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On the growth kinetics of plate-shaped precipitates
Scripta METALLURGICA
ON
School
THE
Vol. 12, pp. 885-891, 1978 Printed in the United States
GROWTH
KINETICS
OF
PLATE-SHAPED
Pergamon Press, Inc
PRECIPITATES
R.D. Doherty, M. Ferrante and Y.H. Chen* of Engineering and Applied Sciences, University Brighton, BNI 9QT, U.K. *PMR/IPD, CTA Sao Paulo,
of Sussex, Brazil.
(Received July 31, 1978)
In our earlier publications we described how our experiments on the precipitate growth kinetics of AIAg~ in AI-Ag at 400°C (I) and of @' in AI-Cu at 200 and 235°C (2) showed close ~greement with the predictions of a model based on volume diffusion control using the equations of Ham (3,4) and of Horvay and Cahn (5). We also pointed out (2) that previously published data by Aaronson and Laird (6) and Sankaran and Laird (7) was also in reasonable agreement with the equations of Ham, Horvay and Cahn. It was pointed out (1,2) that the aspect ratios of the plate like precipitates (A = length/thickness) was large (A > I00) and much larger than the aspect ratios expected on the basis of anisotropy of surface energy (A % 3-10)(8). To account for this discrepancy we postulated (I) a partial interfacial inhibition at the early stages of thickening,giving an increase of aspect ratio until volume diffusion control occurred at a precipitate thickness large enough to be measured easily. Sankaran and Laird (9) in their current discussion of our analysis, following their earlier discussion (i0) and our reply (II) criticize our analysis and argue for the original analysis (6,7). The crucial differences between the analyses are the following:The
original
analysis
suggested:-
(i)
a constant lengthening velocity of @' plates diffusion would allow for a linear lengthening enhanced diffusion postulated to account for
(ii)
a constant thickening the reaction would be
Our
analysis
velocity of slower than
a rate faster than normal model (6,7) with stress the discrepancy (7); and
@' plates at allowed with
a rate that for most classical t ½ kinetics.
of
suggested:-
(i)
thickening and lengthening the observed reaction is control; and
(ii)
the high thickening
at close
t½ kinetics at a rate which for most of to that predicted for volume diffusion
aspect ratios found indicate some reaction before the bulk of the
We will try to answer the criticisms of Sankaran we think that the current evidence supports our refer mainly to their data (and that of Aaronson consideration of all the data including our own between the two analyses. I.
at
Dube-Zener
Analysis
of
Volume
Diffusion
initial inhibition of the experimental observations. and Laird (9) and indicate why analysis. Sankaran and Laird (6,7)) but we think that (1,2) is required to decide
Controlled
Thickening
(12,13)
For the very high aspect ratio @' plates observed, Sankaran and Laird test the model for volume diffusion controlled thickening developed by Dub~ (12) and Zener (13). For A > I00 there is almost no difference between the predictions of Dub~-Zener and the model used by us - the oblate spheroid discussion of Ham (3,4) and Horvay and Cahn (5). So the test of data against Dube-Zener used by
885
886
GROWTH KINETICS OF PRECIPITATES
Sankaran and Laird data (6,7) against
(9) the
is equivalent Ham, Horvay
to Cahn
Vol.
our earlier analysis.
test
(2)
of
the
12, No. I0
same
Figure 1 of Sankaran and Laird (9) shows close agreement to within a factor of less than 2 in thickness with the Dub~-Zener predictions. This involves a time exponent in the log-log graphs of 0.5 and the correct half thickness-time values. We came to the same conclusions (2) via plots of half thickness against time to the power, ½, for the same data (6). Sankaran and Laird's objection to this interpretation of their data is based on the interesting anomaly at 200°C for 4 hours shown by Figure 1 of reference 9. We also drew attention to this point though without comment (2). Sankaran and Laird (9) propose a reasonable explanation of the enhanced thickness found at 200°C - that at early times of aging a quenched aluminium copper alloy might show an enhanced diffusion coefficient due to quenched-in vacancies. On the basis of this idea Sankaran and Laird carried out the following procedure on their experimental data, at all temperatures, to achieve the data points in their Figure 2 (9). (a)
They plot olate the
(b)
They subtract Figure 2 the
(c)
They then claim Zener predictions control.
Arguments (i)
the thickness data to obtain
against
values against an intercept
this intercept from corrected thicknesses
time to of finite
the power thickness
their measured against time.
thicknesses
that as these corrected thicknesses the thickening reaction is slower this
procedure
1 and back extropat t = 0 (6,7). and
plot
in
lie below the Dubethan volume diffusion
include:-
As shown in Figure 1 of (9) a time exponent of ½ is clearly indicated by the experimental data. Plotting the same data against time to the power I, and back extrapolating, inevitably leads to a positive thickness at t = 0. These intercepts are artifacts of the use of the wrong plot. Equivalently the uncorrected data if plotted in their Figure 2 shows reasonable agreement (to better than a factor of 2) with the Dub~-Zener predictions.
ii)
Vacancy enhanced diffusion has been demonstrated only for data at 200°C for 8' (7) - it has not been found for e' at 250°C or 300°C (7), for @' in our rather different experiments at 200°C and 235°C (2) or for our experiments in AI-Ag (I). It is therefore unlikely to be a universal phenomenom
iii)
Consideration of vacancy enhanced diffusion for the thickening reaction gives the following picture:if the reaction occurs at the rate expected by the Dube-Zener analysis for a time t2 but with an enhanced diffusion coefficient D2 it will give precipitates with a half thickness a (I) and a diffusion profile with a characteristic distance (D2t2) ½ . If the diffusion coefficient now falls to the equilibrium value D1 the subsequent thickening will occur with this diffusion profile but with D = DI. The rate of thickening for t > t2 will be identical to that shown at the same half thickness a at the time tl that the precipitate would have achieved this thickness xif the diffusion coefficient had been D1 throughout. (This description was developed in private correspondence between Dr Sankaran and ourselves.) The course of the reaction is plotted in Figures la and Ib for the given assumptions. Figure la is a log thicknes~ log time plot which while not identical to that given by Sankaran and Laird (9) for @' at 200°C is quite close to it. Figure Ib shows the same data on a linear thickness/time plot showing that a constant time difference (t2 - tl) not a constant thickness difference is found.
iv)
A major point of Sankaran and Laird's criticism of our analysis is their rejection of our postulate of volume diffusion controlled thickening of the plate-like precipitates. Nevertheless they use a hypothesis that an increase in volume diffusion coefficient will accelerate the thickening reaction. In a mixed control reaction an increase in the rate of one
Vol.
