Experimental observations of dendritic growth
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A typical d e n d r i t e can be v i s u a l i z e d in t e r m s of the c o m p l e x s t r u c t u r e , s c h e m a t i c a l l y p o r t r a y e d in Fig. 1, which has a p r i m a r y s t r u c t u r e and s e c o n d a r y and h i g h e r o r d e r a r m s o r b r a n c h e s . In an a r r a y of p r i m a r y d e n d r i t e s , or in an alloy which s o l i d i f i e s d e n d r i t i c a l l y , the t e r m " d e n d r i t e s p a c i n g " has no r e a l m e a n i n g , and we m u s t specify e i t h e r the p r i m a r y s p a c ing, o r the s e c o n d a r y a r m spacing, and so forth. E x a m i n a t i o n of the e x t e n s i v e l i t e r a t u r e on d e n d r i t i c growth in alloys r e v e a l s that many i n v e s t i g a t o r s have not d i s c r i m i n a t e d a m o n g the v a r i o u s o r d e r s , but often r e f e r to only "the d e n d r i t e s p a c i n g . " In other r e p o r t e d i n v e s t i g a t i o n s only the s e c o n d a r y a r m s p a c i n g has been m e a s u r e d , s i n c e this m e a s u r e m e n t is often e a s i e r to obtain than the p r i m a r y a r m spacing. Any a p p r o x i m a t e characte~rization of the d e n d r i t e s p a c i n g should take into account at l e a s t the f i r s t two o r d e r s . E l e m e n t a r y theory on the d e v e l o p m e n t of c o n s t i t u t i o n a l u n d b r c o o l i n g shows that some r a t i o of the t e m p e r a t u r e g r a d i e n t in the liquid (GL) and s o l i d i f i c a t i o n velocity (V) i n t e r v e n e s d i r e c t l y in the p l a n e - c e l l d e v e l o p m e n t of i n t e r r a c i a l s u b s t r u c t u r e ~ and this has b e e n e x p e r i m e n t a l l y c o r r o b o r a t e d by m a n y i n v e s t i g a t o r s . 2-5 Some G L / V d e p e n d e n c e has also b e e n noted d u r i n g the m o r e c o m p l e x c e l l - d e n d r i t e t r a n s i t i o n . 2'6'a The p r i m a r y d e n d r i t e velocity and the t e m p e r a t u r e g r a d i e n t at a p r i m a r y tip ought to c o n t r o l the d e v e l o p m e n t of the p r i m a r y s t r u c t u r e as well as the s e c o n d a r i e s which develop a f t e r w a r d s . Latent heat r e l e a s e d as the p r i m a r y grows affects both the s e c o n d a r y tip velocity, which should be lower than that of the p r i m a r y tip, and the g r a d i e n t at the s e c o n d a r y tip, which should approach z e r o . The g r a d i e n t in the liquid at the p r i m a r y tip d e t e r m i n e s the tip morphology as well as the f o r m a t i o n of p e r t u r b a t i o n s away f r o m the tip which eventually lead to b r a n c h i n g and s e c o n d a r y a r m f o r m a t i o n . It should be m a d e c l e a r that final s e c o n d a r y a r m s p a c i n g and morphology depends on what happens d u r i n g the r e s t of f r e e z i n g and c o n s e q u e n t l y o b s e r v a t i o n s made on s e c o n d a r i e s in the a s - c a s t a l loys a r e not the s a m e as those one might o b s e r v e developing at d
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