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University of South Carolina

Scholar Commons

Faculty Publications Chemical Engineering, Department of

1-1-1991

A Mathematical Model of ElectrochemicalReactions Coupled with Homogeneous Chemical

ReactionsKen-Ming YenTexas A & M University - College Station

Taewhan YeuTexas A & M University - College Station

Ralph E. WhiteUniversity of South Carolina - Columbia, white@cec.sc.edu

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Publication InfoJournal of the Electrochemical Society, 1991, pages 1051-1054. Te Electrochemical Society, Inc. 1991. All rights reserved. Except as provided under U.S. copyright law, this work may not bereproduced, resold, distributed, or modied without the express permission of Te Electrochemical Society (ECS). Te archival

version of this work was published in the Journal of the Electrochemical Society.hp://www.electrochem.org/DOI: 10.1149/1.2085714hp://dx.doi.org/10.1149/1.2085714

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A M athem atical M od el o f Electrochemical Reactions Coupledw ith Hom ogeneous C hem ical Reactions

Ken-Ming Yin,* Taewhan Yeu,* and Ralph E. W hite* *C h e m i c a l E n g i n e e r i n g D e p a r t m e n t , T e x a s A & M U n i v e r s i ty , C o ll e ge S t a t i o n , T e x a s 7 7 8 43 -3 1 22

T h e z i n c / b r o m i n e ( Z n/ B r2 ) f l o w b a t t e r y h a s r e c e i v e d c o n -s i d e r a b l e a t t e n t i o n i n r e c e n t y e a r s [e.g., ( 2 - 4 ) ] . A l t h o u g h i ti s a g r e e d t h a t t h e s o l u t i o n c h e m i s t r y i s i m p o r t a n t i n th es y s t e m , m o s t o f t h e w o r k t h a t h a s b e e n d o n e i s c o n c e n -t r a t e d o n t h e d e s i g n v a r i a b l e s . I n t h i s n o t e t h e b a s i c m a s st r a n s f e r - s o l u t i o n a n d s u r f a c e k i n e t i c s a r e s t u d i e d t o f u r-n i s h a b e t t e r u n d e r s t a n d i n g o f t h e s y s t e m . T h e r e s u l t s p r e-s e n t e d a r e f o r e l e c t r o c h e m i c a l r e a c t i o n

2 B r - ~ B r 2 * 2e [1 ]c o u p l e d w i t h t h e h o m o g e n e o u s c o m p ] e x a t i o n r e a c t io n

k fB r - + B r 2 ~ B r 3 - [ 2]kbo n a r o t a t i n g d i s k e l e c t r o d e ( R D E ) ( 4). A d e t a i l e d d i s c u s -s i o n o f t h e m i g r a t i o n e f f e c t is i n c l u d e d .

T h e m i g r a t i o n e f f e ct f or c a s e s w i t h o u t t h e i n t e r f e r e n c eo f h o m o g e n e o u s c h e m i c a l r e a c t i o n s h as b e e n d i s c u s s e dt h o r o u g h l y b y N e w m a n ( R e f. (5 ), C h a p . 1 9) a n d o t h e r r e -s e a r c h e r s [e.g., R e f . (6 -8 )]. A l s o , H a u s e r a n d N e w m a n (9 )p o i n t e d o u t t h e p o s s i b i l i t y o f a n i n t e r e s t i n g p o t e n t i a l m i n i -m u m w i t h i n t h e d i f f u s io n l a y er w h e n t h e s u p p o r t i n g e l ec -t r o l y t e p a r t i c i p a t e s i n t h e e l e c t r o d e r e a c t i o n w i t h o u t h o -m o g e n e o u s c h e m i c a l r e a ct i o n s i n v o l v e d .E l e c t r o c h e m i c a l m e t h o d s h a v e b e e n u s e d f o r t h e d e te r -m i n a t i o n o f h o m o g e n e o u s r e a c t i o n r a t e c o n s t a n t s ( 10 -1 3) .F o r e x a m p l e , t h e l i m i t i n g c u r r e n t d e n s i t y d e p e n d s o n th er a te o f a h o m o g e n e o u s c h e m i c a l r e a c ti o n ; K o u t ec k :~ a n dL e v i c h ( 12 ) a n a l y t i c a l l y d e r i v e d a l i m i t i n g c u r r e n t d e n s i t ye x p r e s s i o n f o r a n e l e c t r o c h e m i c a l r e a c t i o n o n a R D Ec o u p l e d w i t h c h e m i c a l r e a c t i o n s o f t h e t y p e s A ~ B a n d 2 A .B . A l s o , a n e x p e r i m e n t a l d e t e r m i n a t i o n o f t h e d i s so -c i a t i o n r a t e o f a c e t i c a c i d h a s b e e n d o n e o n t h e R D E b y A 1-b e r y a n d B e l l ( R e f . (1 1 ), p . 13 2 ).A l t h o u g h a n a l y t i c a n a l y s i s h a s b e e n d o n e , i t i s s h o w n i nt h is c o m m u n i c a t i o n t h a t a c o m p r e h e n s i v e n u m e r i c a l a na l-y s i s i s r e q u i r e d t o s t u d y t h e s e s y s t e m s . F o r e x a m p l e , t h eb e h a v i o r b e l o w t h e l im i t i n g c u r r e n t w i t h h o m o g e n e o u sc h e m i c a l r e a c t i o n s c a n n o t b e p r e d i c t e d b y a n y a n a ly t i c a p-p r o a c h , b e c a u s e t h e e l e c t r o d e k i n e t i c s n e e d t o b e i n c o r p o -r a te d . A m a t o r e a n d S a v e a n t (1 4) n u m e r i c a l l y i n v e s t i g a t e dt h e e l e c t r o c h e m i c a l , c h e m i c a l , e l e c t r o c h e m i c a l ( E C E ) a n dt h e d i s p r o p o r t i o n a t i o n m e c h a n i s m s . A l t h o u g h el e c t r o d ek i n e t i c s a r e i n c l u d e d , t h e y f o r c e f u l l y s e t o n e o f t h e r e a c -t a n t c o n c e n t r a t i o n s t o z e r o a t t h e i n te r f a c e , e v e n a t t h en o n l i m i t i n g c u r r e n t c o n d i t i o n , w h i c h i s n o t c o r re c t . Y e na n d C h a p m a n (1 5) u s e d t h e o r t h o g o n a l c o l l o c at i o n m e t h o d

