Morphology and chemical nanoanalysis of discontinuous precipitation in MgAl alloys—I. Regular growth

Acta metall, mater. Vol. 42, No. 11, pp. 3843-3854, 1994

Pergamon

0956-7151(94)E0155-A

Copyright © 1994ElsevierScienceLtd Printed in Great Britain. All rights reserved 0956-7151/94 $7.00+ 0.00

MORPHOLOGY A N D CHEMICAL N A N O A N A L Y S I S OF D I S C O N T I N U O U S PRECIPITATION IN Mg-A1 ALLOYS--I. REGULAR GROWTH D. DULY, M. C. C H E Y N E T and Y. B R E C H E T Laboratoire de Thermodynamique et Physico-Chimie M&allurgiques, ENSEEG, B.P. 75, 38402 Saint Martin d'H6res, France (Received 13 September 1993; in revised form 21 January 1994)

Abstract--Discontinuous precipitation in Mg-A1 alloys has been studied both by conventional TEM (morphology) and STEM with local chemical analysis (concentration profiles). Numerous growth defects have been identified. In the case of regular growth, a detailed analysis of concentration profiles using Cahn's solution has been performed, proving the necessity to introduce three velocity scales in order to describe the overall kinetics. The thermodynamical balance of the phenomenon has been derived by applying Hillert's model. R6sum6--La pr6cipitation discontinue dans les alliages Mg-A1 a 6t6 &udi6e ~ la fois par microscopic 61ectronique ~. transmission conventionnelle pour ce qui est de leur morphologie et par microscopie 61ectronique ~ transmission et ~. balayage analytique pour ce qui est des profils de concentration. De multiples d6fauts de croissance ont 6t6 identifi6s. Dans le cas de la croissance r6guli6re de lamelles parall61es, une analyse d6taill6e des profils de concentration par l'6quation de Cahn a montr6 la n6cessit6 d'introduire trios 6chelles de vitesse. Le bilan thermodynamique de la r6action est 6tabli en utilisant le mod61e de Hillert.

1. INTRODUCTION In discontinuous precipitation, also known as cellular precipitation, a supersaturated ~t' phase decomposes into a new fl phase, and an ~t phase structurally identical to a ' but depleted in solute. The transformed zone is separated from the original phase by an interface called the reaction front in which all the diffusion processes that are involved in the reaction take place. In the regions which have been swept by the interface, the reaction is over and ~ ' has been replaced by alternating plates of ~ and fl phase lying parallel to the growth direction. As the diffusion coefficient in the reaction front is finite, the ~t phase is still supersaturated at the end of the reaction, Most theoretical and experimental works on discontinuous precipitation have concentrated on the kinetics of transformation for one nodule, i.e. on the steady state growth process [1-11]. This process is generally modelled as the propagation of the reaction front at a constant velocity in a direction perpendicular to itself, the interlamellar spacing being constant in the whole nodule (Fig. 1) [1-9].The surface tensions at the triple junction between ct, fl and ~ ' are assumed to be at equilibrium, and the contact angles 0 ~ and 0 ~_ between the ct/ct' interface and the ~/fl interface are equal on both sides of the ~lamellae (as are the two contact angles 0 ~+ and 0 ~_). There have been several attempts to relate the speed v of a reaction front, the interlamellar spacing S and the

solute content in the daughter phase. In a classical paper, Cahn established the bases of the methodology which has been adopted by all other workers: (i) A diffusion equation is written and solved analytically. Cahn has shown that when the reaction front is planar and moves in a direction perpendicular to itself at a constant speed v (cf. Fig. 1), the solute content in ct, C ~, is given by (x/~) ch x C ~(x ) = Co - (Co - C ~/~) -~ (1) (_~_~2 a) ch where x is the distance from the middle of the ~tphase, Co is the initial solute content, C ~:# is the solute content in ~ close to t , S ~ is the distance between the two fl lamellae, and a is defined by v (S ~)2 a - KD8 6

(2)

where K is the segregation ratio for the solute between ~ and the grain boundary, D B is the grain boundary diffusion coefficient for the solute, and 6 is the grain boundary width. In most theories, C ~/# is related to the equilibrium concentration corrected by the Gibbs T h o m s o n equation (capillarity effect) [2, 4]. But as pointed out by Hillert, C ~/a is likely to depend on kinetic considerations [5, 6].

3843

3844

DULY

et al.:

DISCONTINUOUS PRECIPITATION IN Mg-A1 ALLOYS--I

t

~

.

.~/ '

~ Y 0

2. E X P E R I M E N T A L M E T H O D S

a' ~

T;l

2.1. Sample preparation and experimental techniques

~ ~

x

~x sa Fig. 1. Schematics of a precipitation front showing the contact angles and surface tensions,

(ii) A global or local thermodynamical equation iswritten, which determines v as a function of S. At this stage, it appears that several interlamellar spacings and speeds could be adopted by the system: the solution is degenerated, (iii) A criterion is then proposed, in order to select the speed among the possible ones. This criterion traditionally corresponds to the optimisation either of the rate of growth or of the rate of entropy production, but no convincing derivation exists. Such approaches have been discussed by Sundquist, who has postulated that a whole range of interlamellar spacings may coexist in a nodule and that the largest spacing encountered should be twice as large as the smallest one [10]. A similar conjecture has been proposed for eutectic structures [12, 13]. Recently, B6gel and Gust have suggested that the grain boundary does not progress regularly but advances via a stop and go motion [11]. Though discontinuous precipitation is a very common phenomenon, and though the growth of a nodule has been the subject of much experimental work, it has not been possible so far to discriminate among all the models. STEM measurements of the concentration profiles between two neighbouring fl lamellae carried out by Porter and Edington in Mg-A! [14, 15], and by Solorzano in AI-Zn [9, 16], have shown that Cahn's diffusion equation is well obeyed. However, the direct test of the thermodynamical equations proposed by Cahn, Hillert, or B6gel and Gust could not be attempted from their data, and it was not possible to determine whether C ~/p depends on thermodynamical or kinetic considerations. The scope of these two papers is to re-address these two problems by studying the growth kinetics of a precipitation nodule in Mg-AI. The main morphological characteristics of discontinuous precipitation in this system as deduced from TEM investigations are first described. The first paper deals with the kinetic study of the migration of the reaction front when lamellae grow parallel. In the companion paper, we consider the case of irregular growth, when a fl lamella changes its habit plane and we show that the morphological changes can be correlated with the shape of the concentration profile between the fl lamellae.

