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Dissolution kinetics of grain boundary allotriomorphs
DISSOLUTION
KINETICS A.
OF GRAIN
PASPARAKISt
EtOtikDARY
ALLOTRIOMORPHS*
and L. C. BROWNt
Observations havebeenmade of the dissolution kinetics of individual grain boundary allotriomorphs in the Al_Cu and Al-Ag systems using the electron probe microenalyzer. It is found that the length and width of the precipitates decrease exponentially with time. This behavior is explained on the besis of a model in which it is considered that impingement of the diffusion fields from adj8cent precipitates takes piece early in the dissolution process. The diffusion field in the solid solution surrounding an allotnomorph is thus Ch8r8Ct&StiC of 8 finite system. The model gives values of the diffusion coefficient that are m edequate agreement with literature values. Experimental observations show that the axial ratio of the precipitates generally changes by less than 300/Lduring dlssolutlon. At high homologous temperatures, there is an increase in axial ratio, whilst et 1OWtemper8tUreS the 8Xi81 ratio remains essentially constctnt or decreases slightly. These variations in 8Xi81 ratio depend only on the homologous tempersture and are independent of the degree of supersaturation. At high homologous temperatures, It is proposed that impingement of the diffusion fields attenuates the point effect of diffusion 8t the tip of the preclpltate. This attenuation causes a decrease in the dissolution rate of the tip and thus Increases the axial r8tio of the precipitate. At low homologous temperatures, it is proposed that grain boundary diffusion always allows the sllotriomorph tip to dissolve preferentially, thus maintaining the axial ratlo. CINETIQUE DE LA DISSOLUTION DES ALLOTRIOMORPHES AUli JOINTS DE GRAINS Des observations de la cin&ique de dissolution des allotriomorphes isoles 8UX joints de grains, effect&es dans les systemes Al-Cu et Al-Ag par microanelyse B sonde Blectronique, ont montr6 que la longueur et la largeur des p&ipit& d6croft exponentiellement en fonction du temps. Ce comportement est interpr6t.6 en B partir d’un modble supposant que l’empi%ment des champs de diffusion des prCcipit6.q adjacents se produit d&s le debut du processus de dissolution. Le champ de diffusion de la solution solide entourrtnt un allotriomorpheest ainsi caract&istique d’un systhme flni. Le modble donne pour le coefficient de diffusion des veleurs qui sont en bon accord 8vec les valeurs donn6es dans la litt&ature. Lea observations exp&imenteles montrent que le rapport axial des pr6cipit6s varie en g&&al de moms de 30% au tours de la dissolution. Aux temp&atures homologues 6levBes, il se produit une augmentetion du rapport axial, alors qu’aux basses temperatures il reste Con&ant ou diminue lbg&ement. C%sverlations du rapport axial dependent seulement de la temp&ature homologue et sont independantes du degr6 de sursaturation. Les auteurs proposent une interpr6tation suivant lequelle, 8UX temp6ratures homologues Blevbs, l’empi8tement des champs de diffusion attbnuerait l’effet lOC81de 18 diffusion sur le bord du pr6cipiti. Cette ett&uation produirait une diminution de la vitesse de dissolution B cet endroit, et de cette faqon le rapport axial du pr6cipit6 augmenterait. Pour les temp&etures homologues basses, lea auteurs proposent que la diffusion aux joints de grains permettrait toujours une dissolution p&f&entielle sur le bord de l’allotriomorphe, ce qui maintiendreit constant le rapport axial. DIE AUFL6SUNGSKINETIK VON ALLOTRIOMORPHEN AN KORNGRENZEN In den Systemen Al-Cu und Al-Ag wurden mit Hilfe der Elektronenstrahl-Mikroanalyse Beobachtungen iiber die Au%sungskinetik einzelner Allotriomorphe an Korngrenzen engestellt. Es zeigt sich, daa die Liinge und Breite der Ausscheidungen exponentiell mit der Zeit abnimmt. Dieses Verhalten wird unter der Annahme erkliirt, da13eine Einwirkung der Diffusionefelder benechberter Ausscheidungen in einem friihen Stadium des Auflijsungaprozesses ststtfindet. Das Diffusionsfeld in der Legierungsumgebung des Allotriomorphs ist somit charakteristisch fiir ein endliches System. Das Model1 liefert Werte des Diffusionskoeffizienten, die m befriedigender nbereinstimmung mit Werten aus der Literatur smd. Experimentelle Beobechtungen zelgen, dell sich das Achsenverhgltnis der Ausscheidungen wiihrend der AuflGsung im ellgemeinen urn weniger 81s 30% iindert. Bei hohen homologen Temperaturen nimmt das Achsenverh<nis.zu, wrihrend es bei tiefen Temperaturen im wesentlichen konstant bleibt oder leicht abnimmt. Diese Anderungen des AchsenverhrZltnisses hiingen nur von der homologen Temperetur 8b und sind unabhiingig vom Grad der ubersiittigung. Bei hohen homologen Temperaturen verringert nach unserer Interpret8tion die Einwirkung der Diffusionsfelder die Punktwirkung der Spitze der Ausscheidung bei der Diffusion. Diese Abschwiichung fiihrt zu einer Erniedrigung der Aufl6sungsgeschwindigkeit an dieser Spltze und somit zu emem gr69eren Achsenverhiiltnis der Ausscheidung. Bei tlefen homologen Temperaturen lst, nach unserem \70rsch18g eine bevorzugte Aufliisung der Spike des Allotriomorphs aufgrund der Korngrenzendiffuslon mdglich, wodurch das AchsenverhSltnis aufrechterhalten werden kann.
growth of both allotriomorphs
INTRODUCTION
When
a solid solution
extensively
is supercooled
phase field the new phase precipitates tially on the parent grain boundaries. having a lenticular grain boundaries,
into a two-
shape tend to form on high angle whilst primary
low angle boundaries. needles may also form
sideplates
form on
Widmanstiitten plates or in the grain interiors. The
l Received January 22, 1973; revised March 12, 1973. t Department of Metallurgy, Umversity of British
Columbia, Vancouver ACTA
8, B.C., Canada.
METALLURGICA,
VOL. 21, SEPTEMBER
1973
Most observations made in the ALCu solubility
of the
temperature,
and sideplates has been
However,
been done on precipitate
out preferenAllotriomorphs
studied.
much less work has
dissolution.
of dissolution and Al-Ag
a-Al
phase
to date have been
systems changes
because the rapidly
with
thus giving large changes in supersatu-
ration with change in temperature. Furthermore diffusion in the a phase is fairly rapid, thus allowing the growth of relatively large precipitates. Thomas and Whelan(l) used high temperature electron microscopy to study precipitate dissolution in 1259
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1260
METALLURGICA,
8n Al4 wt. % Cu alloy. They observed that the engular precipitates initially spheroidized and then in the final stctges of dissolution followed 8 parabolic dissolution law with radius v8rying according to the square root of the dissolution time. Hall and Haworth(2) studied composition profiles adjecent to dissolving Widmanstlitten plates in Al-5 wt. % Cu using the electron probe micro8nalyser. They found that at high homologous temper8tures* (T,= 0.91-0.95) the plates dissolved at 8 rate controlled by volume diffusion. Pasparekis et al. t3) have reported similar experiments on grain boundary allotriomorphs in Al-Cu and Al-Ag alloys. They presented results covering the first stsges of dissolution, before there w8s impingement of the diffusion fields from adjacent precipitates and before there was any detecteble movement of the phase boundaries. Volume diffusion was found to be dominant at Th above 0.93. At lower temperatures the contribution of grain boundsry diffusion to dissolution increased, and below Th = 0.77 there was extensive grain boundary diffusion before volume diffusion could even be detected. The present paper is an extension of the work of Pasparakis et al. to longer dissolution times and involves measuring the dissolution kinetics of grain boundary ellotriomorphs in Al-Cu and Al-Ag alloys. Rather than relying on less precise statistical techniques, the dissolution kinetics of individurtl precipiTwo main parameters are tates are measured. examined-the rate of precipitate dissolution and the variation of axial ratio during dissolution. EXPERIMENTAL
Four Al-Cu alloys of composition Al4.83,2.81, 1.37 and 1.22 wt. % Cu, designated A, B, C, D respectively, were prepared, together with one Al-Ag alloy of composition Al-15.75 wt. ‘A Ag. Large grain boundary allotriomorphs of CuAl, or Ag,Al were grown by cooling the alloys into the two phase field at 8 rate of 3%/day and equilibrating for seven days to remove any composition gradients in the matrix. This procedure gave precipitates with lengt,hs varying from 2&50 ,um and axial ratios from 2 :l to 10 :l. Further deteils of the heat treatment procedure are given in the psper by P8sparakis et u1.u’) Specimens from t,he equilibrated alloy, approximately 10 x 6 x 0.25 mm, were carefully polished to 1 pm diamond 8nd the precipitates were examined in 8 JEOLCO JXA-3A electron probe using the back scattered electron image. This eliminated having to l For an alloy of specified composition, the homologous temperature (!Z’*)is the alloy temperature (“K) divided by the corresponding solidus temperature (OK).
