfacial diffusion coefficients are obtained from analysis of the data using the equation [30]: n n oðn1Þ k fðn1Þ=ng ao DX c=c0 hzi2 eAl;Cr T DIi ¼ ; ð14Þ n1 jTAl;Cr rmmin where ao is the lattice constant of the coherent phases, assumed to be equivalent, rmin is expected to be the order of 10ao [34] and m is an exponent that satisfies the condition n = m + 2. The parameters ao and rmin are involved in an equation describing the interface width, d, on particle size [18,33,34,43], namely:  m r d ¼ ao ; ð15Þ rmin which presumes that d increases as r increases. Support for this presumption is found in the analyses [45] of experimental interface profiles in the ternary Ni–Al–Cr system [7]. The equivalent equation for MDC coarsening in binary alloys [46] is recovered by taking n = 3 (m = 1), hzi = 1 and letting rmin = ao. Using ao = 0.3554 nm [4], taking rmin = 10ao and substituting the other relevant parameters into Eq. (14) produces the results for DIAl and DICr shown in Table 8 (2nd and 6th rows). As noted previously [30], the values of DIAl and DICr cannot be compared directly with results on equivalent diffusivities because there are none available for comparison. However, they can be compared with intrinsic diffusion coefficients measured for the c and c0 phases of equivalent composition at the aging temperature of 873 K, if the data exist.

A.J. Ardell / Acta Materialia 61 (2013) 7828–7840

7837

Table 7 Interfacial free energies, r (mJ m2), extracted from the data on coarsening using the TIDC (n = 2.4) and MDC (n = 3) theories. The data reported previously [5–7] are also shown, designated as rAl and rCr. Alloy

1 2 3 a

r

r

rAl

n = 2.4

n=3

Published values

19.00 15.76 14.51

30.39 28.26 23.99

22.3 22.0a 23.6

rCr

Ref.

25.1

[5] [7] [6]

26.6

Values based on the Al and Cr data were not reported.

Table 8 Diffusion coefficients DIAl and DICr extracted from the data on coarsening using the TIDC theory, Eq. (14). Also shown are the intrinsic diffusion coefficients DAlAl and DCrCr (with superscripts c or c0 ) calculated from published data on diffusion in the c and c0 phases of ternary Ni–Al–Cr alloys (see text for details of the calculations). All the diffusion coefficients are in m2 s1, calculated for T = 873 K.

Fig. 7. Plots of the product of the number density of precipitates, Nv, multiplied by the aging time, t, raised to the 1.25 power vs. t raised to the 1/2.4 power for Alloys 1 (5.2% Al), 2 (6.5% Al) and 3 (7.5% Al), taken from the tabulated data of Sudbrack et al. [4] and Booth-Morrison et al. [7] and [6], respectively. The numbers in the bottom right corner of each figure represent the correlation coefficient, R2.

While there are no data for conditions comparable to all the conditions of the experiments used to generate the data on coarsening, it is nevertheless possible to estimate the relevant diffusion coefficients for the c and c0 phases in the ternary alloy. Nesbitt and Heckel [47] measured intrinsic

Alloy

1

2

3

DIAl DcAlAl 0 DcAlAl DICr DcCrCr 0 DcCrCr

3.354  1022 5.375  1021 4.467  1024 3.107  1022 67.201  1021 65.198  1024

3.937  1022 4.584  1021 – 3.715  1022 75.172  1021 –

3.621  1022 5.154  1021 – 2.431  1022 85.841  1021 –

diffusion coefficients of Al and Cr from diffusion couples in Ni–Al–Cr alloys at 1100 and 1200 K and provided an empirical equation in their Table V that enables extrapolation to 873 K for alloys of compositions equivalent to Alloys 1–3. The results of those extrapolations are shown in the 3rd and 7th rows of Table 8. Cermak et al. [48] measured the interdiffusion coefficients in the c0 phase of several ternary Ni–Al–Cr alloys and reported the results in the form of Arrhenius equations. Using the Arrhenius equation for their alloy designated O/C II, which is closest in composition to the c0 phases in Alloys 1–3 (5% Al, 18% Cr, cf. the compositions in Table 5) produces the intrinsic diffusivities in the 3rd and 6th rows of Table 8. On comparing the DIAl,Cr calculated using Eq. (14) with those estimated from the literature, it is evident that they are greater than the intrinsic diffusivities of the components in the c0 phase, but smaller than those in the matrix. 3. Discussion It is obvious from the correlation coefficients shown in Figs. 3–5 that the equations of the TIDC-T theory with n = 2.4 describe the experimental data at least as well as,