Magnesium alloys with 7.7at.% A1, 10 at.°/o AI and 18.8 at.% A1 were provided as 16 mm diameter bars by Pechiney. These alloys, previously melted using conventional alloying techniques, had been extruded at 250°C under a pressure of 250 bars. The alloys are 99.1% pure, and the principal impurities are Mn (480 ppm), Si (350 ppm) and Fe (30 ppm). The materials were solution heat treated for 128 h at 420°C. After this treatment, the average grain diameter was 95 pro. At this stage, Electron Probe Microanalysis (EPMA) on a polished sample of an alloy with 7.7 at.% A1 showed that the Mg-AI solid solution was approximately homogeneous, local variations of the A1 solute content around the average value being less than 0.3 at.%. Samples were then aged for 16 h at 22°C in order to yield the fl precipitation and then water quenched to room temperature. Samples for TEM and STEM observations were first thinned mechanically to a thickness of approx. 60 #m and then ion thinned at low temperature. Conditions for thinning were an accelerating voltage of 4 kV and an ion current of 1 mA. The conventional TEM observations were carried out on a Jeol 200 CX "side entry". Nanoanalysis measurements were performed with a STEM with field emission gun (VG HB 501.) equipped with an X-ray detector and an energy loss spectrometer which are both connected to a Tracor Northern 5400 computer using commercial softwares for data processing. As the K~ X-ray signals for A1 and Mg are strongly convoluted, the solute concentration in the or-phase was determined by using electron energy loss spectroscopy (plasmon losses). Calibration using several alloys of known composition showed that if the AI concentration is lower than 15 at.%, the volume plasmon loss energy can be well approximated by AEp--kCAl+

k'

with k = 6.2 _ 0.4 eV and k ' = 10.29 _ 0.02 eV. For each experimental point, the plasmon loss spectrum was recorded between 0 and 50 eV, corresponding to an acquisition of 0.048 eV per channel. The typical standard deviation calculated from the measured energy of the reference alloys is equal to 0.024 eV, corresponding to a standard deviation for aluminium concentration of 0.39 at.%.

2.2. Statistical analysis of the concentration profiles In order to determine whether an experimental concentration profile closely follows Cahn's diffusion equation, it is necessary to define a quantitative criterion. For each concentration profile it is possible to determine by iteration the couple (C ~/~,~/a) that minimizes the quantity tr (C) =

I~ 1

~[Ci--Ccal¢(Xi)]2]112

i=

(3)

DULY et al.: DISCONTINUOUS PRECIPITATION IN Mg-A1 ALLOYS--I where x i is the position of the i th point where the AI concentration Ci is measured, and n is the n u m b e r of measurements. The calculated concentration Ccalc is given by Cahn's solution [equation (1)]. The uncertainty due to the measurement of the concentration can be estimated from a reference sample to 0.39 at.%. The uncertainty on the position is of the order of 0.01S ~ which corresponds to an uncertainty of 0.1 at.% in the concentration, Therefore the error bars are about 0.40 at.%. We conclude that Cahn's equation is obeyed when tr(C) is less than 0.40at.%.

3845

3. MORPHOLOGICAL FEATURES 3.1. Different morphologies o f a precipitation nodule

In the Mg-A1 alloys, as in most alloys that exhibit discontinuous precipitation, the aspect of a precipitation nodule never quite resembles the "theoretical one" described in Fig. 1. Figure 2(a) gives an illustration of the different possible morphologies of a domain. The initial position of the reaction front is marked by a string of spherical precipitates. Inside the precipitation nodule, zones in which the lamellae are almost parallel are visible (zones A and B). According to theoretical works, these zones

Fig. 2. Electron microscopy images of the different morphologies observed in a discontinuous precipitation nodule: (a) SEM image of a nodule with zones in which the lamellae grow parallel (zones A and B) and with bush-like zones (zone C). A string of spherical precipitates marks the initial position of the reaction front; white arrows mark the final position. (b) TEM image at high magnification, showing numerous instabilities are revealed in zones with parallel lamellae. (c) SEM image of a typical bush-like precipitation nodule. On (a) and (c), note that the lamellae are perpendicular to the reaction front.

