The analysis of the evolution of particle size distribution during microstructural change
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~'(~-) n,(R,l)- C(R,'r) where ~ is the n u c l e a t i o n t i m e for p a r t i c l e s that grow to size R at t i m e t, and G(R,'c) is a p r o p e r l y e v a l u a t e d growth r a t e for these p a r t i c l e s . It is shown that this equation p r o v i d e s a b a s i s for c o n s t r u c t i n g m o d e l s of the d y n a m i c s of m i c r o s t r u c t u r a l change in g e o m e t r i c a l l y complex s y s t e m s . m o s t d eH t a i l e d d e s c r i p tE i o n of a m i c r o s t r u c t u r a l T change is contained in the m e a s u r e m e n t of the v a r i a tion of the p a r t i c l e size d i s t r i b u t i o n with t i m e . P r o c e d u r e s for a n a l y z i n g such i n f o r m a t i o n to d e t e r m i n e growth r a t e s , for e x a m p l e , have b e e n p r o p o s e d by Spektor, 1 Lifshitz, 2 and Heckel. 3'4 T h e s e a n a l y s e s tend to be e i t h e r t h e o r e t i c a l and a b s t r u s e , or n u m e r i c a l and inflexible. It is the p u r p o s e of the p r e s e n t p a p e r to develop a s i m p l e , flexible p r o c e d u r e for a n a l y z i n g the evolution of p a r t i c l e size d i s t r i b u t i o n d u r i n g m i c r o s t r u c t u r a l change, and r e l a t i n g the r e s u l t to the r a t e of n u c l e a t i o n (or a n n i h i l a t i o n ) or p a r t i c l e s , and to the d i s t r i b u t i o n i n t e r f a c e v e l o c i t i e s (growth r a t e s ) in the s y s t e m . This p u r p o s e is a c c o m p l i s h e d by 1) i n t r o d u c i n g the concept of the growth path enevelope for a m i c r o s t r u c t u r a l change; 2) d e v e l o p i n g a s i m p l e g r a p h i c a l p r o c e dure for deducing the growth path envelope and n u c l e ation r a t e s f r o m a s e r i e s of m e a s u r e m e n t s of the p a r t i c l e size d i s t r i b u t i o n at d i f f e r e n t t i m e s ; 3) d e r i v i n g an a n a l y t i c a l r e l a t i o n between the p a r t i c l e size d i s t r i bution function and the n u c l e a t i o n and growth r a t e s for the s y s t e m . It is i m p o r t a n t to e m p h a s i z e that any a n a l y s i s of the evolution of p a r t i c l e size d i s t r i b u t i o n is l i m i t e d to app l i c a b i l i t y to s y s t e m s for which the p a r t i c l e size d i s t r i b u t i o n has r e a l g e o m e t r i c s i g n i f i c a n c e , and m a y be m e a s u r e d . Specifically, these t e c h n i q u e s a r e l i m i t e d to s y s t e m s c o n s i s t i n g of i s o l a t e d p a r t i c l e s of r e l a t i v e l y s i m p l e shape d i s p e r s e d in a m a t r i x . All e x p e r i m e n t a l p r o c e d u r e s c u r r e n t l y a v a i l a b l e for m e a s u r i n g size d i s t r i b u t i o n s m e t a l l o g r a p h i c a l l y contain the a s s u m p t i o n that all of the p a r t i c l e s a r e the s a m e shape, and that this shape is known, or can be i n d e p e n d e n t l y d e t e r m i n e d , or at l e a s t modeled, s's However, it will be shown in a l a t e r p
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