Table 6 Values of the parameters g and P and the contributions to the curvatures of the molar Gibbs free energy of mixing G00AlAl ; G00CrCr and G00AlCr (all in J mol–1) used to substitute into Eq. (6) to calculate G00 . Alloy

g

P

G00AlAl

G00CrCr

G00AlCr

G00

1 2 3

1.4207 2.6703 2.3586

1.5359 2.8320 3.5150

438254.16 354774.53 349755.60

141459.88 177957.22 187362.66

145682.00 157610.07 157337.19

667001.94 1993693.89 2162842.09

7838

A.J. Ardell / Acta Materialia 61 (2013) 7828–7840

if not slightly better than, those of the MDC theory (n = 3). The two discrepancies involve the variations of f and Nv with aging time. As noted by the authors [4,6,7], the number of precipitates included in the measurements of these two quantities decreased significantly after the maximum number of precipitates was detected, typically at around 4 h of aging. At the longest aging times the number of precipitates was quite small, e.g. around 100 at 16 h and in single digits at the longest aging time for Alloy 1, less than 50 at 64 h to 13.5 at 4096 h of aging for Alloy 2, and less than 50 at 16 h to single digits for Alloy 3. Thus, despite the evident care taken with all the experimental measurements, the statistical samplings involved in taking the data were quite small, leading to the relatively large reported errors. The same can be said for the very poor agreement between the predictions of Eq. (8) and the plots shown in Fig. 7. The poor agreement is completely unrelated to the asymptotic dependencies of Nv at long aging times; plots of Nvt vs. t1/3 are also highly non-linear and do not approach the time-independent value that would be expected if the equation Nv / t1 were correct. Instead, the disagreement between the TIDC theory and experiment is certainly due to the meager statistical sampling. The temporal evolution of both f and Nv is expected to be most accurately predicted by Eqs. (7) and (8) at increasingly long aging times. Unfortunately, this is where the reported experimental results are least accurate. It is immediately apparent that the interfacial free energies extracted from data on coarsening are smaller using the TIDC-T theory than the MDC theory. For the ternary alloys the difference is about 40%. This result is comparable to what was found for c0 precipitates in Ni–Al, Ni–Ga, Ni– Ge, Ni–Si and Ni–Ti alloys, where r was between 25% and 40% smaller using the TIDC theory [43]. It is also apparent that the values of r obtained using the MDC (n = 3) theory differ from those reported by the authors of the original work [5–7]. The main reason for this is that the values of G00AlAl ; G00CrCr and G00AlCr in Table 6 are significantly larger than those reported by Sudbrack et al. [5] for Alloy 1 (365,962.25, 117,511.87 and 113,606.62 J mol–1, respectively) and Booth-Morrison et al. [6] for Alloy 3 (306,697.80, 166,708.50 and 140,244.90 J mol–1, respectively); this information was not reported for Alloy 2. Neither Sudbrack et al. nor Booth-Morrison et al. provided any details of their calculations of the G00ij , so the source of this discrepancy cannot be determined. Another factor contributing to the discrepancies is that g = P in the calculations of Sudbrack et al. [5] and Booth-Morrison et al. [6,7]. This is why their calculated values of rAl and rCr differ (see Table 7). The origin of this problem can be traced to Eq. (25) in the paper by Kuehmann and Voorhees (K–V) [29], where the denominator of the coefficient of t1/3 is correct only if the intrinsic diffusion coefficients of the two solute components are equal [30]. Marquis and Seidman [49] derived their equations for r from Eq. (25) in the K–V paper, apparently without realizing this, and Sudbrack et al. [5] and Booth-Morrison

