Growth kinetics of dispersed thoria in Ni and Ni-Cr alloys

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[1]

w h e r e ~ is the m e a n r a d i u s a f t e r a t i m e of t r e a t m e n t t, and ~o is the i n i t i a l m e a n r a d i u s . 7 is the p a r t i c l e m a t r i x i n t e r f a c i a l e n e r g y , V m is the m o l a r volume of the d i s p e r s e d phase. D and Co a r e the diffusion coeff i c i e n t and the c o n c e n t r a t i o n in e q u i l i b r i u m with the m a t r i x , at a plane i n t e r f a c e , of the d i s p e r s i o n s p e c i e s P. K. FOOTNER and C. B. ALCOCKare Research Associate, and Professor and Chairman, respectively, Department of Metallurgy and Materials Science, University of Toronto, Toronto, Canada. Manuscript submitted October 18, 1971. METALLURGICALTRANSACTIONS

which is the s l o w e s t moving through the m a t r i x . R and T a r e the gas c o n s t a n t and a b s o l u t e t e m p e r a t u r e . The a l t e r n a t i v e step in the m a s s - t r a n s f e r m e c h a n i s m that could c o n t r o l the growth p r o c e s s is the t r a n s f e r of the d i s p e r s o i d s p e c i e s a c r o s s the p a r t i c l e - m a t r i x i n t e r f a c e . W a g n e r 2 has shown that u n d e r these c o n d i t i o n s there e x i s t s a l i n e a r r e l a t i o n s h i p b e t w e e n the s q u a r e of the m e a n r a d i u s and the t i m e of heat t r e a t m e n t as e x p r e s s e d below. [2]

y3 = 6 4 C o S ' / ( V m )2 t 81RT

w h e r e S is a n u m e r i c a l c o n s t a n t . In the case of the growth of a p r e c i p i t a t e , after a given p e r i o d of time the shape of the p a r t i c l e size d i s t r i b u t i o n c u r v e always has a specific f o r m which can be p r e d i c t e d f r o m the m a s s t r a n s f e r a n a l y s i s . 1 In the c a s e of the growth of a d i s p e r s e d p h a s e , there a l r e a d y e x i s t s an i n i t i a l d i s t r i b u t i o n of p a r t i c l e s i z e s due to p r o c e s s e s involved in the m a n u f a c t u r e of the m a t e r i a l . T h i s u s u a l l y p r e cludes the p r e c i s e a p p l i c a t i o n of the L i f s h i t z - W a g n e r equation to an a n a l y s i s of d i s p e r s o i d growth r a t e s . Eq. [1] r e q u i r e s that the p a r t i c l e size d i s t r i b u t i o n has a s s u m e d the specific f o r m which is c h a r a c t e r i s t i c of the c o a r s e n i n g of p r e c i p i t a t e s at the time t = 0 and that the s u p e r s a t u r a t i o n of the m a t r i x has d e c r e a s e d to a l e v e l w h e r e e q u i l i b r i u m e x i s t s at the m a t r i x / p a r t i c l e i n t e r f a c e . T h e s e conditions give r i s e to the n u m e r i c a l 8 factor of ~ in the equation. It would appear r e a s o n a b l e that, as long as a d i s p e r s i o n is in e q u i l i b r i u m with the m a t r i x , Eq. [1 ] can be r e w r i t t e n a s ~3 _ ~

_ Kq~DCo(Ym)2t RT

[3]

for the case of d i s p e r s o i d growth w h e r e K is a c o n s t a n t the value of which is dependent on the p a r t i c l e size d i s t r i b u t i o n . T h i s m e