VOL.
21,
1973
etch the specimen and so change the surface conditions. The largest allotriomorphs were used for kinetic studies in order to ensure that the precipitates were sectioned close to their maximum diameter. In general P6 precipitates were selected for dissolution experiments. The precipitates were normally oriented perpendicular to the specimen surface because the grain boundaries generally ran from top to bottom of the specimen. Dissolution was carried out by annealing the specimen for progressively increasing periods of time in 8 60 % KNOs-40 % NaNO, salt pot held to within f 1°C of the desired dissolution temperature. Contamination of the polished surface w8s avoided by wrapping the specimen in 120 pm thick Ta foil. Table 1 shows the temperatures and aging times involved. Aging was generally continued until the precipitates had completely dissolved. The precipitates were examined in the electron probe between heat treatments using the backscattered electron image. This eliminated having to etch the specimen in order to see the microstructure. A typical set of photographs is shown in Fig 1. Preliminary experiments showed that the absorbed electron image gave poor agreement with the precipitete size measured optically, since 8 change in contrast gave a significantly different precipitate size. However, the back-scattered image could be picked up at all settings without distortion, and at all contrast levels gave precipitate sizes equal to those obtained optically. Measurements of the precipitate dimensions were made directly from photographs of the backscattered electron image. Thickness measurements were made et a point half way between the tips of the precipitate. There is some possibility that the dissolution results might have been affected by surface diffusion. This w8s checked by examining four precipitcttes in alloy A after a 12 min. dissolution heat treatment at 527°C. This resulted in 8 decrease of N40 per cent from the initial thickness. The precipitate sizes were determined using the back-scattered electron image. A thin surface leyer was polished away and the specimen w8s then etched in order to allow measurements to be made optically. In all four cases the optical measurements were in good agreement with the backscattered electron observations, thus elimineting surface diffusion as 8 contribution to dissolution. Individual precipitates were observed in the present work. No statistical analysis was carried out, as is more common, because of the relatively small number of allotriomorphs and the large range of precipitate shapes.
PASPARAKIS
AND
BROWN:
DISSOLUTION
OF GRAIN
BOUNDARY
ALLOTRIOMORPHS
1261
TABLE 1. Dissolution experiments carried out
(TA)
No. of precipitates
Max. dissolution time (min)
424 460
0.76 0.79 0.83
2 6
1003 6960 700
421 421 298
489 460 600 489
0.84 0.86 0.84 0.88
t6 t
1123 297 144 82
D Al-Ag B A A z
298 421 421 460 460 298 421
500 500 513 620 634 634 534
0.86 0.88 0.90 0.95 0.97 0.92 0.89
5 4 5 5 9 :
12 90 48 66 60 20
Al-Ag
421
534
0.92
6
7.:
Equilibration temperature
Dissolution
Temperature
(“C)
(“C)
D :
298 421 361
F-A” :
MOY
DisaoZution kinetics The microprobe studies of Pasparakis et al.(*) have shown that impingement of the diffusion fields from adjacent precipitates takes place before there is any detectable movement of the phase boundaries due to dissolution. Accordingly, models developed in the literature for dissolution in semi-infinite systems~ks)
are not applicable in the present case. These models all predict a parabolic dissolution rate, i.e. a graph of precipitate half length (8) or half width (R) vs tlie should give a straight line. In fact, such plots of the present data are not linear, indicating that a semiinfinite model is not appropriate. In an attempt to find the correct dissolution law, the data was plotted in the form log Sl&, vs t and log R/R, vs t where Is,,and R, are the initial half length and width of the precipitate. At all temperatures a good approximation to a linear relationship was obtained, Figs. 2 and 3. Hence an equation of the form S = SOexp (--kjt) is applicable to the change in half length and R = R,, exp (-k&) is valid for the change in half thickness, where kr and k, are constants. Figures 2 and3 show that there is a significant variation in dissolution rates from precipitate to precipitate. At some values of Th the rate varied by a factor of 3. The reason for this variation will be discussed shortly. In the following section the observed exponential nature of the dissolution kinetics is explained on the basis of an analytical model which aooounts for diffusion field impingement. This model is similar to one developed by Singh and Fleminga for dissolution of second phase structures in aluminum castings. In the present model /l precipitates are considered to form at temperature Tl (Fig. 4(a)) with a supersaturation given by f, =
G P
-___-_
-.
.--
--
FIG. 1. Back scattered electron photographs used to observe the dissolution kinetioaof an allotromorph in alloy A at TI = 0.97.
O
For all cases considered here f,,< 1. After equilibration, the alloy is upquenched to temperature T, so that the /l phase starts to dissolve. Schematio composition profiles during dissolution are shown in Fig. 4(b). The alloy will ultimately have a uniform composition CO. Hence the concentration at the center of the a
1262
ACTA
METALLURGICA,
VOL.
21, 1973
conoentrations C, and 2C, respectively is given by(s) c = c, + 2(Cs -
C,, at x = 0 and x = 2
C&r/l -
cc, -
Cs)
x $;iuyexp(
nF).
After significant impingement only the leading term of the series need be considered. Differentiating to give the composition gradient at x = 0 leads to
0.3 -
= (2/W,
-
Cl) -
x (CJ-
(4/l)
C,) exp (-
GDt/12).
This is substituted into the interface mass balance -(C,
-
Cl)
$ = -D ($)
(2)
Z-0
to give --=
dR dt
D G @f, +
wheref, = (Co -
4f,,exp(--r2Dt/Z2))
(3)
C,)/(C, - C,) is the supersaturation
Time(min)
FIQ. 2. Plots of log [S/S,] and log [RI&] vs time for two ellotriomorpb in alloy A at Th = 0.97 (see Fig. 1).
solid solution will gradually approach 0, but will attain it only after the second phase has completely disappeared and composition gradients in the matrix have been eliminated. A planar model is oonsidered first of all. There exists no analytical solution to the diffusion equation for the situation shown in Fig. 4(b). Therefore an approximate solution is used in which it is considered that the displacement of the phase boundary is negligible compared with the separation, 1, between precipitates (i.e. R,,, S,,< I). It is necessary that the concentration at the mid-point, l/2, slowly approach C,, and attain it after infinite time. This condition is obtained by considering that there is a oonstant concentration Cs = 2C, - C, at x = 1 (Fig. 4(c)). This model is deficient in that it predicts a finite gradient at l/2 after infinite time, whereas there should be no flux across this point. However, it should be emphasised that the problem is essentially to determine the gradient at x = 0, the phase interface, and so the approximation should involve little error. The solution to the diffusion equation for a plate of thickness 1, initial concentration Cs and surface
Q
\ L
0
20
‘0
ri.$?@
80
IK)
FIG. 3. Plate of log [S/S,] and log [R/R,] vs time for three allotriomorphs in alloy D at T,, = 0.84.
