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Coarsening of θ′ plates in Al-Cu alloys—II. influence of ledge mechanism
COARSENING OF 0’ PLATES IN Al-01 ALLOYS-II, INFLUENCE OF LEDGE MECHANISM P. MERLE and .J. MERLIN Groupe d’Etudes de Mttallurgie Physique et de Physique des Materiaux (ERA 463) Insa de Lyon-Bat. 502. 69621 Villeurbanne. Cedex. France (Rrceired 11 Decrmht~r 1980: in rerised.fiwm 7 Muy 1981)
Abstract-A theoretical study was made of the two coarsening mechanisms of 0’ plates: 1. morphological evolution and 2. coarsening at constant aspect ratio. In both of them ledge migration on the broad face of prectpitates is found to be the controlling process. The analysis of the first mechanism leads to an evaluation of the mean diffusion coefficient of copper m the a’-matrix interface. For the second mechanism. the hypothesis of an heterogeneous nucleation of ledges due to the impingement of precipitates leads to a linear theoretical relation between the mean diameter D and r* I1 This relation agrees rather well with the experimental results. R&urn&-Nous avons etudie th~oriquement les deun processus de coaiescence des precipites 0’: 1. evolution morphologique et 2. coalescence a rapport d’aspect constant. Nous trouvons que dans les deux cas la migration des marches existant sur I’interface large prtcipiti-matrice est le processus controlant I’evolution. L’analyse du premier mecanisme condun a une evaluation du coefficient moyen de diffusion du cuivre a I-interface matrice-pricipite Q’. Dans le cas du deuxieme mtcanisme. I’hypothese dune nucleation htttrogine des marches due a la rencontre entre precipitis nous conduit a une relation theorique lintaire enfre le diamttre moyen ~ et rz *I. Cette relation est en assez bon accord avec les r&hats ex~rimentaux. Zusammanfassung7Zwei Vergroberungsmechanismen von 0’-Platten wurden theoretisch untersucht: 1. morphologische Anderung und 2. Vergroberung bei konstantem Formfaktor. Bei beiden Mechanismen besteht der bestimmende Prozess im Wandern von Vorsphingen iiber die breite FlPche der Ausscheidungen. Die Analyse des ersten Prozesses fiihrt zu einer Auswertung des mittleren Diffusionskoeffizienten von Kupfer in der Grenzfliche zwischen Ausscheidung und Matrix. Bei dem zweiten Prozess fiihrt die Annahme. da8 Vorspriinge sich helerogen bilden. zu einem Iinearen theoretischen Zusammenhang zwischen dem mittieren Durchmesser D und t “I’ Dieser Zusammenhang stimmt ziemhch gut mit den experimentellen Ergebnissen iiberein.
I. INTRODUCTION The aim of this paper is to give an interpretation
of the evolution kinetics corresponding to the two basic coarsening processes of 0’ plates as shown in our previous experimental study (Part 1 Cl]). -Morphological evolution at constant volume for each precipitate. -Coarsening with the equilibrium aspect ratio for all precipitates.
2. MORPHOLOGICAL
EVOLl_;TIOIV OF 8’ PLATES
Two alternative assumptions may be made; (a) the evolution of the aspect ratio .4 is due to a progressive modification of the interracial energy of the broad faces of 6’ plates i.e., at any time. the precipitates have their equilibrium shape, but this shape changes with time. (b) High values of A are characteristic of an out of equilibrium population of precipitates. According to Aaronson’s hypothesis [2] such a situation is possible, owing to the inhibition of thickness growth due to the ledge mechanism.
An electron microscope study on an Al-4 wt”,, Cu CR]. allows us to discard the first assumption, as no correlation exists between the evolution of the broad interface coherence and that of the mean aspect ratio (Fig. I) (see Fig. 8, part I for 2 variations). At high temperatures. we observe a progressive loss of coherency and a simultaneous decrease of 2. For an aging of 24 h at 275 C. edge interracial dislocations of Burger’s vector b = I:2 i 100) begin to appear on most of the precipitates [Fig. l(d)}. A complete network is formed for higher aging times [Fig. l(e) - 120 h at 275 CJ. But, for short aging times [2 h at 275-C. Fig. l(c)] 2 has already decreased while the precipitates have not lost their coherency. Besides. at low temperatures and for high aging times (100 h a 225’C. 540 h at 200 C) broad faces of precipitates are still coherent [Fig. I (a) and (b)]. At these aging temperatures x decreases sooner (10 h at 225-C and 50 h at 200 C) and for times cited above, 2 is about half its maximum value. The decrease of the aspect ratio of a precipitate is thus due to its evoiution to its equilibrium shape.
defined by A,,, = CF,:~~op and or, being the interfacial energies of the peripheral and of the broad interfaces. If A > A,,. the solute concentration near the peripheral interface due to the Gibbs-Thomson effect, is
19‘9
1930
AND MERLIN:
MERLE
COARSENING
OF D PLATES
IN AI-CU ALLOYS-II
b
a
d
e Fig. 1. Coherency state of matrix-precipitates interface Al -Cu 4 wt”,, [R]: (al 540 h at X0 C:(b) 100 h at 22s C: (cl 2 h at 275 C: (d) 24 h at 275 C: (e) I20 h at 275 C (dark tield g [EOJL
greater than that near the broad face and a flux of solute will take place between the two interfaces. The corresponding driving force must be greater than the one deriving from the Gibbs-Thomson effect associated to the size of the precipitate and giving rise to a flux of solute between the matrix and the precipitate. The supersaturation (Ok near a precipitate is defined by
(’ COO
z
-
CO
:
t -.
-
CO
(1) CP Where co. i: and cP are the solute concentrations near the precipitate-matrix interface. in the matrix and in the precipitate. By analogy the supersaturation associated to an out-of equilibrium morphology may cg - co
be written c, -
(01
=
CD
-------
cp - cr
c, -
CD
(‘P
(2)
where ct. c,,and cD are the solute concentrations near a precipitate of infinite size, near the peripheral and the broad interface. Calculation of o). and 01~ for an Al-4 wt”; Cu aged at 225 C leads to (Appendix A):
(‘JO
2 D*
(3)
MERLE
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COARSENING
1931
OF 0’ PLATES IN AI-Cu ALLOYS-II
If being the precipitate diameter and D* the critical diameter; for growing precipitates (I) > D*), D/D* has values between 1 and 3 (see shape of size-distributions-Part I). For out-of equilibrium precipitates A may be 20&250. A,, is about 15. Thus w,/oO may be greater than unity. Moreover, the diffusion fields relative to these supersaturations are very different: the effective diffusion distance between the precipitate and the matrix is of the order of (0#‘2, i.e. a few hundred of angstroems for coarsening stage, while it is much less between the two faces of the same precipitate. So. even when the two supersaturations too and (ttr are of the same order, the con~ntration gradient associated to w, is the most important. This explain why a deviation from the equilibrium shape may mask the growth kinetic of the mean diameter due to the size effect.
where h is the height of ledges, ii their interspacing. ??+the diffusion coefficient of the solute. [(nD*/‘2)(h;i)] is approximately the total area of ledge risers. 2 is a proportionality coefficient between h and the width of the diffusion field. A non-anaiytica1 dependence of z with the supersaturation has been found by Jones and Trivedi [6] but with boundary conditions slightly different to those of our problem.
