Diffusion in the semi-solid region during dendritic growth
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AND
J. D. HUNT
Diffusion in the liquid d u r i n g the l a t e r s t a g e s of c e l l u l a r and d e n d r i t i c s o l i d i f i c a t i o n is d e s c r i b e d . A s i m p l e m o d e l is c o n s t r u c t e d , i n c o r p o r a t i n g a d a p t a t i o n s of the e a r l i e r t r e a t m e n t s by B o w e r , B r o d y , and F l e m i n g s 1'2 and R o h a t g i and A d a m s : a the a p p a r e n t conflict b e t w e e n t h e s e t r e a t m e n t s is d i s c u s s e d . Behind the g r o w t h front, the diffusion f i e l d s of n e i g h b o r i n g p r i m a r y a r m s s p r e a d l a t e r a l l y , m e e t and m e r g e in an i n i t i a l t r a n s i e n t . T h i s can take up a s i g n i f i c a n t f r a c t i o n of the s o l i d i f i c a t i o n t i m e , e s p e c i a l l y at high growth r a t e s . A f t e r the i n i t i a l t r a n s i e n t has b e e n c o m p l e t e d , the s u p e r c o o l i n g of the i n t e r d e n d r i t i c liquid is r e d u c e d , but c o n c e n t r a t i o n g r a d i e n t s in the liquid p e r s i s t throughout s o l i d i f i c a t i o n .
1. INTRODUCTION
DIFFUSIONAL segregation during dendritic solidification is commonly the main cause of microsegregation in finished metal products. There have been several recent studies of diffusion at the dendritic growth f r o n t . 4-~3 T h i s p a p e r , by c o n t r a s t , d e a l s with diffusion in the liquid d u r i n g the s o l i d i f i c a t i o n of i n t e r d e n d r i t i c liquid pools behind the growth f r o n t . Two m o d e l s of the s e m i s o l i d r e g i o n a p p e a r in the l i t e r a t u r e . The " c o m p l e t e m i x i n g " m o d e l of Schei114 and o t h e r s is b a s e d on the a s s u m p t i o n that the liquid b e t w e e n d e n d r i t e s is c o m p l e t e l y h o m o g e n e o u s (bec a u s e of r a p i d d i f f u s i o n a l o r c o n v e c t i v e mixing). B o w e r , B r o d y and F l e m i n g s ',2 d i s c u s s e d s e v e r a l of the a p p r o x i m a t i o n s involved in a p p l y i n g this m o d e l to d e n d r i t i c growth. T h e y found (Ref. 2 p. 627) that the c o m p l e t e m i x i n g a s s u m p t i o n is j u s t i f i e d if: az =~
>> 1.
[1]
x2
H e r e , D l is the diffusion c o e f f i c i e n t in the liquid; Of is the " l o c a l s o l i d i f i c a t i o n t i m e " , i . e . the t i m e b e t w e e n s t a r t and f i n i s h of s o l i d i f i c a t i o n within a l o c a l i z e d r e g i o n , and X i s the r e l e v a n t d e n d r i t e a r m s p a c ing. B o w e r e t al took X to be the s e c o n d a r y a r m s p a c ing (X2). T h e y c a l c u l a t e d that a t v a r i e d b e t w e e n 300 to 5000 for a wide r a n g e of e x p e r i m e n t a l a l u m i n u m c a s t i n g s , and d e d u c e d that the c o m p l e t e m i x i n g a s s u m p tion is g e n e r a l l y v a l i d . In the s e c o n d model, a d i f f e r e n t a p p r o a c h was a d o p t e d by R o h a t g i and A d a m s , 3 a s d i s c u s s e d
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