PASPARAKIS
AND
BROWN:
DISSOLUTION
OF
GRAIN
BOUNDARY
ALLOTRIOMORPHS
1263
diffusion equation islg)
crc,ad
r
-
(r -
+ [C,(a, + 9 - 2Cpd
E CC, TV
Q(a,
r(b - a,)
+ b) .
4~ - a&
’ ‘*Xl b
-COMPOSITION-
a,)
_
-exp
ao)2
(b -
a,
(5)
since this gives C = C, at r = a, for all times, and at r = (a, + b)/2, th e mid-point between precipitates, the concentration approaches Cc after infinite time. Differentiating (5) and taking b > a, gives the gradient at r = a as (dc/dr), = (Cc -
Q/a -
(2/o)(Cc -
Cs)
x exp (-n2Dt/(b lb)
aJ2).
Substituting this into the interface mass balance, equation (2), which is applicable to spherical shapes as well, and integrating gives a2 = -2f,Dt
+ (4/?r2)f,(b -
aJ2
x exp (-n2Dt/(b
Taking a0
[
(cl Fxa. 4. Dissolution of p precipitates at temperature T, after growth at temperature T,. (a) Sohematio phase diagram. (b) Composition profiles during dissolution. (0) Composition profile assumed in the planar model.
for dissolution. For planar growth the lever rule gives f, = 2Rcll. Substituting in equation (3) and integrating leads to R = R,, exp ( --rr2Dt/12) -
-
2Df,t/l
(4)
taking S/r2 N 1. It is possible to derive an equation similar to equation (4) for dissolution of spherical precipitates since a solution to the diffusion equation is available for a spherical shell a,, < r < b. The precipitate of interest is considered to have a radius a,, whilst the adjacent precipitates are considered on average to be located at a radius b. An expression for b will be developed shortly. This procedure will give meaningful composition profiles close to a,,, but close to b the profiles will represent average concentrations only, and will not give actual concentrations adjacent to the precipitates. It is assumed as before that impingement takes place early in the dissolution process and also that movement of the precipitate interface can be considered negligible. The appropriate solution to the
J
b-_a
2
-
4fll =>
aJ2).
(6)
gives a = a, at t = 0 as required and leads to an equation for dissolution of spherical precipitates a2 = ao2 exp (-GDt/(b
-
ao)2) -
2fdDt,
(7)
very similar to that for dissolution of planar precipitates. Equation (6) enables a value of b to be found if a0 is known. If one substitutes values of the parameters D, 1, (b - ao) and fd which are typical of the present experimental conditions into equations (4) and (7), it is found that the linear term in both equations becomes important only in the last stages of dissolution. Thus, for most of the dissolution time the kinetics should follow an equation of the form R = R, exp (-kt) with k = r2D/12 for planar symmetry and k = ?r2D/2(b - ao)2 for spherical precipitates. Tanxilli and Heckel(‘c) have developed a numerical model for following the dissolution of planar, cylindrical and spherical precipitates in finite systems. This analysis has been applied by Baty, Tanzilli and HeckeW to the dissolution of spherical &Al, precipitates. If this numerical data is plotted in the form log a/a0 vs t an excellent fit to a straight line is obtained (Fig. 5). The value of diffusion coefficient obtained from the slope of this line is 3.3 x 10-S m2/s, in satisfactory agreement with the literature value of
ACTA
1264
METALLURGICA,
VOL.
21,
1973 ,
&Al2
at 520 ‘%
0.3 -
0.1 + 0
, 5
lo
20
25
30
35
zllew
FIQ. 5. Data from Baty et al.“”
for the dissolution of spherical CuAl, preoipitates in Al-4 wt % Cu at 520°C plotted in the form log R/R, VB t.
6.0x 1O-6 m3/s.(12) Hence the data of Baty et al. provides a good confirmation of the theory developed here. The models developed here are for dissolution of planar and spherical precipitates and are not directly applicable to grain boundary allotriomorphs which have a ‘spheroidal shape. Furthermore, Pasparakis et uZ.(~) have shown that significant diffusion takes place along grain boundaries at temperatures below T,,= 0.84and again this is not considered in the present models. At high homologous temperatures, there is no grain boundary diffusion and impingement of the diffusion fields from adjacent precipitates will occur, both along the grain boundary from allotriomorphs and perpendicular to the broad faces of the allotriomorphs. This impingement will give exponential laws for decrease in the allotriomorph length and width, as shown in Figs. 2 and 3. At low homologous temperatures (T,, < 0.84), there is very rapid grain boundary diffusion. Hence solute atoms from the tip diffuse very quickly along the grain boundary and then outward from it into the matrix (Fig. 6). Thus, the rate controlling step for dissolution of the allotriomorph tip is volume diffusion in the matrix. This corresponds to diffusion in a finite system and so again an exponential law would be expected for decrease of the precipitate length. Dissolution at the center of the broad face of the precipitate will not be affected much by grain boundary diffusion at the tip and again impingement of the
diffusion fields from adjacent precipitates will give an exponential law. Values of the diffusion coefficient were calculated from the experimental data using both the planar and
FIG. 6. Schematic composition profiles around dissolving grain boundary allotriomorpbs at low T,,.
PASPARAKIS
AXD
BROWN:
DISSOLUTION
OF
spherical models. Data for precipitate width (R) was used in the planar model since there was relatively little curvature on the broad face of the allotriomorph. In the spherical model an effective radius a,, was used given by ao3 = S2R since the volume of an oblate spheriod is given by +S2R. This implies that the tip of the allotriomorph dissolves faster than the sphere since there is a large point effect of diffusion at the tip, whilst the broad face dissolves more slowly as might be expected. The spherical model would be expected to be very inaccurate at low T,, where there is grain boundary diffusion at the tip. Mean values of the diffusion coefficient obtained from the experimental data are shown in Fig. 7. There is a very large difference in the diffusion coefficient values calculated according to the planar and spherical models, indicating the significance of divergence of the diffusion flux under the conditions of low supersaturation used in the present work. The literature values for the diffusion coefficient? lie intermediate between the two sets of results but closer to the planar ones. The allotriomorphs have an axial ratio of approximately 3.5 :l and the approach of the broad surface to a plane is much closer than the approach of the precipitate as a whole to a sphere. Hence the relative values of diffusion coefficient calculated according to the spherical and planar models are qualitatively correct. Axial ratio variation during dissolution
The other area of importance in the present study is the change in precipitate shape during dissolution. In general it was observed that a constant axial ratio
I”
PLANAR < “E
,
SPHERICAL LITERATURE
MODEL
Eq.(4)
0
MODEL Eq(7) 0 VALUES REF (12
l
I4 I/T
onoK-’
X IO4
FIG. 7. Values of diffusion coefficient calculated from the planar and spherical dissolution models compared with literature values.
GRAIN
BOUNDARY
ALLOTRIOMORPHS
(S/R) was not maintained during dissolution.