2.2 Mechanism qf the rcolution
The differential equation arising from F, = F, can of the thickness of precipitates [S]. In a similar way. we find the evolution of the aspect ratio of precipitates which is given by:
Weatherley [33 has considered the problem of the thermal nucleation of growth ledges at the precipitate surface. He has shown that the driving force necessary for this nucleation is about 0, l-l kcal*mole- t and that such a driving force is only found during the early stages of precipitation from a supersaturated solid solution or during a phase change. Thus, the nucleation rate of ledges is considerably reduced for low supersaturations as those involved with out-of equilibrium precipitates which. according to Ref. 143 are of the order 10e4. The ledges nucleation rate is thus the mechanism which controls the evolution towards the equilibrium. An electron microscope study confirm this assump tion. At the beginning of the 8’ phase formation (stages I and II). there are numerous growth ledges at the surface of precipitates [Fig. 2(a)]. On the other hand, micrographs in the coarsening stage [Fig. 2(b)] show a very low density of ledges: many precipitates exhibit no ledges at their interface. Some cases of homogeneous nucleation are observed but most of the ledges seem to be created by heterogeneous nucleation. Two mechanisms are possible: interactions between matrix dislocations and precipitates [4] or impingement between growing adjacent precipitates [33 [Fig. 2(c)]. 2.3 ~t.#~ltrj(~nqf the aspect ratio The previous discussion leads us to choose a model which takes into account the existence of ledges and their migration to describe the shape change of precipitates. Such a model has been developed by Shiflet et 01. [S]. They write the equality between the flux of solute F, due to the Gibbs-Thomson effect and the flux of solute FZ arising from the growth of the precipitate. For plate shape precipitates, we can write as in [S] :
1. _I__ -DRT A,,, I 1
2KncExcrP A
is the difference between equilibrium solute concentrations near the peripheral and the broad interface. Fz can be written: nD2
F2
2
kp
-(.&---5(C 4dt
de ’
-c,)---.
nD’
dr>
4 dr
(51
be used to find the evolution
(61
Br = @(A) - &&I.
A is the aspect ratio ai time t and A0 the initial one and :
where V, is the molecular volume of the precipitate. V its volume and 1 A’ ’ &A) = ,4’:3(A _ A,,,) - AdA - A,,) x ,n (A - A,,.)‘~” A’3 _ A”3
_ Ltan-
c9.
+---
1
2A5e*.’ A’ ’
1
.41”-
\3
18) 2.4 Quantitatit‘e
~1Ir~~~~itiot~
To verify equation (8) it is necessary to take into account the size distribution of precipitates. For each size class i, we define a mean volume Vb and the initial aspect ratio Ah. With these data each evolution A#,,, is calculated by means of relation (6) written: 6t = &A)
with
J+jp$23
(91
By interpolation on the obtained curves. isochronous values of Ai and corresponding Di and ci values are determined. The mean aspect ratio 2. thickness 2 and diameter B are derived from them. The experimental cases for which the shape evolution is preponderant are those of the Al-4 wt”,, Cu [T.I.] and the AI-3 wt”,, Cu [R] aged at 225 C (Part I). As the size-distribution of the Al-3 wt”,, Cu [R] exhibits an anomalous behaviour of @’plates. probably due to the beginning of the equilibrium 0 phase formation. [Fig. lo(c). part I] the most suitable experiment seems to be the case of the AHwtY, Cu Cf.1.J. With the following initial conditions: to = 20 h, A0 = 64. e. = 90 A. 5, = 6500.4 and the
MERLE
1932
AND
MERLIN:
COARSENING
OF 8’ PLATES
IN AI-&
ALLOYS-
a
b
C
Fig. 2. Lt :dges on the broad interfaces of precipitates: (a) Srugr I--.41-4 Sroqc If- -AI-3 WI”,, Cu [R] 250 h ;LI 225 C: IC) Sro
sii !e distribution.
with ;3 compu ter the evolutions are st lown on Fii g. 3. Theoretical
we
hate
calculated
of fi. L; and 2. Results curves are plottrd
as
functi Ion of I(:‘P. h(r - r,). On the same diagrams we results and tried to have plotted the: experimental reach the best:
fit between theoretical
and experimen-
ut”,, Cu [R] 15 h : ntrckatron by iml
_. “. (b)
:nt or
tal curves. Such a situation
is COI-rectly
obtain .ed
&r -
rO) = 10 h.
which
ro) 1 3.10’
with equations
and
(r -
for
ei ves
(7) and (9):
‘1 -_ z 2.10-s 21.
cm s-I
at ! 12s c.
10)
MERLE
AND
MERLIN:
COARSENING
OF 8’ PLATES IN AI-Cu ALLOYS-II
1933
Fig. 5. Diffusion paths along 8 precipitates of ledge sup-
ply: tat homogeneous nucleation: nucleation.
--
Fig. 3. Theoretical and experimental evolution of A. D. P. 41--4\\t”,, Cti [T I.] - r, = 125 C
The quantitative verification of relation (IO) is difficult as there are many uncertainties about the value of the different parameters. The discussion of their order of magnitude will allow us to say if this relation is physically meaningful. Jones and Trivedi‘s results [6] have been extrapolated by Shiflet ef ul. f7] towards low values of super‘saturation (Fig. 41. For (‘Jo: 5.10-* r is greater than IO&.The maximum supersaturation encountered during shape evolution is of this order. The minimal height of ledges being equal to the c-parameter of B t58A). the width of the diffusional field rh should then be several microns. This value is not reasonable for the case studied as the source of solute is localized at the periphery of the precipitate. Thus. the width of the diffusion-field is at the most of the order of D/2. Here the limits for the use of Jones and Trivedi’s model appear. Their boundary conditions are not those of the problem considered and we cannot take for I values deduced from Fig. 4. but rather values of the order of 1).‘2h i.e. 5.102 to 10”. A statistical measurement of i with a su~cient number of ledges is a rather difficult problem as interledge spacing is very erratic: several ledges may be observed on a single precipitate (Fig. 2 b). while they can be missing on adjacent precipitates. However qualitative electron microscope observations allow us
(bt heterogeneous
to say that the mean i value is of the order of the precipitates diameter. i.e. about lo4 A. Sankaran [S] gives i. z 1700 A for aging times near tr and aging temperatures of 200 and 250 C. These values agree with our estimates owing to the difficulty of ledge nucleation during coarsening. From our experimental results at 225 C and with z = lo3 and /. 2 lo4 A we obtain: 9 2 2.10-’ cmLls-t
(11)
This value is much greater than the self diffusivity of copper in alum~n~um which according to Anand et al. [93 is about LO- i4 cm’s_’ at this same temperature. This can be explained if we consider that the flux of soluie is directed from the periphery of the plate to the ledge along the matrix-precipitate interface. When ledges are homogeneously nucleated. the solute diffusion takes place mainly along the semi-coherent broad interface [Fig. 5(a)] but the diffusion may also take place along the peripheral interface in the case of heterogeneous nucleation for example [Fig. 5(b)]. The structure of this last interface is not well known but. owing to the great difference between the r-parameter of 0’ and that of the matrix, it is probably very similar to an incoherent one. So. 9 in this case might be close to p,,,,. interphase boundary diffusion coefficient between the matrix and the equilibrium P precipitate. According to Aaron and Aaronson ir,,,, is about 5.10-- cm2 s-r at 225-C [lo]. Thus. the order of magnitude that we find for 9 corresponds effectively to a mechanism of interfacial diffusion. The agreement between the model and the experimental results is rather convenient. owing to the assumptions and simplification we must make and the rather comptex character of the real diffusion mechanisms. 2.6 Remarks
Fig. 4. Variation of r with supersaturation Fig. 2 of Ref. [7]1.
