On the increase of the precipitated volume fraction during Ostwald ripening, exemplified for aluminium–lithium alloys

Materials Science and Engineering A268 (1999) 197 – 201

On the increase of the precipitated volume fraction during Ostwald ripening, exemplified for aluminium–lithium alloys Avraam Kalogeridis, Josef Pesicka 1, Eckhard Nembach * Institut fu¨r Metallforschung der Uni6ersita¨t Mu¨nster, Wilhelm-Klemm-Straße 10, 48149 Mu¨nster, Germany Received 23 November 1998; received in revised form 21 January 1999

Abstract During Ostwald ripening of precipitates, their average radius r and their volume fraction f may increase concurrently with aging time t. In contrast to the majority of investigations in this field, the emphasis of the present study is on the function f(r) and not on r(t). Experimental data f(r) of seven aluminium-rich aluminium – lithium alloys are analyzed in the light of a relation recently published by Ardell (Mater. Sci. Eng. A238 (1997) 108). In agreement with this relation, f is found to be a linear function of 1/r. The slope #f/#(1/r) varies strongly with the overall lithium-concentration of the alloys. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Aluminium–lithium alloys; Ostwald ripening; Particle growth; Precipitation processes; Transmission electron microscopy; d%-Particles

1. Introduction In the final stage of precipitation processes, the precipitate population coarsens: large particles grow and small ones shrink. The driving force derives from the energy stored in the particle-matrix interface. After its discoverer Ostwald [1], this process has been named ‘Ostwald ripening’. Models describing it are based on the Gibbs–Thomson equation, which is given here in its most simple form:



cs(r)= cs(r“ ) 1 +

2GV rRgT

n

(1)

where cs(r) is the thermodynamic equilibrium atomic fraction of solute in the matrix next to the interface of a spherical precipitate of radius r, cs(r “ ) =cs is the corresponding atomic fraction at a planar interface; G its specific energy; V the molar volume; T the temperature; and Rg the gas constant. The subscript s of cs stands for solid ‘solution’ matrix. Evidently, small particles are surrounded by a dense cloud of solute, * Corresponding author. Tel.: +49-251-8333570; fax: + 49-2518338346. E-mail address: [email protected] (E. Nembach) 1 Permanent address: Department of Physics, Charles University, Ke Karlovu 5, 12116 Prague 2, Czech Republic.

whereas large ones have a more dilute cloud. Hence solute diffuses from small to large particles and the latter ones grow at the expense of the former ones, which dissolve. Hence the average radius r of the particles increases with aging time t. In Section 3, a more advanced form of Eq. (1) will be referred to. The first treatments of the diffusion problems involved have been published by Greenwood [2], Wagner [3], and Lifshitz and Slyozov [4]. These authors studied only the limiting case in which the precipitated volume fraction f is much smaller than unity ( f1.0) and found that r 3 is a linear function of t: r 3(t)− r 3(t=0)= kt

(2)

where t=0 marks the start of the coarsening process; and k is governed by the parameters appearing on the right side of Eq. (1), by the particles’ composition, and by the diffusion coefficient. Subsequently, large volume fractions have been treated; the respective publications have been reviewed in Refs. [5,6]. The result was that Eq. (2) is maintained, but k increases with f. From Eq. (1) it is evident that as r increases during aging, the solute concentration of the matrix decreases; consequently f increases [6–17]. Though one was well aware of the fact that r and f may increase concurrently during aging, the main interest was focused on the

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A. Kalogeridis et al. / Materials Science and Engineering A268 (1999) 197–201

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Table 1 Results for the parameters f (Eq. (4)), f l (Eq. (3)) a (Eq. (4)) and k (Eq. (2))a Alloy

ct

f

f l

a (nm)

k (10−30 m3/s)

Reference

L68 L74 L75 L76 L80 L84 L108

0.0683 0.0735 0.0753 0.0763 0.0804 0.0836 0.1078

0.072 0.086 0.066 0.148 0.159 0.134 0.254

0.049 0.079 0.090 0.096 0.120 0.139 0.281

0.038 0.051 0.019 0.125 0.123 0.053 0.282

2.09

[13] [16] Present [13] [13] [13] Present

a

1.82 2.54 3.59 1.56 2.72

Except alloy L74, which has been aged at 473 K, all others have been aged at 433 K. Alloy L84 contains additional elements besides Al and

Li.