12, No.
I0
GROWTH KINETICS OF PRECIPITATES
887
p r o c e s s (in this case diffusion) will increase the p r o p o r t i o n of control exerted by the other p r o c e s s (in this "case the i n t e r r a c i a l reaction). An argument exists (and was d e v e l o p e d by Dr S a n k a r a n in p r i v a t e c o r r e s p o n ~ ence) that at a ledged interface an increase of D will a c c e l e r a t e the v e l o c i t i e s of individual ledges v I (18). The v e l o c i t y of the i n t e r f a c e is given by (18):dax/dt = Vlh/B
equation 1
where h is the ledge height and B the ledge spacing. If the rate of ledge f o r m a t i o n (the inherent 'interface control' process) remains at a constant value of if (ledges per interface per second) then the value of B will be given by:B = Vl/l f so
equation 2
dax/dt = ifh
so that no a c c e l e r a t i o n of t h i c k e n i n g w o u l d be e x p e c t e d unless i~ is itself d i f f u s i o n controlled. In that case we seem again to have an overall volume d i f f u s i o n c o n t r o l l e d t h i c k e n i n g reaction, though the k i n e t i c s need not be identical to those p r e d i c t e d by the D u b e - Z e n e r analysis. For all these reasons we do not accept that the p r o c e d u r e adopted by S a n k a r a n and L a i r d (9) in d e r i v i n g their Figure 2 is valid. In c o n s e q u e n c e we cannot accept the c o n c l u s i o n they derive from this figure that "the overall rate of m i g r a t i o n of the b r o a d faces of 8' is m u c h less than diffusion controlled". The u n c o r r e c t e d data w h i c h indicates near volume d i f f u s i o n control is we think more reliable. The argument at this point can we think best be s e t t l e d by more data on p r e c i p i t a t e sizes at the shortest time over a range of t e m p e r a t u r e s and supersaturations. 2.
Ham, Horvay~ Cahn A n a l y s i s of P r e c i p i t a t e s which have acquired a Large Aspect Ratio.
The analysis using e q u a t i o n s la and Ib of r e f e r e n c e (9) requires 3 parameters D, 8 and A where 8 is given by a function of A and the s u p e r s a t u r a t i o n ~ (I). D and ~ are fixed by the e x p e r i m e n t a l c o n d i t i o n s and the p r e c i p i t a t e aspect ratio A can be e x ~ e r i m e n t a l l y d e t e r m i n e d at the start of the p e r i o d of observation. There are no a d j u s t a b l e parameters. S a n k a r a n and Laird (9,10) are right to c r i t i c i z e our o r i g i n a l m e t h o d of m e a s u r i n g A, taken from the ratio of the slopes of t h i c k n e s s and length against t½ (1,2). If the r e a c t i o n were p a r t i a l l y i n t e r f a c e inhibited, an incorrectly large value of A w o u l d be o b t a i n e d by this method. This error w o u l d have been r e v e a l e d however by c o m p a r i s o n of actual values of A ( l e n g t h / t h i c k n e s s ) m e a s u r e d directly with the incorrect value of A. The value of A = 160 o b t a i n e d from the slopes in (i) can be c o m p a r e d with A = 143 ~ 12 o b t a i n e d by direct o b s e r v a t i o n s of the p r e c i p i t a t e shapes during p r e c i p i t a t i o n . As we have p r e v i o u s l y argued (1,2,11) the fact that at the b e g i n n i n g of the p e r i o d of o b s e r v a t i o n A is found to be larger than the e q u i l i b r i u m value of aspect ratio, A , has s i g n i f i c a n c e for earlier i n h i b i t i o n of t h i c k e n i n g (see point 3 eqbelow) but it is the larger actual value of A that needs to be u s e d in the Ham, Horvay, Cahn analysis of p r e d i c t ion of the subsequent p e r i o d of growth, if the r e a c t i o n is volume d i f f u s i o n control. Ham (3) has d i s c u s s e d the growth of oblate spheroids of a finite size and w i t h v a r i o u s solute distributions, w h i c h then develop w i t h volume d i f f u s i o n control. For small supersaturations (< 1%) the e x i s t i n g aspect ratios are conserved. In the e x p e r i m e n t s under d i s c u s s i o n the s u p e r s a t u r a t i o n is 2 ~ 4% but what appears to be o c c u r r i n g is a t r a n s i t i o n to volume diffusion control from m i x e d control so the use of the a n a l y s i s seems not likely to be too greatly in error.
888
GROWTH KINETICS OF PRECIPITATES
V01.
12, No,
i0
We would completely agree with Sankaran and Laird's comment (9) that "kinetics of thickening as well as lengthening have to be treated in a consistent manner". This is, it seems to us, the great merit of using the Ham, Horvay, Cahn analysis for plate-shaped precipitates. The success of the Ham, Horvay, Cahn analysis in p r e d i c t i n g both the thickening as well as lengthening kinetics for much of the p r e c i p i t a t i o n reaction in A1-Ag and A1-Cu studies (1,2) is to us the strongest evidence in favour of this analysis. Previously the analysis of thickening and lengthening has been carried out separately (e.g. 7). Sankaran and Laird (9) point out that in their data the aspect ratio A is varying throughout the period of lengthening (7,10) and so the lengthening kinetics should show this effect For 8' at 30~°C A increases only from 175 to 270, at 250°C only from 125 to 275 while at 200 C A increases from 50 at 1 hour. to 275 at I0 hours - Figure 1 of (I0). At the higher temperatures the chan~es in A will have only a small effect on the predicted velocities. It might however be noted that the values of A deduced from the ratio of the slopes (2) are A = 200 at 300°C and A = 260 at 250°C close to the directly observed values of A. For the lowest temperature the smaller value of A at around t = 1 hour would be expected to give detectably slower lengthening on a t~ plot - that no such effect is found (2,6,7) is possibly additional evidence for the vacancy enhanced diffusion effect at short times at 200°C in AI-Cu, postulated by Sankaran and Laird (9). The observed lengthening kinetics are compatable with a normal diffusion coefficient and A of 250 (2). 3.