et al. [6,7] simply adapted the equations derived by Marquis and Seidman for their work on Ni–Al–Cr alloys. It turns out that the difference between g and P is relatively small for Alloys 1 and 2, around 10% or less, but for Alloy 3 the difference is close to 50% (Table 6). The discrepant values of r in Table 7 in this work and the published values [5–7] are also influenced by the differences in the rate constants k and j in this work and theirs obtained from analyzing the same sets of data (see Tables 2 and 3). Regarding the DIAl,Cr calculated using Eq. (14) and their comparisons with the intrinsic diffusion coefficients estimated from the literature, the fact that they are greater than the intrinsic diffusivities of the components in the c0 phase, but smaller than those in the matrix, is a particularly important outcome supporting the TIDC-T theory. It is important to point out here that Eq. (14) and its MDC counterpart are completely independent of the thermodynamics of the alloy system, i.e. of G00 . This is not the case for the equations for the diffusion coefficients published by Marquis and Seidman [49] and Sudbrack et al. [5], which include explicit contributions from the G00ij . The reason for this can also be traced back to their use of the coefficient of t1/3 in Eq. (25) of K–V, which is valid only if g = P. 4. Summary and conclusions Data on the coarsening behavior of c0 precipitates in ternary Ni–Al–Cr alloys have been reanalyzed using the equations of the TIDC-T theory. The absence of an effect of equilibrium volume fraction on the kinetics of coarsening and the PSDs is consistent with theoretical expectation, supporting the hypothesis that diffusion through the diffuse c/c0 interface controls coarsening behavior in these ternary alloys. The temporal exponent n = 2.4 was determined by fitting the equations of the theory to published PSDs and subsequently used to re-examine published data on the kinetic behavior of all the timedependent variables. For the kinetics of hri, X1Al and X1Cr the fit to the experimental data is just as good, and indeed actually better in several cases, using n = 2.4 compared to using n = 3. The dependence of f on aging time is in fair agreement with the published data, but the dependence of Nv on aging time agrees poorly with the predictions of the theory. It is argued that the very limited counting statistics adversely affect the accuracy of the data on both f and Nv, especially at longer aging times where the counting statistics are poorest and the theory is expected to be most reliable. The data on particle growth and solute redistribution during coarsening were analyzed to extract values of the interfacial free energy, r. The analysis requires the use of a thermodynamic model of the c/c0 alloy system, and with the model chosen the values of r range from about 14.5 to 19 mJ m–2. These numbers are significantly smaller than those obtained from analyses using the K–V theory of MDC coarsening (n = 3).

A.J. Ardell / Acta Materialia 61 (2013) 7828–7840

The calculated values of the interfacial diffusion coefficients are greater than the intrinsic diffusivities of the components in the c0 phase, but smaller than those in the matrix. This finding is completely consistent with theoretical expectation, since the TIDC-T theory demands that the trans-interfacial diffusivities should be some average of the intrinsic diffusivities of the solute atom components in the matrix and precipitate phases. In closing, it can be stated unequivocally that the data on coarsening of c0 precipitates in Ni–Al–Cr alloys are completely consistent with the predictions of the TIDC theory. A possible exception is the data on Nv, but these data are not well constrained statistically, so the disagreement between theory and experiment can hardly be considered a failure. Coupled with the complete absence of an effect of volume fraction on the kinetics of coarsening, the TIDC theory is the only one that is able to rationalize all the experimental observations on the kinetics and PSDs, as well as the magnitudes of r and the DIAl,Cr. It is difficult to imagine what more could be expected of a theory. Acknowledgments and dedication The author expresses his appreciation to Dr. Nathalie Dupin, Calcul Thermodynamique, for helpful discussions. The author owes a large debt of gratitude to David J. Chellman, whose untimely passing leaves a large void in my heart. His exacting experiments on precipitation processes in binary Ni–Al and ternary Ni–Al–Cr alloys laid the foundation for significant discoveries in this field of research. This paper is dedicated to his memory.

7839

Table A1 Equations for the coefficients, Lfcc;p i;j;k , in the Redlich–Kister [44] polynomial 1 equations for the Gibbs free energy of mixing. The Lfcc;p i;j;k (J mol , with T in K) are taken from the thermodynamic model of Dupin et al. [42]. Lfcc;0 Al;Cr ¼ 45; 900 þ 6T Lfcc;0 Al;Ni ¼ 162; 408 þ 16:213T Lfcc;1 Al;Ni ¼ 73; 418  34:914T Lfcc;2 Al;Ni ¼ 33; 471  9:837T Lfcc;3 Al;Ni ¼ 30; 758 þ 10:253T Lfcc;0 Cr;Ni ¼ 8; 030  12:880T Lfcc;1 Cr;Ni ¼ 33; 080  16:036T Lfcc;0 Al;Cr;Ni ¼ 30; 300

from the Appendix of the paper by Dupin et al. [42]; they are summarized in Table A1. The second derivatives in Eq. (12) are obtained by differentiation of Eqs. (A1) and (A2) and are given by the equations: @ 2 Gm-exc ¼ 2X Cr Lfcc;1 Cr;Ni  2LAl;Ni @X 2Al þ 2ð1  2X Al  X Cr Þ  2X Cr Lfcc;0 Al;Cr;Ni ; @ Gm-exc fcc;1 ¼ 2Lfcc;0 Cr;Ni þ 6ð1  X Al  2X Cr ÞLCr;Ni @X 2Cr  2X Al

