Kinetics of precipitation in aluminum alloys during continuous cooling
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d u r i n g the quench on p r o p e r t i e s , r e g a r d l e s s of the shape of the cooling c u r v e . I s o t h e r m a l p r e c i p i t a t i o n k i n e t i c s f o r a l u m i n u m a l l o y s a r e defined b y the equation: [1]
= 1 - exp (-t/k),
where = fraction transformed, k = t e m p e r a t u r e dependent c o n s t a n t which is p r o p o r t i o n a l to the t i m e r e q u i r e d to p r e c i p i t a t e a c o n s t a n t amount of s o l u t e , t = time. The v a l u e of the c o n s t a n t k, and hence p r e c i p i t a t i o n r a t e , d e p e n d s p r i n c i p a l l y on the d e g r e e of s u p e r s a t u r a tion and the r a t e of diffusion. It can be e s t i m a t e d using a r e c i p r o c a l f o r m of an equation d e s c r i b i n g n u c l e a t i o n r a t e :s Ct
k =--
kak ~
=k aexp
ks
exp - -
R T (k 4 - T) 2
kl
[2]
RT'
where C t = c r i t i c a l t i m e r e q u i r e d to p r e c i p i t a t e a c o n s t a n t
amount (the l o c u s of the c r i t i c a l t i m e s is the C-curve), k 1 = c o n s t a n t which e q u a l s the n a t u r a l l o g a r i t h m of the f r a c t i o n u n t r a n s f o r m e d (1 - f r a c t i o n defined by the C - c u r v e ) , k z = c o n s t a n t r e l a t e d to the r e c i p r o c a l of the n u m b e r of n u c l e a t i o n s i t e s , k s = c o n s t a n t r e l a t e d to the e n e r g y r e q u i r e d to f o r m a nucleus, k 4 = c o n s t a n t r e l a t e d to the s o l v u s t e m p e r a t u r e , k s = c o n s t a n t r e l a t e d to the a c t i v a t i o n e n e r g y for diffusion, R = 8.3143 J . K - l . m o 1 - 1 , T = t e m p e r a t u r e d e g r e e s K. Consequently, Eq. [1] can be r e w r i t t e n a s :
(k,t
= 1 - exp \ - ~ t / .
[3]
Using t h i s m o d e l f o r i s o t h e r m a l p r e c i p i t a t i o n , f r a c tion of s o l u t e p r e c i p i t a t e d d u r i n g the quench can be c a l VOLUME 5, JANUARY 1974-43
l
culated. Cahn 4 has shown, for t r a n s f o r m a t i o n s where r e a c t i o n r a t e is a function only of the amount t r a n s f o r m e d and t e m p e r a t u r e , that a m e a s u r e of the a m o u n t t r a n s f o r m e d d u r i n g continuous cooling is given by the integral:
o 6oo. i,o33,1 ;.,i
2//X
7~
.
tf
f dt _ to Ct(T ) 7:,
0
[4]
where
"
001
T = quench factor, t = t i m e f r o m the cooling c u r v e , t o = t i m e at the s t a r t of the quench, Ct(Ttf.) = t i m e at the end of the quench, = c r i t i c a l t i m e f r o m the C - c u r v e .
7075
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rl,tl
f
I i lltlt
I
I
, i Ir,,ll
[
I
I rf,~l]l
I0 TIME
I
i ,11111
I00
I000
, SEC,
Fig 1.-Fit of Fink and Willey's data 1 to Eq. [7] where (r is yield strength. i.
exp (k~r),
-
-T6
[
P r e c i p i t a t i o n k i n e t i c s for continuous cooling, t h e r e fore, can be e x p r e s s e d by the equation: = 1
0
-1
i
I
IIIIii
I
[
I
[5]
IIIII1[
f
700~
where "r s u b s t i t u t e s for t/C t tn Eq. [3]. When "r = 1, the f r a c t i o n t r a n s f o r m e d , ~,
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