1265
Some predipitates had an increase in axial ratio whilst a much smaller number had a decrease. In general, however, the final axial ratio (SJR,) was less than 30 per cent different from the initial ratio (&JR,). Several methods were tested for presenting the data clearly. The best procedure was to determine the fraction, F, of precipitates in a given experiment showing an increase in axial ratio of over 10 per cent, i.e. the fraction of precipitates for which (S,/R,)/ (&JR,) > 1.1. This procedure is crude for several reasons : 1. The use of 10 per cent as the fiducial axial ratio increase is admittedly arbitrary. 2. The comparison is made on the basis of the last axial ratio measurement taken. However, there may well have been further increases in axial ratio before the precipitate completely dissolved. These could not be observed due to resolution limitations of the technique used. 3. The number of precipitates studied in any experiment was typically 3-8, which is insufficient for unquestionably meaningful statistical studies. Despite these deficiencies, certain correlations could be clearly established. Figure 8 shows a plot of F vs supersaturation and indicates that the change in axial ratio in dissolution is independent of supersaturation. In Fig. 9 the same function F is plotted vs the homologous temperature and shows a definite increase in F with increasing T,,. This is in agreement with the observations of Pasparakis et aZ.t3) which indicate that the shape of the concentration contours during early stages of dissolution depends only on T,. At T,, > 0.95, all the precipitates studied showed an increase in axial ratio, whilst when T,, < 0.80 the precipitates either showed no change or else a slight decrease in axial ratio. A simple explanation for these observations can be developed. At high homologous temperatures (T,, > 0.95) when the contribution of grain boundary diffusion to the dissolution process may be neglected, the precipitate can be considered to be an oblate spheriod dissolving by volume diffusion into the matrix. In order for the axial ratio to be maintained. it is necessary that the tip dissolve faster than the flat faces. For dissolution under semi-infinite conditions this occurs because of the more efficient dispersion of solute at the tip due to the point effect of diffusion. However, after a significant amount of impingement, the composition gradients in the matrix become small and the point effect operates to a smaller extent. Thus, the tip and the broad face of the allotriomorph will
ACTA
1266
METALLURGICA,
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21,
Alby
1~
Alby’
1~
. A 0 A oB OAI-ALJ
a97 a95 a92 a92
. D VB AAl+
a86 a&5 0.84
oD
089
oB m AI-Ag
0.8B Oss
*A
0
.
D
1973
Alloy oc 0
0.79 p
076
a.84 as4
0
0.
0
a75 -
I3
.
.
L
v a50
.
-
.
.
FIG. 8. Fraction of precipitates showing an inoreasein axial ratio greater than 10 per oent vs supersaturation. The results are grouped according to ranges of homologous temperature. *Dissolution below the solvus resulting in spheroidization (see Fig. 10).
h
AI-CU
Alloys
o
Al-Ag
Aloyr
l.oo-
a50
-
a25
-
0.75
0.80
0.85
0.90
0.95
1.00
T,
FIG. 9. Fraction of precipitates showing an increase in axial ratio greater than 10 per cent (F) VY Th.
tend to dissolve at more nearly comparable rates, thus giving an increase in axial ratio as observed experimentally. At low homologous temperatures (T,< 0.80) grain boundary diffusion is much faster than volume diffusion. As discussed before, the tip of the allotriomorph will dissolve preferentially with solute from the tip moving quickly along the grain boundary and then outward into the matrix. As seen in Fig. 6 the composition gradient in the matrix next to the grain boundary is greater than that next to the broad face
of the allotriomorph. Thus, the tip will continue to dissolve preferentially even after significant impingement. This model might suggest a spheroidizing tendency for the precipitates and indeed some of the results do indicate a slight tendency for decrease in axial ratio at the lowest homologous temperatures. However this decrease in axial ratio is generally less than 10 per cent. This model predicts the increase in axial ratio on dissolution at high homologous temperatures and the approximate constancy of axial ratio at low Th. The
PASPARAKIS
BROWN:
AND
DISSOLUTION
OF
effect will depend entirely on T,, since this controls the relative contributions from volume diffusion and grain boundary diffusion (cf. Fig. 9). The model would thus predict the variation of axial ratio to be independent of supersaturation as observed experimentally. Dissolution
below the solves
A few experiments were carried out to examine precipitate dissolution below the solvus. In this case the particles will shrink but never completely disappear. The problem is difficult experimentally owing. to the long times required for the precipitates to reach their final dimensions. A theory can be developed which is very similar to that presented for dissolution above the solvus and yields the following equdions: R -
R,
= (R, -
R,)
exp ( -r2Dt/Z2)
(8)
for planar symmetry, and a -
a,
=
(a, -
a,) exp (-x2Dt/(b
-
aJ2)
(9)
for spherical symmetry, where R, and a, are the precipitate dimensions after ageing for infinite time. The experimental data (i.e. upper curve) appear to follow an exponential law, Fig. 10, as indicated by equations (8) and (9). However, at long times there is a large spheroiclizing effect and after 7000 min. the axial ratio has decreased by a factor of 1.8. This spheroidization has an origin different from that discussed earlier. In this case the decrease in SIR takes place after most of the normal dissolution is complete and composition gradients in the matrix have been largely eliminated. Thus, the spheroidization is due to the requirements for minimization of the interfacial energy, a reaction with a very small driving force.
BOUNDARY
ALLOTRIOMORPHS
1267
This will only become significant after the volume free energy change associated with precipitate dissolution has largely been dissipated. CONCLUSIONS
It has been shown theoretically that planar and spherical precipitates dissolve according to an exponential law R/R, = exp (- kt), provided impingement of the diffusion fields from adjacent precipitates takes place at an early stage in the dissolution process. The numerical calculations of Batp et al. indicate that such a law is obeyed in dissolution of spherical precipitates of CuAl,. Furthermore, all the experimental observations in the present work indicate that an exponential law holds for dissolution of grain boundary allotriomorphs. Measurements of the change in precipitate shape during dissolution show that the axial ratio generally changes by less than 30 per cent. At high homologous temperatures there is a significant increase in axial ratio during dissolution. This is apparently caused by a limitation of the point effect of diffusion at the precipitate tip due to impingement’ of diffusion fields. This causes the tip to dissolve less rapidly and so give an increase in axial ratio. At low homologous temperatures the axial ratio remains essentially constant or decreases slightly. This is due to grain boundary diffusion, which even after impingement of the diffusion fields, still allows the tip to dissolve preferentially. Most observations of dissolution were made above the solvus. Below the solvus an exponential law is still obeyed during the early stages of dissolution. However, at late stages of ageing, spheroidisation becomes important and large decreases in axial ratio of the precipitate take place at this time. ACKNOWLEDGEMENTS
The authors wish to thank Dr. D. E. Coates for many helpful discussions. Financial assistance from the National Research Council is gratefully acknowledged.
0 R/R. l
GRAIN
SIR
REFERENCES 1. G. THOMAS and M. J. WHELAPI, Phil. Afag. 6, 1103 (1961). 2. M. G. HALL and C. W. HAWORTH. Acta iVet. 18,331 (1970). 3. A. PASPARAKIS, D. E. COATES and L. C. BROWN, B&u Met., to be published. 4. M. J. WHELAS. Met. Sci. J. 3, 95 (1969). 5. H. B. AARON. D. FAINSTEIN and G. R. KOTLER. J. a~&. 1X
Phya. 41, 4404 (1970). 6. H. B. AARON and G. R. KOTLER, Met.
Trans. 2, 393
(1971). 7. S. N. SINGH and M. C. FLEIMINGS, Trans. Metall. Sot. Am.
Inst. Min. Engrs. 245, 1803 (1969). 8. J. CRANK, The Mathematics of Difuusios.
C,
I 25
1 50 -TIME
I 75
I 100
a 15 125
in HOURS+
p. 47. Oxford University Press (1957). 9. H. S. CARSLAW and J. C. JAEGER, Contlccctiot~of Heat in Solids, p. 246. Oxford Umversity Press (1959). 10. R. A. TANZILLI and R. W. HECKEL, Tram. MetaU. Sot. Am. Inet. Min. Engrs. 245, 1383 (1969).
11. D. L. BATY, FIQ. 10. Plot of [R/I?,] vs time and [R/R] vs time for an allotriomorph m alloy B at Th = 0.84, below the solvus.
R. A. TANZILLI and T. W.
Trans. 1, 1651 (1971). 12. J. B. MURPHY, Acta Met. 9, 563 (1961).