(deduced from
2.6.1 Influence of elastic interactions. Study of the change in Young’s modulus during the 0’ phase precipitation shows an important change of the interaction energy matrix-precipitate during the coherency loss of precipitates [ 1I]. We have seen that no correlation exists between the variation of A and that of the coherency. The elastic interactions seem thus to have no preponderant influence on the evolution kinetics of precipitates. 2.6.2 Changes qf the size distribution. The changes we have studied seems to have no important influence
MERLE AND MERLIN:
1934
COARSENING OF B PLATFS IN Al-& ALLOYS-II
2
Fig. 6. Size distributions modifications due to shape evolution of precipitates; (1) Initial size distribution r0 = 20 h: (2) r - to = 500 h: (3) r = r.
2 65h
t
3. COARSENING
OF 0’ PLATES WITH EQUILIBRIUM ASPECT RATIO
During coarsening of Al-4 wt”; Cu [e5] and Al-5 wt9; Cu [RI, i is low but still varies (between 30 and 22 respectively. Part I Fig. 13). We have tried to obtain the lowest value of 2 and studied the coarsening of an Al-5 wt?I; Cu [es] aged at 225’C. The experimental results are the following: (a) the aspect ratio is the smallest that we have observed (between 20 and 17-Fig. 7). (b) Size-distributions are rather identical to those already obtained but they are particularly stable (until 1000 hr of aging Fig. 8). This fact means that the 0’ + 0 transformation is hindered. in as much as the 0’ precipitates have a lower aspect ratio. Therefore the observation_ domain of coarsening is extended. For high aging times the greatest precipitates take an octagonal shape but this does not appreciably influence the measure of the precipitates mean diameter assuming that they have a circular shape [12].
I‘
lo
170h
‘fIlTbtL& I
on the precipitate size-distributions which stay rather identical to those observed in other processes. This would mean that the shape evolution does not lead to important modifications of these distributions. To verify these assumptions we have calculated the theorFor etical evolution of the size distribution. I - t,, = 500 h. i.e. for a larger time than our last experimental observation, the size-distribution is very close to the initial one (Fig. 6). If we assume that all precipitates have reached their equilibrium aspect ratio (I = =z) the size distribution is shifted towards high values of D/B and the distribution is steeper but these modifications are still rather weak.
t
Fig.
2
3
2
8. Size distributions during coarsening AI-5 wt”; Cu [ES], TV = 225’C.
3
of an
(c) The mean thickness and the mean diameter grow approximately according a I’ 5 law (Fig. 9) as for the AI-Cu 4”,, [c5] and the AI-Cu 5”” [R]. Such a law is really representative of the behaviour of a population all the precipitates of which could have the equilibrium aspect ratio. since variations of 2 do not lead to important changes in the kinetics. (d) The aspect ratio are spread on both sides of the mean value x (Fig. 10). This results from the initial distribution of aspect ratio during growth which has not completely disappeared. That distribution is however rather stable during coarsening times observed and the growth kinetics of D and Z are thus identical. Such is not the case for the size distributions which are different. (e) The thickness distributions have been studied on populations of about 400 precipitates (Fig. 11). As a consequence of the aspect ratio distribution. they are closer to the theoretical distributions #h,,,, and 11’h,,,,., than the diameter distributions and they are limited to near e,‘r - 2 as for I” II,,,, , which is relative to an interfacial rate limited coarsening.
_ 10’
Fig. 7. Evolution of the aspect rat&-Al-5 T” = 225 c.
l@
‘h,
wt”. Cu [~5] -
Fig. 9. Evolution of D and E-AI-5 wt”,,Cu [es] - TV 225 c.
MERLE
AND
MERLIN:
COARSENlNG
OF 0’ PLATES IN AI-G
1935
ALLOYS-11
may be greater than that between the two precipitates. In that case growth ledges will be created on the surface of the two precipitates and will migrate under the influence of the sursaturation between the precipitates and the matrix. The effective diffusion may also be accelerated by short-circuit diffusion paths at the precipitate matrix interface. Situations where a precipitate dissolves to the advantage of the other may be encountered. However. owing to the size-distribution shape which rapidly decreases for great sizes. we will consider that the probability of dissolution is lower than that of growth. So. the behaviour of a precipitate (growth or dissolution\ is not fundamentally modified by the mechanisms we have considered and we will keep to the critical size (which is defined by the mean concentration in the matrix) its meaning of limit between growing and dissolving plates. This will be introduced mathematically in the growth equation of a precipitate by taking as driving force of the ledge migration. the difference between the equilibrium solute ~ncentration near the precipitate and the mean concentration C. We will take into account phenomenologically the acceleration of growth and dissolution due to the impingement of precipitate and the short-circuit diffusion paths created by the precipitates-matrix interfaces. by considering an effective diffusion coefficient 9,. the value of which might be intermediate between the interfacia1 diffusion coefficient Qi, and the volume diffusion coefficient ‘/? Moreover. assuming that all precipitates have the same aspect ratio A. allows us to consider in the same way both diameter or thickness growth. The equation of the solute flux F, and F2 derived in a way analog to that used to establish (4) and (5). cipitates
A 1
b
Fig. 10. Aspect-ratio distribution--Al-5 wt“, Cu [es] - & = 225 C: (a) I = 500 h: (b) t = 1000h. Ivery low aspect ratio {near .;? 31 are due in fact to anomalous contribution introduced by experimental error [12]).
(i) These experimental results lead us to consider with particular attention the behaviour of the precipitates thickness. Owing to the difficulty of ledge nucleation we can assume that coarsening is controlled by the thickness growth kinetic. As the diameter growth does not require any particular mechanism. this growth has just to ‘follow’ that of the thickness. (ii) Weatherley [13] has shown that the ledge nucleation is easier at the impingement of two precipitates as precipitate edges supply natural nucleation sites where no activation energy is necessary. So. as supersaturation, which can be the other driving force. is very low. we assume that ledge nucleation is essentially heterogeneous and due to the impingement of precipitates. The mean distance between ledges thus varies with the precipitate size.
Owing to the second hypothesis. the evolution of a precipitate is connected to that of its neighbours. So. the problem has no rigorous mathematical interpretation. We will try to simplify it with the following remarks, We consider an alloy state for which the mean solute concentration in the matrix is 2. This is the solute concentration in equilibrium with the critical precipitate of diameter D* and thickness e* such that D* = A c*.We assume that all the precipitates have the same aspect ratio. D, is the diameter of a growing precipitate (D, > D*)which impinges with a precipitate of diameter D2~c, and c2 are the equilibrium solute concentrations near these precipitates (cl < 7). > 7 and the precipitates of such a If D2 < D*.c'2 diameter should dissolve. The impingement with the growing precipitate of diameter Dt does not change this behaviour but the nucleation of dissolution ledges may be accelerated by the concentration gradient between the two precipitates. Growth and dissolution may also be accelerated by the diffusion of solute along the precipitate-matrix interfaces. If Dz > D*.czand c, may have similar values as D, and D2 are now both greater than D* and thus. the supersaturation between the matrix and the pre-
‘: SOOh
1
Fig. 11. Thickness distributions-Al-S 225 c.
:‘:lO@h
wt’, Cu [es] - X, =
MERLE
1936
MERLIN:
AND
COARSENING OF U’ PLATES IN AI-Cu ALLOYS--II Dependence between x und D*. At large precipitates sizes c( is proportional to D [7]. Moreover, owing to the impingement process. it is consistent to assume that the diffusion field is of the order of magnitude of
are :
I). Thus:
where
z = m2D*.