f o r t h e E C E m e c h a n i s m a n d s h o w e d t h a t t h e v a l u e o f t h eh o m o g e n e o u s r a te c o n s t a n t m o d i f i e s th e l i m i t in g c u r r e n td e n s i t y tr e m e n d o u s l y . T h e m e t h o d t h e y u s e d m a y s a v ec o m p u t a t i o n t i m e , b u t a c c u r a c y i s s a c ri f ic e d b e c a u s e t h ec h e m i c a l r e a c t i o n o c c u r s c l o s e t o t h e i n t e r f a c i a l r e g i o n i nw h i c h n o c o l l o c a t i o n p o i n t s a r e a ll o c a te d . A d a n u v o r e t a l .(1 6) c o n s i d e r e d t h e s a m e s y s t e m a s t h a t s t u d i e d h e r e w i t ht h e m i g r a t i o n e f f ec t ; h o w e v e r , t h e b u l k c o n c e n t r a t i o n st h e y u s e d d o n o t s a t is f y th e e q u i l i b r i u m c o n d i ti o n , w h i c hl e a d s to q u e s t i o n a b l e p r e d i c t e d c u r r e n t d e n s i t i e s . I n t h i sn o t e , t h is d i s c r e p a n c y i s r e m o v e d . R e c e n t l y , H a u s e r a n dN e w m a n (1 7) s t u d i e d t h e c o m p t e x a t i o n r e a c t io n r a t e o f c u -p r o u s i o n s b y t h e s i n g u l a r p e r t u r b a t i o n m e t h o d a n d d e m -o n s t r a t e d a s t r a t e g y f o r t h e d a t a a n a l y s i s b y l u m p i n g r e l e -v a n t v a r i a b l e s .

T h e o r yFor genera l i t y , t he mode] equat ions a re deve loped fo rm u l t i p l e r e a c t i o n s , a l t h o u g h t h e s t u d i e d s y s t e m i s f o r s i n-* E l e c t r o c h e m i c a l S o c i e t y S t u d e n t M e m b e r .9 * E l e c t r o c h e m i c a l S o c i e t y A c t i v e M e m b e r .

d . E l e c t r o c h e m . S o c . , Vol. 138, No. 4, Apr i l 1991 9 The Electrochemical S ociety, Inc.

g l e e l e c t r o c h e m i c a l - c h e m i c a l r e a c t i o n s , A o n e - d i m e n -s i o n a l m o d e l i s u s e d h e r e , i . e . , t h e m o d e l i s s t r i c tl y v a l i d a tt h e c e n t e r o f t h e R D E . A d d i t i o n a l a s s u m p t i o n s u s e d a r e :( i ) D i l u t e s o l u t i o n t h e o r y a p p l i e s , i . e . , t h e d r i v i n g f o r c e f o rt h e f l u x o f s p e c i e s i s r e l a t e d t o t h e i o n - s o l v e n t f r i c t i o n . I o n -i o n i n t e r a c t i o n s a r e n o t c o n s i d e r e d . ( i t ) D o u b l e - l a y e rc h a r g i n g i s n o t c o n s i d e r e d . ( i i i ) T h e s o l u t i o n i s i s o t h e r m a l .( i v ) T h e p h y s i c a l a n d t r a n s p o r t p a r a m e t e r s a r e c o n s t a n tw i t h i n t h e d i f f u s i o n l a y e r .

