Modeling diffusion through nonuniform concentrations of traps
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V V H E N i n t e r s t i t i a l a t o m s diffuse in solids they a r e often t r a p p e d by l a t t i c e i m p e r f e c t i o n s such as v a c a n cies, voids, d i s l o c a t i o n s , s u b s t i t u t i o n a l and other i n t e r s t i t i a l i m p u r i t i e s . In 1963 McNabb and F o s t e r 1 a t t e m p t e d to e x p r e s s the d i f f u s i o n - a n d - t r a p p i n g p r o c e s s by a m o d i f i c a t i o n of the second o r d e r diffusion equation ( F i c k ' s equation) and the i n t r o d u c t i o n of a new f i r s t o r d e r d i f f e r e n t i a l equation. They a t t e m p t e d to find a n a l y t i c a l s o l u t i o n s to t h e s e equations for a few s p e c i fic c a s e s in which the t r a p s were u n i f o r m l y d i s t r i b u t e d . L a t e r O r i a n i 2 obtained an e x p r e s s i o n for an a p p a r e n t diffusion coefficient a s s u m i n g local e q u i l i b r i u m b e tween the t r a p s and the diffusing s p e c i e s . A g a i n a u n i f o r m d i s t r i b u t i o n of t r a p s was a s s u m e d . Caskey and P i l l i n g e r 3 applied a finite d i f f e r e n c e a p p r o a c h in t h e i r a t t e m p t to solve the M c N a b b - F o s t e r equations for the case of t i m e dependent p e r m e a t i o n through a u n i f o r m d i s t r i b u t i o n of t r a p s . R e c e n t l y R o b e r t s o n 4 a l s o used a finite d i f f e r e n c e a p p r o a c h to solve the M c N a b b - F o s t e r equations for u n i f o r m d i s t r i b u t i o n s of t r a p s in n i c k e l . P r e s s o u y r e and B e r n s t e i n 5 have used an e l e c t r i c a l analogue c o m p u t a t i o n method to solve the M c N a b b F o s t e r equations and have included c a s e s involving s u r face t r a p p i n g . It is p o s s i b l e , h o w e v e r , to conceive of e x p e r i m e n t a l conditions in which the t r a p s a r e not u n i f o r m l y d i s t r i b u t e d throughout the solid or e x i s t e n t j u s t on the s u r f a c e . Two c a s e s come to m i n d . 1) Somet i m e s the t r a p s c o n s i s t of lower mobility s u b s t i t u t i o n a l or l a r g e r i n t e r s t i t i a l i m p u r i t i e s that have b e e n diffused into the s u r f a c e a c c i d e n t a l l y or i n t e n t i o n a l l y p r i o r to e x p o s u r e of the solid to the higher m o b i l i t y solute which is to be s t u d i e d . 2) A second case would be that in which the s u r f a c e of a solid had b e e n b o m b a r d e d by ions p r i o r to c a r r y i n g out a study of i n t e r s t i t i a l diffusion in that m a t e r i a l . The i n t e r s t i t i a l a t o m s might even be i n s e r t e d by ion b o m b a r d m e n t . The ion b o m b a r d m e n t , in any c a s e , would c r e a t e a l a y e r of l a t t i c e i m p e r f e c t i o n s b e n e a t h the s u r f a c e that could act as t r a p s d u r i n g the i n t e r s t i t i a l diffusion of a r e l a t i v e l y mobile i m p u r i t y . In the p a r a g r a p h s that follow we d e
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