Anomalous coarsening behavior of small volume fractions of Ni3Al precipitates in binary NiAl alloys

Acta metall, mater. Vot. 40, No. 10, pp. 2661-2667, 1992 Printed in Great Britain. All rights reserved

0956-7151/92 $5.00 + 0.00 Copyright ,c: 1992 Pergamon Press Ltd

ANOMALOUS COARSENING BEHAVIOR OF SMALL VOLUME FRACTIONS OF Ni3A1 PRECIPITATES IN BINARY Ni-A1 ALLOYS A, M A H E S H W A R I and A. J. ARDELL Department of Materials Science and Engineering, University of California, Los Angeles, CA 90024, U.S.A. (Received 10 February 1992) Abstract--The kinetics of coarsening of ?' precipitates in binary Ni AI alloys containing nominally 5.72, 5.74 and 5.78 wt% A1 and aged at 630 C were investigated by transmission electron microscopy and magnetic analysis. The alloys were each pre-aged at a temperature just below its solvus prior to re-aging at 630uC. The volume fractions of 7' were low, between 0.02 and 0.03. The pre-aging treatment produced microstructures containing precipitates that nucleated heterogeneously on dislocations and grain boundaries, leaving empty matrix areas in which coherent 7' precipitates subsequently nucleated and coarsened. Measurements were made only on these precipitates. The coarsening kinetics are anomalous in that the rate constants decrease with increasing volume fraction, in contradiction with all the theories of this process. Furthermore, the increase is quite large, exceeding a factor of four over a range of volume fractions that increased by less than 40%, Additionally, the rate constants exceed the values expected from the literature by factors of 10~40. The distributions of particle sizes were measured, and with few exceptions were found to agree with those of earlier investigations. It is suggested that elastic interactions among the 7,' precipitates play a pivotal role in decelerating the kinetics of coarsening, overpowering the expected accelerating effect of increasing volume fraction. R4sum~--On &udie, par microscopie 61ectronique en transmission et par analyses magn6tiques, la ein6tique de grossissement de pr&ipit6s 7' dans des alliages binaires Ni A1 de teneur nominale 5,72, 5,74 et 5,78% AI en poids vieillis fi 630:'C. Les alliages sont tous pr6vieillis fi une temperature imm4diatement inf6rieure ft celles du solvus avant un revieillissement fi 630 C. Les fractions volumiques de ;, ', situ4es entre 0,02 et 0,03, sont faibles. Le traitement de prhvieillissement produit des microstructures contenant des pr6cipiths qui germent d'une mani&e h4t6rog4ne sur les dislocations et les joints de grains laissant des r4gions de matrice vides dans lesquelles des pr6cipit6s coh6rents 7' germent ensuite et grossissent. On effectue des mesures seulement sur ces pr6cipit&. Les cin&iques de grossissement sont anormales par le fair que les constantes de vitesse dkcroissent lorsque la fraction volumique croit en contradiction avec routes les th6ories de ce processus. De plus, la diminution est tr4s importante, exc6dant un facteur quatre sur une gamme de fractions volumiques qui croit de moins de 40%. Enfin, les constantes de vitesse dhpassent les valeurs prhvues fi partir de la litt6rature de facteurs 10-40. Les distributions de tailles des particules sont mesur6es et, fi quelques exceptions pr6s, sont en accord avec celles des exp4riences ant6rieures. On sugg~re que les interactions ~lastiques entre les pr6cipit~s ?' jouent un r61e essentiel dans la d&roissance de la cin&ique du grossissement, surpassant l'effet prhvu d'acc614ration dfi fi la fraction volumique croissante. Zusammenfassung--Die Kinetik der Vergr6berung von 7 '- Ausscheidungen in bin/iren Ni-A1-Legierungen mit nominell 5,72, 5,74 und 5,78 Gew.% A1, die bei 630=C ausgelagert worden sind, wird mittels Durchstrahlungselektronenmikroskopie und magnetischer Analyse untersucht. Jede der Legierungen wird ausgelagert bei einer Temperatur gerade unterhalb ihrer Solvustemperatur, bevor sie bei 630rC ausgelagert wird. Der Volumanteil von 7' ist klein und liegt zwischen 0,02 und 0,03. Bei der vorausgehenden Auslagerung entstehen Mikrostrukturen, die an Versetzungen und Korngrenzen heterogen gebildete Ausscheidungen enth/ilt; dadurch bleiben freie Zonen in der Matrix zurfick, in denen daraufhin koh/irente ).'-Ausscheidungen entstehen, die sich vergr6bern. Nut diese Ausscheidungen werden gemessen. Die Vergr6berungskinetik ist dahingehend anomal, dab die Ratenkonstanten mit ansteigendem Volumanteil abnehmen, welches allen Theorien ffir diesen ProzeB widerspricht. Des weiteren ist die Zunahme sehr groB und iibersteigt den Faktor vier in einem Bereich von Volumanteilen, dernur 40% fiberstreicht. AuBerdem fibersteigen die Ratenkonstanten die literaturbekannten Werte um den Faktor 10-40. Die Verteilungen der Teilchengr6Ben werden ermittelt, mit wenigen Ausnahmen stimmen sie mit denen frfiherer Untersuchungen fiberein. Als Erkl/irung wird vorgeschlagen, dab elastische Wechselwirkungen zwischen den ~,,'-Ausscheidungen eine Hebelwirkung in der Verz6gerung der Vergr6berungskinetik spielen, indem der erwartete Beschleunigungseffekt des zunehmenden Volumanteils fiberspielt wird.