G o v e r n i n g e q u a t i o n s . - - S t e a d y - s t a t e m a s s c o n s e r v a t i o no f s p e c i e s i c a n b e w r i t t e n a s ( R e f . (5 ), p . 2 18 )0 = - V 9 N, + R i [3 ]

w h e r e N i i s t h e m o l a r f l u x o f s p e c i e s i a n d R , i s t h e n e t g e n -e r a t io n r a t e o f i b y h o m o g e n e o u s c h e m i c a l r e a c t io n s . F o rt h e h o m o g e n e o u s c h e m i c a l r e a c t i o n s h o w n i n E q . [2], R B r -= R B r 2 = - - R B r 3 - = - k f C B r - C Sr2 ~- kbC B r3- . N i i n c l u d e s m i g r a -t i o n , d i f f u s i o n , a n d c o n v e c t i o n ( R e f . (5 ), p . 30 1)

N i = - z i u ~ F c ~ V d P - - D i V c ~ + V C , [4 ]w h e r e t h e m o b i l i t y u, c a n b e a p p r o x i m a t e d b y D J R T a c -c o r d i n g t o t h e N e r n s t - E i n s t e i n r e l a t i o n ( R e f . (5 ), p . 2 29 ).F o r a o n e - d i m e n s i o n a l m o d e l , E q . [3 ] b e c o m e s

0 = D , ~ y 2 - v , = - + z i u y l c i + R , [5 ]d y \ d y 2 d y d y /T h e n o r m a l v e l o c i t y at a s m a l l d i s t a n c e f r o m t h e d i s k s u r -f a c e c a n b e e x p r e s s e d b y t h e f i rs t t e r m o f a p o w e r s e r i e sa p p r o x i m a t i o n

v~ = -a 12 y2 [6 ]

w h e r e a i s a c o n s t a n t w i t h t h e v a l u e 0 .5 1 02 3 26 2 ( R e f . (5 ),p . 2 8 2) , f~ i s t h e r o t a t i o n s p e e d ( r ad / s) , a n d v i s t h e k i n e t i cv i s c o s i t y . A l s o , t h e s o l u t i o n c o n c e n t r a t i o n s o f t h e v a r i o u ss p e c i e s m u s t s a t i s fy t h e e l e c t r o n e u t r a l i t y c o n d i t i o nnion0 -= ~ ciz, [7 ]i= l

B o u n d a r y c o n d i t i o n s . - - T h e i o n i c c o n c e n t r a t i o n s a p -p r o a c h t h e u n i f o r m b u l k c o n d i t i o n s a f t e r a c e r t a i n c h a r a c -t e r i s t ic d i s t a n c e o r t h e d i f f u s i o n l a y e r t h i c k n e s s (5) f r o mt h e i n t e r f a c e

= \ a ~ / \ ~ / [8]w h e r e D R r e p r e s e n t s t h e d i f f u s i v i t y o f t h e l i m i t i n g s p e c i e s( B r -) . I t i s c o n v e n i e n t t o s e t t h e b u l k b o u n d a r y c o n d i t i o n sa t y = 2~

cL(2~) = Ci,b~tk [9 ]w h e r e t h e b u l k i o n i c c o n c e n t r a t i o n s s a t i s fy t h e e q u i li b -r i u m c o n d i t i o n ( K e q = kr /kb = CB,-3 b~Ik/CB,..bu~kCm2.bu,k)a n dt h e e l e c t r o n e u t r a l i t y c o n d i t i o n ( E q . [7 ]). T h e s o l u t i o n p o -t e n t i a l a t 2 5 c a n b e a s s i g n e d a n a r b i t r a r y c o n s t a n t f o r c o n -v e n i e n c e ; s i n c e t h e v a l u e o f c P( 25 ) i s i m m a t e r i a l , i t o n l ys e r v e s a s a b a s i s o f c o m p u t a t i o n (1 8). N o t e t h a t t h e s eb o u n d a r y c o n d i t i o n s d o n o t i n c l u d e t h e e x a c t p o s i t io n o ft h e r e f e r e n c e e l e c t r o d e ( YR E) b e c a u s e t h e o h m i c d r o p b e -t w e e n 2 5 a n d YRE c a n b e e a s i l y c o m p e n s a t e d i f t h e a p p l i e dc u r r e n t d e n s i t y a n d s p e c if i c c o n d u c t i v i t y o f t h e s o l u t i o n

