Initiation of dendrites by crystal imperfections
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T H E f a c t o r s which d e t e r m i n e w h e t h e r or not a p l a n a r c r y s t a l s u r f a c e growing into a liquid will b r e a k down into a c e l l u l a r o r d e n d r i t i c s t r u c t u r e have b e e n of inc r e a s i n g i n t e r e s t in r e c e n t y e a r s . An i m p o r t a n t i n i t i a l s t e p in u n d e r s t a n d i n g t h e s e f a c t o r s was the d e v e l o p m e n t b y R u t t e r and C h a l m e r s of the concept of c o n s t i t u t i o n a l supercooling. 1 However, constitutional supercooling is a n e c e s s a r y but not s u f f i c i e n t condition for the growth of c e l l s and d e n d r i t e s , and m o r e d e t a i l e d c r i t e r i a a r e n e e d e d to p r e d i c t w h e t h e r c e l l s or d e n d r i t e s will a c t u a l l y a p p e a r . R e c e n t d y n a m i c a l a n a l y s e s of t h i s p r o b l e m have u s e d the t e c h n i q u e s of p e r t u r b a t i o n a n a l y s i s a s i n i t i a l l y a p p l i e d by M u l l i n s and S e k e r k a 2 and b y V o r o n k o v s to the s t a b i l i t y p r o b l e m in c r y s t a l growth. Such a n a l y s e s m a y include e f f e c t s of s o l i d / l i q u i d s u r face e n e r g y , i n t e r f a c e r e a c t i o n k i n e t i c s , fluid flow, and t i m e - d e p e n d e n t h e a t flow. The s t a b i l i t y of p l a n a r i n t e r f a c e s is i n v e s t i g a t e d t h e o r e t i c a l l y by c a l c u l a t i n g w h e t h e r an i n f i n i t e s i m a l p e r t u r b a t i o n of the p l a n a r i n t e r f a c e g r o w s or s h r i n k s . T h e s e a n a l y s e s r e s u l t in r a t h e r c o m p l i c a t e d e x p r e s s i o n s for the c o n d i t i o n s s e p a r a t i n g s t a b l e p l a n a r growth f r o m u n s t a b l e growth; when the a p p r o p r i a t e l i m i t s a r e t a k e n , t h e s e e x p r e s s i o n s r e d u c e to the c o n s t i t u t i o n a l s u p e r c o o l i n g c r i t e r i a . P e r t u r b a t i o n a n a l y s i s d e a l s only with the i n i t i a l s t a g e s of d e n d r i t e growth when the s o l i d / l i q u i d i n t e r f a c e is just b e g i n n i n g to d e v i a t e f r o m p l a n a r i t y . When the a m p l i t u d e of the p e r t u r b a t i o n s on the s o l i d / l i q u i d s u r f a c e b e c o m e s c o m p a r a b l e to t h e i r wavelength the m a t h e m a t i c s of an e x a c t t i m e - d e p e n d e n t a n a l y s i s b e c o m e h o p e l e s s l y c o m p l e x . U n d e r such c o n d i t i o n s , the d e t a i l e d d e v e l o p m e n t of d e n d r i t e s cannot be p r e d i c t e d t h e o r e t i c a l l y in any q u a n t i t a t i v e way. J a c k s o n 4 h a s shown that c e r t a i n t r a n s p a r e n t o r g a n i c m a t e r i a l s c r y s t a l l i z e with u n f a c e t e d s o l i d / l i q u i d i n t e r f a c e s , a s do t y p i c a l m e t a l s . T h e s e t r a n s p a r e n t m a t e r i a l s can t h e r e f o r e be u s e d a s a n a l o g s of m e t a l
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