Homogenization of random concentration profiles by diffusion
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F e b r u a r y 1971 i s s u e of Metallurgical Transactions c o n t a i n s s e v e r a l a r t i c l e s b a s e d on i n v i t e d t a l k s p r e s e n t e d at a s y m p o s i u m on H o m o g e n i z a t i o n of A l l o y s . It i n c l u d e s a review of s o m e m a t h e m a t i c a l f o r m u l a t i o n s of the h o m o g e n i z a t i o n p r o b l e m by P u r d y and K i r k a l d y . 1 T o i l l u s t r a t e the h o m o g e n i z a t i o n p r o b l e m for twoc o m p o n e n t s y s t e m s , we m a y u s e , following t h e s e a u t h o r s , the s i m p l e s t model i n which the component u n d e r c o n s i d e r a t i o n , s o l u b l e in the s e c o n d c o m p o n e n t , i s a s s u m e d to have, at the i n i t i a l t i m e t = 0, the s i n u soidal c o n c e n t r a t i o n p r o f i l e THE
C(x, 0) = C + q-2-7(0) s i n (Trx/l)
[1]
[The a m p l i t u d e of the s i n u s o i d a l c o n c e n t r a t i o n d e v i a t i o n s f r o m the c o n s t a n t v a l u e C- is ~ - - t i m e s l a r g e r than the r o o t - m c a n - s q u a r e ( r m s ) d e v i a t i o n 7(0).] A s i s known, diffusion p r o c e s s e s in s y s t e m s like a l l o y s a r e r a t h e r complex, i n c l u d i n g diffusion along i n t e r faces b e t w e e n a d j a c e n t c r y s t a l l i n e g r a i n s or m i c r o scopic p h a s e s whose b o u n d a r i e s m a y m o v e . But if one n e g l e c t s t h e s e d e t a i l s and t a k e s s o m e s u f f i c i e n t l y high t e m p e r a t u r e , then v o l u m e diffusion m a y be c o n s i d e r e d a s p r e d o m i n a n t . We will a s s u m e , a s P u r d y and K i r k aldy did, that the t e m p e r a t u r e is kept c o n s t a n t ; f u r t h e r m o r e , the v o l u m e diffusion coefficient will be t a k e n a s i n d e p e n d e n t of the c o n c e n t r a t i o n C, i.e., D = const. T h e n the c o n c e n t r a t i o n p r o f i l e at t i m e t > 0 i s d e t e r m i n e d by the diffusion equation 0 C(r, t) = Dye C(r, t) ~t
[2]
The s o l u t i o n s a t i s f y i n g the i n i t i a l condition [1] i s
C(x, t) = C + x/2 7(t) s i n (~x//l)
[3]
where 7(t) = 7(0) exp (-mr2Dt/l 2)
[4]
That i s , the r m s d e v i a t i o n 7(t) of the c o n c e n t r a t i o n p r o file d e c a y s , due to the diffusion, t o w a r d z e r o with a relaxation time
T--
12
~-~
[5]
This result can at once be generalized to the case when the initial concentration has a wave-like distribution in three perpendicular directions, VIKTOR BEZAKis with the Electrotechnical Institute, Slovak Academy of Sciences, Bratislava,Czechoslovakia. Manuscript submitted May 17, 1971. METALLURGICALTRANSACTIONS
~z 7rx C(r, 0) = C- + f8-7(0) s i n ~ - sin Try l s i n - I-
[6]
T h e n Eq. [2] has the s o l u t i o n
~x s i n C(r, t) = C + ~-77(t) s i n -/-
~
~z s i n -~-
[7]
for t > 0, w h e r e 7(t) = 7(0) exp (-37r2Dt/l z)
[8]
In g e n e r a l , h o w e v e r , the decay of the r m s deviation 7(t) need not be a s quick a s the exponential one, a l t
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