3846

DULY et al.: DISCONTINUOUS PRECIPITATION IN Mg-A! ALLOYS--I Table 1. Differenthabit planes and associatedmorphologiesencounteredfor discontinuous precipitation in Mg-A1 Epitaxial plane for Epitaxialplane the precipitate for the matrix Morphologyof the precipitates El (01l)fl (0001)~t Long parallelprecipitates E2 (2T~)fl (01I~)~ Long parallelprecipitates (231)3 (0112)ct E3 (ll~)/~ (011])ct In bush like precipitationonly (121)/~ (0113)a E4 (l]])fl (01"f])~t In bush-likeprecipitationonly (132)fl (0115)~ E5 (10T)fl (01Tl)~ In bush-likeprecipitationonly (110)# (01Tl)~ E6 (31'~)fl (03~1~) In bush-likeprecipitationonly E7

(32"I')fl

(03~I)~

(21T)fl

(01"i'0)ct

correspond to a steady state growth process. However, their observation at a higher magnification reveals the existence of numerous defects, such as the appearance or disappearance of lamellae [Fig. 2(b)]. Occasionally, the steady state growth process is interrupted, and lamellae stop growing parallel, leading locally to a bush like (or tree like) morphology. This situation is illustrated by the zone labelled C in Fig. 2(a). In some nodules, stationary growth never takes place and the whole nodule exhibits a "bush like" morphology, as the lamellae immediately begin to diverge [Fig. 2(c)].

3.2. Crystallographic characteristics However, it appears that even in nodules with a "bush like" morphology, precipitates do not grow randomly but, on the contrary, satisfy well-defined epitaxial relationships. The different relationships that we have identified are summarized in Table 1. In nodules where the lamellae are parallel, the habit plane is most frequently the matrix basal plane (E~) and the crystallographic orientation relationship is that of Burgers, i.e.

In bush-likeprecipitationonly

A more complete description of the morphological and crystallographic features of discontinuous precipitation can be found elsewhere [17].

3.3. Morphology of the reaction front The reaction front has been observed for samples of the Mg-7.7 at.% AI alloy annealed at 220 and 140°C. In order to avoid the coarsening of the fl lameliae tips and to limit the displacement of the reaction front during quenching, care has been taken in order to quench the samples in less than 1 s. As the average migration speed of the reaction front is about 15nm/s at 220°C ([18]; and see also companion paper) its observed position corresponds to that during the precipitation reaction within an accuracy of + 15 nm.

[1T1]fl l l[2IIO]~t (o 1 l)~//(ooo l )~. Occasionally, parallel arrays of lamellae are found to grow on the (231)////(01T2)~ crystallographic planes (E2). They then exhibit the orientation relationship of Potter, i.e. [1T1]////[2IT0]~ (011)//2.4 _+ 0.2 ° from (0001)~ i.e. (231)fl//(Ol'f2)~. Other habit planes (E3-ET) are found only in nodules that exhibit a bush like morphology and the interface expands only along short distances along these planes. It appears that the habit plane is always a plane of high atomic density for the precipitate and a relatively low-index plane for the matrix. Such a parallelism is possible as t h e / / l a t t i c e rotates slightly each time the epitaxial plane changes,

Fig. 3. Morphology of the reaction front at 220°C in annular dark field image (STEM).

DULY et al.: DISCONTINUOUS PRECIPITATION IN Mg-A1 ALLOYS--I On the annular dark field image shown in Fig. 3, /3 lamellae are brighter than the ct matrix because their A1 content is much higher. The reaction front is also brighter than the ct phase, which indicates that Ca, the solute content in the boundary, is larger than C ~ and C 0. The segregation of AI in the grain boundary is thus positive, i.e. the segregation coefficient Kis larger than 1. The observation at low magnification shows that the reaction front is rarely regular and that it exhibits steps whose height can be as large as 0.1/~ m. The number of lamellae between two steps is of the order of 20-30. Between two steps, the grain boundary is much more regular (it is concave almost everywhere): the majority of the contact angles 0 ~ and 0 ~_ are equal to 45 + 5° (Fig. 3).

3847

The growth of the lamellae is affected by many defects. These may be classified in three categories:

the Potter's to the Burgers' ones, has been observed several times. The reverse change was seldom observed. Such collective deviations are much less frequent than the change of epitaxial plane for only one lameila (see companion paper). This induces a rapid variation of the interlamellar spacing in the neighbourhood of the defect as growth proceeds: generally this defect will be followed by the nucleation or the stoppage of a lamella [defect (ii)]. (ii) The variation of the total number of lamellae: the stoppage of the growth process for one lamella takes place when the interlamellar spacing becomes too small [Fig. 4(a)]. Conversely, when the spacing becomes too large, a new /~ lamella nucleates. In precipitation that results in a bush like morphology, the precipitates generally diverge and this phenomenon thus tends to occur very frequently [Fig. 4(b)]. The nucleation of a precipitate of the fl phase can be assimilated to the branching of an ct lamella. The branching of a ~ lamella can also be observed at all

(i) The change in habit plane of one or several lameUae: the collective change of habit plane from the ct pyramidal plane to the basal plane, with the corresponding change of the orientation relationship from

temperatures between 140 and 300°C, though it is much less frequent. Branching can affect simultaneously several lamellae and one lamella may exhibit several branchings [Fig. 4(d)].

3.4. Different types of growth defects

m ~

w

¸

Fig. 4. Bright field images of the different types of growth defects which affects the number of lamellae: (a) stoppage of growth and disappearance of one lamella; (b) nucleation of new lamellae during growth (white arrows); (c) sympathetic nucleation associated with a change of habit plane; (d) simultaneous branching of several lamellae, and a lamella exhibiting several branchings,

3848

DULY et al.: DISCONTINUOUS PRECIPITATION IN Mg-A1 ALLOYS--I

(iii) The occurrence of sympathetic nucleation: this corresponds to the nucleation of a new lamella on a preexisting one [Fig. 4(c)]. The morphology obtained is very different from that which results from branching:

4. CHEMICAL ANALYSIS OF PARALLEL ARRAYS OF LAMELLAE 4.1. Shapes o f the concentration profiles Six different precipitation nodules labelled A.! to A.VI in the M g - 1 0 a t . % A1 alloy, and one nodule

