On the time exponent in isothermal grain growth
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in p u r e m N O g r a i nRgrowth M Ae t a l s andL s o l i d - s o l u tion a l l o y s can be d e s c r i b e d a p p r o x i m a t e l y by the e m p i r i c a l e q u a t i o n f i r s t e s t a b l i s h e d by Beck and c o w o r k e r s :a-3 D = Kt n
[1]
w h e r e D is the a v e r a g e g r a i n d i a m e t e r , t is the i s o t h e r m a l a n n e a l i n g t i m e , and K a n d n a r e p a r a m e t e r s d e p e n d i n g on m a t e r i a l and t e m p e r a t u r e . T h u s , in a log D v s log t plot, g r a i n growth data c a n be r e p r e s e n t e d by a s t r a i g h t line for each t e m p e r a t u r e , with n b e i n g the slope and K b e i n g the i n t e r c e p t . F o r v e r y s h o r t p e r i o d s of a n n e a l i n g , the t i m e r e q u i r e d for r e c r y s t a l l i z a t i o n is no l o n g e r n e g l i g i b l e ; d e v i a t i o n s f r o m the s t r a i g h t - l i n e r e l a t i o n s h i p a r e c o m m o n l y o b s e r v e d . Eq. [1] r e q u i r e s t h a t D = 0 at t = 0, which is not s t r i c t l y the c a s e in g r a i n growth, but with i n c r e a s i n g a n n e a l i n g t i m e s u c h d e v i a t i o n s b e c o m e u n d e t e c t a b l e . The e q u a tion for g r a i n growth has a l s o b e e n e x p r e s s e d f r e q u e n t l y in the f o r m : nt D n'-D O =Kit [2] w h e r e D O d e n o t e s the i n i t i a l g r a i n s i z e . With inc.reasing a n n e a l i n g t i m e , D Ob e c o m e s n e g l i g i b l e in c o m p a r i son with D . Hence, Eq. [2] is e s s e n t i a l l y e q u i v a l e n t to Eq. [1].* T h i s g r a i n - g r o w t h law also has b e e n found *The equivalencebetweenEqs. [1] and [2] has recentlybeenshownby Murty.33 v a l i d for n o n m e t a l s , such a s s i n t e r e d c e r a m i c c o m pounds .4, s The v a l u e of n in Eq. [1], the t i m e exponent for g r a i n growth, u s u a l l y i n c r e a s e s with t e m p e r a t u r e , and a p p r o a c h e s 0.5 by e x t r a p o l a t i o n to the m e l t i n g point. In o t h e r c a s e s , n m a y be c o n s t a n t o v e r a r a n g e of t e m p e r a t u r e s , but its value is a l w a y s l e s s than, o r at m o s t equal to 0.5. P o s s i b l e f a c t o r s that m a y affect the v a l u e of n w e r e d i s c u s s e d e x t e n s i v e l y by F u l l m a n f l His a n a l y s e s led to the c o n c l u s i o n that r e s i d u a l s t r a i n s r e m a i n i n g in the m e t a l a f t e r r e c r y s t a l l i z a t i o n , o r l o c a l p r e f e r r e d o r i e n t a t i o n s r e s u l t i n g f r o m g r a i n growth, could p o s s i b l y a c c o u n t for the wide v a r i a b i l i t y of the n v a l u e s . N i e l s e n 7 a s s e r t e d that the c o m b i n e d effects of s m a l l - g r a i n d i s a p p e a r a n c e s , i m p u r i t y b u i l d - u p on the b o u n d a r i e s , and c o a l e s c e n c e of g r a i n s d u r i n g growth HSUN HU and B. B. RATH are with the E. C. Bain Laboratory for Fundamental Research, U. S. Steel Corp., Resea
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