Variation
of
A durin~
Precipitation
In the study of the precipitation of Ag2AI at 400°C (I) a constant value of A was found from the first observations, to the onset of soft impingement of the ends of the plates. In some of the other studies for 8' precipitation (2,7,10) an increase of A was reported in the early stages of the thickening reaction.
A = ay/a x ". dA = 1 (axda - ayda x) -5 Y a x
".
dA = 1
d-Y
( da y
Adax -
a-~ dt
For volume diffusion
--dY-)
control,
day = A(~D/t)½ dt
and
the Ham, Horvay,
Cahn analysis predicts:-
dax = (8D/t) ½ dt'
so dA = 0 dt a result obtained by Ham (3). interface inhibited dax = f(~D/t) ½ dt dA = A(SD/t)½(I dt
with - f)
However,
if the thickening
reaction
is partially
0 < f < 1 equation
3
At the start of the reaction with t ~ 0, the thickening reaction would, with volume diffusion control, go at a velocity tending to an infinite rate and unless there were an unlimited supply of ledges (a rough interface) this would be impossible, giving f < I and, with a and t small, a rapid initial rate of increase in A. This idea has been pre~iously used in a qualitative form (1,2) to account for the large values of A, much larger than the possible e q u i l i b r i u m values of aspect ratio, Aeq, found during precipitation of p l a t e - s h a p e d precipitates in aluminium alloys. This model for highly elongated W i d m a n s t t a t e n precipitate shapes is of course merely a reformulation of the original Aaronson hypothesis (14) as proposed in the first paper (I). The argument between
Vol.
12, No. i0
GROWTH KINETICS OF PRECIPITATES
ourselves and both Sankaran and Laird, and f o r how l o n g d u r i n g the precipitation..reaction
Aaronson is f
(15,16) < 1?
889
therefore
concerns
The constant v a l u e o f A f o r Ag2A1 p r e c i p i t a t i o n (1) for values of half thickness greater than 80nm, indicates that for that study all the observed reaction was volume diffusion controlled. However for several of the observations of e' precipitation, an increase of A was reported at the earliest stages of the observed reaction. Since in all these cases the value of A seen during the reaction was greater than any possible value of A , an initial increase of A is therefore required on our analysis as on the eqearlier Aaronson (14) analysis. We ( 2 ) r e p o r t e d an increase of A in the study of 0' precipitation i n A 1 - 2 w t % Cu u p t o h a l f t h i c k n e s s of 6nm. Sankaran and Laird (9,10) report increasing values of A up to half thicknesses of 0' of about 40nm (A1-4wt%Cu); 2 0 0 ° C f o r 10 h o u r s , 250°C for 1 hour and 300°C for 20 mins. These times and half-thicknesses occur at the beginning of the periods of agreement with volume diffusion control of Figure 1 of (9) and Figure 1 of (2), and so do not conflict with the observed agreement with volume diffusion control as determined from direct observation of the thickening reaction. One interesting anomaly, already discussed (2) is that in the observations reported by us (3) an increase in A was found at the early stages of precipitation even though the thickening rate was close to that calculated for a volume diffusion control reaction. We ( 2 ) p o s t u l a t e d that this anomaly might be due to an underestimate of the diffusion coefficient that had to be obtained by lengthy extrapolation of D from high temperature data - that i s we p o s t u l a t e some small inhibition, h i d d e n b y u s e o f t o o s m a l l a v a l u e o f D. An a l t e r n a t i v e hypothesis could be that the increase in A is due, at least in part, to an enhancement of the lengthening reaction by the stress enhanced diffusion model as proposed by Sankaran and Laird (7). In general h o w e v e r we f e e l it safer to conclude that observations of increasing A do indicate some interfacial inhibition of the thickenin~ reaction; however, this only occurs at the earliest part of the precipitation reaction (1,2). For much of the subsequent react'ion, A is found to be constant (1) and the observed thickening and lengthening reactions occur at the rates expected for a volume diffusion controlled reaction. Some evidence is available for a decrease in A but this cannot be due to inhibition of the thickenin~ reaction; it is almost certainly due to a slow down of the lengthenin~ reaction, due to the early 'soft impingement' of the diffusion fields at the rims of the plates. This was predicted b y Ham ( 3 ) a n d f o u n d f o r example in the A1-Ag system (1). For investigations of inhibited thickening, observations at the earliest times in the precipitation reaction seem to be required. Such observations o n Ag2A1 p r e c i p i t a t i o n have been carried out and are in course of publication (17). Inhibition of thickening and an increase of A were found for precipitates with a < 30nm. X
4.
Success
of
the
Ledge
Kinetics
Model
The agreement (7,9,10) between the observed interface velocities and the velocities predicte~ using the Jones-Trivedi analysis (18) for isolated ledges, is striking. However the deductions drawn from this by Sankaran and Laird about the rate of migration o f t h e b r o a d f a c e s o f 0 ' a n d t h e l e d g e s u p p l y we d o not accept for the reasons previously given here and in our earlier reply (11). The only additional comments that seem appropriate are the following:(a)
The results given (7,9,10) indicate the utility of the Jones-Trivedi analysis (18) despite some diffusional interaction between ledges (19) a point made originally by Sankaran and Laird (7).
(b)
Table on 0'
(c)
It would be of great interest to see if it were possible to link the apparent volume diffusion controlled kinetics and the Jones-Trivedi ledge analysis, possibly by ledge observations at the earliest stages of thickening, and by the search for the possible correlation between the ledge density and the thickness of individual precipitates.
-
3 of reference precipitates.
7 shows
the
wide
variability
of
ledge
spacings
890
GROWTH KINETICS OF PRECIPITATES
Some o f t h e o t h e r c o n c l u s i o n s a c c e p t e d by u s , n o t a b l y t h o s e 5.
Ostwald Ripening
made by S a n k a r a n concerning:-
and L a i r d
Vol.
(9,10)
are
12,
No.