(18)
Putting (17) and (18) in (14). we obtain isthe difference between the equilibrium concentration near a precipitate of diameter D and P. F2 =
(13)
with f 19)
which gives dD .---= dt
8AVmc,a,g,
1
(14)
RT
Dependance of R on D
If we consider a unit volume of matrix containing N, precipitates, A varies as i/r, if rV is the number of imp~n~ments and as V& if S, is the total ‘area of broad interface in the unit volume. We assume that r, is the most probable number of impingements between N, precipitates of mean diameter D randomly distributed in the (100) planes of the matrix. An absolute determination of rv is complex owing to the non-random orientation of 0’ precipitates, but the variation law of ry with B may be determined. As the volume fraction is constant N, *+
The integration of equation (14) has been made by using Kahlweit’s method’[ 143. During coarsening the life story of a precipitate may be represented by the evolution of the quantity p = D/D* = eie*. Each curve of Fig. 12 is relative to a precipitate and. except for that of the largest particle, they ali pass through a maximum, decrease again and when they cross the line p = 1 the particle dissolves and disappears. Only the curve for the largest particle approaches the line P = I i.e. it is the only precipitate which remains when time leads to infinity. Kahlweit has shown that the asymptotic growth rate calculated by L.S.W. is in fact the growth rate for the largest particle for I = I,. (Fig. 12). (14) May be written.
and as the fraction of impingements ru/Nv is not a function of the number of precipitates if N, is suiTiciently great, we have r, x
dD dr=
D-’
121) Which gives, after integration DN11’2
;
(16)
/i 3; ,+i)+s!z
(17)
to each other.
P
1 ___+__ i
t
I t
‘”
_
@“2
=
llAVm,c,a, ‘I, -~ (f - fg). tcp - cx )RT mlmt
(22)
The growth equation of a spherical precipitate in the case of control by an interfacial reaction is written:
(15) and (16) lead to i, x bsiz which can be written
as D* and B are proportional
120)
For t = t, it can be easily shown [lZ’J that this equation becomes
1 F
this result is valid whatever may be the repartition of plates and the size-distribution, provided that they remain unchanged during coarsening. As, at constant volume fraction we have S,r
C p-l D*9’z _-. P
‘i
Fig. 12. Evolution during coarsening (Kahlweit [14]).
dr c’p-1 - = -dr r* p’
(23)
(20) and (23) present some analogy. The exponent of the critical size is different but for a given time. i.e. for a given critical size. growth or dissolution rates are identical functions of the deviation from this criticaf size. So. the size distribution might be the same in the two cases. Direct caiculation confirm this qualitative explanation [ 12). So. owing to the properties of the &It,,,, , distribution, we have b = 8!9D*
(24)
MERLE
AND
MERLIN:
COARSENlNC
OF 0’ PLATES
IN AI-Q
1937
ALLOYS-II
the comparison between the slope of experimental curves and their theoretical expressions (25 a and b) gives ‘YI, (26)
ftiZ
is about ;; i.e. 10R to IO9 m-
I
(27)
j. is comprised between I and 10 iurn (this maximum limit means that there is 1 ledge for 20 precipitates). As b is about 2500 A : 3.10-lnm-“?
< ni, < 3.10-”
mv3 2.
(28 t
With (26). (27) and (28) give: lo- ii < I/,. d 10-9cm’s-i.
Fig. 13 Comparison
of experimemai (27).
results
with equation
which gives with (22):
x (I - f(,J n7,m2
(25a)
or for thickness
(29)
This estimation is coherent with our results as its upper limit has a value of the same order as that estimated for Y, ,, and as its lower value is much greater than Y,.: acceieration of coarsening due to the process of impingement had led us to predict this. 3.4.2 Size, tksrrihurior~~. The difference between experimental and theoretical size distribution probably arise from the straightforward assumption we have made. It must however be noticed that the upper theoretical limit agrees rather well with the experimental one. 3.4.3 Rtwd crhn~rrrhici\rir.us
4. coYcLL:s1oIv 3.4.1 Grow& lore. Figure 13 shows that plot of D1’j2 - D”” or P1l z - e11’2 as function of (f - rob are effectively linear. The agreement is better for thickness than for diameter: this can be explained by the existence of the aspect ratio distribution. At 22S’C and with Vm2 2.84~10-5 ~37~ mole-‘. A - i7. CC = 0.0014, cp - (Ix z 0.33 and 6,. 2 0.3 Joule m -‘.
A coherent interpretation of the coarsening mechanisms of 0’ plates can be given if we consider that the precipitates thickness growth is controlled by the ledge migration on their broad faces. The study of the morphological evolution of plates has allowed us to evaluate the mean interfacial diffusion coefficient ‘2, z,j. Our experimental results on coarsening at con-
stant aspect ratio can be explained if we consider that the ledge nucleation is mainly heterogeneous and due to the impingement of precipitates. Results also show the importance of interfacial diffusion in coarsening kmetics. REFERENCES
m
E 1, to
1. P. Merle and F. Fouquet. Acra metall. 29, 1919 (1981). 11.FL I. Aaronson. C. Laird and K. R. Kinsman. Mechan00
too
500 *
Fig. 14. Comparison between thickness growth due to a shape evolution (1 I and a rz I’ law (21.
isms of dlffuslonal growth of precipitate crystals. ~rff~~s~~~~~ur~~fl.4..s.M. f 1970). 3. G. f. Weatherly. AUG mefali. 19. 181 (1971).
4. C. Laird and W. 1. Aaronson. A.I.M.E. 242. 1393 (1968).
Phase
Truns. mernll. Sot.
1938
MERLE
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MERLIN:
COARSENING
5. G. J. Shiflet, H. I. Aaronson and T. H. Courtney, Scripru metall. 1I, 677 ( 1977). 6. G. J. Jones and R. Trivedi. J. J. oppl. Phys. 42, 4299 (1971). 7. G. J. Shiflet. H. I. Aaronson and T. H. Courtney, Acre metal/. 27, 377 (1979). 8. R. Sankaran. Ph. D. Thesis. University of Pennsylvania (1973). 9. M. S. Anand. S. P. Murarka and R. P. Agarwala, 1. oppl. Phys.34, 3860 (1965). 10. H. B. Aaron and H. I. Aaronson. Acta metoll. 16, 789 (1968). t 1. F. Fouquet, Thesis Lyon (1977). 12. P. Merle, Thesis Lyon (1980). 13. G. C. Weatherly. Acra meroll. 18, 15 (1970). 14. M. Kahlweit, Adr. Colloid Inr. Sci. 5, I (1975). 15. M. Ferrante and R. D. Doherty, Acta meroll. 27, 1603 (1979). 16. J. P. Hirth and J. Lothe. The0r.v of ~isioc~ri~~s. McGraw-Hill, New York (1968).
OF ETPLATES IN Al-&
ALLOYS-II
The corresponding concentration of solute are:
(A31
(A41 The mean solute concentration r in the matrix is:
r-c,(l+gg). D* being the critical diameter. From (Al) and (AZ) we have:
(A61
APPENDIX As shown by Ferrante and Doherty[lQ the chemical potential 9. of the solute in equilibrium with the peripheral interface of a precipitate of diameter D and aspect ratio A iS
if pr is the chemical potential of the solute in equilibrium with a precipitate of infinite radius. In the same way. the chemical potential p. of the solute in equilibrium with the broad face of the precipitate is:
(A21
(AS)
002--F
c,4u,v,
!
1 1 E--F.