HECKEL, Met.
KINETICS A.
OF GRAIN
PASPARAKISt
EtOtikDARY
ALLOTRIOMORPHS*
and L. C. BROWNt
Observations havebeenmade of the dissolution kinetics of individual grain boundary allotriomorphs in the Al_Cu and Al-Ag systems using the electron probe microenalyzer. It is found that the length and width of the precipitates decrease exponentially with time. This behavior is explained on the besis of a model in which it is considered that impingement of the diffusion fields from adj8cent precipitates takes piece early in the dissolution process. The diffusion field in the solid solution surrounding an allotnomorph is thus Ch8r8Ct&StiC of 8 finite system. The model gives values of the diffusion coefficient that are m edequate agreement with literature values. Experimental observations show that the axial ratio of the precipitates generally changes by less than 300/Lduring dlssolutlon. At high homologous temperatures, there is an increase in axial ratio, whilst et 1OWtemper8tUreS the 8Xi81 ratio remains essentially constctnt or decreases slightly. These variations in 8Xi81 ratio depend only on the homologous tempersture and are independent of the degree of supersaturation. At high homologous temperatures, It is proposed that impingement of the diffusion fields attenuates the point effect of diffusion 8t the tip of the preclpltate. This attenuation causes a decrease in the dissolution rate of the tip and thus Increases the axial r8tio of the precipitate. At low homologous temperatures, it is proposed that grain boundary diffusion always allows the sllotriomorph tip to dissolve preferentially, thus maintaining the axial ratlo. CINETIQUE DE LA DISSOLUTION DES ALLOTRIOMORPHES AUli JOINTS DE GRAINS Des observations de la cin&ique de dissolution des allotriomorphes isoles 8UX joints de grains, effect&es dans les systemes Al-Cu et Al-Ag par microanelyse B sonde Blectronique, ont montr6 que la longueur et la largeur des p&ipit& d6croft exponentiellement en fonction du temps. Ce comportement est interpr6t.6 en B partir d’un modble supposant que l’empi%ment des champs de diffusion des prCcipit6.q adjacents se produit d&s le debut du processus de dissolution. Le champ de diffusion de la solution solide entourrtnt un allotriomorpheest ainsi caract&istique d’un systhme flni. Le modble donne pour le coefficient de diffusion des veleurs qui sont en bon accord 8vec les valeurs donn6es dans la litt&ature. Lea observations exp&imenteles montrent que le rapport axial des pr6cipit6s varie en g&&al de moms de 30% au tours de la dissolution. Aux temp&atures homologues 6levBes, il se produit une augmentetion du rapport axial, alors qu’aux basses temperatures il reste Con&ant ou diminue lbg&ement. C%sverlations du rapport axial dependent seulement de la temp&ature homologue et sont independantes du degr6 de sursaturation. Les auteurs proposent une interpr6tation suivant lequelle, 8UX temp6ratures homologues Blevbs, l’empi8tement des champs de diffusion attbnuerait l’effet lOC81de 18 diffusion sur le bord du pr6cipiti. Cette ett&uation produirait une diminution de la vitesse de dissolution B cet endroit, et de cette faqon le rapport axial du pr6cipit6 augmenterait. Pour les temp&etures homologues basses, lea auteurs proposent que la diffusion aux joints de grains permettrait toujours une dissolution p&f&entielle sur le bord de l’allotriomorphe, ce qui maintiendreit constant le rapport axial. DIE AUFL6SUNGSKINETIK VON ALLOTRIOMORPHEN AN KORNGRENZEN In den Systemen Al-Cu und Al-Ag wurden mit Hilfe der Elektronenstrahl-Mikroanalyse Beobachtungen iiber die Au%sungskinetik einzelner Allotriomorphe an Korngrenzen engestellt. Es zeigt sich, daa die Liinge und Breite der Ausscheidungen exponentiell mit der Zeit abnimmt. Dieses Verhalten wird unter der Annahme erkliirt, da13eine Einwirkung der Diffusionefelder benechberter Ausscheidungen in einem friihen Stadium des Auflijsungaprozesses ststtfindet. Das Diffusionsfeld in der Legierungsumgebung des Allotriomorphs ist somit charakteristisch fiir ein endliches System. Das Model1 liefert Werte des Diffusionskoeffizienten, die m befriedigender nbereinstimmung mit Werten aus der Literatur smd. Experimentelle Beobechtungen zelgen, dell sich das Achsenverhgltnis der Ausscheidungen wiihrend der AuflGsung im ellgemeinen urn weniger 81s 30% iindert. Bei hohen homologen Temperaturen nimmt das Achsenverh<nis.zu, wrihrend es bei tiefen Temperaturen im wesentlichen konstant bleibt oder leicht abnimmt. Diese Anderungen des AchsenverhrZltnisses hiingen nur von der homologen Temperetur 8b und sind unabhiingig vom Grad der ubersiittigung. Bei hohen homologen Temperaturen verringert nach unserer Interpret8tion die Einwirkung der Diffusionsfelder die Punktwirkung der Spitze der Ausscheidung bei der Diffusion. Diese Abschwiichung fiihrt zu einer Erniedrigung der Aufl6sungsgeschwindigkeit an dieser Spltze und somit zu emem gr69eren Achsenverhiiltnis der Ausscheidung. Bei tlefen homologen Temperaturen lst, nach unserem \70rsch18g eine bevorzugte Aufliisung der Spike des Allotriomorphs aufgrund der Korngrenzendiffuslon mdglich, wodurch das AchsenverhSltnis aufrechterhalten werden kann.
growth of both allotriomorphs
INTRODUCTION
When
a solid solution
extensively
is supercooled
phase field the new phase precipitates tially on the parent grain boundaries. having a lenticular grain boundaries,
into a two-
shape tend to form on high angle whilst primary
low angle boundaries. needles may also form
sideplates
form on
Widmanstiitten plates or in the grain interiors. The
l Received January 22, 1973; revised March 12, 1973. t Department of Metallurgy, Umversity of British
Columbia, Vancouver ACTA
8, B.C., Canada.
METALLURGICA,
VOL. 21, SEPTEMBER
1973
Most observations made in the ALCu solubility
of the
temperature,
and sideplates has been
However,
been done on precipitate
out preferenAllotriomorphs
studied.
much less work has
dissolution.
of dissolution and Al-Ag
a-Al
phase
to date have been
systems changes
because the rapidly
with
thus giving large changes in supersatu-
ration with change in temperature. Furthermore diffusion in the a phase is fairly rapid, thus allowing the growth of relatively large precipitates. Thomas and Whelan(l) used high temperature electron microscopy to study precipitate dissolution in 1259
ACTA
1260
METALLURGICA,
8n Al4 wt. % Cu alloy. They observed that the engular precipitates initially spheroidized and then in the final stctges of dissolution followed 8 parabolic dissolution law with radius v8rying according to the square root of the dissolution time. Hall and Haworth(2) studied composition profiles adjecent to dissolving Widmanstlitten plates in Al-5 wt. % Cu using the electron probe micro8nalyser. They found that at high homologous temper8tures* (T,= 0.91-0.95) the plates dissolved at 8 rate controlled by volume diffusion. Pasparekis et al. t3) have reported similar experiments on grain boundary allotriomorphs in Al-Cu and Al-Ag alloys. They presented results covering the first stsges of dissolution, before there w8s impingement of the diffusion fields from adjacent precipitates and before there was any detecteble movement of the phase boundaries. Volume diffusion was found to be dominant at Th above 0.93. At lower temperatures the contribution of grain boundsry diffusion to dissolution increased, and below Th = 0.77 there was extensive grain boundary diffusion before volume diffusion could even be detected. The present paper is an extension of the work of Pasparakis et al. to longer dissolution times and involves measuring the dissolution kinetics of grain boundary ellotriomorphs in Al-Cu and Al-Ag alloys. Rather than relying on less precise statistical techniques, the dissolution kinetics of individurtl precipiTwo main parameters are tates are measured. examined-the rate of precipitate dissolution and the variation of axial ratio during dissolution. EXPERIMENTAL
Four Al-Cu alloys of composition Al4.83,2.81, 1.37 and 1.22 wt. % Cu, designated A, B, C, D respectively, were prepared, together with one Al-Ag alloy of composition Al-15.75 wt. ‘A Ag. Large grain boundary allotriomorphs of CuAl, or Ag,Al were grown by cooling the alloys into the two phase field at 8 rate of 3%/day and equilibrating for seven days to remove any composition gradients in the matrix. This procedure gave precipitates with lengt,hs varying from 2&50 ,um and axial ratios from 2 :l to 10 :l. Further deteils of the heat treatment procedure are given in the psper by P8sparakis et u1.u’) Specimens from t,he equilibrated alloy, approximately 10 x 6 x 0.25 mm, were carefully polished to 1 pm diamond 8nd the precipitates were examined in 8 JEOLCO JXA-3A electron probe using the back scattered electron image. This eliminated having to l For an alloy of specified composition, the homologous temperature (!Z’*)is the alloy temperature (“K) divided by the corresponding solidus temperature (OK).