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1 0 5 2 J . E l e c t ro c h e m . S o c . , V o l . 1 3 8 , N o . 4 , A p r i l 1 9 9 1 9 Th e Electrochemical So ciety, Inc.a r e k n o w n b y a s s u m i n g t h a t O h m ' s l a w (/T = - -~ ddp/dy)a p p l i e s b e t w e e n 2 5 a n d YRE-T h e b o u n d a r y c o n d i t i o n s a t th e e l e c t r o d e s u r fa c e a r et h a t t h e f l u x o f e a c h i o n i c s p e c i e s i s e q u a l t o t h e a s s o c i a t e ds u r f a ce e l e c t r o c h e m i c a l r e a c t i o n s

nr , i -- - Z s i z~ j = N i [ 1 0 ]j = l n yw h e r e Ni = - ziuiFc~ dcP/dy - D~ dc /d y a t t h e s u r f a c e a n d s , ji s t h e s t o i c h i o m e t r i c c o e f f i c i e n t o f s p e c i e s i in e l e c t r o -c h e m i c a l r e a c t i o n j . T h e p a r t i a l c u r r e n t d e n s i t y i j i s e x -p r e s s e d b y t h e B u t l e r - V o l m e r e q u a t i o n ( 18, 19)

e x p c~a ' jnjF " '" - - Uj , 'ef) lJ

( c 0 t - rl l \ C i ,r ef / e xp L ~ ( Y d - (~I) - , [11]T h e e x c h a n g e c u r r e n t d e n s i t y i0j.~ef s b a s e d o n t h e c h o s e nr e f e r e n c e c o n c e n t r a t i o n s ; p ,.j a n d q i j a r e t h e r e a c t i o n o r -d e r s f o r a n o d i c a n d c a t h o d i c r e a c t i o n s . T h e p o t e n t i a l -d e p e n d e n t t e r m , V a - (P0 - U ~.~ ,f, s t h e o v e r p o t e n t i a l a t t h ei n t e r f a c e f o r r e a c t i o n j , w h i l e a , , j a n d (~ c4 a r e t h e c o r r e s -p o n d i n g t r a n s f e r c o e f f i ci e n t s f o r a n o d i c a n d c a t h o d i c r e a c -t i o n s . T h e e x p r e s s i o n U j ~ i s s h o w n i n E q . [ 8] o f R e f . (18 ).F i n a l l y , t h e e l e c t r o n e u t r a l i t y c o n d i t i o n ( E q . [7 ]) a n d t h e e x -p r e s s i o n o f t o t a l a p p l i e d c u r r e n t d e n s i t y

i T = ~ i j [12]j = ls e r v e a s t h e l a s t t w o e q u a t i o n s f o r t h e i n t e r f a c e s o l u t i o np o t e n t i a l q)0 a n d t h e e l e c t r o d e p o t e n t i a l ltd . I n t h i s c u r r e n t -c o n t r o l l e d m o d e l , t o t a l c u r r e n t d e n s i t y ( iT ) r a t h e r t h a n t h ea p p l i e d p o t e n t i a l ( Vd - 4PR E) s s e t .

S o l u t i o n T e c h n i q u eT h e g o v e r n i n g e q u a t i o n s a n d r e l a t e d b o u n d a r y c o n d i-

t i o n s a r e c a s t i n t h e f i n i t e d i f f e r e n c e f o r m . T h e s e e q u a t i o n sa r e s o l v ed us i n g N e w m a n ' s B A N D s u b p r o g r a m w i t h a m a -t r i x s o l v e r M A T I N V ( R e f . (5 ), p . 41 9) . T h e u n k n o w n s d e t e r -m i n e d a r e io n i c c o n c e n t r a t io n s , q(y), s o l u t i o n p o t e n t i a l ,(I )(y ), a n d e l e c t r o d e p o t e n t i a l , V d . T h e e l e c t r o d e p o t e n t i a l ,V d, w h i c h h a s o n l y a s i n g l e v a l u e a t t h e e l e c t r o d e , i s t r e a t e da s a n u n k n o w n c o n s t a n t i n B A N D (6 , 7 , 1 9).P a r a m e t e r s

T h e e x a m p l e c h o s e n i s t h e B r - /B r 2 e le c t r o d e r e a c t io nc o u p l e d w i t h t h e t r i b r o m i d e c o m p l e x r e a c t i o n i n t h ea q u e o u s s o l u ti o n s . T h e c h e m i c a l s i n t r o d u c e d i n t o t h e b u l ks o l u t i o n a r e B r 2 (0 .3 m o l f l i t e r ) a n d N a B r ( 0.3 m o l / l i t e r ).N o t e t h e e q u i l i b r i u m b u l k c o n c e n t r a t i o n s s h o u l d b e c al c u -l a t e d b a se d o n t h e e q u i l i b r i u m c o n s t a n t a n d t h e a m o u n t o fc h e m i c a l s t h a t a r e a d d e d . T h e a s s o c i a t e d c h e m i c a l a n de l e c t r o c h e m i c a l r e a c t i o n s a r e s h o w n i n T a b le I . T h en e e d e d t r a n s p o r t d a t a a r e l i s t e d in T a b l e I I .

