Solution kinetics of CuAl 2 in an Al-4Cu alloy
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The r e s u l t s of a r e c e n t l y - d e v e l o p e d , s p h e r i c a l , finite g e o m e t r y m a t h e m a t i c a l m o d e l u s i n g num e r i c a l m e t h o d s a r e c o m p a r e d to e x i s t i n g c l o s e d - f o r m m o d e l s . The g e o m e t r y and b o u n d a r y conditions i m p o s e d by the c l o s e d - f o r m m o d e l s a r e shown to be r e s t r i c t i v e in the application of these m o d e l s to the solution k i n e t i c s of second p h a s e s . A q u a n t i t a t i v e m e t a l l o g r a p h y study of the solution k i n e t i c s of CuA12 in an A1-4 wt pet Cu alloy was u n d e r t a k e n to provide data on the changes in s e c o n d - p h a s e p a r t i c l e size d i s t r i b u t i o n s as a function of solution t r e a t m e n t t i m e . The s i z e d i s t r i b u t i o n s were shown to v a r y a c c o r d i n g to the p r o p o s e d s p h e r i c a l , finite model formulated using numerical methods.
AARON 1 and Whelan 2 have r e c e n t l y developed m a t h e m a t i c a l m o d e l s for the s o l u t i o n k i n e t i c s of s e c o n d p h a s e s u n d e r c o n d i t i o n s where c o n c e n t r a t i o n g r a d i e n t s w e r e a b s e n t f r o m the two p h a s e s p r i o r to the solution treatment. Aaron's model assumes one-dimensional flow, l i n e a r g r a d i e n t s d u r i n g solution, and a s e m i i n f i nite c o n c e n t r a t i o n - d i s t a n c e profile. Whelan m o d i f i e d A a r o n ' s o n e - d i m e n s i o n a l flow model by p r o v i d i n g an e r r o r function profile r a t h e r than a l i n e a r g r a d i e n t . In addition, Whelan developed a s p h e r i c a l flow solution model assuming a semiinfinite concentration-distance profile. T a n z i l l i and Heckel 3'4 have shown the a p p l i c a b i l i t y of n u m e r i c a l methods and c o m p u t e r t e c h n i q u e s to the p r o b l e m of solution k i n e t i c s of second p h a s e s . The use of n u m e r i c a l m e t h o d s e l i m i n a t e s the n e c e s s i t y for s i m p l i f y i n g a s s u m p t i o n s such as l i n e a r g r a d i e n t s and s e m i i n f i n i t e g e o m e t r i e s . The c a l c u l a t i o n s of T a n z i l l i a n d Heckel 3'4 have shown that, in g e n e r a l , the c o m p o s i t i o n at the midpoint of two p a r t i c l e s u n d e r g o i n g solution b e g i n s to change e a r l y in the p r o c e s s . T h u s , for m a n y i n s t a n c e s , the s e m i i n f i n i t e g e o m e t r y is a v e r y r e s t r i c t i v e a s s u m p t i o n and the m e a n c o m p o s i t i o n of the s y s t e m m u s t be specified. F o r a s e m i i n f i n i t e s y s t e m , the m e a n c o m p o s i t i o n i s m e a n i n g l e s s and, t h e r e f o r e , cannot be specified. A c o m p a r i s o n of s o l u t i o n k i n e t i c s m o d e l s of A a r o n , ~ W h e l a n , 2 and T a n z i l l i and Heckel 3 is p r e s e n t e d in Fig. 1 in t e r m s of the n o r m a l i z e d second phase size ~/l a s a f
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