Activation energies for the coarsening of compound precipitates
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A fine p a r t i c l e d i s p e r s i o n is one of the m o s t effective s t r e n g t h e n i n g m e c h a n i s m s in h i g h - t e m p e r a t u r e a l l o y s . A c c o r d i n g l y , the s t a b i l i t y of such d i s p e r s i o n s to c a p i l l a r i t y - d r i v e n c o a r s e n i n g (Ostwald r i p e n i n g ) is a m a j o r c o n c e r n in the d e v e l o p m e n t of such m a t e r i a l s . The t h e o r y for d i f f u s i o n - c o n t r o l l e d c o a r s e n i n g of p r e c i p i t a t e p a r t i c l e s in b i n a r y s o l i d s i s r e a s o n a b l y well developed, z-3 The p a r t i c l e size d i s t r i b u t i o n h a s been shown to take on a c h a r a c t e r i s t i c shape which is i n d e p e n d e n t of the s t a r t i n g size d i s t r i b u t i o n and has a long tail at s m a l l s i z e s , but i s s h a r p l y t r u n c a t e d at l a r g e s i z e s . The d i s t r i b u t i o n d r o p s s h a r p l y to z e r o for p a r t i c l e s with r a d i i 1.5 t i m e s the a v e r a g e . The a v e r a g e p a r t i c l e r a d i u s , r , is given ''2 by
7 ~= 7~ + 8DC~V~t 9RT
[1]
w h e r e ro is the a v e r a g e r a d i u s at z e r o t i m e , D i s diffus i v i t y , C is s o l u b i l i t y in the m a t r i x , (r is p r e c i p i t a t e m a t r i x i n t e r r a c i a l e n e r g y , V = m o l a r v o l u m e , R T is gas c o n s t a n t t i m e s a b s o l u t e t e m p e r a t u r e , and t is t i m e . Eq. [1] is in good a g r e e m e n t with e x p e r i m e n t a l data for c o a r s e n i n g of p r e c i p i t a t e s in s e v e r a l b i n a r y m e t a l lic s y s t e m s . 4-~ W a g n e r 2 a l s o c o n s i d e r e d the c o a r s e n ing of compound p r e c i p i t a t e s , but was m a i n l y i n t e r e s t e d in fluid m e d i a , w h e r e all d i f f u s i v i t i e s tend to be much the s a m e . His r e s u l t was e s s e n t i a l l y e q u i v a l e n t to Eq. [1], differing only in a slightly modified s o l u b i l i t y and m o l a r v o l u m e . W a g n e r was not c o n c e r n e d with a c t i v a tion e n e r g i e s , or with the d e p e n d e n c e of c o a r s e n i n g r a t e on e n v i r o n m e n t . The coefficient of t i m e in Eq. [1] is the r a t e c o n s t a n t for c o a r s e n i n g and m a y be e x p r e s s e d a s a p r e e x p o n ential t e r m and an a c t i v a t i o n e n e r g y . Such a f o r m u l a tion m a k e s for s i m p l e p r e s e n t a t i o n , i n t e r p o l a t i o n and e x t r a p o l a t i o n of e x p e r i m e n t a l data. Some s t u d i e s on c o a r s e n i n g of compound p r e c i p i t a t e s (SiO2 in C u r A1203 in Ni 8) have a t t e m p t e d to a s s o c i a t e the a c t i v a t i o n e n e r g y for c o a r s e n i n g with that for s o m e S. K. BHATTACHARYYAand K. C. RUSSELL are Staff Member, Division of Sponsored Research and Associate Professor, respectively, Department of Metallurgy and Materials Science, Massachusetts Institute of Technology, Cambridge, Mass. This
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