effects of finite values of f on the growth parameter k and on the distribution function of particle radii [5]. Recently Ardell [17] again emphasized the variation of f with t and hence with r. This author’s derivations are sketched in Section 3. Depending on the specific parameters (Section 3) of the system under investigation the increase of f with r may be noticeable or not. Examples for both types of systems are: 1. f is virtually independent of r: g%-precipitates in nickel-base superalloys [13,18,19]. 2. f increases with r: g%-precipitates in Ni-rich Ni-Al-alloys, e.g. [7,8,11,17], and d%-precipitates in Al-rich Al-Li-alloys, e.g. [9,10,12,13,15,16]. Aluminium-rich aluminium-lithium alloys are strengthened by spherical, coherent precipitates of the metastable d%-phase [9,10,12,15,16,20]. Its composition is close to Al3Li and it has the long-range ordered L12 crystal structure [9,10,12,13,15,16,20 – 22]. In the progress of studies of the dependence of the critical resolved shear stress of Al-Li-alloys on their d%-precipitate dispersion, a broad data base for r(t) and f(t) has been accumulated [13,15,16]. This affords a welcome opportunity to subject Ardell’s [17] recently published relation for the function f(r(t)) to extensive tests; they will be presented in Section 3. Experimental data f(r) taken for seven Al-rich Al-Li-alloys will be analyzed. Part of the data have been taken from the literature [13,15,16] and part are so far unpublished.

and hence leads to small, widely spaced plate-like precipitates, which are rich in Al and Fe [15]. The preparation of the specimens has been described elsewhere: L74 has been aged at 473 K [16] and all others at 433 K [13,15]. The data of alloys L75 and L108 are new, the others have been taken from the literature [13,15,16]. The average particle radius r and the volume fraction f have been determined by dark-field transmission electron microscopy; the details have been published elsewhere [13,15,16,19,20]. The results for r and f have been subjected to the standard correction procedures described in Refs. [13,16,19,20]. d%-particles in alloy L75 are shown in Fig. 1. Following Ardell [17] (Section 3),

2. Experimental data All alloys studied are listed in Table 1. Here, ct is their overall atomic fraction of Li. The alloy designation Ln involves ct: n : 1000 ct. With the exception of L84, all alloys are binary. L84 contains minor atomic fractions of additional elements: 46×10 − 4 Fe, 3× 10 − 4 Si, 2 ×10 − 4 Zr, 10 − 4 Mg. At the aging temperature, the Fe-content of L84 exceeds the solubility limit

Fig. 1. Dark-field transmission electron micrograph of d%-particles in alloy L75: f= 0.058, r =27 nm.

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Fig. 2. Experimental data taken for seven Al-rich Al-Li-alloys: volume fraction f of the d%-precipitates versus 1/r where r is their average radius. The straight lines represent Eq. (4) least squares fitted to the data. The alloys’ overall atomic fraction of Li is indicated. (a), (b) Data from Ref. [13], (c) data from Ref. [16], (d) present data.

in Fig. 2 f has been plotted versus 1/r. The data f(1/r) have not been smoothed. Consequently, due to experimental errors, f does not always vary monotonically with 1/r. In the former evaluations of critical resolved shear stress data [15], the f-data had been smoothed. The error limits of r and f given in Ref. [13] and of those measured presently are estimated to be below about 5% (r) and 20% ( f ) [13,19]. Evaluating r and f from different micrographs taken for the same alloy and the same aging treatment resulted in ranges of scattering which in most cases were well below the

quoted error limits. Evidently the data f(1/r) shown in Fig. 2 are quite well represented by the straight lines; the deviations are within the error limits. Trinckauf et al. [13] analyzed the functions r(t) of their Al-Li-alloys L68, L76, L80, and L84. Eq. (2) had been least squares fitted to the data. In order not to give large radii too much weight in the fitting routine, the data were weighted with 1/r 3. They were well represented by Eq. (2)—provided r exceeded 9.4 nm. The data of the present alloys L75 and L108 were subjected to the same analyses. No systematic devia-

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tions of the latter data from Eq. (2) have been found; but for r below 9 nm they showed some minor scatter. The results for the growth parameters k are listed in Table 1. Since alloy L74 had been aged at a different temperature, no k-value is given for it. There is some tendency of k to increase with the alloy’s overall atomic fraction ct of Li and hence with the thermodynamic equilibrium d%-volume fraction f . k of alloy L84, which contains additional elements, is, however, rather low. As mentioned in Section 1, current theories of Ostwald ripening [5,6] predict such an increase of k with f . Trinckauf et al. [13] published their results for r(t= 0); the present values of r(t = 0) are 4.2 and 6.6 nm for alloys L75 and L108, respectively. Trinckauf et al.’s and the present r(t= 0)-values are of similar size. 3. Discussion Recently Ardell [17] re-analyzed Ostwald ripening of L12-long-range ordered g%-precipitates in a binary Ni0.1323Al alloy [8] paying special attention to the variation of f with r. In the following the small difference between the average and the critical [3 – 5,17] radius is disregarded. On the basis of a modified Gibbs–Thomson equation [14] and the lever rule ct − cs (3) f= cp −cs a handy expression for the function f(r) was obtained [17]: 1− f 1 f(r)=f −a (4) cp −cs r f is the thermodynamic equilibrium volume fraction reached for infinitely large precipitates, i.e. f = f(r“ ). cs is the atomic fraction of solute in the matrix, cp that in the d%-particles, ct the overall atomic fraction of solute, and cs equals cs(r “ ). The parameter a is the capillary length [14,17]; it is governed by G, V (Eq. (1)), cp, cs , and the matrix’ free enthalpy [14]. The parameter a in Eq. (4) stems from the application of more advanced thermodynamics. Evidently the slope of the function f(1/r) equals [− a(1 −f )/(cp −cs )]. The derivation of Eq. (4) involved linearizations; hence cs must be close to cs , i.e. the supersaturation of the matrix is supposed to be low. This is also necessary because the lever rule applies to homogeneous phases only and as long as cs differs strongly from cs there are concentration gradients, which make the matrix inhomogeneous. The lattice mismatch of the d%-precipitates is very small, probably less than 0.3% [21]; hence it is negligible. Several authors (e.g. [6,7,11]) considered the function f(t) instead of f(r). Since the derivation of the latter function is more direct, it is presently used.