VOL.
21,
1973
etch the specimen and so change the surface conditions. The largest allotriomorphs were used for kinetic studies in order to ensure that the precipitates were sectioned close to their maximum diameter. In general P6 precipitates were selected for dissolution experiments. The precipitates were normally oriented perpendicular to the specimen surface because the grain boundaries generally ran from top to bottom of the specimen. Dissolution was carried out by annealing the specimen for progressively increasing periods of time in 8 60 % KNOs-40 % NaNO, salt pot held to within f 1°C of the desired dissolution temperature. Contamination of the polished surface w8s avoided by wrapping the specimen in 120 pm thick Ta foil. Table 1 shows the temperatures and aging times involved. Aging was generally continued until the precipitates had completely dissolved. The precipitates were examined in the electron probe between heat treatments using the backscattered electron image. This eliminated having to etch the specimen in order to see the microstructure. A typical set of photographs is shown in Fig 1. Preliminary experiments showed that the absorbed electron image gave poor agreement with the precipitete size measured optically, since 8 change in contrast gave a significantly different precipitate size. However, the back-scattered image could be picked up at all settings without distortion, and at all contrast levels gave precipitate sizes equal to those obtained optically. Measurements of the precipitate dimensions were made directly from photographs of the backscattered electron image. Thickness measurements were made et a point half way between the tips of the precipitate. There is some possibility that the dissolution results might have been affected by surface diffusion. This w8s checked by examining four precipitcttes in alloy A after a 12 min. dissolution heat treatment at 527°C. This resulted in 8 decrease of N40 per cent from the initial thickness. The precipitate sizes were determined using the back-scattered electron image. A thin surface leyer was polished away and the specimen w8s then etched in order to allow measurements to be made optically. In all four cases the optical measurements were in good agreement with the backscattered electron observations, thus elimineting surface diffusion as 8 contribution to dissolution. Individual precipitates were observed in the present work. No statistical analysis was carried out, as is more common, because of the relatively small number of allotriomorphs and the large range of precipitate shapes.
PASPARAKIS
AND
BROWN:
DISSOLUTION
OF GRAIN
BOUNDARY
ALLOTRIOMORPHS
1261
TABLE 1. Dissolution experiments carried out
(TA)
No. of precipitates
Max. dissolution time (min)
424 460
0.76 0.79 0.83
2 6
1003 6960 700
421 421 298
489 460 600 489
0.84 0.86 0.84 0.88
t6 t
1123 297 144 82
D Al-Ag B A A z
298 421 421 460 460 298 421
500 500 513 620 634 634 534
0.86 0.88 0.90 0.95 0.97 0.92 0.89
5 4 5 5 9 :
12 90 48 66 60 20
Al-Ag
421
534
0.92
6
7.:
Equilibration temperature
Dissolution
Temperature
(“C)
(“C)
D :
298 421 361
F-A” :
MOY
DisaoZution kinetics The microprobe studies of Pasparakis et al.(*) have shown that impingement of the diffusion fields from adjacent precipitates takes place before there is any detectable movement of the phase boundaries due to dissolution. Accordingly, models developed in the literature for dissolution in semi-infinite systems~ks)
are not applicable in the present case. These models all predict a parabolic dissolution rate, i.e. a graph of precipitate half length (8) or half width (R) vs tlie should give a straight line. In fact, such plots of the present data are not linear, indicating that a semiinfinite model is not appropriate. In an attempt to find the correct dissolution law, the data was plotted in the form log Sl&, vs t and log R/R, vs t where Is,,and R, are the initial half length and width of the precipitate. At all temperatures a good approximation to a linear relationship was obtained, Figs. 2 and 3. Hence an equation of the form S = SOexp (--kjt) is applicable to the change in half length and R = R,, exp (-k&) is valid for the change in half thickness, where kr and k, are constants. Figures 2 and3 show that there is a significant variation in dissolution rates from precipitate to precipitate. At some values of Th the rate varied by a factor of 3. The reason for this variation will be discussed shortly. In the following section the observed exponential nature of the dissolution kinetics is explained on the basis of an analytical model which aooounts for diffusion field impingement. This model is similar to one developed by Singh and Fleminga for dissolution of second phase structures in aluminum castings. In the present model /l precipitates are considered to form at temperature Tl (Fig. 4(a)) with a supersaturation given by f, =
G P
-___-_
-.
.--
--
FIG. 1. Back scattered electron photographs used to observe the dissolution kinetioaof an allotromorph in alloy A at TI = 0.97.
O
For all cases considered here f,,< 1. After equilibration, the alloy is upquenched to temperature T, so that the /l phase starts to dissolve. Schematio composition profiles during dissolution are shown in Fig. 4(b). The alloy will ultimately have a uniform composition CO. Hence the concentration at the center of the a
1262
ACTA
METALLURGICA,
VOL.
21, 1973
conoentrations C, and 2C, respectively is given by(s) c = c, + 2(Cs -
C,, at x = 0 and x = 2
C&r/l -
cc, -
Cs)
x $;iuyexp(
nF).
After significant impingement only the leading term of the series need be considered. Differentiating to give the composition gradient at x = 0 leads to
0.3 -
= (2/W,
-
Cl) -
x (CJ-
(4/l)
C,) exp (-
GDt/12).
This is substituted into the interface mass balance -(C,
-
Cl)
$ = -D ($)
(2)
Z-0
to give --=
dR dt
D G @f, +
wheref, = (Co -
4f,,exp(--r2Dt/Z2))
(3)
C,)/(C, - C,) is the supersaturation
Time(min)
FIQ. 2. Plots of log [S/S,] and log [RI&] vs time for two ellotriomorpb in alloy A at Th = 0.97 (see Fig. 1).
solid solution will gradually approach 0, but will attain it only after the second phase has completely disappeared and composition gradients in the matrix have been eliminated. A planar model is oonsidered first of all. There exists no analytical solution to the diffusion equation for the situation shown in Fig. 4(b). Therefore an approximate solution is used in which it is considered that the displacement of the phase boundary is negligible compared with the separation, 1, between precipitates (i.e. R,,, S,,< I). It is necessary that the concentration at the mid-point, l/2, slowly approach C,, and attain it after infinite time. This condition is obtained by considering that there is a oonstant concentration Cs = 2C, - C, at x = 1 (Fig. 4(c)). This model is deficient in that it predicts a finite gradient at l/2 after infinite time, whereas there should be no flux across this point. However, it should be emphasised that the problem is essentially to determine the gradient at x = 0, the phase interface, and so the approximation should involve little error. The solution to the diffusion equation for a plate of thickness 1, initial concentration Cs and surface
Q
\ L
0
20
‘0
ri.$?@
80
IK)
FIG. 3. Plate of log [S/S,] and log [R/R,] vs time for three allotriomorphs in alloy D at T,, = 0.84.