• sympathetic nucleation does not modify the total number of lamellae, but it can be assimilated to the stoppage of the growth process of a lamella and the simultaneous nucleation of a new precipitate; • when branching takes place, the growth plane remains unchanged. Sympathetic nucleation leads to a change in the epitaxial and orientational relationships. In Fig. 4(c), the initial lamella obeys the orientational relationship of Potter and therefore grows in the pyramidal plane of the matrix. The fl crystal which has nucleated sympathetically follows the orientational relationship of Burgers, and lies in the ~t basal plane,

labelled B.I in the Mg-18.8 at.% Al alloy, exhibited parallel arrays of lamellae and were therefore chosen for further investigations. The concentration of aluminium in the , ' phase ahead of the boundary was first checked for all nodules. When the distance perpendicular to the reaction front was larger than 5 nm, the Al content in the ~t phase was equal to the nominal concentration Co. This shows that there is a negligible volume diffusion ahead of the reaction front. In each nodule, concentration profiles were measured in several neighbouring a lamellae or in a given ~ lamella at different distances from the reaction front. The values of C ~/~ and x ~ were determined using the procedure described in Section 2 and are given in Tables 2 and 3. As shown in both Tables 2 and 3, Cahn's solution of the diffusion equation is well obeyed between parallel lamellae, since ~ (C) is significantly largerthan 0.40 at.% for only seven concentration profiles out of 20. When equation (1) applies, the values of C ~/° and x/~ do not depend on the distance

3.5. Interlamellar spacing We have shown in the two preceding paragraphs that even in domains where the lamellae grow parallel, it is difficult to define a perfectly steady state growth process, due to the numerous growth defects and the lack of regularity of the reaction front,

Another deviation from the stationary growth state described in the models is the possibility for the interlamellar spacing to vary considerably within a single precipitation nodule. We have observed by SEM, in a sample of Mg-7.7 at.% AI annealed at 220°C, different nodules where lamellae were parallel. In each nodule, we measured the ratio S / S o for several lamellae, where S is the interlamellar spacing and S Ois the average value of S in the nodule. We then assembled all the data in a single histogram. This procedure has two main advantages:

from the reaction front. For instance, the concentration profiles measured in the a lamella B.I.1 at distances of 200 and 380nm from the reaction front are identical, with a = 1.85_+0.1 and C ~j~= 3.0 _+ 0.2 at.% (Fig. 6): this shows that bulk diffusion below the reaction front is also negligible. When the concentration profiles are not described by equation (1), the deviation may have three different origins: • C ~/~is not the same on both sides of the ~t lamella,

• the ratio S I S o is the same, whether the lamellae are viewed in edge on or not; • the measurements obtained for different nodules are normalized and can thus be reported in a single histogram. Figure 5 shows the histogram obtained from a study of 18 nodules which together contained more than 950 lamellae. It has a gaussian shape with an average value of 1 and a standard deviation of 0.2. In a precipitation nodule, S may thus essentially vary between 0.6 So and 1.4 S o. Other measurements at different temperatures between 140 and 300°C showed that the limits of this relative interval do not depend on temperature. This existence of a factor close to 2 between the smallest and the largest interlamellar spacing is in agreement with Sunquist's theory (which allows for a band of possible spacings, the lower bound being given by energetic reasons and the upper bound by stability considerations) [10].

25

v-

[

! 20 ~

15 -

~ 10 u. ~ 5 0

IL

i 1.0 1.5 s So Fig. 5. Histogram giving the frequency of the interlamellar spacing S divided by the average spacing in the same nodule So. In this histogram, data corresponding to 12 nodules and 950 lamellae has been gathered. 0.5

DULY

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D I S C O N T I N U O U S P R E C I P I T A T I O N IN Mg-AI A L L O Y S - - I

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Table 2. Results of the local chemical analysis for the Mg-10 at.% A1 alloy, for different nodules (A.I-A.VI) and different lamellae inside a given nodule (zones 1 to 18) Nodule ~ zone No. Sa (nm) C ~/p (%at) v/-a ~r(C) (at.%) A.I 1 1220 5.05 + 0.2 2.3 +_0.3 0.77 A.I 2 960 4.8+0.2 2.4+0.15 0.30 A.I 3 520 3.9 -+ 0.25 2.15 + 0.2 0.40 A.II 4 935 4.05 + 0.4 2.05 +_0.3 0.65 A.II 5 525 3.8+0.15 2,3_+0.1 0.26 A.II 6 465 4.35_+0.2 2.1 +0.15 0.49 A.III 7 420 4.2 -+ 0.2 2.0 -+ 0.15 0.30 A.III 7 420 4.2 -+ 0,2 2.0 -+ 0.15 0.32 A.IV 8 395 3.65 -+ 0.3 2.35 -+ 0.2 0.41 A.IV 9 395 3.95 -+ 0.2 2.15 -+ 0.2 0.42 A.V 10 455 4.3 -+ 0.3 1.9 -+ 0.3 0.48 A.V 11 630 4.3-+0.2 2.2-+0.2 0.29 A.VI 12 290 3.95 -+ 0.1 2.35+__0.1 0.38 A.VI 13 445 3.9-+0.15 2.15-+0.1 0.32 A.VI 14 680 4.25-+0.1 2.1 -+0.1 0.20 S ~ is the spacing between two /~ precipitates. C ~/~ (the solute concentration at the interface) and x/~ (Cahn's parameter) are calculated from the best fit of a profile to Cahn's solution, o (C) represents the accuracy of the agreement [see equation (3)].

b u t still t h e a s y m m e t r i c a l v e r s i o n o f C a h n s o l u t i o n is fulfilled a n d e q u a t i o n (1) m u s t be r e p l a c e d b y ~(x) C ~