1. INTRODUCTION T h e r e has been considerable interest in recent years on the effect of precipitate volume fraction, f, o n the AMM 40AO~M

kinetics of coarsening. All the theories of this p h e n o m e n o n predict that the rate constant, k ( f ) , in the well-known coarsening law ( r ) 3 o c k ( f ) t , where ( r ) is the average precipitate radius a n d t is the aging

2661

2662

MAHESHWARI and ARDELL: ANOMALOUS COARSENING OF PRECIPITATES

time, increase as f increases. The various theories predict different dependencies of k(f) on f [1], but there is no question that under conditions of purely diffusive transport of solute from the shrinking to the growing precipitates in the polydisperse assembly, k(f) must increase as f increases. The experimental evidence reporting this type of dependence is at best sporadic, as demonstrated in a recent review [2]. In particular, there now seems little doubt that coarsening of the 7' (Ni3 A1) precipitate in binary Ni-A1 alloys is entirely independent of its volume fraction [2, 3]. Having noted this, however, it must also be pointed out that most of the measurements of the kinetics of coarsening of this precipitate, which is udoubtedly the most thoroughly investigated of any, have been made in alloys containing 7' volume fractions exceeding 0.05. This is relatively large compared to the values of f a t which the theories predict the largest increase in k(f) per unit change in f. Specifically in the limit of small values of f, k(f) is predicted theoretically [1] to approach eitherp/2 or fl/3, so that in any case the magnitude of dk(f)/df increases as f approaches zero. Another important reason for investigating the coarsening behavior at small values o f f is the influence of elastic interactions among the 7' precipitates. These are due to the fact that the elastic constants of the 7' particles are smaller than those of the matrix [4], which leads to preferential alignment along the elastically soft <100) directions and the well-known phenomenon of "rafting". Since the interaction is attractive and a strong inverse function of the centerto-center spacing of the precipitates, it follows that its influence can also be reduced by lowering the volume fraction. For these reasons we decided to undertake an investigation of the coarsening kinetics of the 7' precipitate at smaller volume fractions than those normally encountered in studies of this type. We will show that the kinetics of coarsening of the 7' precipitates in our alloys are highly unusual in that k(f) decreases with increasing f over the range of volume fractions encountered. Because these results were totally unexpected, the experimental procedures are described in greater detail than is customary in traditional investigations of precipitation phenomena. This is necessary because some of the procedures used were uncommon, and a clear description is required to establish the credibility of the conclusions. 2. EXPERIMENTAL