@LAl;Ni @ 2 LAl;Ni þ X Al X Ni @X Cr @X 2Cr

 2X Al Lfcc;0 Al;Cr;Ni

ðA1Þ

where Rg is the gas constant. Gm-exc in the model of Dupin et al. [42] is expressed as the Redlich–Kister [44] polynomial expansion: fcc;0 Gm-exc ¼ X Al X Cr Lfcc;0 Al;Cr þ X Cr ð1  X Al  X Cr ÞLCr;Ni

@ 2 Gmexc fcc;0 fcc;1 ¼ Lfcc;0 Al;Cr  LCr;Ni þ 2ð1  X Al  3X Cr ÞLCr;Ni @X Al @X Cr @LAl;Ni  LAl;Ni þ ð1  2X Al  X Cr Þ @X Cr 2 @ LAl;Ni þ 2X Al X Ni þ ð1  2X Al  2X Cr ÞLfcc;0 Al;Cr;Ni : @X 2Cr ðA6Þ

fcc;1 þ X Cr ð1  X Al  X Cr Þð2X Cr þ X Al  1ÞLCr;Ni

þ X Al ð1  X Al  X Cr ÞLAl;Ni ;

ðA2Þ

where fcc;1 LAl;Ni ¼ Lfcc;0 Al;Ni þ ð2X Al þ X Cr  1ÞLAl;Ni

þ ð2X Al þ X Cr  1Þ2 Lfcc;2 Al;Ni 3

þ ð2X Al þ X Cr  1Þ Lfcc;3 Al;Ni ;

ðA3Þ Lfcc;p i;j;k

ðA5Þ

and

Gm-id ¼ Rg T fX Al ‘nX Al þ X Cr ‘nX Cr þ ð1  X Al  X Cr Þ‘nð1  X Al  X Cr Þg;

ðA4Þ

2

Appendix A In ternary alloys Gm-id is given by the equation:

@LAl;Ni @ 2 LAl;Ni þ X Al X Ni @X Al @X 2Al

and only the non-zero coefficients, (p = integer) in the model of Dupin et al. [42], have been retained in Eq. (A2). The coefficients Lfcc;p i;j;k are temperature-dependent only, and for the face-centered cubic (A1) solid solution are taken

The contributions to the curvature of the Gibbs free energy from the ideal free energy of mixing are:   @ 2 Gm-id 1  X Cr ¼ Rg T ; ðA7aÞ X Al X Ni @X 2Al   @ 2 Gm-id 1  X Al ¼ Rg T ; ðA7bÞ X Cr X Ni @X 2Cr and @ 2 Gmid Rg T ¼ : @X Al @X Cr X Ni

ðA7cÞ

7840

A.J. Ardell / Acta Materialia 61 (2013) 7828–7840

A few other useful expressions simplify the calculations. They are: @LAl;Ni @LAl;Ni fcc;2 ¼2 ¼ 2Lfcc;1 Al;Ni þ 4ð2X Al þ X Cr  1ÞLAl;Ni @X Al @X Cr 2

þ 6ð2X Al þ X Cr  1Þ Lfcc;3 Al;Ni ;

ðA8Þ

and @ 2 LAl;Ni @ 2 LAl;Ni ¼ 4 @X 2Al @X 2Cr fcc;3 ¼ 8Lfcc;2 Al;Ni þ 24ð2X Al þ X Cr  1ÞLAl;Ni ;

ðA9Þ

which lead to the equations: @ 2 LAl;Ni @ 2 LAl;Ni 1 @ 2 LAl;Ni ¼2 ¼ : 2 @X 2Al @X Al @X Cr @X 2Cr