Whether for a given system the variation of f with r is actually significant or not, depends primarily on the capillary length a. Examples for both types of systems have been given in Section 1. Ardell [17] demonstrated the increase of f with r for a Ni-rich Ni-Al alloy. The variation of f with 1/r of the seven presently analyzed Al-rich Al-Li alloys is clearly discernible in Fig. 2. The straight lines represent Eq. (4) least squares fitted to the data f(1/r). As stated in Section 2, the data are quite well represented by Eq. (4). The thermodynamic equilibrium volume fraction f is given by the ordinate intercept. The results for f are listed in Table 1. Also the capillary lengths a are given. They have been calculated from the slopes #f/#(1/r). The concentrations cs and cp at 433 K have been taken from the literature: 0.06 and 0.23, respectively [6,21–23]. An alternative derivation of f is based on the lever rule expressed by Eq. (3): cs is inserted for cs. Since this alternative value for f involves the lever rule it is referred to by f l, where l stands for ‘lever’. The results obtained for f l have also been compiled in Table 1. The ratios ( f l/f ) range from 0.65 to 1.36; their overall average is 0.93. Even the ratio of alloy L74, which had been aged at a 40 K higher temperature, deviates by only 1.5% from the average. The volume fraction f l is rather sensitive to the value inserted for cs : lowering it from 0.06 to 0.055, which is still well within the range of published values, raises f l of e.g. alloy L84 by 18%. The Li-concentration of L84 is close to the average over all alloys analyzed. Moreover the error limits of ct are estimated to be not less than 0.002; this leads to an error limit of 9% for f l of L84. Hence the presently observed agreement between f and f l is considered as quite satisfactory. In Fig. 3, the capillary length a, which forms part of the slope of the function f(1/r) (Eq. (4)), is plotted versus f . Evidently the function a( f ) changes by a factor of 7.4 in the concentration range investigated.

Fig. 3. Capillary length a appearing in Eq. (4) versus the equilibrium d%-volume fraction f . Open symbol: alloy L84; L84 has been disregarded in fitting the straight line to the data a( f ), because it contains several additional elements.

A. Kalogeridis et al. / Materials Science and Engineering A268 (1999) 197–201

Since the term (1-f ) appearing in Eq. (4) varies only by 24%, the variation of the slope #f/#(1/r) with f is primarily governed by the variation of a with f . Depending on the overall Li-concentration ct of the alloy and hence on f , the variation of f with r may be drastic: f of alloy L108 increases from 0.08 to 0.25. The experimental data f(1/r) of Ni-Al- [8,11,17,18] and of Al-Li-alloys increase strongly even if r exceeds several nm. Recently Poduri and Chen [6] simulated Ostwald ripening of Al-Li alloys in a computer. In their ranges of f and r, the increase of f with r was below 10%. In general the precipitation of particles of a new phase from a supersaturated solid solution involves three processes: (i) nucleation; (ii) growth of the nuclei; and (iii) coarsening. All three processes may occur concurrently. So far it has been assumed that in the present Al-Li-specimens, processes (i) and (ii) were no more operative and that there was only coarsening. This can, however, not be taken for granted. If processes (i) and (ii) were actually still operative, systematic deviations should be noticeable in Fig. 2. To check further on this, the present data f(1/r) have been reevaluated. Eq. (4) has been fitted only to the data taken for long aging times; only those data f(1/r) have been evaluated which simultaneously meet the two conditions: f\f /2 and f\ f l/2. This limitation brought about the following changes: Alloys L74 (Fig. 2(c)), L75 (Fig. 2(d)), L84 (Fig. 2(b)): No significant change. Alloy L68 (Fig. 2(a)): a is raised from 0.038 to 0.064. Alloy L80 (Fig. 2(b)): a decreases from 0.123 to 0.087. Alloy L76 (Fig. 1(a)): since only the outermost left four data shown in Fig. 2(a) meet the above limitation, no meaningful value is obtained for a. Alloy L108 (Fig. 2(d)): only the outermost left three data shown in Fig. 2(d) are kept; a is more than doubled. Reducing the evaluated f-range naturally raises the error limits of a; this holds especially for alloys L76 and L108. In spite of this, the f-limitation leads to no systematic overall changes of the five other reevaluated a values. This f-limitation also ensures that the assumption made in the derivation of Eq. (4), is fulfilled: [(cs − cs )/(cp − cs )] does not exceed 0.14. In Section 1 it has been stated that for the binary alloy systems Al-rich Al-Li and Ni-rich Ni-Al the increase of f with r is pronounced. In both systems the minority elements — Li and Al, respectively—have rather high solubilities in the matrices [6,21,22,24]. In contrast the function f(1/r) of superalloys is virtually constant and the solubility of the g%-forming minority elements Al and Ti is low [18]. Evidently there is a correlation between the variations of f with 1/r and the solubility of one of the precipitating elements. .