Three alloys of Ni and A1 containing nominally 5.72, 5.74 and 5.78 wt% A1 were made from 99.999% pure starting materials. These alloys will henceforth be referred to as A, B and C, respectively. Samples weighing approximately 20 g were prepared by arc melting. All the alloys were remelted 4 times to ensure compositional homogeneity. Small changes in weight were observed; these were increases in the alloys containing 5.72 and 5.74%, but a decrease in the

other one. In all cases the changes were less than 0.01%. After melting the alloys were annealed at 1100°C for 72 h in a Ti-gettered Ar atmosphere and furnace-cooled. The annealed alloys were then coldrolled to thicknesses ranging from 250/~m to 2.5 mm. The thinner sheet material was used for electron microscopy studies and the thicker material for magnetic measurements. The cold-rolled samples were solution treated in a Ti-gettered Ar atmosphere at 1000°C for 1 h and then quenched into refrigerated brine cooled to -15°C. An interesting complication in investigations of 7' coarsening at low volume fractions is an "unfavorable" phase diagram. Since 7' happens to be a solvent rich intermetallic compound rather than a terminal solid-solution phase, a specific undercooling from the solvus temperature produces a relatively large volume fraction of precipitate compared to that obtained when terminal solid solutions are in equilibrium. This is an obvious consequence of the lever rule, the denominator of which is a factor of about eight smaller in Ni-A1 alloys than in the case of dilute terminal solid-solution phases. Because of this, spatial fluctuations in composition which linger after melting and mechanical fabrication can greatly exaggerate the variations in the spatial distributions of the volume fraction of precipitates. This effect becomes increasingly pronounced whenever the degree of undercooling is small, such as those required to produce small volume fractions of precipitates. Effects of this sort have been observed experimentally by Ardell and Nicholson [4] and Hirata and Kirkwood [5], and are vividly illustrated by the following calculation. Consider a Ni-Al alloy containing 5.375 %A1. According to the coherent solubility data of Rastogi and Ardell [6] the solvus temperature of this alloy is 577.8°C. The equilibrium volume fraction of 7' at an aging temperature of 565.7°C is 0.011 ~., = 1.1% at an undercooling just exceeding 12°C). Suppose the solute concentration varies by +0.0002 (i.e. the solute content in the various regions of the alloy is 5.375 + 0.02 %AI), which is not at all unreasonable. The volume fraction of 7' in the solute-rich regions of the alloy (those containing 5.395 %A1) will be 0.014 while that in the solute-poor region will be 0.008. There will thus be a + 27% variation off~, from its average value, and the difference betweenf~, in the solute-rich and solute-poor regions will be close to a factor of two! The variation of solute is not the only source of difficulty. Temperature control can also lead to problems at very small undercoolings. For example, a temperature fluctuation of +0.5°C will produce an approximately + 16% variation off., in a perfectly homogeneous alloy containing 5.375 %AI on aging at 575°C (an undercooling of 2.8°C). The uncertainties associated with temperature control are obviously less serious at larger undercoolings, but it is important to make these estimates in order to appreciate how difficult it can be to produce the required small

MAHESHWARI and ARDELL: ANOMALOUS COARSENING OF PRECIPITATES volume fractions of 7' precipitates. We should also keep in mind that the accuracy with which the solvus temperature of a particular alloy needs to be known to obtain really small values off~, is far greater than the accuracy with which solubility limits are usually measured. The difficulties associated with compositional inhomogeneities are alleviated to a considerable extent at larger undercoolings, but this defeats the purpose of this investigation. In an attempt to overcome this problem, a special pre-aging treatment was employed for each of the alloys, which consisted of pre-aging each alloy a few degrees below the best estimate of the coherent solvus temperature. The purpose of this treatment was to eliminate the effects of compositional fluctuations by stimulating the precipitation of 7' in Al-rich regions of the alloy, thereby producing a more uniform matrix during subsequent aging. The data of Rastogi and Ardell [6] on the coherent solubility of AI in Ni were used to calculate the coherent solvus temperature, the pre-aging temperatures and the expected volume fractions of 7' in the alloys on subsequent re-aging. Alloys A, B and C were pre-aged at 635 __.2, 641 __ 2 and 648 +_ 2°C for 159.6, 159.3 and 120h, respectively, and quenched into refrigerated brine after the pre-aging treatment. Aging was carried out at 630°C in chloride salt baths to optimize the uniformity and control of the temperature, the importance of which has already been noted earlier. The detrimental effect of direct contact of the molten salt with the samples was recognized during preliminary experiments, so the samples were placed in evacuated quartz tubes to protect them. The bath temperature was controlled to +0.5°C by a digital controller. The temperature of the samples was measured using a type K thermocouple calibrated against the melting temperatures of AI, Pb and H20. The variation of A1 content of the matrix, w, during aging, was monitored by a magnetic technique using a modification of earlier apparatus [7]. Samples were cooled to 77 K, inserted into the apparatus and allowed to warm naturally. The ferromagnetic Curie temperature, Oc, was determined from the magnetic transition curves by extrapolating the steepest portion of the transition region to the region at which there is no further change in output signal on heating. The values of Oc of the solution-treated and quenched alloys were in excellent agreement with the previously determined [7] calibration curve of Oc vs w. The variation of w with aging time was obtained by referring the measured values of Oc to this curve. Thin foils were prepared from 3 mm discs of the aged samples for observations by transmission electron microscopy (TEM) using a JEOL model 100 CX T E M S C A N microscope operating at 100 keV. The 7' superlattice reflections of the type {100} and {110} in foils oriented (001) were used to take dark-field TEM images of the precipitates. Figure 1 illustrates the precipitate microstructure in Alloy C produced by