= Co- (C0-

\

C ~(x = + 0 . 5 S ~) + 2

C~(x

= - 0 . 5 S ~) '~ J

( x//~ x ) × ch

~-~ ( 4 )

C ~(x = + 0 . 5 S ~) 2

x / ~ varies b e t w e e n 1.9 a n d 2.65 for all the profiles measured in b o t h the Mg-10at.% A1 a n d M g - 1 8 . 8 a t . % AI alloys. T h e a v e r a g e v a l u e o f C ~/~ d e p e n d s o n the initial solute c o n t e n t : it is a b o u t 4at.% in Mg-10at.% AI and 5at.% in M g - 1 8 . 8 a t . % A1 a n d is t h u s h i g h e r t h a n the t h e r m o d y n a m i c e q u i l i b r i u m v a l u e C~ = 3.1 a t . % . W e c o m p a r e d these values w i t h t h o s e o b t a i n e d by

C~(X

= - 0 . 5 S ~) ] /

sh (v/~ x ) ×

T h e last t w o types o f defects c a n n o t be a c c o u n t e d for by a s t r a i g h t f o r w a r d m o d i f i c a t i o n o f C a h n " s e q u a t i o n (in c o n t r a s t w i t h the first t y p e o f d e v i a t i o n ) .

4"2" Values °f the parameters x/a and C~l~

ch

4

• the c o n c e n t r a t i o n profile is n o t s y m m e t r i c a l a n d C ~ is n o t a m a x i m u m for x = 0 , T h u s the c u r v a t u r e C~(x) mayn°t bethesame°nb°thsides °fthe m a x i m u m [Fig. 7(c)].

Porter et al. in Mg-9at.% Al at different tempera(4)

tures [14, 15]. W e h a v e a n a l y s e d their e x p e r i m e n t a l profiles u s i n g t h e m e t h o d d e s c r i b e d in Section 2.2. F o r T < 260°C, their ratio C ~/~/Ceis a b o u t 1.2, w h i c h

T h i s s i t u a t i o n is e n c o u n t e r e d in t h e ~ l a m e l l a A . I V . 8 [Fig. 7(a)]: t r ( C ) is e q u a l to 0.41 a t . % a n d to 0.24 a t . % w h e n e q u a t i o n s (1) a n d (4) are used to fit t h e e x p e r i m e n t a l d a t a respectively:

is in g o o d a g r e e m e n t w i t h t h e v a l u e o b t a i n e d in s a m p l e A, B u t their values o f v / a are smaller t h a n o u r s at all t e m p e r a t u r e s except at 212°C, w h e r e they o b t a i n x / ~ = 2.15 for the l a m e l l a 11.2. W h e n t h e /3 lamellae g r o w parallel a n d w h e n C a h n ' s diffusion e q u a t i o n is o b e y e d , t h e values o f x / ~ a n d C ~/~ d o n o t c h a n g e d u r i n g g r o w t h , w h i c h p r o v e s t h a t v o l u m e diffusion is n o t efficient in e l i m i n a t i n g the residual s u p e r s a t u r a t i o n . T h e C ~/~ value is generally

(~2a) sh

• the AI c o n c e n t r a t i o n a b o u t x = 0 ( h a l f w a y b e t w e e n t w o /~ precipitates) is larger t h a n t h a t o b t a i n e d f r o m t h e regression, i.e. t h e c o n c e n t r a t i o n c u r v e h a s a " h a t like" s h a p e [Fig. 7(b)];

Table 3. Results of the local chemical analysis for the Mg-18.8 at.%AI alloy, for different nodules (A.I-A.V.I) and different lamellae inside a given nodule (zones 1 to 18) Nodule ct zone No. Y (nm) S ~ (nm) C~/~(at.%) x/a-~ a(C) (at.%) B.1 1 -- 380 550 3.0 _+0.2 1.85 + 0.1 0.39 B.I I -200 550 3.0 ± 0.2 1.85 + 0.1 0.39 B.I 2 -350 I100 3.65+0.3 2.1 +0.1 0.58 B.I 2 -175 1115 3.9+0.4 2.1 +0.15 0.84 B.I 3 - 120 480 3.3 + 0.35 2.2 -+ 0.1 0.65 S ~ is the spacing between two fl precipitates. C ~/a (the interface concentration) and x/a(Cahn's parameter) are calculated from the best fit of Cahn's solution to a profile, a (C) represents the accuracy of the agreement [see equation (3)].

3850

DULY

DISCONTINUOUS PRECIPITATION IN Mg-A1 ALLOYS--I

et al.:

~ -

(b) T 9

-0.5 -0.4 -0.3 -0.2 -0.1

0

0.1

0.2

0.3 0.4 0.5

-0.5 -0.4 -0.3 -0.2 -0.1

x

--

0

0.1

0.2

0.3 0.4 0.5

x

Sa Sct Fig. 6. In the Mg-18.8 at.% A1 alloy, the concentration profiles between two parallel lamellae but at different distances from the reaction front: (a) at 380 nm, (b) at 200 nm, are well described by Cahn's equation, with the same values for C ~/# and ~//-a.

7.5

--

m

(a) _T "[

~

I I I I 13.5 I I I -0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5

~

\

I [ I I 13.5 [ I I I -0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5

xlS a

x/S a

7.5

B

L I I I 13.5 -0.5-0.4-0.3-0.2-0.1

I I I I I 0.1 0.2 0.3 0.4 0.5 xlS a

Fig. 7. In the Mg-10 at.% A1 alloy, different examples of concentration profiles between parallel lamellae showing significant deviation from Cahn's equation: (a) different values of C ~/# on both sides of the at lamella, A.IV.8; (b) "hat like" profile; (c) identical value of C ~/# on both sides of the ~t lamella but asymetrical profile.

