Desorption Rates at Electrode/Electrolyte Interfaces

The u n d e r s t a n d i n g of t he m i c r o s c o p i c de ta i l s of processes o c c u r r i n g a t m e t a l a n d s e m i c o n d u c t o r e l ec t rodes h a v e g rea t i m p o r t a n c e in t he d e s i g n a n d i m p r o v e m e n t of e l e c t r o c h e m i c a l cells, fuel cells, etc., as wel l as in t he fund a m e n t a l s t u d y of so l id / l iqu id in ter faces . T he bas ic a n d m o s t i m p o r t a n t s tep in a n y s y s t e m t h a t c o n t a i n s a so l id / l iqu id in t e r f ace is t he c h a r g e t r a n s f e r process . Alt h o u g h a la rge a m o u n t of w o r k h a s b e e n d e v o t e d to t he s t u d y of c h a r g e t r a n s f e r at e lec t rodes , re la t ive ly l i t t le is k n o w n a b o u t t he de ta i l ed m i c r o s c o p i c d y n a m i c s of t h e s e p rocesses . The ra te of c h a r g e t r a n s f e r at an e l ec t rode can b e r ega rded , in general , as a c o m b i n a t i o n of t he ra tes of t h r e e bas ic s teps ; (i) a d s o r p t i o n to t he e l ec t rode surface , (ii) c h a r g e or a t o m t r a n s f e r b e t w e e n t he a d s o r b a t e a n d t he e lec t rode , a n d (iii) d e s o r p t i o n of t he n e w l y f o r m e d a d s o r b a t e f rom the sur face to t h e so lvent . We c a n wr i te t h e s e t h r e e s teps in t h e fo l lowing s c h e m a t i c fo rm Xsol + e l ec t rode ~ Xad + e l ec t rode --~ Yad + e lec t rode-~ Yso, + e l ec t rode [1] w h e r e t he s u b s c r i p t s sol a n d ad s t a n d for so lva t ed a n d adso rbed , respec t ive ly , a n d it was a s s u m e d t h a t par t ic le X was t r a n s f o r m e d in to Y af te r t he c h a r g e t r a n s f e r t akes place. In genera l , it is i m p o s s i b l e to m e a s u r e e x p e r i m e n ta l ly t h e i n d i v i d u a l ra tes of t h e s e t h r e e processes . Theore t ica l d e t e r m i n a t i o n of t he c h a r g e t r a n s f e r rate , Xad + e l ec t rode --> Yad + e lec t rode , is pos s ib l e p r o v i d e d t h a t t h e a d s o r b a t e e l e c t r o d e i n t e r a c t i o n po t en t i a l s a n d t he corres p o n d i n g c o u p l i n g t e r m s are k n o w n . However , s ince the a d s o r p t i o n a n d d e s o r p t i o n s t eps are c o m p l i c a t e d m a n y b o d y p r o c e s s e s t h a t t ake p lace on a v e r y l o n g t i m e scale c o m p a r e d to a m o l e c u l a r v i b r a t i o n a l per iod , a t heo re t i c a l e s t i m a t e of t he c o r r e s p o n d i n g ra t e s is ve ry difficult . R e c e n t l y we h a v e d e s c r i b e d a n e w m e t h o d to eva lua t e r a t e s of d e s o r p t i o n for a t o m s a n d m o l e c u l e s a d s o r b e d on so l id sur faces , w h i c h is b a s e d on c lass ical s t o c h a s t i c diffu s ion t h e o r y (CSDT) (1). U s i n g t he C S D T f o r m a l i s m (in t h e fo l lowing we shal l re fer to l~ef. ( lb ) as p a p e r I), we h a v e de r i ved s imp le ana ly t ic e x p r e s s i o n s for a t o m s a n d m o l e c u l e s d e s o r b i n g f rom sol id sur faces (to vacuum) , w h e r e t he d e s o r p t i o n ra tes were g iven in t e r m s of t he mic ro scop i c p a r a m e t e r s of t he s y s t e m (e.g., De, t h e b i n d i n g energy , ~0, t he s t r e t c h i n