)
(A71
a, is not well known as the tr c-parameter and that of the matrix are very different. it can be estimated to 0.3 J/m’ which is haif the interfacial energy of high-desoriented grain boundary in a aluminium [ 161. Thus we obtain, with values already given for cpr c, and V, and with D in meters: wi =
1.7*10-” B
a2
_A_ _ 1 ( A*+
)
(‘48) (A91
Abstract-A theoretical study was made of the two coarsening mechanisms of 0’ plates: 1. morphological evolution and 2. coarsening at constant aspect ratio. In both of them ledge migration on the broad face of prectpitates is found to be the controlling process. The analysis of the first mechanism leads to an evaluation of the mean diffusion coefficient of copper m the a’-matrix interface. For the second mechanism. the hypothesis of an heterogeneous nucleation of ledges due to the impingement of precipitates leads to a linear theoretical relation between the mean diameter D and r* I1 This relation agrees rather well with the experimental results. R&urn&-Nous avons etudie th~oriquement les deun processus de coaiescence des precipites 0’: 1. evolution morphologique et 2. coalescence a rapport d’aspect constant. Nous trouvons que dans les deux cas la migration des marches existant sur I’interface large prtcipiti-matrice est le processus controlant I’evolution. L’analyse du premier mecanisme condun a une evaluation du coefficient moyen de diffusion du cuivre a I-interface matrice-pricipite Q’. Dans le cas du deuxieme mtcanisme. I’hypothese dune nucleation htttrogine des marches due a la rencontre entre precipitis nous conduit a une relation theorique lintaire enfre le diamttre moyen ~ et rz *I. Cette relation est en assez bon accord avec les r&hats ex~rimentaux. Zusammanfassung7Zwei Vergroberungsmechanismen von 0’-Platten wurden theoretisch untersucht: 1. morphologische Anderung und 2. Vergroberung bei konstantem Formfaktor. Bei beiden Mechanismen besteht der bestimmende Prozess im Wandern von Vorsphingen iiber die breite FlPche der Ausscheidungen. Die Analyse des ersten Prozesses fiihrt zu einer Auswertung des mittleren Diffusionskoeffizienten von Kupfer in der Grenzfliche zwischen Ausscheidung und Matrix. Bei dem zweiten Prozess fiihrt die Annahme. da8 Vorspriinge sich helerogen bilden. zu einem Iinearen theoretischen Zusammenhang zwischen dem mittieren Durchmesser D und t “I’ Dieser Zusammenhang stimmt ziemhch gut mit den experimentellen Ergebnissen iiberein.
I. INTRODUCTION The aim of this paper is to give an interpretation
of the evolution kinetics corresponding to the two basic coarsening processes of 0’ plates as shown in our previous experimental study (Part 1 Cl]). -Morphological evolution at constant volume for each precipitate. -Coarsening with the equilibrium aspect ratio for all precipitates.
2. MORPHOLOGICAL
EVOLl_;TIOIV OF 8’ PLATES
Two alternative assumptions may be made; (a) the evolution of the aspect ratio .4 is due to a progressive modification of the interracial energy of the broad faces of 6’ plates i.e., at any time. the precipitates have their equilibrium shape, but this shape changes with time. (b) High values of A are characteristic of an out of equilibrium population of precipitates. According to Aaronson’s hypothesis [2] such a situation is possible, owing to the inhibition of thickness growth due to the ledge mechanism.
An electron microscope study on an Al-4 wt”,, Cu CR]. allows us to discard the first assumption, as no correlation exists between the evolution of the broad interface coherence and that of the mean aspect ratio (Fig. I) (see Fig. 8, part I for 2 variations). At high temperatures. we observe a progressive loss of coherency and a simultaneous decrease of 2. For an aging of 24 h at 275 C. edge interracial dislocations of Burger’s vector b = I:2 i 100) begin to appear on most of the precipitates [Fig. l(d)}. A complete network is formed for higher aging times [Fig. l(e) - 120 h at 275 CJ. But, for short aging times [2 h at 275-C. Fig. l(c)] 2 has already decreased while the precipitates have not lost their coherency. Besides. at low temperatures and for high aging times (100 h a 225’C. 540 h at 200 C) broad faces of precipitates are still coherent [Fig. I (a) and (b)]. At these aging temperatures x decreases sooner (10 h at 225-C and 50 h at 200 C) and for times cited above, 2 is about half its maximum value. The decrease of the aspect ratio of a precipitate is thus due to its evoiution to its equilibrium shape.
defined by A,,, = CF,:~~op and or, being the interfacial energies of the peripheral and of the broad interfaces. If A > A,,. the solute concentration near the peripheral interface due to the Gibbs-Thomson effect, is
19‘9
1930
AND MERLIN:
MERLE
COARSENING
OF D PLATES
IN AI-CU ALLOYS-II
b
a
d
e Fig. 1. Coherency state of matrix-precipitates interface Al -Cu 4 wt”,, [R]: (al 540 h at X0 C:(b) 100 h at 22s C: (cl 2 h at 275 C: (d) 24 h at 275 C: (e) I20 h at 275 C (dark tield g [EOJL
greater than that near the broad face and a flux of solute will take place between the two interfaces. The corresponding driving force must be greater than the one deriving from the Gibbs-Thomson effect associated to the size of the precipitate and giving rise to a flux of solute between the matrix and the precipitate. The supersaturation (Ok near a precipitate is defined by
(’ COO
z
-
CO
:
t -.
-
CO
(1) CP Where co. i: and cP are the solute concentrations near the precipitate-matrix interface. in the matrix and in the precipitate. By analogy the supersaturation associated to an out-of equilibrium morphology may cg - co
be written c, -
(01
=
CD
-------
cp - cr
c, -
CD
(‘P
(2)
where ct. c,,and cD are the solute concentrations near a precipitate of infinite size, near the peripheral and the broad interface. Calculation of o). and 01~ for an Al-4 wt”; Cu aged at 225 C leads to (Appendix A):
(‘JO
2 D*
(3)
MERLE
AND
MERLIN:
COARSENING
1931
OF 0’ PLATES IN AI-Cu ALLOYS-II
If being the precipitate diameter and D* the critical diameter; for growing precipitates (I) > D*), D/D* has values between 1 and 3 (see shape of size-distributions-Part I). For out-of equilibrium precipitates A may be 20&250. A,, is about 15. Thus w,/oO may be greater than unity. Moreover, the diffusion fields relative to these supersaturations are very different: the effective diffusion distance between the precipitate and the matrix is of the order of (0#‘2, i.e. a few hundred of angstroems for coarsening stage, while it is much less between the two faces of the same precipitate. So. even when the two supersaturations too and (ttr are of the same order, the con~ntration gradient associated to w, is the most important. This explain why a deviation from the equilibrium shape may mask the growth kinetic of the mean diameter due to the size effect.
where h is the height of ledges, ii their interspacing. ??+the diffusion coefficient of the solute. [(nD*/‘2)(h;i)] is approximately the total area of ledge risers. 2 is a proportionality coefficient between h and the width of the diffusion field. A non-anaiytica1 dependence of z with the supersaturation has been found by Jones and Trivedi [6] but with boundary conditions slightly different to those of our problem.