2663

Fig. 1. Dark-field TEM micrograph illustrating the microstructure in a sample of Alloy C aged for 171 h after pre-aging for 120 h at 648 _+2°C. 7' precipitates that nucleated heterogeneously on a dislocation are seen to be flanked by regions of homogeneously nucleated precipitates.

the double aging treatment. Many 7' precipitates nucleated heterogeneously 7' on grain boundaries and dislocations during pre-aging; 7' precipitates on dislocations are visible in Fig. 1. On re-aging, new 7' precipitates nucleated homogeneously in the regions adjacent to the heterogeneous 7'. The homogeneous 7' particles were always separated from the heterogeneous precipitates by a precipitate-free zone, as is also illustrated in Fig. 1. The microstructures of the other alloys were similar. It is seen in Fig. 1 that the ?' particles on the periphery of the homogeneously nucleated group of precipitates, i.e. at the edge of the precipitate-free zone, are slightly larger than those in the interior. This was usually, but not always, the case. The reasons for this behavior are not known, but since the coarsening of these peripheral precipitates is unique, we felt that they should be excluded when measuring the sizes of the 7' precipitates in the interior of the homogeneously nucleated group. Only the kinetics of coarsening of these interior particles are reported herein. The areas of the projected images of individual 7' particles were measured using a digitizer. Assuming that the particles are cubic in shape and that the projected areas are squares, the edge lengths, a, of the particles were calculated as the square root of this projected area. This method of measurement differs from that used in previous studies, but was convenient for measuring the projected area fractions, and ultimately volume fractions, of the 7' precipitates. If the particles are perfectly cubic in shape there is no difference between the square root of the projected area and the projected edge length. However, it has been known for some time that when 7' precipitates are small they are nearly spherical in shape, becoming cuboidal as they grow [4, 8]. We have shown recently [9] that when the volume fractions are small the individual precipitates grow to large enough sizes that their faces are actually

2664

MAHESHWARI and ARDELL: ANOMALOUS COARSENING OF PRECIPITATES

concave. The particles are therefore never perfect cubes, of course, but the difference in the values of a calculated by direct measurement and by taking the square root of the projected area is negligible. The number of homogeneous 7' particles available for measuring the distributions of particle size was limited, varying from a minimum of 52 to a maximum of 333. Particle size distributions were nevertheless obtained for purposes of comparison, although the statistical significance in some instances is quite limited. 3. RESULTS

Table I. Summary of the parameters relevant to the measurement of the volume fraction of ),' precipitates, f~,,, and the measured values of the rate constants for coarsening, k(f), at 630°C

k(f) × 103o

Alloy

% AI (post-pre-aging)

f/

(m3/s)

A B C --

5.704 5.710 5.757 6.350

0.0245 0.0255 0.0332 0.1295

126.22 57.40 28.81 1.89

the tendency of k ( f ) to decrease with increasing f~, is a real phenomenon.

