Diffusion controlled precipitate growth in ternary systems: II

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I N a r e c e n t a r t i c l e , 1 h e n c e f o r t h r e f e r r e d to a s I, the p r e s e n t a u t h o r d e v e l o p e d a t h e o r e t i c a l a n a l y s i s of d i f f u s i o n c o n t r o l l e d p r e c i p i t a t e growth in i s o t h e r m a l t e r n a r y s y s t e m s of a r b i t r a r y c o n s t i t u t i o n . To d e l i n e a t e the e s s e n c e of the p r o b l e m , it is c o n v e n i e n t to c o n s i d e r a t e r n a r y s y s t e m 1 - 2 - 3 and a r b i t r a r i l y s e l e c t 1 and 2 a s the i n d e p e n d e n t c o m p o n e n t s . It i s a s s u m e d that a s p e c i m e n of bulk c o m p o s i t i o n (CIB , CaB ) h a s b e e n quenched f r o m a s i n g l e - p h a s e ~ f i e l d into a t w o - p h a s e c~ +/3 field, a s shown s c h e m a t i c a l l y in Fig. 1. The p r o b l e m is to i n v e s t i g a t e the i s o t h e r m a l , d i f f u s i o n a l growth of i s o l a t e d c~ p r e c i p i t a t e p a r t i c l e s in the/3 m a t r i x . In p a r t i c u l a r , one has to l o c a t e the p h a s e d i a g r a m t i e - l i n e , (C1p , C2p)-(C~f , C2M), which d e f i n e s the s t a t e of l o c a l e q u i l i b r i u m a s s u m e d to e x i s t at the p r e c i p i t a t e - m a t r i x i n t e r f a c e and, s i m u l t a n e o u s l y , to d e t e r m i n e the v e l o c i t y of t h i s i n t e r f a c e . In I, the c o n c e p t s of i n t e r f a c e c o m p o s i t i o n (IC) and i n t e r f a c e v e l o c i t y (IV) c o n t o u r s on the t e r n a r y i s o t h e r m w e r e i n t r o d u c e d a s i n s t r u m e n t s f o r the s y s t e m a t i c s o l u t i o n of the p r o b l e m j u s t outlined. To m i n i m i z e the m a t h e m a t i c a l c o m p l e x i t i e s , t e r n a r y d i f f u s i o n a l i n t e r a c t i o n was n e g l e c t e d in that treatment. U n f o r t u n a t e l y , t h e r e a r e a n u m b e r of t e r n a r y s y s t e m s f o r which the a s s u m p t i o n of n e g l i g i b l e d i f f u s i o n a l i n t e r a c t i o n is u n r e a l i s t i c . P a r t i c u l a r l y i m p o r t a n t e x a m p l e s a r e s y s t e m s of the f o r m F e - C - X w h e r e X r e p r e s e n t s a s u b s t i t u t i o n a l a l l o y i n g e l e m e n t such a s C r , Mn, Mo, Ni, Si, and s o f o r t h . A c c o r d i n g l y , the p u r p o s e of the p r e s ent c o n t r i b u t i o n is to r e c o n s i d e r the p r e c i p i t a t e growth p r o b l e m , taking a c c o u n t of t e r n a r y d i f f u s i o n a l i n t e r a c tion. It t r a n s p i r e s that, although the e n s u i n g c a l c u l a t i o n s a r e s o m e w h a t t e d i o u s , the r e s u l t a n t equations have a s u r p r i s i n g l y s i m p l e f o r m f r o m which k i n e t i c and t h e r m o d y n a m i c i m p l i c a t i o n s a r e e a s i l y e x t r a c t e d . F u r t h e r m o r e , the IC and IV c o n t o u r s a p p r o a c h d e v e l oped in ! e m e r g e s a s the m o s t s u i t a b l e m e a n s of d e a l ing with t h

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