2.2 Mechanism qf the rcolution
The differential equation arising from F, = F, can of the thickness of precipitates [S]. In a similar way. we find the evolution of the aspect ratio of precipitates which is given by:
Weatherley [33 has considered the problem of the thermal nucleation of growth ledges at the precipitate surface. He has shown that the driving force necessary for this nucleation is about 0, l-l kcal*mole- t and that such a driving force is only found during the early stages of precipitation from a supersaturated solid solution or during a phase change. Thus, the nucleation rate of ledges is considerably reduced for low supersaturations as those involved with out-of equilibrium precipitates which. according to Ref. 143 are of the order 10e4. The ledges nucleation rate is thus the mechanism which controls the evolution towards the equilibrium. An electron microscope study confirm this assump tion. At the beginning of the 8’ phase formation (stages I and II). there are numerous growth ledges at the surface of precipitates [Fig. 2(a)]. On the other hand, micrographs in the coarsening stage [Fig. 2(b)] show a very low density of ledges: many precipitates exhibit no ledges at their interface. Some cases of homogeneous nucleation are observed but most of the ledges seem to be created by heterogeneous nucleation. Two mechanisms are possible: interactions between matrix dislocations and precipitates [4] or impingement between growing adjacent precipitates [33 [Fig. 2(c)]. 2.3 ~t.#~ltrj(~nqf the aspect ratio The previous discussion leads us to choose a model which takes into account the existence of ledges and their migration to describe the shape change of precipitates. Such a model has been developed by Shiflet et 01. [S]. They write the equality between the flux of solute F, due to the Gibbs-Thomson effect and the flux of solute FZ arising from the growth of the precipitate. For plate shape precipitates, we can write as in [S] :
1. _I__ -DRT A,,, I 1
2KncExcrP A
is the difference between equilibrium solute concentrations near the peripheral and the broad interface. Fz can be written: nD2
F2
2
kp
-(.&---5(C 4dt
de ’
-c,)---.
nD’
dr>
4 dr
(51
be used to find the evolution
(61
Br = @(A) - &&I.
A is the aspect ratio ai time t and A0 the initial one and :
where V, is the molecular volume of the precipitate. V its volume and 1 A’ ’ &A) = ,4’:3(A _ A,,,) - AdA - A,,) x ,n (A - A,,.)‘~” A’3 _ A”3
_ Ltan-
c9.
+---
1
2A5e*.’ A’ ’
1
.41”-
\3
18) 2.4 Quantitatit‘e
~1Ir~~~~itiot~
To verify equation (8) it is necessary to take into account the size distribution of precipitates. For each size class i, we define a mean volume Vb and the initial aspect ratio Ah. With these data each evolution A#,,, is calculated by means of relation (6) written: 6t = &A)
with
J+jp$23
(91
By interpolation on the obtained curves. isochronous values of Ai and corresponding Di and ci values are determined. The mean aspect ratio 2. thickness 2 and diameter B are derived from them. The experimental cases for which the shape evolution is preponderant are those of the Al-4 wt”,, Cu [T.I.] and the AI-3 wt”,, Cu [R] aged at 225 C (Part I). As the size-distribution of the Al-3 wt”,, Cu [R] exhibits an anomalous behaviour of @’plates. probably due to the beginning of the equilibrium 0 phase formation. [Fig. lo(c). part I] the most suitable experiment seems to be the case of the AHwtY, Cu Cf.1.J. With the following initial conditions: to = 20 h, A0 = 64. e. = 90 A. 5, = 6500.4 and the
MERLE
1932
AND
MERLIN:
COARSENING
OF 8’ PLATES
IN AI-&
ALLOYS-
a
b
C
Fig. 2. Lt :dges on the broad interfaces of precipitates: (a) Srugr I--.41-4 Sroqc If- -AI-3 WI”,, Cu [R] 250 h ;LI 225 C: IC) Sro
sii !e distribution.
with ;3 compu ter the evolutions are st lown on Fii g. 3. Theoretical
we
hate
calculated
of fi. L; and 2. Results curves are plottrd
as
functi Ion of I(:‘P. h(r - r,). On the same diagrams we results and tried to have plotted the: experimental reach the best:
fit between theoretical
and experimen-
ut”,, Cu [R] 15 h : ntrckatron by iml
_. “. (b)
:nt or
tal curves. Such a situation
is COI-rectly
obtain .ed
&r -
rO) = 10 h.
which
ro) 1 3.10’
with equations
and
(r -
for
ei ves
(7) and (9):
‘1 -_ z 2.10-s 21.
cm s-I
at ! 12s c.
10)
MERLE
AND
MERLIN:
COARSENING
OF 8’ PLATES IN AI-Cu ALLOYS-II
1933
Fig. 5. Diffusion paths along 8 precipitates of ledge sup-
ply: tat homogeneous nucleation: nucleation.
--
Fig. 3. Theoretical and experimental evolution of A. D. P. 41--4\\t”,, Cti [T I.] - r, = 125 C
The quantitative verification of relation (IO) is difficult as there are many uncertainties about the value of the different parameters. The discussion of their order of magnitude will allow us to say if this relation is physically meaningful. Jones and Trivedi‘s results [6] have been extrapolated by Shiflet ef ul. f7] towards low values of super‘saturation (Fig. 41. For (‘Jo: 5.10-* r is greater than IO&.The maximum supersaturation encountered during shape evolution is of this order. The minimal height of ledges being equal to the c-parameter of B t58A). the width of the diffusional field rh should then be several microns. This value is not reasonable for the case studied as the source of solute is localized at the periphery of the precipitate. Thus. the width of the diffusion-field is at the most of the order of D/2. Here the limits for the use of Jones and Trivedi’s model appear. Their boundary conditions are not those of the problem considered and we cannot take for I values deduced from Fig. 4. but rather values of the order of 1).‘2h i.e. 5.102 to 10”. A statistical measurement of i with a su~cient number of ledges is a rather difficult problem as interledge spacing is very erratic: several ledges may be observed on a single precipitate (Fig. 2 b). while they can be missing on adjacent precipitates. However qualitative electron microscope observations allow us
(bt heterogeneous
to say that the mean i value is of the order of the precipitates diameter. i.e. about lo4 A. Sankaran [S] gives i. z 1700 A for aging times near tr and aging temperatures of 200 and 250 C. These values agree with our estimates owing to the difficulty of ledge nucleation during coarsening. From our experimental results at 225 C and with z = lo3 and /. 2 lo4 A we obtain: 9 2 2.10-’ cmLls-t
(11)
This value is much greater than the self diffusivity of copper in alum~n~um which according to Anand et al. [93 is about LO- i4 cm’s_’ at this same temperature. This can be explained if we consider that the flux of soluie is directed from the periphery of the plate to the ledge along the matrix-precipitate interface. When ledges are homogeneously nucleated. the solute diffusion takes place mainly along the semi-coherent broad interface [Fig. 5(a)] but the diffusion may also take place along the peripheral interface in the case of heterogeneous nucleation for example [Fig. 5(b)]. The structure of this last interface is not well known but. owing to the great difference between the r-parameter of 0’ and that of the matrix, it is probably very similar to an incoherent one. So. 9 in this case might be close to p,,,,. interphase boundary diffusion coefficient between the matrix and the equilibrium P precipitate. According to Aaron and Aaronson ir,,,, is about 5.10-- cm2 s-r at 225-C [lo]. Thus. the order of magnitude that we find for 9 corresponds effectively to a mechanism of interfacial diffusion. The agreement between the model and the experimental results is rather convenient. owing to the assumptions and simplification we must make and the rather comptex character of the real diffusion mechanisms. 2.6 Remarks
Fig. 4. Variation of r with supersaturation Fig. 2 of Ref. [7]1.
