Surface diffusion from a needle source including the effect of volume diffusion into the bulk
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J A E G E R 1 h a s g i v e n a n a p p r o x i m a t e s o l u t i o n to t h e p r o b l e m of d e t e r m i n i n g the c o n c e n t r a t i o n of a diffusing s p e c i e s to b e f o u n d in a t h i n l a y e r a t t h e s u r f a c e o f a s o l i d w h e n t h e s o u r c e i s c i r c u l a r , of f i n i t e r a d i u s a n d constant strength. His analysis treated the equivalent h e a t flow p r o b l e m a s s u m i n g t h e r e i s no f l o w p e r p e n d i c u l a r to t h e s u r f a c e . In t h a t c a s e t h e p r o b l e m i s m a t h e m a t i c a n y t h e s a m e a s t h a t of d e t e r m i n i n g t h e t e m p e r a t u r e d i s t r i b u t i o n in a m a t e r i a l of u n i f o r m h e a t conductivity surrounding a completely embedded circ u l a r c y l i n d e r w h i c h is m a i n t a i n e d at a c o n s t a n t t e m p e r a t u r e . T h u s t h e t h i c k n e s s of t h e s u r f a c e l a y e r d o e s not a p p e a r in t h e s o l u t i o n . T h e s o l u t i o n i s in t h e f o r m of a n i n t e g r a l w h i c h w a s e v a l u a t e d n u m e r i c a l l y a n d the r e s u l t s p r e s e n t e d ' a s a table w h o s e e n t r i e s , in t e r m s of t h e d i f f u s i o n p r o b l e m , g i v e t h e c o n c e n t r a t i o n of the diffusing s p e c i e s in the s u r f a c e m a t e r i a l d i v i d e d by its c o n c e n t r a t i o n in the s o u r c e . T h e s o l u t i o n i s g i v e n in t e r m s of two d i m e n s i o n l e s s p a r a m e t e r s , T and R, w h i c h a r e d e f i n e d by T -
Dst a
~
a n d R = r/a
In t h e s e e x p r e s s i o n s , D s i s the s u r f a c e d i f f u s i o n c o e f ficient, t is the time, a is the source radius and r is the radial distance from the source center. This sol u t i o n to t h e d i f f u s i o n p r o b l e m , w h i c h w i l l b e c a l l e d H(R, T) in t h i s p a p e r , n e g l e c t s t h e r e m o v a l o f t h e d i f fusing s p e c i e s f r o m the s u r f a c e by diffusion into the v o l u m e of t h e s a m p l e . S u c h a p r o c e s s b r i n g s t h e d i f f u s i n g a t o m s i n t o a r e g i o n w h e r e t h e y w i l l m o v e by a much slower volume diffusion mechanism. In w h a t f o l l o w s , u s e w i l l b e m a d e o f H(R, T) b u t it w i l l b e c o n v e n i e n t to h a v e i t s v a l u e s o v e r a g r e a t e r r a n g e of t h e p a r a m e t e r ~- t h a n w a s g i v e n b y J a e g e r . The necessary computations were made and the res u l t s a r e g i v e n in T a b l e I w h i c h i s to b e r e g a r d e d a s a n e x t e n s i o n of t h e t a b l e in J a e g e r ' s p a p e r . S e v e r a l w o r k e r s 2-4 h a v e d e r i v e d e x p r e s s i o n s t h a t a p p r o x i m a t e t h e s u r f a c e c o n c e n t r a t i o n of d i f f u s a n t which would r e s u l t when " l e a k a g e " f r o m the s u r f a c e i n t o t h e b u l k b y v o l u m e d i f f u s i o n o c c u r s . N o n e of J. H. ROSOLOWSKI is a Physicist with the Cor
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