Cross terms in the thermodynamic diffusion equations for multicomponent alloys
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The kinetic e q u a t i o n s for the atom fluxes in a m u l t i c o m p o n e n t alloy a r e d e r i v e d in t e r m s of a t o m i c d r i v i n g f o r c e s for a r a n d o m alloy model. In this model, the c o n t r i b u t i o n to each atom flux f r o m the v a c a n c y wind effect is d i r e c t l y p r o p o r t i o n a l to the v a c a n c y flux. The kinetic equations when w r i t t e n in the s a m e form a s the t h e r m o d y n a m i c diffusion equations allow the Lik c o e f f i c i e n t s to be identified. The c r o s s t e r m s r e l a t i n g the flux of s p e c i e s i to the c h e m i c a l potential g r a d i e n t s of other s p e c i e s k a r e found to be n o n z e r o . These c r o s s t e r m s can have an a p p r e c i a b l e effect on the atom fluxes and can lead to negative i n t r i n s i c diffusion coefficients even in ideal s y s t e m s . N e g l e c t i n g the c r o s s t e r m s can lead to s e r i o u s i n a c c u r a c i e s . The Lik in an n - c o m p o n e n t s y s t e m depend on the n t r a c e r diffusion coefficients. In this r e s p e c t , it s t i l l is p o s s i b l e to e x p r e s s the n atom fluxes in t e r m s of only n coefficients. W H E N diffusion o c c u r s by a v a c a n c y m e c h a n i s m and t h e r e is a flux of v a c a n c i e s , the jump p r o b a b i l i t i e s of the i n d i v i d u a l a t o m s will be a l t e r e d by the v a c a n c y flux. T h i s effect, which has b e e n d i s c u s s e d in detail for b i n a r y a l l o y s , ~,2 l e a d s to a d d i t i o n a l t e r m s in the diffusion e q u a t i o n s . T h e s e a d d i t i o n a l t e r m s a p p e a r p a r t i a l l y as c r o s s t e r m s in the g e n e r a l t h e r m o d y n a m i c diffusion equations.2's In the p r e s e n t p a p e r , the a n a l y s i s applied p r e v i o u s l y to b i n a r y a l l o y s i s extended to t e r n a r y and o t h e r m u l t i c o m p o n e n t a l l o y s . The method used is f i r s t to c a l c u l a t e k i n e t i c e x p r e s s i o n s for a r a n d o m alloy m o d e l , finding the atom flux of each s p e c i e s in t e r m s of atom and v a c a n c y jump f r e q u e n c i e s . T h e s e e q u a t i o n s then a r e exp r e s s e d in t e r m s of the t h e r m o d y n a m i c d r i v i n g f o r c e s , and the c r o s s c o e f f i c i e n t s Lik a r e identified. The r e s u l t i n g kinetic e x p r e s s i o n s for the Lik a r e valid for a l l a l l o y s , e i t h e r b i n a r y or m u l t i c o m p o n e n t , for which the r a n d o m alloy model a p p l i e s . Effects of the c r o s s t e r m s in s e v e r a l s i m p l e c a s e s a r e d i s c u s s e d . It is found that the c r o s s t e r m s can be v e r y i m p o r t a n t . KINETIC EQUATIONS The g e n e r a l a t o m i c - k i n e t i c equation for the flux Ji of s p e c i e s i in the x d i r e c t i o n i s
~c i Fici D* Ji = - D * ~ + kT
[1]
where D* i s the t r a c e r diffusion coefficient of s p e c i e s i, ci is the c o n c e n
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