(deduced from
2.6.1 Influence of elastic interactions. Study of the change in Young’s modulus during the 0’ phase precipitation shows an important change of the interaction energy matrix-precipitate during the coherency loss of precipitates [ 1I]. We have seen that no correlation exists between the variation of A and that of the coherency. The elastic interactions seem thus to have no preponderant influence on the evolution kinetics of precipitates. 2.6.2 Changes qf the size distribution. The changes we have studied seems to have no important influence
MERLE AND MERLIN:
1934
COARSENING OF B PLATFS IN Al-& ALLOYS-II
2
Fig. 6. Size distributions modifications due to shape evolution of precipitates; (1) Initial size distribution r0 = 20 h: (2) r - to = 500 h: (3) r = r.
2 65h
t
3. COARSENING
OF 0’ PLATES WITH EQUILIBRIUM ASPECT RATIO
During coarsening of Al-4 wt”; Cu [e5] and Al-5 wt9; Cu [RI, i is low but still varies (between 30 and 22 respectively. Part I Fig. 13). We have tried to obtain the lowest value of 2 and studied the coarsening of an Al-5 wt?I; Cu [es] aged at 225’C. The experimental results are the following: (a) the aspect ratio is the smallest that we have observed (between 20 and 17-Fig. 7). (b) Size-distributions are rather identical to those already obtained but they are particularly stable (until 1000 hr of aging Fig. 8). This fact means that the 0’ + 0 transformation is hindered. in as much as the 0’ precipitates have a lower aspect ratio. Therefore the observation_ domain of coarsening is extended. For high aging times the greatest precipitates take an octagonal shape but this does not appreciably influence the measure of the precipitates mean diameter assuming that they have a circular shape [12].
I‘
lo
170h
‘fIlTbtL& I
on the precipitate size-distributions which stay rather identical to those observed in other processes. This would mean that the shape evolution does not lead to important modifications of these distributions. To verify these assumptions we have calculated the theorFor etical evolution of the size distribution. I - t,, = 500 h. i.e. for a larger time than our last experimental observation, the size-distribution is very close to the initial one (Fig. 6). If we assume that all precipitates have reached their equilibrium aspect ratio (I = =z) the size distribution is shifted towards high values of D/B and the distribution is steeper but these modifications are still rather weak.
t
Fig.
2
3
2
8. Size distributions during coarsening AI-5 wt”; Cu [ES], TV = 225’C.
3
of an
(c) The mean thickness and the mean diameter grow approximately according a I’ 5 law (Fig. 9) as for the AI-Cu 4”,, [c5] and the AI-Cu 5”” [R]. Such a law is really representative of the behaviour of a population all the precipitates of which could have the equilibrium aspect ratio. since variations of 2 do not lead to important changes in the kinetics. (d) The aspect ratio are spread on both sides of the mean value x (Fig. 10). This results from the initial distribution of aspect ratio during growth which has not completely disappeared. That distribution is however rather stable during coarsening times observed and the growth kinetics of D and Z are thus identical. Such is not the case for the size distributions which are different. (e) The thickness distributions have been studied on populations of about 400 precipitates (Fig. 11). As a consequence of the aspect ratio distribution. they are closer to the theoretical distributions #h,,,, and 11’h,,,,., than the diameter distributions and they are limited to near e,‘r - 2 as for I” II,,,, , which is relative to an interfacial rate limited coarsening.
_ 10’
Fig. 7. Evolution of the aspect rat&-Al-5 T” = 225 c.
l@
‘h,
wt”. Cu [~5] -
Fig. 9. Evolution of D and E-AI-5 wt”,,Cu [es] - TV 225 c.
MERLE
AND
MERLIN:
COARSENlNG
OF 0’ PLATES IN AI-G
1935
ALLOYS-11
may be greater than that between the two precipitates. In that case growth ledges will be created on the surface of the two precipitates and will migrate under the influence of the sursaturation between the precipitates and the matrix. The effective diffusion may also be accelerated by short-circuit diffusion paths at the precipitate matrix interface. Situations where a precipitate dissolves to the advantage of the other may be encountered. However. owing to the size-distribution shape which rapidly decreases for great sizes. we will consider that the probability of dissolution is lower than that of growth. So. the behaviour of a precipitate (growth or dissolution\ is not fundamentally modified by the mechanisms we have considered and we will keep to the critical size (which is defined by the mean concentration in the matrix) its meaning of limit between growing and dissolving plates. This will be introduced mathematically in the growth equation of a precipitate by taking as driving force of the ledge migration. the difference between the equilibrium solute ~ncentration near the precipitate and the mean concentration C. We will take into account phenomenologically the acceleration of growth and dissolution due to the impingement of precipitate and the short-circuit diffusion paths created by the precipitates-matrix interfaces. by considering an effective diffusion coefficient 9,. the value of which might be intermediate between the interfacia1 diffusion coefficient Qi, and the volume diffusion coefficient ‘/? Moreover. assuming that all precipitates have the same aspect ratio A. allows us to consider in the same way both diameter or thickness growth. The equation of the solute flux F, and F2 derived in a way analog to that used to establish (4) and (5). cipitates
A 1
b
Fig. 10. Aspect-ratio distribution--Al-5 wt“, Cu [es] - & = 225 C: (a) I = 500 h: (b) t = 1000h. Ivery low aspect ratio {near .;? 31 are due in fact to anomalous contribution introduced by experimental error [12]).
(i) These experimental results lead us to consider with particular attention the behaviour of the precipitates thickness. Owing to the difficulty of ledge nucleation we can assume that coarsening is controlled by the thickness growth kinetic. As the diameter growth does not require any particular mechanism. this growth has just to ‘follow’ that of the thickness. (ii) Weatherley [13] has shown that the ledge nucleation is easier at the impingement of two precipitates as precipitate edges supply natural nucleation sites where no activation energy is necessary. So. as supersaturation, which can be the other driving force. is very low. we assume that ledge nucleation is essentially heterogeneous and due to the impingement of precipitates. The mean distance between ledges thus varies with the precipitate size.
Owing to the second hypothesis. the evolution of a precipitate is connected to that of its neighbours. So. the problem has no rigorous mathematical interpretation. We will try to simplify it with the following remarks, We consider an alloy state for which the mean solute concentration in the matrix is 2. This is the solute concentration in equilibrium with the critical precipitate of diameter D* and thickness e* such that D* = A c*.We assume that all the precipitates have the same aspect ratio. D, is the diameter of a growing precipitate (D, > D*)which impinges with a precipitate of diameter D2~c, and c2 are the equilibrium solute concentrations near these precipitates (cl < 7). > 7 and the precipitates of such a If D2 < D*.c'2 diameter should dissolve. The impingement with the growing precipitate of diameter Dt does not change this behaviour but the nucleation of dissolution ledges may be accelerated by the concentration gradient between the two precipitates. Growth and dissolution may also be accelerated by the diffusion of solute along the precipitate-matrix interfaces. If Dz > D*.czand c, may have similar values as D, and D2 are now both greater than D* and thus. the supersaturation between the matrix and the pre-
‘: SOOh
1
Fig. 11. Thickness distributions-Al-S 225 c.
:‘:lO@h
wt’, Cu [es] - X, =
MERLE
1936
MERLIN:
AND
COARSENING OF U’ PLATES IN AI-Cu ALLOYS--II Dependence between x und D*. At large precipitates sizes c( is proportional to D [7]. Moreover, owing to the impingement process. it is consistent to assume that the diffusion field is of the order of magnitude of
are :
I). Thus:
where
z = m2D*.