10
largely
o.f e '
S a n k a r a n a n d L a i r d ( 9 ) m a i n t a i n some o f t h e i r e a r l i e r criticisms of the conclusions r e a c h e d by Boyd a n d N i c h o l s o n who i n v e s t i g a t e d the growth of 8' (20,21). The c r i t i c i s m c o n c e r n e d t h e h i g h v a l u e o f t h e a s p e c t r a t i o (A = 4 0 ) c l a i m e d t o be t h e e q u i l i b r i u m v a l u e , A , and i n i t i a l l y u s e d a s s u c h by u s ( 2 ) . In so f a r as t h i s v a l u e o f A i s l e s s eqthan the values of A found during precipitation s t u d i e s o f 8 ' (A > 100) t h e n e v e n i f Ae < 40 t h e n t h i s e r r o r h a s only quantitative and not q u a l i t a t i v e relevance to th~ argument. However since Boyd an d N i c h o l s o n r e p o r t t h a t A s t a y e d c o n s t a n t a t 40 d u r i n g a r e a c t i o n , that was i n p a r t a t l e a s t t h a t o f O s t w a l d R i p e n i n g , t h e n S a n k a r a n an d L a i r d a r e c o r r e c t t o p o i n t o u t t h a t w i t h Aea < 40 t h e n t h e r e s u l t s o f t h e Boyd an d Nicholson study (20,21) indicate ~ome interracial inhibition of thickening during Ostwald Ripening of e' Recent studies by Purdy and Doherty (22) involving direct observation of 0' growth by high voltage electron microscopy have found a value of A for 8' of 17. This confirms Sankaran and Laird's original argument that Ae~ for 0' is indeed less than 40, and confirms some inhibition of the thickenfng reaction during Ostwald Ripening (20,21). Similar observations of inhibited thickening during Ostwald Ripening of Ag2AI precipitates have recently been made at Sussex (17) and will be reported shortly. Ostwald Ripening has a much smaller driving force than has precipitation and so the observation of inhibited thickening in Ostwald Ripening need not conflict with observations that much of the thickening reaction in precipitation is volume diffusion controlled. For example a small but finite supersaturation might be needed to nucleate ledges (I0). With respect to the comments by Professor Aaronson (16) we are happy to accept that the distinction between us lies in our essentially phenomenological' view based on reaction kinetics and his view that volume diffusion control is to be defined as that occurring only at atomically rough interfaces. We would however point out that his definition does not seem to be a very useful one, since as we have argued (II) once there is a sufficient density of ledges to capture most of the approaching solute, then the equations developed for a rough interface (3-5) seem theoretically and experimentally appropriate to growth of precipitates with ledged interfaces. However again more work is needed to check this suggestion. 6. (i)
Conclusions The correspondence on this topic has led us to further studies of
precipitation kinetics at the earliest stages of precipitate thickening where A is found to be increasing and to studies of inhibited thickening during Ostwald Ripening (17). (ii)
On the basis of the current evidence however we remain confident of our original suggestion ('2-4) that for most of the precipitation reaction in aluminium alloys, the observed kinetics are close to those predicted for volume diffusion control.
(iii) Further electron microscope studies of the reaction kinetics, ledge densities and ledge velocities as pioneered by Professor Aaronson and Laird would seem the most profitable way of finally settling the details of this dispute, particularly if attention is paid to values of the aspect ratios. REFERENCES 1, 2 3 4
5 6
7
M. F e r r a n t e a n d R.D. D o h e r t y , S c r i p t a M e t . , 10, 1059 ( 1 9 7 6 ) . Y.H. Chen a n d R.D. D o h e r t y , S c r i p t a M e t . , 1 1 , 725 ( 1 9 7 7 ) . F . S . Ham, J . P h y s . C h e m . S o l i d s , 6, 335 ( 1 9 5 9 ) . F . S . Ham, Q u a r t . A p p . M a t h s . , 1 7 , 137 ( 1 9 5 9 ) . G. H o r v a y and J . W . C a h n , A c t a M e t . , 9, 695 ( 1 9 6 1 ) . H . I . A a r o n s o n and C. L a i r d , TMS-AIME, 2 4 2 , 1437 ( 1 9 6 8 ) . R. S a n k a r a n a n d C. L a i r d , A c t a M e t . , 22, 957 ( 1 9 7 4 ) .
Vol.
8. 9. i0. II. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.
12, No.
I0
GROWTH KINETICS
OF PRECIPITATES
891
H.I.
Aaronson, J.B. Clark and C. Laird, Metal Science Journal, 2, 155 (1968). R. Sankaran and C. Laird, Scripta Met., 12, 877 (1978). R. Sankaran and C. Laird, Scripta Met., II, 383 (1977). R.D. Doherty, M. Ferrante and Y.H. Chen, Scripta Met., II, 733 (1977). C.A. Dube, Ph.D. Thesis, Carnegie Institute of Technology (1948). C. Zener, J.Appl.Phys., 20, 950 (1949). H.I. Aaronson, D e c o m p o s i t i o n of Austenite by Diffusional Processes, p387, Interscience, N.Y. (1962). H.I. Aaronson, Scripta Met., II, 731 (1977). H.I. Aaronson, Scripta Met., II, 741 (1977). M. Ferrante and R.D. Doherty. To be published. G.J. Jones and R.K. Trivedi, J.Appl.Phys., 42, 4299 (1971). G.J. Jones and R.K. Trivedi, J.Crystal Growth, 29, 155 (1975). J.D. Boyd and R.B. Nicholson, Acta Met., 19, 1101 (1971). J.D. Boyd and R.B. Nicholson, Acta Met., 19, 1379 (1971). G.R. Purdy and R.D. Doherty. To be published.
i
, ,
l/
2 f'n nz
,~,.s t,.o.o,. .....
/ / ' / __~___/
==-~ /J
I
-2
,"
Figure
la
Figure
Ib
~,,,
....
~
/ , v
,
0
I
I
2 tn t
4
,
,
i
o=.,,tt
O.=
2
'1
hm
t
Figure I: E n h a n c e d Diffusion Model. P r e c i p i t a t e s are assumed to thicken with half thickness a = ~t½=(Dt)½. In condition I, the normal diffusion coefficient D1 Xapplies for all times tl > 0 (a I = 1). In condition 2, an e n h a n c e d diffusion coefficient D2 is assumed upto a = a_(1) (t = t2), subsequent thickening occurs at the rate expected f~r th~ same thickness (and diffusion profile) as in condition I. Two values of me are considered one with a2 = 5 (D2/DI = 25) and the other ~2 = /5 (D2/DI ~ 5). In Figure la, the data are p l o t t e d on a log av/log t graph while in Figure Ib the same data are shown on linear ax/t grap'h.