g f r e q u e n c y of t he a d s o r b a t e su r f ace i n t e r ac t i on , k~, t he force c o n s t a n t for f r u s t r a t e d rota t ion , a n d T, t he t empe ra tu r e ) . S imi l a r e x p r e s s i o n s for t h e ra te of d e s o r p t i o n of b o t h a t o m s (2) a n d m o l e c u l e s (3) f r o m sol id sur faces h a v e also b e e n o b t a i n e d u s i n g t rans i t i on s ta te t h e o r y (TST). T h e s e ra te e x p r e s s i o n s h a v e t he fo rm w h e r e fiT) = 1 for a tomic a d s o r b a t e a n d f(T) c~ k~/kT for molecu les . A n i n t e r e s t i n g fea tu re of Eq. [2] is t h a t t he des o r p t i o n ra te does no t d e p e n d on t he cha rae t e r i s t i e s of t he sol id sur face (e.g., D e b y e f r equency ) e x c e p t t h r o u g h t he a d s o r b a t e s u r f a c e i n t e r a c t i o n potent ia l . Th i s b e h a v i o r is a r e su l t of t h e t e r m a l averag ing , i n e o r p o r a t e d in b o t h t he C S D T a n d T S T m e t h o d s , w h i c h s u p p r e s s e s t h e indiv idua l d y n a m i c a l e v e n t s o c c u r r i n g d u r i n g t he d e s o r p t i o n process . I n th i s paper , we der ive a n e x p r e s s i o n for t h e ra te of d e s o r p t i o n of an a t o m (or ion) f rom a n e l ec t rode su r face in to a c o n d e n s e d phase , namely , t he solut ion. I n so doing, we shal l fol low a d e r i v a t i o n s imi la r to t he one e m p l o y e d in gas p h a s e d e s o r p t i o n (1). A n e x p r e s s i o n for t h e deso rp t ion ra te is de r ived be low fo l lowed b y t he p r e s e n t a t i o n a n d d i s c u s s i o n of the r e su l t s o b t a i n e d for s o m e m o d e l calcu la t ions . Final ly , in t he las t pa r t of t he p a p e r we ou t l i ne s o m e c o n c l u s i o n s r e l a t ed to t he ca lcu la ted rates . CSDT for Desorption from Electrode Surfaces We shal l s tar t by c o n s i d e r i n g a s y s t e m c o m p o s e d of a d e s o r b i n g par t ic le (a tom or ion), w i t h mass M, i n t e r a c t i n g w i t h a sol id sur face (e lec t rode) a n d a n u m b e r of s o l v e n t par t ic les , w i t h m a s s M1, as i l l u s t r a t ed s c h e m a t i c a l l y in Fig. 1. Fo r t h e de sc r i p t i on of t he surface , we shal l fo l low t h e ideas i n t r o d u c e d by A d e l m a n a n d Doll (4, 5) a n d d iv ide t he e l ec t rode a t o m s in to two g roups : (i) a sma l l n u m b e r of a t o m s w h i c h are s t rong ly i n t e r a c t i n g w i t h t he a d s o r b e d par t ic le , t h i s g roup will be ca l led t he p r i m a r y zone, a n d (ii) t h e r e s t of t he sol id w h i c h is a s s u m e d to act as a h e a t ba th . W i t h o u t loss of genera l i ty we shal l a s s u m e t h a t t he p r i m a r y zone c o n t a i n s on ly one a t o m w h o s e m a s s wil l b e d e n o t e d b y Ms. A f u r t h e r s impl i f i ca t ion is i n t r o d u c e d b y l im i t i ng b o t h t he d e s o r b i n g par t ic le a n d t he p r i m a r y zone a t o m to a o n e d i m e n s i o n a l m o t i o n n o r m a l to t he sol id surface. Thus , t he m o t i o n of t he adpa r t i c l e will be d e s c r i b e d b y a N e w t o n i a n e q u a t i o n of m o t i o n MR 0V(R,~,Z) [3] ~R w h e r e ~ r e p r e s e n t s t he d i s p l a c e m e n t of t h e su r f ace pri-