Generalization of the nucleus shape-dependent parameters in the nucleation rate equation
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where J* = nucleation rate, Z = Zeldovich nonequilib r i u m f a c t o r , ~3" = r a t e at which s i n g l e a t o m s i m p i n g e upon the c r i t i c a l nucleus, N = d e n s i t y of a t o m i c nuc l e a t i o n s i t e s , ZxG* = f r e e e n e r g y of a c t i v a t i o n f o r c r i t i c a l n u c l e u s f o r m a t i o n , r = incubation t i m e and t = i s o t h e r m a l r e a c t i o n t i m e . J o h n s o n et al e d e r i v e d e x p r e s s i o n s f o r ~G*, fl*, Z and r f o r h o m o g e n e o u s nuc l e a t i o n and f o r h e t e r o g e n e o u s n u c l e a t i o n at d i s o r d e r e d g r a i n and i n t e r p h a s e b o u n d a r i e s , both in the a b s e n c e and p r e s e n c e of a f a c e t on the c r i t i c a l n u c l e i . D e s p i t e the r a n g e of n u c l e u s m o r p h o l o g i e s and n u c l e a t i o n s i t e s c o n s i d e r e d , the e x p r e s s i o n obtained f o r e a c h p a r a m e t e r d i f f e r s f r o m that for a h o m o g e n e o u s l y n u c l e a t e d s p h e r e only t h r o u g h the i n c o r p o r a t i o n of e i t h e r o r both of two m u l t i p l y i n g f a c t o r s . T h e four e q u a t i o n s p r e s e n t e d below T a b l e I a r e n o r m a l i z e d v e r s i o n s of the r e l a t i o n s h i p s obtained by J o h n s o n et al. T h e y have b e e n r e a r r a n g e d to y i e l d the n u m e r i c a l c o e f f i c i e n t of each, e x p r e s s e d as K A y * , K f l . , K Z and K T. T h e v a l u e s of t h e s e c o e f f i c i e n t s f o r s p h e r e - b a s e d n u c l e i a r e g i v en in the s e c o n d c o l u m n of T a b l e I. R e c e n t l y , S h i f l e t e t al ~ r e p o r t e d the r e s u l t s of s i m i l a r c a l c u l a t i o n s on the h o m o g e n e o u s n u c l e a t i o n k i n e t i c s of n u c l e i shaped as p a r a l l e l e p i p e d s of s q u a r e c r o s s s e c t i o n . T h e r a t i o of the i n t e r r a c i a l e n e r g y of the b a s e s of the p a r a l l e l e p i p e d s , Vb, to that of t h e i r f a c e s , 7f, and h e n c e t h e i r a s p e c t r a t i o , U, w a s t a k e n as a v a r i a b l e (see F i g . l(a)). T h e shape d e p e n d e n t eq u ati o n s obtained by Shiflet et al have the s a m e f o r m as t h o s e r e p o r t e d by J o h n s o n et al for s p h e r e - b a s e d nuc l e i . T h e two s e t s of e q u a t i o n s d i f f e r only in t h e i r n u m e r i c a l c o e f f i c i e n t s . T h e c o e f f i c i e n t s of the eq u ati o n s fo r p a r a l l e l e p i p e d s , in which the a s p e c t r a t i o has b e e n included, can be c o n s i d e r e d as l i s t e d in the t h i r d c o l u m n of T a b l e I for the c a s e of a four (equal) sided p r i s m . Shiflet el al 3 e x c l u d e d n u c l e a t i o n at g r a i n and i n t e r p h a s e b o u n d a r i e s f r o m c o n s i d e r a t i o n b e c a u s e that p o r t i o n of a p a r a l l e l e p i p e d - b a s e d n u c l e u s which e x tends a c r o s s a d i s o r d
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