2 .0 . . . . , . . . . | . . . . , . . . . , . . . .

k t = 1 0 ? t = l O 9k~ l O s / ~ /" ' ^ke'0~ / X

~ L 6

kt_~lOHI. e - - w i t h m i g r a t io n

. .. .. .. .. .. . w i t h o u t m i g r a t i o n1 . 0 . . . . ' . . . . ' . . . . t . . . . , . . . .o.o o., o.z o.', o.4 o .~i T ( A / a m e )

F i g . 1 . P o l a r i z a t i o n c u r v e s w i t h d i f f e r e n t r o t e c o n s t a n t s : k f - < l O 3 ,= | 0 s , = 1 0 7 , = | 0 9 , -> | 0 u , f r o m l o w e r t o h i g h e r c u r r e n t d e n s i t i e s .

R e s u l t s a nd D i s c us s i onF i g u r e 1 s h o w s t h e s i m u l a t e d a n o d i c p o l a r iz a t i o n c u r v e sw i t h d i f f e r e n t h o m o g e n e o u s r a t e c o n s t an t s . T h e m i g r a -

t i o n - e x c l u d e d c a s e is i n c l u d e d f o r c o m p a r i s o n , w h i c h i sd o n e b y t a k i n g o u t t h e m i g r a t i o n t e r m i n E q . [5 ] a n d b y r e-m o v i n g t h e e l e c t r o n e u t r a l i t y c o n d i t i o n ( E q . [7 ]). F o r k f _ 1 07 , t h e p r e d i c t e d l i m i t i n g c u r r e n t d e n s i t i e s i n -c l u d i n g m i g r a t i o n a r e s i g n if i c a nt l y l a rg e r t h a n t h o s e w i t h -o u t i n c l u d i n g t h e m i g r a t i o n e f fe c t. O b v i o u s l y , th e m i g r a -t i o n e f f e c t i s m a g n i f i e d b y i n c r e a s i n g k f ( o r k b). T h ec o n c e n t r a t i o n p r o fi l es w i t h i n t h e d i f f u si o n l a y e r a re s h o w ni n F ig . 2 a. T h e m u c h l a r g e r B r2 c o n c e n t r a t i o n a t t h e i n t e r -f a c e w h e n c o n s i d e r i n g m i g r a t i o n r e f le c t s t h e l a r g e r li m i t -i n g c u r r e n t d e n s i t y i n F i g . 1 . T h e g e n e r a l l y l a r g e r p r e -d i c t e d c u r r e n t d e n s i t i e s w h e n m i g r a t i o n i s i n c l u d e d c a n b ee x p l a i n e d b y t h e c o n c e n t r a t i o n p r o f il e s o f B r - a n d B r 3-b e i n g v e r y c l o s e t o t h e e l e c t r o d e s u r f a c e a s s h o w n i nF i g . 2 b . A t t h e i n t e r f a c e C N,+ = C B~3-, s i n c e e l e c t r o n e u t r a l i t ym u s t b e p r e s e r v e d w h e n m i g r a t i o n i s c o n s i d e r e d , th i sc a u s e s a h i g h e r C Br3- n e a r t h e s u r f a c e . I n o t h e r w o r d s , t h ea n o d e a t t r a c t s m o r e n e g a t i v e B r 3 - i o n s to t h e e l e c t r o d e ,w h i c h t h e n r e l e a s e s m o r e r e a c t a n t B r - b y t h e c h e m i c a l r e-a c t i o n [2] n e a r t h e e l e c t r o d e c o m p a r e d t o t h e c a s e w i t h o u tm i g r a t i o n . B e c a u s e B r3 - p l a y e d t h e r o l e a s t h e s u p p l i e r o fB r , B r - h a s a h i g h e r c o n c e n t r a t i o n n e a r t h e e l e c t r o d ew h e n c o n s i d e r i n g m i g r a t i o n , a n d, c o n s e q u e n t l y , h a s ah i g h e r g r a d i e n t a t t h e i n t e r f a c e a t t h e l i m i t i n g c u r r e n t c o n -d i t i o n . F r o m t h e l i m i t i n g c u r r e n t d e n s i t y a t kf = 109 i nF i g . 1, i t s h o u l d b e c l e a r t h a t t h e g r a d i e n t i s a b o u t t w i c e a sm u c h w h e n c o n s i d e ri n g m i g r a t io n c o m p a r e d t o th e o n ew i t h o u t c o n s i d e r i n g m i g r a t i o n .A n i n t e r e s t i n g b e h a v i o r i s o b s e r v e d f o r t h e e l e c t r i c fi e ld(E = -d,b/dy). A s s h o w n i n F i g . 3 a , t h e e l e c t r i c f i e l d is a l-

Ta ble I . Elect rochemical and chemical react ions.E l e c t r o c h e m i c a l r e a c t i o n U ~a ( V )2/~r- - Br2 --> 2e 1.087 b