PASPARAKIS
AND
BROWN:
DISSOLUTION
OF
GRAIN
BOUNDARY
ALLOTRIOMORPHS
1263
diffusion equation islg)
crc,ad
r
-
(r -
+ [C,(a, + 9 - 2Cpd
E CC, TV
Q(a,
r(b - a,)
+ b) .
4~ - a&
’ ‘*Xl b
-COMPOSITION-
a,)
_
-exp
ao)2
(b -
a,
(5)
since this gives C = C, at r = a, for all times, and at r = (a, + b)/2, th e mid-point between precipitates, the concentration approaches Cc after infinite time. Differentiating (5) and taking b > a, gives the gradient at r = a as (dc/dr), = (Cc -
Q/a -
(2/o)(Cc -
Cs)
x exp (-n2Dt/(b lb)
aJ2).
Substituting this into the interface mass balance, equation (2), which is applicable to spherical shapes as well, and integrating gives a2 = -2f,Dt
+ (4/?r2)f,(b -
aJ2
x exp (-n2Dt/(b
Taking a0
[
(cl Fxa. 4. Dissolution of p precipitates at temperature T, after growth at temperature T,. (a) Sohematio phase diagram. (b) Composition profiles during dissolution. (0) Composition profile assumed in the planar model.
for dissolution. For planar growth the lever rule gives f, = 2Rcll. Substituting in equation (3) and integrating leads to R = R,, exp ( --rr2Dt/12) -
-
2Df,t/l
(4)
taking S/r2 N 1. It is possible to derive an equation similar to equation (4) for dissolution of spherical precipitates since a solution to the diffusion equation is available for a spherical shell a,, < r < b. The precipitate of interest is considered to have a radius a,, whilst the adjacent precipitates are considered on average to be located at a radius b. An expression for b will be developed shortly. This procedure will give meaningful composition profiles close to a,,, but close to b the profiles will represent average concentrations only, and will not give actual concentrations adjacent to the precipitates. It is assumed as before that impingement takes place early in the dissolution process and also that movement of the precipitate interface can be considered negligible. The appropriate solution to the
J
b-_a
2
-
4fll =>
aJ2).
(6)
gives a = a, at t = 0 as required and leads to an equation for dissolution of spherical precipitates a2 = ao2 exp (-GDt/(b
-
ao)2) -
2fdDt,
(7)
very similar to that for dissolution of planar precipitates. Equation (6) enables a value of b to be found if a0 is known. If one substitutes values of the parameters D, 1, (b - ao) and fd which are typical of the present experimental conditions into equations (4) and (7), it is found that the linear term in both equations becomes important only in the last stages of dissolution. Thus, for most of the dissolution time the kinetics should follow an equation of the form R = R, exp (-kt) with k = r2D/12 for planar symmetry and k = ?r2D/2(b - ao)2 for spherical precipitates. Tanxilli and Heckel(‘c) have developed a numerical model for following the dissolution of planar, cylindrical and spherical precipitates in finite systems. This analysis has been applied by Baty, Tanzilli and HeckeW to the dissolution of spherical &Al, precipitates. If this numerical data is plotted in the form log a/a0 vs t an excellent fit to a straight line is obtained (Fig. 5). The value of diffusion coefficient obtained from the slope of this line is 3.3 x 10-S m2/s, in satisfactory agreement with the literature value of
ACTA
1264
METALLURGICA,
VOL.
21,
1973 ,
&Al2
at 520 ‘%
0.3 -
0.1 + 0
, 5
lo
20
25
30
35
zllew
FIQ. 5. Data from Baty et al.“”
for the dissolution of spherical CuAl, preoipitates in Al-4 wt % Cu at 520°C plotted in the form log R/R, VB t.
6.0x 1O-6 m3/s.(12) Hence the data of Baty et al. provides a good confirmation of the theory developed here. The models developed here are for dissolution of planar and spherical precipitates and are not directly applicable to grain boundary allotriomorphs which have a ‘spheroidal shape. Furthermore, Pasparakis et uZ.(~) have shown that significant diffusion takes place along grain boundaries at temperatures below T,,= 0.84and again this is not considered in the present models. At high homologous temperatures, there is no grain boundary diffusion and impingement of the diffusion fields from adjacent precipitates will occur, both along the grain boundary from allotriomorphs and perpendicular to the broad faces of the allotriomorphs. This impingement will give exponential laws for decrease in the allotriomorph length and width, as shown in Figs. 2 and 3. At low homologous temperatures (T,, < 0.84), there is very rapid grain boundary diffusion. Hence solute atoms from the tip diffuse very quickly along the grain boundary and then outward from it into the matrix (Fig. 6). Thus, the rate controlling step for dissolution of the allotriomorph tip is volume diffusion in the matrix. This corresponds to diffusion in a finite system and so again an exponential law would be expected for decrease of the precipitate length. Dissolution at the center of the broad face of the precipitate will not be affected much by grain boundary diffusion at the tip and again impingement of the
diffusion fields from adjacent precipitates will give an exponential law. Values of the diffusion coefficient were calculated from the experimental data using both the planar and
FIG. 6. Schematic composition profiles around dissolving grain boundary allotriomorpbs at low T,,.
PASPARAKIS
AXD
BROWN:
DISSOLUTION
OF
spherical models. Data for precipitate width (R) was used in the planar model since there was relatively little curvature on the broad face of the allotriomorph. In the spherical model an effective radius a,, was used given by ao3 = S2R since the volume of an oblate spheriod is given by +S2R. This implies that the tip of the allotriomorph dissolves faster than the sphere since there is a large point effect of diffusion at the tip, whilst the broad face dissolves more slowly as might be expected. The spherical model would be expected to be very inaccurate at low T,, where there is grain boundary diffusion at the tip. Mean values of the diffusion coefficient obtained from the experimental data are shown in Fig. 7. There is a very large difference in the diffusion coefficient values calculated according to the planar and spherical models, indicating the significance of divergence of the diffusion flux under the conditions of low supersaturation used in the present work. The literature values for the diffusion coefficient? lie intermediate between the two sets of results but closer to the planar ones. The allotriomorphs have an axial ratio of approximately 3.5 :l and the approach of the broad surface to a plane is much closer than the approach of the precipitate as a whole to a sphere. Hence the relative values of diffusion coefficient calculated according to the spherical and planar models are qualitatively correct. Axial ratio variation during dissolution
The other area of importance in the present study is the change in precipitate shape during dissolution. In general it was observed that a constant axial ratio
I”
PLANAR < “E
,
SPHERICAL LITERATURE
MODEL
Eq.(4)
0
MODEL Eq(7) 0 VALUES REF (12
l
I4 I/T
onoK-’
X IO4
FIG. 7. Values of diffusion coefficient calculated from the planar and spherical dissolution models compared with literature values.
GRAIN
BOUNDARY
ALLOTRIOMORPHS
(S/R) was not maintained during dissolution.