ON
School
THE
Vol. 12, pp. 885-891, 1978 Printed in the United States
GROWTH
KINETICS
OF
PLATE-SHAPED
Pergamon Press, Inc
PRECIPITATES
R.D. Doherty, M. Ferrante and Y.H. Chen* of Engineering and Applied Sciences, University Brighton, BNI 9QT, U.K. *PMR/IPD, CTA Sao Paulo,
of Sussex, Brazil.
(Received July 31, 1978)
In our earlier publications we described how our experiments on the precipitate growth kinetics of AIAg~ in AI-Ag at 400°C (I) and of @' in AI-Cu at 200 and 235°C (2) showed close ~greement with the predictions of a model based on volume diffusion control using the equations of Ham (3,4) and of Horvay and Cahn (5). We also pointed out (2) that previously published data by Aaronson and Laird (6) and Sankaran and Laird (7) was also in reasonable agreement with the equations of Ham, Horvay and Cahn. It was pointed out (1,2) that the aspect ratios of the plate like precipitates (A = length/thickness) was large (A > I00) and much larger than the aspect ratios expected on the basis of anisotropy of surface energy (A % 3-10)(8). To account for this discrepancy we postulated (I) a partial interfacial inhibition at the early stages of thickening,giving an increase of aspect ratio until volume diffusion control occurred at a precipitate thickness large enough to be measured easily. Sankaran and Laird (9) in their current discussion of our analysis, following their earlier discussion (i0) and our reply (II) criticize our analysis and argue for the original analysis (6,7). The crucial differences between the analyses are the following:The
original
analysis
suggested:-
(i)
a constant lengthening velocity of @' plates diffusion would allow for a linear lengthening enhanced diffusion postulated to account for
(ii)
a constant thickening the reaction would be
Our
analysis
velocity of slower than
a rate faster than normal model (6,7) with stress the discrepancy (7); and
@' plates at allowed with
a rate that for most classical t ½ kinetics.
of
suggested:-
(i)
thickening and lengthening the observed reaction is control; and
(ii)
the high thickening
at close
t½ kinetics at a rate which for most of to that predicted for volume diffusion
aspect ratios found indicate some reaction before the bulk of the
We will try to answer the criticisms of Sankaran we think that the current evidence supports our refer mainly to their data (and that of Aaronson consideration of all the data including our own between the two analyses. I.
at
Dube-Zener
Analysis
of
Volume
Diffusion
initial inhibition of the experimental observations. and Laird (9) and indicate why analysis. Sankaran and Laird (6,7)) but we think that (1,2) is required to decide
Controlled
Thickening
(12,13)
For the very high aspect ratio @' plates observed, Sankaran and Laird test the model for volume diffusion controlled thickening developed by Dub~ (12) and Zener (13). For A > I00 there is almost no difference between the predictions of Dub~-Zener and the model used by us - the oblate spheroid discussion of Ham (3,4) and Horvay and Cahn (5). So the test of data against Dube-Zener used by
885
886
GROWTH KINETICS OF PRECIPITATES
Sankaran and Laird data (6,7) against
(9) the
is equivalent Ham, Horvay
to Cahn
Vol.
our earlier analysis.
test
(2)
of
the
12, No. I0
same
Figure 1 of Sankaran and Laird (9) shows close agreement to within a factor of less than 2 in thickness with the Dub~-Zener predictions. This involves a time exponent in the log-log graphs of 0.5 and the correct half thickness-time values. We came to the same conclusions (2) via plots of half thickness against time to the power, ½, for the same data (6). Sankaran and Laird's objection to this interpretation of their data is based on the interesting anomaly at 200°C for 4 hours shown by Figure 1 of reference 9. We also drew attention to this point though without comment (2). Sankaran and Laird (9) propose a reasonable explanation of the enhanced thickness found at 200°C - that at early times of aging a quenched aluminium copper alloy might show an enhanced diffusion coefficient due to quenched-in vacancies. On the basis of this idea Sankaran and Laird carried out the following procedure on their experimental data, at all temperatures, to achieve the data points in their Figure 2 (9). (a)
They plot olate the
(b)
They subtract Figure 2 the
(c)
They then claim Zener predictions control.
Arguments (i)
the thickness data to obtain
against
values against an intercept
this intercept from corrected thicknesses
time to of finite
the power thickness
their measured against time.
thicknesses
that as these corrected thicknesses the thickening reaction is slower this
procedure
1 and back extropat t = 0 (6,7). and
plot
in
lie below the Dubethan volume diffusion
include:-
As shown in Figure 1 of (9) a time exponent of ½ is clearly indicated by the experimental data. Plotting the same data against time to the power I, and back extrapolating, inevitably leads to a positive thickness at t = 0. These intercepts are artifacts of the use of the wrong plot. Equivalently the uncorrected data if plotted in their Figure 2 shows reasonable agreement (to better than a factor of 2) with the Dub~-Zener predictions.
ii)
Vacancy enhanced diffusion has been demonstrated only for data at 200°C for 8' (7) - it has not been found for e' at 250°C or 300°C (7), for @' in our rather different experiments at 200°C and 235°C (2) or for our experiments in AI-Ag (I). It is therefore unlikely to be a universal phenomenom
iii)
Consideration of vacancy enhanced diffusion for the thickening reaction gives the following picture:if the reaction occurs at the rate expected by the Dube-Zener analysis for a time t2 but with an enhanced diffusion coefficient D2 it will give precipitates with a half thickness a (I) and a diffusion profile with a characteristic distance (D2t2) ½ . If the diffusion coefficient now falls to the equilibrium value D1 the subsequent thickening will occur with this diffusion profile but with D = DI. The rate of thickening for t > t2 will be identical to that shown at the same half thickness a at the time tl that the precipitate would have achieved this thickness xif the diffusion coefficient had been D1 throughout. (This description was developed in private correspondence between Dr Sankaran and ourselves.) The course of the reaction is plotted in Figures la and Ib for the given assumptions. Figure la is a log thicknes~ log time plot which while not identical to that given by Sankaran and Laird (9) for @' at 200°C is quite close to it. Figure Ib shows the same data on a linear thickness/time plot showing that a constant time difference (t2 - tl) not a constant thickness difference is found.
iv)
A major point of Sankaran and Laird's criticism of our analysis is their rejection of our postulate of volume diffusion controlled thickening of the plate-like precipitates. Nevertheless they use a hypothesis that an increase in volume diffusion coefficient will accelerate the thickening reaction. In a mixed control reaction an increase in the rate of one
Vol.
