Coarsening of grain-boundary precipitates
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R E C E N T L Y the coarsening of precipitates has a t t r a c t e d c o n s i d e r a b l e i n t e r e s t . A f t e r the p i o n e e r i n g w o r k o f L i f s h i t z a n d S l y o z o v , 1 W a g n e r2, h a s t r e a t e d e x t e n s i v e l y the p r o b l e m o f O s t w a l d r i p e n i n g o f s e c o n d p h a s e p a r t i c l e s in a m a t r i x f o r diffusion a n d i n t e r f a c e c o n t r o l l e d r e a c t i o n s . E a c h of the a l t e r n a t i v e m e c h a n i s m s p r o d u c e s a c h a r a c t e r i s t i c s t e a d y s t a t e s i z e distribution, independent o f t h e size distributions at t h e b e g i n n i n g of c o a r s e n i n g , w h i c h i s s u b s e q u e n t l y only s c a l e d in t i m e . G r e e n w o o d a h a s r e v i e w e d t h e s u b j e c t and d i s c u s s e d experimental evidence, w h i c h indicates that c o a r s e n i n g o f c o b a l t p a r t i c l e s d i s p e r s e d i n a c o p p e r - m a t r i x , c o p p e r p r e c i p i t a t e s i n i r o n , Ni3A1 (7') i n A 1 - N i a n d m a n g a n e s e p r e c i p i t a t e s in M g - M n a r e diffusion controlled. Greenwood3 noted the a b s e n c e of a detailed statistical a n a l y s i s for the coarsening of prec i p i t a t e s a t g r a i n b o u n d a r i e s , w h i c h a r e f e d by i n t e r f a c i a l r a t h e r than v o l u m e d i f f u s i o n . A n a p p r o x i m a t e t r e a t m e n t of t h i s p h e n o m e n o n h a s b e e n g i v e n by Speight. 5 A more detailed a n a l y s i s of this phenomenon s e e m s d e s i r a b l e , b e c a u s e M u k h e r j e e e t a l .4 h a v e sugg e s t e d o v e r l a p p i n g o f two characteristic distribution c u r v e s (one f o r g r a i n - b o u n d a r y p r e c i p i t a t e s a n d o n e f o r p a r t i c l e s within t h e grains) to e x p l a i n t h e i r r e g u l a r i t i e s i n t h e d i s t r i b u t i o n s of c a r b i d e s i n c h r o m i u m s t e e l s , w h i c h they f o u n d . T h e p u r p o s e o f t h i s p a p e r is to p r o v i d e a f o r m a l t r e a t m e n t of t h i s p r o b l e m . T h e r e l a t i o n s a n d i n t e r a c t i o n s b e t w e e n d i f f e r e n t c o a r s e n i n g m e c h a n i s m s w i l l be d i s c u s s e d in a subsequent p a p e r . F o r t h e s a k e o f b r e v ity t h e n o t a t i o n of W a g n e r2 w i l l be c l o s e l y f o l l o w e d .
f r o m F i g . 1.* T h e r e l e v a n t p o i n t f o r t h e d i s c u s s i o n i s *The volume of a precipitate as shown in Fig. 1 is 27r[~ -- cos 0 + ~- cos30] P = 27tAr3 = 21r[~- (~,g/2~')+ ~-(Vg/2V) 3 ]r3 where
y)/r
cos0=(r
that g r a i n - b o u n d a r y p r e c i p i t a t e s o f a n y s i z e , i f t h e y h a v e a t t a i n e d t h e i r e q u i l i b r i u m s h a p e , c a n be c h a r a c t e r i z e d by t h e i r c h e m i c a l p o t e n t i a l , i . e . a r a t i o ( 2 7 / r ) . P a r t i c l e s of different s i z e s have different interracial e n e r g y / r a d i u s r a t i o s of this
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