1265
Some predipitates had an increase in axial ratio whilst a much smaller number had a decrease. In general, however, the final axial ratio (SJR,) was less than 30 per cent different from the initial ratio (&JR,). Several methods were tested for presenting the data clearly. The best procedure was to determine the fraction, F, of precipitates in a given experiment showing an increase in axial ratio of over 10 per cent, i.e. the fraction of precipitates for which (S,/R,)/ (&JR,) > 1.1. This procedure is crude for several reasons : 1. The use of 10 per cent as the fiducial axial ratio increase is admittedly arbitrary. 2. The comparison is made on the basis of the last axial ratio measurement taken. However, there may well have been further increases in axial ratio before the precipitate completely dissolved. These could not be observed due to resolution limitations of the technique used. 3. The number of precipitates studied in any experiment was typically 3-8, which is insufficient for unquestionably meaningful statistical studies. Despite these deficiencies, certain correlations could be clearly established. Figure 8 shows a plot of F vs supersaturation and indicates that the change in axial ratio in dissolution is independent of supersaturation. In Fig. 9 the same function F is plotted vs the homologous temperature and shows a definite increase in F with increasing T,,. This is in agreement with the observations of Pasparakis et aZ.t3) which indicate that the shape of the concentration contours during early stages of dissolution depends only on T,. At T,, > 0.95, all the precipitates studied showed an increase in axial ratio, whilst when T,, < 0.80 the precipitates either showed no change or else a slight decrease in axial ratio. A simple explanation for these observations can be developed. At high homologous temperatures (T,, > 0.95) when the contribution of grain boundary diffusion to the dissolution process may be neglected, the precipitate can be considered to be an oblate spheriod dissolving by volume diffusion into the matrix. In order for the axial ratio to be maintained. it is necessary that the tip dissolve faster than the flat faces. For dissolution under semi-infinite conditions this occurs because of the more efficient dispersion of solute at the tip due to the point effect of diffusion. However, after a significant amount of impingement, the composition gradients in the matrix become small and the point effect operates to a smaller extent. Thus, the tip and the broad face of the allotriomorph will
ACTA
1266
METALLURGICA,
VOL.
21,
Alby
1~
Alby’
1~
. A 0 A oB OAI-ALJ
a97 a95 a92 a92
. D VB AAl+
a86 a&5 0.84
oD
089
oB m AI-Ag
0.8B Oss
*A
0
.
D
1973
Alloy oc 0
0.79 p
076
a.84 as4
0
0.
0
a75 -
I3
.
.
L
v a50
.
-
.
.
FIG. 8. Fraction of precipitates showing an inoreasein axial ratio greater than 10 per oent vs supersaturation. The results are grouped according to ranges of homologous temperature. *Dissolution below the solvus resulting in spheroidization (see Fig. 10).
h
AI-CU
Alloys
o
Al-Ag
Aloyr
l.oo-
a50
-
a25
-
0.75
0.80
0.85
0.90
0.95
1.00
T,
FIG. 9. Fraction of precipitates showing an increase in axial ratio greater than 10 per cent (F) VY Th.
tend to dissolve at more nearly comparable rates, thus giving an increase in axial ratio as observed experimentally. At low homologous temperatures (T,< 0.80) grain boundary diffusion is much faster than volume diffusion. As discussed before, the tip of the allotriomorph will dissolve preferentially with solute from the tip moving quickly along the grain boundary and then outward into the matrix. As seen in Fig. 6 the composition gradient in the matrix next to the grain boundary is greater than that next to the broad face
of the allotriomorph. Thus, the tip will continue to dissolve preferentially even after significant impingement. This model might suggest a spheroidizing tendency for the precipitates and indeed some of the results do indicate a slight tendency for decrease in axial ratio at the lowest homologous temperatures. However this decrease in axial ratio is generally less than 10 per cent. This model predicts the increase in axial ratio on dissolution at high homologous temperatures and the approximate constancy of axial ratio at low Th. The
PASPARAKIS
BROWN:
AND
DISSOLUTION
OF
effect will depend entirely on T,, since this controls the relative contributions from volume diffusion and grain boundary diffusion (cf. Fig. 9). The model would thus predict the variation of axial ratio to be independent of supersaturation as observed experimentally. Dissolution
below the solves
A few experiments were carried out to examine precipitate dissolution below the solvus. In this case the particles will shrink but never completely disappear. The problem is difficult experimentally owing. to the long times required for the precipitates to reach their final dimensions. A theory can be developed which is very similar to that presented for dissolution above the solvus and yields the following equdions: R -
R,
= (R, -
R,)
exp ( -r2Dt/Z2)
(8)
for planar symmetry, and a -
a,
=
(a, -
a,) exp (-x2Dt/(b
-
aJ2)
(9)
for spherical symmetry, where R, and a, are the precipitate dimensions after ageing for infinite time. The experimental data (i.e. upper curve) appear to follow an exponential law, Fig. 10, as indicated by equations (8) and (9). However, at long times there is a large spheroiclizing effect and after 7000 min. the axial ratio has decreased by a factor of 1.8. This spheroidization has an origin different from that discussed earlier. In this case the decrease in SIR takes place after most of the normal dissolution is complete and composition gradients in the matrix have been largely eliminated. Thus, the spheroidization is due to the requirements for minimization of the interfacial energy, a reaction with a very small driving force.
BOUNDARY
ALLOTRIOMORPHS
1267
This will only become significant after the volume free energy change associated with precipitate dissolution has largely been dissipated. CONCLUSIONS
It has been shown theoretically that planar and spherical precipitates dissolve according to an exponential law R/R, = exp (- kt), provided impingement of the diffusion fields from adjacent precipitates takes place at an early stage in the dissolution process. The numerical calculations of Batp et al. indicate that such a law is obeyed in dissolution of spherical precipitates of CuAl,. Furthermore, all the experimental observations in the present work indicate that an exponential law holds for dissolution of grain boundary allotriomorphs. Measurements of the change in precipitate shape during dissolution show that the axial ratio generally changes by less than 30 per cent. At high homologous temperatures there is a significant increase in axial ratio during dissolution. This is apparently caused by a limitation of the point effect of diffusion at the precipitate tip due to impingement’ of diffusion fields. This causes the tip to dissolve less rapidly and so give an increase in axial ratio. At low homologous temperatures the axial ratio remains essentially constant or decreases slightly. This is due to grain boundary diffusion, which even after impingement of the diffusion fields, still allows the tip to dissolve preferentially. Most observations of dissolution were made above the solvus. Below the solvus an exponential law is still obeyed during the early stages of dissolution. However, at late stages of ageing, spheroidisation becomes important and large decreases in axial ratio of the precipitate take place at this time. ACKNOWLEDGEMENTS
The authors wish to thank Dr. D. E. Coates for many helpful discussions. Financial assistance from the National Research Council is gratefully acknowledged.
0 R/R. l
GRAIN
SIR
REFERENCES 1. G. THOMAS and M. J. WHELAPI, Phil. Afag. 6, 1103 (1961). 2. M. G. HALL and C. W. HAWORTH. Acta iVet. 18,331 (1970). 3. A. PASPARAKIS, D. E. COATES and L. C. BROWN, B&u Met., to be published. 4. M. J. WHELAS. Met. Sci. J. 3, 95 (1969). 5. H. B. AARON. D. FAINSTEIN and G. R. KOTLER. J. a~&. 1X
Phya. 41, 4404 (1970). 6. H. B. AARON and G. R. KOTLER, Met.
Trans. 2, 393
(1971). 7. S. N. SINGH and M. C. FLEIMINGS, Trans. Metall. Sot. Am.
Inst. Min. Engrs. 245, 1803 (1969). 8. J. CRANK, The Mathematics of Difuusios.
C,
I 25
1 50 -TIME
I 75
I 100
a 15 125
in HOURS+
p. 47. Oxford University Press (1957). 9. H. S. CARSLAW and J. C. JAEGER, Contlccctiot~of Heat in Solids, p. 246. Oxford Umversity Press (1959). 10. R. A. TANZILLI and R. W. HECKEL, Tram. MetaU. Sot. Am. Inet. Min. Engrs. 245, 1383 (1969).
11. D. L. BATY, FIQ. 10. Plot of [R/I?,] vs time and [R/R] vs time for an allotriomorph m alloy B at Th = 0.84, below the solvus.
R. A. TANZILLI and T. W.
Trans. 1, 1651 (1971). 12. J. B. MURPHY, Acta Met. 9, 563 (1961).
HECKEL, Met.