DULY et al.: DISCONTINUOUS PRECIPITATION IN Mg-A1 ALLOYS--I Table 4. For the Mg-10at.%Al alloy the value of (a/S ~2) which is proportional to vi,s. is not constant in a given nodule ~t lamella 106a/(S~)2(nm-2) A.I.I 3.6 +_0.9 A.1.2 6.3 + 0.8 A.I.3 17.2 _+3.2 A.VI.12 66.8 _+5.6 A.VI.13 23.4 _+2.2 A.VI.14 9.6 + 0.9

increases. It can be expected that r and S ~ have the same behaviour. Experimentally (see for instance nodule A.I) C ~/a can increase when S a increases. Therefore, the departure of C ~/~ from the equilibrium concentration cannot be explained by the sole consideration of capillarity forces: kinetic effects are dominant, as proposed by Hillert [5, 6].

5.3. Velocity of migration of the reaction front

the same on both sides of a fl lamella. However, it may also present differences larger than 1 at.%. Let us, for instance, consider the fl precipitate that separates the two ct zones A.I.2 and A.I.3: C ~/~ is equal to 4.8 + 0.2 at.% on one side and 3.9 + 0.25 at.% on the other side. 5. INTERPRETATION AND DISCUSSION

5.1. Volume diffusion in front of the grain boundary We have seen that when the aluminium concentration is measured in ~ ', more than 5 n m away from the reaction front, it is equal to the initial solute content Co. The solute content in ~t' would be different from Co if continuous precipitation or volume diffusion took place ahead of the grain boundary. The latter p h e n o m e n o n affects the A1 concentration over a distance 1 = Dv/v where D v is the volume diffusion coefficient of A1 in Mg and where v is the average migration speed of the reaction front. According to Moreau et al.

Dv = Do exp

a - ~

] /

with D O= 12 x 1014nm2/s and Q = 144 kJ/mol [19]. Thus, at 220°C, Dv = 0.68 nmZ/s; it has been shown in a previous work [18] that at T = 2 2 0 ° C in the M g - 7 . 7 a t . % AI alloy, v = 15nm/s. As v increases with the initial solute content, 1 is smaller than 0.045 n m in both the Mg-10 at.% A1 and Mg-18.8 at.% A1 alloys. Volume diffusion is thus negligible ahead of the reaction front. As Hillert points out, this is a necessary condition for Cahn's diffusion equation to be valid [5, 6].

5.2. Does the C ~/~value only depend on thermodynamical considerations? The values of C ~/~ obtained in the Mg-10 at.% AI and M g - 1 8 . 8 a t . % AI alloys are higher than the equilibrium value C~ = 3.1 at.%. The capillarity correction would give [2]

\

C ~/p = C~ exPk(~V~rKl )

3851

Table 4 shows the value of the velocity v determined from the experimental values of a and S ~ using equation (2). In a given precipitation nodule, this velocity may change by an order of magnitude from one lamella to another one. Continuous growth at this speed would lead to a highly corrugated reaction front, with steps several micrometres high. Such steps are not observed. Moreover, the reaction front exhibits steps about 0.1 # m high every 25 or 30 lamellae, which shows that the growth of a whole nodule is not uniform over this distance. These different observations have thus led us to propose that it is necessary to consider three speeds averaged on three different time scales, in order to describe the grain boundary migration: •vins, which describes the growth of a lamella over a distance of the same order of magnitude as 6, and which is represented in Cahn's equation through a and S" [equation (2)]: this is the velocity shown in Table 4. • f, which describes the growth of a lamella over a distance of about 0.1/~ m (typical height of the steps), i.e. about 100 6. • , which is the average rate of growth of a lamella between its nucleation and the final stop of the reaction front. It is also the rate of growth for the whole nodule, averaged over a distance of about 10/~m, i.e. typical grain size. As suggested by B6gel and Gust [ll], it is likely that the migration takes place by a stop and go motion. When the reaction front between two fl lamellae effectively moves, it has the speed v~. However, it remains immobile most of the time, so that its average migration speed on a distance of 100 6 is f.

5.4. Testing of the thermodynamic equations To relate the v speed of a reaction front, the interlamellar spacing S ' , and the concentration C" at the end of the reaction, a thermodynamical equation is needed. Two different approaches, respectively by Hillert [5,6] and Bogel and Gust [11], have been proposed. (i) According to HiUert [5, 6], the thermodynamical

(5)

where y a is the surface energy of the fl/~' interface, Vm the molar volume of the ct and fl phases (equal to 13.6 x 10-6 m3), and r the radius of curvature of the fl lamella at the triple junction between ct, 0t' and ft. According to equation (5) C ~/a should decrease when r

iseqUati°nwhich describes the reaction between ct and ct' v 2), ~VmCOS0 ~ (AG~h~)~ = ~ - ~ - t S~

(6a)

where (AG~h~m), is the average chemical free energy dissipated in the reaction front situated between

3852

DULY

et al.: DISCONTINUOUS PRECIPITATION IN Mg-A1 ALLOYS--I

and ~ ' and where 0 ~= 0 + = 0 ~_. (AG ~h~)a can be calculated from a knowledge of the exact thermodynamic functions [20]; however Hillert's approximation for the average free energy (Ref. [5]) leads to very similar results, but Cahn's approximation (Ref. [4]) is much less accurate for the Mg-A1 system. M ~ is the mobility of the grain boundary between ~ and ~ ' (i.e. v/M ~ is the friction energy of the grain boundary) and y ~ is the energy of the ~t/ct' interface. An equivalent equation may be written to describe the reaction that takes place between fl and ~ '