E l e c t r o c h e m i c a l k i n e t i c p a r a m e t e r s(~a., (x~.~ n~ i0j.~ef A /c m2) U~.,'~f v )0.5 0.5 2 0.001861 ~ 1.0466dC h e m i c a l e q u i l i b r i u m E q u i l i b r i u m c o n s t a n t ( cm 3 /m o l )

k fB r 2 + B r ~ B r 3 - 17 x 10~kb

Table I I . Mass t ransport data.Dal I0 s Cbi,bulkX103 c%,.,,r I0 r

( ~ ) ( m o l~ ( m o l~S p e c i e s z t \ ~ m 3 ] \ c m a /

Na + 1 1.334 0.3 1.0~r - - 1 2.084 0.106647 1.08Br2 0 1.31 0.106647 0.05Br3- -1 1.31 0.193353 0.92

v = 0 .0 1 2 3 cm 2 / s T - 2 9 8 K p o = l . 0 g / c m 3 1 2= 1 0 4. 7r ad /s= 2 . 2 3 1 0 s c ma A l t h o u g h t h e e x a m p l e i s f o r a s i n g l e e l e c t r o c h e m i c a l r e a c t io n ,t h e s u b s c r i p t j i s k e p t f o r c o n s i s t e n c y .b Ref. (3).c The exc han ge c u r re n t d ens i t y co r resp ond s to c ,.r ,.r n Ref . (3) .d Ca lcu la ted f rom Eq . [8 ] in Ref - (18) .

a DNa% DB from Ref. (5), p . 230; Din2, DBr3 from Ref. (3).b T h e e q u i l i b r i u m b u l k c o n c e n t r a ti o n s a r e c a l c u l a t e d a c c o r d i n gt o t h e i n t r o d u c e d N a B r a n d B r 2 c o n c e n t r a t i o n s .c T h e c o n c e n t r a t i o n s a r e f r o m R e f . ( 3) ; t h e y c o r r e s p o n d t o ir =0.001861 A/c m 2.

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J . E l e c t r o c h e m . S o c . , Vol. 138, No. 4, Apri l 1991 9 The Electrochemical Society, Inc. 1053

An

1 0 . 09 . 0 ~ . . . . ' . . . . ' . . . . ' . . . .kOz o ~ w i t h m i g r a t i o ns.O4.o~.0 ""...". . . ~ . .. .. .. .. .. . w i t h o u t m i g r a t i o n

~.o . . .. . .. x ~ ,: : i ' : : : . , ~ ~ ~ _ _ _z.o '.., x ~~ . ~ , B .r~ -, o

0 . 0 ~ , , , IO. O 0 . 5 1 . 0 ! . 5d i m e n s i o n l e s s d i s t a n c e (0 2 ,0

2 5 . 0N a +

2 0 . 0

-l'a.O ~ w i t h . m i g r a t i o n /. .. .. .. .. w i t h o u t m i g r a t i o n /~- 1o.o . . . .. . . .. . ..

5 ,0B r -

o.o ~ ; ' " " : ' : ' ? : : ' 7 . " 7 . ' " I ' F " 7 " . " ' : ' " ' : ' : ' I I I I I0 . 0 0 0 0 . 0 0 0 0 . 0 1 0 0 . 0 1 5 O .O B Qd i m e n s i o n l e s s d i s t a n c e ( 0

F i g . 2 . ( a , t o p ) T h e c o n c e n t r a ti o n p r o f il e s w i t h i n t h e d i ff u s i o n ! a y e ru n d e r l i m i t in g c u r r e n t d e n s i t y f o r k ~ - ! 0 9 . ( b , b o t t o m ) C o n c e n t r a ti o np r o f i le s o f N a + , B r - , a n d B r 3 - c l o s e t o th e e l e c t r o d e u n d e r l i m i t in g c u r -r e n t d e n s i t y f o r k f = 1 0 9 .

most co nstant near the outer region of the diffusion layerwhere the concentr ations are close to the bulk concentra-tions. However, higher homogeneous rate constants cor-respond to higher E values because of the larger limitingcurrent densities. E increases rapidly when approachingthe electrode bec ause the ionic species are depleted there.There is a maxi mu m in E close to the electrode as can beseen more clearly in Fig. 3b. Note that the m axi mum for kf= 10~ occu rs at ~ ~ 0.04, as sho wn in Fig. 3a. Such phe nom -enon can be ex plaine d by the significant difference in corncentration gradients in two regions, i .e . , the outer diffusionregion and the inner chemical reaction region. Figure 4shows the concentration profiles of Br- within the diffu~sion layer. It is clear th at for kf ~ 107, a sign ific ant differ-ence occurs between the concentration gradients in thetwo regions. The larger the r~te consta nt, the narrowe r theinn er regi on (see Fig. 2b for the inn er region at kf = 10~).The total curren t density, which is constant across the dif-fusion layer, can be e~;pressed by the respective contribu-tions from Ohm's law and the diffusion current (Ref. (5),p. 221)

dcP dciiT -~ - K - - - d y - F~ . ziD~ d Y [13]where K = F ~ z ~ u i c ~ . As shown in Fig. 5a and b, Ohm's law( iOhm = --K dq)/dy ) applies well for ~ > 1.5 until the di ffusioncurrent ( i D i f f = ~ F Z i z iD i d c J d y ) becomes importa nt closer tothe interface. For kf -< 103, whe n th e chemica l reacti on canbe neglected, the diffusion current density (Fig. 5b) in-creases mono tonica lly toward the electrode because con-centration gradients increase steadily when approachingthe electrode. As kr increases, the diffusion-migrationmechan ism induced diffusion current is important when-< 1.5. Unli ke the no-ch emica l reaction case, there is a drop

+

$SO.O . . . . , . . . . , . . . . , . . . .