ðA10Þ

The contributions G00AlAl ; G00CrCr and G00AlCr to the curvature of the Gibbs free energy of mixing are readily calculated using Eqs. (A4-A7), with the help of (A8)–(A10), using the expressions for the Lfcc;p i;j;k in Table A1. Substitution into Eq. (12) produces the results shown in Table 6. References [1] Schmuck C, Danoix F, Caron P, Hauet A, Blavette D. Appl Surf Sci 1996;94–95:273. [2] Schmuck C, Caron P, Hauet A, Blavette D. Philos Mag A 1997;76:527. [3] Blavette D, Danoix F, Cadel E, Geandier G, Menand A. J Phys IV 1999;9:113. [4] Sudbrack CK, Yoon KE, Noebe RD, Seidman DN. Acta Mater 2006;54:3199. [5] Sudbrack CK, Noebe RD, Seidman DN. Acta Mater 2007;55:119. [6] Booth-Morrison C, Weninger J, Sudbrack CK, Mao Z, Noebe RD, Seidman DN. Acta Mater 2008;56:3422. [7] Booth-Morrison C, Zhou Y, Noebe RD, Seidman DN. Philos Mag 2010;90:219. [8] Mao Z, Booth-Morrison C, Sudbrack CK, Martin G, Seidman DN. Acta Mater 2012;60:1871. [9] Taylor A, Floyd RW. J Inst Metals 1953;81:451. [10] Hornbogen E, Roth M. Z Metallkd 1967;58:842. [11] Booth-Morrison C, Noebe RD, Seidman DN. Acta Mater 2009;57:909.

[12] Sudbrack CK, Isheim D, Noebe RD, Jacobson NS, Seidman DN. Microsc Microanal 2004;10:355. [13] Sudbrack CK, Ziebell TD, Noebe RD, Seidman DN. Acta Mater 2008;56:448. [14] Yoon KE, Noebe RD, Seidman DN. Acta Mater 2007;55:1159. [15] Harada H, Ishida A, Murakami Y, Bhadeshia HKDH, Yamazaki M. Appl Surf Sci 1993;67:299. [16] Srinivasan R, Banerjee R, Hwang JY, Viswanathan GB, Tiley J, Dimiduk DM, et al. Phys Rev Lett 2009;102:086101. [17] Mishin Y. Acta Mater 2004;52:1451. [18] Ardell AJ, Ozolins V. Nat Mater 2005;4:309. [19] Janssen MMP. Metall Trans 1973;4:1623. [20] Watanabe M, Horita Z, Sano T, Nemoto M. Acta Metall Mater 1994;42:3389. [21] Fujiwara K, Horita Z. Acta Mater 2002;50:1571. [22] Ikeda T, Almazouzi A, Numakura H, Koiwa M, Sprengel W, Nakajima H. Acta Mater 1998;46:5369. [23] Ma Y, Ardell AJ. Mater Sci Eng A Struct Mater 2009;516:259. [24] Lifshitz IM, Slyozov VV. J Phys Chem Solids 1961;19:35. [25] Wagner C. Z Elektrochem 1961;65:581. [26] Umantsev A, Olson GB. Scripta Metall Mater 1993;29:1135. [27] Morral JE, Purdy GR. Scripta Metall Mater 1994;30:905. [28] Morral JE, Purdy GR. J Alloys Compd 1995;220:132. [29] Kuehmann CJ, Voorhees PW. Metall Mater Trans A 1996;27:937. [30] Ardell AJ. Acta Mater; 2013 [in press]. [31] Chellman DJ, Ardell AJ. Acta Metall 1974;22:577. [32] Jayanth CS, Nash P. Mater Sci Technol 1990;6:405. [33] Ardell AJ, Kim DM, Ozolins V. Z Metallkd 2006;97:295. [34] Ardell AJ. Acta Mater 2010;58:4325. [35] York D. Can J Phys 1966;44:1079. [36] Cantrell CA. Atmos Chem Phys 2008;8:5477. [37] Ardell AJ. Precipitate coarsening in solids: modern theories, chronic disagreement with experiment. In: Lorimer GW, editor. Phase transformations ‘87. London: The Institute of Metals; 1988. p. 485–94. [38] Voorhees PW. Annu Rev Mater Sci 1992;22:197. [39] Baldan A. J Mater Sci 2002;37:2171. [40] Ardell AJ. Phys Rev B 1990;41:2554. [41] Ardell AJ. Mater Sci Eng A Struct Mater 1997;238:108. [42] Dupin N, Ansara I, Sundman B. Calphad 2001;25:279. [43] Ardell AJ. J Mater Sci 2011;46:4832. [44] Redlich O, Kister AT. Ind Eng Chem 1948;40:345. [45] Ardell AJ. Scripta Mater 2012;66:423. [46] Ardell AJ. Interface Sci 1995;34:119. [47] Nesbitt JA, Heckel RW. Metall Trans A 1987;18:2075. [48] Cermak J, Gazda A, Rothova V. Intermetallics 2003;11:939. [49] Marquis EA, Seidman DN. Acta Mater 2005;53:4259.