3.2. Measurement of solute concentration

3.1. Kinetics of coarsening Conventional plots of (a/2) 3 vs t, where (a/2) is the average particle "radius", are shown in Fig. 2; experimental values of k ( f ) were determined from the slopes of the curves. The values of k ( f ) are listed in Table l, where they are seen to decrease systematically by more than a factor of four over a range of f~, that increases by less than 40% (the measurement off~, is discussed in Section 3.2). A value of k ( f ) of approximately 2.9 x l0 3°m3/s at 630°C was estimated by interpolation from the large volume of data on 7' coarsening in N i - A l binary [2, 3]. The measured values o f k ( f ) are larger than this by factors ranging from 10 to 40. These extraordinary results show a clear increase in the rate constant by coarsening with decreasing volume fraction of precipitates. This is quite contrary to the predictions of all the theories of coarsening. By way of confirming the unexpected results described above, several aging experiments were performed on the Ni-A1 alloy containing 6.35 wt% AI that was used previously by Ardell and Nicholson [4]. Samples were aged at 630°C in the same salt baths as the other samples, the only difference being that there was no pre-aging treatment. The rate constant for coarsening obtained from this experiment, 1.89 x 10 3°m3/s (Table 1), is very close to the value 2.9 x 10-3°m3/s expected from interpolation of the data in the literature. This result helps confirm that

Representative magnetic transition curves for the three alloys at different aging times are shown in Fig. 3. In the early stages of aging the transition from paramagnetic to ferromagnetic behavior is very sharp, occurring over a narrow range of temperature (all the curves on the left in Fig. 3). Observation of the microstructures of these alloys in the TEM indicated the complete absence of homogeneously nucleated 7' precipitates. Further aging, however, marked the appearance of a "knee" in all of the magnetic transition curves. Examples of this behavior in various stages of development, which was accompanied by the appearance of homogeneously nucleated 7' particles in the microstructure, are shown in Fig. 3.

268 h

363 h

(a) I

~

171h

I

,

I

i

I

268h

8 t-

(b)

12

I

w IO

f

E

a Alloy A

8

A Alloy B

o 6

o AlloyC

54h

/ / /~

°~

~

~2 (c) I

200 5

10

15

20

25

t x I O"s is)

Fig. 2. Plots of the cube of the average particle sizes, (a/2) 3, vs aging time, t, for the three alloys.

~

I

I

250 2,50 300 Temperature (K)

I

350

Fig. 3. Magnetic transition curves for samples of alloys A, B and C, respectively, in (a), (b) and (c). The aging times are indicated.

MAHESHWARI and ARDELL:

ANOMALOUS COARSENING OF PRECIPITATES

Separation of the magnetic transition curves in two parts, as seen in Fig. 3, has been observed and analyzed by Yellup and Parker [10]. They showed that this feature indicates the presence of a bimodal distribution of precipitate phase in the matrix. In our case the bimodal distribution consists of the homogeneous ,;' precipitates in the matrix, surrounded by the 7' particles that nucleated heterogeneously on the dislocations and grain boundaries. The composition of the matrix associated with each distribution differs, hence the separation of the magnetic transition curve into two parts. When this happens the matrix is characterized by two values of Oc, and the value of w of interest in this investigation, i.e. the value of w in equilibrium with a homogeneously nucleated 7' precipitate of average size, can be difficult to determine, depending on the degree of separation of the curves. For example, in Fig. 3(a, b) there is significant separation and the two values of Oc are relatively easy to measure: the lower value of Oc corresponds to w. In Fig. 3(c), however, the separation is indistinct, and w is uncertain. Unfortunately, the quantity of data that we were able to obtain from curves such as those in Fig. 3(a, b) was restricted because the homogeneously nucleated precipitates were consumed by those that nucleated heterogeneously during pre-aging. The variation of our best estimates of w as a function of aging time, plotted in the form w vs t - 13 in accord with the predictions of theory [2], is shown in Fig. 4. The pre-aging treatment lowered the concentrations of AI in alloys A, B and C to 5.704, 5.710 and 5.757 %AI, respectively. These values, which are also presented in Table 1, correspond to the plateaus in Fig. 4, and w represents the concentration of the matrix prior to the appearance of homogeneously nucleated 7' precipitates. Despite the uncertainty in measuring w when magnetic transition curves start to separate, there is no doubt that w begins to drop sharply as the homogeneous 7' precipitates nucleate and coarsen. The plot of w vs t 1,3 should be linear at long aging times, and should extrapolate to the same value of We at infinite aging time (which denotes the equilibrium matrix composition), since all the aging experiments were performed at the same temperature. However, since the data at longer aging times were limited by the disappearance of the homogeneous 7' precipitates, it was not possible to obtain reliable values of w~ from data on the doubly aged samples't. Since accurate estimates of f., are crucial for interpreting the data generated in this study, we obtained the necessary value of we ( = 5.555% A1) from a plot of w vs t ],3 for the alloy containing 6.35% AI investigated in earlier work [4, 7], but aged in this case at tThe values of f reported by us previously [9] were, in fact, calculated by the lever rule using values of we determined from plots of w vs t-In extrapolated to t ] ~- 0. This was the best we could do at the time.