~ . O . O

1 8 0 . 0

S 4 0 . O

2 0 0 . 0

ISO.O

k f = 1 0 I tk f = 1 0 9

. . . . . . . . . . . . k f f fi 1 0 7

........................ k f = 1 0 5k f = 1 0 3

+ i 10 .0 0.5 1,0 I.w S.O

d i m e n s i o n l e s s d i s t a n c e (~' )

8 0 0 . 0 . . . . , . . . . , . . . . , . . . . , . . . . , . . . .

~.0.0

S S O . O

u o . o k r = l o l O ~ ~ _ _ _ _ _ _ _ _ . . . . ~

2 0 0 . 0 ~1 0 0 , 0

1 2 0 , 0k f f f i I 0 O

6O.O . ~ ~4 0 . 0 k f = 107

0 .0 . . . . ' " " ' ' ' . . . . ' . . . . . ' . . . . ' . . . .o.oo o o.oo e o.~1o o.oso O.OZO o.o ~ o.osod i m e n s i o n l e s s d i s ta n c e ( 0F i g . 3 . T h e p o t e n t i a l g r a d i e n t p r o f il e s ( a , t o p ) w i t h i n a n d ( b , b o t t o m )n e a r t h e d i f f u s i o n l a y e r a t d i f f e r e n t r a t e c o n s t a n t s u n d e r l i m i t in g c u r -r e n t c o n d i t i o n s .

in iDiff after a max imu m, the n iDiff increases rapidly nearthe surface. The drop in ibm.must be due to the decrease inthe concen tration gradients ( d c , / d y , especially for Br-), i .e . ,concentrat ions in this region are somewhat flattened. Theflattening should come from the conti nuous release of Br-from Br3-. The flatten ed con cent rati on region is reflectedby the m inim um of i D i f f in Fig. 5b. It should now be clearthat i D i f f will rise again at the interface because of the largerdCBr- /dy there due to the Br- co ncentrati on passingthrough a flattened region and then sudden ly dropping tozero at the li mitin g cur rent co ndition. Note that for kr =1011, the diffusio n curre nt inc reases after a m in im um at ~

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1054 J . E l e c t r o c h e m . So c . , Vol. 138, No. 4, Apri l 1991 9 T h e E l e c t r o c h e m i c a l S o c i e t y , Inc.0 . 5 00 . 4 50 . 4 00 . 3 50.00O . 2 5

2 0 . 2 00 . 1 00.I00 . 0 0O.OO

. . . . I . . . . I . . . . I . . . .

} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . k f = l O 1 1............... k f = l O ?

kr= lO 3

, , , . , . . . . , . . . . , . . . .0.o 0.0 l.O 1.$

d i m e n s i o n l e s s d i s t a n c e (~)2. 0

0 . 1 00.140.150 . 1 2O . I IO.IOO.OS

i 0.060.07~, O.OS

0 . 0 50 . 0 40 . 0 3O.OS0.01O.OO

. . . . . . . . k f = l O 1 1

............... k f = l O ?k t = l O 3

o.o 0.0 I . O I ~d i m e n s i o n l e s s d i s t a n c e (~ ) 2 .0

F i g . 5 . ( a , to p ) T h e c o n t r i b u t i o n f ro m o h m i c c u r r e n t w i t h i n t h e d i f fu -s io n l a y e r a t d i f fe r e n t r a t e c o n s t an t s u n d e r t h e l i m i t i n g c u r r e n t c o n d i -t io n s . ( b , b o t t o m ) T h e c o n t r i b u t io n f r o m d i f f u s i o n c u r r e n t w i t h in t h e d i f -f u s i o n l a y e r a t d i f f e r e n t r a t e c o n s t a n ts u n d e r th e l im i t in g c u r r e n tc o n d i t i o n s .

S u m m a r yA one-dimensional model of electrochemical reactionscoupled with homogeneous chemical reactions is given.An exampl e of the oxidati on of Br- coupled with Br + BrzBr3- shows the imp orta nce of including the migrati oneffect. The homo gen eou s chemic al reaction causes a maxi-mum in the electric field within the reaction layer.

A c k n o w l e d g m e n t sThe authors are grateful for the support of this projectgiven by Sandia National Laboratories, the Texas Ad-vanced Technology and Research Program, and the TexasA&M University Board of Regents through the Available

University Fund.Manuscri pt submit ted July 23, 1990; revised manuscr iptreceived Nov. 30, 1990.T e x a s A & M U n i v e r s i t y a s s i s t e d i n m e e t i n g t h e p u b l i c a -t i o n c o s t s o f t h i s a r t i c l e .