12, No.
I0
GROWTH KINETICS OF PRECIPITATES
887
p r o c e s s (in this case diffusion) will increase the p r o p o r t i o n of control exerted by the other p r o c e s s (in this "case the i n t e r r a c i a l reaction). An argument exists (and was d e v e l o p e d by Dr S a n k a r a n in p r i v a t e c o r r e s p o n ~ ence) that at a ledged interface an increase of D will a c c e l e r a t e the v e l o c i t i e s of individual ledges v I (18). The v e l o c i t y of the i n t e r f a c e is given by (18):dax/dt = Vlh/B
equation 1
where h is the ledge height and B the ledge spacing. If the rate of ledge f o r m a t i o n (the inherent 'interface control' process) remains at a constant value of if (ledges per interface per second) then the value of B will be given by:B = Vl/l f so
equation 2
dax/dt = ifh
so that no a c c e l e r a t i o n of t h i c k e n i n g w o u l d be e x p e c t e d unless i~ is itself d i f f u s i o n controlled. In that case we seem again to have an overall volume d i f f u s i o n c o n t r o l l e d t h i c k e n i n g reaction, though the k i n e t i c s need not be identical to those p r e d i c t e d by the D u b e - Z e n e r analysis. For all these reasons we do not accept that the p r o c e d u r e adopted by S a n k a r a n and L a i r d (9) in d e r i v i n g their Figure 2 is valid. In c o n s e q u e n c e we cannot accept the c o n c l u s i o n they derive from this figure that "the overall rate of m i g r a t i o n of the b r o a d faces of 8' is m u c h less than diffusion controlled". The u n c o r r e c t e d data w h i c h indicates near volume d i f f u s i o n control is we think more reliable. The argument at this point can we think best be s e t t l e d by more data on p r e c i p i t a t e sizes at the shortest time over a range of t e m p e r a t u r e s and supersaturations. 2.
Ham, Horvay~ Cahn A n a l y s i s of P r e c i p i t a t e s which have acquired a Large Aspect Ratio.
The analysis using e q u a t i o n s la and Ib of r e f e r e n c e (9) requires 3 parameters D, 8 and A where 8 is given by a function of A and the s u p e r s a t u r a t i o n ~ (I). D and ~ are fixed by the e x p e r i m e n t a l c o n d i t i o n s and the p r e c i p i t a t e aspect ratio A can be e x ~ e r i m e n t a l l y d e t e r m i n e d at the start of the p e r i o d of observation. There are no a d j u s t a b l e parameters. S a n k a r a n and Laird (9,10) are right to c r i t i c i z e our o r i g i n a l m e t h o d of m e a s u r i n g A, taken from the ratio of the slopes of t h i c k n e s s and length against t½ (1,2). If the r e a c t i o n were p a r t i a l l y i n t e r f a c e inhibited, an incorrectly large value of A w o u l d be o b t a i n e d by this method. This error w o u l d have been r e v e a l e d however by c o m p a r i s o n of actual values of A ( l e n g t h / t h i c k n e s s ) m e a s u r e d directly with the incorrect value of A. The value of A = 160 o b t a i n e d from the slopes in (i) can be c o m p a r e d with A = 143 ~ 12 o b t a i n e d by direct o b s e r v a t i o n s of the p r e c i p i t a t e shapes during p r e c i p i t a t i o n . As we have p r e v i o u s l y argued (1,2,11) the fact that at the b e g i n n i n g of the p e r i o d of o b s e r v a t i o n A is found to be larger than the e q u i l i b r i u m value of aspect ratio, A , has s i g n i f i c a n c e for earlier i n h i b i t i o n of t h i c k e n i n g (see point 3 eqbelow) but it is the larger actual value of A that needs to be u s e d in the Ham, Horvay, Cahn analysis of p r e d i c t ion of the subsequent p e r i o d of growth, if the r e a c t i o n is volume d i f f u s i o n control. Ham (3) has d i s c u s s e d the growth of oblate spheroids of a finite size and w i t h v a r i o u s solute distributions, w h i c h then develop w i t h volume d i f f u s i o n control. For small supersaturations (< 1%) the e x i s t i n g aspect ratios are conserved. In the e x p e r i m e n t s under d i s c u s s i o n the s u p e r s a t u r a t i o n is 2 ~ 4% but what appears to be o c c u r r i n g is a t r a n s i t i o n to volume diffusion control from m i x e d control so the use of the a n a l y s i s seems not likely to be too greatly in error.
888
GROWTH KINETICS OF PRECIPITATES
V01.
12, No,
i0
We would completely agree with Sankaran and Laird's comment (9) that "kinetics of thickening as well as lengthening have to be treated in a consistent manner". This is, it seems to us, the great merit of using the Ham, Horvay, Cahn analysis for plate-shaped precipitates. The success of the Ham, Horvay, Cahn analysis in p r e d i c t i n g both the thickening as well as lengthening kinetics for much of the p r e c i p i t a t i o n reaction in A1-Ag and A1-Cu studies (1,2) is to us the strongest evidence in favour of this analysis. Previously the analysis of thickening and lengthening has been carried out separately (e.g. 7). Sankaran and Laird (9) point out that in their data the aspect ratio A is varying throughout the period of lengthening (7,10) and so the lengthening kinetics should show this effect For 8' at 30~°C A increases only from 175 to 270, at 250°C only from 125 to 275 while at 200 C A increases from 50 at 1 hour. to 275 at I0 hours - Figure 1 of (I0). At the higher temperatures the chan~es in A will have only a small effect on the predicted velocities. It might however be noted that the values of A deduced from the ratio of the slopes (2) are A = 200 at 300°C and A = 260 at 250°C close to the directly observed values of A. For the lowest temperature the smaller value of A at around t = 1 hour would be expected to give detectably slower lengthening on a t~ plot - that no such effect is found (2,6,7) is possibly additional evidence for the vacancy enhanced diffusion effect at short times at 200°C in AI-Cu, postulated by Sankaran and Laird (9). The observed lengthening kinetics are compatable with a normal diffusion coefficient and A of 250 (2). 3.