(AG cflhim)a= ~

v "]

2? #Vmcos 0 # S#

lO A. VX. 18 ~ A. VI 17

~ ~ ~ ~

(6b)

A. Vl. 16 5 -

[/__ZJ

0

I 0

5 10 5 (sa)--------T(nm_2)

(AG cflhim)ais difficult to estimate. Hillert has proposed

I 10

the following expression (AGc~him)a

= R T I (C°-C~/#)22(7o

+ C,(1--(C#-Ce)---x--c' ..... (t'Ce,"~--Ce) 1. (7)

With this estimate of (AG ~ ) ~ , it can be seen that the ratio S~(AG~chim)a/S#(AG#ehim)a is generally less than 0.5: most of the chemical free energy dissipated during the reaction is thus produced in the reaction front that separates fl and ~ '. Hillert's treatment assumes that the chemical energy lost by volume diffusion ahead of the grain boundary is negligible, which is justified, as D~/v is much smaller than the atomic distance (cf. Section 5.1). Also, the solute drag energy is neglected. The triple point between ~, fl and ct' is assumed to be at equilibrium, i.e. y ~cos 0 ~ + y # cos 0 # = tr and ?~ cos 0" = ? # cos 0 # (8) where tr is the ct/fl interface energy. A question arises: which one of the three speeds v~,,, 6 or ( t r ) has to use in Hillert's equation? (ii) According to B6gel and Gust, the two speeds vi., and 6 are related by the equation I

2crVm ]

1

2trVm ] 1 S ~ + S # ] ~-~

=v~, k/ (9) where (AG~am)a is the average chemical energy dissipated during the reaction in the grain boundary that separates a transformation cell (ct + fl lamellae) and ~ ' . order

to

discriminate

calculated from equation (6a) using ? ~= 0.28 J/m2,

and 0 ~ = 4 5 °. E~=45 ° is the most frequently observed angle (see Section 3.3). The value of the surface energy 7 ~ is obtained by simulating the whole shape of the ct/ct' interface (see companion paper). Due to the uncertainty in the values of C ~/# and x/a, a confidence area of rectangular shape is defined for each zone a/(S ~)2 is proportional to vin,. which varies by a factor of 7 within the nodule. On the opposite, ( 6 ) is by definition constant along the nodule. Therefore if one can draw a horizontal straight line through all the confidence area,(6) is the velocity which has to be used. If a straight line going through the origin can be drawn, it is/)ins which has to be used. Figure 8 shows that for nodule A.VI, it is not possible to draw a horizontal straight line nor a line starting from the origin across all the confidence domains: both V~n,and ( ~ ) have to be rejected. However, the values of /)/M~RT are very close for the three ct lamellae. Equation (6a) might thus apply with t~. Data (6a) is compatible with the experimental results if the

[-S'(AG~h~)~ + S#(AGcahtm)~

In

[a/(S~)2; v/M~RT] is reported in a diagram. a/(S~) 2 is calculated from Table 2 and v/M~RT is

couple

f r o m t h e o t h e r n o d u l e s give t h e s a m e result: e q u a t i o n

t~ = V~n, (AG~him)~ S ~ + S # R T

--S~+S #

Fig. 8. Test of Hiller's thermodynamic equation (6a) for the nodule A.VI. For each ~-lamella, the uncertainty in the determination of a and C ~/# leads to the existence of a confidence area for thev/(MPosition~RT),.ofthe point (a/(S~)2,

between

the

two

approaches, we h a v e selected n o d u l e A . V I w h i c h is well described by Cahn's equation. To test the validity of equation (6a), and decide which of the velocities is to be considered in v/M ~RT, we proceeded as follows: for each • lamella, the

growth speed t~ is considered. The velocities entering equations (6a) and (6b) have to be identical. Therefore the ratio (v/M'RT)/ (v/M#RT) should be a constant inside a given Table 5. For the nodule A.VI in the Mg-10 at.%Al alloy, the values of V/M" and v/M # are computed from equations (6a) and (6b),

respectively. ~ lamella IO000/M#RT(v/M#RT)/(v/M~RT) 106X(rim-2) A.VI.12 A.VI.13

91 _+ 11 9 9 _ 18

19 + 5 13 +_2

1.0 _+0.2 0.4 + 0.1

A.VLI4 147 + 12 2O_+6 O.25_+O.O5 The ratio M#/M ~ is roughly constant in agreement with Hillert's theory. Conversely the value of X [see equation (10)l used to test B6gel and Gust's hypotheses varies by a factor of 4 within a

given nodule

DULY et al.: DISCONTINUOUS PRECIPITATION IN Mg-AI ALLOYS--I nodule. Using the values 0 ~ = 45 ° and y a = 0.28 J/m 2, ( v / M a R T ) can be estimated from equation (6b). It is found that the ratio (v/Mt~RT)/(v/M~RT) is almost constant and equal to 15. This result, shown for nodule A.VI (Table 5), has also been confirmed for other nodules. Equations (6a) and (6b) are thus compatible. The ratio M ~ / M ~ is close to 15, which means that the reaction front that separates ~ and ct ' has a much larger mobility than the reaction front between fl and ~ '. Such a value of M ~ / M ¢ is not surprising: when the atoms cross the grain boundary to be enclosed in a phase with the same crystallographic structure, their displacement is only a fraction of the interatomic distance. On the other hand, the integration of atoms in the fl phase requires a complete modification of the crystal structure, which is much more compelling, Having calculated (AG c~him)aand (AG ~him)a, we may now test the model proposed by Brgel et al., i.e. equation (9), which relates v~n~and f. According to this equation, the parameter X defined by \

X-

( S ~~ a

1

(AGchim)a

S~-~-S fl20"Vm)

(10)

should have the same value for all the ~ lamellae of a given nodule. The value of a obtained from the fit of the local shape of the moving grain boundary (see companion paper)is a = 0.4 J/m 2. With this value of a for nodule A.VI (see Table 5) the value of X varies by a factor of 4 from one lamella to another. Such a variation is observed in all the other nodules. If we change the value of a to 0.1 J/m 2, the variation still remains important. Therefore the relation between V~nsand f proposed by Gust does not apply to the Mg-AI system. Among the two approaches proposed by Hillert and B6gel-Gust, the only one compatible with our results is Hillert's equation where the velocity is f.