LIST OF SYMBOLSa 0.51023262c~ con cent rat ion of speci es i, mol/ cm3cj~ concen tratio n of species i at the solid-solution inter-face, mol/c m3ci.b~m bulk solution concentration of species i, mol/cm3ci.~r refe ren ce concen tra tio n of specie s i, mol/ cm 3D~ diffus ion coeff icient of specie s i, cm2/sDR diffusion coefficient of the limiting species (Br-),cm2/sE elect ric field ( = - d ( P / d y ) , V/cmF Fara day 's constan t, 96,487 C/molij partial current density due to reaction j, A/cm ~iD~,~ diffus ion curren t densit y ( = -F ~ i z iD i d ci /d Y) , A/cm ~ioh~ ohmic con tribut ion to total current density (= -Kd~P/dy), A/cm 2

i0j.re~ exchange current density at reference concentra-tions for reaction j, A/cm 2iT total current density, A/cm2kb backwa rd homo gene ous rate constant, 1/skf forward homo gene ous rate constant, cm3/s molKeq equil ibriu m constant, cm3/molnlon number of ionic speciesn r number of electrochemical reactions (equal to 1here)nj numb er of electrons transferred in reaction jN1 flux vec tor of specie s i, mol /cm 2 9 sN i flux of species i in y direction, mol/cm 2 - sp,j anodic reaction order of ionic species i in reaction j,dimensionlessqi4 cathodic reaction order of species i in reaction j, di-mensionlessR univ ersa l gas constan t, 8.3143 J/mo l 9 KRi the homogen eous reaction rate of species i,mol/s 9 cm 3sij stoichiome tric coefficient of ionic species i in elec-trochemical reaction j, dimensionlessT absolute temper ature, Kui mobility of species i ( = D i /R T ) , tool - cm2/J 9 sUj.r~f theoretic al open-circui t potentia l of reaction evalu-ated at reference concentrations, VU~ standard electro de potential for reaction j, Vv solution velocity vector, cm/svy normal solution velocity, cm/sVd potential of the workin g electrode, Vy normal coordinate, cmYRE posit ion of refe renc e elect rode, cmzi charge numb er of species iGreek letters~a,j anodic transfer coefficient for reaction j, dimen-sionless~c.j cathodic transfer coefficient for reaction j, dimen-sionlesscharacteristic layer thickness according to Eq. [8],

c mK solution conductivit y, 1/~ 9 cmv kin emat ic viscosit y, cm2/sdimensionless distance ( = y /g )po pure solv ent density, g/cm 3solution potential, V~0 solution potential adjacen t to electrode surface, Vdisk rotation speed, rad/s

RE FE RE NCE S1. J. Lee and J. R. Selman, T h i s J o u r n a l , 129, 1670 (1982).2. M. J. Mad er an d R. E. White, i b id . , 133, 1297 (1986).3. T. I. Eva ns a nd R. E. White, i b id . , 134, 866 (1987).4. Yu. V. Pleskov and V. Yu. Filinovskii, "The RotatingDisc Electrode," Consultants Bureau, New York(1976).5. J. S. Newman, "Elec troch emica l Systems," PrenticeHall, Inc., En gle woo d Cliffs, NJ (1973).6. S. Chen, K.-M. Yin, and R. E. White, T h i s J o u r n a l , 135,2193 (1988).7. K.-M. Yin and R. E. White, A I C h E J ., 36, 187 (1990).8. C. Y. Mak and H. Y. Cheh, T h i s J o u r n a l , 135, 2262(1988).9. A. K. Hauser and J. Newman, i b id . , 136, 3320 (1989).10. E. F. Caldin, "Fas t Reacti ons in Solut ions, " John Wileyand Sons, N ew Y ork (1964).11. J. Albery, "Electro de Kinetics," Clarendon Press, Ox-ford (1975).12. V. G. Levich, "P hysi coch emic al Hydrody namics, "Pre nti ce Hall, Inc., Eng le woo d Cliffs, NJ (1962).13. A. J. Bard and L. R. Faulk ner, "Ele ctr och emi cal Meth-ods: Fundamentals and Applications," John Wileyand Sons, Ne w York (1980).14. C. Amatore and J. M. Saveant, J . E l e c t r o a n a l . C h e m . ,85, 27 (1977).15. S.-C. Yen and T. W. Cha pma n, C h e m . E n g . C o m m u n . ,38, 159 (1985).16. P. K. Adanuvor, R. E. White, and S. E. Lorimer, T h i sJ o u r n a l , 134, 1450 (1987).17. A. K. Hauser and J. Newman, T h i s J o u r n a l , 136, 3250(1989).18. R. E. White, S. E. Lorimer, and R. Darby, i b id . , 130,1123 (1983).19. R. E. White, Ph.D. Thesis, University of California atBerkeley (1977).