Numerical calculation of concentration distribution and dimensional change for constant-rate diffusion of a solute into
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T H E g e n e r a l f o r m of F i c k ' s s e c o n d law for o n e - d i m e n s i o n a l diffusion of c o m p o n e n t A in a b i n a r y s o l u t i o n of A and B is
OCA at
a [--aCA'~ - 8x \O - - ~ /
[1]
w h e r e CA is the c o n c e n t r a ~ o n of c o m p o n e n t A; x, the d i s t a n c e ; t, the time; and D, the m u t u a l diffusion c o e f f i c i e n t which m a y depend on CA . H a r t l e y and C r a n k x'2 have p o i n t e d out that Eq. [1] with c o n v e n t i o n a l length units for m e a s u r i n g the d i s t a n c e x holds only if the t o t a l v o l u m e of the diffusing m a t e r i a l s does not change d u r i n g the diffusion p r o c e s s . That is, the p a r t i a l m o l a r v o l u m e s of c o m p o n e n t s A and B, VA and VB, r e s p e c t i v e l y , a r e not functions of c o n c e n t r a t i o n . P r a g e r s showed that if VA and VB a r e c o n c e n t r a t i o n dependent an e x t r a t e r m m u s t be added to Eq. [1] to c o m p l e t e l y d e s c r i b e the t i m e r a t e of change of c o n c e n t r a t i o n :
a ic x
/aVB caCA
]
[2]
The m u t u a l diffusion coefficient /~ is
= DA VBCB + DB VACA
[3]
w h e r e DA and DB a r e the i n t r i n s i c diffusion coeffic i e n t s for c o m p o n e n t s A and B. The value of D for a p a r t i c u l a r c o n c e n t r a t i o n CA c a n n o t be found by the s t a n d a r d B o l t z m a n n - M a t a n o method, as this method a p p l i e s only to equations of the f o r m of Eq. [1]. H a r t l e y and C r a n k x'2 s u g g e s t e d use of the modified
c o o r d i n a t e ~ in o r d e r to r e t a i n the g e n e r a l f o r m of Eq. [1] and thus allow for the use of the B o l t z m a n n Matano technique for d e t e r m i n i n g / ~ . This p a p e r e x p l o r e s the use of the m o d i f i e d c o o r d i nate to s i m p l i f y the m a t h e m a t i c s i n v o l v e d in the c a l c u l a t i o n of c o n c e n t r a t i o n d i s t r i b u t i o n s and d i m e n s i o n a l c h a n g e s in the c a s e where a solute is diffused into the s u r f a c e of a s u b s t r a t e s o l v e n t at a c o n s t a n t r a t e . I) RELATION BETWEEN MODIFIED COORDINATE ~ y AND UNITS OF CONCENTRATION The modified c o o r d i n a t e ~ y g r e a t l y s i m p l i f i e s c a l c u lation of c o n c e n t r a t i o n and d i m e n s i o n a l changes of diffusion c o a t i n g s . Equal i n c r e m e n t s of ~ y c o n t a i n an equal n u m b e r of m o l s of component B, e.g.
dn B = ( N B S / V ) d x = d ~ y
[4]
w h e r e S is a unit c r o s s - s e c t i o n a l a r e a in sq cm, V is the local m o l a r v o l u m e of the solution, and N B is the local mol f r a c t i o n of component B. In o r d e r to use this modified c o o r d i n a t e , c o n c e n t r a t i o n CA is m e a s u r e d in t e r m s of m o l a r r a t i o of c o m p o n e n t s A and B (C~ = YA = n A / n B ) . A diffusion s y s t e m d e s c r i b e d by the ~y c o o r d i n a t e cannot have a c o n c e n t r a t i o n g r a
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