Variation
of
A durin~
Precipitation
In the study of the precipitation of Ag2AI at 400°C (I) a constant value of A was found from the first observations, to the onset of soft impingement of the ends of the plates. In some of the other studies for 8' precipitation (2,7,10) an increase of A was reported in the early stages of the thickening reaction.
A = ay/a x ". dA = 1 (axda - ayda x) -5 Y a x
".
dA = 1
d-Y
( da y
Adax -
a-~ dt
For volume diffusion
--dY-)
control,
day = A(~D/t)½ dt
and
the Ham, Horvay,
Cahn analysis predicts:-
dax = (8D/t) ½ dt'
so dA = 0 dt a result obtained by Ham (3). interface inhibited dax = f(~D/t) ½ dt dA = A(SD/t)½(I dt
with - f)
However,
if the thickening
reaction
is partially
0 < f < 1 equation
3
At the start of the reaction with t ~ 0, the thickening reaction would, with volume diffusion control, go at a velocity tending to an infinite rate and unless there were an unlimited supply of ledges (a rough interface) this would be impossible, giving f < I and, with a and t small, a rapid initial rate of increase in A. This idea has been pre~iously used in a qualitative form (1,2) to account for the large values of A, much larger than the possible e q u i l i b r i u m values of aspect ratio, Aeq, found during precipitation of p l a t e - s h a p e d precipitates in aluminium alloys. This model for highly elongated W i d m a n s t t a t e n precipitate shapes is of course merely a reformulation of the original Aaronson hypothesis (14) as proposed in the first paper (I). The argument between
Vol.
12, No. i0
GROWTH KINETICS OF PRECIPITATES
ourselves and both Sankaran and Laird, and f o r how l o n g d u r i n g the precipitation..reaction
Aaronson is f
(15,16) < 1?
889
therefore
concerns
The constant v a l u e o f A f o r Ag2A1 p r e c i p i t a t i o n (1) for values of half thickness greater than 80nm, indicates that for that study all the observed reaction was volume diffusion controlled. However for several of the observations of e' precipitation, an increase of A was reported at the earliest stages of the observed reaction. Since in all these cases the value of A seen during the reaction was greater than any possible value of A , an initial increase of A is therefore required on our analysis as on the eqearlier Aaronson (14) analysis. We ( 2 ) r e p o r t e d an increase of A in the study of 0' precipitation i n A 1 - 2 w t % Cu u p t o h a l f t h i c k n e s s of 6nm. Sankaran and Laird (9,10) report increasing values of A up to half thicknesses of 0' of about 40nm (A1-4wt%Cu); 2 0 0 ° C f o r 10 h o u r s , 250°C for 1 hour and 300°C for 20 mins. These times and half-thicknesses occur at the beginning of the periods of agreement with volume diffusion control of Figure 1 of (9) and Figure 1 of (2), and so do not conflict with the observed agreement with volume diffusion control as determined from direct observation of the thickening reaction. One interesting anomaly, already discussed (2) is that in the observations reported by us (3) an increase in A was found at the early stages of precipitation even though the thickening rate was close to that calculated for a volume diffusion control reaction. We ( 2 ) p o s t u l a t e d that this anomaly might be due to an underestimate of the diffusion coefficient that had to be obtained by lengthy extrapolation of D from high temperature data - that i s we p o s t u l a t e some small inhibition, h i d d e n b y u s e o f t o o s m a l l a v a l u e o f D. An a l t e r n a t i v e hypothesis could be that the increase in A is due, at least in part, to an enhancement of the lengthening reaction by the stress enhanced diffusion model as proposed by Sankaran and Laird (7). In general h o w e v e r we f e e l it safer to conclude that observations of increasing A do indicate some interfacial inhibition of the thickenin~ reaction; however, this only occurs at the earliest part of the precipitation reaction (1,2). For much of the subsequent react'ion, A is found to be constant (1) and the observed thickening and lengthening reactions occur at the rates expected for a volume diffusion controlled reaction. Some evidence is available for a decrease in A but this cannot be due to inhibition of the thickenin~ reaction; it is almost certainly due to a slow down of the lengthenin~ reaction, due to the early 'soft impingement' of the diffusion fields at the rims of the plates. This was predicted b y Ham ( 3 ) a n d f o u n d f o r example in the A1-Ag system (1). For investigations of inhibited thickening, observations at the earliest times in the precipitation reaction seem to be required. Such observations o n Ag2A1 p r e c i p i t a t i o n have been carried out and are in course of publication (17). Inhibition of thickening and an increase of A were found for precipitates with a < 30nm. X
4.
Success
of
the
Ledge
Kinetics
Model
The agreement (7,9,10) between the observed interface velocities and the velocities predicte~ using the Jones-Trivedi analysis (18) for isolated ledges, is striking. However the deductions drawn from this by Sankaran and Laird about the rate of migration o f t h e b r o a d f a c e s o f 0 ' a n d t h e l e d g e s u p p l y we d o not accept for the reasons previously given here and in our earlier reply (11). The only additional comments that seem appropriate are the following:(a)
The results given (7,9,10) indicate the utility of the Jones-Trivedi analysis (18) despite some diffusional interaction between ledges (19) a point made originally by Sankaran and Laird (7).
(b)
Table on 0'
(c)
It would be of great interest to see if it were possible to link the apparent volume diffusion controlled kinetics and the Jones-Trivedi ledge analysis, possibly by ledge observations at the earliest stages of thickening, and by the search for the possible correlation between the ledge density and the thickness of individual precipitates.
-
3 of reference precipitates.
7 shows
the
wide
variability
of
ledge
spacings
890
GROWTH KINETICS OF PRECIPITATES
Some o f t h e o t h e r c o n c l u s i o n s a c c e p t e d by u s , n o t a b l y t h o s e 5.
Ostwald Ripening
made by S a n k a r a n concerning:-
and L a i r d
Vol.
(9,10)
are
12,
No.
