An extension of the dehoff growth path analysis
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DEHOFF 1-s h a s d i s c u s s e d an a n a l y s i s of p h a s e t r a n s f o r m a t i o n k i n e t i c s which he has d e s i g n a t e d Growth Path A n a l y s i s (GPA). The growth path of a p a r t i c l e * i s a c o m p l e t e s p e c i f i c a t i o n of i t s g e o m e t r y *The analysis is very general and may be applied to voids, bubbles, dislocation loops, etc., as welt as particles.
a s a function of t i m e , including i t s n u c l e a t i o n t i m e . F o r e x a m p l e , s p h e r i c a l p a r t i c l e s m i g h t grow with r a d i i , R = R (~-, t), w h e r e ~- _< t is the t i m e a given p a r t i c l e n u c l e a t e s , and t i s t i m e . The c o m p l e t e f a m i l y of growth p a t h s d e f i n e s a s u r f a c e c a l l e d the growth path envelope. DeHoff's c o n t r i b u t i o n was b a s i c a l l y a f o r m u l a t i o n , involving p a r t i c l e n u c l e a t i o n and growth r a t e s , of the quantity nv(R , t). This q u a n t i t y is a n u m b e r d e n s i t y function, such that nv(R, t) dR i s the n u m b e r of p a r t i c l e s p e r unit v o l u m e in the s i z e band R to R + dR at t i m e t. B a s i c to DeHoff's a n a l y s i s and that which follows a r e the a s s u m p t i o n s that a l l p a r t i c l e s u n d e r a n a l y s i s have the s a m e , t i m e i n d e p e n d e n t s h a p e and that a l l p a r t i c l e s n u c l e a t i n g in the s a m e i n s t a n t of t i m e f o l low i d e n t i c a l g r o w t h p a t h s . DeHoff's t h i r d a s s u m p t i o n was t h a t g r o w t h p a t h s do not c r o s s . If a p a r t i c l e n u c l e a t e s at t i m e ~'a and g r o w s to a s i z e R = R ( T a , t) which is l e s s than the s i z e of a y o u n g e r p a r t i c l e which n u c l e a t e s at t i m e rb > ~'a, t h e s e growth p a t h s a r e s a i d to have c r o s s e d . The p u r p o s e of t h i s w o r k i s f i r s t to i l l u s t r a t e that nv(R , t) s a t i s f i e s the c o n t i n u i t y e q u a t i o n which i m p l i e s that the p a r t i c l e population b e h a v e s l i k e a fluid in n u m b e r d e n s i t y s p a c e . This r e s u l t is due to D e H o f f ' s f o u r t h and i m p l i c i t a s s u m p t i o n that the t o t a l n u m b e r of p a r t i c l e s p e r unit v o l u m e is a d e t e r m i n i s t i c quantity. An a l t e r n a t e t e c h n i q u e for the f o r m u l a t i o n of nv(R , t) w i l l b e p r e s e n t e d . A l s o , two t e c h n i q u e s f o r c a l c u l a t i n g v o l ume fraction transformed, heretofore considered dist i n c t , w i l l be shown to y i e l d i d e n t i c a l r e s u l t s . Secondly, by e m p l o y i n g a t e c h n i q u e v e r y s i m i l a r to DeHoff's, we s h a l l g e n e r a l i z e G P A to i n c l u d e s t o c h a s t i c n u c l e a t i o n . The d e t e r m i n i s t i c and s t o c h a s t i c t r e a t m e n t s a r e then c o m p a r e d using a s i m p l e n u c l e a t i o n and growth m o d e l . B e f o r e p r o c e e d i n g ,
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