2665

5.8 oO0

5.7

~[~6

5.6

o

0

0

0

o

8

B

a Alloy A

r'l Alloy B

5.5

o Alloy C 5.4 0

' l l "1* i '

4

I l 2i ' 6*

I t8 J I t lO I * ' 1 12

t-1/3 x 10 2 (s't/3)

Fig. 4. The variation of the wt fraction A1 in the matrix, w, as a function of t ],3, where t is the total aging time. 630=C. The values off.. are reported in Table l: they are only slightly smaller than those we reported previously [9]. They are also smaller than those reported [9] as having been measured using TEM, but we believe that the volume fractions measured using the TEM serve mainly as a rough check of the consistency of the data. 3.3. Particle size distributions Figure 5 shows the size distributions of the 7' precipitates in all three alloys. The distribution function predicted by the theory of Lifshitz and Slyozov [11] and Wagner [12] (LSW) has been superimposed on the curves for comparison. The number of distributions available for Sample A was restricted to only three because data on only three aging times were available before the homogeneous coherent particles completely disappeared from the matrix. A few of the distributions are broader than the typical result, but there is no correlation with either volume fraction or aging time. These results are in general agreement with earlier observations [13, 14]. 4. DISCUSSION

It has been noted previously [2, 3] that there are other data in the literature suggesting that k ( f ) decreases with increasing f.. in Ni-A1 alloys. This trend is present in the data of Hirata and Kirkwood [5], Ardell and Nicholson [13], Kirkwood [15], and Irrisari et al. [16], whose data are reproduced in Fig. 6. Here k ( f ) is normalized by k(0) (the rate constant predicted by the LSW theory), using previously calculated values [2]. This trend had been noted earlier [2, 3], but the possibility that it might be real was dismissed and attributed to experimental uncertainty because the data were obtained at several different temperatures, rather than at a single temperature, and the absolute values off,., are not known with precision. In light of the present results, however, it appears that the observed trends are very real, and must now be taken seriously. These results provide additional evidence that at small volume fractions, the coarsening of ),' precipitates in Ni AI

2666

MAHESHWARI and ARDELL: ANOMALOUS COARSENING OF PRECIPITATES n=22]

n=80

n=98

. . . .

n 126

L.,-'J,. ~ ,

n

=

= 181

=

I;

,;

•

.i

.

i

.

J

.~

'~i~I I~. 3"01~n'333 1.5~" i.~l/~, In'92 ~ /'i ~ I 0.5 0

•

.f,.% i

~.¢

Alloy B

~ j

n.2~61

ml.t'%

.~

.~

.

,

n=252 /~ ,

. i

~

,

n-22F1~

n-208 •

Alloy,

. ,

0 0,4 0.81.21,6 2,0 U

Fig. 5. The distributions of particle sizes for all the samples investigated. The distribution function, g(u), where u = a/(a ~, is normalized to unity. The theoretical distribution of the LSW theory is superimposed for comparison, and the aging times and numbers of precipitates measured, n, are indicated. alloys is characterized by a rate constant that decreases with increasing volume fraction. It is important to demonstrate that the anomalous coarsening kinetics reported herein are a true phenomenon and not somehow a consequence of the

(a)

4.0

~

3.0

I

"',..,..m

'*'"""'........ • i .........,,•. .

....................

2.0I ] 0/

I

0

(b)

22I

'!

,

0.05

o

1.8

1.4 1.2

1.0

I

,

O. lO

I

,

0.15

f-c

I

0.20

o Hirata and Kirkwood [5] • Kirkwood [15] [] Ardell and Nieholson [13] • Irissariet al. [16]

2.0

1.6

o ................5-

[] []

',,,.: 0"-.. ".... 0

0 "" . . . . . . . . . . .