10
largely
o.f e '
S a n k a r a n a n d L a i r d ( 9 ) m a i n t a i n some o f t h e i r e a r l i e r criticisms of the conclusions r e a c h e d by Boyd a n d N i c h o l s o n who i n v e s t i g a t e d the growth of 8' (20,21). The c r i t i c i s m c o n c e r n e d t h e h i g h v a l u e o f t h e a s p e c t r a t i o (A = 4 0 ) c l a i m e d t o be t h e e q u i l i b r i u m v a l u e , A , and i n i t i a l l y u s e d a s s u c h by u s ( 2 ) . In so f a r as t h i s v a l u e o f A i s l e s s eqthan the values of A found during precipitation s t u d i e s o f 8 ' (A > 100) t h e n e v e n i f Ae < 40 t h e n t h i s e r r o r h a s only quantitative and not q u a l i t a t i v e relevance to th~ argument. However since Boyd an d N i c h o l s o n r e p o r t t h a t A s t a y e d c o n s t a n t a t 40 d u r i n g a r e a c t i o n , that was i n p a r t a t l e a s t t h a t o f O s t w a l d R i p e n i n g , t h e n S a n k a r a n an d L a i r d a r e c o r r e c t t o p o i n t o u t t h a t w i t h Aea < 40 t h e n t h e r e s u l t s o f t h e Boyd an d Nicholson study (20,21) indicate ~ome interracial inhibition of thickening during Ostwald Ripening of e' Recent studies by Purdy and Doherty (22) involving direct observation of 0' growth by high voltage electron microscopy have found a value of A for 8' of 17. This confirms Sankaran and Laird's original argument that Ae~ for 0' is indeed less than 40, and confirms some inhibition of the thickenfng reaction during Ostwald Ripening (20,21). Similar observations of inhibited thickening during Ostwald Ripening of Ag2AI precipitates have recently been made at Sussex (17) and will be reported shortly. Ostwald Ripening has a much smaller driving force than has precipitation and so the observation of inhibited thickening in Ostwald Ripening need not conflict with observations that much of the thickening reaction in precipitation is volume diffusion controlled. For example a small but finite supersaturation might be needed to nucleate ledges (I0). With respect to the comments by Professor Aaronson (16) we are happy to accept that the distinction between us lies in our essentially phenomenological' view based on reaction kinetics and his view that volume diffusion control is to be defined as that occurring only at atomically rough interfaces. We would however point out that his definition does not seem to be a very useful one, since as we have argued (II) once there is a sufficient density of ledges to capture most of the approaching solute, then the equations developed for a rough interface (3-5) seem theoretically and experimentally appropriate to growth of precipitates with ledged interfaces. However again more work is needed to check this suggestion. 6. (i)
Conclusions The correspondence on this topic has led us to further studies of
precipitation kinetics at the earliest stages of precipitate thickening where A is found to be increasing and to studies of inhibited thickening during Ostwald Ripening (17). (ii)
On the basis of the current evidence however we remain confident of our original suggestion ('2-4) that for most of the precipitation reaction in aluminium alloys, the observed kinetics are close to those predicted for volume diffusion control.
(iii) Further electron microscope studies of the reaction kinetics, ledge densities and ledge velocities as pioneered by Professor Aaronson and Laird would seem the most profitable way of finally settling the details of this dispute, particularly if attention is paid to values of the aspect ratios. REFERENCES 1, 2 3 4
5 6
7
M. F e r r a n t e a n d R.D. D o h e r t y , S c r i p t a M e t . , 10, 1059 ( 1 9 7 6 ) . Y.H. Chen a n d R.D. D o h e r t y , S c r i p t a M e t . , 1 1 , 725 ( 1 9 7 7 ) . F . S . Ham, J . P h y s . C h e m . S o l i d s , 6, 335 ( 1 9 5 9 ) . F . S . Ham, Q u a r t . A p p . M a t h s . , 1 7 , 137 ( 1 9 5 9 ) . G. H o r v a y and J . W . C a h n , A c t a M e t . , 9, 695 ( 1 9 6 1 ) . H . I . A a r o n s o n and C. L a i r d , TMS-AIME, 2 4 2 , 1437 ( 1 9 6 8 ) . R. S a n k a r a n a n d C. L a i r d , A c t a M e t . , 22, 957 ( 1 9 7 4 ) .
Vol.
8. 9. i0. II. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.
12, No.
I0
GROWTH KINETICS
OF PRECIPITATES
891
H.I.
Aaronson, J.B. Clark and C. Laird, Metal Science Journal, 2, 155 (1968). R. Sankaran and C. Laird, Scripta Met., 12, 877 (1978). R. Sankaran and C. Laird, Scripta Met., II, 383 (1977). R.D. Doherty, M. Ferrante and Y.H. Chen, Scripta Met., II, 733 (1977). C.A. Dube, Ph.D. Thesis, Carnegie Institute of Technology (1948). C. Zener, J.Appl.Phys., 20, 950 (1949). H.I. Aaronson, D e c o m p o s i t i o n of Austenite by Diffusional Processes, p387, Interscience, N.Y. (1962). H.I. Aaronson, Scripta Met., II, 731 (1977). H.I. Aaronson, Scripta Met., II, 741 (1977). M. Ferrante and R.D. Doherty. To be published. G.J. Jones and R.K. Trivedi, J.Appl.Phys., 42, 4299 (1971). G.J. Jones and R.K. Trivedi, J.Crystal Growth, 29, 155 (1975). J.D. Boyd and R.B. Nicholson, Acta Met., 19, 1101 (1971). J.D. Boyd and R.B. Nicholson, Acta Met., 19, 1379 (1971). G.R. Purdy and R.D. Doherty. To be published.
i
, ,
l/
2 f'n nz
,~,.s t,.o.o,. .....
/ / ' / __~___/
==-~ /J
I
-2
,"
Figure
la
Figure
Ib
~,,,
....
~
/ , v
,
0
I
I
2 tn t
4
,
,
i
o=.,,tt
O.=
2
'1
hm
t
Figure I: E n h a n c e d Diffusion Model. P r e c i p i t a t e s are assumed to thicken with half thickness a = ~t½=(Dt)½. In condition I, the normal diffusion coefficient D1 Xapplies for all times tl > 0 (a I = 1). In condition 2, an e n h a n c e d diffusion coefficient D2 is assumed upto a = a_(1) (t = t2), subsequent thickening occurs at the rate expected f~r th~ same thickness (and diffusion profile) as in condition I. Two values of me are considered one with a2 = 5 (D2/DI = 25) and the other ~2 = /5 (D2/DI ~ 5). In Figure la, the data are p l o t t e d on a log av/log t graph while in Figure Ib the same data are shown on linear ax/t grap'h.