201

4. Conclusion During isothermal aging of Al-rich Al-Li-alloys the average radius r of the d%-precipitates and their volume fraction f increase concurrently. The data f(1/r) taken for seven Al-rich Al-Li-alloys are well represented by Eq. (4), which has recently been published by Ardell [17]. The capillary length appearing in the latter equation increases drastically with the thermodynamic equilibrium d%-volume fraction f .

Acknowledgements Financial support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged.

References [1] [2] [3] [4] [5]

[6] [7] [8] [9]

[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

[22] [23] [24]

W. Ostwald, Z. Phys. Chem. 34 (1900) 495. G.W. Greenwood, Acta Metall. 4 (1956) 243. C. Wagner, Z. Elektrochem. 65 (1961) 581. I.M. Lifshitz, V.V. Slyozov, J. Phys. Chem. Solids 19 (1961) 35. R. Wagner, R. Kampmann, in: R.W. Cahn, P. Haasen, E.J. Kramer (Eds.), Materials Science and Technology, vol. 5, VCH Verlagsgesellschaft mbH, Weinheim, 1991, p. 213. R. Poduri, L.-Q. Chen, Acta Mater. 46 (1998) 3915. A.J. Ardell, Acta Metall. 15 (1967) 1772. D.H. Kirkwood, Acta Metall. 18 (1970) 563. Y. Miura, A. Matsui, M. Furukawa, M. Nemoto, in: C. Baker, P.J. Gregson, S.J. Harris, C.J. Peel (Eds.), Aluminium–Lithium Alloys III, Proc. 3rd Int. Al-Li Conf, The Institute of Metals, Oxford, 1986, p. 427. W.A. Cassada, G.J. Shiflet, E.A. Starke Jr, Acta Metall. 34 (1986) 367. S.Q. Xiao, P. Haasen, Acta Metall. Mater. 39 (1991) 651. V. Gerold, H.-J. Gudladt, J. Lendvai, Phys. Stat. Sol. (a) 131 (1992) 509. K. Trinckauf, J. Pesicka, C. Schlesier, E. Nembach, Phys. Stat. Sol. (a) 131 (1992) 345. H.A. Calderon, P.W. Voorhees, J.L. Murray, G. Kostorz, Acta Metall. Mater. 42 (1994) 991. C. Schlesier, E. Nembach, Acta Metall. Mater. 43 (1995) 3983. S.M. Jeon, J.K. Park, Acta Mater. 44 (1996) 1449. A.J. Ardell, Mater. Sci. Eng. A238 (1997) 108. E. Nembach, G. Neite, Prog. Mat. Sci. 29 (1985) 177. C. Schlesier, E. Nembach, Mater. Sci. Eng. A119 (1989) 199. E. Nembach, Particle Strengthening of Metals and Alloys, Wiley, New York, 1996. D.B. Williams, in: T.H. Sanders Jr, E.A. Starke Jr (Eds.), Aluminium – Lithium Alloys, Proc. 5th Int. Aluminium–Lithium Conf., Williamsburg, 1989, vol. 2, Materials and Component Engineering, Birmingham, UK, 1989, p. 551. G. Cocco, G. Fagherazzi, L. Schiffini, J. Appl. Cryst. 10 (1977) 325. B. Noble, S.E. Bray, Acta Mater. 46 (1998) 6163. T.B. Massalski, J.L. Murray, L.H. Bennett, H. Baker (Eds.), Binary Alloy Phase Diagrams, vol. 1, American Society for Metals, Metals Park, OH, 1986.