•

I

0.02

0.04

0.06

Fig. 6. The variation of the rate constant for coarsening,

k(f), with volume fraction of the ?' precipitate, fr', for several sets of data taken from the literature. The data of Hirata and Kirkwood [5] and Kirkwood [15] are shown in (a), and the data of Ardell and Nicholson [13] and Irissari et al. [16] are shown in (b). The individual values of k ( f ) are normalized by k(0), the rate constant of the LSW theory.

bimodal distribution of 7' precipitates produced by the pre-aging treatment. The most convincing evidence that this is so is provided by the results of Chellman and Ardell [17], who investigated the kinetics of coarsening in concentrated Ni-A1 alloys which were doubly aged to produce two size classes of homogeneously nucleated 7' precipitates. The alloys were aged first at 800°C for several different aging times, producing large (Class I) precipitates, then re-aged at 600°C to produce small (Class II) precipitates. The volume fractions of 7' in both size classes were considerably larger than those in this investigation. The kinetics of coarsening of the Class II particles at 600 ° were unaffected by the presence of the large Class I particles produced during aging at 800°C and were also in excellent agreement with the kinetics normally observed in conventionally aged Ni-AI alloys. The measurements in this work were made on the equivalent of Class II precipitates, and we are therefore confident that their coarsening kinetics are completely unaffected by the heterogeneous 7' precipitates introduced during the pre-aging treatment. We believe that elastic interactions among neighboring particles must be responsible. There are theoretical calculations which predict that elastic interactions stabilize the array of 7' precipitates against coarsening [18, 19], but until now the principal experimental manifestation of this prediction is the observation that at yet larger values off~, there is no effect of volume fraction on k(f)[3]. The implication here is that elastic interactions retard the kinetics in such a way that the expected accelerating effect of volume fraction is nearly exactly compensated. At lower volume fractions, however, elastic interactions overpower the expected effect of volume fraction and cause k ( f ) to decrease with increasing f~,. As we have recently suggested [9], elastic energy plays another crucial role in the coarsening behavior

MAHESHWARI and ARDELL:

ANOMALOUS COARSENING OF PRECIPITATES

of 7' precipitates. It influences their equilibrium shape, enabling them to acquire concave interfaces at large sizes ( > 50 nm). The volume fraction is important in this process because when it is too large the particles interact elastically, leading to rafting. This in turn prevents individual precipitates from growing to large enough sizes to acquire concave interfaces. None of the theories of coarsening includes the effect of elastic energy on the equilibrium shape of the precipitates and the way this influences the concentration of solute at the particle-matrix interface. At this stage we can only speculate that this contributes to the observed effect of volume fraction on the kinetics of coarsening, and note that it is yet another factor that must be included in a comprehensive theory of coarsening. It would appear surprising that the effect off., on k ( f ) is so large over the relatively small range off~.. investigated. It must be realized, however, that the energy of elastic interaction is very strongly dependent on the center-to-center distance, R, between the precipitates, varying approximately as R 6 [4]. A rough estimate using the calculations of Bansal and Ardell [20] indicates that the interaction energy would nearly double over the range o f f , encountered in this investigation. It is also important to bear in mind that the greatest uncertainty in our experiments involves our estimates of the absolute values off., reported in Table 1. We are confident thatf., increases from alloy A to B to C, but cannot vouch for the absolute accuracy of the values measured. 5. CONCLUSIONS Despite the difficulty in determining f,, with the precision that we would like, the major conclusion of this research is inescapable. At small values off~., in the approximate range 0.0245 to 0.0332, k ( f ) decreases with increasing f . . This is evident not only from the data generated in this investigation, but must now be regarded as supported by the earlier data of Ardell and Nicholson [13], Kirkwood [15], Hirata and Kirkwood [5] and Irisarri et al., [16]. In the range of volume fractions found in this investigation the k ( f ) is larger than the expected rates at this aging temperature by factors between 10 and 40.

2667

The distributions of particle sizes are for the most part comparable to those measured in earlier investigations. There are some examples of excessively broad distributions at early aging times, but the uncertainties in the statistics are sufficient to eliminate a correlation. It is concluded that elastic interactions among the precipitates play a crucial role in the observed phenomenon. It is suggested that they overcompensate for the expected accelerating effect of the increase in the volume fraction of ~,' precipitates. Acknowledgement--We gratefully acknowledge the financial

support of this research by the National Science Foundation under Grant No. DMR-8901555. REFERENCES

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