(18)
Putting (17) and (18) in (14). we obtain isthe difference between the equilibrium concentration near a precipitate of diameter D and P. F2 =
(13)
with f 19)
which gives dD .---= dt
8AVmc,a,g,
1
(14)
RT
Dependance of R on D
If we consider a unit volume of matrix containing N, precipitates, A varies as i/r, if rV is the number of imp~n~ments and as V& if S, is the total ‘area of broad interface in the unit volume. We assume that r, is the most probable number of impingements between N, precipitates of mean diameter D randomly distributed in the (100) planes of the matrix. An absolute determination of rv is complex owing to the non-random orientation of 0’ precipitates, but the variation law of ry with B may be determined. As the volume fraction is constant N, *+
The integration of equation (14) has been made by using Kahlweit’s method’[ 143. During coarsening the life story of a precipitate may be represented by the evolution of the quantity p = D/D* = eie*. Each curve of Fig. 12 is relative to a precipitate and. except for that of the largest particle, they ali pass through a maximum, decrease again and when they cross the line p = 1 the particle dissolves and disappears. Only the curve for the largest particle approaches the line P = I i.e. it is the only precipitate which remains when time leads to infinity. Kahlweit has shown that the asymptotic growth rate calculated by L.S.W. is in fact the growth rate for the largest particle for I = I,. (Fig. 12). (14) May be written.
and as the fraction of impingements ru/Nv is not a function of the number of precipitates if N, is suiTiciently great, we have r, x
dD dr=
D-’
121) Which gives, after integration DN11’2
;
(16)
/i 3; ,+i)+s!z
(17)
to each other.
P
1 ___+__ i
t
I t
‘”
_
@“2
=
llAVm,c,a, ‘I, -~ (f - fg). tcp - cx )RT mlmt
(22)
The growth equation of a spherical precipitate in the case of control by an interfacial reaction is written:
(15) and (16) lead to i, x bsiz which can be written
as D* and B are proportional
120)
For t = t, it can be easily shown [lZ’J that this equation becomes
1 F
this result is valid whatever may be the repartition of plates and the size-distribution, provided that they remain unchanged during coarsening. As, at constant volume fraction we have S,r
C p-l D*9’z _-. P
‘i
Fig. 12. Evolution during coarsening (Kahlweit [14]).
dr c’p-1 - = -dr r* p’
(23)
(20) and (23) present some analogy. The exponent of the critical size is different but for a given time. i.e. for a given critical size. growth or dissolution rates are identical functions of the deviation from this criticaf size. So. the size distribution might be the same in the two cases. Direct caiculation confirm this qualitative explanation [ 12). So. owing to the properties of the &It,,,, , distribution, we have b = 8!9D*
(24)
MERLE
AND
MERLIN:
COARSENlNC
OF 0’ PLATES
IN AI-Q
1937
ALLOYS-II
the comparison between the slope of experimental curves and their theoretical expressions (25 a and b) gives ‘YI, (26)
ftiZ
is about ;; i.e. 10R to IO9 m-
I
(27)
j. is comprised between I and 10 iurn (this maximum limit means that there is 1 ledge for 20 precipitates). As b is about 2500 A : 3.10-lnm-“?
< ni, < 3.10-”
mv3 2.
(28 t
With (26). (27) and (28) give: lo- ii < I/,. d 10-9cm’s-i.
Fig. 13 Comparison
of experimemai (27).
results
with equation
which gives with (22):
x (I - f(,J n7,m2
(25a)
or for thickness
(29)
This estimation is coherent with our results as its upper limit has a value of the same order as that estimated for Y, ,, and as its lower value is much greater than Y,.: acceieration of coarsening due to the process of impingement had led us to predict this. 3.4.2 Size, tksrrihurior~~. The difference between experimental and theoretical size distribution probably arise from the straightforward assumption we have made. It must however be noticed that the upper theoretical limit agrees rather well with the experimental one. 3.4.3 Rtwd crhn~rrrhici\rir.us
4. coYcLL:s1oIv 3.4.1 Grow& lore. Figure 13 shows that plot of D1’j2 - D”” or P1l z - e11’2 as function of (f - rob are effectively linear. The agreement is better for thickness than for diameter: this can be explained by the existence of the aspect ratio distribution. At 22S’C and with Vm2 2.84~10-5 ~37~ mole-‘. A - i7. CC = 0.0014, cp - (Ix z 0.33 and 6,. 2 0.3 Joule m -‘.
A coherent interpretation of the coarsening mechanisms of 0’ plates can be given if we consider that the precipitates thickness growth is controlled by the ledge migration on their broad faces. The study of the morphological evolution of plates has allowed us to evaluate the mean interfacial diffusion coefficient ‘2, z,j. Our experimental results on coarsening at con-
stant aspect ratio can be explained if we consider that the ledge nucleation is mainly heterogeneous and due to the impingement of precipitates. Results also show the importance of interfacial diffusion in coarsening kmetics. REFERENCES
m
E 1, to
1. P. Merle and F. Fouquet. Acra metall. 29, 1919 (1981). 11.FL I. Aaronson. C. Laird and K. R. Kinsman. Mechan00
too
500 *
Fig. 14. Comparison between thickness growth due to a shape evolution (1 I and a rz I’ law (21.
isms of dlffuslonal growth of precipitate crystals. ~rff~~s~~~~~ur~~fl.4..s.M. f 1970). 3. G. f. Weatherly. AUG mefali. 19. 181 (1971).
4. C. Laird and W. 1. Aaronson. A.I.M.E. 242. 1393 (1968).
Phase
Truns. mernll. Sot.
1938
MERLE
AND
MERLIN:
COARSENING
5. G. J. Shiflet, H. I. Aaronson and T. H. Courtney, Scripru metall. 1I, 677 ( 1977). 6. G. J. Jones and R. Trivedi. J. J. oppl. Phys. 42, 4299 (1971). 7. G. J. Shiflet. H. I. Aaronson and T. H. Courtney, Acre metal/. 27, 377 (1979). 8. R. Sankaran. Ph. D. Thesis. University of Pennsylvania (1973). 9. M. S. Anand. S. P. Murarka and R. P. Agarwala, 1. oppl. Phys.34, 3860 (1965). 10. H. B. Aaron and H. I. Aaronson. Acta metoll. 16, 789 (1968). t 1. F. Fouquet, Thesis Lyon (1977). 12. P. Merle, Thesis Lyon (1980). 13. G. C. Weatherly. Acra meroll. 18, 15 (1970). 14. M. Kahlweit, Adr. Colloid Inr. Sci. 5, I (1975). 15. M. Ferrante and R. D. Doherty, Acta meroll. 27, 1603 (1979). 16. J. P. Hirth and J. Lothe. The0r.v of ~isioc~ri~~s. McGraw-Hill, New York (1968).
OF ETPLATES IN Al-&
ALLOYS-II
The corresponding concentration of solute are:
(A31
(A41 The mean solute concentration r in the matrix is:
r-c,(l+gg). D* being the critical diameter. From (Al) and (AZ) we have:
(A61
APPENDIX As shown by Ferrante and Doherty[lQ the chemical potential 9. of the solute in equilibrium with the peripheral interface of a precipitate of diameter D and aspect ratio A iS
if pr is the chemical potential of the solute in equilibrium with a precipitate of infinite radius. In the same way. the chemical potential p. of the solute in equilibrium with the broad face of the precipitate is:
(A21
(AS)
002--F
c,4u,v,
!
1 1 E--F.
)
(A71
a, is not well known as the tr c-parameter and that of the matrix are very different. it can be estimated to 0.3 J/m’ which is haif the interfacial energy of high-desoriented grain boundary in a aluminium [ 161. Thus we obtain, with values already given for cpr c, and V, and with D in meters: wi =
1.7*10-” B
a2
_A_ _ 1 ( A*+
)
(‘48) (A91