3853

accepted: optimizing either the speed [4, 6], or the rate of entropy production / 2aVm ",~ Vins~(AGchim)a~---- ) (11) • Optimizing the average speed f would lead to an infinite interlamellar spacing; • The criterion of maximum entropy production has been successfully applied to Ni-In alloys [22]. In our case, optimizing the rate of entropy production assuming that M ' = 15 M a leads to ~/a = 1.8 and Soot = 180 nm for the Mg-10 at.% A1 alloy. Although x/a is close to the observed value, Sopt is about three times smaller than the values of S observed in this alloy. Adjusting Sopt to the experimental value would necessitate an ~//3 interface energy of about 1.2 Jm z, which is much too large. The difficulty in finding an optimization which fits the experimental results in the Mg-AI system may come from the fact that the coupling mechanism between the lamellae has not been taken into account. Indeed, none of the thermodynamic equations describes the interaction that causes all the lamellae to grow with the same average speed, though their instantaneous speeds can be different. The writing of such an equation should be the next step in modelling the propagation of discontinuous precipitation, and will allow us to obtain an expression for t5 as a function of the characteristics of the system. The optimization, if any, should therefore be o f " a kinetic nature", taking into account the distribution of spacings and the coupling of lamellae.

6. SUMMARY AND CONCLUSION

From STEM measurement of the concentration profiles, it has been shown that Cahn's diffusion equation is generally well obeyed when the/3 lamellae 5.5. Does a principle of optimization apply? grow parallel, and that volume diffusion ahead of the mobile grain boundary can be neglected. From Several experimental results seem to indicate that TEM observations and the analysis of STEM the system selects some sort of "optimal state": measurements, it appears that the growth of the (i) the value of a does not depend on the initial precipitation nodule is not a steady state process, and concentration and hardly changes from one lamella that three speeds (Vins, /~ and ( f ) ) defined on three to another. Typically, we have found x/a ~ 2, and time scales (the time necessary for the reaction front this value is very close to that which optimizes the to migrate over a distance of about 6, 0.1 p m and rate of entropy dissipation. 10#m respectively) must be defined in order to (ii) the distribution of interlamellar spacing has a describe it. Gaussian shape, which means that the average The concentration C ~/~ at the interface is not the spacing So tends to be favoured. Moreover, it has equilibrium one and the deviation cannot come from recently been demonstrated that in A1-Zn alloys, SO capillarity corrections alone: the thermodynamical does not depend on the thermal history of the equation proposed by Hillert is compatible with all material, i.e. that So is a function of T and COonly, the experimental results, provided that the speed used no matter the nature of the transients (changes in T, to calculate the friction loss energy is f. The mobility e t c . . . ) [21]. of the reaction front would be about 15 times smaller From the preceding remarks, it seems that the between/3 and ~ ' than between 7 and ~'. Conversely, optimization probably has to be carried out with the model proposed by B6gel and Gust in order to respect to a and S. Two optimization principles have relate vmsand ~ is not compatible with the experimenbeen proposed, but neither has been universally tal results from our system.

3854

DULY et al.: DISCONTINUOUS PRECIPITATION IN Mg-A1 ALLOYS---I

Acknowledgements---4)ne of the authors wants to thank Pechiney, Centre de recherches de Voreppe, for financial support and for providing the alloys. The authors greatly acknowledge J. C. Rouveyre for his assistance in the preparation of TEM foils. They also want to thank Professors G. Purdy and W. Gust, and Dr J. P. Simon for enlightening discussion, and M. Gharghouri for careful reading of the manuscript.

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8. J. M. Shapiro and J, S. Kirkaldy, Acta metall. 16, 1239 (1968). 9. I. G. Solorzano and G. R. Purdy, Metal. Trans. 15A, 1055 (1984). 10. B. E. Sundquist, Metal Trans. 4, 1919 (1973). 11. A. B6gel and W. Gust, Z. Metallk. 79, 296 (1988). 12. J.D. HuntandK. A. Jackson, Trans. Metall. Soc. A I M E 236, 843 (1966). 13. K.A. Jackson and J. D. Hunt, Trans. Metall. Soc. A I M E 236, 1129 (1966). 14. D. A. Porter, Ph.D thesis, Cambridge Univ. (1975). 15. D. A. Porter and J. W. Edington, Proc. R. Soc.Lond. A 358, 335 (1977). 16. I. G. Solorzano, Ph.D thesis, McMaster Univ. (1983). 17. D. Duly and M. Audier, Phil. Mag. Submitted. 18. D. Duly, Y. Br6chet and B. Chenal, Acta metall. 40, 2289 (1992). 19. G. Moreau, J. A. Cornet and D. Calais, J. nucl. Mater. 38, 197 (1971). 21. N. Ness, Masters thesis, McMaster Univ. (1991). 22. A. B6gel, W. Gust and B. Predel, Z. Metallic. 83, 11 (1992). 20. N. Saunders, Calphad 14, 61 (1990).